History of Vector Analysis

M i c h a e l J. C r o w e

A HISTORY OF

VEC O R ANALYSIS

T h e Evolution of t h e I d e a of a Vectorial System

A

HISTORY

OF

VECTOR ANALYSIS T h e E v o l u t i o n of the Idea of a V e c t o r i a l System

M I C H A E L J. CROWE University

of Notre

Dame

D o v e r Publications, I n c . New York

To

Copyright ©

MARY

ELLEN

1967 b y U n i v e r s i t y o f N o t r e D a m e Press

N e w m a t e r i a l C o p y r i g h t © 1985 by M i c h a e l J. C r o w e A l l rights reserved u n d e r Pan A m e r i c a n a n d International C o p y r i g h t Conventions. Published Lesmill Road,

in

Canada by

General

Publishing Company,

Ltd.,

30

D o n Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, This

London W C 2 H 7EG.

D o v e r edition, first published in

1985,

is an unabridged and

corrected republication of the work first published by the University of N o t r e D a m e Press i n 1967. A n e w Preface has b e e n a d d e d t o this e d i t i o n . M a n u f a c t u r e d i n t h e U n i t e d States o f A m e r i c a D o v e r Publications, Inc., 31 East 2 n d Street, Mineola, N.Y.

L i b r a r y of Congress Cataloging in Publication D a t a C r o w e , M i c h a e l J. A h i s t o r y of v e c t o r analysis. Originally published: Notre D a m e : University of N o t r e D a m e Press,

1967. W i t h c o r r e c t i o n s a n d n e w pref.

Includes bibliographies and index. 1.

Vector analysis—History.

QA433.C76 ISBN

1985

0-486-64955-5

515'.63

I.

Title. 85-13081

11501

Preface

Shortly before becoming President of Harvard in ematician the

Thomas

best k n o w n

Newton have dynasties

Hill

made

vectorial

done

the

system

following of his

1862, the math-

statement concerning

day:

"The

discoveries

of

m o r e for E n g l a n d a n d for the race than w h o l e

of British

monarchs;

and we

d o u b t not that in the great

mathematical birth of 1853, the Quaternions of H a m i l t o n , there is as much

real

promise

of benefit to

mankind as

in any event of Vic-

toria's r e i g n . " L o r d K e l v i n , w r i t i n g w h e n V i c t o r i a was v e r y o l d a n d the

modern

vector

"Quaternions been

done;

system

came

and,

from

very

new,

Hamilton

though

took

a very

different view.

really

good work had

after his

beautifully

ingenious, have been an

m i x e d evil to those w h o have t o u c h e d t h e m in any w a y .

.

un-

. vectors

. . . have n e v e r b e e n of the slightest use to any creature." T h o u g h Kelvin's barbs quaternion

in this attack of the 1890's w e r e d i r e c t e d against the

system,

methods since the

he

had

been

waging

w a r against all

vectorial

1860's.

If a scientist of the present day w e r e forced to take sides in this dispute

on the value of vectorial

methods, he might view Hill

as

overly enthusiastic, but he w o u l d not side w i t h Kelvin. T h e v i e w of vectorial m e t h o d s c h a m p i o n e d by this great physicist has b e e n refuted by the thousands

of uses

that have b e e n f o u n d for vectorial

m e t h o d s . N e a r l y all b r a n c h e s o f classical p h y s i c s a n d m a n y areas o f m o d e r n physics are n o w p r e s e n t e d i n the language o f vectors, a n d the benefits d e r i v e d t h e r e b y are m a n y . V e c t o r analysis has l i k e w i s e proved omy,

a valuable

Despite little

aid

for

many problems

in engineering, astron-

and geometry. the

studied.

importance Not a single

of vector book

analysis,

and

its

not more

history has than

been

a handful of

s c h o l a r l y p a p e r s h a v e u p t o n o w b e e n w r i t t e n o n its h i s t o r y . C o n s e quently m a n y historical errors m a y be f o u n d in the relevant literaiii

Preface

ture. T h e present study was not w r i t t e n in the expectation that all or even

most historical

swered;

questions a b o u t vector analysis w o u l d b e an-

rather it was

written

in the h o p e of presenting an essen-

t i a l l y c o r r e c t o u t l i n e o f t h e h i s t o r y o f t h i s i m p o r t a n t area. I n u n d e r t a k i n g this study I have f r e q u e n t l y b e e n h i n d e r e d by the scarcity of scholarly

studies

numbers,

linear algebra, tensors, theoretical

of the

history

teenth-century mechanics.

This

of such

r e l a t e d areas

as

complex

electricity, and nine-

is of course the c o m m o n plight of

historians of science, a n d I have b e e n consoled by the hope that the present

study

m a y s h e d l i g h t o n t h e h i s t o r y o f o t h e r areas

o f sci-

ence, such as those m e n t i o n e d above. In

this

pects

study I

h a v e c o n c e n t r a t e d o n t h e m o r e f u n d a m e n t a l as-

of vectorial

treated in detail:

analysis;

the

history of the

following topics

is

vector a d d i t i o n a n d subtraction, the forms of vec-

tor multiplication,

vector division

(in

those

systems

where

i t oc-

curs), a n d t h e s p e c i f i c a t i o n o f v e c t o r types. Less a t t e n t i o n has b e e n given to the history of vector differentiation and integration, and the operator V a n d the associated transformation theorems, since these were

for the

work.

No

most part developed originally in a Cartesian frame-

detailed

presentation

of the

complicated

history of the

l i n e a r vector f u n c t i o n has b e e n a t t e m p t e d . T h o u g h the above statement indicates the materials included, it does not sufficiently specify the approach taken in this study. For a n u m b e r o f r e a s o n s I h a v e c h o s e n t o f o c u s (as t h e s u b t i t l e i n d i c a t e s ) on t h e h i s t o r y of t h e i d e a of a v e c t o r i a l system. It s h o u l d not be forgotten that the m o d e r n

system

many

vectorial

created

these

systems e m b o d i e d an

systems

of vector analysis is but one of the in

the

course

of history.

Each of

idea or conception of the form that a

vectorial system can have and should have. A n d it is the history of these ideas that I have tried to describe. To do this I have discussed each

of the

tempted to led

to

these

vectorial

systems

created

before

1 9 0 0 a n d at-

creation,

development,

and acceptance

or rejection of

systems.

The the

the

major

d e t e r m i n e w h a t ideas (mathematical a n d motivational)

history of vectorial analysis m a y in one sense be v i e w e d as

history of systems of abbreviation, since any p r o b l e m that can

b e s o l v e d b y vectorial m e t h o d s can also b e s o l v e d ( t h o u g h usually less

conveniently) by the

vectorial

analysis

older Cartesian methods. T h e history of

may equally w e l l be v i e w e d as the history of a

w a y of looking at physical a n d geometrical entities. Consideration of these

two

aspects

chosen to focus

of the

history

will

help explain w h y I

have

on the evolution of the idea of a vectorial system,

rather than on the history of the major theorems in vectorial analy-

iv

Preface

sis, m a n y o f w h i c h w e r e i n a n y case d i s c o v e r e d b e f o r e a n d o u t s i d e of the vectorial traditions. Concerning

the

references.

The

reader

will

find

that a simple

and

not u n c o m m o n system o f reference has b e e n e m p l o y e d i n t h e text. T h e notes for each chapter are located at the e n d of that chapter; w i t h i n each chapter ordinary note numbers w i l l be f o u n d as w e l l as r e f e r e n c e s t o t h e s e n o t e s o f t h e f o r m (3,11,1; 2 7 ) . T h e l a t t e r a r e r e a d as follows: the first n u m b e r always refers to a note at the e n d of the chapter;

the

numbers to the right of the semicolon always refer to

the page n u m b e r s in the publication indicated in that note. In some c a s e s (as a b o v e ) o n e o r t w o o t h e r n u m b e r s a r e i n c l u d e d t o t h e l e f t of the semicolon; these numbers ( w h e n they occur) refer to the volume

number and

part n u m b e r of the

the reference above is read:

publication

indicated.

Thus

see v o l u m e I I , part I, of i t e m 3 in t h e

notes; consult page 27. T h r o u g h this m e t h o d it has b e e n p o s s i b l e to provide the reader w i t h

many references

that otherwise

could be

i n c l u d e d only t h r o u g h a substantial increase in the size of the book. Concerning sources

quotations

for this

and

translations.

study were books

Since

many

and journals

of

the

of limited circula-

tion I have used quotations rather liberally. All quotations from documents written in foreign languages (French, German, Italian, Russian,

and

cases

Danish)

where

have

previously

been

translated

published

into

English.

translations

In

were

the

few

available,

I

have used these after c h e c k i n g t h e m against the original a n d n o t i n g deviations. T h e sole e x c e p t i o n to this statement occurs in t h e case of Wessel's D a n i s h ; here I have c h e c k e d Nordgaard's E n g l i s h translation

against

translations Concerning

the

French

translation

of Zeuthen.

The

remaining

(the majority) are my o w n . bibliography.

No

formal

bibliographical

been i n c l u d e d in this book. T h e reader w i l l

find

section

has

h o w e v e r that the

sections of notes at the e n d of each chapter w i l l serve rather w e l l as a bibliography raphy

is

for that

greatly

chapter.

Moreover the

d i m i n i s h e d by the

existence

n e e d for a bibliogof a book

that

lists

nearly all relevant p r i m a r y d o c u m e n t s p u b l i s h e d to about 1912; this is

Alexander

Systems

of

Macfarlane's

Mathematics

(Dublin,

Bibliography 1904).

of

Quaternions

Supplements

to

and

Allied

this

un-

c o m m o n l y accurate bibliography w e r e p u b l i s h e d up to 1913 in the Bulletin

of

Quaternions The

the

International

and

Allied

author

wishes

Society Systems

to

express

for of

his

Promoting

the

Study

of

Mathematics.

gratitude

to those

who

have

aided h i m in preparing this study. P u b l i s h e d and u n p u b l i s h e d materials have b e e n o b t a i n e d f r o m libraries too n u m e r o u s to m e n t i o n , a n d this through the kindness of the librarians of the universities of v

Preface

Notre D a m e , Wisconsin, C a m b r i d g e , a n d Yale. Assistance at important

points

has

come

from

Professor Stephen J.

Rogers

of Notre

D a m e University and from Professor D e r e k J. Price of Yale University. Sincere thanks are e x t e n d e d to Professors C. H. B l a n c h a r d a n d William

D.

sor James

Stahlman of the W.

U n i v e r s i t y of W i s c o n s i n a n d to Profes-

B o n d o f P e n n s y l v a n i a State U n i v e r s i t y . T h e s e three

scholars (a physicist, an historian of science, a n d a mathematician) gave generously of their time (in reading the entire manuscript) and of t h e i r w i s d o m (in s a v i n g t h e m a n u s c r i p t f r o m a n u m b e r of errors). To

Professor E r w i n N.

most

sincere

thanks

Hiebert, of the University of Wisconsin, my

for

his

numerous,

detailed,

and

perceptive

c o m m e n t s on the entire manuscript. Portions of the research for this book

w e r e carried out w i t h f i n a n c i a l assistance p r o v i d e d b y funds

administered by C o m m i t t e e on Grants for the Arts and Humanities of the

University of Notre

Notre

Dame,

Dame. Michael J. Crowe

March,

vi

1967

Indiana

Acknowledgments

Grateful lishers

acknowledgment is hereby made to the following pub-

and

libraries

published B.

G.

for permission

to

quote

from

books

and un-

materials:

Teubner Verlag,

Friedrich

Engel,

Hermann

Grassmanns

kalische

Stuttgart,

Grassmanns

for

Leben,

permission

contained

Gesammelte

in

to

quote

Vol.

mathematische

from

Ill

und

of physi-

Werke.

Cambridge

University

published material

in

Library

for permission to quote from

un-

the correspondence of James Clerk M a x w e l l

and Peter Guthrie Tait. C a m b r i d g e U n i v e r s i t y Press for p e r m i s s i o n to quote f r o m Cargill Gilston

Knott,

Life

and

Scientific

Work

of

Peter

Guthrie

Tait.

Ernst B e n n L i m i t e d , L o n d o n , for permission to quote f r o m O l i v e r Heaviside,

Electromagnetic

Macmillan Oliver

&

Co.,

Theory,

Ltd.,

Heaviside,

Vols.

London,

Electrical

I

for

and

III.

permission

to

quote

from

Papers.

T h o m a s N e l s o n a n d Sons, Ltd., L o n d o n , for p e r m i s s i o n to quote from and

Sir

Edmund

Electricity,

Vol.

Yale

Whittaker,

A

History

of

the

Theories

o f Aether

I.

University Library for permission to quote from the u n p u b -

lished material

in the correspondence of Josiah W i l l a r d Gibbs.

Yale U n i v e r s i t y Press for p e r m i s s i o n to q u o t e f r o m L y n d e P h e l p s Wheeler,

Josiah

Willard

Gibbs:

The

History

of

a

Great

Mind.

vii

List

Graph I

Graphs

and

Tables

Q u a t e r n i o n P u b l i c a t i o n s f r o m 1841 to 1900

Graph I I Graph

of

III

Q u a t e r n i o n Books from 1841 to 1900 Annual

N u m b e r of Titles

of Mathematical

Articles and Books, 1868-1909 Graph I V

Grassmannian

Analysis

Publications

113 from

1841 to 1900 Graph V Graph V I Graph VII Graph

VIII

113

G r a s s m a n n i a n A n a l y s i s B o o k s f r o m 1841 to 1900 Q u a t e r n i o n P u b l i c a t i o n s by C o u n t r y

Analysis

Publications

115 by

Country Graph I X Chronology

viii

114 114

Q u a t e r n i o n B o o k s by C o u n t r y Grassmannian

111 112

G r a s s m a n n i a n Analysis Books by C o u n t r y

115 116 256

Contents

Chapter

One

THE

EARLIEST

TRADITIONS

I. I n t r o d u c t i o n II.

1

The Concept of the

Parallelogram

of Velocities

and

Forces

2

III.

L e i b n i z ' C o n c e p t of a G e o m e t r y of Situation

IV.

The

Concept of the

Geometrical

3

Representation

of

Complex Numbers V.

Chapter

5

Summary and Conclusion

11

Notes

13

Two

SIR

WILLIAM

ROWAN

HAMILTON

AND

QUATERNIONS I.

Introduction: Hamiltonian Historiography

17

Hamilton's Life and Fame

19

III.

Hamilton and Complex Numbers

23

IV.

Hamilton's Discovery of Quaternions

27

Quaternions until H a m i l t o n ' s D e a t h (1865)

33

II.

V. VI.

Chapter

Summary and Conclusion

41

Notes

43

Three

OTHER

EARLY

ESPECIALLY

VECTORIAL

GRASSMANN'S

SYSTEMS, THEORY OF

EXTENSION I. Introduction II.

III. IV.

47

August Ferdinand Mobius and His

Barycentric

Calculus

48

Giusto Bellavitis and His Calculus of Equipollences

52

Hermann

Grassmann

and His

Calculus

Introduction

of Extension: 54

V.

G r a s s m a n n ' s Theorie der Ebbe und Flut

60

VI.

G r a s s m a n n ' s Ausdehnungslehre o f 1 8 4 4

63 ix

Contents

VII. VIII.

IX.

T h e Period from 1844 to 1862 Grassmann's

77

Ausdehnungslehre

of

1862

and

the

Gradual, Limited Acceptance of His Work

89

Matthew O'Brien

96

Notes

Chapter

102

Four

TRADITIONS THE

I. II.

IN

MIDDLE

VECTORIAL PERIOD OF

ANALYSIS ITS

FROM

HISTORY

Introduction

109

Interest in Vectorial Analysis

in Various

Countries

from 1841 to 1900 III.

110

Peter Guthrie Tait:

Advocate

and

Developer of

Quaternions IV.

V. VI.

Chapter

Benjamin

117

Peirce:

Advocate

of Quaternions

in

America

125

James Clerk Maxwell: Critic of Quaternions

127

William Kingdon Clifford: Transition Figure

139

Notes

144

Five

GIBBS

AND

HEAVISIDE

DEVELOPMENT

OF

AND

THE

THE

MODERN

SYSTEM OF VECTOR ANALYSIS I.

Introduction

150

II. Josiah W i l l a r d Gibbs III.

Gibbs' Early W o r k in Vector Analysis

152

IV.

G i b b s ' Elements o f Vector Analysis

155

Gibbs' Other W o r k Pertaining to Vector Analysis

158

V. VI. VII. VIII. IX. X.

Chapter I. II. III.

x

150

Oliver Heaviside

162

Heaviside's Electrical Papers

163

H e a v i s i d e ' s Electromagnetic Theory

169

T h e Reception Given to Heaviside's Writings

174

Conclusion

177

Notes

178

Six

A

STRUGGLE

FOR

EXISTENCE

Introduction

IN

THE

1890'S 182

T h e "Struggle for Existence"

183

Conclusions

215

Notes

221

Contents

Chapter

Seven

THE

EMERGENCE

OF

THE

OF VECTOR ANALYSIS: I. II.

III.

SYSTEM

Introduction Twelve

Major Publications

225 in Vector Analysis

from

1894 to 1910

226

Summary and Conclusion

239

Notes

243

Chapter Eight Notes

Index

MODERN

1894-1910

SUMMARY AND CONCLUSIONS

247 255

260

xi

Preface

to

the

Dover

Edition

It is v e r y gratifying that interest in the materials presented in this v o l u m e i s s u f f i c i e n t t o j u s t i f y a s e c o n d e d i t i o n . T h i s has p e r m i t t e d t h e c o r r e c t i o n o f a n u m b e r of small errors and, more importantly, provides an opportunity to b r i n g t o r e a d e r s ' a t t e n t i o n s o m e o f t h e r e l e v a n t s t u d i e s o f s p e c i f i c areas w h i c h h a v e a p p e a r e d s i n c e t h e b o o k ' s first p u b l i c a t i o n i n 1967. Recent researches have shed light particularly on the history of algebra d u r i n g the nineteenth century. T h e most broadly conceived of such works is

Lubos

Novy's

Origins

o f Modern

Algebra.l

British

developments

in

algebra have received most attention, i m p o r t a n t studies having b e e n p u b lished by Harvey W.

Becher,

J.

M.

Dubbey,

Philip C.

Enros,

Elaine

K o p p e l m a n , L u i s M . Laita, a n d Joan L . Richards.2 Interest i n Sir W i l l i a m R o w a n H a m i l t o n ' s a c h i e v e m e n t s i n a l g e b r a has b e e n especially intense. R e s e a r c h i n t h i s a r e a has b e e n a i d e d b y t h e a p p e a r a n c e i n 1967 u n d e r t h e editorship of H. Hamilton's

Halberstam and R.

Mathematical Papers,

tions in algebra.3 T h o m a s L.

E.

Ingram of the third volume of

that volume

being devoted

H a n k i n s has e n r i c h e d

to his publica-

H a m i l t o n i a n schol-

arship by various publications, most notably his e n g a g i n g b i o g r a p h y of the great Irish m a t h e m a t i c i a n a n d scientist.4 T h e scholar most actively engaged in assessing H a m i l t o n ' s place in t h e history of B r i t i s h algebra is H e l e n a M. Pycior,

w h o s e d o c t o r a l d i s s e r t a t i o n i n t h i s a r e a has b e e n f o l l o w e d b y a

number

of studies

of the

contemporaries.5 Jerold algebraic/analytic

researches

W a e r d e n has p r o v i d e d quaternions.7

algebraic

David

ideas

of Hamilton

and

M a t h e w s has p u b l i s h e d a p a p e r o n during

the

a n e w analysis

1830's,6 w h i l e

of Hamilton's

B l o o r has b r o a d l y c o n s i d e r e d

B.

his

British

Hamilton's L.

van

der

1843 d i s c o v e r y o f

Hamilton's algebraic

a p p r o a c h in r e l a t i o n to t h e social, political, a n d p h i l o s o p h i c a l context o f his times,8 whereas T. the genesis and

L.

H a n k i n s a n d John H e n d r y have focused studies on

importance of Hamilton's conception of algebra as the

"Science of Pure Time."9 Arnold R.

N a i m a n i n his d o c t o r a l dissertation

surveyed the role of quaternions

the overall development of mathe-

matics.10 xii

in

Preface to the D o v e r Edition

The fascination

I felt for

Hamilton while

researching this b o o k was

rivaled, if not surpassed, as I l e a r n e d m o r e of his r e m a r k a b l e c o n t e m p o r a r y Hermann Grassmann.

M a n y issues I e n c o u n t e r e d i n s t u d y i n g his m a t h e -

matical creations have b e e n treated in d e p t h by A l b e r t C. doctoral dissertation

is

Lewis whose

a c a r e f u l a n a l y s i s o f G r a s s m a n n ' s Ausdehnungslehre

o f 1844 a n d its sources. D r . L e w i s has n o w p u b l i s h e d s o m e o f his r e s u l t s i n papers on the influence of Grassmann's father and of Schleiermacher on his mathematical system.11 M o r e o v e r , Jean D i e u d o n n e a n d D e s m o n d Fearnl e y - S a n d e r h a v e e a c h p u b l i s h e d essays o n G r a s s m a n n ' s p l a c e i n t h e h i s t o r y of linear algebra.12 R e c e n t r e s e a r c h e s h a v e also d e v e l o p e d n e w p e r s p e c t i v e s o n f i g u r e s less c e n t r a l t h a n H a m i l t o n a n d G r a s s m a n n i n t h i s h i s t o r y . H e l e n a M . P y c i o r has presented ciative

a fresh

Algebra,13

analysis

and

of B e n j a m i n

Hubert

Kennedy

Peirce's has

p i o n e e r i n g Linear Asso-

investigated

James

Mills

Peirce's place in t h e " c u l t of q u a t e r n i o n s " that arose in late n i n e t e e t h century America.14 G.

C . S m i t h i n a r e c e n t p a p e r has u r g e d t h a t M a t t h e w

O ' B r i e n deserves significantly m o r e credit as a pioneer of the

modern

v e c t o r i a l s y s t e m t h a n has t r a d i t i o n a l l y b e e n a c c o r d e d h i m , 1 5 a n d t h e late B. R.

Gossick has

provided

n e w insights on

the contrasting views

of

vectorial methods espoused by O l i v e r Heaviside and L o r d Kelvin.16 T h e first systematic study of the history of Stokes' T h e o r e m a n d of the associated theorems n a m e d after Gauss and G r e e n is d u e to V i c t o r J.

Katz.17 T h e

i n v o l v e m e n t o f R u s s i a n m a t h e m a t i c i a n s w i t h v e c t o r i a l m e t h o d s has b e e n treated b y W . D o b r o v o l s k i j , 1 8 a n d studies o f t h e history o f v e c t o r analysis i n general have b e e n u n d e r t a k e n by A d a l b e r t A p o l i n a n d James W. Joiner, b o t h of w h o m seem to have w r i t t e n w i t h o u t k n o w l e d g e of my book.19 F o r t h e sake o f c o m p l e t e n e s s , m e n t i o n s h o u l d b e m a d e o f t h r e e p u b l i c a tions w h i c h a p p e a r e d before m y book, b u t w h i c h escaped m y b i b l i o g r a p h i c searches. T w o o f these, b o t h f r o m t h e 1930 s , t r e a t t h e h i s t o r y o f c o m p l e x n u m b e r s ; the earlier was w r i t t e n by J. B u d o n whereas the second is E r n e s t Nagel's " I m p o s s i b l e N u m b e r s . " 2 0 T h e latter omission is especially regrettab l e b e c a u s e t h a t essay p r o v i d e s a h i s t o r i o g r a p h i c p e r s p e c t i v e w h i c h w o u l d h a v e e n r i c h e d m y p r e s e n t a t i o n . T h i s i s e v e n m o r e t r u e o f t h e t h i r d essay, that p u b l i s h e d in 1 9 6 3 - 6 4 by the late I m r e Lakatos.21 A l t h o u g h c o n t a i n i n g essentially n o t h i n g r e l e v a n t to the h i s t o r y of vector analysis, Lakatos' n o w f a m o u s essay i s r i c h i n p h i l o s o p h i c a n d h i s t o r i o g r a p h i c i n s i g h t s w h i c h , w e r e I to rewrite this book, w o u l d certainly be included. Some hints as to the d i r e c t i o n I w o u l d t a k e a r e p r o v i d e d i n a h i s t o r i o g r a p h i c essay I p u b l i s h e d i n 1975.22 Persons w h o m a y acquire f r o m this book an interest in f u r t h e r readings in the history of mathematics

may wish

to consult

the

excellent

general

histories of mathematics w r i t t e n by Carl B. Boyer and M o r r i s Kline,23 or for articles on i n d i v i d u a l mathematicians, o f Scientific

Biography,24

Bibliographic

t h e m a n y v o l u m e s o f t h e Dictionary

searches

which

required

days

in xiii

Preface to the D o v e r Edition

t h e e a r l y 1960's c a n n o w b e a c c o m p l i s h e d b y s p e n d i n g a f e w h o u r s w i t h t h e late

Kenneth

Mathematics

O.

Mays

and

with

Bibliography

the

ISIS

and

Research

Manual

of the

History

of

Bibliography.25

Cumulative

In c o n c l u d i n g this u p d a t e d preface, I extend w a r m e s t thanks to the two persons

w h o have

made

this

new edition

possible:

John

W.

Grafton,

Assistant to t h e P r e s i d e n t of D o v e r Publications, a n d James R. L a n g f o r d , D i r e c t o r o f t h e U n i v e r s i t y o f N o t r e D a m e Press. T h e e n c o u r a g e m e n t o f t h e f o r m e r a n d t h e cooperation of the latter are chiefly responsible for this book b e i n g once again available to readers. M i c h a e l J. C r o w e University of Notre

Dame

January, 1985

Notes 1

LuboS

2

Harvey

Novy,

Mathematica, Historia

W.

Origins

7 (1980),

Mathematica,

(1812-1813): (1983),

"Woodhouse,

3 8 9 - 4 0 0 ; J. 4

(1977),

Elaine

Archive for

M.

by Jaroslav T a u e r

Dubbey,

Renewal

Koppelman,

History

trans, Babbage,

295-302;

Precursor of t h e

24-47;

Algebra,"

o f Modern Algebra,

Becher,

Peacock,

and

"Babbage,

Philip

C.

1973).

A l g e b r a , " Historia

Peacock a n d M o d e r n A l g e b r a , "

Enros,

of C a m b r i d g e

(Leyden,

Modern

"The

Analytical

M a t h e m a t i c s , " Historia

Society

Mathematica,

10

" T h e C a l c u l u s o f O p e r a t i o n s a n d t h e Rise o f Abstract

o f Exact

Sciences,

8

(1971),

155-242;

Luis

M.

Laita,

"The

Influence of Boole's Search for a Universal M e t h o d in Analysis on the Creation of His Logic," Annals

o f Science,

Controversy

34

(1977),

between

163-176;

William

Luis

Hamilton

M.

Laita,

and

"Influences

Augustus

De

on

Boole's

Logic:

The

M o r g a n , " Annals o f Science,

36

(1979), 4 5 - 6 5 ; L u i s M . L a i t a , " B o o l e a n A l g e b r a a n d Its E x t r a - l o g i c a l Sources: T h e T e s t i m o n y of M a r y

Everest

Boole,"

History and Philosophy o f Logic,

" T h e A r t and the Science of British Algebra: Truth," 3

H.

Hamilton, 4

Historia

Mathematica,

Halberstam vol.

Ill:

Thomas

L.

7

and

Algebra

(1980),

R.

E.

Hankins,

Sir

William

(1980),

3 7 - 6 0 ; Joan

L.

Richards,

343-365.

Ingram,

(Cambridge,

1

A Study in the Perception of Mathematical

The

Mathematical

England, Rowan

Papers

of

Sir

William

Rowan

1967).

Hamilton

(Baltimore,

1980).

For

a

shorter

b i o g r a p h y w h i c h discusses H a m i l t o n ' s m a t h e m a t i c s o n a m o r e e l e m e n t a r y l e v e l , see: Sean O'Donnell, 5

William

Helena

Algebra ( A

M.

Rowan

Hamilton:

Pycior,

1976 C o r n e l l

The

Portrait

Role

of Sir

of a

Prodigy

William

Hamilton

University doctoral dissertation);

(Dublin,

1983).

in

the

Development

H.

M.

Pycior,

and the

B r i t i s h O r i g i n s o f S y m b o l i c a l A l g e b r a , " Historia Mathematica, 8 ( 1 9 8 1 ) ,

Pycior,

"Early

Criticisms

(1982), 3 9 2 - 4 1 2 ; H . Isis, 6

74 ( 1 9 8 3 ) ,

7

L.

available

in

Approach

to

Algebra,"

23-45;

Historia

H.

M.

Mathematica,

9

211-226. "William Rowan

Archive for

B.

Symbolical

Modern

M . Pycior, " A u g u s t u s D e Morgan's Algebraic W o r k : T h e T h r e e Stages,"

Jerold Mathews,

Analysis, "

of t h e

of British

" G e o r g e Peacock

van

History der

English

of Exact

Waerden,

as

H a m i l t o n ' s P a p e r o f 1837 o n t h e A r i t h m e t i z a t i o n o f

Sciences,

19

Hamiltons

"Hamilton's

(1978),

Entdeckungder

Discovery

177-200. Quaternionen(Gottingen,

of Q u a t e r n i o n s , "

1974).

Mathematics

Now

Magazine,

49

(1976), 2 2 7 - 2 3 4 . 8

David

Nineteenth

Bloor,

Century

"Hamilton

Mathematics,

and

ed.

by

Peacock

on

Herbert

the

Essence of A l g e b r a "

Mehrtens,

Henk

Bos

in

and

Social History o f Ivo

Schneider

( B o s t o n , 1981), p p . 2 0 2 - 2 3 2 . 9

Thomas L. Hankins, "Algebra as Pure Time: William Rowan Hamilton and the Founda-

tions

xiv

of

Algebra"

in

Motion

and

Time,

Space

and

Matter:

Interrelations

in

the

History

of

Preface to the D o v e r Edition

Philosophy and Science,

ed.

by P.

J.

Machamer and

R.

G.

Turnbull (Columbus,

1976),

pp.

327-359. 10

Arnold

R.

Naiman,

The

Role

of Quaternions

in

the

History

of

Grassmanns

of Mathematics

(A

1974

New

York University doctoral dissertation). 11

Albert

C.

Lewis,

An

Historical

Analysis

1975 U n i v e r s i t y o f T e x a s a t A u s t i n d o c t o r a l dissertation); A . Ausdehnungslehre A.

and

Schleiermacher's

Dialektik ,"

Annals

Ausdehnungslehre

of 1844

(A

C . L e w i s , " H . G r a s s m a n n ' s 1844

of Science,

34

(1977),

103-162;

C. Lewis, "Justus Grassmann's School Programs as Mathematical Antecedents of H e r m a n n

Grassmann's in

the

Early

1844

Ausdehnungslehre"

Nineteenth

Century,

ed.

in

by

Epistemological

Hans

Niels

and

Jahnke

Social and

Problems

Michael

of the

Otte

Sciences

(Dordrecht,

1981), p p . 2 5 5 - 2 6 7 . 12

Jean

1-14;

Dieudonne,

Desmond

gebra,"

American

"Hermann Monthly, 13

"The

Mathematical

Grassmann

89

(1982),

Helena

Tragedy

Fearnley-Sander, Monthly,

and

of G r a s s m a n n , ' '

"Hermann 86

the

Linear and

Grassmann

(1979),

809-817;

Prehistory

of

Multilinear Algebra,

and

the Creation

see

also

Universal

D.

8

(1979),

of Linear Al-

Fearnley-Sander,

Algebra,"

American

Mathematical

161-166.

M.

Pycior,

"Benjamin

Peirce's

Linear

Associative

Algebra,"

I sis,

70

(1979),

537-551. 14

Hubert

Kennedy,

"James

Mills

Peirce and

t h e C u l t o f Q u a t e r n i o n s , " Historia

Mathemat-

ica, 6 ( 1 9 7 9 ) , 4 2 3 - 4 2 9 . 15

G.

C.

Smith,

Mathematica, 16

B.

9

R.

"Matthew

(1982),

O'Brien's

Anticipation

of Vectorial

Mathematics,"

Historia

172-190.

Gossick,

"Heaviside

and

Kelvin:

A

Study

in

C o n t r a s t s , " Annals

o f Science,

33

(1976), 2 7 5 - 2 8 7 . 17

Victor

J.

Katz,

"The

History

of Stokes'

Theorem,"

Mathematics

Magazine,

52

(1979),

146-156. 18

W . D o b r o v o l s k i j , " D e v e l o p p e m e n t d e l a t h e o r i e des v e c t e u r s e t des q u a t e r n i o n s dans les

travaux des

mathematiciens

russes

du X I X e

siecle,"

Revue d'histoire des sciences,

21

(1968),

345-349. 19

Adalbert

ralis,

12

Apolin,

(1970),

"Die

357-365;

geschichtliche James

Entwicklung der Vektorrechnung,"

W a l t e r Joiner,

A

History of Vector Analysis (A

Historia Natu-

1971

doctoral

dissertation at G e o r g e Peabody C o l l e g e for Teachers). 20

J.

Budon,

quelques (1933), Ideas, the

175-200; 3

(1935),

Philosophy 21

" S u r la r e p r e s e n t a t i o n g e o m e t r i q u e s des n o m b r e s imaginaires (Analyse de

memoires

Imre

parus

de

220-232; 427-474;

and

History

Lakatos,

1795

Ernest reprinted

of Science

"Proofs

a

1820),"

Nagel, in

(New

and

Bulletin

des sciences

"Impossible

Ernest York,

Nagel,

1979),

Refutations,"

mathematiques,

Numbers,"

ser. the

2,

57

History

of

Teleology Revisited and Other Essays

in

pp.

Studies i n

166-194.

British Journal for

the

Philosophy

of Science,

14

( 1 9 6 3 - 1 9 6 4 ) , 1 - 2 5 ; 1 2 0 - 1 3 9 ; 2 2 1 - 2 4 5 ; 2 9 6 - 3 4 2 . T h e s e e s s a y s h a v e n o w b e e n r e p u b l i s h e d as: Imre

Lakatos,

Proof and

Refutations:

The

Logic

of

Mathematical

Discovery,

ed.

by

John

W o r r a l l a n d E l i e Z a h a r ( C a m b r i d g e , E n g l a n d , 1976). 22

M.

Historia 23

ical 24

J. Crowe, " T e n ' L a w s ' C o n c e r n i n g Patterns ofChange in the History of Mathematics," Mathematica,

Carl

B.

2

Boyer,

Thought from

(1975), A

Ancient

161-166.

History o f Mathematics ( N e w Y o r k , to

Modern

Dictionary o f Scientific Biography,

14

Times

(New

vols.,

ed.

1968) a n d

York, by

Morris

Kline,

Mathemat-

1972).

Charles

Coulston

Gillispie

( N e w York,

Science

Formed from

1970-1980). 25

ISIS

ISIS

Cumulative

Critical

1 9 7 1 - 1 9 8 2 ) ; ISIS 1980);

and

Bibliography.

Bibliographies

1-90:

Cumulative

Kenneth

O.

A

Bibliography

1913-1965,

Bibliography May,

5

of

vols.,

1966-1975, Bibliography

vol. and

the

ed.

History by

I,

Research

of

Magda

ed.

by

Manual

Whitrow

John of the

Neu History

(London, (London, of Mathematics

( T o r o n t o , 1973). XV

CHAPTER

The

I.

Earliest

ONE

Traditions

Introduction The

early

history

of

vectorial

analysis

is

most

viewed within the context of two broad traditions

appropriately

in the history of

science. O n e of these traditions relates to mathematics, the other to physical

science.

T h e first tradition, that w i t h i n the history of mathematics, extends from the time of the Egyptians and Babylonians to the present and consists

in

the

progressive

Throughout time to include

the

broadening of the

concept of number.

c o n c e p t of n u m b e r has b e e n e x p a n d e d so as

not only positive

integers, but negative

numbers,

frac-

tions, and algebraic and transcendental irrationals. E v e n t u a l l y complex

and

higher complex

numbers

(including

vectors)

were intro-

duced. T h e activities of some of the figures in the history of vector analysis m a y be v i e w e d as b e l o n g i n g to this tradition. The also

second tradition, that w i t h i n the history of physical science,

extends

back

mathematical

to

ancient

entities

and

times

a n d consists

operations

that

in the

represent

search for aspects

of

physical reality. This tradition p l a y e d a part in the creation of G r e e k geometry, and the inherited problems.

from

natural philosophers of the seventeenth century

the

Greeks

However

in the

the

geometrical

course

of the

approach

seventeenth

to

physical

century the

physical entities to be represented passed t h r o u g h a transformation. This

transformation

consisted

in

the

shift

in

emphasis

from

such

scalar q u a n t i t i e s as p o s i t i o n a n d w e i g h t to s u c h v e c t o r i a l q u a n t i t i e s as velocity, force, m o m e n t u m , and acceleration. T h e transition was neither abrupt nor was it confined to the seventeenth century. Later developments to

transform

with

in

the

electricity, space

magnetism,

of mathematical

and

optics

physics

into

acted

further

a space filled

vectors.

These two broad traditions converged at a n u m b e r of periods history;

one

such

period

was

in the

in

nineteenth century, and this

1

A H i s t o r y of V e c t o r Analysis

convergence

is

marked

by

the

creation

and

d e v e l o p m e n t of vec-

torial methods. T h e first major three-dimensional vectorial systems were

created in

portant ideas

1840,s.

the

Before

were put forth w h i c h

this

time, however, three im-

l e d t o t h e m a j o r v e c t o r i a l sys-

tems. T h e s e t h r e e ideas are the subject of the present chapter; t h e y are the c o n c e p t of a p a r a l l e l o g r a m of forces, L e i b n i z ' c o n c e p t of a geometry of situation, and the concept of the geometrical representation

of imaginary

II.

The and

Concept

numbers.

of

the

Parallelogram

of

Velocities

Forces

O n e of the most f u n d a m e n t a l m a t h e m a t i c a l ideas in vector analysis

is

the

idea of the addition of vectors.

T h e sum of t w o vectors

w h i c h have a c o m m o n point of origin is defined as the vector originating at the same point and extending to the opposite corner of the parallelogram d e f i n e d by the t w o original vectors. Certain physical entities,

such

as

velocities

a n d forces,

m a y be c o m p o u n d e d

in

a

similar w a y , a n d f r o m this c o r r e s p o n d e n c e stems m u c h of the usefulness

of vector analysis.

T h e idea of a parallelogram of velocities may be found in various ancient forces

authors,8 *

Greek was

not

uncommon

and in

the the

concept

of a

parallelogram

of

sixteenth and seventeenth cen-

turies.9 By the early n i n e t e e n t h century parallelograms of physical entities

frequently

appeared

in

treatises,

and

this

usage

indirectly

led to vector analysis, for this idea p r o v i d e d a striking example of how

vectorial

entities

could

be

used

for physical

applications.

It

s h o u l d not be inferred, h o w e v e r , that all of those w h o used the concept of a parallelogram of physical entities were aware of the idea of a vector or of vector addition.

T h e essential

idea in the parallelo-

gram of physical entities is the construction of a diagram in terms of w h i c h the operations involved in determining the resultant become evident.

The

i d e a o f adding t h e l i n e s n e e d n o t b e i n t r o d u c e d o r w a s

i t (to m y k n o w l e d g e ) e v e r i n t r o d u c e d b e f o r e t h e c r e a t i o n o f vectors. T h u s this

i d e a a l o n e c o u l d n o t a n d a l m o s t c e r t a i n l y d i d n o t directly

s t i m u l a t e a n y o n e to t h e c r e a t i o n of a vectorial system. Its i n f l u e n c e was

indirect b u t i m p o r t a n t , f o r i t w a s t h e f i r s t a n d m o s t o b v i o u s c a s e

in w h i c h vectorial methods could be brought to the aid of physical science. ° T h e system used for n u m b e r i n g notes is described in the preface.

2

The

III.

Leibniz'

Concept

of

a

Geometry

of

Earliest

Traditions

Situation

Gottfried W i l h e l m Leibniz (1646-1716) made many contributions to

mathematics;

among

geometry of situation.

the

In

less

this

well

regard

known

is

Leibniz

his

concept

discussed

of a

the possi-

bility of creating a system w h i c h w o u l d serve as a direct m e t h o d of space

analysis.

Although

the

details

of this

idea were

never fully

w o r k e d out by L e i b n i z , he a d v a n c e d far e n o u g h to be r a n k e d as a conceptual

forerunner of the

essay, w h i c h w a s

history of vectorial Leibniz' main 8,

first

vectorial

first p u b l i s h e d in

analysts.

M o r e o v e r his

1833, p l a y e d a part in the later

analysis.

ideas w e r e contained in a letter dated S e p t e m b e r Huygens.1

1679, a n d w r i t t e n to Christian

In this

letter L e i b n i z

wrote: I am still not satisfied w i t h algebra, because it does not give the shortest methods or the most beautiful constructions in geometry. This is w h y I believe that,

so far as

geometry is

concerned, we

need still

another

analysis w h i c h is distinctly geometrical or linear a n d w h i c h w i l l express situation

[situs]

directly

as

algebra

expresses

magnitude

directly.

And

I

believe that I have f o u n d the w a y a n d that we can represent figures a n d even

machines

and

movements

by

characters,

as

algebra represents

n u m b e r s or m a g n i t u d e s . I am s e n d i n g y o u an essay w h i c h seems to me t o b e i m p o r t a n t . (1; 3 8 2 ) I n his essay, w h i c h was c o n t a i n e d i n the letter, L e i b n i z d e s c r i b e d his

system further: I

have discovered certain e l e m e n t s of a n e w characteristic w h i c h is

entirely different f r o m algebra a n d w h i c h w i l l have great advantages i n r e p r e s e n t i n g t o t h e m i n d , e x a c t l y a n d i n a w a y f a i t h f u l t o its n a t u r e , e v e n w i t h o u t f i g u r e s , e v e r y t h i n g w h i c h d e p e n d s o n sense p e r c e p t i o n . A l g e b r a is the characteristic for u n d e t e r m i n e d n u m b e r s or m a g n i t u d e s o n l y , b u t it does not express situation, angles, a n d m o t i o n directly. H e n c e it is often difficult to analyze the properties of a figure by calculation, and still more difficult to find very convenient geometrical demonstrations and constructions,

even

w h e n the algebraic calculation

is completed.

But

this n e w characteristic, w h i c h follows the visual figures, cannot fail to give the solution, the construction, and the geometric demonstration all at the

same time,

and

in

a natural

w a y and in

one

analysis,

t h a t is,

through determined procedure.

B u t its c h i e f v a l u e l i e s i n t h e r e a s o n i n g w h i c h c a n b e d o n e a n d t h e c o n clusions could

which

not

be

c a n b e d r a w n b y o p e r a t i o n s w i t h its c h a r a c t e r s , w h i c h expressed

in

figures,

and

still

less

in

models,

without

m u l t i p l y i n g these too greatly o r w i t h o u t c o n f u s i n g t h e m w i t h too m a n y points and lines in the course of the m a n y futile attempts one is forced to

make.

This

method,

by contrast, w i l l g u i d e us

surely and without

3

A

History

effort.

of V e c t o r Analysis

I b e l i e v e that by this m e t h o d one c o u l d treat mechanics almost

l i k e g e o m e t r y , a n d o n e c o u l d e v e n test the qualities of materials, because this o r d i n a r i l y d e p e n d s o n c e r t a i n f i g u r e s i n t h e i r s e n s i b l e parts. F i n a l l y , I have no h o p e that we can get v e r y far in physics u n t i l we have f o u n d s o m e s u c h m e t h o d o f a b r i d g m e n t t o l i g h t e n its b u r d e n o f i m a g i n a t i o n . (1; 3 8 4 - 3 8 5 )

His

system

as actually sketched by h i m shows that he by no means

discovered a primitive vector analysis, though the above quotations show

that

he

was

searching

for

s o m e t h i n g a k i n to vector analysis.

L e i b n i z ' s y s t e m c e n t e r e d o n t h e i d e a o f t h e c o n g r u e n c e o f sets o f points.

He used A, B, . . . to represent fixed points and X, Y , . . . to

represent u n k n o w n relation

points.

congruence;

The

thus

he

s y m b o l b was used to express the wrote

A B C b DEF

to

express

that

a

set of t h r e e p o i n t s A, B, C, e a c h of w h i c h w a s a fixed distance f r o m t h e o t h e r t w o p o i n t s , c o u l d b e m a d e t o c o i n c i d e w i t h a n o t h e r set o f similarly fixed points D, E, F. He then discussed locus relations and stated that the locus of points congruent to a

fixed

space

g i v e n t h a t AB

infinite

in

all

directions."

(1;

387)

If it is

point " w i l l be a b

AY,

the p o i n t values of Y w i l l be points on a sphere w i t h center at A and radius

of

whose

points

length

A B C b A B Y

(X)

AB.

The

are

equidistant

determines

relation

AX b BX

from

a circle.

locus

that the

of all

Y's

r e l a t i o n AY

will V

BY

be V

a

determines

and

B,

and

a

the

plane

relation

L e i b n i z then discussed the locus

of points Y satisfying the relation AY "the

A

b BY

straight

CY V

DY

line."

b

CY and c o n c l u d e d that (1;

389)

determines

applied his analysis to four simple problems.

After

showing

a point, Leibniz

O n e of these may be

discussed as typical. T h e p r o b l e m is to show that the intersection of two planes plane,

is a straight line.

and

the

combining before,

relation

these

we

determines

AY

have

T h e relation AY b BY determines one b CY

AY

determines

b BY b CY,

a

second

which,

as

plane.

it was

By

shown

a straight line.10

Proceeding from this s u m m a r y of Leibniz' best-known exposition system,11

of his sis.

First,

w e m a y d i s c u s s its r e l a t i o n t o m o d e r n v e c t o r a n a l y -

Leibniz

deserves

much

credit for suggesting that a n e w

algebra, w h e r e i n geometrical entities are symbolically represented and he

the

symbols

operated u p o n directly, was

desirable.

However,

failed to discover a system in w h i c h geometrical entities could

be a d d e d , subtracted, a n d m u l t i p l i e d . L i k e w i s e he failed to see that A B a n d B A (for e x a m p l e ) can b e v i e w e d a s distinct entities a n d that —AB

could have a significant meaning.

representing runner the

4

of

a

fixed

Mobius

concept

point

and

of a vector.

by

a

T h o u g h his idea of directly

symbol

Grassmann,

he

makes

certainly

him did

a partial forenot

introduce

D e s p i t e the fact that angle considerations

The

Earliest Traditions

d i d not enter into his system, he still m u s t be v i e w e d as h a v i n g constructed a system w h i c h a l l o w e d for the use of co-ordinai^s. L e i b n i z saw that a n e w algebra of the applications failed

to

in

mathematics

develop

Leibniz'

system

practical taken

form

and

sought

in

methods

by

w o u l d have

the physical for these

Couturat,

though

numerous

sciences, but he

tasks.

stated

The in

view

of

relation

to

Grassmann's system, is also a p p l i c a b l e in r e l a t i o n to m o d e r n vector analysis; Couturat wrote:

" I n summary, the calculus of Grassmann

seems to bring fully into reality the geometrical characteristic conceived by Leibniz, and shows that Leibniz' idea was not simply a dream. But there is such a disproportion b e t w e e n Leibniz' conception

of a

system

and

the

very

defective

p r o d u c e d that Grassmann felt a sharp

essay

which

he

actually

distinction should be made

between the ideal conceived and the sketch actually written." Shortly after Jablonowski

1833,

12

w h e n L e i b n i z ' essay was first p u b l i s h e d , the

Gesellschaft expressed their interest in

and enthusi-

asm for the essay by offering a prize for the f u r t h e r d e v e l o p m e n t of Leibniz'

system.

One

mathematician entered the competition and

w o n the prize, e v e n t h o u g h he had created his system before hearing of L e i b n i z ' ideas; this mathematician was Grassmann a n d this incident w i l l be more fully discussed in the third chapter.

IV.

The of

Concept

Complex

Though ence

to

the

term

systems

dimensional

of

the

Geometrical

Representation

Numbers vector

of

space,

analysis

is

mathematics it

should

now

that

not

be

used

may

primarily

be

applied

forgotten

that

in

refer-

in

the

three-

complex

n u m b e r s y s t e m m a y l e g i t i m a t e l y be c o n s i d e r e d as a v e c t o r i a l system.

The

metrical useful

as

two-dimensional representation the

of

vectorial complex

three-dimensional

primary subject of this history. cuss

briefly

system

based

numbers

vectorial

is

on

the

certainly

systems

which

geo-

not are

as the

Nevertheless it is i m p o r t a n t to dis-

the early history of the

geometrical representation

of

complex numbers, not only because the complex n u m b e r system is (broadly

speaking)

a vectorial

system

b u t also

because

Hamilton

discovered quaternions in the course of a search for a t h r e e - d i m e n sional analogue to the complex n u m b e r system. A t least six m e n are c o m m o n l y c r e d i t e d w i t h t h e d i s c o v e r y o f t h e geometrical

representation

of complex

numbers;

t h e y are Wessel,

Gauss, Argand, Buee, M o u r e y , a n d Warren.13 Since the systems crea t e d b y these six m e n are v e r y s i m i l a r a n d are o f l i m i t e d r e l e v a n c e to

the

present

study, they need not all be discussed in

detail.

In

5

A H i s t o r y of V e c t o r Analysis

w h a t follows, the system p u b l i s h e d by Wessel, w h i c h was the earliest a n d a m o n g t h e m o s t i m p r e s s i v e , w i l l b e a n a l y z e d i n s o m e d e p t h ; t h e ideas o f t h e o t h e r f i v e m e n w i l l b e t r e a t e d less f u l l y t h o u g h w i t h special will

be

attention

to

certain

shown

that

some

three-dimensional

aspects of their d e v e l o p m e n t . T h u s it of

vectorial

these

mathematicians

systems

and

that

one

searched

of them

for

influ-

enced Hamilton in an important manner. Although

Hero

of Alexandria

and

Diophantus

in ancient times

h a d encountered the question of the m e a n i n g of the square root of a negative

number,

and although

Cardan

had in his

1 5 4 5 Ars Magna

used complex numbers in computation, nevertheless complex numbers were not accepted by most mathematicians as legitimate mathematical

entities

until

well

into

hardly surprising since numbers

the

nineteenth

such as

century.

This is

V ^ T seem to be neither

less t h a n , greater t h a n , n o r e q u a l to zero. In

modern

mathematics

complex

numbers

are

usually justified

either by representing t h e m in terms of couplets of real numbers or by representing them geometrically. T h e origin of the first method w i l l be discussed in the next chapter. T h e first attempt (which was unsuccessful)

to

represent

complex

numbers

geometrically

was

m a d e in the seventeenth century by John Wallis.14 W h e r e Wallis failed, a Norwegian Caspar Wessel geometrical

(1745-1818)

representation

surveyor succeeded;

in

1799

published the first explanation of the of complex

numbers.15

His

ideas w e r e

p r e s e n t e d before the Royal A c a d e m y o f D e n m a r k i n 1797 a n d publ i s h e d t w o years later in the m e m o i r s of that society.2 Unfortunately, however, Wessel's ematicians until

publication went unnoticed by European math-

1897, w h e n it was r e p u b l i s h e d in a F r e n c h transla-

tion.3 In the first paragraph of his m e m o i r W e s s e l stated: attempt

deals

with

the

question,

"This present

h o w may we represent direction

a n a l y t i c a l l y ; t h a t is, h o w s h a l l w e e x p r e s s r i g h t l i n e s s o t h a t i n a single the

equation length

i n v o l v i n g one u n k n o w n line and others k n o w n , both and

the

direction

of the

unknown

line

may

be

ex-

p r e s s e d . " (2; 55) A s t h i s q u o t a t i o n suggests a n d later passages confirm, Wessel's chief interest was the creation of geometrical methods; his representation of c o m p l e x n u m b e r s was subservient to this aim.

Nonetheless

the

latter p l a y e d a f u n d a m e n t a l role as is indi-

cated by the following statement: treatise]

was

my

" T h e o c c a s i o n for its b e i n g [his

seeking a method whereby I could avoid the im-

possible operations. . .

(2; 57)

After stating that previously only oppositely directed lines could be r e p r e s e n t e d analytically, W e s s e l suggested that it s h o u l d be pos-

6

The

sible to a

find

definition

added

Traditions

methods to represent inclined lines. Wessel then gave of the

if we

Earliest

addition

of straight

lines:

" T w o

unite t h e m in such a w a y that the

right

lines

are

second line begins

w h e r e t h e first o n e e n d s , a n d t h e n pass a r i g h t l i n e f r o m t h e first to t h e last p o i n t o f t h e u n i t e d lines. T h i s l i n e i s t h e s u m o f t h e u n i t e d lines."

(2;

58)

In

the

subsequent

discussion

of addition

Wessel

stated that the same definition can be used in a d d i n g m o r e than t w o (not necessarily coplanar) lines a n d that the order of addition is immaterial.

(2;

59)

Hence Wessel

had

introduced three-dimensional

vector addition and realized the importance of the commutative law for

addition.

Though

Wessel

had

up

to

called the positive unit (our 1

this of x

point •

only

discussed

what

he

1 + yV—I) a n d h a d n o t y e t i n d i -

cated h o w lines in general were to be represented in terms of complex numbers, nevertheless he proceeded to introduce the multiplication of lines.

The

product of two lines

(coplanar with each other

and w i t h the positive unit) was to have a length equal to the product of the lengths of the t w o factors. T h e p r o d u c t l i n e was to be coplanar w i t h t h e t w o factor l i n e s a n d w a s t o h a v e its i n c l i n a t i o n o r d i r e c t i o n angle (defined by reference to the inclination of the positive unit as 0°) e q u a l Wessel

to the

then

L e t 4-1

sum

of the

inclinations

of the

factor lines.

(2; 60)

added:

designate the positive rectilinear unit and +e a certain other

unit perpendicular to the positive unit and having the same origin; then t h e d i r e c t i o n a n g l e o f + 1 w i l l b e e q u a l t o 0°, t h a t o f - 1 t o 180°, t h a t o f + e t o 90°, a n d that o f - e t o - 9 0 ° o r 270°. B y t h e r u l e that t h e d i r e c t i o n a n g l e of the p r o d u c t shall e q u a l the s u m of the angles of the factors, we have: (+1)(+1) = + 1 ;

( + 1 ) ( — 1 ) = —1; ( - 1 ) ( - 1 ) = + 1 ; ( + l ) ( + e ) = +

(-l)(+e) = - e ; ( - l ) ( - € ) = +6;

(+e)(+e) = - 1 ;

e ;

(+e)(-e) = + 1 ;

(+l)(-6) = -

€ ;

(-e)(-e) = - 1 .

F r o m this it is seen that e is equal to V ^ I ; a n d the divergence of the product is d e t e r m i n e d such that not any of the c o m m o n rules of operation are c o n t r a v e n e d . (2; 60) Wessel

stated that any straight line in a plane m a y be represented

analytically showed

how

by

the

expressions

such

expressions

a +

eb

and

are to be

r(cos

v + €

multiplied,

v)

and

divided,

sin

and

raised to powers.

After giving t w o examples of the application of his

methods, Wessel

developed an elementary three-dimensional vec-

tor analysis.

(3; 2 3 - 2 8 )

Wessel began by constructing three mutually perpendicular lines w h i c h passed t h r o u g h the center of a sphere of radius r. W e s s e l specified three

that three radii

of the sphere w h i c h were collinear with the

m u t u a l l y p e r p e n d i c u l a r axes

s h o u l d be

designated by r,

171%

and er and that any point in space c o u l d be designated by a vector of

7

A H i s t o r y of V e c t o r Analysis

t h e f o r m x + r)y + ez.

(3;

23-24)

By a n a l o g y w i t h o r d i n a r y c o m p l e x

n u m b e r s W e s s e l d e f i n e d 1717 a n d e e a s e q u a l t o — 1 . T h e m u l t i p l i c a tion

of vectors corresponded to the rotation and extension of one

v e c t o r by a n o t h e r . T h u s (x

f 171/ + ez)

,,

( c o s u + € s i n u) r e p r e s e n t e d

t h e r o t a t i o n o f t h e v e c t o r x + iqy + e z t h r o u g h t h e a n g l e u a r o u n d t h e 17 or y axis. W e s s e l stated that t h e c o m p o n e n t of t h e v e c t o r that lies on

the

axis

product

of

of t h e

rotation above

is

should 17y +

remain x

cos

unchanged,

u — z

sin

and

thus

u + ex s i n

the

u + ez

cos u . ( 3 ; 2 5 - 2 6 ) T h e s y m b o l , , w a s u s e d t o i n d i c a t e m u l t i p l i c a t i o n . A r o t a t i o n of v d e g r e e s a r o u n d t h e € or z axis w a s e x p r e s s e d in t h e f o l l o w i n g w a y : (x + 1 7 y + ez)

,, ( c o s v + 1 7 s i n v) = ez + x c o s v — y s i n v

+ rjx s i n v + 7)y c o s v . ( 3 ; 2 6 ) R o t a t i o n s a r o u n d t h e 1 7 a x i s c o u l d b e compounded with

rotations

a r o u n d t h e € axis a n d v i c e versa, b u t

r o t a t i o n s a r o u n d t h e axis of t h e p o s i t i v e u n i t (the x axis) w e r e n e v e r discussed by Wessel. T h e reason for this is that serious mathematical

difficulties

were

involved

in

determining how

such

rotations

s h o u l d b e r e p r e s e n t e d , f o r e x a m p l e , t h e p r o d u c t s 17c a n d erj w o u l d h a v e h a d to be defined. W e s s e l p r e s u m a b l y e n c o u n t e r e d these difficulties b u t c o u l d not solve them.16 But even w i t h this limitation on his m e t h o d s Wessel was able to use t h e m to derive a n u m b e r of important results in spherical trigonometry. Wessel's

t h r e e - d i m e n s i o n a l v e c t o r i a l s y s t e m e x h i b i t e d a n a d hoc

character that makes

it appear seriously deficient w h e n compared

to m o d e r n systems; nevertheless, if it is v i e w e d as a creation of the late

eighteenth century, it can only be v i e w e d w i t h awe. Wessel's

treatment of ordinary complex numbers is equally impressive, and it was unfortunate for Wessel a n d for mathematics that his m e m o i r lay b u r i e d for nearly a century. In the early history of complex numbers a striking p h e n o m e n o n occurred:

on three separate occasions t w o m e n i n d e p e n d e n t l y and

simultaneously

discovered

the geometrical representation of com-

plex numbers. In 1806 A r g a n d a n d Buee both p u b l i s h e d independent

treatments

of imaginary

numbers,

and the

same

coincidence

o c c u r r e d i n 1828 w i t h M o u r e y a n d W a r r e n . W h a t i s e v e n m o r e surprising

is

that

Gauss

probably

discovered the

geometrical repre-

sentation of complex numbers at the same time as Wessel. Gauss' tion

first

published

treatment

of the geometrical representa-

of complex numbers appeared in

1831;

17

herein Gauss com-

m e n t e d that h e h a d h a d this i d e a for m a n y years a n d that traces o f i t could Lowell

easily be f o u n d in his Coolidge

investigated

1799 this

"Demonstratio Nova." point

and

showed

18

that

Julian Gauss'

c l a i m was a m p l y s u p p o r t e d b y the fact that s o m e m e t h o d s u s e d i n the

8

1799

paper

seem

"blind

and meaningless"

unless the author

The

Earliest Traditions

already possessed this idea.19 It was t h r o u g h Gauss'

1831 publica-

tion that most mathematicians came into contact w i t h the geometrical

representation of complex numbers, although

of Gauss'

paper

only

in

1852

However Hamilton heard in

(4;

312)

and

Hamilton

Grassmann

1845 that Gauss

heard 1844.20

in

had been

searching

for a "triple algebra" corresponding to the d o u b l e algebra of complex numbers. tion

(4; 3 1 1 - 3 1 2 ) F e l i x K l e i n a r g u e d i n a n 1 8 9 8 p u b l i c a -

that Gauss

Knott

had in

vigorously

accept

the

fact d i s c o v e r e d this.21

denied

geometrical

quaternions,

Ironically

representation

Gauss

but Tait and

himself did

of c o m p l e x

numbers

not

as

a

sufficient justification for them.22 In conclusion it m a y be n o t e d that Gauss' p u b l i c a t i o n w a s t h e shortest, t h e m o s t precise, t h e last, a n d the most influential

o f t h e six i n d e p e n d e n t p r e s e n t a t i o n s .

T h e next p u b l i c a t i o n to be c o n s i d e r e d was the longest, the least precise, the earliest (except for Wessel's), a n d the least influential. On J u n e 20, 1805, a l o n g essay e n t i t l e d " M e m o i r e sur les q u a n t i t e s imaginaires"

was

read before

the

Royal

Society

of London.

The

author was A b b e B u e e a n d his paper was p u b l i s h e d ( w i t h o u t translation)

in

the

1806

Transactions

of

the

Society.23

Royal

treatment of complex numbers was not of high quality; fact

has

expressed

surprise

that

it

was

Buee's

Coolidge in

published.24

The

well-

founded consensus a m o n g those w h o have studied Buee's paper is that

some

ingenuity

mixed

with

much

obscurity

is

to

be

found

there, as w e l l as a near approach to the concept of the m u l t i p l i c a t i o n of directed lines.

H a m i l t o n asserted that B u e e attempted to extend

his m e t h o d s t o space (5; [57]), b u t i f B u e e d i d d o t h i s , h e d i d i t i n a very

unorthodox

manner.

A far s u p e r i o r w o r k also a p p e a r e d in Argand's tites

small

book,

imaginaires

Essai

dans

sur

les

une

1806; this was Jean Robert maniere

constructions

de

representer

les

geometriques.6

quan-

Herein

Argand gave the m o d e r n geometrical representation of the addition and multiplication of complex numbers, and s h o w e d h o w this representation

could

trigonometry,

be

applied to deduce a n u m b e r of theorems

elementary

geometry, and algebra.

in

At this t i m e Ar-

gand d i d not attempt to e x p a n d his m e t h o d s for application to threed i m e n s i o n a l space. F o r seven years A r g a n d shared the fate of W e s sel;

however

unexpected In

in

1813 J.-F.

Gergonne's

1813

attention

was

called to his

book in

a very

way. Frangais published a short m e m o i r in v o l u m e IV of

Annates

de

mathematiques

(6;

63-74),

in

which

Fran-

gais p r e s e n t e d t h e g e o m e t r i c a l r e p r e s e n t a t i o n o f c o m p l e x n u m b e r s . At the conclusion of his paper Frangais stated that the f u n d a m e n t a l ideas in his paper w e r e not his o w n ; he h a d f o u n d t h e m in a letter

9

A

History

of V e c t o r Analysis

w r i t t e n by L e g e n d r e to his (Frangais') brother w h o had died. In this letter L e g e n d r e discussed the ideas of an u n n a m e d mathematician. Frangais

a d d e d that he h o p e d that this mathematician w o u l d make

himself k n o w n and publish his The

unnamed

ideas,

for

Legendre's

Frangais'

paper,

Gergonne

in

results.

mathematician friend

Argand

which

he

had

(6;

in

was Jean

already

p u b l i s h e d his

Robert Argand.

immediately

identified

74)

fact

sent

a

Hearing of

communication

to

himself as the mathematician of

L e g e n d r e ' s l e t t e r , c a l l e d a t t e n t i o n t o h i s b o o k , s u m m a r i z e d its c o n tents, a n d

finally

presented an (unsuccessful) attempt to extend his

system to three-dimensions. lications,

(6; 7 6 - 9 6 ) B e f o r e s e e i n g A r g a n d ' s p u b -

Frangais h a d w r i t t e n a letter to G e r g o n n e containing his

admittedly unsatisfactory attempts to extend the geometrical representation of c o m p l e x n u m b e r s to space.

(6;

96-101) A n d soon after

Argand's publication, Servois published a paper criticizing Argand's attempt and (6;

o u t l i n i n g his o w n ideas on a m e t h o d of space analysis.

101-109)

to

an

Hamilton

anticipation

triplets.

.

. ."

o f the

(5;

[57])

f o l l o w i n g passage

attributed

quaternions,

or

to at

In m a k i n g this

from

Servois least

to

"the an

nearest

approach

anticipation

statement Hamilton

of

had the

Servois' paper in mind.

Analogy w o u l d seem to

indicate that the tri-nominal should be of the

f o r m p cos a + q cos (3 + r cos y, a,

a n d y b e i n g t h e a n g l e s m a d e by a

r i g h t l i n e w i t h t h r e e r e c t a n g u l a r axes, a n d t h a t we s h o u l d h a v e (p cos a + q c o s (3 + r c o s y)(p' c o s a + q' c o s (3 + r' c o s y) = c o s 2 a + c o s 2 /3 + c o s 2 y — 1 . T h e v a l u e s o f p , q , r , p ' , q', r ' s a t i s f y i n g t h i s c o n d i t i o n w o u l d b e absurd;

but

A+

would

B V ^ I ? (7;

they

be

imaginaries,

reducible

to

the

general

form

114-115)

C o n c e r n i n g this

passage

Hamilton

wrote:

T h e s i x N O N - R E A L S w h i c h S e r v o i s t h u s w i t h r e m a r k a b l e s a g a c i t y foresaw,

without

b e i n g a b l e t o determine t h e m ,

the ther; u n k n o w n symbols +i, at

least,

these

latter

him, and furnish

may n o w be

identified with

+ / c , — i , — j , —/c, o f t h e q u a t e r n i o n t h e o r y :

symbols

fulfil

p r e c i s e l y t h e condition p r o p o s e d b y

a n answer t o h i s " s i n g u l a r q u e s t i o n . " I t m a y b e p r o p e r

t o state that m y o w n t h e o r y h a d b e e n c o n s t r u c t e d a n d p u b l i s h e d for a l o n g t i m e , b e f o r e t h e l a t e l y c i t e d passage h a p p e n e d t o m e e t m y eye. (5; [57]) The

series

of articles

in

Gergonne's

letter written by A r g a n d in w h i c h by

Lacroix

(6;

111)

Annales

was

concluded

calling attention to

Buee's

(1806)

this

very

point were

nions

10

is

in

1843.25

strong evidence unknown to The

ideas

that all

Hamilton of the

a

publication.

A r g a n d w r o t e t h a t h e h a d h a d n o k n o w l e d g e o f B u e e ' s w o r k . (6; There

by

he responded to a notice sent in

the

men

123)

discussed up to

w h e n he discovered quater-

next man

to be

discussed were

The

k n o w n to H a m i l t o n as early as thinking,

as

published cal

he

in

repeatedly

1828

Representation

a of

Earliest Traditions

1829 a n d m o r e o v e r i n f l u e n c e d his

acknowledged.

short

book

the

Square

(4;

entitled A Roots

190)

Treatise of

John Warren on

the

Negative

GeometriQuantities.

Warren's

presentation of the geometrical representation of complex

numbers

e x h i b i t e d great care a n d u n d e r s t a n d i n g ; he, u n l i k e B u e e

a n d A r g a n d , was a w a r e o f t h e i m p o r t a n c e o f t h e c o m m u t a t i v e , associative, and distributive laws, t h o u g h he d i d not use these terms.26 W a r r e n discovered his ideas in c o m p l e t e i n d e p e n d e n c e of the other mathematicians

who

wrote

on

the

geometrical

representation

of

complex numbers, b u t he, unlike the majority of t h e m , d i d not discuss t h e e x t e n s i o n o f h i s s y s t e m t o space.27 T h e final independent discoverer of the

geometrical representa-

tion of complex n u m b e r s was the F r e n c h m a n C. V. M o u r e y , w h o in 1828

published

quantites

an

negatives

excellent et

des

the conclusion of his book bra surpassing

treatise

entitled

quantites

La

pretendues

vrai

Theorie

imaginaires.28

des At

M o u r e y stated that there exists an alge-

not o n l y o r d i n a r y algebra b u t also the t w o - d i m e n -

sional algebra created by him.

This

algebra, he stated, extends to

three-dimensions.29 Presumably M o u r e y searched for such an algebra; if he f o u n d it, he d i d not p u b l i s h his discovery.

V.

Summary

and

Conclusion

T h u s w e can say that a t least f i v e m e n , w o r k i n g i n d e p e n d e n t l y o f each other, had by 1831 discovered and p u b l i s h e d the geometrical representation

of

complex

numbers.

These

men

were

Wessel,

Gauss, A r g a n d , W a r r e n , a n d M o u r e y . A t least t w o others, W a l l i s a n d Buee, had c o m e close to the same idea. Wessel, Gauss, Argand, a n d M o u r e y , a s w e l l a s S e r v o i s a n d F r a n g a i s , a n d p e r h a p s B u e e , h a d attempted to

find

higher c o m p l e x n u m b e r s for the analysis of space,

and all had failed. A number of conclusions cussed.

The

first is

m a y be d r a w n f r o m w h a t has b e e n dis-

that the

idea of a graphical representation of

complex numbers was certainly " i n the air" at that time. H o w e v e r , the acceptance of this idea was very slow, and little attention was p a i d to these ideas u n t i l Gauss p u b l i s h e d his p a p e r of 1831.

The

fact that the i d e a was n e g l e c t e d u n t i l Gauss e n t e r e d t h e f i e l d s h o u l d not, I think, be taken as surprising. peatedly

shown

that

radically

new

H i s t o r i a n s of science h a v e reideas

presented only on their

o w n merits are usually neglected. T h e m e n b e f o r e Gauss w e r e all little k n o w n ;

i n d e e d t h e y are n o w k n o w n o n l y because o f t h e i r o n e

great discovery. B u t w h e n Gauss wrote, he w r o t e w i t h the authority

11

A H i s t o r y of V e c t o r Analysis

of one w h o had already acquired fame through impressive work in traditional fields and through his w i d e l y k n o w n prediction of the position of the lost p l a n e t o i d Ceres.

It m a y be n o t e d n o w a n d dis-

cussed later that the pattern exhibited in this instance w i l l recur in the later history of vectorial analysis. S e c o n d , i t has b e e n n o t e d that m o s t o f those w h o w o r k e d o n the geometrical

representation

struct analogous

methods

of c o m p l e x n u m b e r s a t t e m p t e d to confor t h r e e - d i m e n s i o n a l space. T h a t m a n y

e m b a r k e d on this quest illustrates w h a t is probably mathematically o b v i o u s : the search for a system of space analysis was a natural concomitant to the numbers. fore

idea of the

geometrical representation of complex

Up to this point only those w h o m a d e their attempts be-

1831 h a v e b e e n discussed; m a n y others also p u z z l e d over this

p r o b l e m after 1831. A m o n g t h e m was H a m i l t o n , w h o , w o r k i n g precisely in this tradition, discovered quaternions.

12

Notes 1

Gottfried W i l h e l m

Christian Leroy E. first

Leibniz, "Studies

Huygens"

in

Loemker, vol.

published

mathematische

in

und

"Christi.

Leibniz,

I

1833;

Papers

(Chicago, 1956), 3 8 1 - 3 9 6 . the

physikalische

Huygenii

in a G e o m e t r y of Situation w i t h a Letter to

Philosophical

citation

Werke,

as

vol.

aliorumque

given

I,

seculi

pt.

based

Schriften, e d . (which

C.

is

sophic, e d .

I.

translation

Gerhardt, vol.

superior)

as

on II

the

given

are

above)

and

Hermann (Leipzig,

in

and

from

Loemker and

Uylenbroek's

have

text as

trans.

Grassmann, 1894),

Gesammelte

415-416,

celebrium

is

exercitationes

H a g a e c o m i t u m 1833. fasc. I I , p . 6 . " as

1850),

given

in

Leibniz,

Mathematische

17-27, a n d on U y l e n b r o e k ' s text

Leibniz,

Hauptschriften

zur

Griindung

been checked with

the

der

Philo-

1924). Q u o -

Gerhardt's text

g i v e n i n G r a s s m a n n , Werke, v o l .

I, pt.

(cited

I, 417-420.

All quotations have been taken from Martin A. Norgaard's English translation of

the first sixteen sections of Wessel's book; A

in

virorum

text

(Berlin,

ed.

E r n s t Cassirer, trans. A. B u c h e n a u , 2nd. ed., 2 vols. ( L e i p z i g ,

tations

2

his

Letters,

I

XVIII.

mathematicae et philosophiae. Ed. Uylenbroek. Loemker

and

L e i b n i z ' essay a n d letter w e r e

Source

55-66.

Book

I

in

Mathematics,

have also

vol.

used the

I,

ed.

French

see W e s s e l , " O n C o m p l e x N u m b e r s " i n

David

Eugene

Smith

translation of Wessel's

(New

book

York,

1959),

w h i c h is cited in

n o t e (1) a b o v e . T h e t i t l e f o r W e s s e l ' s o r i g i n a l p u b l i c a t i o n i s " O m D i r e c t i o n e n s a n a l v tiske

Betegning,"

Danske

Videnskabernes

by S. Lie

and

it

appeared

Selskabs

Skrifter.

in

vol.

V

Wessel's

(1799)

essay

o f Nye

was

Samling

a f det

rediscovered

Kongelige

in

1895

D. Christensen and C. Juel; it was republished without translation by Sophus in

the

1896

Archiv for

Mathematik

og

Naturvidenskab.

In

this

connection

see

Viggo Brun, "Caspar Wessel et l'introduction geometrique des nombres c o m p l e x e s " in

Revue 3

d'histoire

Caspar

des

Wessel,

sciences, Essai

12

sur

(1959), la

V a l e n t i n e r a n d T . N . T h i e l e , trans. 4

Robert

Perceval

Graves,

20-21.

representation

analytique

de

la

direction,

ed.

H. G. Z e u t h e n and others (Copenhagen,

Life

of

Sir

William

Rowan

Hamilton,

vol.

Ill

H.

1897).

(Dublin,

1889). 5

Sir

William

Rowan

Hamilton,

Lectures

on

Quaternions

(Dublin,

1853).

All

refer-

ences are to H a m i l t o n ' s Preface, w h e r e A r a b i c n u m e r a l s set in p a r e n t h e s e s are u s e d to indicate page numbers. 6

Jean

naires

Robert

dans

contains papers

les

Essai

sur

geometriques,

a reprint of the

on

papers

Argand,

constructions

first

une

2nd

edition

maniere

ed.,

(Paris,

ed.

de J.

representer Hoiiel

les

quantites

imagi-

1874).

This

(Paris,

1806) a l o n g w i t h

selections from the

complex numbers by Frangais, Argand, Gergonne, Lacroix, and Servois,

which

were

originally

published

in

Gergonne's

Annales

des

Mathematiques,

4 ( 1 8 1 3 - 1 8 1 4 ) a n d 5 ( 1 8 1 4 - 1 8 1 5 ) . S e e t h e w o r k c i t e d i n n o t e (7) b e l o w f o r a n E n g l i s h translation series in 7

Hoiiel's

edition.

Jean

Robert

trans. A. S. 8

of Argand's

book;

Hardy

included

less

material

than

Hoiiel

from

the

o f p a p e r s i n G e r g o n n e ' s Annales b u t s u p p l i e d v a l u a b l e c o m m e n t a r y n o t f o u n d

Argand,

Imaginary

Quantities:

Their

Geometrical

Representation,

H a r d y ( N e w Y o r k , 1881).

T h e t h r e e G r e e k a u t h o r s w h o u s e d t h i s c o n c e p t are (1) t h e a u t h o r o f t h e s o - c a l l e d

"pseudo-Aristotelian

Mechanica,"

(2)

Archimedes,

and

(3)

Hero

of Alexandria.

For

13

A H i s t o r y of V e c t o r Analysis the

first

Ages

(Madison,

and

the

nique

Analytique

"On

Spirals"

third

1959),

in

see

4-5,

Marshall

41.

Lagrange,

in

The

On

CEuvres,

Works

Clagett,

The

Archimedes vol.

XI

o f Archimedes,

Science

o f Mechanics

see J o s e p h

(Paris,

trans.

Louis

1888),

Thomas

12,

in

and

Heath

the

Middle

L a g r a n g e , MecaArchimedes,

(New

York,

n.d.),

165. 9

T h e history of this concept is discussed by n u m e r o u s authors; the f o l l o w i n g are

among

the

Maddox J.

most

(New

McCormack

1962);

(4)

tischen 10

(La

A.

For

this

Salle,

Rene

(2)

111.,

Dugas,

Ernst

Mach,

1960);

(3)

"Grundlegung

vol.

example

. .

(1)

1955);

Voss,

Wissenschaften,

plane . has

important:

York,

IV, see

pt.

(1;

I

A

The

History

o f Mechanics,

Science

Max Jammer,

der

Concepts o f Force

Mechanik"

(Leipzig,

390) but note

trans.

o f Mechanics, t r a n s .

in

1901-1908),

J.

R.

Thomas

( N e w York,

Encyklopadie

der

mathema-

43-46.

that L o e m k e r wrote

"AB

b BY for one

w h e r e a s t h e U y l e n b r o e k t e x t ( s e e G r a s s m a n n , Werke, v o l . I , p t . I , 4 2 0 )

(correctly) " A Y « B Y . "

11

There

is

a

fuller

but

similar

vol. V, ed. C. I. Gerhardt (Halle,

exposition

in

Leibniz,

Mathematische

Schriften,

1858), 141-171. M a n y m i n o r statements of L e i b n i z

(for e x a m p l e , statements in letters) are referred to a n d discussed by L o u i s Couturat, La

Logique

de

Leibniz

(Paris,

ideas, particularly as discussed by A. nection

with

Grassmann

E.

Heath,

Leibniz's

in

1901),

they relate

his

which

"The

Geometrical

Characteristic"

Geometrische

Analyse

in

in

12

Louis

Couturat,

There

have b e e n a n u m b e r of studies

that have a i d e d me ruff

Beman,

American

"A

Cajori,

for

"Historical of

1912),

1924);

in

the

Leibniz

(3)

(4)

on

the

in Julian

Analysis

vol.

discussion

27

I,

of

Leibniz'

L e i b n i z ' s y s t e m was also

of Grassmann

Monist,

(Paris,

History of

(1917),

pt.

1901),

I,

a n d Its Con-

36-56,

and

by

321-399.

538.

on the early history of complex numbers

of

Mathematics"

Science,

Graphic

American

46

Monthly,

Coolidge,

Hankel,

der

The

Proceedings

33-50;

(2)

of Imaginaries 19

Geometry

Theorie

in

(1897),

Representation

Mathematical

Lowell

Hermann

full

a m o n g t h e m o s t i m p o r t a n t are (1) W o o s t e r W o o d -

the

Advancement

Note

Wessel"

167-171;

ford,

de

in this study;

Chapter

Association

Time

Logique

a

system.

The

Werke,

13

La

includes

to Grassmann's

o f the

Florian

Before

the

(September-October,

o f the

complexen

Complex

Domain

Zahlensysteme

(Ox-

(Leipzig,

1 8 6 7 ) ; (5) P . S . J o n e s , " C o m p l e x N u m b e r s : A n E x a m p l e o f R e c u r r i n g T h e m e s i n t h e Development

of

263, 340-345; of

Certain

vancement narii

Branches

o f Science

nella

and 36

Mathematics"

of

(1834),

Mathematics

317-345;

Quantities"

Analysis" 185-352;

geometria"

(1898),

Complex

in

Teacher,

47

(1954),

106-114,

257-

(6) G e o r g e P e a c o c k , " R e p o r t o n t h e R e c e n t P r o g r e s s a n d P r e s e n t State in

(7)

in

Battaglini's

(8)

G.

in

Mathematical

of

the

British

Romorino,

Giornale

Windred,

Unfortunately a recent excellent

Report

Angelo

di

Association

"Gli

matematica,

for

the

Elementi

35

(1897),

Ad-

imagi-

242-258;

" H i s t o r y of the T h e o r y of Imaginary and

Gazette,

14

study came

(1929),

533-541.

to my attention too late to take full

a d v a n t a g e o f it. T h i s i s F . D . K r a m a r ' s " V e k t o r n o e i s c h i s l e n i e k o n t s a X V I I I i n a c h a l a XIX

vv"

14

The

matics, e d . 15

be

(in

Russian)

important David

in

passage

Eugene

lstoriko-Matematicheskie

from

Smith,

Wallis

vol.

I

may

Issledovaniia,

be

found

( N e w York,

in

15 A

(1963), Source

Book

225-290. in

Mathe-

1959), 4 6 - 5 4 .

T h e w o r d s " t o p u b l i s h " qualify this statement sufficiently that no m e n t i o n n e e d

made

in

the

text

of Leonard Euler, Charles Walmesley, and Dominique Truel.

T h e basis for attributing the geometrical representation to the first t w o of these m e n is

that it seems f r o m reading their writings on relevant subjects that they probably

had

this

representation.

The

sole

basis

for

mentioning

C a u c h y that T r u e l had this representation as early as Florian Cajori, fore

14

Wessel"

"Historical in

American

Notes on the Mathematical

Truel

is

a

statement

by

1786. F o r f u l l e r discussion see

G r a p h i c Representation of Imaginaries be-

Monthly,

19

(1912),

167-171.

T h e Earliest Traditions 16

Some

of these

difficulties

H a m i l t o n ' s efforts to 17

ische

gelehrte

Friedrich l

will

be

discussed

more

fully

in

Chapter

II, where

a t h r e e - d i m e n s i o n a l vectorial system are treated.

Gauss' untitled publication, w h i c h was a discussion of his " T h e o r i a r e s i d u o r u m

biquadraticum,

Commentatio

Anzeigen

of A p r i l

G a u s s , Werke,

»Ibid.,

19

find

secunda," 23,

vol.

II

1831.

was I

originally

have

(Gottingen,

used

1863),

published

the

text

in

as

the

given

Gotting-

in

Carl

169-178.

175.

Julian

Lowell

Coolidge,

The

Geometry

o f the

Complex

Domain

(Oxford,

1924),

28-29. 20

Hermann

Werke, v o l . 21

Felix

matische

Klein,

o f Edinburgh,

this in

Abbe

Royal

of

24

Coolidge,

25

This

is

VIII

96

Geometry

X,

sur

les

(1806), of

the

implied

Philosophical that

2

see

is

especially

(5;

in

on

the

Gilston

Proceedings

which

Mathe-

Recently

i n Proceedings

Knott,

o f the

Klein

in

Claim

"Pro-

Royal

Society

his

claim,

based

357-362. "Uber

Gauss

(Gottingen,

quantites

Arbeiten

zur

Function-

1922-1933), 55-57.

imaginaires"

the

Domain,

fact

in

Transactions

of

the

24.

that

3rd

in

an

Ser.,

had influenced him;

in

" O n

23-88.

Magazine,

implied

physikalische

Gauss' W e r k e n "

Cargill

criticism"

Complex by

von Tait,

17-23;

1900),

pt.

und

Discovery) of Quaternions"

Schlesinger,

representation of complex

same conclusion Quaternions;

the

(Leipzig,

Ludwig

discussed the authors geometrical

For

"Memoire

in

the

(1900),

document

London,

Quaternions"

Guthrie

24-34.

see

strongly

(not

23

G a u s s , Werke, v o l .

Buee,

Society

Invention

Herausgabe

Peter

a

vol.

point

der

mathematische

397-398.

of Q u a t e r n i o n s :

(1900),

G a u s s , Werke, On

the

Stand

128-133;

o f Edinburgh,

View

23

entheorie" 23

to

Gesammelte

1896), 8 - 9 ,

den

(1898),

Society

Klein's

Grassmann,

(Leipzig,

"Uber

51

Royal

fessor

2 2

II

for Gauss

the

see

pt.

Annalen,

Made of

Giinther

I,

1844

25

paper

(1844),

[31]—[57])

richly historical

as

"On

Hamilton

o f t h e six m e n w h o d i s c o v e r e d t h e

numbers, only Warren was

Hamilton's

(Hamilton,

489-495)

well

as

the

mentioned.

The

p r e f a c e t o h i s Lectures o n work

listed

in

note

(4)

above, w h e r e i n m a n y letters f r o m H a m i l t o n t o D e M o r g a n w e r e p u b l i s h e d i n w h i c h Hamilton

discussed these men.

p a p e r (4; 3 1 2 ) , (2) paper Rowan

(5;

[57]),

Hamilton,

and

vol.

H a m i l t o n e x p l i c i t l y d e n i e d h a v i n g s e e n (1) G a u s s '

M o u r e y ' s b o o k (4; 4 8 9 ) , (3) A r g a n d ' s b o o k (4; 4 3 5 ) , (4) S e r v o i s '

II

(5)

Frangais'

[Dublin,

papers

1885],

(Robert

606).

Perceval

From

the

Graves,

fact

that

Life o f Sir William

Hamilton

had not

read Servois' and Frangais' papers or Argand's book, it seems reasonable to conclude that he had Annales

not read any of the

before

1844.

Hamilton

relevant papers did

not

in

explicitly

volumes IV and V of Gergonne's

deny knowledge

of Wessel,

since

h e never, e v e n after 1843, h e a r d o f W e s s e l , a n d h e d i d not e x p l i c i t l y d e n y k n o w l e d g e of Buee's hand,

paper,

Hamilton

since

he

had already d e n i e d that

attended

the

vancement of Science, and on

the

Recent

(.B.A.A.S.

Report,

Argand's 26

See

Roots

Progress

and

John

o f Negative

the

in

Present

which

papers

Warren, Quantities

it had any merit.

meeting of the

A

from

Treatise

State

Peacock

of

on

the

1828),

Certain

briefly

Gergonne's

(Cambridge,

addition, page 9 for c o m m u t a t i v e

Branches

extent

on

"associative,"

probably

the

the

recognition

first

the

other

discussed

of

(ibid.y

"Report

Analysis" page

228)

Annales.

Geometrical page

Representation

3

for

of

the

commutative

law for multiplication, page

was aware of the associative l a w for m u l t i p l i c a t i o n , a n d page law. T h e importance of this

names

On

British Association for the Ad-

at this m e e t i n g George Peacock presented his and

185-352),

book

1833

Square law

of

18 for a hint that he

13 for the distributive

is that the discovery of quaternions d e p e n d e d to some of the

importance

"commutative,"

historical

and

statement

of these

laws.

"distributive"

was

made

by

in

On a

origin

of the

mathematical

sense

Hermann

the

Hankel,

Theorie

der

15

A H i s t o r y of V e c t o r Analysis complexen

Zahlensysteme

"These have

names

(Leipzig,

have

been

1867),

adopted

footnote

on

universally in

page

3,

not hesitated to transplant t h e m to G e r m a n soil;

was

it

seems

first

i n t r o d u c e d by Sir. W.

R.

he

said,

1840 and hence I

'distributive' and 'commuta-

tive' w e r e introduced by Servois ( G E R G O N N E ' S Ann. vol. V. tive'

where

E n g l a n d since

1814, p . 93); 'associa-

H a m i l t o n . " T h i s statement is re-

peated by both D a v i d E u g e n e S m i t h and Florian Cajori. T h e earliest recognition of the

necessity of proving the commutative

VII, Proposition the term

"associative" is

Connected with lished

in

the

"However, in

l a w for m u l t i p l i c a t i o n is in

Euclid, Book

16. T h e first p u b l i c a t i o n , t o m y k n o w l e d g e , i n w h i c h H a m i l t o n u s e d in the paper " O n a N e w Species of Imaginary Quantities,

a T h e o r y o f Q u a t e r n i o n , " c o m m u n i c a t e d N o v e m b e r 13, 1843, p u b Proceedings

of

the

Royal

Irish

Academy,

2

(1844)

424-434.

He

wrote:

virtue of the same definitions, it w i l l be f o u n d that another important

property of the o l d m u l t i p l i c a t i o n is preserved, or e x t e n d e d to the n e w , namely, that which 27

may

be

called

the

associative

character of t h e

operation.

.

.

."

Ibid., 4 2 9 - 4 3 0 .

At least no extension is suggested in his b o o k or in the t w o s u b s e q u e n t papers

w h i c h he p u b l i s h e d on this subject. Philosophical entitled

Transactions

of

"Considerations

sentation of the Square Geometrical Square

the

H i s t w o later papers w e r e b o t h p u b l i s h e d in the

Royal

of the Roots

Society

Objections

of

London,

119

(1829);

they

were

Raised Against the Geometrical Repre-

of Negative Quantities," pages 2 4 1 - 2 5 4 , and " O n the

Representation of the Power of Quantities Whose Indicies Involve the

Roots

of

251-254) Warren

Negative

Quantities,"

stated that he

pages

had written

339-359.

In

the

first

paper

(ibid.,

his book before he heard of Buee's or

M o u r e y ' s p u b l i c a t i o n ; A r g a n d was not m e n t i o n e d , p r e s u m a b l y because W a r r e n still had not heard of Argand's 28

C.

dues

V.

Mourey,

imaginaires

reprint Buee; have

of his

(Paris, 1828

book.

La 1828).

work.

vrai

Theorie

The

second

Nowhere

des

in

quantites

edition the

work

negatives

et

book

since

quantites

preten-

1861) was used; this was a

does

Mourey mention Argand or

i n fact n o m a t h e m a t i c i a n s are ever m e n t i o n e d i n the book. k n o w n Warren's

des

(Paris,

it was p u b l i s h e d after his o w n .

Mourey could not M o u r e y [ibid., I X )

m a d e the interesting c o m m e n t that his b o o k was an a b r i d g e m e n t of a longer treatise he had written but had not published. 29

16

Ibid.,

95.

CHAPTER

Sir

William

Rowan and

I.

Introduction: The

task

of the

Hamiltonian historian

w o r k of Sir W i l l i a m R o w a n estimates

of his

extreme

dinger

of Hamilton:

While

these

Hamilton Quaternions

Historiography

who

wishes

to

treat any

aspect of the

H a m i l t o n is c o m p l i c a t e d by the fact that

significance for the history of science have varied

between two wrote

TWO

positions.

discoveries

Thus,

(Quaternions,

for example,

etc.)

would

Erwin

suffice

to

Schro-

secure

H a m i l t o n in the annals of both mathematics a n d physics a h i g h l y hono u r a b l e place, s u c h p i o u s m e m o r i a l s can i n his case e a s i l y b e d i s p e n s e d with. For H a m i l t o n is virtually not dead, he h i m s e l f is alive, so to speak, not his m e m o r y . I daresay n o t a d a y passes — a n d s e l d o m an h o u r — w i t h out somebody,

somewhere

writing or printing

on

Hamilton's

this

globe,

name.

pronouncing or reading

or

T h a t is due to his f u n d a m e n t a l

d i s c o v e r i e s i n g e n e r a l d y n a m i c s . T h e H a m i l t o n i a n p r i n c i p l e has b e c o m e the cornerstone of m o d e r n physics, t h e t h i n g w i t h w h i c h a p h y s i c i s t exp e c t s every p h y s i c a l p h e n o m e n o n t o b e i n c o n f o r m i t y . . . . T h e modern development of physics is continually enhancing Hamilton's name. H i s famous analogy b e t w e e n mechanics a n d optics v i r t u a l l y anticipated wave-mechanics,

which

did

not

have to add m u c h to his

ideas, o n l y h a d to take t h e m seriously —a little m o r e seriously t h a n he was

able to take t h e m , w i t h the experimental k n o w l e d g e of a century

ago.

The

central conception of all

modern theory in physics

is

"the

Hamiltonian." If you wish to apply modern theory to any particular probl e m , y o u m u s t start w i t h p u t t i n g t h e p r o b l e m " i n H a m i l t o n i a n f o r m . " T h u s H a m i l t o n i s o n e o f t h e greatest m e n o f s c i e n c e t h e w o r l d has p r o duced.6 In

*

1945 J.

into eclipse.7 above

all

variations.

L.

Synge lamented that Hamilton's fame was passing

Synge cited m a n y aspects of this eclipse b u t stressed

the neglect of Hamilton's contribution to the calculus of He wrote:

" H a m i l t o n was, in fact, a great c o n t r i b u t o r —

probably the greatest single contributor of all t i m e —to the calculus of variations."

(7;

15)

17

A H i s t o r y of V e c t o r Analysis

In

1940 E. T. Whittaker p u b l i s h e d a paper entitled " T h e H a m i l -

tonian ton's

Revival,"

8

reputation

century: verse

in

which

was

he maintained:

touched

about

the

since w h e n , t h e r e has b e e n a steady m o v e m e n t in t h e re-

direction:

one

after

another,

the

i n n o v a t i o n s has b e e n a p p r e c i a t e d . . . ton in

" T h e nadir of Hamil-

beginning of the present

1954:

significance 9

."

of his

" A f t e r Isaac N e w t o n , t h e greatest m a t h e m a t i c i a n of the

E n g l i s h - s p e a k i n g p e o p l e s is W i l l i a m R o w a n H a m i l t o n . . . ." In

1937

titled

great

W h i t t a k e r w r o t e of H a m i l -

E.

T.

Bell

in

his

widely

the

chapter on

Hamilton

presented

Hamilton's

life

as

"An

read

Men

Irish

a tragedy,

o f Mathematics

Tragedy."

10 11

Herein

enBell

in a sense a m a g n i f i c e n t

failure. This

disparity o f v i e w s c o n c e r n i n g H a m i l t o n , w h i c h i n fact dates

back to the nineteenth century, is central to Hamiltonian historiography. T h e m a i n source of this disparity of v i e w s relates to H a m i l ton's

work

on

quaternions.

represented

the

voted

more

than

twenty

quaternions

held

by

Hamilton

mathematics

of the

years

nearly

of his

all

believed

future

and

life

to

that

them.

mathematicians

quaternions

consequently The

of the

de-

view

of

present is

h o w e v e r quite different; the consensus n o w is that the quaternion system

is b u t one of m a n y comparable mathematical systems, and

though value

it for

is

i n t e r e s t i n g as a rather special

application.

system, it offers

little

T h e historian of today must take the above

e v a l u a t i o n of q u a t e r n i o n s as m o s t p r o b a b l y v a l i d , t h o u g h t h e r e remain

sources

of doubt.

Statements qualifying or contradicting this

evaluation — m a d e by such important scientists as E. T. Whittaker,12 George D. Birkhoff,13 and P. A. M. Dirac

14

caution

large

in the

historian,

as

do

the

two

— instill some degree of volumes by Otto F.

Fischer,15 in w h i c h the author attempted to rewrite m u c h of m o d e r n physics

in terms of Hamilton's quaternions.

E. T. Bell's v i e w of H a m i l t o n as a tragic failure certainly s t e m m e d from

the

modern victim

fact that he

felt that

mathematics. of

a

Bell

monomaniacal

deepest tragedy was

quaternions

was

are of little interest to

convinced that

delusion;

he

Hamilton was the

stated

"that

Hamilton's

neither alcohol nor marriage but his obstinate

belief that quaternions h e l d the key to the mathematics of the physical

universe."

much taker

16

of what E. however

The

passed

contributions to

problem

of quaternions

T. Whittaker wrote over

the

mathematical

also stands b e h i n d

concerning Hamilton; Whit-

problem

by

stressing

Hamilton's

physics and by arguing that quater-

nions " m a y even yet prove to be the most natural expression of the n e w physics."

17

T h e present study m u s t stand in the shadow of this dispute con-

18

Sir W i l l i a m R o w a n H a m i l t o n

and Quaternions

cerning Hamilton's greatness; nevertheless it is h o p e d that substantial

progress

toward

following analysis, not possible

to

a

solution

which

argue

may

be

achieved

in

terms

of the

w i l l be d e v e l o p e d m o r e f u l l y later.

that the

quaternion

system

is

the

It is

vectorial

system of the present day; the so-called Gibbs-Heaviside system is the

only system that merits

this distinction.

N o r is it legitimate to

a r g u e (as W h i t t a k e r h a s d o n e ) t h a t t h e q u a t e r n i o n s y s t e m w i l l b e t h e system of a future day.

B o t h of these alternatives are unacceptable;

nonetheless

argued

it

can

be

(though

previously been done) that Hamilton's historically

determinable

path to the

hence to the m o d e r n system.

in

my opinion

this

has

not

quaternion system led by an Gibbs-Heaviside

system

and

In what follows it w i l l be s h o w n that

this was in fact t h e case, a n d thus it w i l l b e c o m e clear that H a m i l ton deserves i m m e n s e credit for his w o r k in quaternions, since this work

led

reasons

to

the

now

w h y this

is

widely

so

used

system

little k n o w n

will

of vector analysis.

also be

discussed.

The

If this

analysis is f o u n d acceptable, it s h o u l d clear up the major p r o b l e m in Hamiltonian

historiography.

II.

s

Hamilton Though

in

Life

and

Fame

general a detailed discussion of a scientist's life n e e d

not be i n c l u d e d in a study such in

regard to

the

fame

Hamilton

attained

and

by

Hamilton

fluenced subsequent events. Hamilton's the

title

The the

during

Some

his

lifetime

indication

of the

strongly

in-

importance of

fame in this history may be attained by a comparison of

pages

title

as this, it is of necessity otherwise

quaternions. T h e reason for this is that

of Hamilton's

page

of

and

Grassmann's

Grassmann's Ausdehnungslehre

first of

major

1844

works.

contained

following: Hermann

Grassmann

Lehrer an der Friedrich-Wilhelms-Schule zu Stettin By

contrast,

the

title

page

of

Hamilton's

Lectures

on

Quaternions

contained: SIR W I L L I A M R O W A N H A M I L T O N , L L . D . , M . R . I . A., F E L L O W OF THE AMERICAN

SOCIETY O F ARTS A N D SCIENCES; O F T H E

SOCIETY

FOR

OF

ARTS

SCOTLAND;

NOMICAL SOCIETY OF LONDON; ERN

SOCIETY

SPONDING

OF

ANTIQUARIES

MEMBER

OF

T H E

HONORARY OR CORRESPONDING OR

ROYAL ACADEMIES

TURIN; LIN;

OF

OF THE THE

OF

ST.

OF

T H E

AND OF T H E AT

ROYAL

ASTRO-

ROYAL NORTH-

COPENHAGEN;

INSTITUTE

OF

CORREFRANCE;

M E M B E R OF T H E IMPERIAL

PETERSBURGH,

BERLIN, AND

ROYAL SOCIETIES OF EDINBURGH A N D DUBCAMBRIDGE

PHILOSOPHICAL

SOCIETY;

T H E

19

A

History

of V e c t o r Analysis

N E W YORK HISTORICAL SOCIETY; T H E SCIENCES CIETIES

AT

IN

LAUSANNE;

BRITISH

A N D

AND

OF

FOREIGN

PROFESSOR OF ASTRONOMY IN T H E AND

ROYAL

William

ASTRONOMER

Rowan

Hamilton

OF

was

was

o r p h a n e d at age

SCIENTIFIC

COUNTRIES;

SO-

ANDREWS'

UNIVERSITY OF DUBLIN;

IRELAND.

born of undistinguished ancestry

on the midnight between August 3 and 4, He

SOCIETY OF NATURAL

OTHER

fourteen,

1805, in D u b l i n , Ireland.

b u t h a d ceased to live w i t h his

parents f r o m the age of three, at w h i c h t i m e he h a d b e e n sent to live with

his

Trim,

uncle,

Ireland.

James

Hamilton,

Hamilton's

uncle,

an a

Anglican

man

clergyman

of education

serving

and

intelli-

gence, d i r e c t e d his n e p h e w ' s p r e u n i v e r s i t y education. T h e success of the uncle as tutor and the brilliance of Hamilton as student were manifested in thirteen, teen

many ways,

Hamilton

"was

languages. . . ."

brew,

Syriac,

French,

Italian,

however

only

18

of w h i c h

in

These

Persian,

the

best k n o w n

is that at age

different degrees acquainted w i t h thirlanguages

Arabic,

were

Sanskrit,

Greek,

Latin,

Hindoostanee,

He-

Malay,

Spanish, and German. T h e study of languages was one

of

Hamilton's

interests,

for

he

also

read

in

geography, religion, mathematics, astronomy, and the best of English

and

foreign

Mecanique but

not

was

in

forces. In

Celeste

literature.

and

significant Laplace's (2,1;

1823

At

detected for this

age

an

sixteen

error

study that the

demonstration

he

began

therein.

It

is

Laplace's interesting

error found by Hamilton

of the law of the parallelogram of

661-662)

Hamilton

entered Trinity College of D u b l i n University.

He h a d placed first in the entrance exam a n d h a d decided that his calling was to science. incredible.

In

knowledge

of Greek,

knowledge optime

was

"became vitations, .

His record at the University bordered on the

second

rare.

celebrity

in

Upon in

the

embarrassing

him.

.

year

the

This

wish

year

Hamilton

was

third

physics.

an

optime

for

another

optime

for his

his

T h e w i n n i n g of even a single

winning

the

intellectual their

awarded year

second

circle

optime,

Hamilton

of Dublin;

number,

poured

and in

in-

upon

. " (2,1; 2 0 9 ) H a m i l t o n r e s o l v e d t o a t t e m p t t o w i n i n h i s f i n a l University was

mer of Ireland. creative

honors

he

his

from

Gold

for

Medals

not fulfilled, was

offered

fessor of A s t r o n o m y at the

in

year

and

of mathematical very

a

his

for the

in

both

classics

during the

and in

science.

s u m m e r after his third

honor of becoming Andrews'

Pro-

University of D u b l i n and Royal Astrono-

H i s s t u d e n t days w e r e also d i s t i n g u i s h e d b y success

endeavors. some

of

He

them.

wrote

numerous

Researches

in

poems science

and

received

begun

in

his

seventeenth year on certain questions in mathematical optics led to

20

Sir W i l l i a m R o w a n H a m i l t o n

his

now famous

1824

and

"Theory

published

of Systems

with

Other important papers

of Rays,"

which

further developments

in

the

same

line

and Quaternions

was

read

four years

in

later.19

of development came in

1830,

1831, a n d 1837. H i s a i m in these papers ( w h i c h e x t e n d to over

three

hundred

science his

in

pages)

terms

mathematical

methods

for

was

of his

methods

use

in

to

reduce

optics

"Characteristic in

optics

dynamics.

to

a

Function."

led

The

Hamilton

mathematical

The

success

of

to extend these

distinguished

historian

of

m e c h a n i c s R e n e D u g a s has s u m m a r i z e d the nature a n d i m p o r t a n c e of Hamilton's

work in

I n short, jealous

optics

and

dynamics:

of the formal perfection w h i c h Lagrange had been

able to give to dynamics, and w h i c h optics lacked, H a m i l t o n undertook the rationalisation of geometrical optics.

He

d i d this by d e v e l o p i n g a

formal theory w h i c h was free of all metaphysics a n d w h i c h , moreover, s u c c e e d e d in a c c o u n t i n g for a l l t h e e x p e r i m e n t a l facts. . . . Then,

returning to

dynamics,

Hamilton presented the

l a w o f varying

action i n a f o r m v e r y l i k e t h a t w h i c h h e h a d d i s c o v e r e d i n o p t i c s . T h u s h e r e d u c e d t h e g e n e r a l p r o b l e m o f d y n a m i c s (for c o n s e r v a t i v e systems) to the solution of t w o simultaneous equations in partial derivatives, or to the determination of a single function satisfying these t w o equations.

Hamilton's

g u i d i n g idea is continuous f r o m his optical w o r k to his

w o r k in d y n a m i c s — i n this fact lies his greatness a n d his p o w e r .

Here

was a synthesis that L o u i s de Broglie was to rediscover a n d t u r n to his o w n account; a synthesis that was, it appears, to be Schrodinger's direct inspiration.20 These

works

certainly

contributed

to

was probably thinking of them w h e n in

Hamilton's

fame;

Jacobi

1842 he referred to H a m i l -

t o n a s " l e L a g r a n g e d e v o t r e p a y s . " (2,111; 5 0 9 ) O f t e n h o w e v e r s u c h highly

mathematical

otherwise

for

basis

new

two

refraction. colleague In

this

At

phenomena

of Hamilton,

he

was and

wrote

siderable

vigorous

and

"perhaps made.

.

the .

most

popular fame.

predicted

on

It was

a theoretical

Humphrey

men

Whewell's

and

Airy

remarkable

friend

in

prediction

predicted (2,1;

635)

a very con-

England and on the

praise

referred

both

time

and

prediction.

predicted.

"excited at the

scientific

636)

finding

had not been

discovery

Lloyd,

to verify Hamilton's

successful,

which

immediate,

he

in optics, internal and external conical

among

(2,1;

21

request,

attempted

third

sensation

not produce

1832

completely a

that this

Continent. . . ."

do

in

Hamilton's

phenomena Graves

papers

Hamilton;

for

to

Hamilton

the

that

was

discovery

has

ever

as

been

. " (2,1; 6 3 7 ) D e M o r g a n w r i t i n g i n 1 8 6 6 s t a t e d : " O p t i c i a n s

had no more i m a g i n e d the possibility of such a thing, than astronomers

had

imagined

the

planet

Neptune,

which

Leverrier

and

21

A

History

of V e c t o r Analysis

A d a m s calculated into existence. t o g e t h e r as,

perhaps,

predictions." No

2 2

the

Pliicker of Bonn

experiment

These two things deserve to rank

t w o most remarkable of verified scientific

i n p h y s i c s has

wrote:

made

such a strong impression on my

m i n d as that of conical refraction. A single ray of light e n t e r i n g a crystal and leaving as a l u m i n o u s cone: this is something unheard of and without

analogy.

wave

Mr.

H a m i l t o n p r e d i c t e d it, starting f r o m the f o r m of the

w h i c h h a d b e e n d e d u c e d b y a l o n g calculation f r o m a n abstract

theory. I confess I w o u l d have h a d little hope of seeing an experimental confirmation theory

of such

which

an

extraordinary

Fresnel's

genius

had

result,

recently

predicted by the created.

But

mere

since

Mr.

L l o y d h a d d e m o n s t r a t e d that the e x p e r i m e n t a l results w e r e i n c o m p l e t e accordance w i t h the predictions of M r . H a m i l t o n , all prejudice against a t h e o r y s o m a r v e l o u s l y l o f t y has b e e n f o r c e d t o d i s a p p e a r . (2,1; 6 3 7 ) T h e fame that came to

Hamilton because

of this

discovery was in-

creased by the fact that it w a s m a d e , l i k e nearly all the discoveries d i s c u s s e d t h u s far, b e f o r e Also was

his

literary

close figures

friendship such as

ridge.

Numerous

worth,

and

had up

H a m i l t o n h a d r e a c h e d his t h i r t i e t h year.

illustrative of, a n d c o n t r i b u t o r y to, H a m i l t o n ' s p o p u l a r fame with

William

Wordsworth

and

other

Maria Edgeworth and Samuel Taylor Cole-

letters

passed

between

Hamilton

and

Words-

each visited the other on m a n y occasions. W o r d s w o r t h

said that H a m i l t o n was one of t w o m e n to w h o m he could look (the

other was Coleridge). To this H a m i l t o n replied:

" I f I am to

look d o w n on you, it is only as L o r d Rosse looks d o w n in his teles c o p e to see t h e stars of h e a v e n r e f l e c t e d . " By

1835

Hamilton's

fame

was

(2,111; 2 3 7 )

established.

In

that year he

was

knighted and received a medal from the Royal Society; in addition he

finished

ton's In

a paper on algebraic couples, w h i c h is the first of Hamil-

publications to be of direct importance for the present study.

1837

h e l d this covery

he

was

elected president of the Royal

position

(1843)

until

1843 to

ment

of quaternions. of

resignation

of quaternions.

from

Honors

his

The

received

Sciences along

Irish Academy and

1845, soon after his dis-

last t w e n t y - t w o years

of his life,

1865, w e r e for the most part devoted to the develop-

all

sorts

continued

these deserves final mention. In ton

in

notice

that the

to

be

bestowed

on

him.

One

of

1865, the year of his death, H a m i l -

newly

founded

National

Academy of

of t h e U n i t e d States h a d e l e c t e d h i m a F o r e i g n Associate,

with

Hamilton's

fourteen

other men.

The

name at the head of the

members

had voted to

place

list of F o r e i g n Associates, pre-

s u m a b l y signifying that in their o p i n i o n he was the greatest living scientist.

In

this

j u d g m e n t does

22

they were

attest to the

probably overly enthusiastic, but their fact,

which

is

very significant for this

Sir W i l l i a m

study,

that

great.

Hamilton's

fame

Rowan

among

Hamilton

his

and Quaternions

contemporaries

had b e e n completed, b u t he was still nearly u n k n o w n . in

was

very

At this same time the majority of Grassmann's scientific w o r k

his

1862

Ausdehnungslehre

now "Professor am

III.

Hamilton

It was

had

Gymnasium

and

Complex

changed zu

only

His subtitle

slightly;

it

was

Stettin."

Numbers

stated previously that H a m i l t o n ' s

was in the tradition of the work done on

discovery of quaternions complex numbers, and in

this regard the history of the geometrical representation of c o m p l e x numbers

and associated ideas

was

given.

But there

was

a second

line of d e v e l o p m e n t in studies on c o m p l e x n u m b e r s that also l e d to quaternions. This line of development was established by H a m i l t o n himself in his titled:

l o n g a n d i m p o r t a n t essay p u b l i s h e d in

1837 and en-

"Theory of Conjugate Functions, or Algebraic Couples; with

a Preliminary Pure Time."

and Elementary Essay on Algebra as the This

paper is

important

in

itself;

Science of

indeed one mathe-

matician referred to it as a greater c o n t r i b u t i o n to algebra than his of quaternions.23

discovery sections:

Hamilton's

paper is divided into three

the first section, w h i c h consists of " G e n e r a l Introductory

Remarks,"

was

written

last;

the

second

section,

an

essay

" O n

Algebra as the Science of Pure T i m e " was w r i t t e n in 1835; a n d the third or

section,

Algebraic

(2,11;

containing Couples,"

his

was

"Theory for

the

of

most

Conjugate part

Functions,

written

in

1833.

144) N e g l e c t of t h e historical s e q u e n c e of t h e c o m p o s i t i o n has

led to

a n u m b e r of historical

Hamilton

began

the

misconceptions.

paper by

writing:

T h e study of Algebra may be p u r s u e d in three very different schools, the Practical, the Philological, or the Theoretical, according as A l g e b r a itself is a c c o u n t e d an I n s t r u m e n t , or a L a n g u a g e , or a C o n t e m p l a t i o n ; according

as

ease

thought,

(the

of operation, agere,

or

the f a r i ,

s y m m e t r y of expression, or

the

sapere,)

is

or clearness

eminently

prized

of

and

sought for. T h e Practical p e r s o n seeks a R u l e w h i c h he m a y a p p l y , t h e P h i l o l o g i c a l person seeks a F o r m u l a w h i c h he m a y w r i t e , the T h e o r e t i c a l p e r s o n seeks a T h e o r e m o n w h i c h h e m a y m e d i t a t e . (3; 2 9 3 ) He then proceeded to

state that t h e a i m o f this p a p e r w a s theoreti-

cal. The

thing

aimed

at,

is

to

improve

the

Science,

not

the

Art

nor

the

L a n g u a g e o f A l g e b r a . T h e i m p e r f e c t i o n s s o u g h t t o b e r e m o v e d , are confusions of t h o u g h t , a n d obscurities or errors of reasoning; riot difficulties of application of an instrument nor failures of symmetry in expression.... F o r i t has n o t f a r e d w i t h t h e p r i n c i p l e s o f A l g e b r a a s w i t h t h e p r i n ciples

of Geometry.

No

candid

and

intelligent

person can

doubt the

23

A H i s t o r y of V e c t o r Analysis truth

of the

chief properties

o f Parallel

Lines,

as

set f o r t h

by E U C L I D

i n his E l e m e n t s , t w o t h o u s a n d years ago; t h o u g h h e m a y w e l l desire t o see t h e m t r e a t e d i n a c l e a r e r a n d b e t t e r m e t h o d . T h e d o c t r i n e i n v o l v e s no

obscurity

nor

confusion

of

thought,

and

leaves

in

the

mind

no

reasonable g r o u n d for d o u b t , a l t h o u g h i n g e n u i t y m a y usefully be exercised in improving the plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and I m a g i n a r i e s , w h e n s e t f o r t h (as i t h a s c o m m o n l y b e e n ) w i t h p r i n c i p l e s like

these:

that

the

numbers plied t h e

that

a

remainder denoting one

greater is

magnitude

less

than

magnitudes

by the

other,

may

be

nothing;

each

less

subtracted from that

two

than

a

negative

nothing,

less,

and

numbers, o r

may be

multi-

a n d t h a t t h e p r o d u c t w i l l b e a positive n u m -

ber, or a n u m b e r d e n o t i n g a m a g n i t u d e greater than nothing; and that a l t h o u g h t h e square o f a n u m b e r , o r t h e p r o d u c t o b t a i n e d b y m u l t i p l y i n g that be

number

by

itself,

is

therefore

always positive,

whether the

number

p o s i t i v e o r n e g a t i v e , y e t t h a t n u m b e r s , c a l l e d imaginary, c a n b e f o u n d

or conceived or d e t e r m i n e d , a n d operated on by all the rules of positive and

negative

they

have

selves

numbers,

negative

squares,

as

i f t h e y w e r e s u b j e c t t o t h o s e r u l e s , although

and

must

therefore

be

s u p p o s e d to

be t h e m -

neither positive or negative, nor yet n u l l numbers, so that the

m a g n i t u d e s w h i c h t h e y are s u p p o s e d t o d e n o t e can n e i t h e r b e greater t h a n n o t h i n g , n o r less t h a n n o t h i n g , n o r e v e n e q u a l t o n o t h i n g . I t m u s t b e h a r d t o f o u n d a S C I E N C E o n s u c h g r o u n d s a s t h e s e . . . . (3; 294)

Hamilton

then

asked

w h e t h e r e x i s t i n g A l g e b r a , i n t h e state t o w h i c h i t has b e e n a l r e a d y u n f o l d e d b y t h e m a s t e r s o f its r u l e s a n d o f its l a n g u a g e , offers i n d e e d n o rudiment which Algebra:

a

may encourage

Science

a hope

properly so called;

of developing a S C I E N C E of strict, pure, a n d i n d e p e n d e n t ;

d e d u c e d b y v a l i d r e a s o n i n g s f r o m its o w n i n t u i t i v e p r i n c i p l e s ; a n d t h u s n o t less a n o b j e c t o f p r i o r i c o n t e m p l a t i o n t h a n G e o m e t r y , n o r less d i s t i n c t , i n its o w n e s s e n c e , f r o m t h e R u l e s w h i c h i t m a y t e a c h o r use, a n d f r o m t h e S i g n s b y w h i c h i t m a y express its m e a n i n g . (3; 2 9 5 ) Hamilton

concluded

ment"

295) a n d elaborated on this idea by writing:

(3;

The folded is

argument into

an

possible,

for

"that

the

chiefly

Intuition

conclusion

independent

rests

the

Pure on

that

the

or

that

Science,

the

of T I M E

existence

notion a

is

such

o f time

Science

of certain

a rudi-

may

be

of Pure

priori

unTime

intuitions,

c o n n e c t e d w i t h that n o t i o n of t i m e , a n d fitted to b e c o m e the sources of a pure Science; and on the actual deduction of such a Science from those p r i n c i p l e s , w h i c h t h e a u t h o r c o n c e i v e s t h a t h e has b e g u n . (3; 2 9 6 - 2 9 7 ) In velop

the

second

the

real

section

of this

number system

concept of time.

paper

on

the

Hamilton attempted to basis

of the

intuition

de-

of the

In this w a y he b e l i e v e d he c o u l d justify the use of

negative n u m b e r s as c o r r e s p o n d i n g to steps in time. It

is

derived

24

generally from

Kant.

believed Such

that

Hamilton's

stress

on

m a y n o t b e t h e case, for Kant's

time

was

name is

Sir W i l l i a m R o w a n

never

mentioned

these

ideas

he

did

Pure

in

mention

Reason

the

paper.

Preface

Kant

(4;

In

to

and

"encouraged

v i e w . . . ." after

in

the

Hamilton's

his

Lectures

wrote

[him]

that

to

[2]) As early as

mentioning geometry:

nected made

became

with by

other."

Hamilton

reading

K a n t four

in

years

Hamilton wrote:

later

exposition

Quaternions

reading

(4;

Kant's

and

of

[2]-[3]) Critique

publish

of this

1827 H a m i l t o n wrote, i m m e d i a t e l y

"The

intimately

each

on

entertain

sciences

adopt here a v i e w of Algebra w h i c h propose)

Hamilton and Quaternions

I

of Space

have

intertwined

(2,1;

229)

letters

it

From

seems

after m a k i n g t h e

and Time

(to

elsewhere v e n t u r e d to and

indissolubly

con-

a n u m b e r of statements

quite

clear that

he

statement.24

above

began

In

1835

"and my o w n convictions, mathematical and meta-

physical, have b e e n so long and so strongly converging to this point (confirmed I

cannot

no

doubt

easily

of late by the

yield

to

the

stare at my strange t h e o r y . "

study of Kant's

authority (2,11;

142)

of those It thus

Pure Reason),

other

friends

seems

that who

that at most

K a n t served as a catalyst for the d e v e l o p m e n t of his ideas a n d as a confirmation

of them.

In the third part of the Conjugate

of his essay is part

is

essay H a m i l t o n p r e s e n t e d his " T h e o r y o f

Functions, or Algebraic Couples." W h i l e the second part generally considered of minor importance, the third

universally

admitted

to

be

of great importance,

for herein

H a m i l t o n d e v e l o p e d c o m p l e x n u m b e r s in terms of ordered pairs of real numbers in almost exactly the same w a y as it is done in m o d e r n mathematics.25 T h e stress

in this section was not on time, although

the interpretation of the couples in terms of time was given. Hamilton

at no point in the paper mentioned Warren or the geometrical

interpretation of complex n u m b e r s ; f r o m this it seems probable that H a m i l t o n believed (like Gauss) that the geometrical representation was

an aid to

intuition,

b u t not a satisfactory justification for com-

plex numbers. Essentially w h a t H a m i l t o n d i d in this section was to set u p o r d e r e d pairs o f r e a l n u m b e r s (a, b ) a n d d e f i n e o p e r a t i o n s o n them.

T h e s e operations w e r e all

numbers.

He

equivalent to In the and

then

complex

THEORY

denotes

showed

an

OF

done

that the

numbers SINGLE

in terms of the rules for real

couples thus considered were

o f t h e f o r m a - h bi.

N U M B E R S , the

IMPOSSIBLE

He

symbol

EXTRACTION,

or a

wrote:

V ^ T is

merely

absurd,

IMAGI-

N A R Y N U M B E R ; but i n the T H E O R Y O F C O U P L E S , the same symbol V - l REAL

is

significant, COUPLE,

square-root

o f the

and

denotes

namely

couple

(—1,

(as 0).

a we In

POSSIBLE have the

just

EXTRACTION, now

latter theory,

seen)

the

or

a

principal

therefore, though

not in the former, this sign V — l may properly be e m p l o y e d ; a n d we m a y write,

if

a, + a2V-l

we

choose,

for

any

couple

(au

a2)

whatever,

(a,,

a2)

=

(3; 4 1 7 - 4 1 8 )

25

A

History

Hamilton

of V e c t o r Analysis

concluded

the

essay b y

writing:

t h e p r e s e n t Theory o f Couples i s p u b l i s h e d . . .

to s h o w .

.

.

that expres-

sions w h i c h seem a c c o r d i n g to c o m m o n v i e w s to be m e r e l y symbolical, and quite

incapable

of being interpreted,

m a y pass

into the w o r l d of

thoughts, and acquire reality and significance, if Algebra be v i e w e d as not

a

mere

Art or

Language,

but as

the

Science

of Pure T i m e .

The

author hopes to p u b l i s h hereafter m a n y other applications of this v i e w ; especially to Equations and Integrals, and to a T h e o r y of Triplets and Sets

of Moments,

Steps, a n d N u m b e r s , w h i c h includes this T h e o r y of

C o u p l e s . (3; 4 2 2 ) The

" T h e o r y of Triplets" that he sought was of course the extension

of the

complex

It is and

number system

clear from

importance

this

to three

paper that

of the

dimensions.

Hamilton

associative,

understood the

commutative,

nature

and distributive

laws.26 T h e majority of mathematicians appreciated the significance of these

laws

had been

only

after

number

developed which

Hamilton's

"Theory"

systems

(especially

quaternions)

did not obey them.

was

poorly received.

Most mathematicians

d i d n o t agree w i t h H a m i l t o n ' s stress o n t i m e , a n d a f e w felt t h e n e e d for

the

d e v e l o p m e n t of c o m p l e x

geometrical

one.

That

Gauss

numbers

and

justification of complex numbers

Bolyai

on

a basis

other than

rejected the

a

geometrical

is almost certainly d u e to the fact

that they both had previously discovered non-Euclidean geometry. W h e n n o n - E u c l i d e a n g e o m e t r y b e c a m e k n o w n (after 1860), mathematicians

then

numbers

became

interested

in

the

in terms of ordered pairs of real

Hamilton's

"Essay"

was

quaternions

numbers.

an

important event in the history of the

for

a

discovery

of

Hamilton

on

direction,

in addition to the

the

development of complex

number

of

quest for higher complex quest in terms

reasons. numbers

First, from

it

set

another

of a m e t h o d of analysis

for t h r e e - d i m e n s i o n a l space. Second, t h r o u g h his m e t h o d of couples at

least

Hamilton

himself became

convinced of the

legitimacy

of

c o m p l e x n u m b e r s , a n d m o r e i m p o r t a n t l y he also o b t a i n e d a m e t h o d that c o u l d be e x t e n d e d in such a w a y as to assure the legitimacy of higher

complex

numbers,

formed

for example by triplets or quad-

r u p l e t s (as i n t h e c a s e o f q u a t e r n i o n s ) . T o p u t i t a n o t h e r w a y , b y t h i s method cantly

H a m i l t o n was prepared, perhaps to discover, m o r e signifi-

to

accept

as

legitimate,

"four-dimensional"

complex

num-

b e r s (as q u a t e r n i o n s ) , e v e n i f n o g e o m e t r i c a l j u s t i f i c a t i o n w e r e t o b e available. from

Support for the above analysis is f o u n d in a letter of 1841

Hamilton to

As

to T r i p l e t s ,

De I

Morgan:

m u s t a c k n o w l e d g e , that t h o u g h I fancied m y s e l f at

one time to be in possession of something w o r t h publishing about them,

26

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

I never could resolve the p r o b l e m w h i c h you have justly signalised as t h e m o s t i m p o r t a n t i n t h i s b r a n c h o f ( f u t u r e ) A l g e b r a : t o assign t w o s y m b o l s O a n d &), s u c h t h a t t h e o n e s y m b o l i c a l e q u a t i o n a + M l + co) = a , +

+ c,&>

shall give the three equations b = bu c = c,

a = au

B u t , i f m y v i e w o f A l g e b r a b e j u s t , i t must b e p o s s i b l e , i n some w a y o r o t h e r , t o i n t r o d u c e n o t o n l y t r i p l e t s b u t polyplets, s o a s i n s o m e s e n s e t o satisfy the s y m b o l i c a l e q u a t i o n a= (a,, a2, . .

.

an);

a being here one symbol, as indicative of one (complex) thought; and f l j , a2, . . . a n d e n o t i n g n r e a l n u m b e r s , p o s i t i v e o r n e g a t i v e ; t h a t i s , i n o t h e r w o r d s , n dates, in t h e c h r o n o l o g i c a l sense of t h e w o r d , o n l y excluding

outward

marks

and

measures,

and

the

notion

of cause

and

e f f e c t . (2,11; 3 4 3 ) Moreover,

after

1843

point of v i e w for his he made this perusal

Hamilton

discovery one

of my

old

stressed

the

importance

discovery of quaternions.

essay,

day w h e n

I

He

of

this

said in fact that

" b e i n g t h e n fresh f r o m a re-

renewed

my

attempts

to

combine

my

g e n e r a l n o t i o n o f sets o f n u m b e r s , c o n s i d e r e d a s s u g g e s t e d b y sets of moments

of time,

with

geometrical

considerations

of points and

lines in t r i d i m e n s i o n a l space. . . . " 2 7

IV. At

Hamilton

s

end

of his

the

Discovery "Essay"

seeking a triplet system. find triplets

as

of

early as

of 1837

not alone

that John system

Hamilton from

at

which

T.

for

in his

1830.

(4;

and

least was

Graves 1836,

similar

(4;

[36]-[37])

De

Morgan,

had

"as

In in

tried

to or

been

1841 which

one

form

in

a

higher

he

was

Morgan

had

received asked

Hamilton

o w n statement

than

correspondence sent

of that year

That

complex

earlier

Graves

Hamilton

Hamilton De

conjecture

property.

perhaps

at w h i c h time to

stated that

shown by Hamilton's

early, had

[39]) T h e

distributive

quest is

Graves space

Hamilton

He h a d , in fact, m a d e d e f i n i t e a t t e m p t s to

entailed abandonment of the was

Quaternions

letter

Hamilton

2 8

on the subject

Hamilton

constructed a

number

myself."

from

a system in

1835.

Augustus

about his

trip-

lets. W i t h this letter w a s a c o p y o f D e M o r g a n ' s 1 8 4 1 p a p e r " O n t h e Foundation

of Algebra,"

2 9

brief discussion of triplets. When lets, t h e

by

1843

Hamilton

framework

within

in (4;

which

De

Morgan

had

included

a

[41]—[42])

began

another intense

which

the

search

search

for trip-

had to be conducted

27

A H i s t o r y of V e c t o r Analysis

was clear to him. T h e following m a y be taken as an outline of the properties that he consciously h o p e d the n e w n u m b e r s w o u l d have. 1. T h e associative property for addition and multiplication. T h u s if

N,

N\

and

(N

+

N')

+

2. T h e N + 3.

N'

N" N"

are

three

and

commutative

=

N'

The

+

N

and

distributive

4. T h e

such

N(N'N")

=

property NN'

=

numbers,

then

N + ( N ' + N") =

and

multiplication.

(NN')N". for

addition

N(N'

+

N'N.

property.

N")

=

NN'

+

NN".

p r o p e r t y that division is u n a m b i g u o u s . T h u s if N a n d N'

are any g i v e n c o m p l e x n u m b e r s , it is always possible to a n d only one n u m b e r X N

and

N')

such

that

(in

general,

NX =

a n u m b e r of t h e

find

one

same f o r m as

N'.

5. T h e property that the n e w n u m b e r s obey the law of the moduli. T h u s if any three triplets c o m b i n e so that {ax

bxi +

+

cj)(a2 +

b2i +

c2j) =

a3 +

b3i +

b22

c 22) =

(a32 +

fo32

c3j,

then (a,2 6. T h e

+

b2

+

c2){a2

+

+

+

c32):

property that the n e w numbers w o u l d have a significant

i n t e r p r e t a t i o n i n t e r m s o f t h r e e d i m e n s i o n a l space. It is well

k n o w n that ordinary c o m p l e x n u m b e r s have all these

properties, w i t h the exception that their geometrical interpretation is for t w o - d i m e n s i o n a l space. In one sense, then, the above is simply

a

detailed

which

statement that

would be

Hamilton

sought for n e w

numbers

directly analogous to ordinary complex numbers.

Of the above properties o n l y the c o m m u t a t i v e property for multiplication h a d to be a b a n d o n e d for quaternions. W i t h limits as restrict i v e a s these H a m i l t o n c o u l d o n l y b e satisfied w i t h quaternions, for, as C. S. Peirce p r o v e d in 1881, " o r d i n a r y real algebra, o r d i n a r y algeb r a w i t h i m a g i n a r i e s , a n d real q u a t e r n i o n s are t h e o n l y associative algebras in w h i c h division by finites always yields an unambiguous quotient." erties

are

30

It is s i g n i f i c a n t to ask at this p o i n t w h i c h of t h e s e p r o p -

retained

for the

scalar (dot)

a n d v e c t o r (cross)

m u l t i p l i c a t i o n s in m o d e r n vector analysis.

product

For the dot product the

associative l a w for multiplication is not relevant, and both the law of the m o d u l i and the unambiguity of division must be abandoned.31 For the must be

cross

product the

associative

and commutative properties

abandoned.32 Moreover division is not unambiguous, and

the l a w of the m o d u l i fails as well.33 F r o m the above comparison of the properties of quaternions and vectors it is e v i d e n t that at least in some ways quaternions not o n l y are s i m p l e r t h a n m o d e r n vectors b u t also entail f e w e r innovations.

28

Sir W i l l i a m R o w a n

In the

period

immediately

after

their

Hamilton

discovery

and Quaternions

quaternions

were

criticized because of the abandonment of the commutative property for m u l t i p l i c a t i o n .

It is an interesting historical speculation in this

regard as to what w o u l d have commutative

and

associative

sion was in general

been

said of a system in w h i c h the

properties

failed,

ent types of multiplication were defined. context of the above attempts to

and

in w h i c h divi-

impossible, and in w h i c h m o r e o v e r t w o differIn any case it was

properties that H a m i l t o n in

in the

1843 r e n e w e d his

triplets.34

find

O n O c t o b e r 16,

1843, H a m i l t o n discovered quaternions.

the best description

Perhaps

of the circumstances surrounding this event is

contained in a letter H a m i l t o n wrote in 1865 to his son A r c h i b a l d H. Hamilton: I f I m a y b e a l l o w e d t o s p e a k o f myself i n c o n n e x i o n w i t h t h e s u b j e c t , I m i g h t d o s o i n a w a y w h i c h w o u l d b r i n g you i n , b y r e f e r r i n g t o a n antequaternionic t i m e ,

when

the

of a V e c t o r ,

conception

you

were as

a

m e r e child, b u t h a d c a u g h t f r o m m e

r e p r e s e n t e d b y a Triplet:

and indeed I

happen to be able to p u t the finger of m e m o r y u p o n the year a n d m o n t h — October, 1843 — w h e n h a v i n g r e c e n t l y r e t u r n e d f r o m visits to C o r k a n d Parsonstown, connected w i t h a M e e t i n g of the British Association, the desire to discover the

laws

of the

multiplication referred to regained

w i t h me a certain strength a n d earnestness, w h i c h h a d for years b e e n dormant, b u t was t h e n on the p o i n t of b e i n g gratified, a n d was occasionally talked of w i t h you. E v e r y m o r n i n g in the early part of the abovecited month, on my c o m i n g d o w n to breakfast, your (then) little brother W i l l i a m E d w i n , a n d y o u r s e l f , u s e d t o a s k m e , " W e l l , P a p a , c a n y o u multiply t r i p l e t s " ? W h e r e t o I of the head:

was always o b l i g e d to r e p l y , w i t h a sad shake

" N o , I c a n o n l y add a n d s u b t r a c t t h e m . "

B u t on the 16th day of the same m o n t h — w h i c h h a p p e n e d to be a M o n day, and a C o u n c i l day of the Royal Irish A c a d e m y — I was w a l k i n g in to attend and preside, a n d your m o t h e r was w a l k i n g w i t h me, along the Royal Canal, to w h i c h she h a d perhaps d r i v e n ; a n d a l t h o u g h she t a l k e d with

me

now

and then,

my m i n d , w h i c h that

I

and a

felt

at

spark

l o n g years

to

gave

once

the

flashed come

yet

a n under-current o f t h o u g h t w a s

going on in

a t l a s t a result, w h e r e o f i t i s n o t t o o m u c h t o s a y importance. forth,

the

An

herald

electric (as

circuit

seemed

to

close;

I foresaw, immediately) o f m a n y

o f d e f i n i t e l y d i r e c t e d t h o u g h t a n d w o r k , b y myself i f

s p a r e d , a n d a t a l l e v e n t s o n t h e p a r t o f others, i f I s h o u l d e v e n b e a l l o w e d to live long enough distinctly to communicate the discovery. Nor could I resist t h e i m p u l s e — u n p h i l o s o p h i c a l as it m a y h a v e b e e n —to cut w i t h a knife on a stone of B r o u g h a m B r i d g e , as we passed it, the f u n d a m e n t a l formula w i t h the symbols, i, j, k; namely i2 = j2 = which

contains

t i o n , has

the

k1 =

ijk = —1,

Solution o f t h e Problem, b u t o f c o u r s e , a s

an

inscrip-

long since m o u l d e r e d away. A more durable notice remains,

however, on the

Council

Books of the A c a d e m y for that day (October

16th, 1843), w h i c h records t h e fact, that I t h e n a s k e d for a n d o b t a i n e d

29

A

History

leave

of V e c t o r Analysis

to

Session:

read

a

Paper

on

Quaternions,

at t h e

First

General

Meeting o f t h e

w h i c h reading took place accordingly, on M o n d a y the 13th of

t h e N o v e m b e r f o l l o w i n g . (2,11; 4 3 4 - 4 3 5 ) Thus

in

a

very

nounced

the

numbers

of t h e

dramatic

discovery form

of

manner

Hamilton

quaternions.

w + ix + jy +

kz,

discovered

These

are

and

an-

hypercomplex

w h e r e w, x, y, a n d z a r e r e a l

n u m b e r s , a n d i, j, a n d /c are u n i t vectors, d i r e c t e d a l o n g the x, y, a n d z axes

respectively. ij = ji

=

The

i, j, a n d /c units obey the f o l l o w i n g laws:

k

jk

- k

kj ii

It is

to be

general

= j j

noted that for t w o

equal

q'q.

=

T h e loss

i

ki

- i

ik

= =

kk

=

= j =

- j

—1

quaternions

q

and

of commutativity in

q',

qq'

does

not in

quaternions, while

it is v e r y i m p o r t a n t h i s t o r i c a l l y , is also significant m a t h e m a t i c a l l y , because this complicates calculations in w h i c h quaternions are used. A l l the other properties discussed above are satisfied by quaternions. T h u s it may be v e r i f i e d that q u a t e r n i o n m u l t i p l i c a t i o n is associative a n d q u a t e r n i o n division

is

unambiguous.

special m e n t i o n ,

These

are

two

important properties

which

bear

since t h e y are not preserved in the algebra of m o d e r n

vectors. There have been a n u m b e r of discussions published on the mathematical present

details

of Hamilton's

purposes

within the Almost

all

that

procedure

need

be

after his

is

his

discovery;

for the

that

Hamilton

worked

discussed above.35

context that has b e e n immediately

in

said

discovery

H a m i l t o n stated that he

"felt that it m i g h t be w o r t h my w h i l e to expend [on quaternions] the labour of at least ten 436)

Hamilton

working

(or it m i g h t be

actually

almost

spent

exclusively on

fifteen)

the

last

y e a r s t o c o m e . " (2,11;

twenty-two

quaternions.

The

years letters

of

his

life

of the first

f e w days after the discovery s h o w that H a m i l t o n felt that his system had

importance

nometry.

for

heat

electricity,36

theory,

sense

I

nions,

from

hope

spherical

trigo-

that

I

am

sions

of their principles, to

me

actually

g r o w i n g modest a b o u t t h e

quater-

m y s e e i n g s o m a n y p e e p s a n d vistas into f u t u r e expan-

pears

to

be

as

I

still

important

must for

century as the discovery of fluxions teenth."

and

(2,11; 4 4 2 ) I n 1 8 5 1 h e w r o t e : " I n g e n e r a l , a l t h o u g h i n o n e

(2,11;

assert that this d i s c o v e r y apthe

middle

of the nineteenth

was for the close of the seven-

445)

In o n e sense at least H a m i l t o n ' s discovery was e p o c h m a k i n g , for quaternions

were

the

number system which His

30

first

well-known

did not obey the

consistent

and

significant

laws of ordinary arithmetic.

" c u r i o u s , a l m o s t w i l d " (as h e c a l l e d i t [2,11; 4 4 1 ] ) d i s c o v e r y m a y

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

be c o m p a r e d to the discovery of n o n - E u c l i d e a n geometry. B o t h discoveries broke bonds

set b y c e n t u r i e s o f m a t h e m a t i c a l t h o u g h t . I m -

mediately

other

after

by Augustus

1843

De

Morgan

new

(who

number

systems

were

(1846).38

T . G r a v e s ( 1 8 4 4 ) (2,11; 4 5 4 - 4 5 5 ) , a n d C h a r l e s G r a v e s This

section

publications 13,

1843,

Irish

will

on

be

concluded

by

a

quaternions through the

Hamilton

Academy,

of

read a paper on which

at

discovered

p u b l i s h e d f i v e n e w systems),37 J o h n

least

discussion

of Hamilton's

year

On

1847.

quaternions

part

was

November

before the Royal

published

in

1844.39

Either this paper or the very similar paper in the July, 1844, issue of the

Philosophical

nions.

(5,25;

Magazine

was

Quaternions,"

delivered

the

In

these

July,

1846,

papers

not analogous (the

w

first

November

Academy and published in in

his

publication

on

quater-

10-13) A m o n g the m o s t i m p o r t a n t papers are his " O n

issue

Hamilton

to

11,

1844,

of the

Philosophical

dealt w i t h

the

the

Royal

Irish

Magazine.

does

(5,29;

26-31)

fact that q u a t e r n i o n s are

ordinary complex numbers

o f w + i x + j y + kz)

to

1847,40 a n d t h e s i m i l a r p a p e r p u b l i s h e d

in that the

scalar part

n o t i n d i c a t e d i s t a n c e on an axis

un-

less, as he h a d s u g g e s t e d earlier, q u a t e r n i o n s be c o n s i d e r e d as f o u r dimensional. A n d on

Thus

account

Hamilton

of the

(writing

facility w i t h

in

the

which

third person)

this

so

stated:

c a l l e d imaginary e x -

p r e s s i o n , o r s q u a r e r o o t o f a n e g a t i v e q u a n t i t y , i s c o n s t r u c t e d b y a right line

having direction

in

space,

and

h a v i n g x,

y,

z f o r its t h r e e r e c t a n g u l a r

c o m p o n e n t s , o r p r o j e c t i o n s o n t h r e e r e c t a n g u l a r axes, h e has b e e n i n d u c e d to call the t r i n o m i a l expression itself, as w e l l as the line w h i c h it

represents,

a

VECTOR.

A

quaternion

g e n e r a l l y o f a real p a r t a n d a vector.

may

The

thus

fixing

be

said

to

consist

a s p e c i a l a t t e n t i o n on

this last part, or e l e m e n t , of a q u a t e r n i o n , by g i v i n g it a s p e c i a l n a m e , a n d d e n o t i n g it in m a n y calculations by a single a n d special sign, appears

to

the

dealing with

author to the

have

subject:

been

although

an

improvement in

the general

his

method

of

notion of treating the

constituents of the imaginary part as coordinates had occurred to h i m in his first researches. Regarded from a geometrical point of v i e w , this algebraically imagin a r y p a r t of a q u a t e r n i o n has t h u s so n a t u r a l a n d s i m p l e a s i g n i f i c a t i o n or representation in space, that the difficulty is transferred to the algebraic a l l y r e a l p a r t ; a n d w e are t e m p t e d t o ask w h a t t h i s last c a n d e n o t e i n g e o m e t r y , o r w h a t i n s p a c e m i g h t h a v e s u g g e s t e d it.41 The

origin

following

of the

word

quotation

in

vector ( a n d t h e w o r d scalar) the

similar

paper

in

is

the

clear from the Philosophical

Maga-

zine. The

a l g e b r a i c a l l y real p a r t m a y r e c e i v e . . . a l l v a l u e s c o n t a i n e d o n t h e

o n e scale o f p r o g r e s s i o n o f n u m b e r f r o m n e g a t i v e t o p o s i t i v e i n f i n i t y ; w e shall

call

it t h e r e f o r e

the

scalar part,

or

simply the

scalar o f t h e q u a t e r -

n i o n , a n d s h a l l f o r m its s y m b o l b y p r e f i x i n g , t o t h e s y m b o l o f t h e q u a t e r -

31

A

History

of V e c t o r Analysis

n i o n , t h e c h a r a c t e r i s t i c S e a l . , o r s i m p l y S., w h e r e n o c o n f u s i o n s e e m s l i k e l y t o arise f r o m u s i n g this last a b b r e v i a t i o n . O n t h e o t h e r h a n d , t h e algebraically

imaginary

part,

being

geometrically

constructed

by

a

straight l i n e or radius vector, w h i c h has, in general, for each d e t e r m i n e d q u a t e r n i o n , a d e t e r m i n e d l e n g t h a n d d e t e r m i n e d d i r e c t i o n in space, m a y be

c a l l e d t h e vector part, o r s i m p l y t h e vector o f t h e q u a t e r n i o n ;

and may

be denoted by prefixing the characteristic Vect., or V. We may therefore say

that

a

quaternion

is

in

general

parts, a n d m a y w r i t e Q = S e a l . SQ + V Q . F r o m the

(5,29;

above

quotations

introduced

cise

mathematical

The than

the

this.

scalar

sense,

analysis.

quotations

In

of its

own

scalar

and

vector

Q = S . Q + V . Q or s i m p l y Q =

it m a y be inferred that it was H a m i l t o n

term

a

sense

Hamilton

also

the

the

term

vector i n

similar

term

its

pre-

radius

vector

before.

however

they

had

and

although

used for m a n y years

above

sum

26-27)

who

had been

the

Q + Vect.

have

mark

the

introduced

a

far

greater

beginning

his

significance

of modern

symbols

S

and

vector

V because

"separation of the real a n d i m a g i n a r y parts of a q u a t e r n i o n is an operation of such frequent occurrence, and may be regarded as so fund a m e n t a l in this theory. . . of his

quaternions xi + yy'

yj + +

(5,29; 26) H a m i l t o n illustrated the use

symbols as applied to the product of the multiplication of two a a n d a',

zk a n d

zz');

V.

in

which

the

a' = x'i + y ' j + z'k, aa'

=

i(yz'

-

z y ' )

scalar parts w e r e 0.

Hamilton wrote: + j(zx'

-

x z ' )

+

Letting a =

" S . aa' = — (xx' +

k(xy'

-

yx').

.

.

(5,29; 30) It is o b v i o u s that these are e q u i v a l e n t to the m o d e r n vector

(cross)

product.42 bols,

product

using them

n o w be used. S.aa'

=

0

negative

of the

modern

scalar (dot)

in

cases

where

the

dot and

cross p r o d u c t w o u l d

Hamilton then proceeded to prove such equations as

when

Another 1847.43

and to the

H a m i l t o n and Tait m a d e very frequent use of these sym-

pair

In these

a

and

of

a'

very

papers

...<],

defined

known

operation

with

are

parallel.

important Hamilton

relation

of partial

to

papers

appeared

introduced the

these

three

differentiation,

in

1846

and

n e w operation

symbols

performed

ijk,

and to

the

w i t h respect to

t h r e e i n d e p e n d e n t b u t r e a l v a r i a b l e s xyz, a s f o l l o w s : id

this

new

expressed

characterestic by

the

will following

have

jd L— the

kd 1 negative

of

its

symbolic

square

formula:

of w h i c h it is clear that t h e a p p l i c a t i o n s to a n a l y t i c a l p h y s i c s m u s t be ext e n s i v e in a h i g h d e g r e e . (5,31; 291)

32

Sir W i l l i a m R o w a n H a m i l t o n a n d Q u a t e r n i o n s

T h e final paper of special importance is that delivered N o v e m b e r 13,

1843, b u t p u b l i s h e d o n l y in f u l l (and certainly w i t h additions) 1848.44 T h i s v e r y l o n g p a p e r w a s m a i n l y d e v o t e d t o d e v e l o p i n g

in

quaternions as

in

analogy to his

number couples,

1837 treatment of complex n u m b e r s

and to relating his

quaternions to the opera-

tions, principles, and elements of traditional mathematics. In s u m m a r y , by the e n d of 1847 H a m i l t o n h a d p u b l i s h e d at least thirty-four papers i n f i v e different journals. M a n y o f these w e r e expository;

some

geometry,

used

quaternions

mechanics,

and

for the

astronomy.

On

solution a

of problems

number

in

of occasions

n e w results w e r e obtained by means of quaternions, b u t n o n e seem to have b e e n of a striking nature.

V.

Quaternions

Until

Hamilton

s

Death

(1865)

T h e aim of this section is to consider H a m i l t o n ' s w o r k on quaternions from

1847 to

1865 and, m o r e i m p o r t a n t to us, to discuss the

reception of quaternions f r o m the t i m e of their d i s c o v e r y u n t i l 1865. In considering the reception of quaternions it must not be forgotten that, u n l i k e m a n y n e w discoveries, the discovery o f q u a t e r n i o n s was made by a m a n having considerable fame before the discovery was

made.

Lagrange 1842

It was in de

votre

1842 that Jacobi referred to H a m i l t o n as " l e pays,"

and

the

Athenaeum,

in

discussing

the

meeting of the British Association, m e n t i o n e d that "peculiar

interest was excited by the presence of the three great astronomers, Bessel, Herschel, a n d H a m i l t o n , w h o w e r e seen seated together on the

platform."45

Hamilton's

fame

acted

as

a

powerful

force

in

spreading knowledge of quaternions and in forestalling criticism of them.

Mathematicians w h o d i d not use quaternions w e r e nonethe-

less c a u t i o u s a b o u t a t t a c k i n g t h e m . The

question

within

of the

reception

of quaternions

should

be

seen

the perspective provided by the history of the reception of

s i m i l a r s y s t e m s . T h e r a p i d i t y w i t h w h i c h n e w s y s t e m s o f i d e a s (as o p p o s e d to n e w e x p e r i m e n t a l or technical results) are r e c e i v e d is often

exaggerated.

Non-Euclidean

geometry,

Boolean

algebra,

Maxwell's theory, and even the calculus were only gradually appreciated

and

numbers,

assimilated

into

scientific

thought.

the Grassmannian system, and the

Moreover complex

Gibbs-Heaviside

(or

modern) system of vector analysis w e r e all s l o w l y received. W h e n H a m i l t o n and the y o u n g Tait express their surprise that the quaternion

system

was

only

slowly being appreciated, their statements

are to be t a k e n w i t h caution, for t h e y express a l a m e n t that is n e a r l y universal a m o n g the p r o p o n e n t s of n e w systems of ideas.

33

A

History

of V e c t o r Analysis

It is also i m p o r t a n t to realize that to a certain extent the groundwork

had

gand,

been

laid for quaternions

representation of complex numbers Gauss, more

before

1843.

The

work

of Ar-

M o u r e y , Warren, Gauss, and others had made the geometrical

Peacock,

De

Morgan,

an accepted idea.

and others

sophisticated v i e w of algebra.

Mobius,

and

Grassmann,

The work of

had helped to produce a

Finally, the work of Bellavitis,

a n d t h e fact that Wessel, Gauss, A r g a n d ,

Servois, Frangais, M o u r e y , John T. Graves, De Morgan, and Hamilton h a d all, a n d for the most part i n d e p e n d e n t l y , searched for triplets, indicates that there was a felt n e e d for such a system as quaternions. Quaternions,

with

their abandonment of commutativity,

were

in

any case s o m e w h a t r e v o l u t i o n a r y i n conception. T h u s , e v e n J o h n T . Graves, such

a

perhaps new

Hamilton's

the

idea,

mathematician

wrote

in

the

best

prepared

following

letter announcing the

manner

for in

accepting

response

to

discovery:

T h e r e is still s o m e t h i n g in the system w h i c h gravels me. I have not yet a n y clear v i e w s a s t o t h e e x t e n t t o w h i c h w e are a t l i b e r t y a r b i t r a r i l y t o create imaginaries, and e n d o w t h e m w i t h supernatural properties. You are c e r t a i n l y j u s t i f i e d by t h e event. Y o u have got an i n s t r u m e n t that facilitates t h e w o r k i n g o f t r i g o n o m e t r i c a l t h e o r e m s a n d suggests n e w ones, a n d i t s e e m s h a r d t o ask m o r e ; b u t I a m g l a d t h a t y o u h a v e g l i m p s e s o n p h y s i c a l a n a l o g i e s . (2,11; 4 4 3 ) It seems that De abandonment were

troubled,

lights?

Morgan and

MacCullagh were troubled by the

of commutativity, what

Probably

the

must initial

and

have

if such

been

reaction

the

was

great

mathematicians

reaction

opposition,

time into curiosity and a greater openness of mind.

among

lesser

passing w i t h

By 1847 quater-

nions h a d to some degree b e c o m e k n o w n , a n d at the British Association

meeting

value.

Hamilton

of that year a debate described the

debate

took

place

concerning their

in a letter to R.

P.

Graves:

B e i n g ready, h o w e v e r , at the c o m m e n c e m e n t of the M e e t i n g , I was told that I n e e d not straiten m y s e l f as to t i m e , the pressure of other Papers h a v i n g not yet b e e n felt; a n d Dr. Peacock said afterwards to me that I had g i v e n " a c a p i t a l e x p o s i t i o n " : w h i l e f r o m m a n y o t h e r q u a r t e r s i t has b e e n t o l d m e t h a t h e has e x p r e s s e d h i m s e l f e v e r y w h e r e a s f a v o u r a b l e t o m y w h o l e system.

M r . Jarrett b r o u g h t f o r w a r d again his C a m b r i d g e objec-

tions, and d w e l t particularly on the possibility of making'mistakes in the u s e o f m y n e w c a l c u l u s . I n r e p l y t o w h i c h I d i s c l a i m e d t h e p o w e r o f sett i n g any l i m i t to the faculty of m a k i n g blunders; but said that the practic a l q u e s t i o n o n t h a t p o i n t w a s , w h e t h e r w i t h a r e a s o n a b l e d e g r e e o f attention a reasonable

security against error c o u l d be attained; w h i c h I

t h o u g h t that my experience

of the

w o r k i n g of the quaternions enabled

me to answer in the affirmative. T h e Rev. Richard Greswell, an Oxford m a n for w h o m I have a great respect, b u t w h o is not particularly scien-

34

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

tific, m a d e some critical remarks in a not u n f r i e n d l y spirit; b u t o b l i g e d me to disclaim a triplet w h i c h he attributed to me, of Positive, Negative, and Imaginary:

a n d t o state that w i t h m e t h e d i s t i n c t i o n b e t w e e n Posi-

tive a n d N e g a t i v e was exactly t h e same a s that b e t w e e n F u t u r e a n d Past, o r b e t w e e n Past a n d F u t u r e . A f t e r s o m e g e n t l e s k i r m i s h e s o f this sort, rose H e r s c h e l ; w h o said that his a d m i r a t i o n o f t h e q u a t e r n i o n s h a d increased w i t h every r e s u m p t i o n of his study of t h e m : a n d that a l t h o u g h it m i g h t be d i f f i c u l t at first to master t h e e x t r e m e l y abstract c o n c e p t i o n s , and the n e w algorithm w h i c h they involve, yet he was w e l l c o n v i n c e d that it was w o r t h the trouble. T h e y appeared to h i m a bag, into w h i c h one n e e d o n l y insert his h a n d to d r a w forth treasures; he m i g h t call t h e m a cornucopia of scientific abundance: and in a w o r d , his earnest advice to mathematicians w o u l d be, to " s t u d y the q u a t e r n i o n s . " Sir J o h n H e r s c h e l had r e m a i n e d at O x f o r d on purpose to make this statement; a n d i m m e d i ately afterwards he started off for C o l l i n g w o o d . M r . Airy, seeing that the subject c o u l d not be c u s h i o n e d , rose t h e n to speak of his o w n acquaintance w i t h it, w h i c h he a v o w e d to be n o n e at all; b u t gave us to u n d e r stand that what he d i d not k n o w c o u l d not be w o r t h k n o w i n g . He w a r n e d all persons, if t h e y s h o u l d use the m e t h o d , to do so w i t h the extremest caution; professing to regard me as b e l i e v i n g it to be a right one, solely o n t h e g r o u n d o f t h e a g r e e m e n t o f its r e s u l t s , s o far a s t h e y h a v e b e e n y e t obtained, w i t h those of the older method. W h a t was obscure was to h i m as if it w e r e erroneous, what was paradoxical was to h i m as if it w e r e f a l s e ; a n d h e t h o u g h t that s y s t e m u s e l e s s a s a n a l g e b r a i c a l g e o m e t r y , o f w h i c h the expressions were so extremely difficult of geometrical interp r e t a t i o n . (2,11; 5 8 5 - 5 8 7 ) Concerning this statement It also

quotation it should be m e n t i o n e d that Herschel's

appeared

seems

in

the

Athenaeum

in

an

even

only partly valid, though forcefully delivered. seem

to

stronger

form.45

that Jarrett's, Greswell's, a n d Airy's objections w e r e

have

changed

around

this

time,

for

MacCullagh's views in

late

1847,

on the

night before he c o m m i t t e d suicide, he told a friend that in his opini o n H e r s c h e l h a d not g o n e too far in his statement at the B r i t i s h Association meeting. In

1846

Academy

(2,11;

Hamilton presumably

searches, a n d in

595-596)

resigned to

obtain

the

presidency

more

time

for

of the his

Royal

Irish

quaternion

re-

1848 both the Royal Irish A c a d e m y and the Royal

Society of E d i n b u r g h a w a r d e d h i m medals for his discovery. It was also in

1848 that H a m i l t o n gave a series of four lectures on quater-

nions at D u b l i n

University.

sketch"

of them.

George

Salmon

cian, besides

and

quaternions

time

who

heard the

Arthur Cayley,

who

was

1845, g i v e n "a short 1848

lectures

were

the first mathemati-

Hamilton, to publish a paper on quaternions (Cayley's

paper appeared in in

He had earlier, in

A m o n g those

and

1845).46 G e o r g e

S a l m o n also b e c a m e interested

a d v a n c e d far e n o u g h i n t h e i r s t u d y that a t o n e

H a m i l t o n thought that Salmon m i g h t b e c o m e his successor in

quaternion

studies.

(2,111;

90)

Salmon

however w e n t on

to

attain

35

A

History

of V e c t o r Analysis

fame in other branches of mathematics, and Peter Guthrie Tait (who will

be

discussed

quaternion

Hamilton's and

detail

lectures

published

described the 1852:

in

as

of

his

1848

Lectures

became

Hamilton's

successor in

were

on

eventually

Quaternions

in

greatly

1853.

expanded

Salmon

aptly

n e e d for such a w o r k w h e n he wrote to H a m i l t o n in

"Your book will

general.

later)

researches.

probably m a k e the use of your m e t h o d more

At present it is a b o w of Ulysses, w h i c h no one can bend

but the owner." Hamilton's

(2,111;

Lectures

346)

on

Quaternions

consisted

of a

series

of

seven

lectures, extending to 737 pages w i t h an additional 64-page preface and a 72-page table of contents. the

history

nions

could

be

interpreted

founded on the numbers.

In the preface Hamilton discussed

of quaternions and showed in in

terms

great detail h o w quater-

of his

concept

concept of time and in terms

These

sections

were

of algebra

of quadruples

as

of real

neither easy nor probably interest-

ing reading for most readers.

Early in the w o r k H a m i l t o n wrote that

he

would

style

not

a

use

happy

a

"metaphysical

choice,

and

the

of e x p r e s s i o n . "

work

(4;

4)

This

was

was further e n c u m b e r e d by his

practice of introducing a multitude of n e w terms.

Early in the work

h e set u p a n elaborate t e r m i n o l o g y b y u s i n g such w o r d s a n d c o m b i nations

as

vector,

vehend,

vection,

vectum,

revector,

revehend,

revection, r e v e c t u m , provector, . . . transvector, . . . factor, . . . profactor, . . . versor, . . . a n d quadrantal semi-inversor. pages

the

mathematical

first seventy-four pages. dition,

v e r s o r , w h i c h is of c o u r s e a

Authors of later quaternion books expressed in three

subtraction,

ideas

that

Hamilton

had

presented in his

H a m i l t o n dealt at s o m e l e n g t h w i t h the ad-

multiplication,

and division of quaternions.

He

showed h o w quaternions were to be applied to spherical geometry and ".

trigonometry

L O G A R I T H M S , T I O N S , .

and

showed

. . C O - O R D I N A T E S ,

.

,"

4 7

It was read. any . . .

SERIES,

He

relations

L I N E A R

D I F F E R E N T I A L S ,

plications

the

of

D E T E R M I N A N T S ,

A N D

A N D

quaternions

to

T R I G O N O M E T R Y ,

Q U A D R A T I C

C O N T I N U E D

E Q U A -

F R A C T I O N S .

c o n c l u d e d w i t h s o m e geometrical a n d astronomical apand the

without

introduction question

of the biquaternion.

a very

long

and a very difficult book to

Herschel in 1853 wrote to H a m i l t o n that the book w o u l d "take man

a

twelvemonth

, " 4 8 a n d in

Your

to

read,

and

near

a

lifetime

to

digest

1859 Herschel wrote:

deduction f r o m Quaternions of Fresnel's W a v e is one of those

things w h i c h I have just knowledge enough to admire without enough to understand.

But

it

set

me

again

on

reading

you

Lectures

on

Quater-

nions, a n d I g o t t h r o u g h t h e t h r e e f i r s t c h a p t e r s o f i t w i t h a m u c h c l e a r e r

36

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

p e r c e p t i o n of m e a n i n g than w h e n I attacked it some three or four years back, b u t I was again o b l i g e d to give it up in despair. N o w I pray y o u to listen to this

cry of distress.

I

f e e l certain t h a t i f y o u p l e a s e d y o u could

put the w h o l e matter in as clear a light as w o u l d make the Calculus itself a c c e s s i b l e a s a n i n s t r u m e n t t o r e a d e r s e v e n o f less " p e n e t r a t i n g p o w e r " than

myself,

who,

having

once

mastered

the

algorithm

a n d t h e conven-

tions s o a s t o w o r k w i t h i t , w o u l d t h e n b e b e t t e r p r e p a r e d t o g o a l o n g w i t h y o u i n y o u r m e t a p h y s i c a l e x p l a n a t i o n s . (2,111; It may be was

recalled that Herschel

a man

of great ability.

three chapters cians

121)

had been a Senior Wrangler and

If he could read no more than the first

(129 pages), h o w far w o u l d t h e less a b l e m a t h e m a t i -

advance?

H a m i l t o n faced some difficulty in getting such a large, n e w w o r k published. Thus in

1853 he wrote to a friend ( M o r t i m e r O'Sullivan):

Y o u w i l l I h o p e b e a r w i t h m e i f I s a y , t h a t i t r e q u i r e d a c e r t a i n capital of scientific reputation, amassed in f o r m e r years, to m a k e it other than d a n g e r o u s l y i m p r u d e n t to hazard the p u b l i c a t i o n of a w o r k w h i c h has, a l t h o u g h at b o t t o m q u i t e c o n s e r v a t i v e , a h i g h l y r e v o l u t i o n a r y air. It w a s a part of t h e o r d e a l t h r o u g h w h i c h I h a d to pass, an e p i s o d e in t h e b a t t l e of life, to k n o w that e v e n c a n d i d a n d f r i e n d l y p e o p l e secretly, or, as it m i g h t happen, openly, censured or r i d i c u l e d me, for w h a t appeared to t h e m m y m o n s t r o u s i n n o v a t i o n s . (2,11; 6 8 3 ) The

longest

Review

for

writing: judge,

1857, "It

is

that

in

review

of the

written

probably by Thomas

confidently the

coming

work

appeared

predicted, centuries

by

in

the

Hill.49

those

Hamilton's

North

Hill

best

American

began

by

qualified

Quaternions

to

will

stand out as the great discovery of our nineteenth century. Yet h o w s i l e n t l y has t h e b o o k t a k e n its p l a c e u p o n t h e s h e l v e s o f t h e m a t h e matician's

library. Perhaps not fifty m e n on this side of the Atlantic

h a v e s e e n it, c e r t a i n l y n o t f i v e h a v e r e a d i t . " 5 0 H e t h e n p r o c e e d e d to compare

Hamilton

and Newton

in

to

Archimedes,

Galileo,

Descartes,

Leibniz,

regard to the p r o b l e m of w h o can j u d g e their ideas.

O t h e r sections p r a i s e t h e b o o k a n d its a u t h o r i n s u c h s t r o n g w o r d s as " T h e discoveries of N e w t o n have done m o r e for E n g l a n d a n d for the

race,

archs;

t h a n has

and

we

been

doubt

done by whole

not

that

in

the

dynasties

great

of British

mathematical

mon-

birth

of

1853, the Q u a t e r n i o n s of H a m i l t o n , there is as m u c h real p r o m i s e of benefit to m a n k i n d as in any event of Victoria's reign." tioning Pythagoras, twenty-three found,

like

Hill

hundred that

with

the

eternal

such

statements

of

stated:

years

longer, the name

Pythagoras,

made

somewhat

had not read the work:

"In

After men-

of Hamilton

immortal

truth first revealed through was

5 1

" A n d if the w o r l d should stand for

weakened

by

him."

when

its 5 2

will be

connection

The

force

of

Hill admitted he

looking over Hamilton's eight hundred

37

A

History

of V e c t o r Analysis

pages

on

Quaternions

them.

.

,"

sketch

.

of the

Hill's one

5 3

in

(for w e

review

of

concluded

with

an extremely brief

of quaternions.

Hamilton's

favorable

Lectures

reviews

mathematics

w o u l d feel

w i l l not p r e t e n d t o say w e h a v e read

was

mathematics

review

of most

ested

The

who

ever

read

on

Quaternions

published;

and

is

any

accepted

certainly

person

Hill's

inter-

statements

inclined, if not obliged, to read Hamilton's book.

review Hill

In the

s h o w e d m o r e ability for rhetoric than for the analysis of

mathematical

ideas;

he

reviewed

Hamilton,

not

his

book,

and

thereby failed to h e e d his o w n advice, for he had argued that f e w if any in

could judge

this

are

case

reviewed

At

various

from

the

significance

book. T h e fate o f H a m i l t o n

times

after

these

were

1843

who

whose

Hamilton

received

letters

or visits

had acquired an interest in quaternions.

Thomas

first-rate mathematician Carmichael,

their books

favorably but uncritically.

mathematicians

A m o n g

of the

is the not u n c o m m o n fate of m e n of fame;

Penyngton

but

interest

lost in

Kirkman,

interest

in

quaternions

who

became

quaternions; was

cut

short

a

Robert by

his

death at age 31, b u t w h o h a d by that t i m e p u b l i s h e d five papers on quaternions;

and J.

w h o wrote to no

"less

P.

Nichol, Professor of Astronomy at Glasgow,

H a m i l t o n that the discovery of quaternions seemed of

value

than

the

momentous

step

shown

us

by

Descartes.

. . ." (2,11; 6 3 5 ) In cal

1860

Nichol

Sciences,

for

published

which

a

Hamilton

work

called

wrote

an

Cyclopaedia

exposition

o f the of

Physi-

quater-

nions.54 T h e interesting aspect of this is that N i c h o l a p p e n d e d to the article a short note w h i c h stated that since these sections were written, t h e n e w b r a n c h o f m a t h e m a t i c s t o w h i c h t h e y r e l a t e a p p e a r s t o h a v e attracted an increasing degree of attention, in our o w n and in foreign countries. of a

The

Lectures

favourable

(Dublin,

article

1853) have for e x a m p l e , f o r m e d the subject

in

the

North American

Review

for July,

1857:

and,

i n d e e d , the Q u a t e r n i o n s h a d b e e n m e n t i o n e d as a m o n g the sources of hope for the future progress of analytical mechanics, in the conclusion to a

very

beautiful

Boston,

volume

(A

System

o f Analytic

Mechanics,

&c.,

page

476.

1855) on that science; by Professor B e n j a m i n Peirce of H a r v a r d

University,

U.S., as f o l l o w s :

. . and much

must soon b e c o m e anti-

quated and obsolete as the science advances, and especially w h e n we shall

have

Hamilton's Benjamin the time tail

the

full

benefit

of the

remarkable

machinery

of

Quaternions."55

Peirce,

who

was

the leading American mathematician of

(and the teacher of T h o m a s Hill), w i l l be discussed in de-

later. At this p o i n t it n e e d o n l y be stated that in

published

38

received

hjs

System

o f Analytical

Mechanics

without

1872 P e i r c e realtering

the

Sir W i l l i a m R o w a n H a m i l t o n a n d Q u a t e r n i o n s

statement

above56

quoted

and

without

introducing

quaternions.

Such statements as those discussed above h e l p e d to lay the foundation for the spread of the quaternion system, b u t in a very indirect way. To advocate the use of a system is not of course equivalent to using the system, a n d the latter is the m o r e important. In the late 1850's a n d 1860's the f e e l i n g g r a d u a l l y arose that q u a t e r n i o n s w e r e o f t e n p r a i s e d b u t s e l d o m used. I n t h e last t h i r d o f t h e c e n t u r y t h e i r use became more frequent, partly as a result of H a m i l t o n ' s second book on quaternions

and partly because elementary presentations

of quaternions b e c a m e available t h r o u g h the efforts of other mathematicians. 1880)

De Volson W o o d in an elementary exposition (written in

perhaps

w e n t t o o f a r i n h i s d e s c r i p t i o n o f H a m i l t o n ' s Lectures

w h e n h e w r o t e o f it: " b u t t h e style i s s o p e c u l i a r , b e i n g diffuse a n d hesitating as

if he

labored t o m a k e h i s r e a d e r s u n d e r s t a n d , a n d t h e

arrangement b e i n g such as to separate parts of essential principles, that, p r o b a b l y , c o m p a r a t i v e l y f e w o f its r e a d e r s , w i t h o u t o t h e r aids, have

m a s t e r e d its

57

principles."

Charles

Graves,

Hamilton's

col-

league at D u b l i n , w r o t e in his eloge for H a m i l t o n : "Students of his lectures

on

claimed

from

Quaternions them

too

have

sometimes

much

attention

c o m p l a i n e d that h e has

to the metaphysics of the

subject, a n d has s t o p p e d t h e m i n t h e i r career o f b u i l d i n g u p , i n order that they m i g h t contemplate afresh the plan of the structure." Some information

is

available concerning the early reception of

quaternions in other countries. that

France

had

been

very

Although

slow

in

in

1861

accepting

Hamilton wrote

quaternions

129), i t w a s t h e

Frenchman Alexandre Allegret who

lished

book

the

first

58

on

H e r m a n n H a n k e l stated in

quaternions

not

written

in

(2,111;

1862 p u b Hamilton.59

by

1867 that H a m i l t o n wrote in a very un-

conventional w a y for continental readers a n d that the only source of information for t h e m at that t i m e was Allegret's w o r k , w h i c h H a n k e l called an "indeed insufficient presentation." lavitis

became

published have

interested

papers

spread to

in

relating to

Russia,

quaternions quaternions.61

for Tait wrote to

60

In Italy Giusto Bel-

and

in

1858

and

1862

Finally, interest must

Hamilton in

1862:

"Pro-

fessor B o l z a n i is h e r e a n d we h a v e h a d l o n g discussions on t h e subject [ q u a t e r n i o n s ] . He is i m m e n s e l y i n t e r e s t e d in it, a n d has g i v e n a c o u r s e o f l e c t u r e s o n i t i n Russia, o f w h i c h h e has p r o m i s e d t o s e n d c o p i e s w h e n p r i n t e d . " (2,111; quaternions

among

the

149) T h u s t h e r e w a s s o m e i n t e r e s t i n

non-English

speaking countries,

although

n a t u r a l l y i t w a s less t h a n t h a t i n B r i t i a n a n d i n A m e r i c a . By

1856

nions,

which

Hamilton

had

begun

work

was p u b l i s h e d in a nearly

year after his

death.62

on

his

finished

Elements form in

H a m i l t o n was aware that his

o f Quater1866, the

e a r l i e r Lectures

39

A

History

of V e c t o r Analysis

had not provided a successful presentation of quaternions. In John

Herschel

encounaged

Hamilton

to

make

troductory work with examples and problems approach

to a f r i e n d ( A .

a

one,

for

be

and

cited,

hope

almost

or m e m o i r s "Book

finished state) is

as

sale,

(2,111;

1859

an

in-

121), a n d this

"I

like

w a n t to

was

was

an

and

elementary

Hamilton

Hamilton

could

a Book o f Reference —

other

writers,

is

that

intention

the

Elements

t h e otoix<eta of E u c l i d , in f u t u r e treatises

fulfilled,

one

finish

it cannot n o w be— but my

regards

n o t altogether sad;

composed only

Hart):

for

o n t h e Q u a t e r n i o n s . " (2,111;

o f Reference"

This

S.

unluckily

myself,

may

Elements

seems to have b e e n H a m i l t o n ' s intention. B u t in 1862 he

wrote short

the

for

139) T h e w i s h to m a k e it a

the

Elements

one-half times

as

(even long

in

as

its

un-

t h e Lectures.63

a n u m b e r of mathematicians could have work

have

(and m a n y later did), b u t probably

written

such

a

work

as

the

Elements.

p a i d m u c h attention to the foundations for the quater-

n i o n system. T h i s was valuable in that later workers in quaternions could the

in

their researches

Elements

for

the

mental property. the

"Science

plex

turn

detailed

Hamilton

to,

the

In

proof

of

any

funda-

of Pure T i m e , " or his ideas on the foundation of comi n t e r m s o f o r d e r e d sets o f n u m -

a letter (1861) to De M o r g a n , H a m i l t o n described m a n y of

important

As

t h e i r p u b l i c a t i o n s refer to,

rigorous

d i d n o t stress his c o n c e p t o f a l g e b r a a s

and higher complex numbers

bers.

or in

and

a book,

Lectures,

I

characteristics

am

with


&c.

the

Elements:

far b e t t e r satisfied w i t h t h e n e w v o l u m e t h a n w i t h t h e

which,

satisfied

of

however,

is

them,

point

in

not saying m u c h , for I am v e r y m u c h disof

composition

and

arrangement—the

I t m u s t , I o w n , r e q u i r e a s o m e w h a t r e s o l u t e p a t i e n c e t o r e a d t h e Lectures

through;

but

I

trust

that,

although

parts

may

conveniently

be

o m i t t e d a t f i r s t r e a d i n g , t h e n e w w o r k i s s o a r r a n g e d , a n d subdivided, a s to be q u i t e e a s i l y a c c e s s i b l e to a n y s t u d e n t w h o has a c o m p e t e n t (not a profound) knowledge of former mathematics, and wishes to understand the

subject.

No

new

notation, f o r i n s t a n c e , i s

i n t r o d u c e d — a n d after all,

I do n o t e m p l o y m a n y such — w i t h o u t a series of e x a m p l e s f o l l o w i n g , in n u m b e r e d s u b - a r t i c l e s , t o r e n d e r its m e a n i n g a n d its u s e f a m i l i a r . A n d generally I have

h o p e that t h e progress is

w e l l graduated t h r o u g h o u t ;

a v o i d e d w h a t m a y b e c a l l e d talk,

with

while I

some care; a n d have o n l y

one m e t a p h y s i c a l r e m a r k , i n f o u r a n d a h a l f l i n e s o f a n o t e ! (2,111; 5 6 8 ) The

work

is

certainly

clearer

than

the

Lectures,

although

be-

labored explanations

occur frequently in both books. A primary aim

of the

do

Elements

ordinary velop the

was

to

for

quaternions

what

others

had

done

for

n u m b e r s a n d to some extent for c o m p l e x n u m b e r s —to de-

them

in terms

of all the major aspects

of mathematics.

Thus

logarithms of quaternions were discussed, quaternions raised to

quaternion powers, the solution of equations in quaternions, and so

40

Sir W i l l i a m R o w a n H a m i l t o n a n d Q u a t e r n i o n s

on.

That

were

the

majority

geometrical

of the

applications

rather than

physical

of quaternion

methods

was a natural result of the

f a c t t h a t H a m i l t o n (as h e h i m s e l f a d m i t t e d [2,111; 1 5 0 ] ) h a d l o s t c o n tact

with

some

many

ways

developments

unfortunate,

in

for

physics.

interest

in

This

partiality

quaternions

was

and

in

vector

analysis was to be stronger a m o n g physicists than mathematicians. It

is

over

noteworthy

later

treated and

by

that

vector

quaternions

included

planned to

many

parts

o f t h e Elements

analysts.

Thus

for

with

zero

a large

section

(hence

differentiation. of the

work

be

taken

the first section

scalar c o m p o n e n t s

a treatment of vector

devote

could

example

vectors)

Hamilton had

to his o p e r a t o r <];

however by the t i m e of his death this section h a d not b e e n written. The

development

of

this

mainly

left

science,

was

note:

. . Professor

other

applications

cluding

some

operator, for

Tait,

Tait.

who

has

of Quaternions,

on

so

important

Hamilton already

physical

in

published

mathematical

Electro-Dynamics,

for

noted

appears

and to

a

foot-

tracts

on

physical,

in-

the

writer

emi-

nently fitted to carry on, happily and usefully, this n e w branch of m a t h e m a t i c a l science: a n d l i k e l y t o b e c o m e i n it, i f t h e e x p r e s s i o n may

be

Five

hundred

202), nions

allowed,

and

one

copies

in

the

appeared.

of the

chief successors

o f t h e Elements next

Tait had

year

were

Tait's

begun

this

to

inventor."64

in

1866

printed

Elementary

book

its

by

Treatise

1859,

on

(2,111; Quater-

but at Hamil-

ton's r e q u e s t h e h a d w i t h h e l d its p u b l i c a t i o n u n t i l after t h e p u b l i cation

VI.

of H a m i l t o n ' s

Summary

and

Elements.

(2,111;

133)

Conclusion

T h e status o f q u a t e r n i o n analysis a t t h e t i m e o f H a m i l t o n ' s d e a t h in

1865 m a y n o w b e s u m m a r i z e d . B y the e n d o f 1865 one h u n d r e d

and fifty papers had b e e n published on quaternions; one h u n d r e d and nine

(or 7 3 p e r c e n t )

were by

Hamilton.

T h e r e m a i n i n g forty-

one were by fifteen other authors, four of w h o m w e r e not British. T w o books had been published; the first, written by Hamilton, was l o n g a n d difficult t o read; t h e second, w r i t t e n b y A l l e g r e t , w a s less than one-tenth as long and written in French. T w o other books had been

written;

thorough, cians; value

the as

of

these

other

(Tait's

(Hamilton's

as

Treatise)

a reference was

short,

Elements) work

was

long,

for mathemati-

readable,

and

of great

a text for those w h o w i s h e d to b e c o m e acquainted w i t h

quaternion other.

one

a n d of great value

methods.

Quaternions

mathematics,

Together the two works complemented each were

not

in

1865

an

established

branch

of

b u t by 1865 a f o u n d a t i o n for this h a d certainly b e e n

laid.

41

A H i s t o r y of V e c t o r Analysis

Hamilton his

Elements

Charles

died on of

September 2,

Quaternions

Graves

wrote

in

until an

a

1865.

He continued to work on

few

days

eloge

that

before

Hamilton's

his

death.

"diligence

of

late was e v e n excessive — i n t e r f e r i n g w i t h his sleep, his meals, his exercise, his social enjoyments. It was, I believe, fatally injurious to his health." the

brilliant

life to on

65

T h e r e is certainly something tragic in the thought of Hamilton

devoting

q u a t e r n i o n s , w h i c h are

Hamilton's

place

in

the

the

last

twenty-two

years

of his

now o f l i t t l e i n t e r e s t . B u t j u d g m e n t s history

of

sophisticated basis than that quaternions

science

require

a

more

n o w seem to be of little

i m p o r t a n c e . T h e j u d g m e n t m u s t b e m a d e o n t w o bases. T h e f i r s t i s whether

or

not

Hamilton

acted

with

insight in

light of what was

k n o w n and could be k n o w n in mathematical and physical science at that t i m e . E n o u g h has b e e n said on this that the reader m a y j u d g e for h i m s e l f . T h e s e c o n d p o i n t i s w h e t h e r i t has h i s t o r i c a l l y t u r n e d out that Hamilton's ideas led in any w a y to fruitful developments. E n o u g h w i l l be said on this that the reader m a y j u d g e for himself.

42

Notes 1

A

Collection

Eugene 2

Robert

an

Addendum

Sir

William

Couples; Pure 4

in

Memory

Mathematica

Perceval

a in

Hamilton,

Transactions Rowan

Sir

William

No.

The

Rowan

II).

Life

Hamilton,

(New

of Sir

York,

William

ed.

David

1945).

Rowan

Hamilton,

3

vols,

1882-1891).

Preliminary

William

of

Studies,

Graves,

(Dublin, Rowan

with

Time"

Sir

Papers

(Scripta

Rev.

and 3

of

Smith

"Theory

of Conjugate

Functions,

and Elementary Essay on Algebra as of

the

Royal

Hamilton,

Irish

Lectures

Academy, on

17

(1837),

Quaternions

or

Algebraic

the

Science of

293-422.

(Dublin,

1853).

Three

types of p a g e r e f e r e n c e are used: (10) m e a n s t h e t e n t h p a g e of the preface, x m e a n s the t e n t h page of the table of contents, a n d 10 m e a n s the t e n t h page of the text. 5

Sir W i l l i a m

naries

in

489-495; 461; 31 34

Rowan

Algebra"

Hamilton,

in

"On

Philosophical

26 (1845), 2 2 0 - 2 2 4 ;

29

Quaternions, or on a N e w System of Imagi-

Magazine,

3rd.

Ser.,

(1846), 2 6 - 3 1 ,

(1847), 2 1 4 - 2 1 9 , 2 7 8 - 2 9 3 , 5 1 1 - 5 1 9 ; 32

(1849),

294-297,

340-343,

425-439;

35

25

(1844),

10-13,

241-246,

1 1 3 - 1 2 2 , 3 2 6 - 3 2 8 ; 30 (1847), 4 5 8 (1848), 3 6 7 - 3 7 4 ; 33

(1849),

133-137,

(1848), 5 8 - 6 0 ;

200-204;

36

(1850),

305-306. 6

Erwin

Schrodinger as q u o t e d in " T h e H a m i l t o n Postage Stamp:

ment by the Irish

An Announce-

M i n i s t e r o f Posts a n d T e l e g r a p h s . " (1; 82)

7

J . L . S y n g e , " T h e L i f e a n d E a r l y W o r k o f Sir W i l l i a m R o w a n H a m i l t o n . " (1;

8

E.

T.

Whittaker,

"The

Hamiltonian

Revival,"

Mathematical

Gazette,

24

17)

(1940),

153-158. 9

Ibid.,

10

154.

E.

T.

Whittaker,

"William

Rowan

Hamilton"

in

Lives

in

Science

(New

York,

1957), 61. 11

E.

12

See for e x a m p l e Whittaker's

13

George D. Birkhoff, "Letter from George D. Birkhoff" included in the papers in

the emy, 14

T.

15

o f Mathematics

Centenary

(1944-1945),

P.

A.

Otto

M. of

Dirac, the

F.

Bell,

alcohol;

York,

Celebration"

"Application Irish

Fischer,

with Men

(New

1937).

"Hamiltonian Revival."

in

Proceedings

of

the

Royal

Irish

Acad-

72-75.

Royal

(Stockholm,

Philosophy 16

Men

"Quaternion 50

Proceedings

cade

Bell,

of Quaternions

Academy, Universal

1951)

Technical

Physical 404.

Five

Lorentz Transformations"

and

it

that

Quaternions,

Structural

(Stockholm, has

Models

A in

writings of Hamilton's contemporaries.

CavalNatural

1957).

Hamilton

overindulged

there is to my k n o w l e d g e no u n a m b i g u o u s , direct e v i d e n c e for this

difficult to interpret; thus

in

261-270.

Hamilton's

Mathematical

Quaternions Tradition

to

(1944-1945),

Mechanics

and

o f Mathematics,

50

in

in the

T h e statements that s e e m relevant are v e r y

Hamilton's chief biographer, R.

P. Graves, both was and

wrote like a Victorian clergyman. It is clear that H a m i l t o n ' s marriage was not h a p p y or was his h o m e life w e l l o r d e r e d , b u t it is far f r o m clear that these factors seriously h i n d e r e d his 17

Whittaker,

18

[Rev.

creativity. "Hamiltonian

Robert

Perceval

Revival," Graves],

158.

"Sir William

Rowan

Hamilton"

in

Dublin

Uni-

43

A H i s t o r y of V e c t o r Analysis versity

Magazine,

(2,11;

344).

19

19

William

(1842),

Rowan

94.

For

proof that

Hamilton,

Graves

Transactions

of

the

J.

R.

wrote

Royal

the

Irish

above

paper,

Academy,

15

see

(1828),

69-174. 20

Rene

land, 2 1

Dugas,

A

History

o f Mechanics,

Sir W i l l i a m R o w a n H a m i l t o n ,

Systems 22

trans.

Maddox

(Neuchatel,

Switzer-

1955), 400-401.

of

Rays"

Augustus

(1866),

in

De

" T h i r d S u p p l e m e n t to an Essay on the Theory of

Transactions

Morgan,

of

"Sir

the

W.

Royal

R.

Irish

Academy,

Hamilton"

17

in

(1837),

Gentleman's

1-144.

Magazine,

220

133. T o v e r i f y that this u n s i g n e d o b i t u a r y n o t i c e w a s w r i t t e n b y D e M o r g a n ,

s e e (2,111; 2 1 6 - 2 1 7 ) . 23

C. C.

24

(2,1;

Critique 25

MacDuffee, "Algebras 478).

o f Pure

For

further

Reason

see

D e b t t o H a m i l t o n . " (1; 25)

information

(2,1;

545,

concerning

582,

585)

and

Hamilton's

(2,11;

87-88,

reading 96-97,

of Kant's 342).

It should be m e n t i o n e d that at nearly the same time Johann Bolyai developed a

similar representation of c o m p l e x n u m b e r s . B o l y a i w r o t e d o w n his ideas in 1837 b u t did not publish them.

H a m i l t o n p r e s e n t e d his ideas i n 1833 a n d p u b l i s h e d t h e m i n

1837. Paul Stackel refers to Bolyai's d e v e l o p m e n t as inferior to Hamilton's. See Paul Stackel,

Wolfgang

und

(Leipzig

and

Berlin,

130-133,

for

Stackel's

Bolyai, 26

pages

Johann

1913).

223-233,

Bolyai,

See

commentary; for

the

Geometrische

Part

I,

Leben

and

document

Part

Untersuchungen,

und Schriften II,

Stucke

der

aus

Beiden

den

2

pts.

Bolyai,

Schriften

der

pages Beiden

itself.

See, for e x a m p l e , e q u a t i o n s 65, 75, 112, 191, 195, a n d 196 in the s e c o n d section

o f t h e w o r k c i t e d i n n o t e (3) a b o v e , a n d e q u a t i o n s 6 , 5 6 , 5 7 , a n d 5 8 i n t h e t h i r d section. 27

to

(2,11; 5 7 4 ) .

the

major

discovery part

them by

It seems

doubtful that Hamilton's m e t h o d of "polytets" led directly

of quaternions.

in

his

acceptance

indicating the

However

it

of quaternions

is

probable

and

that this

moreover

method played a

s u p p o r t e d his

search

for

possibility and legitimacy of higher complex numbers. That

this is an important point is indicated from the history of ordinary complex numbers, where 28

the

(4;

difficult

[35]).

Hamilton's 29

30

De

Morgan,

Philosophical

This

is

"Linear

was

not

discovery

but

acceptance.

biographer.

Augustus

Cambridge

task

J o h n T. Graves was a mathematician a n d the brother of R. P. Graves,

" O n

the

7,

pt.

Society,

proved

in

Associative

Charles

Foundation

Saunders

Algebra"

in

of A l g e b r a "

in

Transactions

o f the

2. Peirce's

American Journal

addenda

to

o f Mathematics,

Benjamin 4

(1881),

Peirce's 97-229.

T h e above q u o t a t i o n f r o m C. S. Peirce is f r o m page 229. 31

T h u s (i + j)

will be it

is

•

( 2 i -I- 2 j ) = 4 , b u t s o a l s o d o e s ( i + j ) • ( 4 i ) . T h e t e r m " d o t p r o d u c t "

used t h r o u g h o u t w i t h o u t qualification; t h o u g h this is legitimate historically,

questionable

generally

not

mathematically.

The

term

"product"

in

current mathematics

is

used w h e n the entity resulting from the combination (by multiplica-

tion) of t w o factors

is

not a m e m b e r of the

s a m e class as t h e factors, that is, w h e n

closure is not preserved. T h e dot product is n o w classified as a bilinear functional. 32

(i

To s e e t h a t t h e a s s o c i a t i v e p r o p e r t y f a i l s , c o n s i d e r i X (i X j) = i X k = —j, w h e r e a s

x i) x j = 0. To s e e t h a t t h e c o m m u t a t i v e p r o p e r t y f a i l s , c o n s i d e r i X j = k ^ j X i =

-k. 33

T h u s , to s h o w t h a t d i v i s i o n is

34

It is historically interesting, b u t probably not historically significant, that in 1841

Hamilton Royal

Irish

for the

44

delivered Academy,

first

2

time)

a

paper

(1844), that " t h e

"On

n o t u n a m b i g u o u s , c o n s i d e r i X (i + j) = i X j = k.

the

166-168.

In

Composition this

paper

resultant force coincides

of F o r c e s " Hamilton

in

Proceedings o f the

p r o v e d (he claimed

i n direction w i t h t h e d i a g o n a l o f

Sir W i l l i a m R o w a n H a m i l t o n a n d Q u a t e r n i o n s

the rectangle

constructed with

lines

r e p r e s e n t i n g x a n d y a s s i d e s . " {Ibid.,

168.)

He

said that the previous (Laplace) proof of the parallogram of forces t h e o r e m h a d dealt only with the magnitude 35

of the resultant force.

The main primary documents

of his

discovery written

ceedings next

of

day

Magazine, his

the

Royal

Irish

(October

3rd

on

25

the

1843)

to

The

C. C. MacDuffee, "Algebra's in

36

in

Proceedings

(2,11; 4 3 9 - 4 4 0 ) .

quaternion may

v

polarized,

v

Morgan's

bridge

-I-

OF

(2)

printed

(only)

Hamilton's

9th

mainly

discussions

ed.,

Sequence

Royal

Irish

kz,

"xyz

vol.

of Ideas

Academy,

may

of s o m e

unpolarized.

38

paper

Philosophical See

(4;

Proceedings

50

.

.

.

such

The

as

preface are

to (1)

160-164;

Discovery

he

(3)

of Quater-

93-98.

wrote in regard to the

direction

of

the

(2) P e t e r G u t h r i e T a i t , (1890),

and

electricity,

Calculus

of

Philosophical

point

(1944-1945),

determine

agent

the

historical

the

Thus

letter

in

of this

XX in

His

x,

intensity; y,

z

Quaternions

while

are

may

t;

electrically turn

out

P O L A R I T I E S . " (2,11; 4 3 9 - 4 4 0 )

[38]).

of

was

8,

Royal

the

in

pt.

Charles

the

with

written

Society,

1844

and

published

in

Transactions

o f the

Cam-

3.

Graves'

Irish

Sir W i l l i a m R o w a n

nected 2

(3)

89-92.

S e e t h e a b s t r a c t o f D e M o r g a n ' s " M e m o i r o n T r i p l e A l g e b r a " i n (2,111; 2 5 1 - 2 5 3 ) .

De

3 9

Graves

historical

Britannica,

"The

quantity

to be a C A L C U L U S 3 7

T.

D e b t t o H a m i l t o n " (1; 2 5 - 3 5 ) ;

the

4- jy

the

electrically

John

T h e s e are vague g l i m m e r i n g s .

+ jx

determine

of

d i s c o v e r y . T h i s i s p r i n t e d ( o n l y ) i n Pro-

(1944-1945),

main

Encyclopaedia

E d m u n d Taylor Whittaker, nions"

of the

50

(1844), 490-495.

Quaternions.

"Quaternions"

d e a l i n g w i t h t h i s p o i n t a r e (1) t h e s h o r t s u m m a r y

day

Academy,

17,

Ser.,

Lectures

on

"On

Academy,

Algebraic Triplets"

3

(1847),

51-54,

was

57-64,

published 80-84,

in

the

105-108.

Hamilton, " O n a N e w Species of Imaginary Quantities Con-

Theory

of

Quaternions"

in

Proceedings

o f the

Royal

Irish

Academy,

(1844), 4 2 4 - 4 3 4 . 40

Sir

Irish 41

Sir

Irish 42

4 3

3

then

44

a

"On

Quaternions"

in

Proceedings

o f the

Royal

"On

Quaternions"

in

Proceedings

o f the

Royal

1-16.

Hamilton, 3.

-

a n d a ' = ( x ' i -I- y ' j -I- z ' k ) a r e t w o v e c t o r s

a' = xx' -1- yy' +

zz'

and

a X

a' = i(yz'

in t h e m o d -

— zy') -I- j(zx'

— xz') +

yx').

Sir W i l l i a m R o w a n Proceedings

in

Rowan (1847),

i f a = ( x i + y j + zk)

sense, -

Hamilton,

(1845-1847),

William

Thus

k(xy'

Rowan

3

Academy,

ern

in

William

Academy,

of

the

Hamilton,

Royal

Irish

" O n Q u a t e r n i o n s " ( c o m m u n i c a t e d July 20,

Academy,

3

(1847),

273-292,

and

(5,31;

Sir W i l l i a m R o w a n H a m i l t o n , "Researches respecting Quaternions. Transactions

of

the

Royal

Irish

for July

7,

45

In

46

In a later part of this

t h e Athenaeum,

Academy, 1847,

21

as

(1848),

quoted

in

1846)

278-293).

First Series"

199-296. (2,11;

587).

w o r k a detailed statistical study is g i v e n on the n u m b e r of

papers published, per five-year period, on quaternions. 47

This

is extracted from the summary of the seventh lecture as given in the con-

tents (4; xxxvi). 48

(2,11; 6 8 3 ) .

4 9

U n s i g n e d article, " R e v i e w of Sir W i l l i a m R o w a n H a m i l t o n ' s

ternions" ing the matics

in

North

American

authorship to and

Ginsburg

state

Review,

Hill

Mathematicians

Jekuthiel 1934)

De M o r g a n m a d e a very similar statement.

is

(New in

that T h o m a s

A Hill

85

(1857),

Ibid.,

223.

51

Ibid.,

228.

My

Lectures on Qua-

evidence

for

attribut-

t h a t t h i s i s d o n e i n R o b e r t E d o u a r d M o r i t z , O n MatheYork, History

1958), of

279.

Mathematics

(1818-1891)

was an early advocate of quaternions). 50

223-237.

(2,11; 6 8 3 ) .

was

in

David America

Eugene before

Smith

1900

a student of Benjamin

and

(Chicago,

Peirce

(who

H i l l b e c a m e p r e s i d e n t of H a r v a r d in 1862.

45

A H i s t o r y of V e c t o r Analysis 52

Ibid.,

226.

53

Ibid.,

229.

54

Sir

Physical

William Sciences,

Rowan

2nd

ed.

55

Ibid.,

726.

56

See

Benjamin

Hamilton, (London

Peirce,

"Quaternions"

and

A

Glasgow,

System

of

in

Nichol,

1860),

Analytical

Cyclopaedia

of

the

706-726.

Mechanics

(New

York,

1872),

in

The

Mathematical

Papers

L.

Synge

476-477. 57

De

58

Charles

of

Sir

Volson

William

England,

Wood,

Graves, Hamilton,

"Quaternions" "Eloge"

vol.

Alexandre

Allegret,

60

Hermann

Hankel,

B1

Giusto

Scienze,

(1862), 62

e

W.

(1880),

33.

Hamilton

Conway

ed

Arti

Fisica

Essai

sur

Theorie

le der

calcul

delle

(1858), della

des

and

quaternions

complexen

" D e l Calcolo dei

metodo

William

Though

my

rough

in

Rowan

the 64

Sir

terms

estimate

figure includes in

J.

(Paris,

Zahlensysteme

(Cambridge,

334-342,

and

Italiana

in

Atti

same

delle

del

7

+

72

pp.

1867),

196.

H a m i l t o n e delle sue Reale

author

Scienze

1862),

(Leipzig,

quaternioni di W. R.

equipollenze"

Societd

Hamilton,

the

of pages

the

453,740

words

Istituto

and

title,

(Modena),

2nd

Veneto

di

Memorie

di

Ser.,

1

Elements

o f Quaternions

(London,

1866),

59

+

long

preface

B5

two

in

books are comparable,

as the

compared Lectures,

to

t h e Elements c o n t a i n s

3 0 4 , 0 0 0 f o r t h e Lectures. T h i s

for w h i c h

there

is

vol.

II,

no

counterpart

Elements. William

Rowan

Hamilton,

Charles Jasper Joly (London,

46

7

pp.

63

by

A.

Analyst,

William

126-186.

Sir

762

Bellavitis, col

Lettere

Matematica

ed.

in

Sir

1931), xiii. T h e e l o g e w a s g i v e n b e f o r e t h e R o y a l I r i s h A c a d e m y i n 1865.

5H

relazioni

I,

on

C. G r a v e s , " E l o g e , " xv.

Elements

1901), 350.

o f Quaternions,

2nd

ed.,

ed.

CHAPTER

Other Systems,

I.

THREE

Early

Vectorial

Especially

Grassmann's

Theory

of Extension

Introduction Hamilton and those w h o w o r k e d w i t h ordinary complex numbers

were not the

only mathematicians

for vectorial systems. tries

were

developing systems

character.

These

Bellavitis,

Hermann

of the time

w h o were searching

I n d e e d , a t least six o t h e r m e n f r o m f o u r c o u n -

men

were

that were August

Giinther

more

o r less v e c t o r i a l i n

Ferdinand

Grassmann,

Mobius,

Adhemar

Barre

Giusto (better

k n o w n as Comte de Saint-Venant), Augustin Cauchy, and Reverend Matthew

O'Brien.

By

far t h e

most

important

of these m e n for the

present study is Grassmann, w h o s e first publication of his system, his

Ausdehnungslehre,

came

in

published his first paper on It should be

1844,

remarked that the

history of vector analysis

is

the

year

in

which

Hamilton

quaternions. significance of these m e n in the

different from that of H a m i l t o n ,

for it

w a s o n l y t h e i d e a s o f H a m i l t o n (as w i l l b e s h o w n ) t h a t h a d a m a j o r influence on the development of the m o d e r n system of vector analysis. T h e f a c t t h a t t h e s e m e n h a d l i t t l e o r n o d i r e c t i n f l u e n c e o n s u b sequent

developments

question,

"Why

vector

should

raises these

the

men

important

be

included

methodological in

the

history

of

analysis?"

To this question at least t w o replies m a y be given. First, the ideas o f these m e n are m a n d attention.

of such

great originality a n d m e r i t that they de-

It could in

fact b e

a r g u e d that Grassmann's crea-

tion surpasses that of H a m i l t o n in p r o f u n d i t y a n d perfection. W h i l e this

reason

reason

for

seems

guished from

their even

inclusion more

is

certainly

decisive.

important,

Chronology

may

a

second

be

distin-

history in that whereas the former presents events in

47

A H i s t o r y of V e c t o r Analysis

isolation,

the

latter aims

at

delineating the "trends of the times"

a n d the causes for events. C o i n c i d e n c e is certainly not the explanat i o n for t h e fact t h a t m o r e t h a n t e n m e n i n six c o u n t r i e s , w o r k i n g i n the period from the

1790's to the 1850's, sought to create vectorial

systems. T h o u g h f e w of these m e n k n e w of the ideas of any of the others, nonetheless some factors in the mathematics a n d physics of this

p e r i o d m u s t have m o t i v a t e d their search. T h u s the discussion

of these m e n w i l l be aimed at describing the form of the trend of w h i c h t h e y w e r e part a n d at s e e k i n g out the causes that l e d to their investigations

a n d results.

II.

Ferdinand

August Though he

decisely nand

Mobius

and

His

Barycentric

Calculus

neither constructed an original vectorial

influenced

Mobius

anyone

holds an

who

did,

system nor

nevertheless August Ferdi-

important position

in the history of vector

analysis for a n u m b e r of reasons. T h e c h i e f of these is that M o b i u s constructed

a

mathematical

system

(his

barycentric

calculus)

w h i c h is in m a n y ways similar to vectorial systems. In his system of space

analysis,

were

dealt

Mobius'

geometrical

with

entities

directly

and

positioned points, w i t h

(as

(points he

which

rather

hoped)

than

vectors)

advantageously.

numerical magnitudes

are

usually associated, are a d d e d in such a w a y that b o t h position a n d m a g n i t u d e are method

his

more his was

included in the addition.

system

has

Thus both in aim and in

k i n s h i p w i t h vectorial systems.10 Further-

system as later i n d e p e n d e n t l y discovered by Grassmann

put f o w a r d by Grassmann as an integral part of a system that

included

vectors.

Mobius

also

carried

on

an

important

corre-

s p o n d e n c e w i t h G r a s s m a n n ( w h i c h w i l l b e discussed later), a n d i n his later years M o b i u s d i d some w o r k directly in the vectorial tradition

(although

lifetime).

only

Thus

the

n u m b e r of points

a

small part of this was p u b l i s h e d d u r i n g his

work

of Mobius

admits

to

discussion from a

of view.

August F e r d i n a n d M o b i u s (1790-1868) was born in Schulpforta, e d u c a t e d at L e i p z i g , G o t t i n g e n , a n d Halle, a n d began his teaching career in

1815 at L e i p z i g University, b e c o m i n g ordinary professor

of mechanics and astronomy in death.

He

chanics, The

made

most

Behandlung

48

1844, a position he h e l d until his

contributions

to

mathematics,

me-

and astronomy.11 significant

barycentrische

Bildung

important

Calcul, der neuer

work ein

Mobius

neues

Geometrie Classen

by

dargestellt von

Aufgaben

for this

Hiilfsmittel und

study zur

insbesondere und

die

is

h i s Der

analytischen auf

die

Entwickelung

Other Early Vectorial

mehrerer

Eigenschaften

in

Herein

1827.

points.

Mobius

and by

1821

der

Kegelschnitte,

Mobius came

he

gave

upon

had

a

the

decided

published

full

exposition

first n o t i o n s to

publish

Systems

at

Leipzig

of his

calculus

of his system in

them

in

of

1818,

form.12

book

In

1823 he p u b l i s h e d his first discussion of his n e w m e t h o d as a short Appendix

to

Sternwarte

his

An idea of the following trische

nature

auf

der

Koniglichen

Mobius

In

pointed

the

out

foreword

that

the

center of gravity, or centroid (Schwerpunkt), since (2;

Archimedes'

iii)

Universitats-

of Mobius' system may be gained from the

exposition.14

brief

Calcul

Beobachtungen

Leipzig.13

zu

time

for

the

discovery

to

his

Der

barycen-

physical

concept

had been

useful ever

of

geometric

of

truths.

He w e n t on to state:

T h e p r e s e n t researches also p r o c e e d f r o m t h e same e l e m e n t a r y a n d purely geometrical concept of the center of gravity. W h a t first stimulated these researches

was

consideration of the fruitfulness of the law, that

e a c h s y s t e m o f w e i g h t e d p o i n t s has o n l y o n e c e n t e r o f g r a v i t y , a n d t h a t thus, in whatever sequence one brings the points into connection, the result is that one a n d the same point must always be found. T h e simple technique,

by means

of w h i c h I

was

able to prove more geometrical

laws, stimulated me to find a suitable algorithm for still greater simplif i c a t i o n o f s u c h i n v e s t i g a t i o n s . (2; iv) E a r l y in his first c h a p t e r M o b i u s stated that in his s y s t e m a l i n e segment

from

a

point

A

to

whereas a line from B then

a

point

B

would

be

designated

indicated h o w collinear segments were to be added.

Mobius

extended

by A B ,

t o A w o u l d b e d e s i g n a t e d b y BA, o r — A B . H e

this

sign

principle

to

and

(2;

3-5)

constructed addition

laws for figures d e t e r m i n e d by m o r e than t w o points, for e x a m p l e , to

triangles

(ABC)

and

pyramids

(ABCD).

(2;

20-23)

It

is

surprising

that M o b i u s n o w h e r e in this w o r k presented addition laws for noncollinear segments a n d also that he p r o c e e d e d f r o m such equations as

BDE

+

BEC

+

BCD

=

0

to

ABDE

+

ABEC

+

ABCD

=

0

(2;

24)

without v i e w i n g this as a multiplication by A, as Grassmann was to do

later. T h e central theorem

in his

book was the following:

G i v e n a n y n u m b e r (v) o f p o i n t s A , B , C , . . . N w i t h c o e f f i c i e n t s a , b , c, . . . n w h e r e t h e s u m of t h e coefficients does not e q u a l zero, t h e r e can always be f o u n d one (and only one) point S —the centroid — w h i c h point has t h e p r o p e r t y that i f o n e d r a w s p a r a l l e l l i n e s ( p o i n t i n g i n a n y d i r e c tion) through the given points and the point S, and if these lines inters e c t s o m e p l a n e i n t h e p o i n t s A ' , B ' , C \ . . . A T , S', t h e n o n e a l w a y s h a s : a.AA' + b.BB' + c.CC' +

.

.

. + n.NN' = (a + b + c +

.

.

. + n)SS\

a n d c o n s e q u e n t l y , if t h e p l a n e goes t h r o u g h S itself, t h e n a.AA'

+

b.BB'

+

c.CC'

+

...

+

n.NN'

=

0.

(2;

9-10)

49

A H i s t o r y of V e c t o r Analysis

He A,

later

stated

B, . . . ,

that

and

fo.B + c . C + .

.

in

place

hence

the

of

AA',

above

.+ n . N = ( a + fo + c + .

BB', . . . h e

expression .

would

becomes

write a. A

. + n)S. T h u s M o b i u s dealt

w i t h w e i g h t e d points or points w i t h numerical coefficients (positive or negative) w h i c h w e r e a d d e d as p o i n t masses are a d d e d in comp u t i n g t h e center of gravity of a b o d y . On such a basis M o b i u s proceeded

to

his

discovery

of h o m o g e n e o u s

n e w t r e a t m e n t of m a n y parts Mobius' ceived, Gauss

highly

with

and to a

and

well-presented

Jacobi,

Dirichlet,

work

was

Steiner,

well

Pliicker,

reand

t a k i n g s o m e i n t e r e s t i n it.16 H o w e v e r h i s m e t h o d s n e v e r

all

attained

original

Cauchy,

15

coordinates

of geometry.

widespread

use,

and

no

second edition

of the

w o r k ap-

peared (except for the republication of the w o r k in 1885 in the first volume

of his

collected works).

Mobius

did however continue to

p u b l i s h papers on this subject t h r o u g h o u t his life. In

1843

ing

Mobius

noncollinear

Himmels.17

In

ducing

this

a

Reinhardt but

23,

vectors

letter

concept

implied

Reinhardt

June

p u b l i s h e d t h e m e t h o d for a d d i n g a n d subtract-

also

1835,

in

Baltzer

in

his

that

that

letter to

cussion mann

of this

Die

Elemente

had

lectures

showed

Mobius

he

Mobius

stated

in

the

the

this

method

was

Mechanik

he

winter

discovered

that

der

that

was

des

intro-

of 1841-1842.

method

himself,

presented

in a

M o b i u s f r o m Bellavitis ( w h o w i l l be dis-

cussed in the next section). 1840's

his

to

came

( 1 , I V ; 7 1 7 - 7 1 8 ) I t w a s also i n t h e early into

contact

relationship will

with

best be

Grassmann;

the

dis-

p o s t p o n e d to the Grass-

sections.

The

final

w o r k by M o b i u s that demands attention is his " U e b e r

geometrische

Addition

und

Multiplication,"

written

in

1862

and

revised in 1865, b u t p u b l i s h e d only in 1887 in the fourth v o l u m e of his collected works. (1,IV; 659-697) This w o r k was mainly derived from

Grassmann's

work as

work,

published in

with

changes

in

notation.

Though

the

1887 c o u l d have had some influence on the

history of vector analysis, such influence is unlikely. Nevertheless his treatment is of interest, since it represents M o b i u s ' final judgm e n t as to w h a t form of vector analysis is most useful. After

the

presented tors AB

usual

his

discussion

"geometrische

of the

addition

Multiplication,"

a n d CD was represented by AB

of vectors, which

Mobius

for t w o vec-

• C D . T h i s product was to

be n u m e r i c a l l y equal to the area of the parallelogram, or twice the area of the triangle, d e t e r m i n e d by these t w o vectors. T h e product of the

multiplication

was

the

parallelogram

or triangle

itself and

was considered to be an entity that could assume any position in space p r o v i d e d that it always r e m a i n e d parallel to the plane of AB

50

Other Early Vectorial

and

CD.

The

usual

sign

conventions

were

used.

(1,IV;

Systems

663-665)

This definition is similar but not identical to the modern definition o f t h e v e c t o r (cross) p r o d u c t , s i n c e i n M o b i u s ' g e o m e t r i c a l m u l t i p l i cation the

product is

a plane

figure,

not another vector. T h e t w o

products are of course e q u a l n u m e r i c a l l y . It

may

be

noted

mutually parallel and

similarly

that

any vector uniquely

planes

any

(the planes

plane

figure

determines

a

set of

perpendicular to that vector)

determines

a vector

(or

a

set

of

mutually parallel vectors) perpendicular to that plane figure. T h u s the

two

products

are

similar but

not identical

(the

two

products

w i l l b e c o m p a r e d i n m o r e detail later). M o b i u s was w e l l a w a r e o f these t w o points of v i e w ; indeed, in treating the a d d i t i o n of plane figures he

s h o w e d that this addition c o u l d be represented by the

addition of vectors

perpendicular to those figures if the lengths of

t h e vectors are e q u a l t o t h e areas o f t h e p l a n e f i g u r e s . ( 1 , I V ; 6 7 1 673) M o b i u s t h e n p r o c e e d e d to the geometrical p r o d u c t of a vector and a parallelogram, and concluded that the

figure

produced was

the parallelepiped d e t e r m i n e d by the vector a n d t w o adjacent sides of the parallelogram. (1,IV; 673 ff) This was f o l l o w e d by his definition of the projective product of t w o vectors, w h i c h was s y m b o l i z e d by AB

•

CD a n d is e q u i v a l e n t to m o d e r n scalar (dot) p r o d u c t . (1,IV;

678 ff) T h e next section treated the relation b e t w e e n the geometric and projective products of vectors in a plane. H e r e i n M o b i u s began by considering four coplanar vectors, u, v, a, a', w h o s e positions w e r e only

limited by

perpendicular.

the He

restriction

then

gave

that u

the

and v

equation

w e r e to be mutually

au

•

a'v 4 - a v

•

a'u = 0 ,

w h i c h equation he attempted to prove. It is clear that the left-hand m e m b e r of this

equation

represents

a number,

but it

seems

that

M o b i u s n e v e r d e f i n e d t h e p r o d u c t o f t w o areas; h e n c e t h e i n t e r p r e tation

of the

right-hand

m e m b e r of the

equation

is

by

no means

clear. (1,IV; 6 8 2 - 6 8 3 ) In fact, M o b i u s at a n u m b e r of points in this m a n u s c r i p t f a i l e d e i t h e r to d e v e l o p a suitable s y m b o l i s m or to see that

a

given

expression

veloping the projective

ambiguous.18

was

He

concluded

by

de-

multiplication of solid figures. (1,IV; 6 8 5 -

697) T h o u g h Grassmann was never mentioned, it is clear that M o b i u s ' treatment was

m a i n l y d e r i v e d f r o m G r a s s m a n n ' s efforts.

Hamilton

i s l i k e w i s e n o t m e n t i o n e d , a l t h o u g h M o b i u s h a d k n o w n o f his system

since

casionally analysis.

at

least

dealt

1859.19

with

It

is

matters

noteworthy that M o b i u s relevant

to

his

system

o n l y ocof

point

In general his treatment is almost completely lacking in

comments that allow one to j u d g e h o w M o b i u s h i m s e l f v i e w e d his

51

A

History

o w n

of V e c t o r Analysis

work;

system

eventually the two sions

came

were

Giusto the time

example,

to

Bellavitis,

in

is

viewed clear

it

appreciate when

of the modern

and

an

the

some

which

as

the

that

a

Mobius

significance

of

limited to three-dimen-

dot a n d cross

His

presenting

however

understand and

Calculus

of

products.

Equipollences

Italian mathematician born in

of his death in

but in

he

It

products,

Bellavitis

senator

interests

for

Hamilton's.

precursors

Giusto

a

to

Grassmann

III.

and

whether,

superior

1803, was at

1880 a professor at the University of Padua

Kingdom cases

of Italy.20

He

was

a

man

of broad

of narrow view, whose fame in general

and relevance for the present study derives from the creation of his calculus of equipollences. W h a t this is m a y be understood from the f o l l o w i n g section of a paper (1835) w h i c h contained his earliest full explanation 4.

l.°

of this

method.

A straight l i n e (retta) expressed as usual by t w o letters is u n d e r stood as taken f r o m the first letter to the second, so that AB a n d BA s h o u l d not be regarded as the same entity, b u t as t w o equal q u a n t i t i e s h a v i n g o p p o s i t e signs.

2.° T w o

straight lines

are

c a l l e d equipollent i f t h e y a r e e q u a l , p a r a l l e l

a n d d i r e c t e d i n the same sense. 3.° I f t w o

or more

straight lines

are

related in such a w a y that the

second extremity of each line coincides w i t h the first extremity of the

following, then the

line w h i c h together w i t h these forms a

polygon (regular or irregular), and w h i c h is drawn from the first extremity

of

the

last

line,

is

called

their

equipollent-sum

(com-

posta-equipollente). T h i s is r e p r e s e n t e d by the signs 4- interposed b e t w e e n the lines to be combined, and w i t h the sign

indicating

the equipollence. Thus we have AB AB

+

BC

+ BC +

±

CD

AC, ±

AD,

etc.

Such e q u i p o l l e n c e s c o n t i n u e to h o l d w h e n one substitutes for the lines

in

t h e m , o t h e r lines w h i c h are r e s p e c t i v e l y e q u i p o l l e n t t o

t h e m , h o w e v e r t h e y m a y b e s i t u a t e d i n space. F r o m this i t can b e understood

how

any

summed,

that

in

and

number

whatever

and order

any

kind

these

of

lines

lines are

may

taken,

be the

s a m e e q u i p o l l e n t - s u m w i l l be o b t a i n e d . . . . 5.° I n e q u i p o l l e n c e s , j u s t a s i n e q u a t i o n s , a l i n e m a y b e t r a n s f e r r e d f r o m o n e side to t h e other, p r o v i d e d that t h e sign is changed. . . . 6.° T h e e q u i p o l l e n c e AB

n. C D , w h e r e n stands for a positive n u m -

b e r , i n d i c a t e s t h a t AB is b o t h p a r a l l e l to a n d has t h e s a m e d i r e c t i o n a s CD, a n d t h a t t h e i r l e n g t h s h a v e t h e r e l a t i o n e x p r e s s e d b y the

equation

AB

=

n.CD.

5. L e t us restrict ourselves n o w to lines situated in the same plane. T h e inclination

52

of the

l i n e AB

i s t h e a n g l e HAB, w h i c h t h i s

line forms with

Other Early Vectorial

Systems

the horizontal AH d r a w n from left to right, w i t h the qualification that positive angles are m e a s u r e d f r o m t h e r i g h t u p w a r d s , a n d f r o m 0° to 360° 2.° T h e a n g l e o r i n c l i n a t i o n o f C D o n A B i s e q u a l t o t h e i n c l i n a t i o n o f CD

minus

3.° T h e

that

o f AB.

equipollence C D ^ F A b

GH

r e q u i r e s n o t o n l y t h a t t h e l e n g t h s A B , CD, e t c . , s h o u l d b e s u c h a s to satisfy the e q u a t i o n into w h i c h t h e e q u i p o l l e n c e is c h a n g e d by c o n v e r t i n g the e q u i p o l l e n c e sign i n t o a n e q u a l sign, b u t also that i n c . AB = i n c .

CD + i n c .

EF — i n c .

GH . . . .

T h e l i n e e q u i p o l l e n t to 1 is c o n s i d e r e d as h o r i z o n t a l , t h a t is, as h a v i n g no i n c l i n a t i o n . . . . 6.

Fundamental

Theorem.

In

equipollences,

terms

are

transposed,

sub-

stituted, a d d e d , subtracted, m u l t i p l i e d , d i v i d e d , etc., i n short, all t h e algebraic o p e r a t i o n s are p e r f o r m e d w h i c h w o u l d b e l e g i t i m a t e i f o n e were

dealing

with

equations,

and

the

resulting

equipollences

are

a l w a y s exact. A s w a s s a i d i n 5.°, n o n - l i n e a r e q u i p o l l e n c e s c a n o n l y b e referred to figures in a single plane.21

T h e first point to be m a d e concerning this passage is that Bellavitis

herein

described

equivalent

in

represented.

geometrical

behavior Numerous

to

It

that

are

numbers

mathematicians,

equivalence.22

recognized this

entities

complex

should

as

in

all

ways

geometrically

including

however be

Bellavitis,

stressed that

Bellavitis d i d not v i e w his system as based on the c o m p l e x n u m b e r system; inary

indeed,

numbers

essentially

throughout his entire life he was o p p o s e d to imag-

as

algebraic entities.23 T h u s he v i e w e d his

geometrical

entities,

rather

than

as

lines as

geometrical

repre-

sentations. O n e major significance of Bellavitis is that in his papers a n d book on

equipollences

applications

he

gave n u m e r o u s , a n d i n m a n y cases i n g e n i o u s ,

of his m e t h o d to mathematical and physical problems,

so that in a l i m i t e d sense he was a rival of Grassmann a n d H a m i l ton.

Bellavitis' chief follower was

Charles

Ange

Laisant, w h o

dur-

i n g the last q u a r t e r o f t h e n i n e t e e n t h c e n t u r y d e v o t e d m u c h e n e r g y to

making the

system

known

in

France.

It

that Bellavitis, as the founder of a vectorial ferent viewed

at

least

as

tendency

to

in

another

spirit

from

other

representative

construct

vectorial

systems of the

systems.

is

of the

early It

also

of significance

system w h i c h was dif-

is

times,

may

be

nineteenth-century interesting

in

this

regard to note that Bellavitis m a d e a long, unsuccessful attempt to extend his system to three-dimensional

space.

(3;

158-159)

53

A H i s t o r y of V e c t o r Analysis

Bellavitis came to his system in connection w i t h attempts to give a g e o m e t r i c a l justification for c o m p l e x n u m b e r s . T h u s he stated: " I t was

while

considering

a

geometrical

representation of imaginary

n u m b e r s p r o p o s e d b y B u e e that t h e r e c a m e t o m e (in 1832) the f i r s t idea of the

m e t h o d of equipollences, b u t t h e n I thought that geo-

m e t r i c a l truths c o u l d not rest on the t h e o r y of i m a g i n a r y n u m b e r s , w h i c h entities I h a d for some years o p p o s e d as u n w o r t h y of belonging to

a science

based on

alone."24

reason

Bellavitis

h a d in fact

p u b l i s h e d articles attacking the views of various mathematicians on i m a g i n a r i e s . (3; 159, 163) H i s f i r s t p u b l i c a t i o n (1832) r e l a t i n g t o his m e t h o d , b u t not containing it in a fully d e v e l o p e d form, contained a m e t h o d of deriving properties of points in a plane from the properties of points in a straight line (for e x a m p l e , he d e d u c e d the Pythagorean t h e o r e m by means of this method).25 In this paper he d i d not use

the

orally A

term

"equipollence," but he seems to have used it in an

presented

communication

second publication

which

appeared

of 1832 in

1833,27

at the

Ateneo Veneto.26

a n d a t h i r d in

1835, in

B e l l a v i t i s u s e d t h e t e r m " e q u i p o l l e n c e " a n d gave a f u l l ex-

position of his system.28 T h e l o n g quotation g i v e n at the b e g i n n i n g of this section was taken from that paper.

IV.

Hermann

Grassmann

and

His

Calculus

of

Extension:

Introduction H e r m a n n G r a s s m a n n was a b r i l l i a n t m a t h e m a t i c i a n , w h o s e creations in vectorial analysis can only be c o m p a r e d to those of Hamilton. It w i l l be s h o w n that his system c o u l d have led to m o d e r n vector analysis, b u t it d i d not.

T h e reasons for a n d the nature of this

failure w i l l be traced. Grassmann presented his system in a n u m b e r of different forms; i n fact h e w r o t e four w o r k s i n w h i c h his system was presented, a n d these

four

differ

substantially

among themselves,

two of Grassmann's most important followers

and

moreover,

(Schlegel and Hyde)

chose a fifth f o r m for their presentation of his system. T h e part of his system that is e n c o u n t e r e d in some mathematics books of today is a small a n d by no m e a n s characteristic part. Because of these c i r c u m stances fully.

a

n u m b e r of Grassmann's

works w i l l be discussed rather

M a n y details concerning Grassmann's life and the reception

accorded his writings have b e e n i n c l u d e d in the hope that they may m a k e clear b o t h h o w a n d w h y his system failed to m a k e as significant an impression on the times as it m i g h t have. What

G r a s s m a n n created was above all a mathematical

not just a n e w mathematical idea or theorem.

54

system,

H i s creative act can-

Other Early Vectorial

Systems

not be compared w i t h such mathematical discoveries as the Pythagorean

theorem

or

Newton's

version

of the

calculus.

Rather it is

best thought of as comparable to such creations as n o n - E u c l i d e a n geometry or Boolean algebra.

Thus

it was natural that Grassmann

chose to introduce his system, not by means of a paper, b u t rather by means of a long and complicated book. Grassmann's system is so b r o a d a n d so general that there are serious difficulties in t r y i n g to s u m m a r i z e it; n e v e r t h e l e s s a s u m m a r y is necessary for t h e p r e s e n t purposes. T h e m a i n stress h o w e v e r w i l l b e p l a c e d o n t h e d i s c u s s i o n of such

ideas

as

Grassmann's

form

of the

scalar (dot)

and vector

(cross) p r o d u c t s , w h i c h h a v e c o u n t e r p a r t s i n m o d e r n v e c t o r a n a l y sis. Hermann

GiAnther

Grassmann

(1809-1877)

was

born and lived

the m a j o r i t y of his life in Stettin (or Szczecin), a t o w n in P o m e r a n i a on the O d e r River a short distance inland from the Baltic.

He was

the third of the t w e l v e c h i l d r e n b o r n to Justus Giinther Grassmann (1779-1852), who, though trained mainly in theology, taught mathematics

and

physical

science

at

the

Gymnasium.29

Stettin

Grass-

m a n n , u n l i k e H a m i l t o n , w a s n o p r o d i g y ; i n fact, q u i t e t h e opposite. Grassmann's

father,

it

is

said,

often

remarked

that he

h a p p y if H e r m a n n b e c a m e a g a r d e n e r or a craftsman. In ing

w o u l d be

(5; 9)

1827 Grassmann e n t e r e d the U n i v e r s i t y of Berlin, w h e r e durhis

six

semesters

he

mainly

studied philology

and theology,

being especially influenced by the Church historian Neander and Schliermacher;

he

attended

no

mathematical

lectures, though he

d i d read some m a t h e m a t i c a l textbooks w r i t t e n b y his father.30 After leaving Berlin, Grassmann returned to Stettin and pursued studies in

mathematics,

physics, natural history, theology, and philology,

preparing himself for the

state e x a m i n a t i o n s r e q u i r e d for teachers

of these subjects. In 1834 he accepted a position at a B e r l i n technical school, a position just vacated to take Grassmann

stayed

w h i c h the great geometer Jacob Steiner h a d

a position at the U n i v e r s i t y of Berlin.

in

(5; 45)

B e r l i n for slightly m o r e than a year, d u r i n g

w h i c h time he seems to have had little contact w i t h Steiner. At the b e g i n n i n g of 1836 he returned to Stettin a n d spent the r e m a i n d e r of his life t e a c h i n g t h e r e i n v a r i o u s schools, i n all cases b e l o w t h e u n i versity

level

despite

his

constant

attempts

to

attain

a university

post. In

order to

improve

his

standing

as

a

teacher

of science

and

mathematics, Grassmann in 1839 w r o t e the B e r l i n scientific examination committee that he w o u l d like to write a w o r k to prove his competence. entitled

Thus

Theorie

der

Grassmann began work on Ebbe

und

Flut31

He

a study of the tides

completed

his

study

in

55

A

History

1840,

of V e c t o r

and

ceived

it

the on

Analysis

chief reader of the

April

26,

1840.

essay,

Carl

L u d w i g

Conrad,

re-

T h e essay was of considerable length

(over t w o h u n d r e d pages as printed in Grassmann's works) and was not carefully read by C o n r a d , for he h a d r e t u r n e d it by M a y 1, 1840, the

day

of Grassmann's examination.

T h u s it is not surprising that

C o n r a d failed to see the i m p o r t a n c e of the work.32 Yet Grassmann's Theorie

der

Ebbe

presentation

Information from

a

letter

cerning

a

und

Flut

is

important,

of a system of spatial concerning written

paper

by

the

origin

Grassmann

published

by

for

analysis

the

it

of this in

contained

work

1847 to

latter

t h e first

based on vectors.

in

may

be

gained

Saint-Venant con-

1845,

in

which

Saint-

V e n a n t c o m m u n i c a t e d results identical to results discovered earlier b y G r a s s m a n n (this p a p e r w i l l b e t r e a t e d later). mann

In this letter Grass-

wrote:

As I was reading the extract f r o m your paper on the geometric sum and difference, the

which

marvelous

was

p u b l i s h e d in the

similarity

between

your

Comptes rendus, results

and

I

was struck by

those

discoveries

w h i c h I m a d e e v e n as e a r l y as 1 8 3 2 . . . . I c o n c e i v e d t h e first i d e a o f t h e g e o m e t r i c s u m a n d d i f f e r e n c e o f t w o o r m o r e lines a n d also o f t h e g e o m e t r i c p r o d u c t o f t w o o r t h r e e lines i n that y e a r (1832). T h i s i d e a i s i n a l l w a y s i d e n t i c a l t o t h a t p r e s e n t e d i n t h e extract f r o m y o u r paper. B u t since I was for a l o n g t i m e o c c u p i e d w i t h entirely different pursuits, I c o u l d not d e v e l o p this idea. It was o n l y in 1839 that I was lead back to that idea a n d p u r s u e d this geometrical analysis up to the p o i n t w h e r e it ought to be applicable to all mechanics. It was possible for me to a p p l y this

m e t h o d of analysis to the theory of tides,

a n d in this I was astounded by the simplicity of the calculations resulting f r o m t h i s m e t h o d . (5; 4 2 - 4 3 ) Thus

in

ideas theory. 1832

1832

which

Grassmann were

attained

developed

Grassmann explained in

and

dehnungslehre

indicated of

the

1844.33

source

in

the the

first work

ideas

for

his

(1839-1840)

system, on

tidal

greater detail w h a t he had f o u n d in thereof in the

Grassmann

foreword to

h i s Aus-

wrote:

T h e first i m p u l s e c a m e f r o m t h e c o n s i d e r a t i o n o f negatives i n g e o m e try;

I

was a c c u s t o m e d to v i e w i n g t h e distances AB a n d BA as o p p o s i t e

m a g n i t u d e s . A r i s i n g f r o m this i d e a was t h e c o n c l u s i o n that if A, B, C are p o i n t s o f a s t r a i g h t l i n e , t h e n i n a l l cases A B + B C = A C , t h i s b e i n g t r u e whether AB

and BC

are d i r e c t e d i n t h e same d i r e c t i o n o r i n o p p o s i t e

d i r e c t i o n s ( w h e r e C lies b e t w e e n A a n d B). I n t h e latter case A B a n d B C were not v i e w e d as merely lengths, but simultaneously their directions were considered since they were oppositely directed. Thus d a w n e d the distinction b e t w e e n the s u m of lengths a n d the s u m of distances w h i c h w e r e fixed in direction. F r o m this resulted the r e q u i r e m e n t for establishi n g this latter c o n c e p t o f s u m , n o t s i m p l y for t h e case w h e r e t h e distances w e r e d i r e c t e d i n t h e same o r o p p o s i t e d i r e c t i o n s , b u t also for any other case. T h i s c o u l d b e d o n e i n t h e m o s t s i m p l e m a n n e r , since t h e l a w that

56

Other Early Vectorial

Systems

AB + BC = AC r e m a i n s v a l i d w h e n A, B, C do n o t l i e in a s t r a i g h t l i n e . T h i s t h e n was t h e first step t o a n analysis w h i c h s u b s e q u e n t l y l e d t o the n e w branch of mathematics, w h i c h is presented here. I d i d not h o w ever t h e n realize h o w fruitful and h o w rich was the field that I h a d o p e n e d up; rather that result s e e m e d scarcely w o r t h y of note u n t i l it was c o m b i n e d w i t h a related idea. W h i l e I was p u r s u i n g the concept of geometrical product, as this idea h a d b e e n e s t a b l i s h e d b y m y f a t h e r ( i n h i s Raumlehre, p t . I , 1 7 4 a n d h i s Trigonometrie, p . 1 0 ) , I c o n c l u d e d t h a t n o t o n l y r e c t a n g l e s , b u t a l s o parallelograms, m a y be v i e w e d as products of t w o adjacent sides, prov i d e d that the sides are v i e w e d not m e r e l y as lengths, b u t rather as directed magnitudes. W h e n I j o i n e d this concept of geometrical product w i t h the previously established idea of geometrical s u m the most striki n g harmony resulted. T h u s w h e n I m u l t i p l i e d the s u m of t w o vectors by a third coplanar vector, the result coincided (and must always coincide) w i t h the result obtained by m u l t i p l y i n g separately each of the t w o original vectors by the t h i r d vector a n d a d d i n g together ( w i t h d u e attention to positive and negative values) the t w o products. [ T h u s A ( B + C) = AB + AC.] F r o m t h i s h a r m o n y I c a m e t o see t h a t a w h o l e n e w area o f a n a l y s i s w a s o p e n i n g u p w h i c h c o u l d lead t o i m p o r t a n t results. T h i s idea r e m a i n e d dormant for some t i m e since the demands of my occupation l e d me to o t h e r tasks; also I was i n i t i a l l y p e r p l e x e d by t h e strange r e s u l t t h a t though the other laws of ordinary multiplication (including the relation of multiplication to addition) were preserved in this n e w type of multiplication, yet o n e c o u l d o n l y exchange factors i f o n e s i m u l t a n e o u s l y c h a n g e d t h e s i g n (i.e. c h a n g e d + to — a n d — to +)• A w o r k on tidal theory, w h i c h I u n d e r t o o k at a later t i m e , l e d me to L a g r a n g e ' s Mecanique analytique a n d t h e r e b y I r e t u r n e d t o t h o s e i d e a s o f analysis. A l l t h e d e v e l o p m e n t s i n that w o r k w e r e t r a n s f o r m e d t h r o u g h the principles of the n e w analysis in such a s i m p l e w a y that the calculations often came out m o r e than ten times shorter than in Lagrange's work. T h i s encouraged me to a p p l y the n e w analysis to the difficult theory of the tides; there were numerous n e w concepts to d e v e l o p and to clothe in the n e w analysis; the concept of rotation l e d to geometrical exponential m a g n i t u d e s , to t h e analysis of angles a n d of t r i g o n o m e t r i c f u n c t i o n s , etc. I was d e l i g h t e d h o w t h o r o u g h the analysis thus f o r m e d a n d e x t e n d e d , not o n l y the o f t e n v e r y c o m p l e x a n d u n s y m m e t r i c f o r m u l a e w h i c h are fundamental in tidal theory, come out as the most simple and symmetric f o r m u l a e , b u t also t h e t e c h n i q u e o f d e v e l o p m e n t parallels t h e concept.

T h u s I feel e n t i t l e d to hope that I have f o u n d in this n e w analysis the only natural m e t h o d according to w h i c h mathematics should be applied t o nature, a n d a c c o r d i n g t o w h i c h g e o m e t r y m a y also b e t r e a t e d , w h e n ever it leads to general a n d to f r u i t f u l results. T h u s I d e c i d e d to m a k e the presentation, e x t e n s i o n a n d a p p l i c a t i o n of this analysis a task of my life. W h e n I c a m e to d e v o t e a l l my free t i m e to this task, m a n y of t h e gaps, left in the m o r e casual earlier d e v e l o p m e n t , w e r e f i l l e d up. T h u s the f o l l o w i n g ideas ( w h i c h are p r e s e n t e d i n this b o o k ) r e s u l t e d : t h e s u m o f several points is their centroid; the product of t w o points is the vector connect-

57

A

History

of V e c t o r Analysis

i n g t h e m ; t h e p r o d u c t o f t h r e e points i s t h e surface area d e t e r m i n e d b y t h e m ; the p r o d u c t of four points is the spatial m a g n i t u d e (a p y r a m i d ) determined by them. The

c o n c e p t i o n o f c e n t r o i d a s s u m l e d m e t o e x a m i n e M o b i u s ' Bary-

centrische

Calcul, a w o r k

of w h i c h

until then

I k n e w only the title; and I

was not a little pleased to find here the same concept of the summation of points to w h i c h I had been led in the course of the development. This was the which

first, a n d as I s u b s e q u e n t l y l e a r n e d , t h e o n l y p o i n t of contact

my new

system of analysis

had w i t h the one that was

already

k n o w n . Since h o w e v e r the concept of a product of points does not occur in that w o r k a n d since w i t h this concept, w h e n it is c o m b i n e d w i t h that of the s u m of points, the d e v e l o p m e n t of the n e w analysis begins, I f o u n d that I

c o u l d not expect a n y a d v a n c e m e n t of my ideas f r o m that source.

(4,1,1; 7 - 1 0 ) From

this

quotation

w e l e a r n first o f all that G r a s s m a n n ' s discus-

sion of the addition of lines is only partly within the "parallelogram of forces" tradition. sum

of two

with the

Thus

directed

what Grassmann

lines

and

was

discussing was the

o n l y that, whereas those w h o dealt

parallelogram of forces w e r e not conceptually adding lines

b u t w e r e taking the geometrically d e t e r m i n e d diagonal as the representation of the resultant of t w o forces. with ing

sums lines

of complex numbers but

representing

Grassmann's stated

that

complex

view he

was

numbers

ferred h i m to It

is

gested the

was

obvious of

unaware

In

Similarly those w h o dealt

not directly conceptually add-

as

a

line.

any

of the

case

Thus

to

some

Grassmann

geometrical

extent

explicitly

representation

G a u s s i n a l e t t e r o f D e c e m b e r 14,

of

1844, re-

p a p e r o f 1831.34

that

Mobius'

Grassmann's

c o n c e p t that the line AB

negative of the

sum

new.

until

Gauss'

some

a

were

Barycentrische

ideas;

thus

Calcul for

could

example

have

sug-

Mobius

had

(the line f r o m A to B) w a s e q u a l to the

line BA (from B to A).

M o b i u s h a d moreover come to

v i e w (before G r a s s m a n n ) not o n l y lines b u t also triangles a n d tetrahedra as positive or negative. mann

It is thus very interesting that Grass-

stated that he had k n o w n of Mobius' w o r k only by name until

he h a d laid the foundation for his to

note

that

Grassmann

concept for his Mobius

wrote

points),

he

things

failed to

In the above

II,

p. Justus

footnote

58

the 194"

not

system.

It is above all important

anticipated by Mobius in the key

system, that of geometrical multiplication. Although such

a s A B = —BA

conceive

quotation

multiplication that the referred

was

of this

(where A

as

Giinther in the

B

represent

Grassmann stated concerning geometrical

idea came from his

father and in a footnote

reader to t w o of his father's books, and

and

a multiplication of points.

"Trigonometric,

p.

Grassmann

first of these

10."

made

books:

"Raumlehre Theil

35

the

following

statements

in

a

O t h e r E a r l y Vectorial Systems

The tion

rectangle

of it,

as

it

itself

is

appears

the in

true

§53,

geometrical is

really

product,

and

geometrical

the

construc-

multiplication.

If t h e

c o n c e p t o f m u l t i p l i c a t i o n i s t a k e n i n its p u r e s t a n d m o s t g e n e r a l sense, then one comes to v i e w a construction as something constructed f r o m elements already c o n s t r u c t e d a n d i n fact c o n s t r u c t e d i n the same w a y . Thus multiplication is only a construction of a higher power. In geometry the point is the original " p r o d u c i n g " element; from it through construction the l i n e emerges; if we take as the basis of a n e w construction the finite line constructed from the point, a n d if we treat it in the same manner as

we formerly treated the point, then the rectangle emerges.

Just as the l i n e c a m e f r o m the p o i n t , so t h e rectangle conies f r o m the line. T h e s i t u a t i o n i s t h e same i n a r i t h m e t i c . I n this case t h e u n i t i s t h e original

"producing''

element. T h e unit must simply be v i e w e d as given.

F r o m this, t h r o u g h c o u n t i n g (arithmetic construction), n u m b e r appears. If this n u m b e r is t a k e n as t h e basis of a n e w c o u n t i n g (setting it in p l a c e of the

unit),

then

arises

the arithmetic

connection to

multiplication,

w h i c h is n o t h i n g else b u t a n u m b e r of a higher order, a n u m b e r of w h i c h the u n i t is also a n u m b e r . T h u s it c o u l d perhaps be said that t h e rectangle is a (finite) line in w h i c h , in place of the " p r o d u c i n g " points, a f i n i t e l i n e has gested: this

A

product

b e e n substituted. T h u s the f o l l o w i n g laws m a y be sug-

rectangle behaves

is in

the the

geometrical same

way

product as

of its

the

base

arithmetic

and

height

product.

and

(4,1,11;

507) In in

the

second

book

(the

Trigonometrie)

the

elder

Grassmann

wrote

a footnote: I f t h e c o n c e p t o f p r o d u c t i s t a k e n i n its m o s t p u r e a n d g e n e r a l sense, then it is v i e w e d as the result of a synthesis in w h i c h an e l e m e n t (prod u c e d f r o m a n e a r l i e r s y n t h e s i s ) i s set i n t h e p l a c e o f t h e o r i g i n a l e l e m e n t a n d treated i n the same w a y . T h e p r o d u c t m u s t arise l i k e w i s e f r o m w h a t r e s u l t e d f r o m the first synthesis, just as this arose f r o m the o r i g i n a l elements. In arithmetic the unit is the element, counting is the synthesis, a n d t h e r e s u l t i s n u m b e r . I f t h i s n u m b e r , a s t h e r e s u l t o f t h e f i r s t s y n thesis, i s set i n t h e p l a c e o f t h e u n i t , a n d t r e a t e d i n t h e s a m e w a y (i.e., counted), then the arithmetic product appears, a n d this m a y be v i e w e d as a n u m b e r of a h i g h e r o r d e r t h a n a n u m b e r of w h i c h u n i t y is a l r e a d y a number. In geometry the point is the element, the synthesis is the motion of the point in some direction, and the result, the path of the point, is t h e line. I f this l i n e , p r o d u c e d b y t h e first synthesis, i s set i n t h e p l a c e o f t h e p o i n t a n d t r e a t e d i n t h e s a m e w a y (i.e., m o v e d i n s o m e o t h e r d i r e c tion), t h e n a surface is p r o d u c e d f r o m the path of the line. T h i s is a true geometrical p r o d u c t of t w o linear factors a n d appears in the first place as a rectangle, insofar as t h e first d i r e c t i o n shares n o t h i n g w i t h t h e second. If t h e surface is set in p l a c e of t h e p o i n t , t h e n a g e o m e t r i c a l s o l i d is p r o d u c e d as t h e p r o d u c t of three factors. T h i s is as far as o n e can go in g e o m e t r y s i n c e space has o n l y t h r e e d i m e n s i o n s ; n o s u c h l i m i t a t i o n a p p e a r s in

arithmetic.

F o r f u r t h e r i n f o r m a t i o n , s e e m y Raumlehre, p t .

II

(Berlin,

1 8 2 4 ) . (4,1,11; 5 0 7 - 5 0 8 ) These quotations justify attributing to Justus Giinther Grassmann the honor for b e i n g the first to p u b l i s h the idea of a p u r e l y geometri-

59

A H i s t o r y of V e c t o r Analysis

cal product. blown

He of course failed to extend the concept into a full-

system;

this

his son d i d , a n d this was clearly a far greater

achievement. It is noteworthy that the first of these statements was p u b l i s h e d b e f o r e 1832, that is, b e f o r e H e r m a n n G r a s s m a n n r e a c h e d the

first

more

insights

leading to his

sophisticated

system, whereas the

statement appeared

after

1832.

second and

This

fact m a y

h o w e v e r be of small importance, for the quotations m a y do no more than suggest w h a t m i g h t have b e e n c o m m u n i c a t e d b e t w e e n a father a n d a son w h o was l i v i n g i n his father's h o u s e i n 1832. N o i n f o r m a tion seems to be available concerning the interesting question as to the source of J. G. Grassmann's concept of a geometrical product. It w o u l d for e x a m p l e b e v e r y e x c i t i n g t o f i n d a l i n k b e t w e e n his statem e n t s a n d t h e g e o m e t r i c a l r e p r e s e n t a t i o n o f c o m p l e x n u m b e r s tradition or the "parallelogram of forces" tradition. As

Grassmann's

important to keep published interest

until

than

lehre o f 1 8 4 4 .

Theorie the

1911;

the

der

Ebbe

und

Flut

following points hence

first

it

is

at

publication

is

considered,

in mind. a

lower level

of his

it

is

First, it was not

system,

of historical

h i s Ausdehnungs-

S e c o n d , it is of great interest as a source of i n f o r m a -

tion concerning h o w Grassmann came to the m u c h fuller system, w h i c h w a s p u b l i s h e d i n 1844. T h i r d , the presentation o f his system was

somewhat biased,

as

he r e p e a t e d l y stated, by the fact that it

was t i e d t o t h e specific p h y s i c a l p r o b l e m o f t i d a l theory. I n this regard it w o u l d be interesting to be able to determine h o w m u c h of Grassmann's full system was developed w i t h physical applications in

mind.

At least this

much

seems

clear:

his

original ideas were

engendered, like those of Hamilton, in the course of a mathematical quest, but, unlike Hamilton, Grassmann seems to have forged many o f h i s m a t h e m a t i c a l tools w i t h p h y s i c a l a p p l i c a t i o n s (to t i d a l t h e o r y ) in

mind.

V.

Grassmann s The

Theorie

der

Ebbe

und

Flut

s t u d y b e g a n w i t h a n i n t r o d u c t i o n i n w h i c h G r a s s m a n n sur-

veyed the

h i s t o r y o f t i d a l t h e o r y a n d i n d i c a t e d p a r t i c u l a r l y his re-

lationship

to

celeste.

Laplace's

new mathematical felt

treatment

of the

subject

in

La

Mecanique

In the first chapter Grassmann stated that he had developed methods

which

forced however to comment:

he w o u l d use in the work.

He

"Since the scope of the present

w o r k does not a l l o w me to present all the principles of the methods of

geometrical

laws

analysis

(though

they

which could

I

have be

employed,

developed

l a t e d l a w s o f a l g e b r a i c a n a l y s i s . " (4,111,1;

60

I

will

independently)

borrow

from

the

the re-

18) T h i s suggests t h a t b y

Other Early Vectorial

1840

he

had

presented

developed

in

1844.

Grassmann

then

pressed

by

that

equality

the

the

a

substantial

considered

equation sign

^

with

the

=

part

law

Const."

the

of the

of

dot above

system

inertia,

(4,111,1; it

Systems

as

which

18),

fully

he

a n d he

signified

ex-

noted

geometrical

e q u a l i t y , t h a t is, t h a t t h e v e l o c i t y r e m a i n e d c o n s t a n t i n b o t h m a g n i tude

and

direction.

velocities, by

the

(P, S

imparted

P 4- Q

Proceeding to

arrived

combined

()) =

he

at

force

by

the

is

the

the

parallelogram

statement:

the

"The

geometrical

individual forces,

or

of forces a n d

velocity

sum

of

expressed

(S)

imparted

the

as

a

velocities formula,

w h e r e 4- is t h e s i g n of g e o m e t r i c a l s u m m a t i o n . " (4,111,1;

19) A f t e r i n d i c a t i n g t h a t m o r e f o r c e s (or v e l o c i t i e s ) c o u l d b e a d d e d in

the

same

way,

Grassmann

turned

a d d i t i o n a n d s u b t r a c t i o n . " (4,111,1; tive

and

vectors

associative (Strecken)

laws

and

to

h o l d for the

argued

"the

laws

of geometrical

19) H e p r o v e d t h a t t h e c o m m u t a -

that

addition

since

all

and

subtraction

laws

for

addition

of in

ordinary algebra can be d e r i v e d f r o m these t w o laws, all those laws could n o w be assumed for vectorial addition.

(4,111,1; 2 0 ) I t s h o u l d

be noted here a n d kept in m i n d for later discussions t h a t G r a s s m a n n was among the first to realize the full well the

as

their correlates

development of the

with

the

sequently analysis, tions

differential

statement: also

of

insofar except

as

sketched the

"All

they

method

of

are

are

and

in

algebraic

then

differentiation

likewise

fact

subtraction."

for taking

Grassmann

began

calculus of vectors, concluding

laws

integration

addition

significance of these laws as

for multiplication.

valid

obtained

from

(4,111,1;

partial

21)

in

con-

geometrical

no

other

After

derivatives

and

opera-

this

he

for vectors a n d

"d2 restated

the

law

of inertia as

p — 0"

( w h i c h is

equivalent, in

2

dp modern

notation,

to

= 0),

sents three o r d i n a r y equations. Grassmann and

gave

gravity. two

for

applied example

F r o m this

his the

and

he

(4,111,1;

methods

to

vectorial

pointed out that this

repre-

21-22) the

physics

formulation

of the of the

problem center

of

he returned to define the geometrical product of

vectors. By

the

geometrical

product

o f two

vectors,

we

mean

the

surface

con-

t e n t o f t h e p a r a l l e l o g r a m d e t e r m i n e d b y t h e s e v e c t o r s ; w e h o w e v e r fix the position o f the p l a n e i n w h i c h the p a r a l l e l o g r a m lies. W e refer t o t w o surface areas a s g e o m e t r i c a l l y e q u a l o n l y w h e n t h e y are e q u a l i n c o n t e n t and

lie

in

we m e a n the

parallel

planes.

By

the

geometrical

product

of

three

vectors

s o l i d ( a p a r a l l e l e p i p e d ) f o r m e d f r o m t h e m . A s t h e sign o f

61

A

History

of V e c t o r Analysis

geometrical

multiplication

while

indicate

we

we

choose

either

the

point

or

the

sign

x,

ordinary algebraic multiplication by writing the two

q u a n t i t i e s n e x t t o e a c h o t h e r o r b y u s i n g t h e s i g n x . (4,111,1; 3 0 ) As

was

natural,

he

s h o w e d that the

product of three vectors w h i c h

lie in the same or in parallel planes is zero. He p r o v e d b o t h forms of the

distributive

by

anticommutativity.

law

and

showed

(4,111,1;

that 30)

commutativity was

Also

the

replaced

expression

for

the

geometrical product was given as the product of the lengths of lines multiplied to

the

by

the

sign),

raume)

was

of the

finally

angle

the

between them (with attention

addition

of directed

areas

(Flachen-

defined.

Grassmann's vector

sine

and

(cross)

geometrical

product is

product.

will

It

be

quite

similar to the modern

advantageous

to

specify

the

similarities a n d differences for these t w o products, since the Grassm a n n product w i l l occur again and since this difference in products constitutes mann's

one

of the

system

identical

and

most significant differences b e t w e e n the

numerically

conventions.

modern

and

system.

likewise

The

two

correspond

in

Grass-

products regard

T h e most significant difference is that in the

are

to

sign

modern

m u l t i p l i c a t i o n t h e p r o d u c t is of t h e s a m e k i n d as the factors (all are vectors),

whereas

in

the

Grassmann

multiplication the

product is

not of t h e same nature as the factors. T h u s in the G r a s s m a n n m u l t i plication of t w o vectors the product is not another vector, but rather a

d i r e c t e d area.

It is t r u e that this area (or t h e

set of g e o m e t r i c a l l y

e q u a l areas) d e t e r m i n e s a v e c t or (or set of vectors) p e r p e n d i c u l a r to that area, a n d that t h e v e c t o r (or vectors) so d e f i n e d is p r e c i s e l y that vector tion.

which

plane

(or

which

the

lard

is

the

Conversely, set

and

has

E d w i n

for other uses, as After

reaping

In

in

there

is

it

was

concepts;

rather

the

true

the

establishment

62

of parallel

which

is

vectors)

precisely

that

defines

a

plane

in

purely

W i l s o n maintained that the vector for physical

applications, whereas

I

harvest

a

be

made

mechanical

came of of

concept

determined

not

foundation

geometrical

g e o m e t r y a n d in m u l t i p l e algebra.36 possible

by

this

new

without

these

stated:

scarcely

that a

of the

can

and in

Bidwell

projective

some

concepts

Nevertheless these

set

G r a s s m a n n f o r m of the p r o d u c t has a superiority

Grassmann

fact,

geometrical

(or

planes)

greater advantages

they h e l d that the

technique,

vector

of parallel

d i r e c t e d area o f t h e G r a s s m a n n p r o d u c t lies. Josiah W i l -

Gibbs

product

product in the modern form of the multiplica-

the

to

these their

context.

in in

purely

concepts (4,111,1;

general

(which 33)

a

which

on

development

which and

considerations

them natural

mechanics a

a I

simple led

geometrical cannot

must

be

way.

me

to

basis,

so

now given

discuss) only

Other Early Vectorial

Soon

after

this

Grassmann

defined

what

he

called

Systems

the

linear

product. By the one

linear product o f t w o

vector

multiplied

by

vectors the

we

mean

the

perpendicular

algebraic product of

projection

of the

second

o n t o it. We s e l e c t t h e s i g n ^ as t h e s i g n of l i n e a r m u l t i p l i c a t i o n , so t h a t by d e f i n i t i o n = cos

b — a b cos (ab). F r o m t h i s d e f i n i t i o n a n d s i n c e c o s ( a b )

(ba), we

s e e t h a t a ->b = b ^

a.

(4,111,1;

40, 212)

He then p r o v e d that the distributive law holds for this product, a n d explained that the where

a,

b,

c

l i n e a r p r o d u c t of (a 4- b 4- c)

a n d au bu cx

are t w o

sets

w i t h (ax 4 - b x 4 - cx)

of m u t u a l l y p e r p e n d i c u l a r

v e c t o r s a n d a i s p a r a l l e l t o a u b t o b u c t o c u i s e q u a l t o aax ccx.

(4,111,1;

40)

Finally

he

stated

that

since

the

+

distributive laws w e r e preserved in this multiplication, all algebraic

multiplication

this product is

to

applicable.

identical to the m o d e r n

Throughout the methods

were

remainder of his

tidal

theory

with

It

fob,

commutative

should

be

+

and

laws

clear

of

that

scalar product.

work Grassmann applied these

some

success.

In

later

portions

he

i n t r o d u c e d o t h e r n e w , b u t less r e l e v a n t , m a t h e m a t i c a l m e t h o d s , including the linear vector function.

(4,111,1;

7 9 ff.)

In summary, Grassmann presented in this w o r k a strikingly large part

of

major

vector kinds

elements

analysis.

of

Vector

vectorial

of the

addition

products,

and

vector

linear vector function

subtraction,

differentiation,

were

all

presented

the

two

and

the

in

forms

either equivalent or nearly equivalent to their m o d e r n counterparts. Grassmann's

work

vector analysis;

was

more

than

that was d a w n i n g at that time. algebra

may

the

first

important

system

of

it was also the first major w o r k in the n e w algebra

be

compared

T h e revolutionary nature of this n e w

with

that

of non-Euclidean

geometry.

Paradoxically Grassmann's d e v e l o p m e n t of his system as applied to three

dimensions

Ausdehnungslehre one

sense

This

was

early

in

led a

him

form

constituted a

a

magnificent

thirties

who

had

to

present

applicable discovery

of

achievement never

his

to

system

in

n-dimensions, non-Euclidean

for

attended

a

a

his

in

geometry.37

mathematician

university

1844

which

in

his

mathematical

lecture.

VI.

Grassmann's

Between textbooks

Ausdehnungslehre

1840 and

and

1844

Grassmann

a long mathematical

his n e w method. A r o u n d Easter, full

energies

to

the

1844 published

paper which

some

language

did not pertain to

1842, G r a s s m a n n b e g a n to t u r n his

composition

and by the fall of 1843 he had

of

of his

finished

Ausdehnungslehre

(5;

91-92),

w r i t i n g it. (5; 9 5 ) T h a t G r a s s -

63

A

H i s t o r y of V e c t o r Analysis

mann

r e q u i r e d such a short t i m e for the compositon of the book is

doubly striking since he wrote it w h i l e involved in numerous other activities

and

since,

presentation

was

according

reworked

to

in

a

remark

many

s e l e c t e d t h e f o r m a c t u a l l y e m p l o y e d . (4,1,1; ing the

best form

fact that

of presentation

sometime

discovered coveries.

new

He

after

1839,

in the

his

was

foreword,

forms

the

before

he

16) D i f f i c u l t y i n c h o o s -

caused in

large

part by the

and probably in

1842,38 G r a s s m a n n

and

of

implications

wrote

in

different

extensions

his

earlier

dis-

foreword:

W h e n I thus proceeded to w o r k out, coherently and from the beginnings, the results that I h a d found, b e i n g careful to appeal to no principle p r o v e n in any b r a n c h of mathematics, it resulted that the analysis w h i c h I d i s c o v e r e d d i d not t o u c h , as it h a d s e e m e d before, on t h e area of g e o m e t r y . R a t h e r I soon r e a l i z e d that I h a d c o m e u p o n a r e g i o n of a n e w science of w h i c h geometry itself is only a special application. It h a d for a l o n g t i m e b e e n e v i d e n t to me that geometry can in no w a y be v i e w e d , like arithmetic or combination theory, as a branch of mathematics; instead g e o m e t r y relates to s o m e t h i n g already g i v e n in nature, n a m e l y , space. I also h a d r e a l i z e d that t h e r e m u s t be a b r a n c h of mathematics

w h i c h yields

in a p u r e l y abstract w a y laws

similar to those in

g e o m e t r y , w h i c h appears b o u n d t o space. B y means o f t h e n e w analysis it

appeared

possible to f o r m such a p u r e l y abstract branch of mathe-

matics; i n d e e d this n e w analysis, d e v e l o p e d w i t h o u t the assumption of any principles

e s t a b l i s h e d o u t s i d e o f its

own

domain and proceeding

p u r e l y by abstraction, was itself this science. T h e essential advantages w h i c h w e r e attained t h r o u g h this conception w e r e 1) in relation to the f o r m — n o w all p r i n c i p l e s w h i c h express v i e w s of space

are e n t i r e l y o m i t t e d , a n d c o n s e q u e n t l y t h e b e g i n n i n g s of the

science w e r e as direct as those of arithmetic — a n d 2) in relation to content—the limitation to three dimensions is omitted. O n l y herein do the laws come to light in their full clarity and generality and reveal themselves

in

their essential

mutual

relationships,

and

m a n y regularities,

w h i c h for three dimensions present themselves either not at all or only obscurely,

now

present

themselves

with

this

generalization

in

full

c l a r i t y . (4,1,1; 1 0 - 1 1 ) Thus Die

in

1844

lineale

dargestellt

und

Mathematik,

wie

Magnetismus tioned

Grassmann's

book

Ausdehnungslehre,

und

before,

durch

neuer

Anwendungen

auch die

appeared

ein

auf

die

had

auf

die

Statik,

Krystallonomie

Grassmann

under Zweig ubrigen

Mechanik,

erlautert.

no

array

the

As of titles

full

der

Zweige

der

Lehre

vom

die it to

title:

Mathematik

was put

menafter his

name, as h a d H a m i l t o n w h e n n i n e years later in 1853 he p u b l i s h e d the

first

the

title

Schule

full treatment of his page zu

appeared

Stettin."

only

Some

system.

Under Grassmann's name on

"Lehrer an

idea

of the

der

Friedrich-Wilhelms-

n u m b e r of copies

of Grass-

m a n n ' s b o o k t h a t w e r e p r i n t e d , a s w e l l a s s o m e i d e a o f its r e c e p t i o n ,

64

Other Early Vectorial

may be

obtained from

written

to

dehnungslehre: print

for

the

Grassmann "Your

some

following

in

1876

book

time.

600 copies w e r e in

Die

Since

by

quotation the

Ausdehnungslehre

your work

taken

publisher

hardly

from a letter

of his

has

Systems

first

been

sold

at

out

all,

Ausof

roughly

1864 used as waste paper and the remainder, a

few o d d copies, have n o w been sold w i t h the exception of one copy which remains

in our library."

3 9

As Grassmann's book is discussed, it is important to keep in m i n d that

not

only

sidered.

its

To put

richness

but

also

its

readableness

must

be

con-

it briefly, the analysis of the w o r k w i l l lead to the

conclusion that the w o r k is a classic, that it is v e r y difficult to read, and

that

it

contains

a

large

portion

of

modern

vector

analysis,

w h i c h was h o w e v e r d e e p l y e m b e d d e d w i t h i n a far b r o a d e r system, and embedded in

such

a w a y that it c o u l d be extracted only w i t h

difficulty. Grassmann i n c l u d e d at the beginning of his w o r k a philosophical introduction

which

is

for it is

of the

major barriers b e y o n d w h i c h m a n y mathemati-

one

cians of his

time

both

interesting and historically significant,

d i d n o t pass.

Grassmann began this

introduction by stating:

T h e primary d i v i s i o n in all the sciences is into the real a n d the formal. T h e former represent in thought; the existent as existing i n d e p e n d e n t l y of thought, existent.

and their truth

The formal

consists

sciences

in their correspondence with the

on the other hand have as their object

w h a t has b e e n p r o d u c e d b y t h o u g h t a l o n e , a n d t h e i r t r u t h consists i n t h e correspondence b e t w e e n the t h o u g h t processes themselves.40 Concerning

proof

in

"Proof in the formal

the

formal

sciences thus

sphere of t h o u g h t itself;

sciences

does

Grassmann

stated:

not proceed outside of the

rather it remains purely in the combination

o f d i f f e r e n t acts o f t h o u g h t . T h u s i t i s that t h e f o r m a l s c i e n c e s n e e d not begin

with axioms;

instead definitions form their foundation."

(4,1,1; 2 2 ) T h e s p e c i a l r e l e v a n c e o f t h i s v i e w b e c o m e s c l e a r w i t h t h e following

two

statements:

P u r e m a t h e m a t i c s i s t h u s t h e s c i e n c e o f t h e p a r t i c u l a r e x i s t e n t w h i c h has come

to be

through

thought.

The

particular existent,

viewed in

this

sense, we n a m e a t h o u g h t - f o r m or s i m p l y a f o r m . T h u s p u r e m a t h e m a t i c s i s t h e t h e o r y o f f o r m s . (4,1,1; 2 3 ) Before we proceed to the division of the theory of forms, we have to s e p a r a t e o u t o n e b r a n c h w h i c h has h i t h e r t o i n c o r r e c t l y b e e n i n c l u d e d i n it.

This

branch

is

geometry.

e v i d e n t that g e o m e t r y as

From

the

concepts

set o u t a b o v e ,

it is

w e l l as m e c h a n i c s refers to a real existent;

for g e o m e t r y this is space. T h i s is clear since t h e c o n c e p t of space can in no w a y be produced by thought, b u t rather comes forth as something g i v e n . (4,1,1; 2 3 )

65

A

History

of V e c t o r Analysis

W h a t Grassmann b e l i e v e d he h a d discovered was thus a system, a purely

formal

system,

which

geometry a n d i n d e e d of all was

a

sort

necessarily

was

algebra41

of universal

above

mathematics in

and

as

independent

of

k n o w n at that time.

which

the

elements

had

It no

real content but could take on geometrically significant

content w h e n

such

deed marvelously

was

desirable.

general

His

" T h e o r y of F o r m s " was in-

a n d abstract.

Grassmann c o n t i n u e d in the introduction to lay the philosophical foundation his

theory

for his

system.

of forms

could

He

specified four concepts with which

develop

all

of mathematics;

these

were

t h e d i s c r e t e a n d t h e c o n t i n u o u s , t h e e q u a l a n d t h e d i f f e r e n t . (4,1,1; 24-25)

According to

of each

different—comes which

Grassmann,

from

the

combination of the

last

o f t h e s e pairs o f o p p o s i t e s — t h a t is, t h e c o n t i n u o u s a n d t h e forth

includes the

dimensions points)

of

vary

the line).

his

"Ausdehnungslehre,"

notion of difference

space)

and

continuously

and

in

which

hence

that

doctrine

(for e x a m p l e , the different

the

elements

produce

different

(for

example,

entities

(as

(4,1,1; 2 6 - 2 7 ) T h u s G r a s s m a n n s t a t e d :

Space theory [die R a u m l e h r e ] m a y again serve as an example. H e r e in t w o d i f f e r e n t d i r e c t i o n s f r o m a n e l e m e n t all t h e e l e m e n t s o f a p l a n e are p r o d u c e d w h e n the p r o d u c i n g e l e m e n t progresses i n any w a y according to two

directions.

T h e totality of points

(elements) producible in this

w a y i s c o m p r i s e d i n o n e e l e m e n t [ t h e p l a n e ] . T h e p l a n e i s t h u s t h e system of the second order; in it there is an infinite collection of directions d e p e n d e n t on those t w o original directions. If a third independent direct i o n i s a d d e d , t h e n b y m e a n s o f t h i s d i r e c t i o n i n f i n i t e s p a c e (as t h e s y s tem of the third order) is produced. Up to n o w one could not go beyond three directions; however in the pure theory of extension the number of d i r e c t i o n s c a n i n c r e a s e t o i n f i n i t y . (4,1,1; 2 9 )

The above mann

gives

attempted

concluded with tion

for the

s o m e idea of the p h i l o s o p h i c a l basis that Grass-

to

construct for his

a discussion

work;

it

aware that a reader's Whatever may

be

system.

T h e introduction was

concerning the best form of presenta-

n e e d only be noted that Grassmann was w e l l entry into his the

absolute

s y s t e m w o u l d not i)e easy.

value of Grassmann's philosophi-

cal u n d e r p i n n i n g of his system, it is important to keep in m i n d that it

seems

new

to

have

a l l o w e d G r a s s m a n n h i m s e l f to feel at ease w i t h

mathematical

ideas,

tions

and n-dimensional

with

Hamilton's

time.

Finally,

general the

66

for example, n o n c o m m u t a t i v e multiplicaspaces.

speculations Grassmann's

on

In this regard it m a y be c o m p a r e d algebra

penchant

d i d certainly lead to m a n y ideas

next section

of his

book.

as

for

the

the

science abstract

of and

pure the

of value, a fact e v i d e n t in

Other Early Vectorial

This

section

Forms."

He

was

began

entitled

with the

"Survey

of

the

General

Systems

Theory

of

statement:

By the general t h e o r y of forms I m e a n that series of truths w h i c h relate to all branches of mathematics in the same w a y , a n d thus p r e s u m e o n l y the general concepts of e q u a l i t y a n d difference, c o n n e c t i o n a n d separation.

The

general

theory

of forms

should

thus

precede

all

special

branches of mathematics. Since h o w e v e r that general b r a n c h does not as such exist yet, a n d since we cannot o m i t it w i t h o u t e n t a n g l i n g ourselves in useless r a m b l i n g s , we have no c h o i c e b u t to d e v e l o p this subject insof a r a s w e n e e d i t f o r o u r s c i e n c e . (4,1,1; 3 3 ) This

statement

tions

of the

introduced what was

book

for

the

reader

one

of the

o f t h e m o s t d i f f i c u l t sectime;

in

this

case

it

was

mainly the richness of content and the newness of the concepts that made

this very important section difficult.

c e e d e d to construct the basis

H e r e i n Grassmann pro-

of his theory of forms by postulating

certain initially contentless f o r m s (or m a g n i t u d e s ) a n d j o i n i n g these forms

by

certain

connections

(Verkniipfungen).

Thus

given

two

f o r m s a a n d b, G r a s s m a n n first i n t r o d u c e d a c o n n e c t i o n s y m b o l i z e d by

^

to

yield

connection could be

a

was

new

form

of such

a ^

(a ^ b) He

Grassmann

that the

specified

that

this

following two equations

set u p : a

s h o w e d that

forms

b.

a nature

were

^

b =

b

^

a

^ c = a

=

if equations

connected,

were

then

one

set u p i n w h i c h a n y n u m b e r o f could

members and omit the parentheses.

in

all cases

rearrange the

Grassmann called such a con-

n e c t i o n a s y n t h e t i c c o n n e c t i o n (4,1,1; 3 4 - 3 6 ) a n d f r o m i t p r o c e e d e d to his analytical connections.

For Grassmann an analytical connec-

tion connects t w o forms in such a w a y that w h e n the resultant form is

synthetically connected with one of the original forms, the other

original f o r m results. cal

connection

thetically

and

connected

Grassmann then

Thus when a

and to

b

b

as

^ is taken as the sign for analytithe

gives

a,

two or,

forms, in

then

short,

(a

(a ^

s h o w e d that the following equations

^ b)

b)

^

synb = a.

are true:

Grassmann further specified this f o r m of analytic connection by the assumption unique b

that

the

result

or u n e q u i v o c a l ;

remains

constant

of

thus,

and

the

an

analytic

for example, other

connection if a

changes,

was

to

be

^ b = c, t h e n if a or the

result

(c)

must

67

A H i s t o r y of V e c t o r Analysis

change.

Finally

he

s h o w e d that these

results

to connections of any n u m b e r of forms. These

statements

must

have

reader

of the

times.

At the

mann

stated

what

should

appeared

very

end

have

could be

extended

(4,1,1; 3 6 - 3 8 ) very

of this

strange

to

the

d e v e l o p m e n t Grass-

considerably

enlightened

the

matter: w h e n these connections obeyed the above laws, they could be called addition and subtraction. Grassmann then s h o w e d that t w o n e w forms, the indifferent and the

analytic

lytically

form,

could

connected

to

be

produced.

itself—for

When

example,

a

any form

was

ana-

w a —then the indif-

ferent f o r m resulted (indifferent in the sense that a c o u l d have any value).

He represented this

n e w f o r m b y (/) a n d s t a t e d t h a t w h e n

t h i s f o r m i s c o n n e c t e d t o b > p r o d u c i n g (
called

the

analytic form,

emerges.

b), t h e n t h e n e w f o r m

Finally Grassmann

ad-

m i t t e d t h a t t h e i n d i f f e r e n t f o r m c o u l d b e c a l l e d " N u l l " (i.e., zero) and

the

analytic

form

could

be

called

the

negative

form.

(4,1,1;

38-40) Grassmann went on to

develop two further connections, one of

w h i c h was

synthetic and the other analytic.

tions

of such

were

a sort that w h e n

a form

Both of these connecwas

applied through

such a connection to two forms connected in either of the previous manners, a transformation w o u l d take place w i t h o u t a change in the result.

His

new

k i n d of synthetic connection he

symbolized by «

and postulated the following: a

connection,

(4,1,1;

41-43)

Grassmann Note

operation that is in his

Such

stated,

could

be

called

multiplication.

that this implies that Grassmann w i l l call any

distributive a multiplication, and w h a t he sought

work was, broadly speaking,

distributive operations, w h i c h

n e e d not be either c o m m u t a t i v e or associative. Grassmann then introduced the second k i n d of analytical connection

(i.e., d i v i s i o n ) . T h i s w a s l i m i t e d b y t h e l a w e x p r e s s e d i n t h e

equation

_

.

?

a • +• b

a _ b

He pointed out that unlike the former type of analytic connection, three

forms

determined

connected (thus,

for

by

division

example,

are

if t

=

c

not in and

all cases

a = 0,

then

uniquely b

is

not

b

uniquely determined). To the

(4,1,1; 4 3 - 4 4 )

above s u m m a r y it n e e d only be a d d e d that according to

Grassmann similar forms (forms of the same order, thus t w o points, t w o vectors, a n d so forth) w h e n connected by either of the first t w o kinds

68

of

connection

give

in

general

forms

of

the

same

order,

Other Early Vectorial

Systems

whereas in multiplication forms of the same or different orders may be c o m b i n e d and in general produce forms of a higher order. T h u s the p r o d u c t of t w o vectors (forms of the first order) is a d i r e c t e d area (a form of the second order). The

section just

discussed

combined

with

the

introduction

to

present a nearly impassable barrier for most mathematicians of the time. Ideas s u c h as these w e r e so n e w , so abstract, a n d at first sight so useless that it is not difficult to u n d e r s t a n d w h y M o b i u s r e g a r d e d the work as simply unreadable, Baltzer f o u n d it m a d e h i m

dizzy,

and H a m i l t o n was led to write to De M o r g a n that to be able to read Grassmann passages they

he

would

have

to

learn

smoke.42

to

However

these

are not o n l y characterized by o b s c u r i t y a n d abstractness;

also

reveal

powerful mind. commutative,

(to

a modern

mathematician)

the

brilliance

of a

Grassmann had c o m e to understand the associative,

and

distributive

laws

more

fully

than

any

earlier

mathematician, a n d this understanding is reflected in the construction of his

system.

Grassmann

allowed

his

initially

many

values —numbers,

forth;

moreover he eventually

plicative

connections.

points,

All

43

these

contentless

vectors,

forms

oriented

to

areas,

assume and

so

developed sixteen kinds of multiforms

and connections

were

un-

d e r s t o o d i n t e r m s o f t h e l a w s set o u t b y G r a s s m a n n i n this section, a n d thus this

section was

one of the most important.

Having laid

the foundation, we m a y turn to the m a i n contents of the book.44 T h e first chapter of part one of Grassmann's book centered on the production

of his

various

systems.

N e w systems

were created by

taking an element from another system and allowing it to undergo a n e w variation; thus for e x a m p l e a point (the element) is m o v e d in a single direction (the variation) to produce a straight line (a system of the first order).

If this

line n o w varies by m o v i n g in some (recti-

linear) direction, a plane (a system of the second order) is produced. According to Grassmann the same process can be e x t e n d e d to f o r m systems o f a n y o r d e r ( h e n c e his " a n y " d i m e n s i o n a l spaces). I n this chapter a n d w i t h i n this f r a m e w o r k Grassmann i n t r o d u c e d vectors, proved that they obey his previously d e v e l o p e d addition a n d subtraction

laws,

and

explained

vectorial

addition.

He

furthermore

proved various

algebraic properties for his systems. T y p i c a l of his

statements

the

order the

can m

pression

is be

following:

expressed

given

independent

is

unique."45

as

the manners

"Every

vector

sum

m

of

of change

of a

mth

system

of the

vectors,

which

belong

to

of the

system.

This

ex-

T h e first chapter was concluded, as w e r e all others, w i t h a section on application of the theoretical ideas presented.

For this chapter

69

A

History

the

applications

ideas

were

mainly

two:

Grassmann

showed

how

his

c o u l d be u s e d to lay a secure foundation for geometry a n d he

gave a

of V e c t o r Analysis

the

body

vectorial

as

well

method

as

the

of representing

laws

of force

the

center of gravity of

a n d velocity for the center of

gravity. The tion

second

chapter

of Vectors."

lent to

what had been

work, and hence Grassmann only

of this

in

it is

called the

entitled

"Outer Multiplica-

product is

essentially equiva-

geometrical

product in

linear product

(renamed

the first published

uct,

we shall

Besides

definition

give it in full.

of his

1840

Since this

o f t h e m o d e r n scalar (dot) p r o d -

After G r a s s m a n n h a d c o n c l u d e d a dis-

outer product,

this

his

inner product)

offhand manner in the foreword to the book.

was

cussion

outer

s i m i l a r t o t h e m o d e r n v e c t o r (cross) p r o d u c t .

m e n t i o n e d his

an

part was

Grassmann's

he

wrote:

concept there is another that l i k e w i s e relates distances

to fixed directions. N a m e l y , w h e n I took the perpendicular projection of the one distance onto the other, the arithmetic product of this projection and the distance (onto w h i c h the projection was made) represented the product of those distances,

provided

that

the

multiplicative

relation

to

addition

was

valid. But this product was of an entirely different k i n d than the first, in that the factors c o u l d be e x c h a n g e d w i t h o u t c h a n g i n g the sign a n d the product of t w o

m u t u a l l y perpendicular vectors

first

product

the

latter

outer

has

non-zero

and

values

was zero.

I

called the

the

second

the

inner product,

only

when

the

directions

since

approach

the one

a n o t h e r , that is, w h e n t h e distances lie p a r t l y i n e a c h other.46 Grassmann's lehre

of

1844

outer

in

a

product

manner

was

both

presented

different

than the presentation in his study of the tides. a

simple

way

by

in

from

and

his

Ausdehnungs-

more

general

Grassmann began in

stating:

We shall b e g i n w i t h geometry in order to secure an analogy according to w h i c h the abstract science m u s t proceed, a n d thus obtain a clear idea to

escort us a l o n g the u n k n o w n a n d arduous path of abstract d e v e l o p -

ment. We go f r o m the vector to a spatial form of higher order w h e n we allow the

entire

another vector

vector,

which

is

that

is

each

point of the

heterogeneous

to the

vector, to

first,

describe

so that all points

c o n s t r u c t a n e q u a l v e c t o r . T h e s u r f a c e area p r o d u c e d i n t h i s w a y has t h e form of a parallelogram.

T w o such

surface areas

w h i c h belong to the

same p l a n e are d e s i g n a t e d as e q u a l if the d i r e c t i o n of t h e m o v e d vector lies i n b o t h cases o n t h e s a m e s i d e (for e x a m p l e , o n t h e left side) o f t h e v e c t o r p r o d u c e d t h r o u g h t h e m o t i o n . W h e n i n t h e t w o cases t h e c o r r e sponding

vectors

lie

on

opposite

sides,

then

the

surface

areas

are

designated unequal. Thus we have a simple and general law: If in then the

70

the the

individual

plane total

a

vector

surface

surface

area

elements

moves

successively

along

thereby

(provided

that

given

manner)

produced are

set

in

the

any

series

of vectors,

the

signs

of

is

equal

to

that

area

sum

which

of those

would

vectors.

he

produced

(4,1,1;

Other Early Vectorial

Systems

if

along

the

vector

had

moved

the

77-78)

Statements as concrete as the above were not typical of the work nor always

desirable

geometrical have

hindered

went

beyond

the

we

the

the

above

parallel

if

consider

the

nature

understanding

usual

(cd,

ef

of

later

three-dimensional

statement,

lines

Grassmann and

ab)

lines:

i n g along ac u n t i l

for

developments

were

cut

To

which

illustrate

three by

if

it could

coplanar

three

pairs

of

d

move

L e t t i n g ab r e p r e s e n t

parallel to itself, he p i c t u r e d ab m o v -

i t r e a c h e d l i n e cd.

T h u s t h e a r e a acdb w a s

swept

It w i l l be seen by simple geometry that this area is equal to the

sum

of

areas cd.

work,

geometry.

e c a n d d f e a a n d f b , c a a n d db.

a vector constrained to

out.

the

considered

which

c

parallel

of

intuition had b e e n a l l o w e d to enter too early,

the

areas

of the

swept out w h e n

Moreover the

parallelograms

ab

aefb

and

ecdf

that

is,

the

m o v e s a l o n g ae to ef a n d t h e n a l o n g ec to

r e s u l t i n g areas

w i l l be

the

same,

he

stated,

if ab

m o v e s along ae u n t i l it coincides w i t h ef or if it m o v e s a l o n g ac a n d t h e n ce as l o n g as the reversal in d i r e c t i o n ac is c o m p e n s a t e d for by taking

the

area

Grassmann

resulting

then

noted

from

that

ab's

since

motion for

along

example

ce

the

as

negative.

motion

of

ab

along ac gave the same area as w h e n ab m o v e d a l o n g ae a n d t h e n ec,

this

was

on the basis From 1844

the

was

an

operation

above in

that

should

be

viewed

as multiplication

of his theory of forms.47

all

one ways

might

easily infer that his outer product of

equivalent

to

his

geometrical

product

of

1 8 4 0 . T h i s i s t r u e o n l y i n p a r t , s i n c e (as i t w i l l b e s h o w n p r e s e n t l y ) his

geometrical

general

product

of 1840 was

outer p r o d u c t of 1844.

Thus

only one

species

lar t o t h e m o d e r n v e c t o r (cross) p r o d u c t , w a s i n t h e bedded many

within

of

dehnungslehre

the of

a

far

broader

elements 1844

and

of is

system.

vector an

of the

more

his geometrical product, simi-

This

analysis important

1844 w o r k em-

situation

is

developed reason

w h y

typical in

the it

of Aus-

would

71

A H i s t o r y of V e c t o r Analysis

have b e e n difficult to extract the m o d e r n system of vector analysis f r o m his system.

At the e n d of his introductory section Grassmann

stated:

. . we can n o w return to our science in order to pursue it

according

to

a

purely

abstract

manner

and

independently

of all

s p a t i a l c o n s i d e r a t i o n s . " (4,1,1; 8 0 ) At this point it w i l l be helpful to consider in m o d e r n terms what G r a s s m a n n m e a n t by his outer product. It was a p r o d u c t of base elements

e2,

eu

e3, . . . ,

en

which

obey

the

following

laws:

letting

x, y, z take on a n y v a l u e s of n,

^xi^y

€z

^x^y

I t s h o u l d b e n o t e d that t h e last e q u a t i o n n e e d not h a v e b e e n g i v e n since

the

outer product

must be

distributive,

for this property is

contained for Grassmann in the definition of " p r o d u c t . " T h e product of N such order.

elements

was

considered to be an entity of the N t h

H e r e again Grassmann's outer product differs f r o m the m o d -

e r n cross p r o d u c t , since t h e r e s u l t o f t h e cross m u l t i p l i c a t i o n o f t w o vectors

is

another vector (in

Grassmann's

terms,

an

entity of the

first order), whereas the outer product of t w o vectors is an entity of the

second

especially

order.

This

important

difference

in

that

between

Grassmann's

the

outer

two

products

product

is

allowed

h i m t o g e n e r a t e all sorts o f n e w entities w h i c h h e d u l y c o n s i d e r e d . Grassmann vectors.

proceeded

to

discuss the

product of any number of

G r a s s m a n n s t a t e d t h a t a p r o d u c t s u c h a s a.b.c . . . w a s t o

m e a n t h a t t h e v e c t o r a f i r s t m o v e s a l o n g b (as b e f o r e ) , t h e n t h e r e sultant o r i e n t e d area w o u l d m o v e along c, a n d so on t h r o u g h orders higher than the third. distributive

and

that

Grassmann entities

proved that such relations were

thus

arising could be

added

under

c e r t a i n c o n d i t i o n s . A s e a c h n e w e n t i t y arose, h e c h e c k e d its p r o p e r ties in relation to his general theory of forms. S o m e of Grassmann's applications of his outer p r o d u c t m e r i t discussion. uct

I m m e d i a t e l y after his

Grassmann

showed

full presentation of the outer prod-

how

Varignon's

principle

could

be

represented in terms of the outer product in a very simple manner, and he in general product.48 solving Then

he

n

discussed the representation of m o m e n t s by this

H e also s h o w e d h o w the outer p r o d u c t c o u l d b e used i n first-degree considered

equations what

he

outer division ordinary numbers t e m f o r t h e f i r s t t i m e . (4,1,1;

72

in

called

n

unknowns.

(4,1,1;

outer division,

99-102)

and through

( " Z a h l e n g r o s s e " ) e n t e r e d h i s sys-

118-137)

Other Early Vectorial

In

the

last c h a p t e r of t h e

called "Abschattung," was

primarily

original equation

when

with

could be an

part he

dealt w i t h

a process

or, loosely speaking, projection.

concerned

equation

first

the

conditions

definitely

equation

among

extensive

he

Herein he

under

inferred from

Systems

which

the

the

resulting

magnitudes

was

m u l t i p l i e d by an extensive magnitude. T h e results obtained in this chapter

were

applied

to

spatial

projections

and

the

solution

of

equations. T h e majority of the ideas considered up to this point w e r e given by Grassmann in the five chapters of the first section (Abschnitt) of his

book,

a

section

entitled

"Extensive

Magnitude"

("Die

Aus-

dehnungsgrosse"). T h e second of the t w o sections of the book was entitled

"Elementary

Magnitude"

("Die

Elementargrosse").

This

section began w i t h a lengthy discussion of elementary magnitudes, a term by w h i c h Grassmann referred mainly to points. of Grassmann's was

point analysis

a prominent

part of his

is

relevant both

Discussion

directly, in that it

system, and indirectly, in that it had

n u m e r o u s relationships to his system of vector analysis. H i s system of point analysis, h a v i n g b e e n discovered i n d e p e n d e n t l y of M o b i u s , was presented differently and developed more fully. Grassmann

began

this

section

by

considering the addition and

subtraction of elementary magnitudes, although he seems never to have

stated

precisely

except by implication.

what

he

meant by

elementary

magnitudes

E a r l y in the discussion he stated: " I n order

to b r i n g this result into sharper focus, we w i l l apply it to geometry a n d t h u s t a k e t h e e l e m e n t s as p o i n t . . . ." (4,1,1;

158) G r a s s m a n n

proceeded to write such equations as the following, in w h i c h Greek letters

represent elementary

represent

magnitudes,

and the

i s,

UPA,] + .

.

. + in[pan] = k f o M + .

.

. + kn[pf3n]

Grassmann

mained true

when

then

established

both

the

traction,

any other element a was

commutative that

these

and

members

of such

associative

operations

gave

. are the

laws

unique

substituted for p. an

equation

He

satisfied

for addition a n d subresults,

and that the

satisfied for the m u l t i p l i c a t i o n of the e l e m e n t

p a i r s s u c h a s [a/3] w i t h n u m b e r s . s h o w e d that

.

159)

w e n t f r o m this t o s h o w that t h e e q u a t i o n re-

that the

distributive law was

mann

a n d ris

(4,1,1;

T h i s i s e s s e n t i a l l y a v e c t o r e q u a t i o n ( i n w h i c h pau pa2 . vectors).

/c's

numbers:

in

On t h e basis of t h e s e facts Grass-

the e q u a t i o n g i v e n above, the p's

could be

d e l e t e d to y i e l d an e q u a t i o n s u c h as aa + fo/3 . . . w h e r e a, b, . . . r e p r e s e n t n u m b e r s ; a , f3, . . . e l e m e n t s o r p o i n t s ; a n d a a w a s c o n s i d e r e d a w e i g h t e d p o i n t . (4,1,1;

160-163)

73

A

History

of V e c t o r Analysis

Grassmann, analysis

in

previously He then be

unlike

close

established

of the

Grassmann

by

a

weights to

developed vectorial laws

his

ideas

in

system and

proving

point whose

laws

weight was

individual points.

(4,1,1;

point

in

for

fact

points.

equal

to the

164) T h i s

led

consider a point of 0 weight, w h i c h he s h o w e d was a v e c t o r . (4,1,1;

164-166) T h e final topic in the

part of this chapter concerned the result obtained w h e n

a p o i n t or a m u l t i p l e p o i n t w a s a d d e d to a vector. The

of

used

sum of a n u m b e r of weighted points could

single of the

best interpreted as theoretical

to

vectorial

s h o w e d that the

represented

sum

Mobius,

relation

a p p l i e d parts

of the

chapter dealt with

(4,1,1;

166-167)

such topics

as

geo-

metrical m e t h o d s for constructing s u m points and physical applications

of his

made

reference

point

analysis.

to

In

this

part of the

chapter Grassmann

Mobius:

W e are h e r e a t t h e first a n d o n l y p o i n t a t w h i c h our science touches o n ideas already k n o w n . addition of simple primarily only as the

same

N a m e l y in the barycentric calculus of M o b i u s an

and multiple

points

is

presented.

M5bius used it

an abbreviated m e t h o d of expression but developed

techniques

of calculations

which

we

presented (in

greater

g e n e r a l i t y ) i n t h e first paragraphs o f this chapter. W h a t i s e n t i r e l y l a c k i n g in M o b i u s presentation is the conception of s u m as a magnitude in the case w h e r e t h e w e i g h t s t o g e t h e r total zero. W h a t h i n d e r e d the sagacious author o f that w o r k i n v i e w i n g this s u m as a vector of constant l e n g t h a n d d i r e c t i o n is certainly the strangeness of c o m p o u n d i n g length a n d direction into one concept. If that s u m h a d b e e n d e t e r m i n e d to be a vector, then the concept of the addition and s u b t r a c t i o n o f v e c t o r s f o r g e o m e t r y (as w e h a v e p r e s e n t e d i t i n S e c t i o n I, C h a p t e r I) w o u l d have arisen, a n d thus our w o r k w o u l d have had a s e c o n d p o i n t o f contact w i t h M o b i u s ' w o r k ; also the b a r y c e n t r i c calculus w o u l d h a v e o b t a i n e d a m u c h f r e e r a n d m o r e g e n e r a l d e v e l o p m e n t . (4,1,1; 172) In

the

mann chapter P — a, [a/3]

second

chapter of the

introduced he that

had is,

the

outer

s t a t e d t h a t [a/3] as

second

should be

a vector from

s h o u l d be v i e w e d ,

section

multiplication

a

t o /3.

of

of his

book

points^

In

interpreted as

(4,1,1;

165-166)

Grass-

the first equal

to

However

n o t s i m p l y a s t h e v e c t o r a/3, b u t a s a v e c t o r

l i m i t e d t o p o s i t i o n s i n t h e l i n e t h r o u g h a a n d /3, t h a t i s , ( t o u s e t h e modern (the

term)

line

from

as a

a

line-bound

point

to

the

vector. same

(4,1,1;

point

is

179)* S i n c e no

line)

[pp] = 0

and

since

[ p a ] — — [ a p ] ( b e c a u s e [ p a ] = a — p = — ( p — a) = — [ a p ] ) , [ a / 3 ] s h o u l d be v i e w e d as an outer multiplication. It is mann's

(4,1,1;

175)

not necessary for present purposes to f o l l o w in detail Grasshighly

abstract

and

complicated development of the outer

product for points. Rather it n e e d only be noted that here as before Grassmann continually

74

made

use

of vectors and vector relations in

Other Early Vectorial

developing

the

outer

product.

Finally

this

section

may

Systems

be

sum-

marized through the use of Grassmann's o w n s u m m a r y as presented in

his

der

paper

of

1845

entitled

"Kurze

Uebersicht

iiber das

Wesen

4 9

Ausdehnungslehre."

I f A , B , C , D , are points, t h e n w e m e a n b y (1) A . B , t h e l i n e , w h i c h has A a n d B

as e x t r e m i t i e s , r e g a r d e d as a

definite part of the infinite right line d e t e r m i n e d by A and B; (2) A . B . C , t h e

triangle,

whose

vertices

are A, B, C, r e g a r d e d as a

definite part of the infinite plane d e t e r m i n e d by A, B, C; ( 3 ) A.B.C.D,

the

tetra[h]edron,

whose

vertices

are

A,

B,

C,

D,

re-

g a r d e d as a d e f i n i t e part of i n f i n i t e space. That

is,

we

put A . B = A1.B1,

when

both

products

represent equal

p'ts, w i t h like signs, of the same r i g h t l i n e ; f u r t h e r A . B . C ^ A ^ B ^ C ^

w h e n b o t h triangles are e q u a l parts, w i t h l i k e signs, o f t h e same p l a n e ; and finally A.B.C.D

=

Ax.Bi.C1.D1,

w h e n b o t h t e t r a [ h ] e d r o n s h a v e e q u a l v o l u m e s , w i t h l i k e signs.50 The

applied

section

ordinate

systems,

plane

terms

in

of

this

chapter

treated

such

topics

as

co-

transformations of co-ordinates, the equation of a

of points,

and

cluding the representation

n u m e r o u s applications to statics, in-

o f static

moments and equilibrium con-

ditions. The than

materials

discussed

two-thirds

to

dealt w i t h what Grassmann the

regressive

characterized

product. by

factors is zero;

this

of Grassmann's

the

point constituted

book.

called

The

"Das

that

the

more

last c h a p t e r

eingewandte Produkt," or

He pointed out that the

property

slightly

next to the

product

outer product was of two

dependent

since this property was not essential to the ideas of

a product,

a

restriction.

(4,1,1;

new

product could 206)

Thus

be

defined which was

Grassmann

free

of this

introduced his regressive

product, or, m o r e accurately, he extended the concept of product in such a w a y that the

regressive

and outer products could be treated

as t w o forms of one product. As E n g e l pointed out in his notes, this complicated the presentation. An

idea of Grassmann's

the following.

(4,1,1; 4 0 8 )

regressive

product may be

gained from

It is natural to ask if a n y m e a n i n g m a y be a s s i g n e d to

the p r o d u c t of t w o factors,

some of w h o s e c o m p o n e n t s are m u t u a l l y

dependent

multiplication

as

Similarly

any

o f factors

may be

viewed

taking place w i t h i n a system of some order, for example, three-

or four-dimensional space. W h e n the s u m of the orders of the factors exceeds the order of the space u n d e r consideration, the factors m u s t

75

A

History

be

of V e c t o r Analysis

dependent.

gressive

It was

product.

this further;

for this case that G r a s s m a n n d e f i n e d t h e re-

T w o

quotations

from

Grassmann

will

explicate

the first quotation is from his " K u r z e Uebersicht" and

is m o r e general than the second, w h i c h is f r o m the latter part of the chapter under discussion and w h i c h gives a geometrical interpretation

of the

regressive

product.

Henceforth outer multiplication will be designated by simply writing the

factors

tween where is

together;

the A,

B,

C

as

a

treated

referred time.

to

regressive

factors.

the

(4,1,1;

are

understand

by

the

any

magnitudes,

the

product

co-efficient field

multiplication,

We

belonging

of lowest

to

order

in

A,

by

a point placed be-

regressive provided

which A,

product

ABC .A,

B,

that

and C

in the lie

AB.AC, which

ABC

product at

the

be same

310-311)

T h e product of t w o line magnitudes in the plane is the point of intersection of the t w o lines j o i n e d w i t h a part of that plane as factor; if for example

ab

a n d ac

magnitudes, three

line

then

(where

their

magnitudes

a, b, c r e p r e s e n t points) are t h e t w o l i n e

product in

the

is

abc.a.

plane

is

Furthermore equal

to the

the

product of

d o u b l e d surface

c o n t e n t (set t w i c e a s factor) o f t h e t r i a n g l e e n c l o s e d b y t h e lines, m u l t i plied by the

p r o d u c t o f the three quotients, w h i c h express h o w m a n y

t i m e s e a c h s i d e i s c o n t a i n e d i n its c o r r e s p o n d i n g l i n e m a g n i t u d e ; t h u s if a,

b,

c are

those three points,

a n d mab,

nac, pbc ( w h e n m ,

n,

p are

n u m e r i c a l m a g n i t u d e s ) are the three l i n e m a g n i t u d e s , t h e n their p r o d u c t is e q u a l to mnp.abc.abc T h e p r o d u c t of t w o plane magnitudes in space is part of the intersecting edge

m u l t i p l i e d b y a p a r t o f s p a c e , f o r e x a m p l e , abc.abd —

abed. ab.

Furthermore the product of three plane magnitudes is the intersection p o i n t of t h e t h r e e planes, m u l t i p l i e d by t w o parts of space, for example, abc.abd.acd= The

above

pose, does partly

abed.abed.

in

(4,1,1;

description,

which

243-244) is

sufficient

for

the

present pur-

scant justice to the complexities of this chapter.

mitigated

deleted

a.

by

the

the

fact

Ausdehnungslehre

that of

some 1862,

ideas and

in

this

many

This is

chapter were

of the

develop-

ments were approached from a different point of view. T h e chapter concluded

with

geometrical

applications.

T h e final chapter in the book was devoted mainly to applications of the

previous

ideas

to

geometry and to crystallography. T h e final

section of the chapter was

entitled "Remark on the O p e n Product."

This section is of special importance since Gibbs argued that it contained " t h e key to the theory of matrices .

.

and the linear vector

function.51 Before the reception accorded to Grassmann's w o r k is discussed, it will above

76

be

useful

detailed

to

state t h e m a j o r c o n c l u s i o n s e m e r g i n g f r o m the

discussion

of

his

first

Ausdehnungslehre.

First,

it

Other Early Vectorial

Systems

should be evident that it was a w o r k of great brilliance, a w o r k that along with

much

else

contained

a large

v o l v e d in m o d e r n vector analysis.

n u m b e r of the

ideas

in-

It s h o u l d also be clear that e v e n

if priority disputes are j u d g e d by date of p u b l i c a t i o n (thus e x c l u d i n g Grassmann's

Theorie

der

Ebbe

und

Flut),

Grassmann's

ideas

as

p u b l i s h e d i n 1844 w e r e far m o r e e x t e n s i v e a n d r i c h e r t h a n t h o s e o f H a m i l t o n at the same time. T h e r e is merit to Sarton's remark that Grassmann's Hamilton's

Ausdehnungslehre Lectures

on

of

1844

Quaternions

should

of

1853,

be not

compared with

with

Hamilton's

early researches.52 Grassmann's b o o k is a great classic in t he history of mathematics,

even

though one

of t h e mosc u n r e a d a b l e classics.

T h e w o r k was exceedingly abstract a n d very c o m p l i c a t e d and, perhaps most importantly, it departed from all the then current mathematical traditions. E v e n in the early sections of the book the reader encountered h i g h l y original ideas and at most tenuously

placed in a philosophic setting

attached to

the

mathematical

ideas

of the

time. In order for a m a t h e m a t i c i a n of that t i m e to

derive the

modern

system of vector analysis from Grassmann's book, he w o u l d have to (1) r e a d a n d u n d e r s t a n d t h e b o o k ( n o s m a l l t a s k ) , (2) d e l e t e m a j o r mathematical

portions

of the

book

(such

as

point

analysis),

(3)

l i m i t t h e p r e s e n t a t i o n t o t h r e e - d i m e n s i o n a l s p a c e , (4) r e d e f i n e s o m e o f t h e f u n d a m e n t a l i d e a s ( s u c h a s t h e o u t e r p r o d u c t ) , (5) c h a n g e t h e structure

and

from

philosophical

the

emphasis

of the ideas

work,

(6)

detach

contained in

it,

the

a n d (7)

presentation attach

to

it

ideas already in the literature of the times, b u t u n k n o w n to Grassmann, such as the development of the geometrical representation of complex n u m b e r s and the theorems of G r e e n a n d Gauss. No one at that t i m e or at a later t i m e a c c o m p l i s h e d these seven labors;

that

this was so is not surprising. It w i l l presently be s h o w n that m o r e than t w e n t y years

passed before

someone accomplished even the

f i r s t o f these, t h a t is, b e f o r e a n y o n e r e a d a n d u n d e r s t o o d a n d f u l l y appreciated

VII. In

The the

Grassmann's

Period from period

Grassmann's events,

work

which

1844

from

a n u m b e r of events

book.

to

the

1862

Ausdehnungslehre

of

1844

to

that

of 1862

took place w h i c h shed light on the relation of to

will

the be

mathematical discussed

narrative of Grassmann's activities

ideas of the times.

within

the

framework

These of

the

in this period, range in variety

from the discovery a n d publication by others of ideas contained in Grassmann's

system to the reading of Grassmann's book by Hamil-

77

A History

of V e c t o r Analysis

ton.

The

tion

accorded Grassmann's

primary concern

Friedrich this

Engel

reception Thus

of this section is discussion of the recepwork.

accurately described the magnitude and form of

in the following statement:

Grassmann

experienced

what

must

be

the

most

painful

ex-

p e r i e n c e for the author of a n e w w o r k : his book n o w h e r e r e c e i v e d attention; the p u b l i c was c o m p l e t e l y silent about it; there was no one w h o d i s c u s s e d i t o r e v e n p u b l i c l y f o u n d f a u l t w i t h it. T h e m a t h e m a t i c i a n s t o w h o m h e h a d sent t h e w o r k e x p r e s s e d t h e m s e l v e s a s not u n f r i e n d l y t o it, t o s o m e e x t e n t e v e n a s b e n e v o l e n t t o it, b u t n o o n e r e a l l y s t u d i e d it. (5; 97) Engel's given

statement

the

book

mathematician

is

well

illustrated by considering the reception

by the

man generally esteemed as the most gifted

of

period—Carl

the

Friedrich

Gauss.

apparently sent a copy of his book to Gauss, and the with

a

note

of thanks he

had

dated

worked

December

stated

that

on

before

and had published some

similar of his

14,

1844.

ideas

Grassmann

latter r e p l i e d

Therein

Gauss

nearly a half century

results

in

1831; Gauss was

probably referring to his w o r k on the geometrical representation of numbers.53

complex

Gauss

also

stated that he

was

very busy and

that he h a d c o n c l u d e d that to get at the kernel of the w o r k it w o u l d be

necessary

lichen) Gauss

to

familiarize

terminology never

used

himself with

in

the

book.

the peculiar (eigenthiim-

(4,1,11;

398)

This

it seems

did.

T h e reaction of Mobius to Grassmann's work should prove illuminating,

for of all

mathematicians

best position to judge visited asked

Mobius Mobius

at Leipzig, to

of the

period

Mobius

Grassmann's work. Thus in

write

and in

a review

was in the

1844 Grassmann

a l e t t e r o f O c t o b e r 10,

of his book.

He wrote:

1844, he

"Finally I

take the liberty of asking y o u to w r i t e a r e v i e w of the w o r k for some critical journal,

for

I

am

c o n v i n c e d that just as n o w no one stands

nearer the ideas expressed in the w o r k than you, no one will be in a better position than you to judge the work so fundamentally and to bring to may

light both

contain."

nearly I

(5;

the 99)

four months reply that I

kindred

spirit,

weaknesses

and whatever merits

Mobius' reply was

dated

the book

February 2,

1845,

later:

was but

sincerely pleased to have c o m e to m e e t in y o u a our

kinship

relates

only

to

mathematics,

not

to

p h i l o s o p h y . As I r e m e m b e r t e l l i n g y o u in person, I am a stranger to the area of p h i l o s o p h i c speculation.

The

philosophic element in

y o u r ex-

c e l l e n t w o r k , w h i c h lies at t h e basis of t h e m a t h e m a t i c a l e l e m e n t , I am not prepared to appreciate in the correct manner or even to understand properly.

78

Of this

I

have

become

sufficiently aware

in

the

course

of

Other Early Vectorial

numerous

attempts

to

study

your

work

Systems

without interruption;

in

each

case h o w e v e r I h a v e b e e n s t o p p e d b y t h e g r e a t p h i l o s o p h i c a l g e n e r a l i t y . (5; 100) However named

Mobius

wrote

Drobisch

would write that the

who

a r e v i e w (he

Engel

(5;

recorded

had a

contacted

philosopher

d i d not).

Mobius

a

mathematician

and

hoped

that

he

went on to recommend

Friedrich

100-101)

an

illuminating

Apelt

Jena, and M o b i u s . to

he

also

best procedure m i g h t be for Grassmann h i m s e l f to p u b l i s h

a r e v i e w (he did!).

Ernst

that

was

exchange

(1812-1859),

of

professor

letters of

between

philosophy

at

On September 3,1845, Apelt wrote the following

Mobius: Have only

you

from

read

Grassmann's

Grunert's

mathematics

lies

Archiv;

at

matical knowledge,

its its

it

strange

seems

foundation.

to The

intuitiveness

Ausdehnungslehre?

me that a false essential

I

know

it

p h i l o s o p h y of

character of mathe-

(Anschaulichkeit),

seems

to

have

b e e n e x p e l l e d f r o m the work. Such an abstract theory of extension as he seeks c o u l d o n l y b e d e v e l o p e d f r o m concepts. B u t t h e source o f m a t h e m a t i c a l k n o w l e d g e lies n o t i n c o n c e p t s b u t i n i n t u i t i o n . (5; Mobius

in a letter dated January 5,

You this

I

ask

me

whether

I

have

101)

1846, responded:

read

Grassmann's

Ausdehnungslehre.

To

answered that Grassmann h i m s e l f presented me w i t h a copy and

that I have on n u m e r o u s occasions a t t e m p t e d to study it b u t have never gone

b e y o n d the first sheets,

since,

as

you

mentioned,

intuitiveness

(Anschaulichkeit), the essential character of mathematical thought, is not to be found in the work. H o w e v e r I have been forcing myself through the w o r k by s k i m m i n g it on n u m e r o u s occasions in regard to extension or g e n e r a l i t y a s y o u w o u l d p r e f e r t o c a l l it. F r o m t h i s I h a v e c o m e t o f e e l that it can be influential in a g o o d w a y for mathematics, especially in regard to the

systematic presentation

o f its

elements. To this belongs

the a d d i t i o n a n d s u b t r a c t i o n o f lines w h e n these are c o n s i d e r e d not o n l y i n r e l a t i o n t o t h e i r l e n g t h b u t also t h e i r d i r e c t i o n . (5; Partly on

the

basis of the above letters a n d on the translation of

"Anschaulichkeit" major reason "the

for

as

the

contemporary

intuitiveness

Ernest

Nagel

To

Nagel's

views

suggested that a

poor reception of Grassmann's

scene

was

dominated by

i n d i s p e n s a b i l i t y of i n t u i t i o n for m a t h e m a t i c s .

Writing in

101)

may

be

added

the

work

was

Kantian views .

.

on

that the

."54

opinions

of A.

E.

Heath.

1917, H e a t h a r g u e d that a m a j o r reason for t h e p o o r re-

ception was the philosophical

nature of the book, a condition com-

p o u n d e d by a reaction against a philosophical f o r m of presentation that h a d set in a m o n g m a t h e m a t i c i a n s

as

a result of " t h e exaggera-

tions of the metaphysical unification of k n o w l e d g e in the schools of Schelling

and

Hegel."

5 5

In

any

case

Grassmann's

book

provoked

79

A H i s t o r y of V e c t o r Analysis

the

following

comment

from

Heinrich

Mobius had recommended the book: to enter into those thoughts; fore

my

eyes

Baltzer:

when

I

Richard

Baltzer,

to

whom

" i t is not n o w possible for me

I b e c o m e d i z z y a n d see s k y - b l u e be-

read them."

(5;

102)

Mobius wrote back to

"If, as you write me, you have not relished Grassmann's

Ausdehnungslehre,

I

reply

that

I

have

the

same

experience.

I

like-

w i s e h a v e m a n a g e d to get t h r o u g h no m o r e t h a n the first t w o sheets ( B o g e n ) o f h i s b o o k . " (5;

102)

Grassmann had sent a copy of the w o r k to Johann August Grunert, founder

and

Grunert

editor

encountered

of

the

serious

Archiv

der

difficulties

Mathematik

und

Physik.

in reading the work and

requested Grassmann to write a r e v i e w of his book for publication in

t h e Archiv a n d t o g i v e s o m e s i m p l e e x a m p l e s .

Grassmann did, and in der

appeared.56

Ausdehnungslehre"

book

was

neither

understanding stated,

completely

the

102-103) This

work.

It

Grassmann's

representative

was

s u m m a r y of his

nor

however, as

a

major

other publication

Elektrodynamik,"57 f o u n d through his fate

of

his

contained

of

an

n e w system

1845,

his

important

aid

to

Grassmann himself

the only r e v i e w of his book that was published.

Grassmann's

the

(5;

1845 his " K u r z e U b e r s i c h t u b e r das W e s e n

(4,1,11; 3 )

" N e u e Theorie der electrical

discovery

of mathematics. This paper shared

Ausdehnungslehre;

the

significance

of

Grassmann's

d i s c o v e r y was o n l y r e a l i z e d after C l a u s i u s in the 1870's p u b l i s h e d the same result, w i t h o u t of course k n o w i n g of Grassmann's earlier discovery.

(5;

104-105)

T h e o n l y recognition that Grassmann r e c e i v e d at this t i m e for his mathematical "Die

investigations

Geometrische

February 2,

1845,

to

Jablonowskischen for the creation system

for the

Grassmann prize

in

offered

notify

in

regard

Mobius

him

Gesellschaft

of an

to

a

had written

paper

entitled

Grassmann

on

1844 a n n o u n c e m e n t of the

der Wissenschaft

offering

a prize

of a s y s t e m s i m i l a r to (or for t h e e x t e n s i o n of) t h e

sketched

published

came

Analyse."

by

Leibniz

first

the

time

rather

for the

in in

his

letter

1833.

delightful

completion

(5;

to

Huygens

109)

position

Mobius'

of

1679,

letter put

of being aware

of a

of a task j u d g e d significant by a

scientific g r o u p , w h i c h task G r a s s m a n n h a d already a c c o m p l i s h e d . Grassmann gekniipft teristik. prize

und The to

die

sis;

80

came

to

von

Leibnitz

Jablonowskische

Grassmann's

appendix by

of

thus

write

his

Gesellschaft

work58

and

Die

Geometrische

erfundene in

published

geometrische 1846 it

in

Analyse Charak-

awarded 1847

with

the an

Mobius.59 T h e w o r k does not d e m a n d detailed analy-

it m a y be noted that Grassmann i n c l u d e d a thorough treatment his

inner

product

(the

modern

dot

product),

which

he

had

Other Early Vectorial

neglected

to

do

in

his

Ausdehnungslehre

of

1844.

Systems

Though

this

paper had major defects in presentation, it was more readable than his

Ausdehnungslehre;

neglect. In

(5;

its

fate

however

was

the

same:

colossal

111-118)

1847 Grassmann attempted to secure a university position. He

wrote to with

Eichhorn, the

his

letter

Prussian minister for culture, and i n c l u d e d

copies

of

his

principal

mathematical

Eichhorn asked the noted mathematician report on

the

Kummer's about

significance

Grassmann's

Kummer's

lack

was

of

less

than

others

a

month. as

was

In a n y case

great

mixed

clarity.

negative judgment

and

greatness

of Grassmann's

report scant praise

mathematical

works.

Probably

a

major

K u m m e r joined the who

achievement.

In

with repeated statements cause

that K u m m e r had the

mathematicians

of Grassmann's

writings.

Ernst Eduard K u m m e r t o

failed

Thus

to

of

works

ranks

for

of Gauss

appreciate

the

no professorship was

o f f e r e d , n o r i n fact w a s o n e e v e r o f f e r e d t o G r a s s m a n n . (5; 1 2 3 - 1 3 0 ) Three

mathematicians

take

notice

they

were

of

of this

Grassmann's

Adhemar

period

work

Barre,

were

forced,

because

Comte

de

of

as

it were, to

priority

questions;

Saint-Venant

(1797-1886),

Augustin C a u c h y (1789-1857), a n d Sir W i l l i a m R o w a n H a m i l t o n . Saint-Venant was elasticity, w h o in sommes

et

simplifier ideas

les la

differences

similar

his

or

7

identical

system

paper

engineer,

noted for his researches in

geometriques,

Mecanique."

Grassmannian began

a French

1845 p u b l i s h e d a p a p e r e n t i t l e d " M e m o i r e sur les

by

This to

and to

some

modern

stating:

n u m b e r w h a t e v e r of lines

paper

"I

a,

b,

et

sur

leur

contained

ideas

fundamental

vector analysis.

call

the

usage

to

the

Saint-Venant

geometrical

c, . . . g i v e n

pour

mathematical

sum

of

any

in m a g n i t u d e , direc-

tion, a n d sense a l i n e w h i c h is e q u a l a n d parallel to t h e last side of a polygon

of w h i c h

end,

in

each

its

I = a + b + c . the

.

.

"difference

ferentielle

geometriques."

call

other

."

the

sides are a, b, c, .

sense.

(7;

620)

If

(7;

areas

620)

This

product

or

and

and of the

geometrical

I

is

the

last

. . p l a c e d e n d to side,

then

I

write

S a i n t - V e n a n t t h e n p r o c e e d e d to d e f i n e

geometrique,"

geometrique";

sum of plane I

the

proper

vector the

was

subtraction;

"coefficients

followed

by a

the

"dif-

differentiels

definition

of the

"produit geometrique." (of

a

line

b

multiplied

by

a

line

a,

designated a b) the area obtained, b o t h in m a g n i t u d e a n d d i r e c t i o n , in forming a parallelogram from those t w o lines d r a w n from the same point. T h e positive face is that on w h i c h a is on the left a n d b is on the right. T h u s a a — 0 a n d b a = —a b. I

call

the

geometrical

product

of

an

area

multiplied

by

a

line

the

v o l u m e of t h e p a r a l l e l e p i p e d (or t h e o b l i q u e p r i s m ) h a v i n g t h e area for a base a n d the sides e q u a l a n d parallel to the g i v e n line. T h e v o l u m e is

81

A

History

of V e c t o r Analysis

c o n s i d e r e d n e g a t i v e w h e n t h e sides are o n t h e n e g a t i v e side o f t h e base. a b c w i l l d e s i g n a t e t h e p r o d u c t of t h e area b c m u l t i p l i e d by t h e l i n e a. (7; 6 2 1 ) Thus

Saint-Venant

had

discovered

vectorial

addition,

subtrac-

tion, differentiation, a n d also a m u l t i p l i c a t i o n similar to the m o d e r n cross

product, the major difference b e t w e e n t h e m b e i n g that Saint-

Venant's

product was,

spatially

oriented

equations brief

Grassmann's,

Saint-Venant

not another vector, but a

then

stated

that

geometric

can be added, subtracted, and multiplied and made very

mention

vectors.

like

area.

(7;

of

the

possibility

of

integration

and

division

for

621) T h e ideas m e n t i o n e d to this point w e r e treated in

t w o pages by Saint-Venant; thus his paper is best seen as a sketch of the fundamentals of a vector analysis. pages

of the

paper

suggested

how

The

final

his

ideas

paper

nor

three and one-half

could

be

applied to

mechanics. Though mann

neither

question

as

letter does gan

to

to

out

to

his

what

contain

work

response that

Saint-Venant's

his

letters

to

Grass-

(those p u b l i s h e d by Engel) allow us to answer the interesting motivated

Saint-Venant

to

his

information concerning when

his ideas.

In

a

letter

(quoted

initial

ideas

came

a letter of July earlier)

in

1832,

in

creation,

one

Saint-Venant be-

17,

which

1847, w r i t t e n in

Grassmann

stated

Saint-Venant commented:

"It

was thus around

1832 that I first came to the idea of extending the

use

signs

of algebraic

to

those

geometrical

operations

which

are

p e r f o r m e d on lines a n d areas in m e c h a n i c s , b u t I p u b l i s h e d n o t h i n g until

1 8 4 5 . " (5;

and motivation

122) A l t h o u g h the q u e s t i o n of t h e sources of ideas for Saint-Venant m a y not be a n s w e r e d definitively,

some indication as to the possible sources m a y be given. It is very improbable

that

Grassmann's

he

was

stimulated

by

the

work

of Mobius

or

of

father, since Saint-Venant had great difficulty in read-

ing German.60 Since the writings of Argand, Servois, Buee, Mourey, Warren,

and

numbers

on

the

geometrical

published

before

representation

to

which

presumably

Saint-Venant

had

but

direct

evidence

lacking,

Saint-Venant

is

1832

stimulated by one or more of these men.61

and

of complex

all

languages possible,

Gauss

were

that

were

written

access,

it

in is

was

Saint-Venant's influence

on later writers was at most v e r y small.62 Grassmann April a

18,

letter

had

heard

for

Saint-Venant

taneously

Grassmann

dehnungslehre

and

copy

82

of Saint-Venant's

paper by

1847,

and on

1847, G r a s s m a n n w r o t e to A u g u s t i n C a u c h y a n d enclosed

of the

book

whose

mailed

requested to

address

to

Cauchy

Saint-Venant.

he

Cauchy to (5;

did two

give

not know. copies

the

letter

120-121) T h e

of

Simulhis

and

Ausone

part of Grass-

Other Early Vectorial

Systems

mann's letter w h i c h contained statements concerning the history of his discoveries has already b e e n q u o t e d ; t h e r e m a i n d e r i n c l u d e d a statement

of priority and a discussion

developed inner

in

his

Ausdehnungslehre

of some

(in

materials

particular,

the

not fully linear

or

product).

On July

17,

received the

1847, Saint-Venant w r o t e to G r a s s m a n n that he h a d letter but

send h i m the book.

not the book;

(5;

he requested Grassmann to

122) G r a s s m a n n a s s u m e d that C a u c h y h a d

been delayed in giving the book to Saint-Venant, and on January 27,

1848, he w r o t e to Saint-Venant a n d sent h i m a c o p y of his " D i e

Geometrische Analyse," along w i t h another paper. Grassmann d i d not

send

the

his

copy

mann's

Ausdehnungslehre

sent

"Die

to

him

nor

did

Geometrische

Analyse"

t h e r w i t h G r a s s m a n n a t t h i s t i m e . (5; In

Cauchy

for Saint-Venant.

give

Saint-Venant

Saint-Venant read Grass-

but d i d not correspond fur122)

1853 another priority question was raised by a p u b l i c a t i o n of

that

year

in

entitled "clefs

the

"Sur

Comptes

les

rendus,

clefs

algebriques"

authored by A u g u s t i n

algebriques."

or

algebraic

63

keys

Cauchy,

Cauchy's was

paper

primarily

on

and his

directed

toward p r o v i d i n g m e t h o d s for dealing w i t h algebraic problems, in general, the

finding

of u n k n o w n s in equations. A v e r y s i m p l e ex-

a m p l e w i l l i l l u s t r a t e b o t h h i s m e t h o d a n d its r e l e v a n c e . G i v e n t h e equations

x + 3y = 11

algebraic

keys,

i

and

a n d j,

Ax + 2y = 14,

which

behave

we

so

introduce

two

that i • i = j • j = 0

may

and

i - j = —j - i. We t h e n m u l t i p l y e a c h e q u a t i o n by o n e k e y a n d o b t a i n xi + 3yi = 1 1 i

and 4xj + 2yj =

14j.

we

equation

f o r m Ax + By = K, w h e r e A = i + 4 j ,

obtain

an

B = 3i + 2 j , (Ax + laws

By)B as

and

=

i

K =

K(B).

a n d j,

of t h e

l l i + 14j.

If we n o w a d d these equations,

From

this

we

S i n c e A, B, a n d K o b e y t h e the

above

equation

obtain

the

same

equation

multiplication

b e c o m e s ABx = KB, w h i c h c a n

KB be transformed into x =

=

(lli +

T h u s we obtain

AB

1 4 ; ) ( 3 t + 2jy)

=

(» + 4 j ) ( 3 < + 2 j ) The

significance

equivalent to

of this

2ij + 1 2 j i

is

that

Cauchy's

- 2 0ij

=

nearly

=

-10»j keys

are

Grassmann's extensive magnitudes

latter's outer m u l t i p l i c a t i o n . Ausdehnungslehre

( 2 2 ij + 4 2 \ j i )

algebraically

in regard to the

M o r e o v e r Grassmann d e v e l o p e d in his

identical

algebraic

methods.65

Grassmann probably first learned of Cauchy's publications in the following way. ing

the

B a l t z e r o n J u n e 14, 1853, w r o t e t o M o b i u s c o n c e r n -

identity

those

of Grassmann.

Mobius then wrote to Grassmann on September 2,

of Cauchy's

methods

with

1853, to i n f o r m

83

A H i s t o r y of V e c t o r Analysis

h i m of Cauchy's publication and to suggest that Grassmann make a c l a i m for priority. Grassmann "De

(5;

(5;

172-175) At the same time Mobius informed

172)

of a

Interpretation 66

determinants/' Cauchy's this,

to

those

keys

could

made he

be

des

interpreted

mathematical

Saint-Venant

though

of 1853 by

Saint-Venant entitled

clefs

algebriques

et

des

T h e aim of Saint-Venant's paper was to show h o w

show the

Venant

paper

geometrique

no

had

presented

priority

mentioned

geometrically and,

in

relationship of Cauchy's

claim

in

for

Grassmann's

his

paper

himself linear

or

of

or

1845.

for

inner

doing

ideas

to

Saint-

Grassmann, product

in

a

footnote. On

February

19,

1854, G r a s s m a n n w r o t e to M o b i u s that illness

h a d d e l a y e d h i m in seeing Cauchy's papers, b u t that he h a d finally managed to travel from

Stettin to Berlin to read them. Grassmann

had decided to claim priority through a letter to the F r e n c h Academy and through a publication

i n C r e l l e ' s Journal.

(5;

176-182) His

letter to the French A c a d e m y was read on April 17,1854, and stimulated the A c a d e m y to form a committee to investigate the priority question.

The

C a u c h y (!);

committee

was

no decision was

composed

haps because of Cauchy's death in paper les

for

C r e l l e ' s Journal

differents

genres

Lame,

Binet,

and

M a y , 1857. (5; 198) G r a s s m a n n ' s

appeared

de

of

h a n d e d d o w n by the committee, per-

in

1855

under

multiplication."67

the

Herein

title

"Sur

Grassmann

claimed priority over Cauchy and Saint-Venant and published some new of

results,

in

particular the

definition

of sixteen

different kinds

multiplication. On

December

16,

1856,

S a i n t - V e n a n t ( p r o b a b l y a t C a u c h y ' s re-

quest) wrote to Grassmann that he had never received the copy of the

Ausdehnungslehre

and

had

been

p l a i n e d that this fact was the

unable

to

find

a

copy.

He

ex-

cause of his slighting Grassmann in

his paper of 1853, a n d he requested Grassmann to i n f o r m h i m as to how

a

copy

of

the

Ausdehnungslehre

might

be

obtained.

(5;

199-

200) G r a s s m a n n r e p l i e d in a letter of M a r c h 28, 1857, a letter that is

typical

of Grassmann's

patience

and

good

nature.

Grassmann

stated that his book was in the library of the French Institute, but w r o n g l y classified; copy

of

the

that he had b e l i e v e d Cauchy had passed on the

Ausdehnungslehre

to

Saint-Venant,

so

that

another

copy had not b e e n sent directly to h i m ; and that he was sending to Saint-Venant a copy of his book and of a paper and was translating parts of b o t h so that Saint-Venant c o u l d read t h e m m o r e easily. He also a s k e d S a i n t - V e n a n t t o pass o n t h e l e t t e r t o C a u c h y a n d t o t e l l C a u c h y of his

high respect for him.

from Saint-Venant.

84

(5; 2 0 0 - 2 0 1 )

There

is

no record of a reply

Other Early Vectorial

In

Grassmann's

Cauchy:

Ausdehnungslehre

of

1862

he

stated

in

Systems

regard

to

"I have no intention of accusing the famous mathematician

o f p l a g i a r i s m . . . . " (4,1,11; 9 - 1 0 ) V i c t o r S c h l e g e l h o w e v e r i n 1 8 7 8 explicitly

accused

publication believed Engel

that

he

that or

strongly

of plagiarism

went

too

Cauchy

did

not

criticized

(6;

38-39),

his judgment.68

reversed

Schlegel

argued

Ausdehnungslehre ever

Cauchy

of 1896

far

in

probably

remember

Cauchy

for

his

charge

either its

though

Friedrich

a

of plagiarism.

did

not

contents.

not

in

Engel

read

Engel

answering

the how-

Grassmann's

p r i o r i t y c l a i m . (5; 2 0 2 ) Another alternative may h o w e v e r be suggested. Cauchy was w e l l acquainted

with

Saint-Venant's

paper

of

1845

and

used

results

f r o m it in a p a p e r of 1849 e n t i t l e d " S u r les Q u a n t i t e s g e o m e t r i q u e s , et sur u n e m e t h o d e n o u v e l l e p o u r la resolution des equations algebriques

de degre

quelconque."

69

M o r e o v e r this

paper shows that

Cauchy had a good k n o w l e d g e of the w o r k of Argand, Servois, and Buee

on

complex

numbers;

and papers by C a u c h y of 1853 make M o b i u s ' works.70 T h u s

clear that C a u c h y k n e w of H a m i l t o n ' s

and

it

knowledgeable

can

be

suggested

works

done

charge

brought by

part

of the

"clefs

that Cauchy

independently

of

Grassmann

methods

was

published

algebriques."

was

Grassmann

by

and

that

appropriate Cauchy

in

only his

in

relevant

the in

priority

regard

papers

on

to his

It was perfectly possible for C a u c h y to d r a w

on the sources m e n t i o n e d above for h e l p in creating his methods. A residuum of originality of course w o u l d remain, and it is in regard to that r e s i d u u m that a priority dispute w o u l d be in order. Saint-Venant and Cauchy were not the only mathematicians w h o encountered Grassmann

firmly

lodged in a d o m a i n that they had

previousJy v i e w e d

as

Sir W i l l i a m R o w a n

Hamilton's surprise w h e n in some unrecorded

way

he

heard

of 1844. his

their o w n

of the

Stettin

discovery;

great m u s t have b e e n

schoolmaster

and

his

Ausdehnungslehre

P r o c u r i n g a c o p y of t h e b o o k , H a m i l t o n set to r e a d i n g it;

interesting c o m m e n t s are p r e s e r v e d in letters, in the m a r g i n s

of his copy of Grassmann's book, a n d in the historical preface to his Lectures it,

on

Quaternions.

The

latter

comments,

as

comprised (with the exception of Mobius'

fate

would

have

comments) the only

p u b l i s h e d d i s c u s s i o n o f G r a s s m a n n that a p p e a r e d b e f o r e t h e 1860's. It was

in late

historical a

letter

very

1852 that H a m i l t o n , w h o was t h e n p r e p a r i n g the

preface of October

original

to

his

26,

1852,

work. . . . which

Lectures,

read

Hamilton work,

the

wrote

if any,

the

Ausdehnungslehre. to

De

Morgan:

Germans,

In "a

if t h e y

t h i n k m e w o r t h n o t i c i n g , w i l l p e r h a p s set u p i n r i v a l s h i p w i t h m i n e , but which

I d i d not see till l o n g after m y o w n v i e w s w e r e f o r m e d

85

A H i s t o r y of V e c t o r Analysis

and

published."

book

(8;

424)

Hamilton continued to read Grassmann's

a n d on January 31,

I

1853, w r o t e the

following to

De

Morgan:

h a v e r e c e n t l y b e e n reading ( a n d i t i s c u r i o u s t h a t s o m e t i m e s , w h e n

o t h e r w i s e in m e n t a l activity, I seem to m y s e l f unable to read a page, or almost a sentence of G e r m a n ) m o r e t h a n a h u n d r e d pages of Grassmann's Ausdehnungslehre, only the

with

most

great

admiration

slight and

and

interest.

Previously

I

had

general knowledge of the book, and thought

t h a t i t w o u l d r e q u i r e m e t o l e a r n t o smoke i n o r d e r t o r e a d i t . I f I c o u l d h o p e to be put in rivalship w i t h D e s Cartes on the one hand, and w i t h Grassmann on the other, my scientific ambition w o u l d be fulfilled! But i t i s c u r i o u s t o see h o w n a r r o w l y , y e t h o w c o m p l e t e l y , G r a s s m a n f a i l e d to hit off the Quaternions. He p u b l i s h e d in 1844, a little later t h a n m y self, b u t w i t h t h e m o s t o b v i o u s a n d p e r f e c t i n d e p e n d e n c e . (8; 4 4 1 ) H a m i l t o n ' s c o m m e n t s as

given in a letter of February 2,

1853, are

I am n o t q u i t e so e n t h u s i a s t i c t o - d a y a b o u t G r a s s m a n n as I was w h e n I last w r o t e . B u t I h a v e r e a d t h r o u g h n e a r l y all of w h a t I c o u l d p r o c u r e of his

writings,

Mobius.

including

Grassmann

is

a a

subsequent great

and

commentary

(in

German)

most German genius;

space i s a t l e a s t a s n e w a n d c o m p r e h e n s i v e a s

by

his v i e w of

m i n e o f time; b u t h e h a s

not a n t i c i p a t e d , nor attained t h e c o n c e p t i o n of, the

quaternions, e v e n s o

nearly as I guessed he m i g h t have done, f r o m a notion hastily taken up, of w h a t

might

have

been

know

even

now),

quote

from

memory.

His

and

is

understand;

in

his

that

his

meaning

doctrine outer saying

of

( a n d w h a t i t was,

"eingewandte

products

(aiissere)

something

I

very d i m l y

multiplikation." I

think

that

for a person w h o

I

has

I do

not

l e a r n e d t o s m o k e . A n d e v e n h i s inner p r o d u c t s , p u b l i s h e d s u b s e q u e n t l y to

the

outer

ones

(in

1847),

I

"inner products"

of Grassmann

of a

and

quaternion,

notion

o f combining

his

them

can

swallow

have

much

"outer products"

h a d occurred to

pretty well. analogy to to

him,

my he

In

my

fact,

" v e c t o r parts." might

the

"scalar parts''

have

If the

been

led

to the quaternions; b u t those he seems to me to have altogether failed to perceive. have

a

Yet

I

better

t h i n k that chance

my o w n

of b e i n g

researches,

appreciated i n

or

these

speculations, w o u l d countries,

if readers

h a d first b e e n p u t t h r o u g h a sufficient course (or dose) of G r a s s m a n n . I m u s t say that I s h o u l d n o t fear t h e c o m p a r i s o n . Y o u tolerate e g o t i s m in correspondence . . . . De

Morgan

Christian

had

name

was

(8; 4 4 2 ) asked

Hamilton

the joke a n d e m b e l l i s h e d it in his if you

have

any curiosity

Nebuchadnezzarological shoulder is

at

Nebuchadnezzar.

to

know

reading

amused at the

one

point

if

Grassmann's

(8; 4 2 5 ) H a m i l t o n p i c k e d u p

letter of February 9,

1853:

anything of the result of my recent (my

daughter

looking

over

my

folly of philosophers), it w i l l be quite con-

s i s t e n t w i t h m y h u m o u r t o i n f o r m y o u . T o t h e p u b l i c I a m l i k e l y t o say b u t little a t p r e s e n t a b o u t G r a s s m a n n ; f o r I

find

adding

independently

lines,

whereas

I

except one

86

which

took

he

seems

to

have

that b e y o n d the rule for worked

out,

it from Warren, we have scarcely a result in c o m m o n ,

thing which

is

(in

my view)

important,

namely, the inter-

Other Early Vectorial

pretation He

of B

comes

to

preparations, knowing

— A,

where A and B

this,

in

and

his

page

d e n o t e points, a s t h e directed line AB.

139

of the

ostrich-stomach-needing

nothing

of

this

result,

Systems

as

in

Ausdehnungslehre, iron

any

after

previous

way

arrived

long

doses. at

by

I,

him,

S T A R T E D w i t h the same interpretation i n m y Lectures, i n 1848, h a v i n g printed

the

same

conception

some

years

earlier,

and

f a m i l i a r w i t h i t (see P u r e T i m e ) f o r a long t i m e b e f o r e . Hamilton's

published

his

on

Lectures

It

is

statement

Quaternions

proper

to

state

tion f o r i n c l i n e d l i n e s and

remarkable

Leipzig, from

here, by

which

invention

concerning

1853

is

that

(aiissere

work

1844),

the

of

as

a

having

been

(8; 4 4 4 )

Grassmann

given

in

follows:

species

o f non-commutative

multiplica-

M u l t i p l i k a t i o n ) occurs in a v e r y original

Prof.

H.

Grassmann

(Ausdehnungslehre,

I d i d not m e e t w i t h till after years h a d e l a p s e d

and

communication

of the

quaternions:

in

which

w o r k I have also n o t i c e d ( w h e n too late to a c k n o w l e d g e it e l s e w h e r e ) an

employment

(Strecke),

of

drawn

the

from

symbol

the

point

— a, a

to

to

the

denote

p o i n t ($.

the

Not

directed

line

withstanding

these, a n d perhaps some other coincidences of v i e w , Prof. Grassmann's system and m i n e appear to be perfectly distinct and i n d e p e n d e n t of each other, in their c o n c e p t i o n s , m e t h o d s , a n d results. At least, that the profound and philosophical author of the Ausdehnungslehre was not, at the time

o f its

publication,

in

possession

of the

theory

of the

quaternions,

w h i c h h a d in the p r e c e d i n g year (1843) b e e n a p p l i e d by me as a sort of organ

or

calculus for

spherical

trigonometry,

seems

clear

from

a

passage

o f his Preface ( V o r r e d e , p . xiv.), i n w h i c h h e states ( u n d e r date o f J u n e 28th,

1844),

imaginairies

that

from

he

the

had plane

not to

then

succeeded

space;

and

in

extending

generally

the

use

of

unsurmounted

difficulties had opposed themselves to his attempts to construct, on his principles, moglich,

a

theory

of

vermittelst

des

angles

in

space

Imaginaren

(hingegen

auch

die

nicht

mehr

Gesetze fur den

ist

es

Raum

abzuleiten. A u c h stellen sich iiberhaupt der Betrachtung der W i n k e l im Raume

Schwierigkeiten

entgegen,

zu

deren

allseitiger

Losung

mir

n o c h n i c h t h i n r e i c h e n d e m u s s e g e w o r d e n ist).71

Finally,

a remark

September

30,

mann's

1855

recent

researches,

worthy to and not." By

it

article

have

appears (8;

by

may

in

Hamilton

in

be

After

cited.

a letter to J. mentioning

T.

Graves of

that

Grass-

Crelle's journal had stimulated h i m in some

Hamilton

stated

that

Grassmann

"was

well

anticipated me in the discovery of the quaternions; to

me

a very

remarkable

circumstance

that he

did

70)

1860

one

mathematician one

made

1856,

French

mathematician of

the

of the

British

German-speaking

mathematician

(Saint-Venant)

Isles

(Hamilton), one

countries

(Mobius),

and

had come to appreciate

to some extent Grassmann's work. T w o other mathematicians, both of

Italy,

complete

Cremona and

this

Giusto

small

group.

These

two

men

were

Luigi

Bellavitis.

87

A

History

In

of V e c t o r Analysis

1860

mathematiques had

Luigi a

been

Cremona

note

proposed

a brief exposition

with

the

following cannot

the

in

solution

that journal.72

in

gave

"... I

published

concerning

As

of Grassmann's

Nouvelle

of t w o

part of this

Annales

de

problems

that

note Cremona

ideas, w h i c h was prefaced

statements:

refrain

from

mentioning

a

very

expeditious

and

very

c u r i o u s m e t h o d , o f w h i c h t h e first i d e a seems t o b e l o n g t o L e i b n i z , b u t w h i c h has b e e n t r u l y e s t a b l i s h e d by G r a s s m a n n . E x c e p t for M o b i u s . . . and

Bellavitis . . . I

do

not

know

of any

geometers

who

have

given

Grassmann's researches the attention w h i c h they deserve. I w i l l here r e p r o d u c e the p r i n c i p l e definitions a n d conventions of this ingenious At

a

later

theory time

Grassmann least o n e

which

Bellavitis and

by

one

Bellavitis,

a

names

even

contact

looks

letters that

back

independently arrive

on It

but

9,

their

ideas

1853,

schneider,

who

(5;

and

163n);

copies (4,1,1;

praise

ideas

Italian

and

Grassmann by

went

from

the

of

in at

during the

the

latter

with

on

to

study

the

Bellavitis

in

1860

and

mathematician

intended

the

decades

before

is

striking

first

Grassmann,

to

was

acquaint

imhis

1860,

of

all

a number of that

Mobius,

Saint-Venant, and Cauchy should

a t ideas that w e r e i n m a n y cases similar. T h e

brilliance

was

in nearly every instance

primarily

algebraic.

Though

had already obtained m u c h attention, Grassmann

remained nearly unknown. June

in

73

106-107)

emerge.

Hamilton,

Hamilton's

analysis."

words

paper

Bellavitis

Ausdehnungslehre (5;

with

motivation b e h i n d their investigations was geometric,

geometrical

stronger

geometrical

issue.

learned

the

generalizations

into

was

through

c o u n t r y m e n w i t h it. W h e n

used

came

took

Grassmann

pressed

author

books.74

occasion

Ausdehnungslehre, 1862

Cremona

Bellavitis the

the

335) and i n c l u d e d some of Grassmann's

of his

Giusto 1850's;

(5;

which

that

he

had as

I n d e e d M o b i u s m e n t i o n e d in a letter of

knew

read

of

Grassmann

of

the

first

18)

The

factors

only

one

Grassmann's

completely

himself mentioned,

Ausdehnungslehre that

mathematician,

book

were

caused

this

used

neglect

the

as have

Bret-

through

remaining

wastepaper. been

amply

discussed. Finally a summary will

fill

1844 to

of Grassmann's

1861

in this period

mathematics

From

G r a s s m a n n p u b l i s h e d seventeen scientific papers, in-

cluding important papers textbooks.

also

published

was

done

88

other activities

out the picture of Grassmann as a very active person.

materials

while

he

in physics, and a n u m b e r of language and He

edited

on the

taught

a

a

political paper for a t i m e a n d

evangelization

heavy

load

and

of China.75 All this raised

a

family,

for

Other Early Vectorial

Grassmann children This

had

married

p e r i o d of his

and

from

this

union

eleven

life

was

concluded w i t h the publication of his

Ausdehnungslehre.

VIII.

Grassmanns Limited

On

1849,

came.

second

Ausdehnungslehre

Acceptance

October 31,

high,

in

Systems

sent

of

His

of

1862

and

1861, Grassmann, n o d o u b t w i t h

to

Mobius

a

the

Gradual,

Work.

copy

of

the

hopes running

second

Ausdehnungslehre.

E i g h t years earlier in a letter to the same c o r r e s p o n d e n t G r a s s m a n n had stated his intention of preparing a n e w w o r k ; there is e v i d e n c e that by 1854 or 1855 he h a d b e g u n writing. of the book bore the date mann's

brother,

was

Die

bearbeitet.

and were

paid for by the

Ausdehnungslehre:

Under

the

Vollstanding

Grassmann's

a m G y m n a s i u m TAX S t e t t i n . " In

foreword

(5; 2 2 3 ) T h e 3 0 0 c o p i e s

1862, w e r e p r i n t e d in the shop of Grass-

name

author. und

appeared

(5;

223) Its title

in

strenger

the

title

Form

"Professor

76

Grassmann

discussed

the

poor

reception

ac-

corded his earlier w o r k and stated that the content of the n e w book was presented in

"the

strongest mathematical form that is actually

k n o w n to us; this is the E u c l i d e a n . the that

content of the of Gauss,

book and

Mobius,

.

Bellavitis,

For

I

. " (4,1,11; 4 ) H e e l a b o r a t e d o n relation

Saint-Venant,

final paragraph in the foreword was

pended

.

discussed the

of his

the

science

presented

The

the following:

remain completely confident that the labor w h i c h on

work to

and Cauchy.

here

and

which

has

I

have ex-

demanded a

significant part of my life as w e l l as the most strenuous application of m y p o w e r s , w i l l not b e lost. I t i s t r u e that I a m a w a r e that t h e f o r m w h i c h I have given the science is imperfect and must be imperfect. B u t I k n o w a n d f e e l o b l i g e d to state ( t h o u g h I r u n t h e risk of s e e m i n g arrogant) t h a t even

if this

work

s h o u l d again r e m a i n u n u s e d for another seventeen

years o r e v e n l o n g e r , w i t h o u t e n t e r i n g i n t o t h e actual d e v e l o p m e n t o f science, still that t i m e w i l l come w h e n it w i l l be brought forth f r o m the dust of oblivion

a n d w h e n ideas n o w d o r m a n t w i l l b r i n g forth fruit.

I

k n o w that if I also fail to gather a r o u n d me in a p o s i t i o n ( w h i c h I h a v e up to n o w desired in vain) a circle of scholars, w h o m I c o u l d fructify w i t h these ideas, a n d w h o m I c o u l d stimulate to d e v e l o p a n d e n r i c h further these ideas, nevertheless there w i l l c o m e a t i m e w h e n these ideas, perhaps i n a n e w f o r m , w i l l arise a n e w a n d w i l l enter into l i v i n g c o m m u n i cation w i t h contemporary developments. For truth is eternal and divine, and no phase in the d e v e l o p m e n t of truth, h o w e v e r small m a y be the r e g i o n e n c o m p a s s e d , c a n pass o n w i t h o u t l e a v i n g a t r a c e ; t r u t h r e m a i n s , even

though

the garment in w h i c h poor mortals

clothe

it may fall to

d u s t . (4,1,11; 1 0 )

89

A

History

As the

of V e c t o r Analysis

Grassmann Euclidean.

with

himself pointed out, The

a m i n i m u m

banished (which

consisted

of comment.

(which

was

book

not

was

form of presentation was of theorems

presented

Philosophical commentary had been

helpful),

helpful).

the

o f sets

as

were

Concerning

physical

this

form

of

applications presentation

w h i c h G r a s s m a n n h a d also u s e d in an e l e m e n t a r y textbook of 1861, Lehrbuch

der

Arithmetik,

Engel

commented:

W i t h o u t a d o u b t this was a disastrous mistake. W h a t was perfectly in place in the treatment of a subject so c o m m o n p l a c e for all mathematicians

as a r i t h m e t i c , at least for readers w h o p u r s u e m a t h e m a t i c s as a

science, to

was

which

the

the

most

unsuitable

reader was

form

for the

of presentation

first b e i n g

for

introduced.

a subject

Though

this

form of presentation led to an admirable codification of the n e w concepts a n d laws of his theory of extension, still it was not a presentation likely to

win

followers

wished

to

read

undertaken work

the

because

for his

his

ideas,

first

let

alone

Ausdehnungslehre.

convert those

Grassmann

who

had

had

actually

not only

i m m e n s e task i n v o l v e d i n the c o m p o s i t i o n o f the n e w

he

h o p e d t h a t n o w a t last h e w o u l d

find

readers.

It is

actually puzzling that he could so deceive himself and be so mistaken in his c h o i c e o f t h e m e a n s f o r a t t a i n i n g his a i m . (5; 2 3 1 ) The

second

first: it was most

Ausdehnungslehre

important of these

"Pfaffian

was

more

than

a

reworking

of the

o n e - t h i r d longer a n d c o n t a i n e d m a n y n e w results. T h e

Problem."

appreciated after

was

Grassmann's

According

1877,

to

solution of the

Engel

a neglect which

this

solution

so-called was

only

E n g e l ascribed to the fact

that f e w mathematicians w o u l d seek such a solution in Grassmann's book

and fewer w o u l d be able to disentangle it from the other ma-

terials

s u r r o u n d i n g it.

equal justice be The been

reception a

great

(5;

232-233) The

accorded

the

disappointment

Ausdehnungslehre

Grassmann,

one

must

for at first his

have efforts

he w r o t e in 1877, "this n e w w o r k m e t

e v e n l e s s a t t e n t i o n t h a n t h e f i r s t . " (4,1,1;

a p p e a r e d (5; only

second

to

seemed to have been in vain; with

latter c o m m e n t could w i t h

a p p l i e d in relation to vector analysis.

18) N o r e v i e w s o f i t

231), and it seems probable that Grassmann received f r o m those to w h o m he sent copies.77 T h e

note of thanks

f o l l o w i n g statements of E n g e l m a y serve as a c o n c l u s i o n to the discussion "Thus

of

the

the

maticians

early

second

a book

with

reception

of

Ausdehnungslehre seven

the

also

seals,

and

second

Ausdehnungslehre:

remained the

for

mathe-

abundance of entirely

n e w developments w h i c h it contained was like something buried in the

ground"

matters

and

which

pendently

"As

in

the

Grassmann

rediscovered

by

first

had

Ausdehnungslehre

published

others,

and

in

only

so it

in

were

much

the

second:

later later

indewas

it

r e a l i z e d t h a t G r a s s m a n n h a d d i s c o v e r e d t h e m e a r l i e r . " (5; 2 3 2 - 2 3 3 ) Thus

90

Grassmann's

works

were

almost

totally

neglected

during

Other Early Vectorial

Systems

the first fifty-five years of his life; the first real ray of h o p e a p p e a r e d in the second half of the second 1844

edition

Grassmann

in

1878)

in the foreword to the

of

his

Ausdehnungslehre

of

d e s c r i b e d the events that l e d up to the r e c o g n i t i o n that he re-

ceived in the

last years

Hermann

Hankel

Zahlensysteme of

1860's.

(published

my

was

(Leipzig,

decisive

the

(see

was

life. first

1867),

Ausdehnungslehre

more

of his

who,

stressed

pp.

16,

Clebsch's

in

the

112,

his

Theorie

der

fundamental 119-140,

recognition.

Shortly

complexen

significance

and

140).

Even

his

death

before

Clebsch, in his " z u m Gedachtniss an Julius Pliicker, G o t t i n g e n , 1 8 7 2 " (see t h e

notes

significance

set at t h e

b o t t o m of pages 8 a n d 28), e m p h a s i z e d the

o f m y Ausdehnungslehre

In fact in his

of

1844

second note Clebsch

in

stated:

words

of

strong

praise.

" I n a c e r t a i n sense t h e co-

ordinates of the straight line, as w e l l as a large part of the basis of the newer

algebra,

o f 1844.

The

are

already

more

contained

in

Grassmann's

exact statement of these

relations

Ausdehnungslehre

would however

l e a d too far a t p r e s e n t . " I n v i e w o f his l o v i n g a n d s o c o n s t a n t l y f r u i t f u l participation in the works of others, w h i c h trait distinguished this most eminent of the

more

recent mathematicians, it is certain that Clebsch

w o u l d later have f o u n d space to present these relations and, as was his way,

to

fructify

the

Ausdehnungslehre

with

new

and

far-reaching

ideas.

B u t , alas, h e w a s s n a t c h e d a w a y s o s u d d e n l y i n t h e m i d s t o f his p o w e r f u l efforts on b e h a l f of science. However three

years

earlier (in

1869) V i c t o r S c h l e g e l h a d b e g u n t o

execute the ideas suggested by Clebsch. In his " S y s t e m der R a u m l e h r e nach

den

Prinzipien

Einleitung Teubner," 1875,

in

der

dieselbe

of which

Schlegel

has

independently,

the

Grassmann'

dargestellt

schen

Ausdehnungslehre

Victor

Schlegel,

von

the first part appeared in presented with meaning

als bei

1872 a n d the second in

great clarity, and in large measure

of the

completely appropriate manner.

und

Leipzig

Ausdehnungslehre.

This

he

did

in

a

It is especially to be e m p h a s i z e d that

this b o o k b y S c h l e g e l i s t h e first w h i c h has v i e w e d t h e essential ideas o f the

Ausdehnungslehre

in

p r e s e n t a t i o n . (4,1,1; It was from

in

their

November

Hermann

inner

connections

and

has

given

them

a

18-19)

Hankel

Riemann and w h o was wrote

1866 that Grassmann received a long letter (1839-1873),

who

had

been

a

student

of

at that time privatdozent in mathematics at

Leipzig.

Hankel

numbers

in w h i c h he planned to include a treatment of Hamilton's

that

he

was

w r i t i n g a treatise

on

complex

q u a t e r n i o n m e t h o d s a n d also Grassmann's ideas, w h i c h h e strongly praised. A series of letters passed b e t w e e n and

in

18B7

appeared. mann's

the

Roughly

system,

10

former's

Theorie

percent of this

on which

Hankel

Hankel and Grassmann, der

book

complexen

was

Zahlensysteme

d e v o t e d to Grass-

b e s t o w e d m u c h praise.

book was influential, and if he had not died in

Hankel's

1873, he m i g h t have

done even more to make Grassmann's system known. Clebsch came to

know

Grassmann's

work

in

the

following

way:

Grassmann's

91

A

History

oldest

son,

1869 to

of V e c t o r Analysis

and

be

Justus,

began

brought

with

given

to

Stern

studies him

and

in

mathematics

copies

of the

Clebsch.

(5;

at Gottingen in

s e c o n d Ausdehnungslehre

311)

Stern

became

en-

thusiastic about Grassmann's book a n d passed on his enthusiasm to Felix

Klein,

1872.

(5;

his

who

312)

death in

Grassmann

of the

earliest of his

Because

that

influenced

mentioned,

proponents

life

his

was

Erlanger Program

of

unfortunately stilled by

students

his

system;

and

at

courses,

who

(1843-1905) was one

indeed

Schlegel

developing Grassmann's

isolation

mathematical

any

Victor Schlegel

of

expounding

of Grassmann's

elementary produce

it

enthusiasm

1872.

As

most

said

Clebsch's

would

Stettin

it

was

a n d his

unlikely

develop

his

spent

system.

teaching only that

system.

he

would

However a

n u m b e r of his followers came f r o m his "associates"; these i n c l u d e d his sons Justus a n d H e r m a n n , his brother Robert, a n d Schlegel h i m self,

w h o for t w o years ( 1 8 6 6 - 1 8 6 8 ) t a u g h t w i t h G r a s s m a n n i n Stet-

tin.

Schlegel

Sein

Leben

discussed

und

seine

Werke;

many

conversations

rarely

mentioned

this during

with his

period his

in

time

Grassmann, Ausdehungslehre.

his in

but

Stettin

decided with

to

explaining them. ect

from

der

In

Clebsch,

Raumlehre

dehnungslehre

und

published in Schlegel

study

Schlegel's

Grassmann's

Grassmann's methods 1871

and

works.

Grassmann: had

had

had

only

interest

had

1869 after leaving Schlegel

was

de-

and decided to publish a work

he received e n c o u r a g e m e n t in this proj-

in

1872 appeared the

nach

dem

als

Einleitung

1875.

he

Grassmann

h o w e v e r b e e n stimulated to the point that he in

lighted

Hermann

Stettin

Prinzipien in

den

Dieselbe;

first part of his

Grassmann the

second

System

schen

Aus-

part

was

(6; 6 1 - 6 2 )

attempted to explain Grassmann's

results through their

relations to e l e m e n t a r y g e o m e t r y (part I) a n d to the n e w e r m e t h o d s of h i g h e r g e o m e t r y a n d algebra (part II). E n g e l c o m m e n t e d on the content

and

significance

of Schlegel's

book

in

the

following way:

A c t u a l l y S c h l e g e l ' s w o r k also was n o t a success. T h e a u t h o r h a d definitely

gone

too

one-sidedly

for Grassmannian

methods

w i t h o u t prac-

t i c i n g the necessary criticism. H e t h o u g h t that the p e o p l e w h o a r g u e d for a progress of m o d e r n algebra as opposed to the theory of extension had d o n e it o n l y because t h e y k n e w the theory of extension m a i n l y by hearsay, w h e r e a s h e h a d c l a i m t o w h a t c o n c e r n s k n o w i n g . O n t h e o t h e r h a n d , the

superiority of modern

others

asserted was

algebra over the theory of extension which

no fiction a n d c o n s e q u e n t l y a large part of Grass-

mann's methods had b e c o m e dispensible. Schlegel was not the man to p u t the o l d G r a s s m a n n i a n w i n e in n e w vessels; he was not able to present the ideas c o n t a i n e d in the theory of extension from the point of v i e w o f t h e t h e o r y o f i n v a r i a n t s a n d t o b r i n g t o l i g h t w h a t w a s s t i l l n e w . (5; 324)

92

Other Early Vectorial

Engel

also

published lehre,

commented in

1878,

despite

(or

that

was

Schlegel's

more

perhaps

biography

influential

because

of)

the

went to extremes in praising Grassmann.

than

Systems

of Grassmann,

his

System

fact that the

derRaum-

biography

(5; 3 2 4 ) S c h l e g e l ' s p a r t l y

successful efforts to m a k e Grassmann's ideas k n o w n c o n t i n u e d until Schlegel's death in

1905, b y w h i c h t i m e h e h a d p u b l i s h e d over

twenty-five papers in the Grassmannian tradition. Three other m e n w h o developed an interest in Grassmann's work during the mid-1870's and published related works were H e r m a n n Noth

(1840-1882),

William

Kingdon

Clifford

(1845-1879),

and

W. Preyer (1841-1897); the latter w r o t e to Grassmann that d u r i n g a trip to

England

(made

in

1875

or

1876)

he

h a d discussed Grass-

mann's w o r k w i t h Sylvester, w h o was quite interested in it and had p l a n n e d a p u b l i c a t i o n i n r e g a r d t o it. (5; 3 3 0 ) As

more

ideas,

mathematicians

the

creased;

demand

for

became

copies

of

interested the

first

in

Grassmann's

Ausdehnungslehre

in-

none h o w e v e r were available, since the publisher had in

1864 used the r e m a i n i n g copies for wastepaper. A l t h o u g h the 300 copies

of

second

edition

the

second

Ausdehnungslehre

of the

had

f i r s t Ausdehnungslehre

not was

yet

been

sold,

published

in

a

1878

w i t h three appendices a n d a n e w f o r w a r d (left i n c o m p l e t e by Grassmann and completed by

Schlegel).

Grassmann's activities after 1862 w e r e m a n y a n d diversified. H e , like the y o u n g H a m i l t o n , had a strong interest in a n d great talent for languages. It was only Grassmann h o w e v e r w h o m a d e an important contribution

to

philology,

a contribution

which

in

fact rivals

m a t h e m a t i c a l w o r k . A s t u d y o f S a n s k r i t b e g u n i n 1 8 4 9 (5; minated Rig-Veda

in

the

(1784

pages).79

1870's

with

the

pages)78

and

his

These

achievements

publication translation

of Grassmann

of his of the did

not

his

155) cul-

Worterbuch Rig-Veda

zum

(1123

go unrecog-

nized, for in 1876 he was m a d e a m e m b e r of the A m e r i c a n O r i e n t a l Society and received an honorary doctorate from the University of T u b i n g e n . (5; 3 0 9 ) In the p e r i o d after

1862

Grassmann published textbooks on the

German and Latin languages and on mathematics, as well as numerous religious a n d m u s i c a l w r i t i n g s a n d a b o o k on G e r m a n botanical terminology. time.75

These

He

also

invented

activities,

the

combined

Grassmann w i t h his

Heliostat

at

this

increasing interest in

philology and increasing disappointment at the neglect of his mathematical creations, explain his d i m i n i s h e d mathematical productivity d u r i n g the late In the

1860's.

1870's h o w e v e r G r a s s m a n n p u b l i s h e d a n u m b e r of m a t h e -

matical papers w h i c h Engel described as of inferior quality. E n g e l

93

A H i s t o r y of V e c t o r Analysis

ascribed this to the fact that G r a s s m a n n h a d lost contact w i t h the current

mathematical literature a n d even to some extent w i t h his

o w n e a r l i e r ideas. (5; 3 1 5 - 3 1 7 ) I n h i s " D i e n e u e r e A l g e b r a u n d d i e A u s d e h n u n g s l e h r e " of 1874

80

Grassmann attempted somewhat un-

successfully to relate his ideas to n e w d e v e l o p m e n t s in algebra, particularly 1877

invariant

81

he

previously mann in

theory;

claimed

and

(justly)

mentioned

in

his

"Zur

electrodynamical

law

P r i n z i p i e n der A u s d e h n u n g s l e h r e " of 1872 that

time

of

still

published

by

Grass-

1876.82 " D i e M e c h a n i k n a c h d e n

1845 a n d by Clausius in

at

Elektrodynamik"

priority over Clausius in regard to the

83

contained some of the

results

of his

unpublished

theory.

O n e of the m o r e controversial papers was his " D e r Ort der

dissertation

on tidal

H a m i l t o n s c h e n Q u a t e r n i o n e n i n d e r A u s d e h n u n g s l e h r e " o f 1877,84 in

w h i c h Grassmann attempted to show that quaternions could be

derived from

the

p a p e r " S u r les

units

a n d multiplications discussed in his

differents

genres de multiplication";

1855

unfortunately

his p r e s e n t a t i o n was w e a k e n e d b y the fact that h e k n e w H a m i l t o n ' s ideas only f r o m second-hand sources such as Hankel. Grassmann's richly productive, b u t in ways tragic, life came to an e n d on S e p t e m b e r 26, and

for

Grassmann

schoolmaster were 1870's

1877. It is regrettable b o t h for mathematics

that

the

ideas

appreciated at

of this

brilliant,

but

isolated,

such

a late hour. A l t h o u g h the

brought h i m a measure of fame,

nevertheless by that t i m e

m a n y o f h i s i d e a s a n d m e t h o d s h a d b e e n (as E n g e l c o m m e n t e d [ 5 ; 315]) rediscovered by others and integrated into different formulations a n d systems. M u c h of m o d e r n vector analysis d i d h o w e v e r lie e m b e d d e d w i t h i n a n d as yet unextracted from his system. But the fates d e c r e e d h e r e as e l s e w h e r e :

Grassmann's ideas exerted little

or no influence on the later history of vector analysis. T h e irony is t h a t t h e y c o u l d h a v e ; t h e f a c t i s (as w i l l b e s h o w n ) t h a t t h e y d i d n o t . M u c h has b e e n said i n e x p l a n a t i o n o f the n e g l e c t o f Grassmann's works.

S o m e of the factors that c o n t r i b u t e d to this neglect b o r d e r on

the incredible.

E s p e c i a l l y s t r i k i n g is the fate of G r a s s m a n n ' s ideas

i n r e l a t i o n t o t h e s m a l l g r o u p w h o b e f o r e 1870 r e c o g n i z e d t h e i r significance. T w o (Hankel and Clebsch) w h o "discovered" Grassmann d i e d almost immediately thereafter; three (Mobius, Hamilton, and Bellavitis) and

had

Schlegel

strong

was

allegiances

a man

whose

elsewhere

(their o w n

systems);

enthusiasm exceeded his critical

facilities. Though

these

and

other factors

help to

explain

the

neglect of

Grassmann's w o r k , in a b r o a d sense this neglect needs little explanation, for their discoveries

were

revolutionary and the historical

pattern exhibited here is not uncommon.

94

Other Early Vectorial

Contemporary

analyses

of the

history

of

scientific

Systems

revolutions

have uncovered m a n y important patterns. C o n c e r n i n g discovery, it has

been

quently

shown

that there are

essentially

no

irrelevant,

direct paths

often

to

discovery;

philosophic,

ideas

may

freplay

a decisive role. T h u s it is p r o b a b l y not accidental that b o t h H a m i l ton a n d Grassmann w e r e supported (but not directed) in their discoveries

by

certain

philosophic

assumptions.

While

Hamilton

meditated on the metaphysics of time, Grassmann d e v e l o p e d ideas c o n c e r n i n g space. T h e truth a n d direct r e l e v a n c y of their speculations

in this regard are beside the

these

speculations

plored

domains.

supported

The

point; w h a t is relevant is that

them

discovery

in

acceptance, a n d philosophic notions new

idea frequently

and

Grassmann

bulked

large

hinder

realized

in

their

ventures

of a n e w idea does

its

the

their first books

unex-

that aid in the discovery of a

acceptance.

this;

into

n o t i n s u r e its

Eventually

philosophic

were

Hamilton

discussions

banished

from

that

their later

presentations. Historical analyses of the acceptance of revolutionary ideas s h o w that acceptance by

definition.

scriptive;

it

requires

But

the

explains

much

time;

this

seems

statement that time nothing.

The

real

is

to be

needed

question

is

so almost

is

only

de-

W h y is time

n e e d e d ? W h a t factors operate in t i m e to b r i n g about the acceptance of revolutionary factors

ideas?

that frequently

revolutionary

Listed play

below

are

three

significant roles

in

of the the

numerous

acceptance of

scientific ideas.

(1) A c c e p t a n c e i s p r o m o t e d w h e n t h e d i s c o v e r y i s t a k e n u p (or made)

by

tional

pursuits

someone

who

or w h e n

has

already

it is

attained great fame

associated w i t h

some

in tradi-

event or non-

scientific question of great importance. (2) A c c e p t a n c e made

which

are

is

promoted

more

easily

when

less

startling discoveries are

interpreted in

terms

of the

revolu-

tionary ideas than in terms of traditional doctrines. (3) A c c e p t a n c e i s p r o m o t e d w h e n a n e e d f o r t h e m e t h o d s w h i c h the

revolutionary

ideas

provide

e i t h e r arises

or is recognized for

the first time. Examples those Young,

action with

Lobachevski,

abundance. factors

of the

associated

In

the

of these

factors

Copernicus,

Darwin, present

and

case

in

Einstein we

such

Galileo,

shall

revolutions

Lavoisier,

as

Hutton,

could be supplied in examine

these

three

only in relation to the acceptance of the ideas of H a m i l t o n

and Grassmann.

Hardly a more striking example could be found to

illustrate the first factor: in 1844 H a m i l t o n h a d attained w i d e s p r e a d fame on the basis of important, b u t not r e v o l u t i o n a r y , discoveries,

95

A H i s t o r y of V e c t o r Analysis

a n d his ideas r e c e i v e d i m m e d i a t e t h o u g h l i m i t e d attention; Grassm a n n was u n k n o w n h o w e v e r a n d suffered the fate o f a m a n w h o s e first great

discovery

factor it is were did

is

revolutionary.

not accidental

rediscovered before not

certain

discover coastal

continent.

the

areas

b u t his

continent.

find

this

was

whole

readers

appreciated.

American

subsequent

over a

a continent,

second

the

and

to

the

parts of Grassmann's system

North

Grassmann

Concerning

that m a n y

continent;

generations

discovered

recognized

p e r i o d of t w e l v e years neither expected to

Columbus

he

a

d i d discover

find

nor wanted

C o n c e r n i n g the t h i r d factor it is important

t o r e a l i z e t h a t t h e n e e d for v e c t o r i a l m e t h o d s i n 1800 w a s less t h a n the n e e d in 1900 and that although vectorial methods aid investigations, t h e y are n e v e r i n d i s p e n s i b l e . W h a t can b e d o n e b y vectorial methods was

can

needed

also

be

during

done

by

which

traditional methods, and thus time

scientists

could

see

that

the

labor of

learning vectorial methods was amply compensated. We

may

now

Grassmann's Roughly

150

Grassmann's though

the

briefly

ideas

trace

during

papers

and

collected majority

the

the 9

reception

last

books

works)

of these

two

and

(excluding

appeared

development

decades

from

publications

of the

the

publication

1881

of

1900.85 A l -

to

appeared

of

century.

in

German

journals, a substantial n u m b e r of papers may be found in the American,

British,

French,

and

Italian journals

of the

period.86

Engel

c o m m e n t e d that there was a strong tendency a m o n g the authors of these papers to b e c o m e unrealistically enthusiastic and narrow conc e r n i n g t h e G r a s s m a n n m e t h o d s . (5; 3 4 3 ) F e l i x K l e i n , w h o w a s w e l l acquainted w i t h Grassmann's methods, expressed the same view.87 The works Gibbs

task was 8S

of

collecting

begun

in

the

and

publishing

1890's.

Grassmann's

complete

T h e stimulus for this came from

a n d f r o m Klein.89 In 1892 K l e i n contacted Frederick E n g e l

(1861-1941)

and

requested

that

he

prepare a biography of Grassmann.

edit (4,1,1;

Grassmann's

works

and

vi) T h e works began to

appear in 1894, w i t h the final section containing Engel's biography appearing in

1911.

Though

E n g e l was by no m e a n s an ardent fol-

l o w e r o f G r a s s m a n n ' s i d e a s , a s h e h i m s e l f s t a t e d , (4,1,1; v i ) h e d i d nevertheless

write

a detailed, critical, and sympathetic biography

of that great mathematician.

IX.

Matthew

Writing in

O'Brien 1892, the quaternionist C. G. Knott c l a i m e d " t h e anti-

quaternionic vector analysts of today [Gibbs, Heaviside, and Macfarlane] h a v e b a r e l y a d v a n c e d b e y o n d t h e stage r e a c h e d [ i n

96

1852]

Other Early Vectorial O'Brien. . . ."90

by

Knott's

statement

(which

he

Systems

frequently

re-

peated) is obviously of great importance if it is true. The

Reverend

1854

Professor

Matthew O'Brien of

Natural

(1814-1855) was from

Philosophy

and

Astronomy

1844 to

in

King's

College, L o n d o n . In 1830 he h a d b e e n admitted to T r i n i t y College, Dublin, whence he proceeded to Cambridge to graduate as T h i r d W r a n g l e r i n 1838.91 D u r i n g his short life O ' B r i e n p u b l i s h e d r o u g h l y twenty

papers

and

to vector analysis

a few

elementary

books.

T h e papers relevant

were published during the

last t e n years

of his

life.92 Judgment on the

significance of O'Brien's ideas is complicated,

in part because the presentation of his vectorial ideas g i v e n in his early

papers

differs

longest of his cal

Transactions

with

from

that

of his

was

that

published

papers

1852.9

for

passing

mention

It

is

this

going

to

later

paper

the

papers.

in

the

that

The

will

important

last

London

and

Philosophi-

be

discussed

papers

published

earlier. O ' B r i e n began by stating that his a i m was to p r o v i d e a n e w notation of

for an "the

operation

translation

considered surface

by

duced it

on

.

from

.

three the a

.

the

actual

directed

instances

parallel rigid

The

of constant occurrence

of a

magnitude." of such

motion body

of

by

a the

effect

produced

motion

of its

by point

in mathematics, that

(9;

161)

translations: right

line

.

translation the

.

of

. a

The

of

application."

a

(9;

then

generation

force

translation

of

O'Brien "The

effect acting

of pro-

upon

force

resulting

162)

From

a n analysis o f t h e s e cases h e c o n c l u d e d that t h e essential c o n c e p tion

in

of a

directed

each

case

magnitude

"is .

.

that

of

,"

and

.

effects m a y be r e p r e s e n t e d as magnitude

the

effect

he

produced

noted

that

by

the

each

translation of

these

a product of the translated directed

and the translation undergone.

(9;

162)

He stated that

t h e s e t r a n s l a t i o n s c o u l d be c l a s s i f i e d as " l a t e r a l " (cases 1 a n d 2) or " l o n g i t u d i n a l " (case 3 ) a n d t h a t his n e w n o t a t i o n w o u l d b e a i m e d a t representing these t w o types of translations.

(9;

162-163)

H e t h e n d e f i n e d directed m a g n i t u d e a n d the t w o types o f translation:

t h e t r a n s l a t i o n of v a l o n g u is l o n g i t u d i n a l w h e n a n g l e A is

A 0 ° a n d lateral w h e n a n g l e A i s 90°. (9; duced

the

"symbolical

that they w e r e

forms"

distributive

x.y

163-164) O ' B r i e n then intro-

and

functions

x X y

9;J

;

of x and y.

he

assumed only

H i s choice of the

97

A H i s t o r y of V e c t o r Analysis

s y m b o l s . a n d X was e x p l a i n e d as o n e s t e m m i n g f r o m the fact that they

were

O'Brien

obsolete

defined

together," tion,"

while

used for

"successive

symbols

"addition" he

for

of

multiplication.

directed

differentiated

between

directed magnitudes

addition,"

with

used for t w o

(9;

magnitudes

164-165)

as

"putting

"simultaneous

addi-

coincident origins, and

directed magnitudes,

one

of

w h i c h has its o r i g i n a t t h e e n d p o i n t o f t h e o t h e r . (9; 1 6 5 - 1 6 6 ) A f t e r a

discussion

of equality

and

of the

suitability

of directed magni-

tudes for the representation of physical entities, O ' B r i e n introduced three " d i r e c t e d units" a,

y w h i c h go out from the origin along

t h e x, y, z axes r e s p e c t i v e l y ; a n y d i r e c t e d m a g n i t u d e u w a s thus to be r e p r e s e n t e d in t h e f o r m

" w = x a + yfi + r y . "

(9;

166-167)

O ' B r i e n w e n t on to discuss the significance of such an expression a s da.

He n o t e d that for t w o d i r e c t e d m a g n i t u d e s a a n d a' separated

by an infinitesimal angle, " d a is the expression for an indefinitely small

line

a t right

angles

to

a."

(9;

168)

This

property bears

men-

tion, since it served as the starting p o i n t for his vectorial system as presented results

in

his

obtained

earlier papers published during the in

these

papers

were

very

1840's. T h e

similar to the results

obtained in the paper presently under discussion, but the approach was

quite different.

O'Brien

showed

that

both

lateral

and

longitudinal

effects

of

t r a n s l a t i o n s a r e d i s t r i b u t i v e a n d i d e n t i f i e d u.v a s t h e n o t a t i o n t o b e u s e d to represent the

lateral

effect of the translation of v along u,

a n d u X u t o represent the longitudinal effect of the translation of v along u.

(9;

169-170) He established the rules to be used for ordi-

nary numbers in regard to these t w o multiplications and w e n t from this

to

show

O'Brien

t h a t u.v = v.u

and

u X v = v X u.

(9;

170-172)

Thus

h a d e s t a b l i s h e d t h a t u.v a n d u X v w e r e b o t h d i s t r i b u t i v e

a n d that the latter was a c o m m u t a t i v e multiplication; he never h o w ever the

investigated paper,

tudes

or

planes).

to

directed This

associativity.

state

w h e t h e r u.v magnitudes

ambiguity

was

H e also failed, u p t o this p o i n t i n and u X v were numerical magni(for

only

example, partly

vectors

clarified

by

or

vectorial

his later ex-

planations. O ' B r i e n then i n t r o d u c e d his t e r m "directrix" explaining that in a lateral translation a plane was d e t e r m i n e d a n d that the directrix for a lateral translation was to be a l i n e of a certain l e n g t h p e r p e n d i c u lar to that p l a n e ; for

t h u s t h e d i r e c t r i x f o r a.fi w a s t o b e y ; f o r / 3 . y , a ;

a , —y; a n d s o on. I t t h e n b e c o m e s clear ( t h o u g h o n l y i n part

from O'Brien's direct statements) that the magnitude of the product of t w o d i r e c t e d m a g n i t u d e s l i n k e d by " . " is mn sin 6 w h e r e m a n d n are the lengths of the t w o m a g n i t u d e s a n d 0 is the angle b e t w e e n

98

Other Early Vectorial

the t w o directed magnitudes. symbol

"D,"

D(a.f3)

y.

=

meaning

(9;

O'Brien proceeded to introduce the

directrix of," a n d wrote, for example,

173-176)

Proceeding 0 X 0 = 1 ,

"the

Systems

to

longitudinal

y X y = 1";

a X

translations, a X y,

he

wrote,

0 X y, . . . are

as e q u a l to z e r o . H e n c e "u X v = mn c o s

"a: X a = 1,

naturally taken

w h e r e m a n d n are the

m a g n i t u d e s of u a n d v, a n d 0 is t h e a n g l e b e t w e e n t h e m . (9;

176)

O ' B r i e n then considered the results of the repetition of the operat i o n Da, f o r e x a m p l e , i n t h e e x p r e s s i o n ( D a . ) 2 a ' w h e r e a a n d a ' a r e any t w o unit directed magnitudes. In this regard he c o n c l u d e d that if a n d o n l y if a a n d a'

a r e m u t u a l l y p e r p e n d i c u l a r , t h e n (Da.)

—a',

has

and thus

The tem;

that

(Da.)

one

of the

of

a'

=

V^-I.

s e c o n d p a r t o f t h e p a p e r d e a l t w i t h a p p l i c a t i o n s o f t h e sys-

in his applications to physical optics O ' B r i e n i n t r o d u c e d the

symbol

O

"to

denote

the

operation

used this in the expressions f l u , above his

properties

2

exposition

system;

of O'Brien's

a

+ /3

fiXu,

+ y -J-^. . . . "

a n d CI.v.

thought

He

(9; 2 0 4 - 2 0 5 ) T h e

should give

some

ideas

of

the question that must be dealt w i t h is w h e t h e r Knott

was correct in his belief that O'Brien's system is essentially that of Gibbs

and

Heaviside,

and

hence

is

the

now

current

system.

If

Knott was correct, then O ' B r i e n deserves great credit a n d m u s t be called the father of m o d e r n vector analysis. O n e aspect of this question is w h e t h e r O ' B r i e n attained conceptions

o f t h e scalar (dot) p r o d u c t a n d t h e v e c t o r (cross) p r o d u c t .

He

must

be

his

given

credit for h a v i n g the

scalar p r o d u c t ,

although

justification of it is hardly traditional. Concerning the vector product

the

question

numerical

is

magnitude

far

from

with

simple.

the

His

modern

product

vector

u.v

agrees

product;

in

however

O ' B r i e n never clearly stated w h a t it represented. As we m e n t i o n e d previously, it seems that he v i e w e d it as a n u m e r i c a l quantity; at another point in the paper he seems h o w e v e r to,have v i e w e d it as a directed

and

oriented

parallelogram.

(9;

180)

It

is

clear that he

v i e w e d h i s p r o 4 u c t Du.v a s a d i r e c t e d l i n e s e g m e n t , e q u a l i n m a g nitude, direction, a n d sense to the m o d e r n product. But still

this

spond with failure to y

do

the

product in modern

his

s y s t e m does n o t i n all cases corre-

product.

understand associativity not

correspond

with

the

T h e reason for this

is O'Brien's

a n d the related fact that his a, modern

i, j,

k.

This

becomes

evident from statements made by O ' B r i e n in a footnote to his paper in w h i c h he c o m p a r e d his

system to that of H a m i l t o n .

Herein he

compared

H a m i l t o n ' s i , j , k w i t h h i s a , (3, y a n d n o t e d t h a t H a m i l -

ton's

k are

i, j,

both

units

of direction

and

have

the

property of

99

A H i s t o r y of V e c t o r Analysis

c o r r e s p o n d i n g to square roots of m i n u s one, whereas his a, n o t h a v e t h e l a t t e r p r o p e r t y . (9; property

in

some

cases,

y do

178) T h e m o d e r n i, j, k do h a v e this

specifically

for

example

in

i X (i X v)

w h e r e v has no i c o m p o n e n t . B u t O ' B r i e n r e a l i z e d t h a t he c o u l d express

this

contains

(Da.)2 v = Da. (Da. v)

as

where

a

is

a

unit

vector

and

v

no a c o m p o n e n t .

O'Brien's

failure

to

understand

associativity

was manifested in

t h i s f o o t n o t e w h e n h e s t a t e d t h a t H a m i l t o n ' s i 2 a l w a y s e q u a l s —1, (Da.)2 e q u a l s — 1

whereas his magnitude

perpendicular

o n l y w h e n it is a p p l i e d to a directed

to

a.

(9;

178)

W h a t is

striking

is

that

O ' B r i e n s h o u l d h a v e e x p e c t e d this, for the t w o forms are not comparable,

(i2) =

since

(ii),

(Da.)2 =

whereas

Da.(Da.

.

.

.).

T h e first

means i times i; the latter means a p p l y Da to the quantity attained after Da

has

failed to

been

see

applied

to

some other quantity.

Thus

O'Brien

that his system was not associative for this product;

indeed it seems

not to have b e e n a question for him.

He never in

the course of the paper wrote a triple product except in the form (Da.)2 v .

He h a d in fact no s y m b o l i s m for expressing the e q u i v a l e n t

of (i X j) X k ( w r i t t e n in m o d e r n terms); this c o u l d o n l y be w r i t t e n (Da.Dfi.)y,

which

would

have

been

meaningless

in

his

system.

T h u s one can at most attribute to O ' B r i e n a very near approach to t h e m o d e r n v e c t o r (cross) p r o d u c t . In and

summary,

O'Brien

should

be

viewed

a forerunner of Gibbs

Heaviside, but he d i d not anticipate them in the construction

of the m o d e r n system. His system was rather primitive as compared to theirs, w a s d e v e l o p e d on a d i f f e r e n t a n d less satisfactory f o u n d a tion, a n d d i f f e r e d f r o m theirs in at least o n e m a j o r point. H i s system failed to

include

not ended in

a treatment of associativity.

If O'Brien's life had

1855, this last defect m i g h t h a v e b e e n r e m e d i e d , b u t

in n e i t h e r his last p a p e r n o r in a n y of t h e earlier ones was it in fact remedied. O'Brien

seems to have had no followers; neither Heaviside nor

Gibbs

ever mentioned him,

to

work.

his

Hamilton

It

had

w o u l d be

even

after Knott h a d called attention

interesting to

of O'Brien's

ideas

and

know

what view,

likewise

to

if any,

know whether

O ' B r i e n created his system from Hamilton's system. T h e first question seems unanswerable since H a m i l t o n referred to O'Brien only once, a n d t h e n o n l y to i n c l u d e his n a m e in a list of mathematicians who

had

searches,

"at to

moments notice,

turned

and

in

s p e c u l a t i o n s of m i n e . . . ."

aside

some

from

instances

their

own

to

extend,

original results

reor

94

T h e answer to the second question seems almost certainly to be that O ' B r i e n created his system directly from Hamilton's. He never

100

Other Early Vectorial

Systems

stated t h i s , n o r d i d h e d e n y it. O ' B r i e n c e r t a i n l y k n e w o f H a m i l t o n ; indeed

Hamilton

probably taught O'Brien during the early

1830's

when O'Brien studied at Trinity College, Dublin. Moreover, as we indicated before, Hamilton's w o r k immediately attracted attention. H e h a d p u b l i s h e d f r o m 1844 t o 1846 e i g h t papers (or e i g h t sections of

one

paper95

paper)

in

the

Philosophical

Magazine

appeared in the same journal.

before

But this

O'Brien's

external

first

evidence

does no m o r e than a d d further support to w h a t is obvious f r o m the internal similarities b e t w e e n their systems. If we

put

priority

disputes

aside,

O'Brien

is

found to

make

a

fitting conclusion for this chapter, since he was i n d e e d on the road that lead to

Heaviside and Gibbs, a n d his w o r k further illustrates

the breadth a n d d e p t h of the search for n e w vectorial systems.

101

Notes 1

August

Ferdinand

2

August

Ferdinand

3

Anton

lichen

Favaro,

Mobius,

"Justus

Werkens"

Gesammelte

Mobius,

in

Der

Werke,

4

barycentrische

vols.

(Leipzig,

Calcul

(Leipzig,

Bellavitis, E i n e Skizze seines Schlomilch's

Zeitschrift fur

1885-1887). 1827).

Lebens u n d wissenschaft-

Mathematik

und

Physik,

26

(1881),

Gesammelte

mathe-

153-169. 4

Hermann

matische 5

und

Giinther physikalische

Friedrich

Grassmann,

Werke,

Engel,

3

vols,

Grassmanns

Leben

Hermann

in

6

Grassmanns

pts.

comprises

(Leipzig,

vol.

Ill,

1894-1911).

pt.

II

of the

work

cited

i n n o t e (4) a b o v e . 6

Victor

Schlegel,

Hermann

Grassmann:

Sein

Leben

und

Seine

Werke

(Leipzig,

1878). 7

Adhemar

Barre,

differences Comptes 8

rendus

Rev.

9

10

VAcademie

Saint-Venant, sur

des

Perceval

Matthew

leur

usage

Sciences,

21

Graves,

O'Brien,

Translation

Royal

de et

Life

"Memoire pour

(1845), of

Sir

sur

les

simplifier

sommes

la

et

les

mecanique"

in

620-625.

William

Rowan

Hamilton,

vol.

Ill

1889).

Rev.

the

de

Robert

(Dublin,

Comte

geometriques,

Society

1

a

of London,

Rudolf

rechnung,

of

142

Mehmke Band,

"On

Symbolic

Directed (1852),

in

his

Forms

in

Philosophical

Transactions

of

of the

161-206.

valuable

Punktrechung,

derived from the Conception

Magnitude"

1.

book

Vorlesungen

Teilband

(Leipzig

uber and

Punkt

und

Berlin,

Vektoren-

1913),

346,

argued that G i o v a n n i Ceva could be v i e w e d as a "Vorlaufer," a precursor of M o b i u s in regard to a system of analysis based on points. 11

For information on

Mobius'

life and scientific activity the following were used:

R i c h a r d B a l t z e r , " A u g u s t F e r d i n a n d M o b i u s " (1,1; v - x x ) a n d C u r t R e i n h a r d t , " U e b e r die

Enstehungszeit u n d den Z u s a m m e n h a n g der wichtigsten Schriften u n d Abhand-

lungen von 12

These

Mobius" facts

(1,IV;

were

published writings. See 13

Published at

14

A

much

Calculus

fuller of

(1892), 2 - 2 1 . Gibbs, York, (New

Reinhardt

through

his

study

For this

work

see (1,IV; 4 4 1 - 4 7 6 ) , b u t for the i m p o r t a n t

exposition

is

in

available

Proceedings

in of

R.

E.

the

Allardice,

Edinburgh

Algebra"

91-117;

1963),

in

Julian

148-150;

Papers

H e r m a n n Rothe,

Willard A

ff.

A section of M o b i u s ' book was translated by J.

A

Source

102

Barycentric Society,

10

Book

1959), 6 7 0 - 6 7 6 .

in

vol.

Boyer,

Mathematics,

vol.

of

II

(New

Geometrical

vol.

Methods

"Systeme geometrischer Analyse" in

1956),

B.

Gibbs,

History

1289-1293;

Carl

Wissenschaften,

of J.

Coolidge,

1931[?]),

in

mathematischen

Scientific Lowell

der

lished

"The

Mathematical

Encyklopadie

York,

un-

389-398).

Mobius"

"Multiple

242

of Mobius'

B r i e f expositions w i t h historical c o m m e n t s are g i v e n i n Josiah W i l l a r d

1961), York,

by

(1,IV; 707-710).

Leipzig.

a p p e n d i x see (1,1;

699-728).

established

Ill,

History

II,

pt.

1,2nd

o f Analytical

ed.

David

half,

Geometry P.

(Leipzig,

(New

York,

Kormes and pub-

Eugene

Smith

(New

O t h e r E a r l y Vectorial Systems

1 5

This

discovery was made

Bobillier, Analytical

K.

W.

Geometry

nearly simultaneously and independently by Etienne

Feuerbach, (New

and

York,

Julius

1956),

16

See R i c h a r d Baltzer's remarks

17

Published at Leipzig;

Pliicker.

See

Carl

B.

Boyer,

History

of

240-244.

i n (1,1; x i - x i i i ) .

later p u b l i s h e d in

(1,IV;

1-318).

For Mobius' treatment

of vectorial a d d i t i o n , see (1,IV; 41). 18

Thus,

A'B'C'

is

for

the

of this

example,

projection

equation

is

Mobius

of the

wrote

t r i a n g l e ABC

a triangle

(fixed as

A'B'C'

on

= ABC

some

parallel

-

cos

ABC

A'B'C'

where

plane. T h e right-hand m e m b e r

to

some

plane)

m u l t i p l i e d by the

cosine of the angle b e t w e e n the t w o planes, w h i c h must be a n u m b e r , whereas the left-hand

member

would

is clear that he was

in

general

not

be

parallel

s i m p l y stating that the

to the

projection

right-hand member.

It

of a triangle on some plane

gives an area w h i c h is equal to the original area m u l t i p l i e d by the cosine of the angle between the two 19

In

1859

Lectures

on

planes.

Mobius

H o w e v e r his

symbolism

d i d not express this.

published a paper entitled

" N e u e r B e w e i s des

aufgestellten

Princips

Quaternions

associativen

bei

der

(1,IV; 685)

in

Hamilton's

Zusammensetz-

u n g v o n B o g e n g r o s s t e r K r e i s e a u f d e r K u g e l f l a c h e . " (1,11; 5 5 - 7 0 ) 20

(3)

Biographical above

sciences 2 1

and

mathematiques,

Giusto

analitica

Bellavitis,

22

geometrique

Vavancement himself

des

Lettere, 23

ed

(3;

in the

"Saggio

di

(1835), (3;

Bellavitis"

(1880),

sciences.

Comptes

the

work cited in Darboux's

note

Bulletin

des

di

in

un

nuovo

Annali

delle

metodo Scienze

di

Geometria

del

Regno

Lom-

Laquiere, "Observations sur l'origine naturelle

des rendus,

equivalence

Equipollenze"

Arti,

(1876),

449

the in

343-348.

applicazioni

155) or M. Calcul

19

mainly from

246-247.

delle

155).

drawn "Giusto

equipollenze)"

du

recognized

Metodo

4

5

See for e x a m p l e

been

Laisant,

Ser.,

delle

Padova,

has

Ange

2nd

(Calcolo

bardo-Veneto.

et

information

Charles

Equipollences"

in

(1881),

evidence

see

in

77.

For

Giusto

Memorie

Association

Bellavitis,

del

Reale

Frangaise

that

"Sulle

Istituto

pour

Bellavitis

Origini

Venetoy

di

del

Scienze,

ff.

F o r his v i e w s in 1876 see Bellavitis, " O r i g i n i , " 4 7 6 - 4 7 7 . F o r his v i e w s

1830's see b e l o w . In r e g a r d to his n a r r o w n e s s of v i e w it is i n t e r e s t i n g to n o t e

that Bellavitis e x h i b i t e d the same t e n d e n c y in relation to n o n - E u c l i d e a n g e o m e t r y and,

according

24

Bellavitis,

25

Giusto

to

Favaro,

crusaded

against

its

acceptance.

On

this

see

(3;

159).

Scienze

del

Regno

" O r i g i n i , " 449.

Bellavitis,

Lombardo-Veneto.

Padova,

"Sulla 2

Geometria

(1832),

Derivata"

in

Annali

delle

250-253.

26

Bellavitis,

2 7

Giusto Bellavitis, "Sopra alcune applicasioni di un nuovo metodo di Geometria

Analitica"

" O r i g i n i , " 451.

in

II

Poligrafo.

Giornale

di

Scienze,

Lettere

ed

Arti.

Verona,

13

(1833),

53-61. I have not seen this paper, for the above journal does not s e e m to be available in this 28

country.

Giusto

analitica Veneto.

Bellavitis, (Calcolo

Padova,

5

"Saggio

delle

(1835),

di

applicasioni

equipollenze)"

244-259.

I

have

in

di

un

Annali

inferred

nuovo delle

from

metodo

Scienze some

del

di

Geometria

Regno

Lombardo-

not completely clear

comments m a d e by Bellavitis ( " O r i g i n i , " 451) that the 1833 paper contained neither the term "equipollence" nor the presentation 29

T h e r e are a n u m b e r of g o o d b i o g r a p h i c a l

of his fully d e v e l o p e d system. studies

of Grassmann. T h e definitive

w o r k i s c i t e d i n n o t e (5) a b o v e . A s h o r t e r , v a l u a b l e , b u t s o m e w h a t o v e r l y e n t h u s i a s t i c b o o k i s c i t e d i n n o t e (6) a b o v e . mann" A.

E.

in Heath,

Jahresbericht "Hermann

der

Also used was F r i e d r i c h E n g e l , " H e r m a n n Grass-

Deutschen

Grassmann"

Mathematiker-Vereinigung, i n Monist, 2 7 ( 1 9 1 7 ) ,

19 1-21; A.

(1910), E.

1-13;

Heath, " T h e

103

A H i s t o r y of V e c t o r Analysis

Neglect "The

of the

Work

Geometrical

Characteristic"

in

Monist,

35 (1944), 3 2 6 - 3 3 0 ; sein

Leben

Annalen,

14

ception

R.

und

Grassmann"

27

(1917),

and

27

and

(1917),

Its

George

Mathematik

Victor Schlegel's

work,

und

"Die

22-35;

Connection Sarton,

A.

E.

with

Heath,

Leibniz's

"Grassmann"

Physik,

41

Arbeiten"

valuable

article,

Grassmann'sche (1896),

30

S e e (5; 3 - 7 ) a n d (6;

This was published posthumously in

32

Monist,

36-56;

31

note

in

Grassmann

mathematisch-physikalischen

1-45;

Grassmann's

Z e i t s c h r i f t fur

of

in

Isis,

Sturm, E. Schroder, and L. Sohncke, " H e r m a n n Grassmann:

seine

(1879), of

of H.

Analysis

1-21

in

Mathematische

m a i n l y on the

re-

Ausdehnungslehre"

in

and

41-59.

16-30). 1911

as vol. I l l , pt. I, of the w o r k c i t e d in

(4) a b o v e . C o n c e r n i n g t h e d e t a i l s o f t h i s p e r i o d i n G r a s s m a n n ' s l i f e see (5; 7 2 - 8 5 ) a n d for

C o n r a d ' s s t a t e m e n t s e e (4,111,1; 2 0 9 ) . 33

Though

in

this

foreword Grassmann d i d not give the date

1832, there is every

reason to believe that E n g e l was right in taking the statements from the foreword as referring to the 34

part of Gauss' 35

1832 period.

S e e (4,1,11; 8 - 9 ) f o r G r a s s m a n n ' s s t a t e m e n t a n d (4,1,11; 3 9 7 - 3 9 8 ) f o r t h e r e l e v a n t letter.

(4,1,1; 8). T h e s e b o o k s , t h o u g h i n t e n d e d f o r e l e m e n t a r y i n s t r u c t i o n , w e r e w r i t t e n

f r o m a r a t h e r p h i l o s o p h i c a l p o i n t o f v i e w . O n t h i s see (6; 48) a n d (5; 2 - 7 ) . T h e f u l l references Raumlehre, und

to

Grassmann's

Ebene

raumliche

spharischen

Raumlehrey

father's

Trigonometrie

(Berlin,

first

having

the

part

two

Grossenlehre

been

These

gress, British

Museum, nor Biblioteque

For Wilson's

discussion

seem

to

are

Justus

1824)

The

book

first

published

Combinationslehre.

38

books

1835).

books

(Berlin,

be

in

quite

is

1817

rare;

Giinther

and the

under

neither

Grassmann,

Lehrbuch

der

ebenen

second

part

of his

the

the

title

Geometrische

Library

of Con-

Nationale list copies in their catalogues.

of these t w o products a n d for Wilson's statements con-

c e r n i n g G i b b s ' v i e w s see E d w i n B i d w e l l W i l s o n , " O n Products i n A d d i t i v e F i e l d s " in

Verhandlungen

1904. 37

(Leipzig,

des

C o n c e r n i n g this

nichteuklidischen Engel's

III.

1905),

Internationalen

see Grassmann's

Geometrie

zur

Kongresses

statement of 1877:

im

Heidelberg,

" U b e r das V e r h a l t n i s s d e r

Ausdehnungslehre"

(4,1,1;

293-294),

and

for

n o t e t h e r e o n s e e (4,1,1; 4 1 3 ) .

38

This is the year suggested by Engel.

39

(5;

See (5; 92).

331). T h i s is based on Engel's restoration of one unclearly abbreviated w o r d

in the original 40

Mathematiker

202-215.

(4,1,1;

letter.

22). E r n e s t

of philosophical Formation

of

Geometry"

in

Nagel gave an interesting, but brief, discussion of the degree

sophistication Modern

Osiris,

7

manifested by

Conceptions (1939),

of

142-224.

Grassmann

Formal On

Logic

Grassmann

in this w o r k in his " T h e in

the

Development

see especially pages

of

168-

174. 41

This

is

not

an

inappropriate

cluded many of Grassmann's

indeed

Whitehead's

Universal

Algebra

in-

42

F o r t h e i r s t a t e m e n t s see (5;

43

In a paper published in

44

S o m e discussion of the writings in E n g l i s h that attempt to explain Grassmann's

system

may

Grassmann, Algebra," (2) in

term;

ideas.

be "A

Annals

104

at

this

Brief Account

trans.

Alexander

given

102, 207).

point. of the

There

2

"A

are

Essential

Wooster Woodruff Beman,

Ziwet,

o f Math,

102,

1855, w h i c h w i l l be discussed later.

i n Analyst, 8

Brief Account of H.

(1885-1886),

1-11

and

five

such

Features

(1)

Hermann

of Grassmann's

Extensive

(1881),

Grassmann's

25-34;

(3)

writings:

96-97

and

Geometrical

Edward Wyllys

114-124; Theories" Hyde,

The

O t h e r E a r l y Vectorial Systems

Directional Analysis

Calculus

(Boston,

(London,

1906);

Grassmann's

1890); (5)

(4)

Joseph

Edward V.

'Ausdehnungslehre'"

Wyllys

Collins, in

American

193-198, 2 6 1 - 2 6 6 , 2 9 7 - 3 0 1 , a n d 7 (1900), 3 1 - 3 5 , 258.

I

have

used

all

Hyde,

"An

Mathematical

163-166,

of these, but their usefulness

is

Grassmann

Elementary

s

Space

Exposition

Monthly,

6

of

(1899),

181-187, 207-214, 253-

definitely limited.

They did

h o w e v e r set t h e s t a n d a r d i n g i v i n g E n g l i s h e q u i v a l e n t s for G r a s s m a n n ' s n e w t e r m s , and I have thus used their terms. I t e m 1 is the o n l y m a t h e m a t i c a l w r i t i n g by Grassm a n n ever translated marize

his

scope;

in

Items

2,

into

Ausdehnungslehre fact

3,

only 32

4,

presentation

and 5

English. of

of the all

is

useful

but

188

stem

substantially

It

1844,

an

sections

from

different

in that in it he attempted to sum-

adequate

summary

o f t h e Ausdehnungslehre

t h e Ausdehnungslehre o f

from

that

of the

was were

1862,

which

earlier book.

beyond

its

summarized. embodied

Item 2

is

a

short,

general, a n d h e n c e l i m i t e d . Items 3 a n d 4 contain Grassmann's ideas in a s i m p l i f i e d f o r m , i.e., l i m i t e d t o t h r e e

d i m e n s i o n s a n d w i t h t h e stress

on

applications.

Item 5

stays f a i r l y c l o s e to t h e o r i g i n a l text of 1862, as 3 a n d 4 do n o t ; it h o w e v e r is m a r r e d by misprints a n d a f e w mathematical errors and, t h o u g h surpassing the above works, still

falls

Finally and

far

short

Alfred

Henry

of b e i n g

North

an

adequate

Whitehead's

George

Forder's

summary

Universal

Calculus

of

Algebra

of the

Ausdehnungslehre

(Cambridge,

Extension

(Cambridge,

of

1862.

England,

1898)

England,

1941)

should be mentioned, though both depart considerably from Grassmann's presentation and content, and hence were used sparingly. 45

(4,1,1;

analysis

62).

This

books

that

may be compared with the any

statement found in

current vector

v e c t o r m a y be e x p r e s s e d in t h e f o r m a\ + bj + ck w h e r e a,

b, c a r e u n i q u e . 46

(4,1,1;

veloped

11-12). in

appeared. 47

the

In

a footnote

second

Grassmann

volume

of

the

stated that this Ausdehnungslehre,

product w o u l d be de-

but

no

second

volume

He d i d h o w e v e r develop this product in other writings.

This was so because for Grassmann any distributive operation was by definition

a multiplication.

F o r G r a s s m a n n ' s p r e s e n t a t i o n o f t h e a b o v e s e c t i o n s e e (4,1,1;

77-

79). 48

C o n c e r n i n g t h i s s e e (4,1,1; 9 3 - 9 9 ,

49

This

was

published

in

114-118).

Grunert's

Archiv

der

Mathematik

und

Physik,

6

(1845),

3 3 7 - 3 5 0 , a n d r e p u b l i s h e d i n (4,1,1; 2 9 7 - 3 1 2 ) . I t w i l l b e d i s c u s s e d l a t e r . H e r e a s e l s e w h e r e in the s u m m a r y Grassmann focused on s u m m a r i z i n g the results m o r e than the methods. 5 0

tial 51

As translated by W. W. Features Josiah

Willard

York, 1961), 52

George

53

The

pointed

Sarton, is

Extensive The

Algebra"

Scientific

"Grassmann"

given

previously,

representation Gauss

Gibbs,

Papers

i n Analyst, o f j .

wrote,

in

Seite

der Ternaren

Carl

Freidrich

in

in

Isis, 3 5 (4,1,11;

in

numbers July,

through

1831,

Formen." This

Gauss,

(1944),

Werke,

vol.

a

was II

this

very

117.

vol.

II

(New

162-167.

It

may be

n o t e d that,

as

we

acquainted with the geometrical letter.

short

however

(1881), Gibbs,

327.

397-398).

Grassmann first became

of complex probably

part

8

Willard

167. F o r a f u l l e r d i s c u s s i o n see pages 9 4 - 1 1 3 ,

letter out

Beman and published as "A Brief Account of the Essen-

of Grassmann's

It m a y also be

essay

entitled

noted that

"Geometrische

only published posthumously

(Gottingen,

1863),

305-310;

therein

in

Gauss

d e v e l o p e d an expression r o u g h l y e q u i v a l e n t to t h e scalar product. 5 4

Ernest

Nagel, " T h e Formation

Development 55

A.

E.

of G e o m e t r y "

Heath,

"The

in

Neglect

of Modern Conceptions of Formal

Osiris,

7

(1939),

of the W o r k

Logic

in the

173-174.

of H.

G r a s s m a n n " i n Monist, 2 7 ( 1 9 1 7 ) ,

24.

105

A H i s t o r y of V e c t o r Analysis 56

Hermann

Giinther

dehnungslehre" 350. 5 7

in

Grassmann, Grunert's

"Kurze

Archiv

Ubersicht

der

Mathematik

iiber

und

das

Wesen

Physik,

6

der

Aus-

(1845),

337-

R e p u b l i s h e d i n (4,1,1; 2 9 7 - 3 1 2 ) . H e r m a n n Giinther Grassmann, " N e v e Theorie der Elektrodynamik" in Poggen-

dorff's

Annalen

der

Physik

und

Chemie,

64

(1845),

1-18.

Republished

in

(4,11,11;

147-160). 58

Strictly speaking it was not a competition, since Grassmann's work was the only

essay s u b m i t t e d ! O n t h i s see (5; 59

Hermann

von

Giinther

Leibnitz

see

erfundene

(4,1,1;

Lehre (1,1;

geometrische

321-398).

von

119).

Grassmann,

For

Mobius'

Punktgrossen

Die

Geometrische

charakteristik essay,

und

den

Analyse

(Leipzig,

which

davon

gekniipft

1847).

For

und

this

die work

was entitled " D i e Grassmannische

abhangenden

Grossenformen,"

see

613-633).

60

See for e x a m p l e Saint-Venant's e x p l i c i t s t a t e m e n t to this effect.

61

I n t h i s r e g a r d i t s h o u l d b e n o t e d t h a t (1) o n e o f G a u s s ' t w o p u b l i c a t i o n s o f 1 8 3 1

was

i n L a t i n (the o t h e r i n G e r m a n ) , a n d h e n c e S a i n t - V e n a n t m i g h t h a v e h a d access

to this w o r k ;

(5; 200).

(2) S a i n t - V e n a n t d e s i g n a t e d v e c t o r s b y a s t r a i g h t l i n e a b o v e t h e l e t t e r

s y m b o l i z i n g the vector (thus

a),

and Argand was the first and only one of the m e n

m e n t i o n e d w h o u s e d t h e s a m e s y m b o l ; a n d (3) S a i n t - V e n a n t k n e w o f W a r r e n ' s w o r k by

at

least

trique des

1853, w h e n he

des

clefs

Sciences,

62

36

mentioned it in

algebriques

(1853),

et

des

an article,

" D e

determinants"

in

l'interpretation geome-

Comptes

rendus

de

VAcademie

583.

A l t h o u g h Saint-Venant is rarely m e n t i o n e d by later authors, he d i d influence his

countryman

Henri

Resal,

whose

Traite

de

Cinematique

pure

(Paris,

1862)

included

a

presentation of elementary vector addition, subtraction, and differentiation as well as

one

vectorial

product.

Resal's

product

was

however not Saint-Venant's;

it was

instead the m o d e r n scalar (dot) p r o d u c t w h i c h h e p r e s u m a b l y created b y g o i n g f r o m S a i n t - V e n a n t ' s a b s i n 6 to a b cos 6. N e i t h e r G r a s s m a n n n o r H a m i l t o n is m e n t i o n e d , and

although

references

to

sufficient to show that the use

of vectorial

Mechanik x-xi

.

methods

Kinematik,

Saint-Venant latter was

in

trans,

are

mechanics

from

few

his source. by

Russian

(see

Joseph by

pages

18

a n d 64), t h e y are

Resal was f o l l o w e d in his l i m i t e d Somoff.

Alexander

See

Somoff,

Ziwet

Theoretische

(Leipzig,

1878),

and 46-51.

63

Augustin

des

Sciences,

Cauchy,

1st

Cauchy, 36

"Sur

(1853),

Ser.,

vol.

les

Clefs

70-75

XI

and

(Paris,

algebriques"

129-136.

1899),

439-445,

in

See and

Comptes

also vol.

rendus

CEuvres

XII

de

VAcademie

completes

d'Augustin

(Paris,

1900),

12-21.

C a u c h y also p u b l i s h e d " S u r les avantages q u e presente, dans

un grand nombre de

questions

36

or

in

1'emploi

CEuvres,

ferentielles rendus,

36

46-63; sur

les

64

et

a

des

Ser., les

(1853),

and clefs

1938),

1st

clefs vol.

variations

38-45

memoir

algebriques,

algebriques"

XII

and

(Paris,

title

57-68,

as

Comptes 21-30;

employees

published

its

in

1900),

as

or a

in

comme CEuvres,

separate

republished

in

rendus, and

clefs

1st

(1853),

"Memoire

algebriques"

Ser.,

vol.

XII

161-169,

sur in

(Paris,

les

dif-

Comptes 1900),

publication

under the

t i t l e Memoire

CEuvres,

Ser.,

XII

2nd

vol.

(Paris,

417-466.

Because of the simplicity of the above illustration, the usefulness of the m e t h o d

for the solution of a large n u m b e r of simultaneous equations is not readily apparent. Cauchy's keys

for

CEuvres,

method the

is

directly analogous

solution

1st Ser., v o l .

of n

XII

equations.

(Paris,

1900),

65

C o n c e r n i n g t h i s s e e (4,1,1; 9 9 - 1 0 2 ;

66

Comte

106

and For 13

involves Cauchy's

the

introduction

more

elaborate

of n distinct example

see

ff.

156-157).

de Saint-Venant, " D e l'interpretation g e o m e t r i q u e des clefs algebriques

O t h e r E a r l y Vectorial Systems

et

des

determinants"

in

Comptes

rendus

de

VAcademie

des

Sciences,

36

(1853),

582-585. 67

Hermann

in

Crelle's

Giinther Journal

Grassmann,

fiir

die

R e p u b l i s h e d i n (4,11,1; 6 8

Victor

Schlegel,

Geschichte schrift 69

fiir

der

VAcademie

und

1st S e r . , v o l .

XI

the

interpretation

developed in Cauchy,

no

in

differents

29

letzten

41 les

(1896),

genres

Mathematik,

de

44

multiplication"

(1855),

123-141.

1899),

Ein

Beitrag

zur

fiinfzig

Schlomilch's

Zeit-

152-160.

Jahren"

The

or

in

in

Comptes

CEuvres

rendus

d'Augustin

de

Cauchy,

method of solving equations given in

later m e t h o d ;

numbers

in

geometriques"

250-257,

w i t h his

complex

Ausdehnungslehre.

6.

Quantites

(1849),

connection of

den

Physik,

(Paris,

paper has

les

angewandte

Grassmann'sche

"Sur

Sciences,

this

70

"Die

Cauchy,

des

"Sur und

199-217).

Mathematik

Mathematik

Augustin

reine

according

the paper deals to

the

mainly with

geometrical

methods

1845 by Saint-Venant for g e o m e t r i c lines. "Sur

les

Clefs

443-445, for

Hamilton;

les

lineaires

moments

algebriques"

and des

Cauchy,

in

CEuvres,

1st

Ser.,

vol.

XI

(Paris,

1899),

" S u r la T h e o r i e des m o m e n t s lineaires et sur

divers ordres"

i n CEuvres,

1st S e r . , v o l . X I I ( P a r i s , 1 9 0 0 ) ,

6, for M o b i u s . 71

Sir

William

Rowan

Hamilton,

Lectures

on

Quaternions

(Dublin,

1853),

(62)

of

Grassmann

et

preface. 72

Luigi

Cremona,

propriete

de

la

"Solution

cubique

des

questions

gauche"

in

494

Nouvelle

et 499,

Annales

methode

de

mathematiques,

19

(1860),

356-361. 73

Ibid.,

74

C o n c e r n i n g this see V i c t o r Schlegel, " D i e G r a s s m a n n ' sche A u s d e h n u n g s l e h r e "

in

357

Zeitschrift

fiir

Mathematik

und

Physik,

41

(1896),

48.

75

See t h e list of his p u b l i c a t i o n s as g i v e n in (5; 3 5 6 - 3 6 7 ) .

76

Hermann

strenger

Form

Giinther

bearbeitet

Grassmann,

(Berlin,

1862).

Die

Ausdehnungslehre:

Republished

as

vol.

Vollstandig

I,

pt.

II,

und

of

the

in work

c i t e d i n n o t e (4) a b o v e . 7 7

G r u n e r t wrote this note a n d M o b i u s m a y have w r i t t e n a note.

See (5; 2 3 1 - 2 3 2 ,

223-224). 78

Hermann

Giinther

Grassmann,

Worterbuch

zum

Rig-Veda,

6

parts

(Leipzig,

1872-1875). 79

Rig-Veda,

George in

his 8 0

die 82

in

reine

Mathematische

und

This

Giinther

angewandte is

Grassmann,

work.

Annalen,

dehnungslehre"

in

"Die

7

83

significant

Willard

parts

(Leipzig,

1876-1877).

philological achievement

Gibbs

Grassmann,

Algebra

538-548,

"Zur (1877),

see

57-64,

Gibbs'

and

Haven,

1962),

Mechanik

nach

(1877),

Ausdehnungs-

(4,11,1; in

256-267).

C r e l l e ' s Journal fur

(4,11,11;

first

came

letter to Schlegel

"Die

12

die

in

in

it Gibbs

(New

Annalen,

und

and

Elektrodynamik"

in that through

this

Mathematische

neuere

(1874),

Grassmann,

Concerning

Giinther

2

of Grassmann's

(1944), 326-330.

Mathematik,

also

W h e e l e r , Josiah

Hermann

Isis, 3 5

Grassmann,

Giinther

paper

Grassmann's

8 3

in

Giinther

Hermann

Phelps

Hermann

"Grassmann"

Hermann

lehre" 81

trans.

Sarton discussed the significance

203-210). to

given

know in

of

Lynde

108. den

Prinzipien

222-240,

and

in

der Aus(4,11,11;

46-72). 8 4

der

Hermann

(4,11,1; 85

Giinther Grassmann,

Ausdehnungslehre"

in

"Der Ort

Mathematische

der

Annalen,

Hamiltonschen 12

(1877),

Quaternionen

375-386,

and

in in

268-282).

F o r f u r t h e r details see the next chapter.

107

A H i s t o r y of V e c t o r Analysis 86

For

a

fuller

discussion

Ausdehnungslehre"

in

of

period

see

Zeitschrift

fur

Victor

Schlegel,

Mathematik

und

"Die

Physik,

Grassmann'sche

41

(1896),

1-21,

41-59. 87

Felix

Klein,

hundert ( i n 2

Vorlesungen

uber

die

Entwicklung

parts, p u b l i s h e d as o n e v o l u m e ) , pt.

der

1, ed.

Mathematik

R.

im

19.

Jahr-

C o u r a n t a n d O. N e u g e -

b a u e r ; p t . 2 , e d . R . C o u r a n t a n d St. C o h n - V o s s e n ( N e w Y o r k , 1 9 5 6 ) , p t . 1 , 1 8 1 - 1 8 2 . 88

See

the

Gibbs,

letter to

Gibbs

from

Hermann

89

S e e E n g e l ' s r e m a r k s i n (4,1,1; v i ) .

90

Cargill

of

the 9 1

in

G r a s s m a n n , Jr.,

as

given by Wheeler,

116-117.

Gilston

Royal

Knott,

Society

"Recent

of Edinburgh,

Innovations

19

(1892),

in

Vector

Theory"

in

Proceedings

212.

T h e a b o v e i n f o r m a t i o n has b e e n t a k e n f r o m C h a r l e s Parish, " M a t t h e w O ' B r i e n " Dictionary

o f National

Biography,

vol.

XIV

(Oxford,

be the best of the extremely f e w sources 92

T h e following constitutes

to vector analysis. bolical

and

Mechanics"

Elastic

Medium,

whether

seems

to

31

of

paper

IV),

Philosophical

Magazine,

" O n the Symbolical Vibratory Motion of an

Crystallized

Ser.,

which

"Contributions towards a System of Sym-

(Abstract

3 r d Ser., 3 1 ( 1 8 4 7 ) , 1 3 9 - 1 4 3 ; (2)

Philosophical Magazine, 3 r d

794,

a complete bibliography of O'Brien's papers relevant

(1) O ' B r i e n , M a t t h e w .

Geometry

1917),

on O'Brien.

(1847),

or

Uncrystallized"

376-380;

(Abstract

(3)

of paper

"On

V),

a N e w Nota-

t i o n for expressing various C o n d i t i o n s a n d E q u a t i o n s in G e o m e t r y , Mechanics, a n d Astronomy," 428;

Transactions

(4)

Mechanics," 507;

of

the

Cambridge

"Contributions Transactions

(5)

"On

Medium,

Product

the

whether

Philosophical

Society, of

8

a

3 9 7 ; (7) 491-495;

of

towards

the

Symbolic

(1849),

" O n

Symbolical

(8)

a

"On

Symbolical

tion

of

Royal (9)

the

Society

"On

Translation o f London,

142

of

a

(1852),

Magazine,

Philosophical

is

of an of

the

Elastic Cambridge

Interpretation 1

4th

of t h e

(1851), Ser.,

Philosophical

and 497-

2

394-

(1851),

Magazine,

4th

derived from the Concep-

Magnitude," This

415-

(1849),

Ser.,

Magazine,

Mechanics,"

161-206.

8

Motion

4th

Symbolic Forms

Directed

(1849), Geometry

Transactions

" O n the

Statics,"

Ser., 2 ( 1 8 5 1 ) , 1 2 1 - 1 2 5 ; (9)

Society,

of Vibratory

Philosophical

8

Symbolical

Uncrystallized,"

(6)

Force,"

Society,

of

Philosophical

Equation

or

508-523;

and

System

Cambridge

Crystallized

Line

Philosophical

a

Philosophical

the

work

Transactions

referred

to

o f the

in

note

above.

93

O ' B r i e n w o r k e d w i t h i n a tradition c o m m o n a m o n g British algebraists of the day

w h o w e r e interested in treating algebra as a science of symbols. A n u m b e r of important D.

mathematicians F.

worked

Gregory, and De

within

this

tradition,

including

Peacock,

Hamilton,

Morgan. T h e points of v i e w of these m e n of course differed,

a n d O ' B r i e n s h a r e d m o s t s y m p a t h i e s w i t h G r e g o r y , as he e x p l i c i t l y stated. 94

Sir

William

of the preface. "In

one

of

praise."

Rowan

Hamilton,

Lectures

on

Quaternions

(Dublin,

1853),

(9;

163)

p.

(64)

See the footnote on that page for O ' B r i e n ' s name. C. G. K n o t t stated:

his

early

Knott,

papers

"Recent

Hamilton

Innovations,"

refers p.

to

213.

O'Brien's

Knott

work

however

in

terms

gave

no

of high

reference

to this paper, a n d a search of all H a m i l t o n ' s papers f r o m 1844 to 1858 — i n c l u d i n g a check

of

his

biography

Hamilton,

3

vols.

[Dublin,

(Robert

Perceval

Graves,

1882-1889]) —produced

no

The

Life

of

information

Sir

William

Rowan

whatsoever con-

c e r n i n g such a paper a n d i n d e e d m a d e it doubtful that H a m i l t o n was even familiar w i t h O'Brien's w o r k , since in his papers a n d letters he often d i d discuss papers by his 95

contemporaries relating to quaternions. This

paper

Philosophical

108

was

Society

a

synopsis and

of a

published

paper in

the

delivered Transactions

in

1846

of that

to

the

Society

Cambridge in

1849.

CHAPTER

Traditions Analysis

I.

from

FOUR

in

the

Vectorial

Middle

Period

of Its

History

Introduction As

a

first

approximation

divided into three

the

periods.

history

of vector analysis

The first period may be

may

be

characterized

as the t i m e w h e n mathematicians searched for, discovered, a n d developed

systems

of hypercomplex

numbers

which

could be

used

for space analysis. T h i s p e r i o d began in the late e i g h t e e n t h century with Wessel; Hamilton when

its c o n c l u s i o n i s b e s t set a s

died.

some

The

of the

1865, the year in w h i c h

second period may be

vectorial

systems

described as

of the

first

period

the time were

dis-

cussed, tested, a n d i n s o m e cases e x t e n d e d . T h i s p e r i o d w a s m o r e a

time

of

during

"recognitions"

this

vectorial

time

a

methods

than

number and

in

of discoveries.

of scientists

s o m e cases

to be looked for in a vectorial

Thus

for

recognized

the

example need

for

specified the characteristics

system. T h e year 1880 m a y be taken

as the terminus for this p e r i o d as w e l l as the b e g i n n i n g of the third period,

which

modern

system

extended of

vector

to

1910.

analysis

During was

the

created,

third

period

discussed,

the and

accepted. T h e a b o v e p e r i o d i z a t i o n i s o f c o u r s e n o t e n t i r e l y satisfactory; its value

and

accuracy

as

well

as

its

shortcomings

will

become

ap-

parent f r o m later sections.

Its i m m e d i a t e relevance is that in terms

of

present

it

the

function

of

the

chapter

may

be

defined:

this

chapter attempts an analysis of the second, or m i d d l e , period of the history

of vector analysis.

T h e central figures in this chapter are Tait, Peirce, M a x w e l l , a n d Clifford.

The

selection

of the

year

1880

as

the

terminus

of this

109

A H i s t o r y of V e c t o r Analysis

period should be v i e w e d as somewhat arbitrary, even though it is true that M a x w e l l a n d C l i f f o r d d i e d i n 1879 a n d Peirce i n 1880, a n d that Gibbs' first presentation of m o d e r n vector analysis appeared in 1881. N e v e r t h e l e s s the transitions that o c c u r r e d in these m e n in the p e r i o d 1865 to 1880 took place in the m i n d s of lesser m e n at a later date. More

specifically,

this

chapter is a i m e d at presenting a discus-

sion of the degree and k i n d of reception accorded to the two major traditions

that

emerged

from

the

first

period;

these

were

the

H a m i l t o n i a n and the Grassmannian traditions. T h e second of these has b r i e f l y b e e n d i s c u s s e d for this p e r i o d o f t i m e i n t h e last c h a p t e r , w h i c h was natural since G r a s s m a n n d i e d i n 1878. H o w e v e r there i s m o r e to be said in this regard. Evidence tion

will

became

be

supplied to s h o w that the

H a m i l t o n i a n tradi-

the more vigorous, and activities and transformations

w i t h i n this tradition w i l l be especially analyzed, since it was from this t r a d i t i o n that t h e m o d e r n s y s t e m o f v e c t o r analysis arose. T h e four m e n emphasized in this chapter may at first sight seem strange after to

bedfellows.

Hamilton's

quaternions.

Tait

death;

was

the

Maxwell,

Similarly

Peirce

great

champion

of quaternions

on the other hand, was opposed shared Tait's

enthusiasm,

while

C l i f f o r d c o u l d not. Be that as it m a y , in this chapter it w i l l be s h o w n that

these

tradition;

men

represent

various

traditions

within

the

vectorial

i n later c h a p t e r s i t w i l l b e s h o w n t h a t s o m e o f t h e s e tra-

ditions p l a y e d a decisive role in the history of vector analysis.

II.

Interest 1841

to

in

Vectorial

Analysis

in

Various

Countries

from

1900

T h e r e a r e n u m e r o u s w a y s i n w h i c h a n h i s t o r i c a l s t u d y o f t h e acceptance of a mathematical system m a y be conducted. A m o n g these various

methods

is

the

quantitative

study of the number of works

p u b l i s h e d in the system as related to certain variables, for instance, time

and

methods

country have

of

been

publication. used

in

an

In

the

present

attempt to

analyze

section the

such

level

of

interest in the Grassmannian and the Hamiltonian traditions in the period from

1841

to

1900.

Works

b e e n c l a s s i f i e d i n t e r m s o f (1) publication, intervals

from these two traditions

have

n a t u r e o f p u b l i c a t i o n , (2) s u b j e c t o f

(3) t i m e o f p u b l i c a t i o n a s b r o k e n d o w n i n t o f i v e - y e a r

from

1841

to

1900,

(4)

country

of publication,

and

(5)

a u t h o r in s o m e cases. It is t h e a u t h o r ' s b e l i e f t h a t s u c h a s t u d y does not replace analysis

110

historical

which

can

interpretation;

complement the

it is

simply another form of

more traditional

techniques.9

Traditions in Vectorial Analysis

84

84

84

76

68

63

1841

» 1851 GRAPH

I.

•

1 8 6 1 — 1 8 7 1

Quaternion Publications from

T h e first major

finding

>1881

»

1891

1841 to 1900.

d e r i v e d f r o m this study is that d u r i n g the

period from 1841 to 1900 there w e r e 594 quaternion publications as c o m p a r e d to 217 Grassmannian analysis publications.10 H e n c e 73.2 percent there

of the

were

publications

2.73

quaternion

were

in

the

quaternion

publications

for

each

tradition,

or

Grassmannian

publication. The

results

obtained

strikingly similar. published from

when

only

books

were

considered

was

By actual count there were 38 quaternion books

1841

to

1900, whereas there w e r e

16 books pub-

lished d u r i n g this p e r i o d in the G r a s s m a n n i a n tradition. T h u s 70.4 percent

of the

books

dealt w i t h

quaternions,

or there

were

2.37

quaternion books for each book in the Grassmannian tradition. T h e quaternion

books

Grassmannian

averaged 281

tradition,

249

pages in length;

pages.11

the books of the

F r o m these numbers it may

111

A H i s t o r y of V e c t o r Analysis 8

6

6 5 4

4 3

1841 GRAPH

II.

1871

1861

1851

Quaternion Books from

1841 to

1891

1881

1900.

be inferred that interest in the tradition b e g u n w i t h H a m i l t o n was far greater t h a n that b e g u n w i t h G r a s s m a n n . T h e s e n u m b e r s have been broken d o w n into five-year intervals in Graphs I, II, IV, and V. Graph I shows the n u m b e r of quaternion publications in terms of five-year intervals Hamilton the

from

h i m s e l f are

1841 t o 1900. T h e p u b l i c a t i o n s w r i t t e n b y

indicated by

s o l i d areas.

n u m b e r of quaternion books for the

books by

Hamilton

Graph II

presents

same time intervals.

The

(including a translation and a second edition)

are i n d i c a t e d b y s o l i d areas. From

Graphs

I

and II the following conclusions may be drawn.

I n t e r e s t i n q u a t e r n i o n analysis w a s a t its h i g h e s t l e v e l d u r i n g t h e 1876-1900

period.

The

decrease

in

interest for the period

1881-

1885 indicated by G r a p h I is balanced by the peaking of G r a p h II for the same interval. It is important to note that H a m i l t o n wrote 73 percent of the pre-1866 quaternion publications and 19 percent of all

quaternion

publications.

It w o u l d of course be significant to compare the form of Graph I w i t h a graph s h o w i n g the rate of increase of m a t h e m a t i c a l publications

d u r i n g this

quaternion

time.

publications

Some after

idea of h o w the 1870

compares

rate with

of increase the

rate

of

of in-

crease in mathematical publications in general m a y be obtained by means of the study m a d e by H. S. W h i t e in 1915 based on an analysis

of

Graph

works III

listed

in

the

journal

Fortschritte

der

Mathematik.12

shows the n u m b e r of titles of mathematical articles a n d

b o o k s p u b l i s h e d i n t h e p e r i o d 1868 t o 1909.13 W h e n G r a p h s I a n d I I I are c o m p a r e d , it seems at first sight that interest in evident

quaternions

that

the

was

percentage

declining from

1876 to

of mathematical

1900, for it is

literature

that

was

d e v o t e d to quaternions decreases slightly. B u t this seems to be an

112

Traditions in Vectorial Analysis

1870

GRAPH

III.

Annual

'SO

N u m b e r of Titles

'90

1900

1910

of Mathematical Articles

and Books,

1868-

1909.

erroneous conclusion, for even m o r e

striking than the increase in

the n u m b e r of mathematical publications d u r i n g this period is the increase in the n u m b e r of fields of mathematical research. N u m e r ous fields —such as

non-Euclidean

geometry, mathematical logic,

group theory, as w e l l as m a n y branches of a p p l i e d mathematics — came into p r o m i n e n c e in this period. 68

48

28

16

1841

• GRAPH

IV.

1851

»

1861

>

1871

16

» 188T

Grassmannian Analysis Publications from 1841 to

»

1891

1900.

113

A H i s t o r y of V e c t o r Analysis 4

1841

M 851 GRAPH V.

1861

>1871

Grassmannian Analysis Books from

Graphs mannian

^ 1 8 8 1 — — •

1841

to

1 8 91

1900.

IV a n d V present the results of a s i m i l a r study of Grassanalysis

publications

and

books.

The

G r a s s m a n n are i n d i c a t e d b y t h e s o l i d areas.

works

written by

F r o m these graphs it

becomes clear that the b e g i n n i n g of the m a i n period of interest in Grassmannian

analysis

similar

for

period

other hand, the

came

roughly

quaternions

(1891

fifteen

years

compared

to

later than the 1876).

On the

n u m b e r of Grassmannian analysis publications for

the p e r i o d 1896 to 1900 was approaching the n u m b e r of quaternion publications

for the

same interval.

In regard to Grassmann's per-

358

88

52

52 44

American

British

GRAPH

114

VI.

Quaternion

Publications

French

by Country.

German

Other

Traditions in Vectorial Analysis

10

10 Spanish Russian

8

8

Polish Czech

Portuguese

Japanese

2

British

GRAPH

VII.

Quaternion

German

French

American

Dutch

Books by Country.

sonal contribution it is

noteworthy that he published 25 of the 33

(or 7 6 p e r c e n t ) o f t h e p u b l i c a t i o n s u p t o 1875, a n d 3 3 o f t h e total o f 2 1 7 (or 15 p e r c e n t ) of G r a s s m a n n i a n analysis p u b l i c a t i o n s . A

study

of the

two

fields

in

terms

of interest by country

is

of

significance. T h u s Graphs V I and V I I represent quaternion publications and books respectively as classified into five groups: British, American,

French,

German,

and those

of other countries. Quater-

nion books appeared in ten languages as follows (the n u m b e r after 125

32 28

16

16

GRAPH

French

American

British

VIII.

Grassmannian given

height

Analysis on

this

height on Graph VI

Publications

scale

indicates

indicates

2x

Other

German

by x

Country.

(Note

publications,

that

then

an

if any equal

publications.)

115

A H i s t o r y of V e c t o r Analysis

12

1 American GRAPH

each

IX.

Grassmannian Analysis

language

which

French

indicates

the

German

Italian

Books by Country.

number

of books):

English

(12,

of

1 0 w e r e p u b l i s h e d i n B r i t a i n a n d 2 i n A m e r i c a ) , F r e n c h (8),

German

(8),

Dutch

(2),

Japanese

(2),

P o r t u g u e s e (2), C z e c h o s l o -

v a k i a n (1), P o l i s h (1), R u s s i a n (1), a n d S p a n i s h (1). T h e r e w e r e i n a d d i t i o n a n u m b e r of papers in Italian, at least one in D a n i s h , a n d at least one tive

study

paper was p u b l i s h e d in Australia. F r o m this quantita-

it

may

be

inferred that 60

(books and papers) w e r e British,

percent

of the publications

15 percent w e r e American, 9 per-

cent were French, 8 percent German, with the remaining 8 percent c o m i n g from other countries. This should be considered in relation to

the

fact

that

26

percent

of the

books

on

quaternions

were of

British origin, 5 percent of A m e r i c a n origin (the British books w e r e of course

often

used by Americans), 21

percent

of German

guages.

These

origin,

statistics

and

the

percent of French and 21

remaining were

in

other

lan-

point out that interest in quaternions was

strongest in Britain but was substantial in America, Germany, and France, a n d that it e x t e n d e d to most of the then intellectually productive

countries of the

Graphs

VIII

and

IX

world. represent

Grassmannian

analysis

publica-

tions a n d books r e s p e c t i v e l y that are classified on the same basis of country

of publication.

four countries:

Grassmannian analysis books

each in A m e r i c a and Italy. the

appeared in

12 w e r e p u b l i s h e d in G e r m a n y , 2 in France, and 1 T h e study revealed that 57 percent of

Grassmannian analysis publications appeared in Germany, 18

p e r c e n t i n A m e r i c a , 10.5 p e r c e n t i n b o t h B r i t a i n a n d F r a n c e , w i t h a

116

Traditions in Vectorial Analysis

few

works

appearing

in

Polish,

Italian,

Spanish,

Russian,

and

analysis

was

Czechoslovakian. Thus

it

appears

centralized

in

that

Germany,

interest with

in

less

Grassmannian proportionate

interest outside

Germany than the interest in quaternions outside Britain. It is notew o r t h y that for b o t h systems the country in w h i c h the most interest d e v e l o p e d after the m o t h e r country of the analysis was America.14 This study m a y be summarized by the statement that the interest in quaternion analysis was roughly t w o and one-half times as great as interest in Grassmannian analysis and extended to m o r e countries, w i t h greater interest p r o p o r t i o n a t e l y d e v e l o p i n g i n c o u n t r i e s outside the country in w h i c h the system originated. To this m a y be added the observation that there was substantial interest in quaternions from 1876 to 1900 and that although interest in Grassmannian analysis came s o m e w h a t later, it d i d by the p e r i o d 1 8 9 1 - 1 9 0 0 attain substantial It is

magnitude.

the author's belief that this

quantified study tells no m o r e

than part of the story. It does h o w e v e r s u p p l y a v a l u a b l e perspective i n t o w h i c h d e v e l o p m e n t s d i s c u s s e d i n l a t e r s e c t i o n s m a y b e set.

III.

Peter

Guthrie

Tait:

Advocate

and

Developer

of

Quaternions

T h e i m p o r t a n c e o f T a i t f o r t h i s h i s t o r y i s f o u r f o l d . (1) H e w a s t h e a c k n o w l e d g e d leader of the quaternion analysts f r o m 1865 until his death in

1901. I n d e e d e i g h t b o o k s o n q u a t e r n i o n s ( i n c l u d i n g later

editions, translations, and coauthorships) carried his title page.

name on the

(2) T a i t d e v e l o p e d q u a t e r n i o n a n a l y s i s a s a t o o l f o r re-

s e a r c h i n p h y s i c a l s c i e n c e (as H a m i l t o n h a d n o t ) a n d c r e a t e d m a n y n e w theorems in quaternion analysis w h i c h w e r e capable of b e i n g t r a n s l a t e d i n t o m o d e r n v e c t o r a n a l y s i s . (3) I t w a s p r o b a b l y t h r o u g h Tait that M a x w e l l b e c a m e interested in quaternions.

(4) T a i t w a s

the most important o p p o n e n t of m o d e r n vector analysis. Peter Guthrie Tait was born in

1831 near E d i n b u r g h , Scotland.

In 1841 he entered E d i n b u r g h A c a d e m y w h e r e one year earlier the young

James

Clerk

Maxwell

had

enrolled.

Playmates

in

their

y o u t h , t h e t w o b e c a m e fast f r i e n d s a n d f r e q u e n t c o r r e s p o n d e n t s i n their maturity. was

M a x w e l l ' s 1846 entrance into E d i n b u r g h University

followed by Tait's in

1847, w i t h the o r d e r o f e n t r y b e i n g re-

versed w h e n Tait left for C a m b r i d g e

after one

year at E d i n b u r g h

University, w h i l e M a x w e l l stayed for three. After Tait's graduation in

1852

elected

as a

Senior

Fellow

Wranger

and

of Peterhouse

First

Smith's

College,

writing the first of his m a n y books.

Prizeman

Cambridge,

and

he

was

began

This was coauthored by W. J.

117

A H i s t o r y of V e c t o r Analysis

Steele of a

(who

died

Particle.

Lectures

In

on

ordered

before

1853

Quaternions. your

Athenaeum, caught my

its

Tait As

book,

c o m p l e t i o n ) a n d w a s e n t i t l e d Dynamics

ordered

Tait

on

later

a

copy

wrote

account

of

an

of t h e j u s t - p u b l i s h e d

to

Hamilton:

"when

advertisement

in

I

the

I had NO I D E A w h a t it was about. T h e startling title

eye

in

August

'53,

and as

I

was

going off to shooting

quarters I took it and some scribbling paper w i t h me to beguile the t i m e . . . . H o w e v e r as I t o l d y o u in my first letter I got easily e n o u g h t h r o u g h t h e first six L e c t u r e s . . . ." (1; 126) On

his

return

quaternions, he

to

Cambridge

primarily

was writing.

In

at Queen's

volved

in teaching as Thomas

of the

study of

labor involved in the book

1854 he accepted the Professorship in Mathe-

matics

league

Tait d i d not continue his

because

College, well

Andrews.

Belfast,

Ireland.

Here

he became in-

as in e x p e r i m e n t a l w o r k w i t h his colHe

also

pursued

the

study

of

the

' T h e o r i e s o f H e a t , E l e c t r i c i t y a n d L i g h t . " F i n a l l y i n A u g u s t , 1857, his

interest

in

quaternions

returned

as

a result of reading H e l m -

holtz' famous paper on vortex motion. T h e n as Tait wrote to Hamilton:

"I

suddenly bethought me of certain formulae I had admired

years ago at p. 610 of y o u r Lectures — a n d w h i c h I t h o u g h t (and still think) likely to serve my purpose exactly." to

which

Tait

referred

"<1 = i ^ + j ^ + k

'

were 15

merous

other

statements

was

their

physical

for

those

It is by

on

Tait's

that his

applications.

behalf to

127) T h e f o r m u l a e with

the

operator

clear from the above (and from nu-

Tait)

From

quaternions c o n t i n u e d to increase, and in Hamilton

(1;

associated

request

interest in quaternions 1857 Tait's

interest in

1858 A n d r e w s wrote to

Hamilton

to

allow

Tait

to

correspond w i t h him. Hamilton responded cordially, and a voluminous

correspondence

ensued.

Perhaps

fifty

letters

o n e o f w h i c h w a s n i n e t y - s i x p a g e s i n l e n g t h . (1;

were

written,

141)

H a m i l t o n soon realized that Tait was a gifted mathematician and in

1859 encouraged Tait to p u b l i s h a paper based on a quaternion

investigation of the F r e s n e l w a v e surface that T a i t h a d carried out. This Tait did, a n d thus the first of his numerous quaternion papers appeared in

1859.

In the same year Tait was encouraged by some

Cambridge friends to write a book on quaternions. Since Hamilton had

c o m m u n i c a t e d u n p u b l i s h e d results to Tait, it was natural for

Tait

to

though

ask

Hamilton's

Tait

intended

permission that

Hamilton

asked

T a i t to

Quaternions,

which

was

at

to publish

it

would

wait

for the

that

time

consist

such

publication planned

a book,

mainly

as

of his an

even

of examples. Elements

of

elementary

work. Tait agreed to this request, a n d thus his book on quaternions

118

Traditions

was

not

same

published

time John

elementary

until

1867.

It

is

in Vectorial

interesting

that

at

Analysis

nearly

the

H e r s c h e l (at a g e 72) w a s e n g a g e d i n p r e p a r i n g a n

work

on

quaternions,

but

it was

never published.

(1;

141-142) In

1860 Tait was

at Edinburgh

appointed

to

the

U n i v e r s i t y after he

Chair of Natural

had been

Philosophy

selected for this

posi-

tion

from a noteworthy group of candidates that included Maxwell.

Tait

remained

quished scientific written

books by

Edinburgh as

an

365

until

excellent

(twenty-two

him),

published nions.

at

himself

and

as

papers,

his

books

a

death

lecturer, were

in

an

1901

and

important

distin-

writer

of

either wholly or partially

productive

scientific

researcher.

of which approximately 70 were

on

He

quater-

T h e c o m m o n l y accepted v i e w of his importance as a scientist

places h i m b e l o w the group consisting of such m e n as Kelvin, Maxwell,

and

Stokes,

professors Soon liam

after

arriving

Thomson

Originally

in

referred pared

to

authors

as

the

and the

work

on

readers opposed

entitled

for

became

Principia

works

a r r a n g e d for its portunity

Tait began as

1892

to

Lord

Treatise survey

writing

Kelvin) on

all

of the

translation

mathematical

to

introduced.

into

physics

Tait's

century

and Laplace. German.

would

methods.

introduction

After

nineteenth

This

In was

of

death

have

Kelvin

in

science, was

pub-

was

often

was

com-

and

1871

Helmholtz

the

golden op-

such an

important

acquainted

numerous

however

quaternions, Kelvin

it

on

Philosophy.

This

success;

Wil-

work

of physical

for their inclusion in

quaternion the

immediate

of Lagrange

quaternions,

with

an

with

a

Natural

volume on mechanics was completed.

1867 to

Edinburgh,

planned

scientific

Britain.

from

physics

the

but only the

at

(known

mathematical

lished

but certainly above the majority of the

of nineteenth-century

and

1901

was

they

wrote

strongly were

to

not

George

Chrystal: We

[Kelvin

nions.

He

and had

Tait] been

have

h a d a thirty-eight years'

captivated

by

the

originality

war over quaterand

extraordinary

beauty of H a m i l t o n ' s genius in this respect, a n d h a d accepted, I believe, definitely f r o m H a m i l t o n to take charge of quaternions after his death, w h i c h h e has m o s t l o y a l l y e x e c u t e d . T i m e s w i t h o u t n u m b e r I o f f e r e d t o let

quaternions

into

Thomson

and Tait

[the

Treatise],

if he

could

only

s h o w that i n a n y case o u r w o r k w o u l d b e h e l p e d b y t h e i r use. Y o u w i l l see that f r o m b e g i n n i n g t o e n d t h e y w e r e n e v e r i n t r o d u c e d . 1 6 Similarly latter's

Kelvin Algebra

wrote of

in

Coplanar

1892

to

Vectors

R. and

B.

Hayward concerning the Trigonometry:

"I

do

think,

h o w e v e r , that you w o u l d find it w o u l d lose n o t h i n g by o m i t t i n g the word

Vector'

throughout.

It adds

nothing to

the

clearness or sim-

119

A H i s t o r y of V e c t o r Analysis

plicity

of the

mensions.

geometry,

Quaternions

whether came

of two-dimensions

from

or

three-di-

H a m i l t o n after his really good

work had been done; and, though beautifully ingenious, have been an u n m i x e d evil to those w h o have t o u c h e d t h e m in any way, including Clerk Kelvin's

Maxwell."

strong

17

T h e f o l l o w i n g quotation indicates that

opposition

to

quaternions

probably

extended to

a n y sort of vector analysis. In 1896 K e l v i n w r o t e to G. F. F i t z G e r a l d in

regard

to

equations":

FitzGerald's

mention

"Symmetrical

of

equations

"vectors

are

and

good in

symmetrical

their place, but

V e c t o r ' is a useless survival, or offshoot, f r o m q u a t e r n i o n s , a n d has never

been

shunted hilism.

it,

of

the

but

slightest

unwisely

he

use

to

adopted

any

creature.

temporarily

Hertz

wisely

Heaviside's

ni-

He even tended to nihilism in dynamics, as I warned you

soon after his death. He w o u l d have g r o w n out of all this, I believe, 18

if he had lived." rejecting vectors the

feeling

W h a t e v e r m i g h t have b e e n Kelvin's reasons for

and quaternions, his opposition is illustrative of

of many

other physical

scientists

of the

time.

Kelvin

was p r o b a b l y the most influential British physicist of the late nineteenth century, and his opposition to vector methods, t h o u g h never put forward in a published form, was almost certainly well known. K n o t t p o i n t e d o u t (1; 188) t h a t T a i t c o u l d , a n d d i d , t a k e a d v a n t a g e of

the

Treatise

on

Natural

Philosophy

by

developing

in

quaternion

f o r m i n h i s o t h e r w r i t i n g s s o m e o f t h e a r e a s i n c l u d e d i n t h e Treatise, to show in this w a y the compactness that resulted from quaternion treatment.

T h u s at one point Tait covered in his quaternion book

in a matter of lines what had been covered in the jointly authored w o r k i n p a r a g r a p h s o r pages. (1; Tait

acted

in

development.

a

surprising

188)

manner on

one

aspect of quaternion

Although he published numerous papers and books

on quaternions, he d r e w a sharp line b e t w e e n his quaternion works and

his

other publications.

Thus

despite the fact that in m a n y of

his books h e c o u l d have u s e d the q u a t e r n i o n m e t h o d , h e d i d not. O n e example of this edition pect

of

the

that Tait

is his

l o n g article " M e c h a n i c s " for the n i n t h

Encyclopaedia

in

his

Britannica.

mathematical

w o u l d have used quaternions do.

O n e of Tait's

"Tait,

so

far

as

students I

know,

since

one

courses

w h e r e v e r possible.

would

ex-

at Edinburgh

This he did not

wrote in a biographical sketch of Tait: never

lectured

nions] at the University of Edinburgh." unnecessary

Similarly

physics

19

on

the

subject [quater-

To do so was in one way

K e l l a n d , the Professor of M a t h e m a t i c s , d i d dis-

cuss q u a t e r n i o n s in his courses; on the other h a n d , this m a y actually make it more surprising that Tait probably made no use of t h e m in his lectures on physics. Nevertheless at least t w o of Tait's students,

120

Traditions in Vectorial Analysis

Knott and Macfarlane, became important contributors to quaternion analysis. Although matics

Tait

possessed

had

received

and

a high order of ability in incomparable

training

pure mathe-

in

quaternions

through his correspondence w i t h H a m i l t o n , his strongest interests were in physics rather than pure mathematics. This is reflected in the fact that his c h a n g e f r o m than a change in

location:

Belfast to E d i n b u r g h i n v o l v e d more

he had been Professor of Mathematics;

h e w a s n o w ( a n d h a p p i l y so) P r o f e s s o r o f N a t u r a l P h i l o s o p h y . T a i t referred to this change in professional chair and to a consequence thereof

in

(1867):

"The

Professor

the

of

preface

to

Elementary

Treatise

present work was c o m m e n c e d in Mathematics,

and

far

more

on

Quaternions

1859, w h i l e

ready

at

I

was

Quaternion

analysis that I can n o w p r e t e n d to be. . . . T h e d u t i e s of a n o t h e r Chair,

and

appear

till

Sir

W.

after

Hamilton's

the

wish

publication

already extensive preparations."

that

my

his

Elements,

of

volume

should

not

interrupted

my

20

As it was m e n t i o n e d before, eight books on quaternions (includi n g later editions, translations, a n d coauthorships) are c r e d i t e d to Tait.

His

Elementary

Treatise

on

Quaternions

of

1867

was

re-

p u b l i s h e d in a " S e c o n d E d i t i o n , E n l a r g e d " in 1873, a n d in a " T h i r d Edition,

Much

Enlarged"

1890.21

in

Tait's

Treatise w a s

translated

into G e r m a n b y v o n Scherff (1880) a n d into F r e n c h b y Plarr (1884). D e s p i t e its t i t l e t h e w o r k ( e s p e c i a l l y i n t h e later e d i t i o n s ) w a s n o t really elementary. by

the

book

The

n e e d for an e l e m e n t a r y w o r k was fulfilled

Introduction

to

Quaternions

by

Philip

Kelland

and

Tait. This jointly authored w o r k appeared in 1873 a n d was f o l l o w e d by a second edition in 1882 a n d a t h i r d e d i t i o n in 1904 w i t h C. G. Knott as editor.

Tait essentially wrote only one chapter (the tenth

a n d last) of t h i s book.22 N o n e t h e l e s s it w a s in a r e a l s e n s e a j o i n t work, for as K e l l a n d wrote in his preface, Tait " b e i n g my p u p i l in youth

is

my teacher in

r i p e r years. . . . " 2 3 K e l l a n d also w r o t e :

"I

must confess that my k n o w l e d g e of Quaternions is d u e exclusively to

him.

nions,

The

was

first

very

work

dimly

of Sir

and

Wm.

Hamilton,

imperfectly

Lectures

understood

by

on me

Quaterand

I

d a r e say b y o t h e r s , u n t i l P r o f e s s o r T a i t p u b l i s h e d h i s p a p e r s o n t h e subject Tait's

in

the

Elementary

Messenger Treatise

of on

Mathematics." Quaternions

24

was

written

in

such

a

w a y a s t o stress t h e a p p l i c a t i o n s o f q u a t e r n i o n s t o p h y s i c a l s c i e n c e . Thus

approximately

62

(of 320) pages

in the first edition, 93

(of

298) pages of the second edition, a n d 130 (of 422) pages of the t h i r d edition

were

devoted

to

physical

applications.

Furthermore

the

other chapters were directed towards preparing for the final chap-

121

A H i s t o r y of V e c t o r Analysis

ters

on

Treatise from

physical

for this this

applications.

study is

edition

the

The

most

important edition

second edition,

that both

Gibbs

and

since

it was

Heaviside

of the

probably

learned quater-

nions. Chapter

I

position." addition scalar

of Tait's

In

this

and

Treatise

chapter

subtraction,

quantity.

He

is

entitled

Tait

and the

also treated

"Vectors

treated

and Their Com-

vector

equality,

multiplication

the

vector

of a vector by a

differentiation

of a vector in

t e r m s of a single scalar variable. A n o t e w o r t h y aspect of this chapter is that the q u a t e r n i o n n e v e r enters; the chapter c o u l d in fact serve as the first chapter of a m o d e r n vector analysis book, for the vector part of a quaternion m a n n e r as ample

is

added,

a m o d e r n vector.

of w h a t

is

true

to

subtracted, and so on in the same

This

c h a p t e r is o n l y an e x t r e m e ex-

a certain

extent of all

of Tait's

Treatise;

similarities b e t w e e n Tait's book a n d m o d e r n vector analysis books abound.

And

this

their ancestors. and

is

The

theorems25

to

be

expected:

offspring

usually

resemble

lineage of the vast majority of the basic ideas

found

in

modern

vector

analysis

books

can

be

t r a c e d b a c k t o o r t h r o u g h s u c h q u a t e r n i o n b o o k s a s T a i t ' s Treatise.26 Numerous

illustrations

of this all-important point w i l l be given in

this and the f o l l o w i n g chapter. Chapter

II

of Vectors." vectors

is

of Tait's Since

Treatise

in

general

is

entitled

"Products and Quotients

the product and the quotient of two

a quaternion, this chapter concentrated on quaternions.

H e r e again, b u t naturally to a lesser extent, m a n y of the theorems can

be

translated

into

modern

vector analysis.

M a n y of the simi-

larities d e r i v e f r o m the fact s h o w n before: the p r o d u c t of t w o vectors

in quaternion analysis is equal to the

(dot) p r o d u c t p l u s t h e i r v e c t o r (cross)

n e g a t i v e of t h e i r scalar

product. T h u s , for example,

T a i t ' s s t a t e m e n t s t h a t f o r a n y t w o v e c t o r s a a n d / 3 , " S a / 3 = S/3a" a n d "Va/3 = —Vf3a"

(2;

43)

are

vector analysis that the product is

anticommutative.

Chapter

III

Quaternion are

those

e q u i v a l e n t to the statements in m o d e r n

d o t p r o d u c t i s c o m m u t a t i v e a n d t h e cross

is

entitled

"Interpretations and Transformations of

Expressions." A m o n g the first formulae in the chapter for

a

and

/3

vectors:

"Saf3 = —TaT/3

cos

0"

and "Va/3 =

TaTfB s i n 0 • 77." ( 2 ; 4 9 ) T h e q u a t e r n i o n s y m b o l T a p p l i e d t o a v e c t o r indicates

its

l e n g t h , a n d t h e v e c t o r 17 is d e f i n e d as a u n i t v e c t o r

perpendicular to a and

T h u s the counterparts of these formulae

in m o d e r n vector analysis w o u l d be a • p = | a |

| p | cos 0

a X / } = (| a | | p | s i n 0 ) t j

122

Traditions in Vectorial Analysis

Then

follow numerous

examples

and a number of other theorems

w h i c h m a y also b e translated into the n o w c o m m o n vector analysis. This chapter concludes quaternions assume

in

w i t h a sketch of b i q u a t e r n i o n s , w h i c h are

w h i c h the

four scalar m u l t i p l e s

(of 1,

i, j,

k) m a y

imaginary values.

Chapter IV,

"Differentiation

o f Q u a t e r n i o n s , ' ' i s o n l y six pages

long and applies the processes d e v e l o p e d earlier for vector differentiation to quaternion differentiation. Tait began Chapter V by stating that "aSfBp

+

«iS/3,p +

.

.

.

=

X.aSfBp =

y"

which he abbreviated (following Hamilton) as "<£p

The

reader

=

X . a S p p "

acquainted

a n d

with

"<£p

the

=

y"

(2;

modern

78)

treatment

of the

linear

vector f u n c t i o n b y m e a n s o f dyadics w i l l see i m m e d i a t e l y that these equations are essentially ( w h e n translated) a f i . p

+

OLxpx.px

+

.

.

.

=

X a p . p

=

y

or .p = 2 a / 3 . p

or

.p = y

w h e r e <£> i s t h e m o d e r n d y a d i c . T a i t t h u s d e v o t e d t h i s c h a p t e r t o t h e development of the linear vector function. is

somewhat

means

similar

to

the

modern

His form of presentation

method

of

presentation

by

of dyadics.

The

next

four chapters

deal

with

the application

of quaternion

analysis to p r o b l e m s in geometry. F o r the present purposes all that need be said in regard to these chapters is that in t h e m the linear vector function

is

developed

further and the

operator V is intro-

duced, but not developed. T h e r e m a i n i n g t h i r d of the book consists of t w o chapters entitled "Kinematics" and "Physical Applications." Tait began the first of these

by

a discussion of Hamilton's hodograph

and f o l l o w e d this

by a treatment of the rotation of a rigid b o d y by quaternion methods and of homogeneous

strain by means of the

linear vector function.

It was perhaps in regard to methods of treatment for rotations a n d strains that q u a t e r n i o n analysis was of m o s t value. T h e final long chapter, entitled "Physical Applications," begins with

a

discussion

of some

theorems

in dynamics.

Tait began the

chapter by stating: " W h e n a n y forces act on a r i g i d b o d y , t h e force f3 at the point w h o s e vector is a, &c., then, if the b o d y be slightly displaced,

so

that

a

becomes

a + 8a, t h e w h o l e w o r k d o n e

is

XSfida.

123

A H i s t o r y of V e c t o r Analysis

T h i s m u s t vanish if the forces are such as to m a i n t a i n e q u i l i b r i u m . Hence

the

condition

of

equilibrium

of

a

rigid

body

is

XS^da

=

0."

(2; 2 2 2 ) T h e s e e q u a t i o n s express e x a c t l y t h e e q u i v a l e n t o f t h e e q u a tions

2/3.8a = 0

quaternionists product more

and

well

introduces

into

complicated

pendulum.

2/3.8a = 0

were

To

in

aware

modern

of the

mechanics.

matters

illustrate

such

as

Tait the

quaternionic

vector

analysis.

simplification naturally

proceeded to

treatment of the

applications

The

that the dot

in

Foucault

optics, Tait

treated the Fresnel w a v e surface, f o l l o w e d this w i t h a treatment of the effects of electric currents on o n e another a n d on magnets, a n d invited the reader to compare the mathematical methods w i t h those of

Ampere.

j

The

important

discussion

of t h e

operator

V = i -Jr +

+ k -J-^ i n c l u d e d s u c h t h e o r e m s a s t h o s e o f S t o k e s , G r e e n , a n d Gauss.

The book is

concluded by miscellaneous applications and

problems. From

the

Quaternions as

it

above should

presented

analysis.

by

discussion be

of

clear

that

had

many

Tait,

Vector addition

and

Tait's

quaternion

Elementary

Treatise

analysis,

especially

similarities

subtraction,

to

modern

on

vector

vector multiplication in

b o t h t h e scalar (dot) a n d v e c t o r (cross) p r o d u c t s , v e c t o r d i f f e r e n t i a tion, vector algebra, the properties of V, and even the linear vector function vector

were

present.

analysis,

The

and there

exact

were

form

some

was

not

sections

in

that

of modern

the

quaternion

treatment that c o u l d not be translated into m o d e r n vector analysis, but the over-all similarity is b e y o n d question. T h e function of the above

discussion

deriving

modern

presented

by

for

this

history

vector

analysis

The

possibility

Tait.

is

to

from

indicate

the

quaternion

possibility

analysis

of developing modern

of as

vector

analysis from Grassmannian analysis was previously discussed. T h e problem thus

clearly emerges:

D i d m o d e r n vector analysis emerge

historically from the quaternion or from the Grassmannian methods of

analysis? Tait's

contribution

to

quaternion

analysis

was

not

only

as

ex-

positor of k n o w n m e t h o d s b u t also as creator of n e w m e t h o d s , m a n y of which

were

later transferred to vector analysis. Tait m a d e im-

portant advances in the theory and application of the linear vector function,

but

more

operator V, w h i c h

important than this has

been

is his

called "nabla,"

development of the

"del,"

and "atled."27

H a m i l t o n i n t r o d u c e d t h i s o p e r a t o r , b u t a s M a x w e l l w r o t e : " T h e ext e n s i o n o f t h i s o p e r a t o r [ V ] t o v e c t o r d i s p l a c e m e n t s , a n d m o s t o f its further d e v e l o p m e n t , are d u e to Professor T a i t . "

124

28

Essentially what

Traditions in Vectorial Analysis

T a i t d i d w a s t o state s u c h i m p o r t a n t t h e o r e m s a s t h o s e o f G r e e n , Stokes, and Gauss in q u a t e r n i o n form, s h o w their applications, a n d develop

theorems.29

related

written

by Tait

Theorems"30

in

and

Among

this

regard

"On

Some

are

the

"On

most

important

Green's

Integrals."31

Quaternion

papers

and Other Allied Some

of

Tait's polemical articles on quaternions w i l l be discussed in a later chapter. Tait's up.

importance

Tait,

analysis

as in

direction

the

the

for this history m a y n o w be partially s u m m e d

leading

last t h i r d

advocate of the

and

developer

nineteenth

of

century,

quaternion

changed the

o f e m p h a s i s i n q u a t e r n i o n analysis t o w a r d its u s e f u l n e s s

a s a t o o l for p h y s i c a l science. H e d i d this b y d e v e l o p i n g a n d stressing those parts of the analysis w h i c h w e r e most useful for physical science.

This

transformation

of

quaternion

analysis

was

very

probably a necessary p r e l i m i n a r y for the d e v e l o p m e n t of the GibbsHeaviside

modern

vector analysis

from

quaternion analysis.

One

i m p o r t a n t e x a m p l e of this is the fact that in H a m i l t o n ' s w o r k s o n e can

find

mental

almost no discussion of the operator V, w h i c h is a funda-

part of m o d e r n

discussion could

be

of

this

found

vector analysis,

operator in

any

was

other

while

probably

in

Tait's works

fuller

mathematics

and

books

better of

the

the than

time.

Historically it turns out that this change in direction in quaternion analysis was

IV.

decisive for later developments.

Benjamin

Peirce:

Advocate

of

Quaternions

in

America

Tait was not the only advocate of quaternions nor was enthusiasm for

quaternions

and

most

Peirce,

confined to the

influential

who

did

British

advocates

more

than

of

anyone

Isles.

One

of the

quaternions else

to

was

develop

earliest

Benjamin interest

in

q u a t e r n i o n s i n t h e U n i t e d States. T h i s i s i m p o r t a n t , s i n c e this w a s the

homeland

of both

Gibbs

and Wilson.

Peirce

work directly within the quaternion tradition; achievement

(his

"Linear

Associative

did

no

creative

his greatest creative

Algebra")

is

however

inti-

mately linked to Hamilton's discoveries and can be v i e w e d as an excellent from

example

of the

tendency

of mathematicians

quaternions to important discoveries

outside the

to proceed quaternion

t r a d i t i o n . T h u s P e i r c e w i l l b e d i s c u s s e d a s (1) a f i g u r e i n f l u e n t i a l o n t h e h i s t o r y o f q u a t e r n i o n a n a l y s i s a n d a s (2) a an

important

tendency

that

was

associated

figure with

illustrative of

the

quaternion

tradition. Benjamin great

Peirce

American

is

generally considered to have been the first

mathematician.

Dirk

J.

Struik

referred

to

his

125

A

History

"Linear bution

of V e c t o r Analysis

Associative to

Algebra"

mathematics

as

"the

produced

first

in the

major

original

contri-

3 2

U n i t e d States."

As pro-

fessor of astronomy a n d mathematics at H a r v a r d f r o m 1833 until his death

in

1880

he

exerted

a

vast influence

on

mathematics

in the

U n i t e d States. It w a s s h o w n p r e v i o u s l y that t h e r e w a s a s u r p r i s i n g l y large interest in quaternion analysis in America; the greatest single cause of this was In

the

Mechanics that

of

having

Hamilton

was

for

last

to

this

major

the

quotation

which

from

shows

discovery.

publication

quaternions

mentioned

a

given,

attached

Peirce's

thusiasm

on

1855

Peirce

from

B e n j a m i n Peirce's advocacy of Hamilton's system.

chapter

A

the

Peirce's great

second

quotation

demonstrates

that

did not diminish with time.

symbol

for

the

square

Analytical

importance

root

taken

his

en-

Peirce, after

of minus

one,

wrote: T h i s s y m b o l is restricted to a precise signification as the representative o f p e r p e n d i c u l a r i t y i n q u a t e r n i o n s , a n d this w o n d e r f u l algebra o f space is

intimately

symmetry, has

dependent

elegance,

upon

and

the

power.

special

The

use

of the

immortal

s h o w n that t h e r e are o t h e r significations

symbol

for its

author of quaternions

w h i c h m a y attach to the

s y m b o l i n o t h e r cases. B u t t h e s t r o n g e s t u s e o f t h e s y m b o l i s t o b e f o u n d i n its m a g i c a l p o w e r o f d o u b l i n g t h e a c t u a l u n i v e r s e , a n d p l a c i n g b y its s i d e a n i d e a l u n i v e r s e , its e x a c t c o u n t e r p a r t , w i t h w h i c h i t c a n b e c o m pared and contrasted, and, by means of curiously connecting fibres, form w i t h i t a n o r g a n i c w h o l e , f r o m w h i c h m o d e r n a n a l y s i s has d e v e l o p e d h e r geometry.33

surpassing Peirce's in

enthusiasm

that year he

seems

whether Peirce seems

taught his

probable.

quaternions students

as

One

subject in his mathe-

to

Hill

was

course every year, but such

publish

in

subject,"

quaternion

E. 35

Byerly,

referred to

a n d a n u m b e r of his

analysis.

A m o n g these

(who became president of Harvard in

Lowell

became

quaternion

"favorite

(also

a

chancellor

Charles Santiago Saunders son

back to at least 1848;

student of Peirce, W.

Peirce's

went on

Thomas

Lawrence (who

dates

i n c l u d e d this

courses at Harvard.34 It is perhaps impossible to determine

matics

were

for quaternions to have

president of Harvard), Arnold of

Brown),

and

two

of

1862), A . B.

Peirce's

Chase sons,

Peirce a n d James M i l l Peirce. T h e latter

professor of mathematics

at Harvard from

1869-1906 and

c o n t i n u e d the t e a c h i n g of quaternions at H a r v a r d after his father's death.36 Benjamin

Peirce

published

little

on

quaternion analysis;

H.

A.

N e w t o n w r o t e that Peirce h a d said in regard to quaternion analysis: "I

wish

as

only

I was y o u n g again that I m i g h t get such p o w e r in using it a

young m a n can

mathematics,

126

his

get."37 Peirce's

greatest contribution to

"Linear Associative Algebra,"

did

however stem

Traditions

from

his

interest

lithograph veloping would

1870,

"so

much

enable

ciprocal

in

in

led

in

many

of other such

men

possible

as

or,

and

of it

came as

States.

Cajori

versities

less t h a n

seven units."

to

more

more

by

3 9

among

these

important

to

was

result

a study

Florian

countrymen.

Not

it seems

as

the

as

non-re-

In this

work

In a broad sense his w o r k

the

Cajori

he

first

all

was

the

advocate

that

it

led

systems

are

of algebraic structure. serves

interest

probable

the

discovery

was

to

Peirce's enthusiasm for quaternions

for

in

numbers pure,

study of w h a t mathematical

generally,

him,

appeared

discovery of and interest in quaternions

A a

Analysis

a i m e d at de-

hyper-complex inequivalent,

nevertheless

well

which It was

all

directions;

compiled

from

matics

of

work, 1881.38

different algebras.

Peirce still

Peirce;

in

systems.

degree to w h i c h students

theory

fact that the

Information

from

162

different

similar

in

enumerate

systems

Peirce developed illustrates the

to

This

published

of the

him

number

quaternions. was

in Vectorial

indicate

stemmed

of course

that a large

recognized

measure

leader in

of quaternions

the

spread to his

in the

matheUnited

supplied information on twenty-two colleges and uni-

in nineteenth-century America;

quaternion methods

were

t a u g h t at no less t h a n t w e l v e of these. T h i s is s u r p r i s i n g s i n c e t h e level

of mathematical

instruction

time;

indeed

of the

at

some

was

schools

not

at

where

a

high

level

quaternions

at

that

were

not

taught, no other mathematics courses on a comparable level of complexity

V.

James The

Clerk

Maxwell:

discussion

twofold

of

function,

fluential

figure

represents the

taught.40

were

an

this

the

need

This

Quaternions

presented

in

Scot

this

not

section

only

of vector analysis

development

century.

of the

of

brilliant

history

important

nineteenth

awareness

Maxwell

for

in

Critic

in

the

development

(itself increasing)

was

a

but he

serves very

a in-

also aptly

physical

science

was

increasing

for a

the

vectorial

of

approach

to the solution of physical problems. There is m u c h truth and m u c h relevance with the And

in

Saloman

Bochner's comparison of the space of E u c l i d

space of N e w t o n a n d his

yet,

inwardly,

the

Euclidean

successors: Space that u n d e r l i e s ' t h e

Principia

is

mathematically not quite the same as the E u c l i d e a n Space that underlies Greek

mathematics

geometry of the

(and

Greeks

physics)

from

Thales

emphasized congruences

to and

Apollonius. similarities

The be-

t w e e n f i g u r e s , t h a t is, i n a n a l y t i c a l p a r l a n c e , o r t h o g o n a l a n d h o m o t h e t i c transformations

of the

u n d e r l y i n g space. . . . T h e

t h e Principia c o n t i n u e d t o b e a l l t h i s , b u t i t w a s

E u c l i d e a n Space of

also s o m e t h i n g n e w in

127

A

History

of V e c t o r Analysis

addition.

Several

significant

physical

entities

of the

Principia,

namely,

velocities, m o m e n t a , a n d forces are, by m a t h e m a t i c a l structure, vectors, t h a t is, e l e m e n t s o f v e c t o r f i e l d s , a n d v e c t o r i a l c o m p o s i t i o n a n d d e c o m position of these entities constitute an innermost scheme of the entire theory.41

Thus

with

concept

Newton

of a vector)

associated

with

a

host

of vectorial

entities

entered into physical

Newton

is

paralleled

(though

science.

by

a

not

the

The revolution

development

that

oc-

curred in the nineteenth century; this d e v e l o p m e n t is the introduction

into

physical

Einstein

science of the concept of field, a concept w h i c h

and Infeld have described as the

"most important inven-

.

of this

of the space of N e w t o n were

concept

multiplied in increased. vector

part

concept of

all

entities

arose

in

the

from

as

part from in

creation

."

well as

in

the

from

advances

electrical

in mechanics in

optical

of potential

theory.

was greatly

increased n e e d for

developments

part

and elaboration

successes

W i t h the introduction

n e e d for vector analysis

field

hydrodynamics), from

above

vectorial

number and the

The

analysis

pecially in

the

.

42

tion [in physics] since Newton's time.

(es-

theory,

theory,

and

It should thus come

as no surprise that the eventual emergence of vector analysis should intimately

be

associated

with

teenth-century

theoretical

is

in

symbolized

equations in both It a

in

his in is

the

1873

perhaps of

That

accidental; portantly, physics. physics

1860's,

of the great nine-

especially

Maxwell.

in

Electricity

component and

quaternion

notation,

Magnetism

he

M u c h famous

whereas

wrote

them

notation.

an accident of history that Tait and M a x w e l l spent together

close

both both

The

in

achievements

M a x w e l l first presented his

written on

and

years a

were

fact that

Treatise

component

number

bridge.

the

the

electricians,

were

men

tended

of

to take

approaches alike

studying

friendship

of

at

Edinburgh

developed brilliant

between

intellect,

similar approaches Tait

and

Maxwell

and

at

them

and

Cam-

is

less

more

im-

in mathematical in

mathematical

in that both w i s h e d their mathematics to be as

close a representation of the physical entities as possible. Tait left well deen. nion

in

1854

left C a m b r i d g e t w o

Cambridge

years

In the

Andrews,

Maxwell's

128

forms

his

in these

concept

problems

College,

Belfast;

Max-

1856 to 1865, w h i l e Tait p u r s u e d quater-

Hamilton

composed

published

mathematical published

from

with

Maxwell

discoveries with

decade

researches

for Queen's

later for Marischal College at Aber-

of

and

papers,

the

treatable

experimental

four great electrical

papers.

with The

especially those concerned

electromagnetic by

studies

quaternion

field,

involved

analysis.

In

the

of these papers quaternions were never used, and

Traditions in Vectorial Analysis

it

is

that

clear from Maxwell

the

correspondence

began

to

learn

between Tait and Maxwell

quaternion

methods

only

in

4 3

the

1870's.44 N e v e r t h e l e s s a f e w p e o p l e m a d e t h e a s s u m p t i o n t h a t M a x w e l l had b e e n aided in developing his nion

methods.

Though

this

ideas

assumption

was

by the use of quaternot

correct,

it

was

physics

that

probably influential in leading people to study quaternions. E.

T.

Whittaker

described

the

developments

led to Maxwell's great electrical papers In that p o w e r to w h i c h Gauss

in

in the following way:

attached so m u c h importance, of de-

vising dynamical models and analogies for obscure physical p h e n o m e n a , p e r h a p s n o - o n e has e v e r e x c e l l e d W . T h o m s o n ; a n d t o h i m , j o i n t l y w i t h Faraday, is due the credit of having initiated the theory of the electric medium.

In o n e of his earliest papers, w r i t t e n at t h e age of s e v e n t e e n

w h e n he was a very y o u n g freshman at C a m b r i d g e , T h o m s o n c o m p a r e d the distribution of electrostatic force, in a region containing electrified conductors, w i t h the distribution of the flow of heat in an infinite solid; t h e e q u i p o t e n t i a l surfaces i n t h e o n e case c o r r e s p o n d t o t h e i s o t h e r m a l surfaces in t h e o t h e r , a n d an electric charge c o r r e s p o n d s to a source of heat. It may, perhaps, seem as if the value of such an analogy as this consisted m e r e l y in the prospect w h i c h it offered of comparing, a n d thereby extending, the mathematical theories of heat and electricity. B u t to the p h y s i c i s t its c h i e f i n t e r e s t l a y r a t h e r i n t h e i d e a t h a t f o r m u l a e w h i c h relate t o the electric field, a n d w h i c h h a d b e e n d e d u c e d f r o m laws o f a c t i o n at a distance, w e r e s h o w n to be identical w i t h formulae relating to the theory of heat, w h i c h h a d b e e n d e d u c e d f r o m hypotheses of action bet w e e n contiguous particles.

"This

p a p e r , " as M a x w e l l said l o n g after-

wards, "first introduced into mathematical science that idea of electrical action carried on by means of a c o n t i n u o u s m e d i u m , w h i c h , t h o u g h it' had been announced by Faraday, and used by h i m as the guiding idea of his researches, h a d n e v e r b e e n a p p r e c i a t e d b y other m e n o f science, a n d was supposed by mathematicians to be inconsistent w i t h the l a w of electrical action, as established by C o u l o m b , a n d b u i l t on by Poisson." In 1846 —the year after he h a d taken his degree as second w r a n g l e r at C a m b r i d g e — T h o m s o n investigated the analogies of electric p h e n o m e n a w i t h those of elasticity. F o r this purpose he e x a m i n e d the equations of equilibrium

of an

incompressible

elastic solid w h i c h

is

in

a state of

strain; a n d s h o w e d that the d i s t r i b u t i o n of the vector w h i c h represents the elastic d i s p l a c e m e n t m i g h t be assimilated to the d i s t r i b u t i o n of the electric force in an electrostatic system. This, h o w e v e r , as he w e n t on to show, is not the only analogy w h i c h may be perceived w i t h the equations of elasticity; for the elastic d i s p l a c e m e n t m a y e q u a l l y w e l l be i d e n tified w i t h a vector a, defined in terms of the magnetic i n d u c t i o n B by the r e l a t i o n c u r l a = B. T h e vector a is equivalent to the vector-potential w h i c h had been used in the memoirs of Neumann, W e b e r and Kirchhoff, on the induction of currents; b u t T h o m s o n arrived at it i n d e p e n d e n t l y by a different process, and without being at the time aware of the identification. T h e results of T h o m s o n ' s m e m o i r s e e m e d to suggest a picture of the

129

A

History

of V e c t o r Analysis

propagation of electric or magnetic force: m i g h t it not take place in somewhat the

same

w a y as changes in t h e elastic d i s p l a c e m e n t are trans-

m i t t e d t h r o u g h an elastic solid? These suggestions w e r e not at the time pursued

further

by

young Cambridge

their

author;

m a n to take

but they helped to

up the

inspire

another

m a t t er a f e w years later. James

C l e r k M a x w e l l , b y w h o m the p r o b l e m was eventually solved, was b o r n 45

in

T h i s extensive a n d i m p o r t a n t quotation is significant for the history

of vector

analysis

in

numerous

ways.

The

most important of

these is that M a x w e l l presented an historical interpretation of what led to Thomson's discoveries, and this interpretation indicates what is

very

possibly the

quaternions.

way

in

which

Maxwell

became

interested in

Maxwell's interpretation was given in two papers, the

first of w h i c h

consists

of an

address

to

the

British Association

for

1870 a n d the second of w h i c h is his famous paper entitled " O n the Mathematical

Classification

of

Quantities."46

Physical

The

main

idea developed in both papers is the following: " O f the methods by w h i c h the mathematician m a y make his labours most useful to the student of nature, that w h i c h I think is at present most important is the

systematic

classification

of quantities."

(5;

218)

Maxwell went

on to explain w h y the mathematical classification of physical quantities

is important.

B u t w h e n t h e s t u d e n t has b e c o m e a c q u a i n t e d w i t h s e v e r a l d i f f e r e n t sciences, he

finds

the mathematical processes a n d trains of reasoning on

one science r e s e m b l e those in another so m u c h that his k n o w l e d g e of the one science m a y be m a d e a most useful h e l p in the study of the other. W h e n h e e x a m i n e s i n t o t h e r e a s o n o f t h i s , h e f i n d s t h a t i n t h e t w o sciences

he

has

been

dealing with

systems

of quantities,

in

w h i c h the

m a t h e m a t i c a l f o r m s o f t h e relations o f t h e q u a n t i t i e s are the same i n b o t h systems, t h o u g h the physical nature of the quantities may be utterly diff e r e n t . (5; 218) Maxwell such

wrote: was

supplied t w o historical illustrations of the usefulness of

a classification; "Another

the

originally pointed

between duction

problems of heat,

many of the

second

example,

by

results

in

by no

is

the

means

more

important.

so obvious,

is

Maxwell

that w h i c h

out by Sir W i l l i a m T h o m s o n , of the analogy attractions

the

use

and problems

of which

we

of Fourier for heat in

are

in

the

steady con-

able to make

use

of

e x p l a i n i n g electrical dis-

tribution, a n d of all the results of Poisson in electricity in explaining problems Thomson's

in

heat."

discovery

(6; 2 5 8 )

depends

M a x w e l l w e n t on to c o m m e n t that on

the

fundamental

principle that

t h e m a t h e m a t i c a l r e p r e s e n t a t i o n s o f p h y s i c a l entities m a y b e classified; his

chief example

Hamilton:

130

of such

a

classification

was

that found by

Traditions

in Vectorial Analysis

A most important distinction was d r a w n by H a m i l t o n w h e n he d i v i d e d the quantities w i t h w h i c h h e h a d t o d o into Scalar q u a n t i t i e s , w h i c h are completely represented by one numerical quantity, and Vectors, w h i c h require three numerical quantities to define them. The

invention of the calculus of Quaternions

is

a step towards the

k n o w l e d g e of quantities related to space w h i c h can o n l y be c o m p a r e d , for its i m p o r t a n c e , w i t h t h e i n v e n t i o n o f t r i p l e c o o r d i n a t e s b y D e s c a r t e s . T h e ideas o f t h i s c a l c u l u s , a s d i s t i n g u i s h e d f r o m its o p e r a t i o n s a n d s y m bols, are fitted to be of t h e greatest use in all parts of s c i e n c e . (6; 259) M a x w e l l said in effect that if scientists w o u l d pay closer attention to the classification of physical quantities, such analogies as those discovered such

by

Kelvin

a classification

would be is

that

obvious.

which

His

primary

originated

with

example

Hamilton,

of the

classification o f p h y s i c a l entities i n t o scalars a n d vectors. T h e p o i n t is that M a x w e l l seems to have interpreted the discoveries of T h o m son

(which

have

been

presented

in

the

quotation

from Whittaker

and cited in part by Maxwell) in the following way:

T h o m s o n saw

that distributions of electrostatic forces in a region containing electrified conductors

produced a vector

field

a n d that this vector field

was analogous to the vector field associated w i t h the flow of heat in an

infinite

solid.

Associated with

these

vector

fields

are

the

two

scalar fields, t h e lines of e q u a l p o t e n t i a l in t h e case of electrostatic case

and the

isothermal

T h o m s o n saw that both next

step was

lines in the case of heat flow. T h u s w h e n cases

nearly obvious.

i n v o l v e d scalar a n d vector fields, the If the

above

interpretation

of Max-

well's ideas is correct, it is probable that this realization led h i m to an increased interest in quaternion analysis. In the

second of his

two

papers

Maxwell

classified vectors into

" f o r c e " vectors ( w h i c h are referred to a u n i t of length) a n d flux vectors

( w h i c h are r e f e r r e d to a u n i t of area).46 L a t e r in this p a p e r q u a -

ternions were discussed briefly w i t h special attention to Hamilton's operator

V,

names.

He

=

for the

results

gave

+

~

the

+

of w h i c h

Maxwell

definitions

p r o p o s e d a series of

"'V = i ^ +j ^ + k

and

J?)" (6; 263~264) and then stated: " The dis"

covery of the square root of this operation is

due to

Hamilton;

but

most of the applications here given, and the whole development of t h e t h e o r y o f this o p e r a t i o n , are d u e t o P r o f e s s o r T a i t . . . . " (6; 264) He

proposed

quantity to

"to

which

call

the

it is

result

applied."

of V2 (6;

.

.

.

the

264) W h e n

Concentration V

is

of the

a p p l i e d to a

scalar f u n c t i o n of p o s i t i o n P, t h e result is V P . " T h e q u a n t i t y VP is a vector,

indicating the

direction in w h i c h P decreases most rapidly,

a n d m e a s u r i n g the rate of that decrease. I venture, w i t h m u c h diffi-

131

A H i s t o r y of V e c t o r Analysis

d e n c e , t o c a l l t h i s t h e slope o f P . " ( 6 ; 2 6 4 ) T h e r e a d e r s h o u l d n o t e t h a t a c c o r d i n g t o q u a t e r n i o n m u l t i p l i c a t i o n , i f o r = i t + j u 4 - kv, t h e n V c r = — (— + — + a

\dx

+ i

dy

—

dz)

\dy

+ • ( dt dz)

\dz

dv\

+

^ /du

dx)

dt\

\dx

47

dy)'

M a x w e l l c a l l e d the scalar part of this p r o d u c t , r e p r e s e n t e d in quaternion part,

language

VVcr,

"the

as

SVor,

Curl

o r Version

"the

Convergence

of the

o f or,"

original

and the

vector

vector function."

48

B e f o r e p u b l i s h i n g this p a p e r , M a x w e l l h a d w r i t t e n t o T a i t t o request his

views

on

the

suitability of the names:

" H e r e are some

rough h e w n names. W i l l you like a good D i v i n i t y shape their ends p r o p e r l y so as to m a k e t h e m s t i c k ? " M a x w e l l w e n t on to ask in particular about the names for V a p p l i e d to a vector function. T h e scalar part I w o u l d c a l l t h e C o n v e r g e n c e o f t h e v e c t o r f u n c t i o n , a n d the vector part I w o u l d call the T w i s t of the vector function. H e r e the w o r d t w i s t h a s n o t h i n g t o d o w i t h a s c r e w o r h e l i x . I f t h e w o r d turn o r version w o u l d d o t h e y w o u l d b e b e t t e r t h a n t w i s t , f o r t w i s t s u g g e s t s a screw. T w i r l i s free f r o m t h e s c r e w n o t i o n a n d i s s u f f i c i e n t l y racy. Perhaps it is t o o d y n a m i c a l for p u r e m a t h e m a t i c i a n s , so f o r C a y l e y ' s sake I m i g h t say C u r l (after t h e f a s h i o n o f Scroll).49 T h e letter quoted above is the first letter from the Tait-Maxwell correspondence in w h i c h M a x w e l l s h o w e d some k n o w l e d g e of quaternions.

It is

dated November 7,

1870. T h e f i r s t paper i n w h i c h

M a x w e l l i n c l u d e d a discussion of q u a t e r n i o n ideas is in the aforem e n t i o n e d "Address to the Mathematical and Physical Sections of t h e B r i t i s h A s s o c i a t i o n , " w h i c h w a s r e a d o n S e p t e m b e r 15, 1870. I t is thus probable that Maxwell's interest in quaternions originated at about this

time.

At least as early as

1867 Tait had r e c o m m e n d e d

q u a t e r n i o n s t o M a x w e l l . T a i t w r o t e o n D e c e m b e r 13, 1867, t o M a x well:

"If you

Quaternions

(1st

r e a d t h e last 2 0 o r 3 0 pages o f m y b o o k [ T r e a t i s e o n ed.,

1867)]

I

think

you

will

see

that

4ions

are

w o r t h getting up, for there it is s h o w n that they go into that < business

like

greased

lightning.

Unfortunately

I

cannot

find

t i m e to

50

w o r k s t e a d i l y at t h e m . . .

M a x w e l l ' s letter t o T a i t o f N o v e m b e r 7 , 1 8 7 0 , was f o l l o w e d b y another letter dated have

been

Hamilton Magnetism]

good I

with

want

November news to

for

14;

Tait:

leaven

Hamiltonian

my

ideas

Maxwell's message therein must "With book

regard to my [Treatise

without

casting

on the

dabbling in Electricity

and

operations

i n t o H a m i l t o n i a n f o r m for w h i c h n e i t h e r I nor I t h i n k the p u b l i c are ripe.

N o w the value of Hamilton's idea of a vector is unspeakable,

a n d s o a r e t h o s e o f t h e a d d i t i o n a n d m u l t i p l i c a t i o n o f v e c t o r s . " (1; 144) M a x w e l l ' s letter of O c t o b e r 9, 1872, also b r o u g h t g o o d n e w s to Tait. In the previous year M a x w e l l h a d b e e n chosen as the first Pro-

132

Traditions

fessor

of Experimental

Physics

Cavendish Laboratory.

at

in Vectorial Analysis

Cambridge

and

Director of the

M a x w e l l wrote to Tait, "I am going to try, as 51

I have already tried, to s o w 4 n i o n seed at C a m b r i d g e . " The haps

M a x w e l l letter to Tait of D e c e m b e r 21, Maxwell's

strongest

statement

in

1871, contained per-

favor of quaternions.

well wrote: "Impress on T. [ T h o m s o n ] that —V2

and

not + V

a m o n g us.

2

as

h e v a i n l y asserts

+ { ^ j

is

Max-

(cL;)

=

now commonly believed

Also h o w m u c h better a n d easier he w o u l d have done his

solenoidal and lamellar business

if in addition to what we k n o w is

i n his h e a d h e h a d h a d say, 2 0 years ago, Q n s . t o h u n t for C a r t e s i a n s instead of vice versa. T h e one is a way;

the

other

(downward?)."

is

a

(1;

ram,

s w o r d that burns every

westward

and

northward

and

150)

A n u m b e r of interesting points of November 2,

flaming

pushing

1871;

are c o n t a i n e d in M a x w e l l ' s letter

he wrote:

But try and do the 4nions. T h e unbelievers

are r a m p a n t . T h e y say

" s h o w m e s o m e t h i n g d o n e b y 4 n i o n s w h i c h has n o t b e e n d o n e b y o l d plans. At the best it m u s t rank w i t h a b b r e v i a t e d notations." You should reply to this, no doubt y o u will. B u t the v i r t u e of the 4 n i o n s lies not so m u c h as yet in s o l v i n g h a r d q u e s t i o n s , a s i n e n a b l i n g u s t o see t h e m e a n i n g o f t h e q u e s t i o n a n d o f its solution, instead of setting up the question in x y z, s e n d i n g it to the analytical engine, and w h e n the solution is sent h o m e translating it back f r o m x y z so t h a t it m a y a p p e a r as A, B, C to t h e v u l g a r . (1; 101) T h e last p o i n t w a s e m p h a s i z e d i n a n u n s i g n e d article b y M a x w e l l entitled cle

is

ternions

"Quaternions" actually

published

a

and

review in

published

of

1873.

Kelland's

Maxwell

1873.52 T h e arti-

i n Nature i n and

began

Tait's the

Introduction

article

by

to

Qua-

discuss-

ing the popular image of the mathematician as a calculator. He then wrote:

"But though m u c h of the routine work of a mathematician is

calculation, matician—is

his the

proper w o r k —that w h i c h constitutes invention

of

methods. . . . But

him

the

a mathe-

methods

on

w h i c h a m a t h e m a t i c i a n is content to h a n g his reputation are generally those him

the

which

labour

he

fancies

of thinking

will about

save

him

what

and

has

all w h o c o m e after

cost h i m s e l f so

much

thought."53 A c o m m o n a r g u m e n t for quaternions h a d b e e n that they were just quite

such a labor-saving device, but

Maxwell's

position

was

different f r o m this.

Now

Quaternions,

or

the

doctrine

of Vectors,

is

a

mathematical

m e t h o d , b u t it is a m e t h o d of t h i n k i n g , a n d not, at least for the present generation, a method of saving thought.

...

It calls u p o n us at e v e r y step

to f o r m a mental image of the geometrical features represented by the symbols, so that in studying geometry by this m e t h o d we have our m i n d s

133

A

History

of V e c t o r Analysis

e n g a g e d w i t h g e o m e t r i c a l ideas, a n d are not p e r m i t t e d t o fancy ourselves g e o m e t e r s w h e n w e are o n l y arithmeticians.53 M a x w e l l c o n c l u d e d the r e v i e w w i t h praise for Kelland's expository abilities

and

with

stress o n t h e i m p o r t a n c e o f t h e t r e a t m e n t o f t h e

linear vector function The

theme

well's torial

given

i n t h e last c h a p t e r ( w r i t t e n b y Tait).

that dominates

other statements

on

this

review as

well

as

many of Max-

quaternions is that by means of the vec-

approach the physicist attains to a direct mathematical repre-

sentation of physical entities and is thus aided in seeing the physics involved in the mathematics. stress

this

physics,

point

which

by

It is probable that M a x w e l l was led to

pondering

were

obtained

the

great

without

successes the

use

of Faraday

of formal

in

mathe-

matics. Faraday's success presented a c o n u n d r u m to the mathematical

physicists

drum which which

of the

latter half of the

nineteenth century, a conun-

M a x w e l l tried to resolve w h e n he wrote:

Faraday

made

use

of his

"The way in

lines of force in co-ordinating the

phenomena of magneto-electric induction shews h i m to have been in

reality

a

mathematician

of a very high

order—one

from

w h o m

mathematicians of the future m a y derive valuable and fertile methods."

(4,11;

360)

Thus

Maxwell's

enthusiasm

for

quaternions

s t e m m e d from v i e w i n g t h e m as a m e t h o d that allows the physicist to

keep

the

physical

entities

before

his

mind

during calculations,

rather than f r o m v i e w i n g t h e m as a m e t h o d for reducing the labor of thought. T h e most i m p o r t a n t of M a x w e l l ' s p u b l i s h e d works for this history is

his

most tion

Treatise

important

on work

of Maxwell's

it was

influential

Maxwell nary"

Electricity

and

Magnetism

not only because

(1873).54

This

is

the

it gives the fullest concep-

v i e w of quaternions b u t also because historically in leading people to discuss quaternion methods.

began

this

work with

a mainly mathematical

"Prelimi-

chapter. After mentioning Descartes' discovery of co-ordinate

methods,

Maxwell

wrote:

B u t for m a n y purposes

of physical reasoning, as distinguished from

calculation, it is desirable to avoid explicitly introducing the Cartesian c o o r d i n a t e s , a n d t o fix t h e m i n d a t o n c e o n a p o i n t o f s p a c e i n s t e a d o f its three coordinates, and on the magnitude and direction of a force instead o f its

three

components. This mode of contemplating geometrical and

p h y s i c a l q u a n t i t i e s i s m o r e p r i m i t i v e a n d m o r e n a t u r a l t h a n t h e o t h e r , alt h o u g h the ideas c o n n e c t e d w i t h it d i d not receive their full developm e n t till H a m i l t o n m a d e the next great step i n d e a l i n g w i t h space, b y the i n v e n t i o n of his Calculus of Quaternions. As t h e m e t h o d s of D e s Cartes are still t h e m o s t f a m i l i a r to students of s c i e n c e , a n d a s t h e y are r e a l l y t h e m o s t u s e f u l for p u r p o s e s o f calculation, we shall express

134

all our results in the Cartesian form. I am con-

Traditions

vinced,

however,

that the

introduction

in Vectorial Analysis

of the ideas, as d i s t i n g u i s h e d

from the operations and m e t h o d s of Quaternions, w i l l be of great use to us in the study of all parts of o u r subject, a n d e s p e c i a l l y in e l e c t r o d y namics, w h e r e we have to deal w i t h a n u m b e r of physical quantities, the relations o f w h i c h t o each o t h e r c a n b e e x p r e s s e d far m o r e s i m p l y b y a few expressions of Hamilton's, than by the ordinary equations. 11.

O n e of the most important features of H a m i l t o n ' s m e t h o d is the

d i v i s i o n o f q u a n t i t i e s i n t o S c a l a r s a n d V e c t o r s . (3,1; 9 - 1 0 )

After

a

discussion

scalars,

Maxwell

of this

division

of quantities

into

vectors

and

wrote:

T h e r e are p h y s i c a l q u a n t i t i e s o f a n o t h e r k i n d w h i c h are r e l a t e d t o directions i n space, b u t w h i c h are n o t vectors. Stresses a n d strains i n s o l i d b o d i e s are e x a m p l e s of these, a n d so are s o m e of t h e p r o p e r t i e s of b o d i e s considered in the theory of elasticity a n d in the theory of d o u b l e refraction.

Quantities

specifications.

o f t h i s c l a s s r e q u i r e f o r t h e i r d e f i n i t i o n nine n u m e r i c a l

They

are

expressed in the

language of quaternions

l i n e a r a n d v e c t o r f u n c t i o n s o f a v e c t o r . (3,1;

Maxwell then represent

by

10)

stated that he w o u l d use G e r m a n capital letters to

vectors.

The

reason

for

this

innovation

was

that

he

n e e d e d the G r e e k letters ( w h i c h w e r e used to designate vectors by H a m i l t o n a n d Tait) for other uses in the book. After a discussion of line and surface integrals a n d the introduction of the operator V a n d other

mathematical

nate

preliminaries,

form the theorems

these

names).

Maxwell

presented

of Gauss a n d Stokes

Maxwell

mentioned

that

in

(Maxwell

he

would

co-ordi-

d i d not use

use

the

right-

h a n d e d s y s t e m o f co-ordinates axes a n d p o i n t e d o u t that i n this h e differed

from

ployed

by

phy

and

by

section

Tait

was

theorems terms

into

(in

the

practice

Thomson

the

in

his

Tait

with

quaternion

dropped

"Slope"

also

a

and

("with

was

dropped

but

their

agreed Treatise

publications.

translation

notation

edition)

and

of

with

5 5

great

with

on

that

Natural

(3,1;

and

or

This

Gauss'

explanation

"curl"

em-

Philoso-

25-26)

Stokes'

an

"concentration,"

"convergence," and "slope." was

in

quaternion

concluded

first

of Hamilton

and

of his

"version,"

In later editions " c u r l " or " v e r s i o n " diffidence")

"rotation"

and "space-variation"

substituted.

substituted.

(3,1;

30-31) In the text of the book quaternion notations

were used in a sub-

stantial n u m b e r o f cases. T h e s e m a y b e c l a s s i f i e d i n t o t h r e e g r o u p s : to t h e first b e l o n g those cases in w h i c h at t h e e n d of a s e c t i o n M a x well

stated that he w o u l d give the quaternion f o r m of an equation;

to the

s e c o n d g r o u p b e l o n g those cases i n w h i c h q u a t e r n i o n nota-

tion was used in such an elementary form that little importance can b e a t t r i b u t e d t o it; a n d t o t h e t h i r d class b e l o n g t h e m o s t i m p o r t a n t

135

A

History

of V e c t o r Analysis

cases, those

in w h i c h M a x w e l l used quaternion expressions almost

in such a w a y as to integrate t h e m into the development. T h e m o s t i m p o r t a n t e x a m p l e o f t h e f i r s t g r o u p (of w h i c h there are perhaps

fifteen

cases) c o m e s i n t h e c h a p t e r " G e n e r a l E q u a t i o n s o f

t h e E l e c t r o m a g n e t i c F i e l d . " T h e last section o f this c h a p t e r was entitled

"Quaternion

tions."

Herein

Expressions

Maxwell

for

wrote:

the

"In

this

Electromagnetic treatise

we

have

Equaendeav-

oured to avoid any process d e m a n d i n g from the reader a k n o w l e d g e of

the

Calculus

of Quaternions.

scrupled to introduce the do

so."

(3,11;

257)

Maxwell

most important equations

At

the

same

time

we

have

not

idea of a vector w h e n it was necessary to then

wrote in quaternion notation the

d e v e l o p e d in his book.

(3,11;

257-259)

Examples of the second k i n d of use that M a x w e l l m a d e of quaternion

notation

are v e r y n u m e r o u s , especially in the second v o l u m e .

Repeatedly he wrote vectorial quantities as vectors; this was done e s p e c i a l l y t o stress t h e n a t u r e o f t h e q u a n t i t y r e p r e s e n t e d , a n d also as an a b b r e v i a t i o n . At o t h e r t i m e s o n e v e c t o r w a s g i v e n as a scalar multiple symbol when

of another vector. V

but

wrote

presenting

Cases sections ucts.56

out

from the

t h i r d class

which

he

The

vector

and

that

quaternion expressions

full

he

m a d e frequent use of the

Cartesian

form

readers

are

used the scalar

probably

s o m e w h a t rare:

quaternion

examples are the

products

were

used

noticed

their

importance.

sufficiently freThe

full

product was never employed, but it at times influenced that were

used.

Thus

d e f i n e d as either plus or m i n u s sign, whereas

" T h e negative sign sions

consistent

(3,11;

255)

The

equations

scalar a n d vector prod-

in

Cartesian

/

positive

of his

proofs.

in

quently

Similarly

the

following

analysis

V2 can be

\

+

+

Kelvin used the

M a x w e l l used the negative

sign and noted:

is e m p l o y e d here in order to m a k e our expres-

with

those

three

in

which

quotations

Q u a t e r n i o n s are e m p l o y e d . "

are

of considerable

importance.

T h e theory of the c o m p l e t e system of equations of resistance and of conductivity is that of linear functions of three variables, and it is exemp l i f i e d i n t h e t h e o r y o f Strains, a n d i n other parts o f physics. T h e most appropriate m e t h o d of treating it is that by w h i c h H a m i l t o n a n d T a i t treat a linear a n d vector function of a vector. We shall not, however, expressly i n t r o d u c e Q u a t e r n i o n n o t a t i o n . (3,1; 4 2 2 ) T h i s analysis of the forces acting b e t w e e n t w o small magnets was first g i v e n i n t e r m s o f t h e Q u a t e r n i o n A n a l y s i s b y P r o f e s s o r T a i t i n t h e Quarterly

Math. Journ.

for Jan.

442-443, 2nd Edition.57

136

1860.

See

also

his

work

on

Quaternions,

Arts.

Traditions

Finally,

in

Maxwell's

Mutual Action

chapter

on

in Vectorial Analysis

"Ampere's

of Electric Currents"

he

Investigation

of the

stated:

T h e o n l y e x p e r i m e n t a l fact w h i c h w e have m a d e use o f i n this investigation is the fact established by A m p e r e that the action of a c l o s e d c i r c u i t on any portion of another circuit is perpendicular to the direction of the latter.

E v e r y other part of the investigation depends on p u r e l y mathe-

matical considerations

depending on the properties

of lines

i n space.

T h e reasoning therefore may be presented in a m u c h more condensed and appropriate f o r m by the use of the ideas a n d language of the mathematical m e t h o d specially adapted to the expression of such geometrical relations —the This

has

ematics,

Quaternions

been

1866,

done

and

in

of by

his

Hamilton.

Professor T a i t treatise

on

in

the

Quaternions,

Quarterly Journal §

399,

o f Math-

for Ampere's

original investigation, and the student can easily adapt the same m e t h o d t o t h e s o m e w h a t m o r e g e n e r a l i n v e s t i g a t i o n g i v e n h e r e . (3,11; 1 7 1 - 1 7 2 ) M a x w e l l ' s use o f q u a t e r n i o n expressions a n d notations w a s sufficiently

frequent

and

the

above

quotations

praise of quaternions that it seems of his

Treatise

strongly

left

it w i t h

recommended

the

the

impression

wished

natural

to

use

explanation

quaternions

more

in

Maxwell

had

rather

To those readers

not more frequently used, a

available: frequently

Maxwell but

had

had not

in

fact

done

ilarly

Maxwell's one explicit statement (he had r e c o m m e n d e d quaas

distinguished from

o f Q u a t e r n i o n s " [3,1; The

conjecture

thereby

to

Treatise w e r e v e r s e d i n s u c h m e t h o d s .

so

few

"ideas,

of his

were

was

strong

since

ternion

readers

that

study of quaternions.

who wondered w h y quaternions perfectly

sufficiently

probable that not a f e w readers

the operations and methods

9]) w a s easily m i s s e d or easily m i s i n t e r p r e t e d .

that

study

Sim-

readers

of

quaternions

the

gains

Treatise

may

support

have

from

been

evidence

led that

strongly indicates that b o t h G i b b s a n d H e a v i s i d e d i d in fact go f r o m the

Treatise

this

conjecture

Maxwell

of M a x w e l l to the works of H a m i l t o n and Tait.

would

themselves the

operator

is

incorrect,

tend

to

still

become

readers

views

in

order

to avail

he

did

in

1878 gives the clearest pic-

on quaternions. T h e statements contained

are sufficient t o explain that

of Tait

V.

ture of Maxwell's therein

if

of the latter's treatment of the linear vector f u n c t i o n a n d

A letter to Tait written by M a x w e l l in

course

Even

it is very possible that readers of

his

Here is another question.

Treatise.

completely w h y M a x w e l l took the Maxwell

wrote:

M a y o n e p l o u g h w i t h a n o x a n d a n ass t o -

gether? T h e like of you may write everything and prove everything in pure 4nions, but in the transition period the bilingual m e t h o d may help to introduce and explain the more perfect.

137

A

History

of V e c t o r Analysis

B u t even w h e n that w h i c h is perfect is come that w h i c h builds over t h r e e axes w i l l b e u s e f u l f o r p u r p o s e s o f c a l c u l a t i o n b y t h e Cassios o f t h e future. N o w in a b i l i n g u a l treatise it is t r o u b l e s o m e , to say t h e least, to find that the square of AB is always positive in Cartesians a n d always negative in 4nions, and that w h e n the t h i n g is m e n t i o n e d incidentally you do not k n o w w h i c h language is being spoken. A r e the Cartesians to be d e n i e d the idea of a vector as a sensible t h i n g i n r e a l l i f e t i l l t h e y c a n r e c o g n i s e i n a m e t r e scale o n e o f a p e c u l i a r syst e m of square roots of —1? I t i s also a w k w a r d w h e n y o u are d i s c u s s i n g , say, k i n e t i c e n e r g y t o f i n d t h a t t o e n s u r e i t s b e i n g +ve y o u m u s t s t i c k a — s i g n t o i t , a n d t h a t w h e n y o u are p r o v i n g a m i n i m u m i n c e r t a i n cases t h e w h o l e a p p e a r a n c e o f t h e proof should be tending towards a maximum. W h a t d o y o u r e c o m m e n d for E l . a n d M a g . t o say i n s u c h cases? Do

you

know

Grassmann's

Ausdehnungslehre? Spottiswoode spoke

of it in D u b l i n as s o m e t h i n g a b o v e a n d b e y o n d 4nions. I have not seen it, b u t Sir W . H a m i l t o n o f E d i n b u r g h u s e d t o say that t h e greater t h e extens i o n t h e s m a l l e r t h e i n t e n t i o n . (1; A.

P.

Wills

has

151-152)

speculated that if M a x w e l l had read Grassmann,

h e m i g h t h a v e b e e n l e d t o a d o p t its m e t h o d s . 5 8 M a x w e l l a l m o s t certainly

did

not

read

Grassmann's

Ausdehnungslehre,

since

Maxwell

d i e d on N o v e m b e r 5, 1879. In relation to this q u e s t i o n it is interesting to note that Grassmann was fore

1878,

Twice

although

in

his

probably

Treatise

on

not u n k n o w n to M a x w e l l even be-

Maxwell Electricity

h a d never seen his and

Magnetism

books.

Maxwell

re-

ferred to an idea presented in one of Grassmann's papers on electricity.59

In

this

paper

Grassmann

mentioned

his

Ausdehnungslehre

a n d g a v e a f e w o f t h e e l e m e n t a r y i d e a s c o n t a i n e d i n it. A l s o i n t h r e e of Maxwell's early (1855-1860) optical papers some discussion was given concerning one of Grassmann's optical papers he

had

mann,

referred

to

his

Maxwell wrote

in

Ausdehnungslehre. 1860:

Of

this

6 0

in w h i c h also

paper

by

Grass-

" I t appears therefore that if colours

are r e p r e s e n t e d in quantity a n d quality by the m a g n i t u d e a n d direction of straight lines, the rule for the composition of colours is identical w i t h that for the c o m p o s i t i o n of forces in mechanics. T h i s analo g y h a s b e e n w e l l b r o u g h t o u t b y P r o f e s s o r G r a s s m a n n i n . . . . " (4,1; 418-419) Of the

four

books

written by Maxwell

after

1873 only one m e n -

tioned vectors; this is his elementary w o r k on mechanics published in

1876

a short vectors.

and

entitled

section

on

the

Quaternions

Maxwell's may now be

strange,

Matter

and

Motion.

idea of and the

were

Maxwell

included

and subtraction of

not mentioned.

unique,

s u m m e d up.

Herein addition

and

Maxwell

important

place

in

this

history

favored q u a t e r n i o n analysis for

t h e n a t u r a l n e s s o f its r e p r e s e n t a t i o n s o f p h y s i c a l e n t i t i e s a n d for t h e

138

Traditions in Vectorial Analysis

abbreviations

stemming from

this.

M o s t of all he favored quater-

nions because the physical entities were kept before the eye of the mathematician.

He

was

particularly impressed by the

V

operator

and the linear vector function. On the other hand, M a x w e l l in general

disliked

quaternion

"methods"

(as

opposed

to

quaternion

"ideas"); thus for example he was t r o u b l e d by the n o n h o m o g e n e i t y of the

quaternion

or full

square of a vector was

vector product and

negative,

which

by the

fact that the

in t h e case of t h e v e l o c i t y

vector m a d e k i n e t i c energy negative. T h e aspects of q u a t e r n i o n analysis that M a x w e l l l i k e d w e r e clearly b r o u g h t o u t i n his great w o r k on electricity; the aspects that he d i d not like w e r e i n d i c a t e d o n l y b y the fact that M a x w e l l d i d not i n c l u d e t h e m . T h u s M a x w e l l ' s importance for this history is twofold:

(1) h e as-

sociated vectorial ideas w i t h electricity in such a w a y that this linka g e w a s m a i n t a i n e d a n d (2) h e t o s o m e e x t e n t o u t l i n e d t h e f o r m t h a t a suitable vector analysis s h o u l d take.

Such a vector analysis was

not k n o w n to M a x w e l l , b u t soon after his death a n d p r o b a b l y in part due

to

his

Heaviside.

inspiration

such

a

system

Thus, justification may be

m e n t that i n

Maxwell's

was

created by Gibbs

seen for Macfarlane's

and

state-

departure from quaternion orthodoxy " w e

have the origin of the school of vector-analysts as opposed to the pure quaternionists."

VI.

William

61

Kingdon

M a x w e l l d i e d in late

Clifford:

Transition

Figure

1879 at age forty-eight; earlier in the same

year B r i t i s h s c i e n c e lost a n o t h e r o f its m o s t b r i l l i a n t r e p r e s e n t a t i v e s w h e n W i l l i a m K i n g d o n C l i f f o r d d i e d a t age thirty-four. C l i f f o r d h a d graduated from Cambridge in

1867 as Second Wrangler, a n d was

elected Professor of A p p l i e d Mathematics versity College, L o n d o n , in

and

Mechanics

at Uni-

1871.

C l i f f o r d ' s s i g n i f i c a n c e f o r t h i s h i s t o r y i s t w o f o l d : (1) h e w a s o n e o f the few mathematicians of the time w h o k n e w both quaternion and G r a s s m a n n i a n a n a l y s i s a n d (2) h e w r o t e a w o r k w h i c h i s i n a s e n s e transitional f r o m q u a t e r n i o n analysis to vector analysis. In

Clifford's

writings

the

first

mention

of Grassmann

came

in

1868 (7; 114); h i s a c q u a i n t a n c e w i t h G r a s s m a n n i a n a n a l y s i s i s p r o b ably

due

1867.

to

Hankel's

Theorie

der

complexen

Zahlensysteme

of

H i s interest in q u a t e r n i o n s p r o b a b l y stems f r o m 1867 or ear-

lier, b u t his first m e n t i o n of quaternions came in an 1873 paper.62 It is clear f r o m his writings that Clifford's interest was greater in the

Grassmannian

1878

published

than in

in the the

H a m i l t o n i a n system.

American

Journal

of

In a paper of

Mathematics

Pure

and

139

A History

of V e c t o r Analysis

Applied

entitled

bra"

and

Clifford

"Applications

of Grassmann's

Extensive

Alge-

wrote:

I p r o p o s e to c o m m u n i c a t e in a b r i e f f o r m s o m e a p p l i c a t i o n s of Grassm a n n ' s t h e o r y w h i c h i t seems u n l i k e l y t h a t I s h a l l f i n d t i m e t o set f o r t h a t p r o p e r l e n g t h , t h o u g h I h a v e w a i t e d l o n g f o r it. U n t i l r e c e n t l y I w a s u n acquainted is

with

t h e Ausdehnungslehre,

c o n t a i n e d in the author's

HankeVs

Lectures

on

Complex

and

knew

geometrical Numbers.

I

only

papers may,

so

much

of it as

i n Crelle's Journal a n d i n

perhaps,

therefore

be

p e r m i t t e d to express my p r o f o u n d admiration of that extraordinary work, a n d m y c o n v i c t i o n t h a t its p r i n c i p l e s w i l l e x e r c i s e a vast i n f l u e n c e u p o n t h e f u t u r e o f m a t h e m a t i c a l s c i e n c e . (7; 2 6 6 ) Clifford delivered in University College. pure

mathematics

1877

a series of lectures on quaternions at

Clifford's

mathematical interests w e r e more in

(especially

geometry)

than

in

applied

mathe-

matics; this b e c o m e s clear w h e n o n e looks at the notes f r o m his lectures

quaternions.63

on

The

most

Elements

important

o f Dynamic,

of Clifford's

which

was

writings

intended

as

for this

the

first

history is in

his

a series

of

e l e m e n t a r y texts. T h e series was n e v e r c o m p l e t e d , for C l i f f o r d d i e d in

1879.

In this w o r k C l i f f o r d i n t r o d u c e d vectors (or " s t e p s , " as he

sometimes called t h e m ) a n d early in the w o r k explained such ideas as the addition of vectors. Near the m i d d l e of the book, in a section entitled

"Product of T w o Vectors,"

On account of the

importance

Clifford wrote:

of the t h e o r e m of moments, we shall

p r e s e n t i t u n d e r y e t a n o t h e r a s p e c t . T h e a r e a o f t h e p a r a l l e l o g r a m abdc m a y b e s u p p o s e d t o b e g e n e r a t e d b y t h e m o t i o n o f a b o v e r t h e s t e p ac, o r b y t h e m o t i o n o f a c o v e r t h e s t e p ab. H e n c e i t s e e m s n a t u r a l t o s p e a k o f i t a s t h e product o f t h e t w o s t e p s ab, ac. W e h a v e b e e n a c c u s t o m e d t o i d e n t i f y a r e c t a n g l e w i t h t h e p r o d u c t o f its t w o sides, w h e n t h e i r l e n g t h s o n l y are t a k e n into account; we shall n o w m a k e just such an extension of the m e a n i n g of a product as we formerly made of the m e a n i n g of a sum, a n d still r e g a r d t h e p a r a l l e l o g r a m c o n t a i n e d b y t w o steps a s t h e i r p r o d uct, w h e n t h e i r d i r e c t i o n s are t a k e n i n t o account. T h e m a g n i t u d e o f this p r o d u c t is directed

ab.ac s i n bac;

l i k e a n y o t h e r area, it is to be r e g a r d e d as a

quantity.

S u p p o s e , h o w e v e r , t h a t o n e o f t h e t w o s t e p s , s a y ac, r e p r e s e n t s a n a r e a p e r p e n d i c u l a r t o i t ; t h e n t o m u l t i p l y t h i s b y ab, w e m u s t n a t u r a l l y m a k e t h a t a r e a t a k e t h e s t e p o f t r a n s l a t i o n ab.

In so d o i n g it w i l l generate a

v o l u m e , w h i c h m a y b e r e g a r d e d a s t h e p r o d u c t o f a c a n d ab. B u t t h e m a g n i t u d e of this v o l u m e is ab m u l t i p l i e d by the area into the sine of the angle that it makes makes

w i t h ab, t h a t i s , i n t o t h e cosine o f t h e a n g l e t h a t a c

w i t h ab. T h i s k i n d o f p r o d u c t t h e r e f o r e h a s t h e m a g n i t u d e ab. a c

c o s bac; b e i n g a v o l u m e , i t c a n o n l y b e g r e a t e r o r l e s s ; t h a t i s , i t i s a scalar quantity. W e a r e t h u s l e d t o t w o d i f f e r e n t k i n d s o f p r o d u c t o f t w o v e c t o r s ab, ac; a

vector

product,

which

may

be

written

V.ab.ac,

and

w h i c h is

the

area

of t h e p a r a l l e l o g r a m of w h i c h t h e y are t w o sides, b e i n g b o t h r e g a r d e d as

140

Traditions

steps;

a n d a scalar product, w h i c h

in Vectorial Analysis

m a y be w r i t t e n S . a b . a c , a n d w h i c h is

the v o l u m e traced out b y a n area r e p r e s e n t e d b y one, w h e n m a d e t o take t h e s t e p r e p r e s e n t e d b y t h e o t h e r . (8; 9 4 - 9 5 ) After s h o w i n g that b o t h these products are

distributive, Clifford

continued: B u t there is a very important difference b e t w e e n a vector p r o d u c t a n d a p r o d u c t of t w o scalar quantities.

N a m e l y , t h e sign o f a n a r e a d e p e n d s

u p o n the w a y it is gone r o u n d ; an area gone r o u n d c o u n t e r - c l o c k w i s e is p o s i t i v e , g o n e r o u n d c l o c k w i s e is n e g a t i v e . we

must

V.ac.ab is

=

have

by

—V. ab.ac,

changed

by

symmetry

or

Vy/3

inverting

the

—

V/3y.

order

N o w i f V . ab. a c = area, abed,

W.ac.ab — a r e a Hence

of the

aedb,

the

terms.

and

sign

It

is

of a

therefore

vector

agreed

product

upon

that

Va/3 s h a l l b e a v e c t o r f a c i n g t o t h a t s i d e f r o m w h i c h t h e r o t a t i o n f r o m a t o P appears to c o u n t e r - c l o c k w i s e . (8; 96). T h i s passage seems to be a definition of the m o d e r n scalar a n d vector products.

Clifford's

definition of the vector product of t w o vec-

tors is essentially e q u i v a l e n t to the m o d e r n d e f i n i t i o n a n d is in o n e sense equivalent to the q u a t e r n i o n vector product.

However, at no

point in the above passage d i d C l i f f o rd indicate w h e t h e r the scalar product was to be taken as positive or negative; he referred only to the

magnitude

of the

product.

Nearly

one h u n d r e d pages

later he

r e t u r n e d to the scalar p r o d u c t a n d d e f i n e d it as " t h e negative s u m of the products thus

of their components along the axes."

used the

modern In

scalar p r o d u c t in

Sciences

given.

(8;

186) C l i f f o r d

quaternionic sense, not in the

sense.

Clifford's

Exact

the

posthumously

the

This

modern

published

definition

appeared in chapter IV,

of

Common

the

section

Sense

scalar 16,

of

product

the was

of the book.64 Be-

fore Clifford died, he h a d w r i t t e n some of the chapters in this book, and

Karl

wrote: ter IV

Pearson

wrote

the

remainder.

In

his

preface

Pearson

" F o r the latter half of C h a p t e r I I I a n d for the w h o l e of Chap(pp.

116-226)

I

am

alone

responsible."

65

Thus

the appear-

ance of the m o d e r n scalar p r o d u c t in this b o o k is to be credited, not to Clifford, but to the

publication

Pearson.

of his

It is possible that in the year b e t w e e n

Elements

o f Dynamic

(1878)

and

his

death

in

1879 Clifford m i g h t h a v e c h a n g e d his definition of the scalar product

and that

this

fact was

seems far m o r e p r o b a b l e first, rather given product

by in

than

the

Clifford its

communicated to

second,

in

modern

his

Pearson.

that Pearson based his discussion

Elements

form

only

of the

o f Dynamic because

and

However,

it

definition on the scalar

product

hence

wrote

as the

of a misunderstanding!

That this is probable is further indicated by the great similarity of the discussion

of the scalar p r o d u c t g i v e n by Pearson as c o m p a r e d

141

A H i s t o r y of V e c t o r Analysis

with

t h e f i r s t d i s c u s s i o n o f t h e t w o g i v e n b y C l i f f o r d i n h i s Elements

o f Dynamic. to this Later tor

V

The

confusing

in

his

(without

divergence also

nature

of Clifford's

first

definition

adds

probability.

as

Elements

o f Dynamic

introducing this

the

employed

negative the

of

linear

Clifford symbol)

made

Maxwell's

vector

use

of t h e

opera-

and introduced the term convergence.

operation

as

(8;

his

209)

He

mathematical

m e t h o d i n his section o n strains. T h e i n c o m p l e t e s e c o n d v o l u m e o f this

work,

published

lines of development.

posthumously

in

1887,

followed

the

same

It was perhaps natural that in an elementary

text C l i f f o r d w o u l d give f e w historical references. I n fact the i n d e x to the t w o v o l u m e s does not refer the reader to a single h e l p f u l historical reference in regard to the origin of Clifford's vectorial methods.66

Because

of this fact T a i t c a m e close to accusing C l i f f o r d of

plagiarism. Writing

a

review

ences., T a i t s t a t e d :

of

Clifford's

Common

Sense

of

the

Exact

Sci-

" T h u s , especially in matters connected w i t h the

d e v e l o p m e n t of recent mathematical a n d kinematical methods, his statements w e r e by no means satisfactory (from the historical point of v i e w ) to those w h o recognized, as their o w n , some of the best 'nuggets' subtitle cially

that shine here of

open

viewed

Clifford's to

this

and there

Elements

in

objection. . . ."

Clifford's

Elements

his pages.

o f Dynamic] (1;

o f Dynamic;

was,

272-273) Tait

H i s Kinematic [ t h e

throughout, In

spe-

1878 T a i t re-

wrote:

"Though

this

p r e l i m i n a r y v o l u m e contains only a small installment of the subject, the m o d e of treatment to be a d o p t e d by Prof. C l i f f o r d is m a d e quite o b v i o u s . . . . F o r , a l t h o u g h (so far as we h a v e s e e n ) t h e w o r d q u a t e r n i o n is not once m e n t i o n e d in the book, the analysis is in great part p u r e l y quaternionic. . . ." that

quaternionic

notations

(1;

270) T a i t w e n t on to c o m p l a i n

and full

quaternionic methods

should

have b e e n used m o r e extensively, although he suggested that Clifford d i d not do this because students had already been offered such t h i n g s as V a n d m i g h t " r e f u s e a l t o g e t h e r to b i t e a g a i n . " (1; 272) T h e significance of Clifford in the history of vector analysis m a y be best understood by considering h i m as a transition figure. Writi n g a t a t i m e a f t e r G r a s s m a n n a n d H a m i l t o n h a d c r e a t e d t h e i r systems and before Gibbs and Heaviside created the modern system of vector analysis, C l i f f o r d c a m e to appreciate the benefits to be derived

from

Moreover defining

the in

use

his

of vector

Elements

separately

his

methods,

o f Dynamic vector

he

and

especially

introduced scalar

in the

products.

mechanics. practice

of

Considered

against the b a c k g r o u n d of the quaternion tradition, this was a major conceptual

142

innovation,

since

quaternionists

never

viewed

Va/3

Traditions in Vectorial Analysis

and

Sap

only as

as t w o separate products of the t w o vectors a a n d ft b u t t w o parts o f t h e f u l l q u a t e r n i o n p r o d u c t a/3.

H e r e a n d else-

w h e r e i n his b o o k w e see C l i f f o r d s e l e c t i n g a n d a l t e r i n g parts o f t h e q u a t e r n i o n s y s t e m a n d f o r m i n g t h e r e b y t h e r u d i m e n t s o f a n e w syst e m of vector analysis. T h i s process of selection a n d alteration begun by Clifford was carried to completion by Gibbs and Heaviside. Neither of these m e n seem to have been influenced by Clifford, nor w e r e later vector analysts. Clifford

died in

This

may be attributed to the

1879, a n d left o n l y an

fact that

unfinished presentation of

the elements he h a d w o r k e d out. If Clifford h a d lived, the history of vector analysis

m i g h t be quite different.

As a conclusion to this chapter the writings of Tait, M a x w e l l , a n d Clifford may be briefly compared. It is obvious that certain similarities o f v i e w r u n t h r o u g h these w r i t i n g s : example,

convinced that the

all three authors were, for

vectorial approach to physical prob-

l e m s p r e s e n t e d m a n y advantages. O n e s i m i l a r i t y w h i c h i s less t h a n obvious is especially important.

If we read the statements of these

three authors w i t h t h e p e r s p e c t i v e o f the present, w e see that f r o m the w r i t i n g s of each the i d e a comes forth that the scalar p r o d u c t a n d the vector p r o d u c t are of p a r a m o u n t i m p o r t a n c e a n d that the quaternion product is

of limited

from

of p r e s e n t a t i o n

the

Maxwell

form the

idea is

w i t h an ox and an quaternion Magnetism.

In

Clifford this is clearly evident

used

in

his

Elements

o f Dynamic.

In

expressed metaphorically — " M a y one plough

ass

product

In

use.

is

Tait's

together?" — as w e l l as indirectly— the full rarely

Treatise

on

found

in

his

Quaternions

Treatise

this

on

idea

Electricity

may

also

and be

f o u n d , b u t o n l y b y o n e w h o ( u n l i k e T a i t ) has e y e s t o see it. Such a m a n was Gibbs, w h o in an 1891 paper pointed out that the c a r e f u l r e a d e r o f T a i t ' s Treatise w i l l

find

that w h a t is essential to the

developments and what occurs most frequently in the treatment of actual

problems

separate

is,

products

not the Saf3

and

q u a t e r n i o n p r o d u c t a/3, b u t r a t h e r t h e Va/3.67

In

making

this

acute

observa-

tion, w h i c h he presented in a paper arguing for the advantages of his system of vector analysis over the quaternion system, G i b b s perhaps revealed m o r e than he intended: his debt to Tait was p r o b a b l y greater than he Treatise

shortly

realized. before

he

It is

no accident that Gibbs read Tait's

created

the

modern

system

of

vector

analysis. A n d it is p e r h a p s n o t too m u c h to say that a l t h o u g h T a i t felt that he was not in accord w i t h the ideas of M a x w e l l a n d Clifford, the three w e r e i n fact m o v i n g i n the same direction.

143

Notes 1

Cargill

Gilston

England, 2

Peter

Guthrie

3

James

Clerk

( N e w York, 4

for

Scientific

Elementary

Treatise

Treatise

on

on

Work

of Peter

Quaternions,

Electricity

Clerk

Maxwell,

The

Scientific

1, ed. W. D. N i v e n ( N e w York,

Advancement

James

ties"

in

matical

in of

(4,11; Science

Clerk

Maxwell,

(4,11;

257-266).

Society,

William

ed. 8

Tait,

Maxwell,

Association"

the

7

and

Guthrie

Tait

(Cambridge,

and

2nd

ed.

Magnetism,

(Oxford, 2

vols.,

1873). 3rd

ed.

Papers

of James

Clerk

Maxwell,

2

vols,

1965).

James Clerk M a x w e l l , "Address to the Mathematical and Physical Sections of the

British

6

Life

1954).

James

b o u n d as 5

Knott,

1911).

3

(London, 9

published

in

British

Association

(1870).

Mathematical Classification of Physical Quanti-

Originally

published

in

Proceedings

o f the

London

Mathe-

Clifford,

Mathematical

Papers

by

William

Kingdon

Clifford,

1882).

Kingdon

second

Originally

40

224-232.

Kingdon

William A

" O n the

(1871),

R. Tucker (London,

1878).

215-229). Report,

Clifford,

incomplete

Elements

volume

of

was

Dynamic:

published

Part

I.

Kinematic

posthumously,

ed.

(London, R.

Tucker

1887).

T h e following note

is aimed at discussing the main techniques and assumptions

i n v o l v e d in this study. I have t i t l e d the various sections for clarity a n d ease of reference.

Definition of the term "publication": A paper or book that appeared during the

interval pages

1841

which

example,

in

to 1900.

Definition of the term " b o o k " : A publication of more than fifty

d i d n o t a p p e a r in a j o u r n a l . C o n c e r n i n g t h e q u e s t i o n of c i r c u l a t i o n (for regard

to

theses

that

seem to have been

published)

no work was

in-

cluded unless it was listed in one of the standard mathematical bibliographies or in the catalogue of at least o n e of the major libraries of the w o r l d . Translations a n d later editions w e r e counted as separate books.

Definition of the term "subject": The pub-

lications have b e e n classified into Grassmannian and H a m i l t o n i a n or quaternionic. Works ods

w e r e i n c l u d e d if they i n c l u d e d some discussion or use of the ideas or meth-

o f e i t h e r o r both

of these traditions.

T h i s classification w a s in m o s t cases b a s e d

o n a n e x a m i n a t i o n o f t h e p u b l i c a t i o n itself; i n t h o s e cases ( f e w i n n u m b e r ) w h e r e this was

impossible,

descriptions

matik,

were

Definition

used.

of

the

of the

work,

term

such

"country

w h i c h the book or journal article was published. use this

classification

rather than

example, the latter basis

one

based on

as

those

in

Fortschritte

of publication":

The

der

Mathe-

country

in

I t has b e e n f o u n d m o r e fruitful t o language of publication, since, for

w o u l d not allow an analysis of the different levels of inter-

est in B r i t a i n a n d A m e r i c a .

By a G e r m a n p u b l i c a t i o n I m e a n a p u b l i c a t i o n that ap-

p e a r e d in a G e r m a n - s p e a k i n g c o u n t r y . By a B r i t i s h p u b l i c a t i o n I m e a n a p u b l i c a t i o n appearing items ternions

and

thereto Study

in

an

English-speaking

classified:

The

Allied

published of

Quaternions

major

Systems in and

of the

Allied

country,

source

was

but

not the

Alexander

Mathematics

(Dublin,

Bulletin

the

Systems

of

United

1904)

International

of Mathematics.

States.

Macfarlane, and

Association for

Supplements

Sources

Bibliography

of

o f Qua-

supplements Promoting appeared

the in

t h e issues of this j o u r n a l for 1905, 1 9 0 8 , 1 9 0 9 , 1910, 1912, a n d 1913. T h i s b i b l i o g r a p h -

144

Traditions in Vectorial Analysis

ical

source,

members

compiled

by the

mathematician

of the Association, was

Alexander

Macfarlane

and

his

fellow

probably extremely complete, since the members

themselves h a d w r i t t e n m a n y of the articles included.

F r o m my experience I w o u l d

e s t i m a t e that n o t less t h a n 9 7 p e r c e n t o f t h e r e l e v a n t articles w e r e i n c l u d e d . T e c h nique used: I have based the study on a r a n d o m l y selected sample consisting of 334 publications w h i c h constitute 25 percent of the o g r a p h y a n d its 10

These

1338 items listed in the above bibli-

supplements.

numbers

sample there were

were

arrived

at

in

the

following

manner.

In

the

25

percent

146 q u a t e r n i o n publications, of w h i c h 35 c a m e in the p e r i o d f r o m

1841

to 1865. T h e r e w e r e 54 G r a s s m a n n i a n p u b l i c a t i o n s in t h e s a m p l e , of w h i c h 8

were

from

the

quaternion

period

from

1841

to

1875.

p u b l i c a t i o n s = 4 x 35 + 4 x

tions = 4 x 8 + 4 x 4 6 = 216.

It

was

Thus

the

111 = 584.

convenient

to

sample

Total

w o u l d predict:

Grassmannian

determine

by

actual

total

publicacount the

publications in the first tradition for the p e r i o d f r o m 1841 to 1865 a n d for the second tradition from

1841

to

1875. T h e n u m b e r s o b t a i n e d w e r e 150 a n d 3 3 for t h e q u a t e r -

nion and Grassmannian traditions respectively.

An accuracy check on the sampling

t e c h n i q u e is p r o v i d e d by c o m p a r i n g these figures. T h u s for the q u a t e r n i o n tradition and the period 1841 to

1865 the sample predicted ten too f e w publications. T h i s is

an error of 6.7 percent.

For the

Grassmannian tradition

the

sample

predicted one

p u b l i c a t i o n less t h a n t h e actual n u m b e r for t h e p e r i o d f r o m 1 8 4 1 t o 1875. T h i s i s a n error of 3.0 percent. period in each

I

have used the numbers

tradition

and used the

later p e r i o d in each tradition. T h u s 594 quaternion 11

If the

publications

four extremely

obtained by actual count for the first

numbers

p r e d i c t e d f r o m the s a m p l e for the

I have c o n c l u d e d that there w e r e 150 + 4x 111 =

a n d 33 + 4

long books

x 46 = 2 1 7

by

Grassmannian

Hamilton

are

not

publications.

included,

the

average

length of q u a t e r n i o n books b e c a m e 208 pages. If the five l o n g G r a s s m a n n books are not included, the average length of Grassmannian books b e c a m e 181 pages. It m a y be

noted

that

Hamilton

and

Grassmann

each

wrote

only

two

books;

the

higher

n u m b e r g i v e n above is a c c o u n t e d for by the fact that translations a n d later editions were 12

included.

H.

S.

White,

" F o r t y Years'

( N e w Series), 42 (1915), 13

Graph

indication

III as

is to

from the

Fluctuations

in

Mathematical

White,

actual

art.

cit.y

106.

Unfortunately White

n u m b e r of publications

in

Science

did

not give

any

in any year. T h e graph is thus

u s e f u l o n l y for c o m p a r i s o n o f its f o r m w i t h t h a t o f G r a p h s 14

Research"

105-113.

I and IV.

It is to be n o t e d that the accuracy of the sample as a criterion of inference on any

g i v e n point is a f u n c t i o n of the size of the sample. T h u s for e x a m p l e the degree of accuracy m a y not be great for inferences in analysis publications w r i t t e n 15

William

16

As

17

As

Rowan

quoted

in

quoted

( L o n d o n , 1910), 18

lbid.y

1070.

Mechanics 19

of

Chrystal, Silvanus

regard to the n u m b e r of Grassmannian

any country except Germany.

Hamilton,

G. in

in

Lectures

on

Quaternions

"Professor Tait" P.

Thompson,

(Dublin,

i n Nature, The

Life

64

1853),

(1901),

of

William

elementary

form

610.

306.

Thomson,

vol.

II

1138. Hertz

used

vectors

Macfarlane,

"Peter

in

a

very

in

his

Principles

of

1894.

Alexander

Guthrie

Tait"

in

Ten

British

Physicists

(New

Y o r k , 1919), 45. S e e also (1; 2 1 - 2 2 ) . 20

Peter

Guthrie

Tait,

21

These

editions

were

manager

of

Cambridge

Elementary both

Treatise

published

University

Press,

on

Quaternions

at Cambridge. has

(Oxford, R. W.

generously

1867),

v.

David, the present

supplied

me

with

the

following information. T h e second edition probably consisted of 1000 copies, since

145

A H i s t o r y of V e c t o r Analysis

796 copies w e r e still in stock at the Press in 1875. T h e t h i r d e d i t i o n consisted of 750 copies, fact

and this edition w e n t out of print in

that

22

500

Philip

copies

of

Kelland

(London,

Hamilton's

and

Peter

1910. T h i s s h o u l d be c o m p a r e d to the

Elements

Guthrie

were

printed.

Tait,

Introduction

to

Quaternions,

2nd

ed.

1882), v.

23

Ibid.,

x.

24

Ibid.,

ix.

25

N o t e that for the purposes of this study I

have defined " m o d e r n vector analysis"

so as to i n c l u d e s u c h areas as v e c t o r a d d i t i o n , s u b t r a c t i o n , a n d m u l t i p l i c a t i o n ; elementary vector algebra;

vector differentiation and integration; and the properties of

the operator V, along w i t h such famous transformation theorems as those of Stokes a n d Gauss.

D y a d i c s a n d the linear vector f u n c t i o n are not i n c l u d e d in this definition,

but some discussion w i l l be g i v e n of their place in this 2 6

Of course

many

s h o w that this vector 27

was

of the not

theorems

history.

w e r e first stated by Grassmann,

in

m o s t cases

the

name

suggested

to

source

from

but we

will

which they entered modern

analysis.

Nabla

was

the

similarity of the

Tait

by

s y m b o l to an Assyrian harp.

Robertson

S e e (1;

Smith

143).

because

of the

Maxwell used the w o r d

only once in his p u b l i s h e d writings, a n d that was in a poem, " T o the C h i e f Musician upon

Nabla,

poem was peared of

in

print

Students

Gibbs

A Tyndallic

Ode."

The

p u b l i s h e d i n Nature a n d

of

in

Edwin

Mathematics

( N e w York,

1901),

"Chief Musician

is

g i v e n i n (1;

Bidwell and

138.

Wilson's

Vector

Founded

upon

Physics

The

upon

Nabla"

was

Tait. T h e

171-174). T h e n a m e d e l first apAnalysis,

A

the

Text-Book for

Lectures

the

of J.

Use Willard

genealogy of atled (which is A, delta inverted to

f o r m V, atled) is u n k n o w n to me. It m u s t have b e e n u s e d as early as 1870, for in that year M a x w e l l asked Tait in a letter if atled was Tait's n a m e for V. 28

(3,1; 29). O n T a i t ' s c o n t r i b u t i o n see also (1;

"Peter 29

Guthrie

The

Tait"

history

in

Physical

of these

Review,

theorems

15

has

(1902),

never

See (1;

143).

142-149) and Alexander Macfarlane,

(to

56.

my

knowledge) been

written.

It

essentially lies outside the p r o v i n c e of the history of vector analysis, for the theorems were

all

developed

w o r k w i t h vectors. called

the

Divergence

Geometrischen schaften, auf

vol.

die

Ill,

im

for Cartesian

Theorem)

Analyse, pt

I,

attributed

Teil"

Gauss

Verhaltnisse

in

im

Jahre

1839,

in his

des

Abstossungskrafte"

Vereins

analysis,

and

by

people

w h o d i d not

may h o w e v e r be made. Gauss' T h e o r e m (often is

Erster

1345) to

verkehrten

Anziehungs-und magnetischen

originally

Some comments

(by

Hermann

Encyklopadie

"Allgemeine

Quadrates

in

1840),

Rothe,

Wissen-

in

Beziehung

Entfernung

wirkenden

aus

den

1-51,

"Systeme

mathematischen

Lehrsatze

der

Resultate

(Leipzig,

der

with

Beobachtungen special

des

attention

t o pages 3 4 t o 35. T h i s w a s t r a n s l a t e d i n t o E n g l i s h a n d p u b l i s h e d i n R i c h a r d T a y l o r ' s Scientific

Memoirs,

stated

that

3

(London,

Gauss'

1843),

Theorem

153-196.

". . . seems

in a p a p e r read in 1828, b u t p u b l i s h e d in burg,

T.

I.

p.

39."

This

note

is

not

James

to

have

Clerk

Maxwell

in

(3,1;

125)

b e e n first g i v e n by O s t r o g r a d s k y

1 8 3 1 i n t h e M e m . d e l ' A c a d . d e St. P e t e r s -

contained

in

the

first

edition

of his

Treatise.

T h i s fact i s d o u b l y i n t e r e s t i n g a s p o s s i b l y i n d i c a t i n g w h e r e M a x w e l l first f o u n d the theorem.

Oliver

1953), 38,

wrote

of

triple

recherches I,

p.

attractionis tractata,

146

integrals sur

263.

Dimon

la

The

Kellogg,

the following in to

nature double

corporum Commentationes

double et

la

integrals propagation

integrals

are

sphaeroidicorum societatis

Foundations

regard to

was du

given

Potential

employed

son, in

t.

more

ellipticorum regiae

of

Theory,

Gauss' T h e o r e m :

II,

by

(New

York,

"A similar reduction L A G R A N G E :

1760-61,

45;

Nouvelles

CEuvres,

t.

d e f i n i t e f o r m b y G A U S S , Theoria homogeneorum

scientiarum

methodo

Gottingensis

novo

recentiores,

Traditions in Vectorial Analysis

vol.

II,

1813, 2 - 5 ;

equivalent the

to

Application

netism,

Werke,

Bd.

V,

pp.

5-7

the divergence theorem of

Mathematical

Nottingham,

1828."

[.]

was

Analysis

The

A

systematic use of integral identities

m a d e by George

to

the

Green

Theory

of

history of Stokes' T h e o r e m

in

his

Electricity

is

Essay o n

and

Mag-

clear but very

com-

plicated.

I t w a s first g i v e n b y Stokes w i t h o u t p r o o f — a s w a s necessary — s i n c e i t w a s

given

an

as

Among

the

origin

examination candidates

of the

Gabriel

theorem,

Stokes,

1905),

320-321.

Kelvin

in

question

for

the

which

by

Mathematical See

also

for the

prize

1870

and the

was

Smith's

Prize

Maxwell,

who

was

Physical

frequently

Papers,

important

used.

vol.

historical

Examination

V

On

this

Stokes the see

(Cambridge,

footnote

which

George

England,

indicates

that

a letter of 1850 was the first w h o actually stated the t h e o r e m , although

others as A m p e r e h a d e m p l o y e d " t h e same k i n d of analysis . . . See

of that year!

later traced to

also

Max

Bacharach,

Abriss

der

Geschichte

and

Other

der

in particular cases."

Potentialtheorie

(Gottingen,

1883). 30

of

Peter

the

Guthrie 31

Guthrie

Royal Tait,

Peter

Royal

of

in

Tait,

32

Dirk

J.

33

Benjamin

matics,4 34

36

On

Peirce.

26

Papers, Yankee

(1870),

I

"On

Some

318-320,

vol.

I,

159-163.

in

the

Making,

Associative

Theorems"

Published

England,

Quaternion

(1870),

"Linear

Allied

169-184.

(Cambridge,

Science

1898),

Integrals"

and

(1872),

rev.

ed.

in

Transactions in

Peter

136-150.

in

Proceedings

784-788.

(New

Algebra,"

in

also

of

the

Published

York,

1962),

American Journal

415.

o f Mathe-

216-217.

Cajori,

The

1890),

Teaching

and

History

of

Mathematics

in

the

United

States

Benjamin

Peirce

137.

"Reminiscences"

(Oberlin,

Green's

vol.

7

Peirce,

(1881),

a

Tait,

Edinburgh, Scientific

Florian

In

"On

Edinburgh,

Papers,

Struik,

(Washington, 35

of

Scientific

Guthrie

Society

also

Tait,

Society

included

in

Raymond

Clare

Archibald,

1925), 6.

Hill, On

Lowell,

the

Chase,

teaching

of

and

the

two

quaternions

at

Peirce

sons

Harvard

see

see

Archibald,

Cajori,

Math

Benjamin

in

the

U.S.,

127-151. 37

H.

A.

Science,

3rd

38

Benjamin

matics,4 39

Newton, Ser.,

"Benjamin

22

(1881),

Peirce,

(1881),

Peirce"

(Obituary

Notice)

in

American

Journal

of

74.

"Linear

Associative

Algebra"

in

American Journal

o f Mathe-

97-229.

As described by H.

Newton's

description

mention.

See

E.

H a w k e s a n d as

of this

Newton,

q u o t e d b y A r c h i b a l d , Peirce, p .

work

is

too

long

to

quote

"Peirce"

in

American Journal

but too

o f Science,

16.

important

3rd

Ser.,

22

H. A. not

to

(1881),

167-178. 40

Quaternion

Harvard,

methods

were

taught

at

Johns

Hopkins,

Wisconsin,

Michigan,

Princeton (then the C o l l e g e of N e w Jersey), D a r t m o u t h , C o r n e l l , Virginia,

South Carolina, Alabama, Tennessee,

and Texas.

Cajori

gave

no

information

as«to

w h e t h e r courses i n q u a t e r n i o n s w e r e (or w e r e not) t a u g h t a t Yale, B o w d o i n , G e o r g e town,

Virginia

Washington noted See

Military

(of Saint

that the

Florian

mathematics

first

Cajori,

Institute,

Louis),

group op.

teaching;

cit.

and

North the

contains Cajori's

nonetheless

Carolina,

Mississippi,

U n i t e d States

Kentucky,

Military Academy.

Tulane,

It is

to be

the majority of the better schools at that time. main his

concern

book

in

includes

this

book

much

was

the

history

information

of

relevant

to the topic at hand. 41

Saloman

American 42

Bochner,

Scientist,

Albert

50

Einstein

"The

(1962), and

Role

of

Mathematics

in

the

Rise

of

Mechanics"

in

301-302.

Leopold

Infeld,

The

Evolution

o f Physics

(New

York,

1961),

244.

147

A H i s t o r y of V e c t o r Analysis 43

M a n y of the most i m p o r t a n t letters

published so

many

and

in

Knott's

book,

of Maxwell's

hence

the

only

cited

letters

way

in

is

in the correspondence for this

note

in

(1)

large

above.

Knott's

part that most of Tait's

of indicating their content

was

history were

reason for p u b l i s h i n g letters

w e r e lost,

by publishing

Maxwell's

replies. 44

S e e f o r e x a m p l e (1;

and in which bling" 45

I

in

Edmund

in quaternions but was "dab-

Whittaker,

A

History

of

the

Theories

of

Aether

and

Electricity,

vol.

1958), 2 4 1 - 2 4 2 . It s h o u l d be n o t e d that W h i t t a k e r s l i p p e d into a n u m b e r

of anachronisms, B."

"unlearned"

Hamilton.

(London,

a =

144), w h e r e letters f r o m M a x w e l l to T a i t of 1870 are q u o t e d

M a x w e l l admitted he was

This

as for e x a m p l e w h e n he w r o t e that T h o m s o n in

anachronism

would

have

been

particularly

1846 f o u n d "curl

displeasing to Thomson,

w h o s e strong feelings against vectors have b e e n discussed. 46

(6;

261).

papers

cited

The in

reader

notes

leading historically.

may

(5)

note

and

(6)

For

this

see

quotations

have

been

Rhetorically this

is

drawn

useful

from

both

and not mis-

A c c o r d i n g t o C l i f f o r d , M a x w e l l w a s t h e first t o d i s t i n g u i s h b e -

t w e e n "force" and "flux" vectors; 47

that

above.

Tait,

Treatise

see (7; 4 9 7 ) .

(1st

ed.),

268,

or

Hamilton,

Lectures,

610.

Note

that

neither H a m i l t o n nor T a i t nor M a x w e l l use the n o w c o m m o n 8 for partial differentiation. 48

(6;

—V

•

265). y,

N o t e that since S Vcr (in q u a t e r n i o n language) is e q u a l to the m o d e r n

Maxwell's

name

" d i v e r g e n c e " for V 49

(1;

143).

letters

has

The

been

"convergence"

required

alteration.

The

modern

name

• a w a s g i v e n by W i l l i a m K i n g d o n C l i f f o r d in (8; 209). following

aided

by

discussion

the

of

availability

Maxwell's

ideas

of unpublished

as

parts

expressed

in

of Maxwell's

his cor-

r e s p o n d e n c e . T h e s e letters are i n the archives o f C a m b r i d g e U n i v e r s i t y L i b r a r y . T h e officials the

of the

Library

collection.

I

of Professor E r w i n 5 0

F r o m the

51

Quaternions

see

A.

R.

(1935), who

permitted

Professor

N.

Hiebert of the

unpublished portion were

probably

Forsyth,

164 a n d

had been

Derek

"Old

172.

[James this

ography

Papers

Clerk

review

not

taught Days

wrote:

at

a

53

Ibid.,

54

James

1873).

at

Cambridge"

"Occasional

edition.

note

A

Clerk second

"Quaternions"

Maxwell The

I

content

doubt. [4]

have

in

On

this

Mathematical

point

Gazette,

19

attention, by individual students

is

This

of

such

paper

was

always w i t h an introduc-

172).

i n Nature, 9

followed

Knott a

(1873), (1;

nature not

137-138.

115) as

and to

included

put

in

In attribut-

Macfarlane the

Bibli-

question

Maxwell's

Scientific

above).

Maxwell,

Treatise

(posthumous,

but

on with

1881, a n d a t h i r d edition in

of the All

contrary is

3rd.

ed.).

quotations

Quotations may

be

(1st

ed.),

Electricity some

1891

from

Magnetism, by

2

vols.

Maxwell)

(Oxford,

edition

ap-

(the b o o k cited in note [3] above is a

the

assumed to

and

revisions

Treatise

will

be identical

be

referred to

in all

editions

the

third

unless the

noted.

55

Maxwell,

56

F o r t h e s c a l a r p r o d u c t s e e (3,11; 2 5 6 a n d 2 7 4 ) .

Treatise,

240, 244, a n d 305).

148

however.

137.

peared in reprint

in

microfilm

of Tait at E d i n b u r g h or pupils of Barker at Manchester, was

reasonable

(cited

to

correspondence.

Cambridge

never as a non-commutative algebra:

Maxwell], to

o f Quaternions).

beyond

Solla Price

University of Wisconsin.

t o r y t e s t i m o n i a l f r o m N a t u r a l P h i l o s o p h y . " (p. 52

de

of the Tait-Maxwell

Tripos

Forsyth

pupils

paid to quaternions:

ing

J.

have seen the letters in a c o p y of this m i c r o f i l m in the possession

I

28-29. F o r t h e v e c t o r p r o d u c t s e e (3,11;

Traditions in Vectorial Analysis 57

(3,11;

13).

edition 58

A.

of P.

troduction 59

Quaternions. in

Tensor 174

dynamik" 60

Wills

to

(3,11;

I n t h e f i r s t e d i t i o n o f M a x w e l l ' s Treatise t h e r e f e r e n c e w a s t o t h e f i r s t

Tait's

in

the

"Historical

Analysis

and

(New

319).

Introduction"

York,

Grassmann's

Poggendorff's

Annalen

der

1958), paper

is

Physik,

74

M a x w e l l referred to Grassmann on pages

to

his

his

"Neue

(1845),

125,

84.

der

This

Farbenmischung"

was

translated

in

into

Poggendorff's

English

and

with

an

In-

Theorie

der

Electro-

1-18.

141-142,

i n t h e f i r s t v o l u m e o f t h e w o r k c i t e d i n n o t e (4) a b o v e . Theorie

Vector Analysis

xxvi.

152, 414, a n d 4 1 8 - 4 1 9

Grassmann's paper is " Z u r

Annalen

der Physik,

published

in

the

99

(1853),

Philosophical

69-

Magazine,

7 (1854), 2 5 4 - 2 6 4 . 61

Alexander

Macfarlane,

"James

Clerk

Maxwell"

in

Ten

British

Physicists

(New

Y o r k , 1919), 18. 62

William

200). are

Kingdon

an

offshoot

Grassmann's 63

Clifford,

"Preliminary

Clifford's b i q u a t e r n i o n s are of Hamilton's

Sketch

not the same as

quaternion

system

of Biquaternions"

in

(7;

181-

Hamilton's biquaternions. They and

Ball's

theory

of screws

and

system.

T h e content of these

lectures was preserved in notes on the lectures made by a

s t u d e n t a n d p u b l i s h e d i n (7; 4 7 8 - 5 1 5 ) . 64

William

Kingdon

Pearson ( N e w York, 65

Ibid.,

66

The

Clifford,

Philosophy,

which

appeared, and this 67

On

Sense

of

the

Exact

Sciences,

ed.

Karl

1885; the above is a reprint.

vii. index for b o t h v o l u m e s

never m e n t i o n e d ; Tait is Natural

Common

1894). T h e b o o k first a p p e a r e d in

this

see

is

contained in the second volume.

referred to three times, always

Tait

wrote

with

Kelvin.

A

Grassmann is

i n r e g a r d t o t h e Treatise o n

single

reference

to

Hamilton

is of no importance. The

Scientific

Papers

o f j .

Willard

Gibbs,

vol.

ii

(New

York,

1961),

162.

149

CHAPTER

Gibbs

and

Development

Heaviside

of the

and

the

Modern

System

Vector

Analysis

of

I.

FIVE

Introduction To

show

that Josiah W i l l a r d

pendently the

and

modern

chapter.

system

It

interest

nearly

will

in

Gibbs

and

simultaneously

of vector analysis

be

argued

electrical

Oliver Heaviside

created is

the

is

chief concern

men,

inde-

essentially

motivated

of this

that

these

and

inspired particularly by Maxwell,

theory

two

what

by

an

forged m o d e r n vector analysis from quaternion (not Grassmannian) elements. spared

Later chapters w i l l show that Gibbs and Heaviside were

the

years

their

much

used.

fate

of Grassmann

system Since

and

Hamilton;

of vector analysis Gibbs'

work

was

came

within

widely

slightly

twenty-five

appreciated and

earlier than

Heavi-

side's, he w i l l be considered first.

II.

Josiah In

the

(1890) little

Willard preface

Peter

Quaternions." must be

monster, mann."

7

150

7

his

the

third

Tait

expressed

recently

edition

been

made

He went on to remark: one

pamphlet

on

Vector

Gibbs

marks

the

with

on

the

Quaternions at

"how

development

of

" E v e n Prof. W i l l a r d G i b b s

Analysis;

Tait's

progress,

Treatise

disappointment

of the retarders of Quaternion progress, in

notations

remark

of his

his

c o m p o u n d e d of the

quaternion Analysis

has

ranked as

of

to

Guthrie

progress

virtue

Gibbs

about for

beginning

his

is

a

sort

correct;

"pamphlet"

of m o d e r n

of h e r m a p h r o d i t e

of Hamilton

vector

a n d of Grass-

Gibbs

did

Elements analysis.

of

retard Vector

D e v e l o p m e n t of the M o d e r n System of Vector Analysis

M a n y of Tait's of Quaternion

readers

progress"

must have was.

w o n d e r e d w h o this "retarder

Though

Gibbs

had

by

1890

made

major scientific discoveries, f e w scientists had noticed t h e m in the pages

of

the

fate was t h e tion.

Transactions

of

the

Connecticut

Academy.

Gibbs'

n o t u n c o m m o n fate of the theoretician:

slow recogni-

Recognition did come in time, and now Gibbs

is frequently

placed in

a class

with

Gauss,

Faraday,

Maxwell, Helmholtz, and

Einstein.

H o w e v e r w h e n he presented his vectorial system in the

1880's, h e l a c k e d that " c a p i t a l " o f r e p u t a t i o n w h i c h H a m i l t o n h a d attained so early in his life a n d w h i c h f a v o r a b l y i n f l u e n c e d the reception of his ideas. in

Gibbs'

life

fluenced its

that

It will thus be important to survey the events led

up

to

the

creation

of his

system

and

in-

reception.

Josiah W i l l a r d G i b b s was b o r n in 1839: his father was at that t i m e a professor of sacred literature at Yale University.8 G i b b s graduated f r o m Yale in 1858, after he h a d c o m p i l e d a d i s t i n g u i s h e d r e c o r d as a student. the

His training in mathematics was good, mainly because of

presence

graduation

of H.

he

A.

Newton

on

the

faculty.

I m m e d i a t e l y after

e n r o l l e d f o r a d v a n c e d w o r k i n e n g i n e e r i n g a n d at-

tained in 1863 the first doctorate in e n g i n e e r i n g given in the U n i t e d States.

(1;

32) A f t e r r e m a i n i n g at Yale as tutor u n t i l

1866, G i b b s

j o u r n e y e d to E u r o p e for three years of study d i v i d e d b e t w e e n Paris, Berlin, and Heidelberg. N o t a great deal of information is preserved c o n c e r n i n g his areas of c o n c e n t r a t i o n d u r i n g t h e s e years, b u t it is clear that his

main

interests w e r e theoretical

matics rather than applied science. became

acquainted with

Mobius'

science and mathe-

It is k n o w n that at this time he work in geometry, but probably

n o t w i t h t h e s y s t e m s o f G r a s s m a n n o r H a m i l t o n . (1; 4 3 ) G i b b s returned to N e w Haven in

1869 a n d t w o years

later was m a d e pro-

fessor of m a t h e m a t i c a l physics at Yale, a p o s i t i o n he h e l d u n t i l his death. H i s m a i n scientific interests in his first year of teaching after his return

seem

to

have

been

mechanics

a n d optics.

(1;

61-62)

His

interest in t h e r m o d y n a m i c s increased at this time, a n d his research i n this area l e d t o t h e p u b l i c a t i o n o f t h r e e papers, t h e last b e i n g his now

classic

published Connecticut

"On

in

the

Equilibrium

1876 a n d

Academy.

This

1878 work

of Heterogeneous

in v o l u m e of over

III

three

of i m m e n s e importance. W h e n scientists

of the

Substances,"

Transactions

hundred

finally

pages

o f the was

r e a l i z e d its s c o p e

and significance, they praised it as one of the greatest contributions of the century. H o w e v e r , for the present purpose the focus m u s t be on

the

immeidate

r e c o g n i z e d its

reception

of the paper.

Although

Maxwell had

importance, f e w others d i d before the early 1890's,

151

A

History

of V e c t o r Analysis

w h e n O s t w a l d arranged for a G e r m a n translation a n d publication of the work.9 T h u s w h e n in the 1880's G i b b s b e g a n to m a k e k n o w n his v i e w s on v e c t o r analysis, his f a m e as a scientist was not great. There

is

good evidence for c o n c l u d i n g that G i b b s

Maxwell's

ideas

Treatise

Electricity

ing

on

interest

Maxwellian

and

until

Magnetism

in

the

appearance

1873.

(1;

62)

did not k n o w of the His

latter's increas-

1 8 7 7 (1;

62), a n d e v e n t u a l l y to p u b l i s h papers in the

tradition.

Gibbs'

The

electricity

in this area led h i m to give a course in electricity and

magnetism in

III.

on

Early

historian

letter written

by

Work

is

in

Vector

extremely

Analysis

fortunate

that

the

draft

of an

1888

G i b b s to V i c t o r S c h l e g e l has b e e n preserved, for

this letter answers a host of interesting questions. T h e introductory paragraph

of the

letter was

followed by:

Your apt characterization of my Vector Analysis in the Fortsch. Math, suggests that y o u m a y be i n t e r s d [interested] to k n o w the precise relation of that p a m p h l e t to the w o r k of H a m .

& Grass,

w i t h r e s p e c t t o its

composition. My E.

&

first M.

acquaintance [Electricity

with and

quaternions

Magnetism]

where

was

in

reading

Quaternion

Maxwell's

notations

are

considerably used. I became c o n v i n c e d that to master those subjects, it was necessary for me to c o m m e n c e by mastering those methods. At the same t i m e I saw, that although the methods we r e called quaternionic the idea of the q u a t e r n i o n was quite foreign to the subject. In regard to the products

o f v e c t o r s , I s a w t h a t t h e r e w e r e t w o i m p o r t a n t f u n c t i o n s (or

p r o d u c t s ) c a l l e d t h e v e c t o r part & t h e scalar part of t h e p r o d u c t , b u t that the u n i o n of the t w o to form what was called the (whole) product d i d not advance the theory as an instrument of geom. investigation. Again w i t h r e s p e c t to t h e o p e r a t o r V as a p p l i e d to a v e c t o r I saw that t h e v e c t o r part &

the

their

scalar union

part

of the

(generally

to

result be

represented important operations, but separated

afterwards)

did

not

seem

a

v a l u a b l e idea. T h i s is i n d e e d o n l y a r e p e t i t i o n of my first o b s e r v a t i o n , since the operator is defined by means of the multiplication of vectors, & a change in the idea of that multiplication w o u l d involve the change in the use to the operator V. I therefore began to w o r k out ab initio, the algebra of the t w o kinds of m u l t i p l i c a t i o n , t h e t h r e e d i f f e r e n t i a l o p e r a t i o n s V a p p l i e d to a scalar, & the t w o operations

to a vector, & those functions or rather integrating

operators w h [ w h i c h ] ( u n d e r c e r t a i n l i m i t a t i o n s ) are t h e i n v e r s e o f t h e said differential operators, & wh play the l e a d i n g roles in m a n y departments of M a t h . Phys. To these subjects was a d d e d that of lin. vec. functions wh is also p r o m i n e n t in M a x w e l l ' s E. & M. This I ultimately printed but never published, although I distributed a g o o d m a n y copies a m o n g such persons as I t h o u g h t m i g h t possibly take a n i n t e r e s t i n it.

152

My delay & hesitation in this respect was principally

D e v e l o p m e n t of the

Modern

System of Vector Analysis

due to difficulty in m a k i n g up my m i n d in respect to details of notation, matters trifling in themselves, b u t in wh it is undesirable to make unnecessary

changes.

M y a c q u a i n t a n c e w i t h Grassmann's w o r k was also d u e t o t h e subject of E. [electricity] & in particular to the note wh he published in Crelle's Jour, in 1877 c a l l i n g attention to the fact that t h e l a w of the m u t u a l action of two elements given in

of current wh

Clausius

had just published had been

1845 by himself. I was the m o r e i n t e r e s t e d in the subject as I

had myself (before seeing Clausius' work) come to regard the same as the simplest expression for the mechanical action, & p r o b a b l y for the same reason as Grassmann, because that l a w is so v e r y s i m p l y expressed by means of the external product. At all events I saw that the methods wh I was using, w h i l e nearly those of Hamilton, were

almost exactly those of Grassmann.

I procured the

t w o E d . o f t h e A u s d . b u t I c a n n o t say t h a t I f o u n d t h e m easy r e a d i n g . I n fact

I have never h a d the perseverance to get t h r o u g h w i t h either of

them, & have perhaps got m o r e ideas f r o m his miscellaneous m e m o i r s than from those works. I

am

not h o w e v e r conscious

that Grassmann's

writings

exerted any

particular influence on my V-A, although I was glad e n o u g h in the introductory names felt

paragraph

to

[Grassmann

would

be

shelter myself b e h i n d one

or two

and Clifford] in m a k i n g changes

distasteful

to

quaternionists.

In

distinguished

of notation

fact

if you

wh

read

I

that

p a m p h l e t c a r e f u l l y y o u w i l l see that i t all f o l l o w s w i t h t h e i n e x o r a b l e logic of algebra from the p r o b l e m wh I

h a d set m y s e l f l o n g b e f o r e m y

acq. w i t h Grass. I have no d o u b t that y o u consider, as I do, the m e t h o d s of G r a s s m a n n to be superior to those of Hamilton. It thus seemed to me that it m i g h t [be] interesting to you to k n o w h o w c o m m e n c i n g w i t h some k n o w l e d g e of Ham's methods & influenced simply by a desire to obtain the simplest algebra for the expression of the relations of G e o m . Phys. &c I was l e d essentially

to

Grassmann's

algebra

of vectors,

independently

of any

i n f l u e n c e f r o m h i m o r a n y o n e else.10 This

letter is so rich in information that from it we m a y establish

Gibbs' relationship to history

each

of the

important

figures

in

the

earlier

of vector analysis.

Concerning Gibbs' relationship to Maxwell: we learn that interest in

electricity

and

magnetism

here he found quaternions usefulness.

Thus

from

Maxwell's

gentle

quaternion

methods

perhaps their

the

quaternion

is

vector analysis, Concerning that Gibbs

probably

and

not

Maxwell

criticisms

practices

theoretical

of the

Gibbs

and did

Gibbs

to

he

Maxwell's

went

to

the

discriminating not go

Treatise;

quaternionists. employment

unnoticed by

quaternionists

rhetorical

needed.

Gibbs'

led

"considerably used" and inferred their

statements

Gibbs.

revealed to could

not

him say:

of Or

what the

In a n y case, w h a t M a x w e l l w a n t e d in a

produced. relationship to

c o m m e n c e d his

Hamilton

and Tait:

we learn

search for a vector analysis " w i t h some

153

A H i s t o r y of V e c t o r Analysis

knowledge

of

Ham[ilton]'s

methods . .

and

methods that were "nearly those of Hamilton. ments

showing historical

ended

up

with

. . . " S t r o n g e r state-

c o n t i n u i t y are s e l d o m found.

It is clear

that Gibbs was no more than a m i n o r heretic in relation to quatern i o n orthodoxy. G i b b s ' a c h i e v e m e n t was great, b u t so was his d e b t to H a m i l t o n and his followers. especially influential Concerning Gibbs him

Gibbs'

created

or any

on

his

one

It w i l l later be argued that Tait was

Gibbs.

relationship

system

else."

to

Grassmann:

"independently

Obviously

Gibbs'

we

of any

learn

that

influence

from

"any one

else" was

not

m e a n t t o i n c l u d e M a x w e l l a n d t h e quaternionists. G i b b s also stated that he

was

not "conscious that Grassmann exerted any particular

influence on my V-A. . . ." This is to be expected since Gibbs had begun

searching

acquaintance] finally began though either

of

priority Gibbs' were

to

Gibbs

for

with

a

read

warmly

statements

vector

Grassmann,

admitted

Grassmann's

and

new

Grass[mann]."

he

had

he

he

his

found

concerning

did ideas

his

"long

(1877 a

never been

books,

praised

system

When

or

before later)

kindred

able

my

Gibbs

spirit.

Al-

to

read through

recognize

Grassmann's

on

numerous

independence

of

occasions. Grassmann

fully accepted by V i c t o r Schlegel, w h o as Grassmann's lead-

i n g d i s c i p l e was perhaps least d i s p o s e d to accept them. In his r e p l y to

Gibbs'

letter Schlegel

stated:

"I

realized from your letter h o w

y o u attained to ideas similar to Grassmann's but i n d e p e n d e n t l y of h i m ; I have repeatedly witnessed this p h e n o m e n o n : w h e n the time has

come —when

discovery —then

the this

development of science discovery

will

be

made

demands by

a

a definite

n u m b e r of in-

11

vestigators."

Concerning Gibbs' relationship to Clifford:

we learn that it was

l i m i t e d to the use of Clifford's n a m e "as a shelter" in the introductory paragraph of his vector analysis book. Thus

from this

letter we can obtain a general

outline of Gibbs'

e a r l y w o r k i n v e c t o r analysis. T h e r e are h o w e v e r a n u m b e r o f facts that w i l l give a fuller picture of G i b b s ' activities. In to

1879 G i b b s gave a course in vector analysis w i t h applications

electricity

private the

and

printing

magnetism,12 of

the

first

second half appeared in

known,

Gibbs

of

the

day

received

copies,

in

of h i s

1884.

sent out copies

entists a n d mathematicians.

and half

1881

he

arranged

Elements

for the

o f Vector Analysis;

In an effort to m a k e his system

of this w o r k to more than

130 sci-

(1; 2 4 7 ) M a n y o f t h e l e a d i n g scientists for

example,

Michelson,

Newcomb,

J. J. T h o m s o n , Rayleigh, FitzGerald, Stokes, Kelvin, Cayley, Tait, Sylvester, G. H. D a r w i n , Heaviside, Helmholtz, Clausius, Kirchhoff,

154

D e v e l o p m e n t of the

Modern

System

of Vector Analysis

L o r e n t z , W e b e r , F e l i x K l e i n , a n d S c h l e g e l . (1; 2 3 6 - 2 4 7 ) T h o u g h t h e work

was

would making

IV.

not

have

the

such

advertisement

a selective

a

regular publication

distribution

that

must have aided in

it known.

Gibbs'

Elements

Some

idea

may be

of

of Vector

the

form

obtained from

The are

given

had,

fundamental

Analysis of

Gibbs'

Gibbs'

Elements

of

Vector

Analysis

13

introductory paragraph:

principles

of the

following

analysis

are

such

as

familiar u n d e r a slightly different f o r m to students of quaternions.

The manner in w h i c h the

subject is

from that f o l l o w e d in treatises writer

does

being

simply to

vectors,

not

require any use

of the

give

notation

or between

and which

developed is somewhat different

on quaternions, since the object of the

a

suitable

vectors

lend themselves

conception

of the

for those

a n d scalars, w h i c h

quaternion,

relations b e t w e e n

seem most important,

most readily to analytical transformations,

a n d to e x p l a i n some of these transformations. As a p r e c e d e n t for such a departure

from

quaternionic

usage,

Clifford's

Kinematic

may

be

cited.

I n this c o n n e c t i o n , t h e n a m e o f G r a s s m a n n m a y also b e m e n t i o n e d , t o whose

system

the

following

method

attaches

m o r e c l o s e l y t h a n t o t h a t o f H a m i l t o n . (2; Although

Gibbs

introductory that his

mentioned

paragraph,

chief debt was

the

quaternionists.

be

illustrated;

the

the

I,

it

edition

of

Tait's the

be

Grassmann

letter makes

in the it clear

or Grassmann but to

suggested form

Treatise

Algebra

Gibbs

on

was

found

Quaternions.

of

Vectors,"

and

"vector

began

with

much

of the symbolism introduced, Gibbs followed the quaternion

(for

by

a

means

vector)

of

and

Tait and

Maxwell, but unlike

ordinate

system.

In

analysis."

dealing

with

vector

i, j,

and

k.

Where

T a i t AB, G i b b s Hamilton,

products

Hamilton

w r o t e AB.

Gibbs,

the

relation be

between

expressed

course the

now-current

never

by

scalar

these

products

writing

combined

quaternion

(dot)

Gibbs

equation

a.fB

=

these <xf3

and and

introduced

—Sa/3

vector the and

products —

Sot(3

H~

had like

u s e d the r i g h t - h a n d e d co-

the

p r o d u c t , " w r i t t e n a./3, a n d t h e " s k e w p r o d u c t , " w r i t t e n a X are

In

for example, G i b b s represented vectors by G r e e k letters,

components

w r i t t e n AB

"scalar,"

that

of presentation

definitions

their

"vector,"

respects

such

traditions;

as

will

some

of Gibbs' book this point w i l l

content and

"Concerning

and

cited

either Clifford

discussion

specifically

second

Chapter

Clifford

previously

not to

In the

strongly influenced by the in

only

itself in

17)

(cross)

a

X

/3

any

Vctfi.

To

== way

These

products.

quaternion

in

"direct

system

Vocfi.

The may

Gibbs

analogous

illustrate

of to

further

155

A H i s t o r y of V e c t o r Analysis

the

closeness

of

Gibbs'

booklet

to

Tait's

Treatise

on

Quaternions

( 2 n d ed.), various e q u i v a l e n t equations f r o m C h a p t e r I of G i b b s a n d from Tait have been listed below: G i b b s (2; 21)

a./3

=

/3.a

T a i t (3; 4 3 )

Sa/3

=

S/3a

G i b b s (2; 21)

a

(3 = - 0 X

T a i t (3; 4 3 )

Vaf3

Gibbs

(2;

X

22)

=

a.p (zx'

Tait

(3;

=

Tait

(2;

(3;

23)

Tait

a

44)

G i b b s (2; (3;

xx'

a/3 = (xy'

Gibbs

—V/3a

- x z ' ) j

43)

+

yy'

+

(xy'

{xx'

-

yx')k

X

\fi

X

VaVpy =

24)

a

+

+

zz' -

and

y] = ySaP

VVapVyS

X

p

=

(yz'

-

zy')i

+

yy'+ zz') + ( y z ' - z y ' ) i + ( z x ' - x z ' ) j +

(a.y)p -

(a.p)y

- pSya

[a X p] X [y x 8] = ( a . y X

44)

a

yx')k

=

-pSaVyS

+

8)p - (p.y X 8 ) a

aSpVyd

M a n y m o r e such pairs of equations could be given, but the above s h o u l d b e sufficient t o suggest s t r o n g l y that t h e t w o w o r k s are q u i t e similar

in

the

symbolisms

employed

and

mathematical

ideas

ex-

pressed. T h e similarities are p r o b a b l y most easily e x p l a i n e d by the conjecture hence

that G i b b s learned quaternionic analysis from Tait and

when

he

came

content, it was

to

write

a

work

natural for h i m to write

had learned from Tait.

with

similar mathematical

in the

" l a n g u a g e " that he

U n f o r t u n a t e l y for the historian, Gibbs, hav-

i n g w r i t t e n his booklet p r i m a r i l y as a teaching aid, m a d e no attempt whatever to indicate the origins of the various theorems. Chapter vectorial

I

concluded

equations,

Differential

and

with

and

Integral

troduced

the

operator

theorems,

and

gave

an

a

treatment

chapter II Calculus V,

was

of methods

of Vectors."

proved

for

the

Herein

Gibbs

in-

related

transformation

extended treatment of the

mathematics of

potential theory.14 T h e part of Gibbs' booklet printed in minated near the

solving

entitled "Concerning the

1881 ter-

e n d of chapter II; the remainder was printed in

1884. Chapters

III

vector functions

156

and

I V c e n t e r e d o n l i n e a r v e c t o r f u n c t i o n s , t h a t is,

of such a nature that a function of the s u m of any

D e v e l o p m e n t of the

t w o vectors treat

is

cepts

"dyad"

and

Gibbs

"dyadic."

A

dyad

for the

is

an

the

terms

expression

To

and con-

of the

form

In these two chapters Gibbs followed the general

treatment

of linear vector functions

w o r k e d out by H a m i l t o n and Tait. C. G.

introduced

a a n d X are vectors; a n d a d y a d i c refers to t h e s u m of a

n u m b e r of dyads. scheme

System of Vector Analysis

equal to the s u m of the functions of the vectors.

linear vector functions

ak, where

Modern

that had been

Because of this the quaternionist

Knott was able to argue:

b u t w h e n w e b e a r i n m i n d that Professor G i b b s d e l i b e r a t e l y set o u t t o construct a system free f r o m the fancied b l e m i s h of the q u a t e r n i o n a n d yet d i d n o t s c r u p l e t o i n t r o d u c e i n its s t e a d a n i n d e t e r m i n a t e p r o d u c t [the dyad] w h i c h w h e n we

find

is without any geometric significance whatever, and

o n c a r e f u l c o m p a r i s o n t h a t practically t h e d y a d i c s y s t e m i s

simply a modification of quaternion methods, in large measure a m e r e difference of notation, we can find no satisfactory reason for a m a n of Professor

Gibbs's

great

powers

leaving

quaternionic

paths

to

invent

n e w notations, n e w names for o l d things, a n d a n i n d e t e r m i n a t e p u r e l y symbolic product to take the place of the determinate real quaternion.15 Although typical

Knott

of a

tradition the

were

relation

notation of

strains,

which in

obtained

somewhat argument

by

far

Gibbs,

these

were

this

The

like

The

to

Tait,

statement, of the

same

sections

main

the

it

is

quaternion

conclusion as to

that

Gibbs The

physical

treatment

covered

quaternionists.17

dealt w i t h

in

proponents

is implicit in the tables of comparative

authors.16

later

functions

by the

booklet

too

that

to b r i n g against Gibbs.

vector

primarily

of

of the systems

given

linear

his

went

type

in

some

went

detail.

beyond

concluding

transcendental

applications

of rotations

functions

It

the

and was

results

brief chapter of

of dyadics, and to

this was a p p e n d e d a short note on b i v e c t o r analysis.18 That of

his

Gibbs'

book

was

contemporaries,

Wilson, vector

Gibbs' analysis

following

in

a

not and

student

and

"founded 1936

highly in

the

upon

article

and

contributions

what

I

believe

to

thirty years ago I would

have

be

Multiple

his

author the

very

for his

view

own

of

"The

Algebra": in

was

they

this

1 9

Page

err.

important

of'

Gibbs,

"If I

field

I

seems

E.

B.

book

on

wrote

the

of Gibbs

have not claimed have but followed

matured judgment.

Leigh

view of most

not

Contributions

should have claimed much

admitted."

the

did

an

lectures

entitled

to Vector Analysis much

original

this

Had

I

written

m o r e for h i m than he to

have come to the

same

conclusion, for he is q u o t e d as w r i t i n g that " t h e value of his

work

lies

more

in

the

formulation

of a convenient and significant

notation than in the development of a n e w mathematical method." Heaviside respects

referred

original

to

little

Gibbs' treatise

booklet on

as

"an

able

vector analysis.

and

. . . " (4;

in

20

some 138)

If

157

A

History

Gibbs he

of V e c t o r Analysis

cannot be

deserves

given

praise

deletions

and

system

order to

in

for

great credit the

alterations make

for originality

sensitivity should

be

a viable

of methods, yet

of his j u d g m e n t as made

system.

in

His

the

to what

quaternionic

symbolism,

n o w in

the m a i n accepted, also constitutes a significant contribution. Gibbs' from his

work

which

is,

historically

modern

vector

considered,

analysis

came

one

of the

two

sources

into

existence.

Though

b o o k l e t w a s d i f f i c u l t t o r e a d b e c a u s e o f its c o m p a c t n e s s ( H e a v i -

side

called

it

a

"condensed

basis

for later writers.

V.

Gibbs'

Other

Gibbs'

Elements

contribution Tait,

Work

and

to

of

Pertaining

Vector

vector

Heaviside

synopsis"),

to

Analysis

analysis.

it

Vector

was

vector

course

e v e r y year.21

ninety

lectures.22 also

the

history

Gibbs'

view

created his had much power and that

no

means

his

Grassmann,

At

least in

and

D u r i n g the 1880's G i b b s

in

later years

analysis.

Grassmann

importance

had

It

may

is

be

the

1890's

taught

the

the course consisted of

primarily

determined.

given

famous as

of Grassmann's

system,

much

Astronomy

from

these

Though

an

when

of value

for the

papers

was

"Address

of the

pure

entitled

before

American

work.

viewed

that

Gibbs

the

M o r e o v e r Gibbs felt

as

a

work

in

multiple

mathematician.23 " O n

Multiple

Section

Association

One

of

Algebra."

of Mathematics

for the A d v a n c e m e n t

of Science, by the Vice-President" and was published in Gibbs'

only

Hamilton,

system i n d e p e n d e n t l y of Grassmann's, the t w o systems

It

all

a

in c o m m o n , a n d it was natural for G i b b s to argue for the

most

was

periodically

of vector of

Gibbs'

and

form

p u b l i s h e d a n u m b e r of articles w h i c h have relevance

Grassmann's

algebra,

did

d i d almost no teaching of their ideas on vector

analysis

Gibbs to

and

Analysis

by

Although

analysis, this was not the case w i t h G i b b s . taught

could

1886.24 O f

w r i t i n g s this essay gives the best picture of his concep-

tion of the place of vector analysis w i t h i n the w i d e r fields of algebra and plea

mathematics for

greater

brief treatment

in

general.

interest of the

in

Broadly

multiple

history

considered, algebra.

of multiple

the

Gibbs

article

is

a

began with a

algebra and

considered

such m e n as Mobius, Grassmann, Hamilton, Saint-Venant, Cauchy, Cayley, Gibbs

Hankel,

Benjamin

Peirce,

and

Sylvester.

By this

analysis

hoped among other things

to illustrate t h e fact, w h i c h I t h i n k is a g e n e r a l one, that the m o d e r n geometry

is

not

only

tending

to

results

which

are

a p p r o p r i a t e l y ex-

pressed in m u l t i p l e algebra, but that it is actually striving to clothe itself

158

Development of the

in

forms

which

are

Modern

remarkably

System

similar

to

the

of Vector Analysis

notations

of multiple

a l g e b r a , o n l y less s i m p l e a n d g e n e r a l a n d far less a m e n a b l e t o a n a l y t i c a l treatment, a n d therefore, that a certain logical necessity calls for t h r o w i n g off t h e y o k e u n d e r w h i c h a n a l y t i c a l g e o m e t r y has s o l o n g l a b o r e d . (2; 103) The

strongest

Grassmann.

praise

and

Considering

fullest in

treatment

detail

a

was

number

bestowed

of the

upon

products

de-

fined by Grassmann, Gibbs discussed their significance and argued that

Grassmann's

of view.

Gibbs

system

provides

concluded the

a rich

paper with

and

encompassing point

a discussion

tions of m u l t i p l e algebra to physical science. First of all, things

geometry,

having

position

and the in

physics, crystallography, position

in

space

is

geometrical

space,

kinematics,

seem to

sciences

w h i c h treat of

mechanics,

demand a method

essentially a multiple

of applica-

T h u s he stated:

quantity

astronomy[,]

of this and

kind, for

can

only be

represented by simple quantities in an arbitrary and cumbersome manner.

For this

reason,

veloped than those

a n d because our spatial i n t u i t i o n s are m o r e de-

o f a n y o t h e r class o f m a t h e m a t i c a l r e l a t i o n s , t h e s e

subjects are especially a d a p t e d to i n t r o d u c e t h e s t u d e n t to t h e m e t h o d s o f m u l t i p l e a l g e b r a . (2; Proceeding then to

113) specifics, G i b b s

noted that through

Maxwell

electricity and magnetism had b e c o m e associated w i t h the methods of multiple (2;

114)

algebra but that

Having

algebra will Gibbs or

noted

astronomy for

had

so

geometrical

far r e m a i n e d aloof.

applications

multiple

generally take the f o r m of a point or a vector analysis,

stated

that

"in

crystallography,

wanted."

that

(2;

115)

mechanics,

kinematics,

Grassmann's

point

astronomy,

analysis

will

physics,

rarely

be

H o w e v e r arguments were given on behalf of the

usefulness of point analysis for investigations in pure mathematics. The

concluding

ticular have

paragraph

now become

and

the

concluding

sentence

in

par-

classic.

But I do not so m u c h desire to call your attention to the diversity of t h e a p p l i c a t i o n s o f m u l t i p l e a l g e b r a , a s t o t h e s i m p l i c i t y a n d u n i t y o f its principles. T h e student of m u l t i p l e algebra s u d d e n l y finds h i m s e l f freed from various doubtless,

restrictions

this

t o w h i c h h e has b e e n a c c u s t o m e d . T o m a n y ,

liberty seems

like an invitation to

license.

H e r e is

a

boundless f i e l d in w h i c h caprice m a y riot. It is not strange if s o m e look w i t h distrust for t h e result o f such a n e x p e r i m e n t . B u t t h e farther w e advance, the m o r e e v i d e n t it becomes that this too is a r e a l m subject to law. T h e more we study the subject, the more we find all that is most useful and beautiful attaching itself to a few central principles. We begin by studying

multiple

algebras;

A L G E B R A . (2;

117)

Gibbs'

may

paper

be

we

end,

taken

as

I

think,

by

symbolic

studying

of the

MULTIPLE

ever-increasing

interest expressed at that time by pure a n d applied mathematicians

159

A

History

of V e c t o r Analysis

in m u l t i p l e algebra. a

part,

in

one

was able, as

The

way

the

every and

interested two

that

in

or three

part,

of that trend.

Gibbs

offered by m u l t i p l e algebra. That Gibbs was multiple

years

is

shown

course

in

the

facts

that

algebra25 multiple 118)

papers

on

additional

by

multiple on

five

publish

a

(1;

of his

to

gave

writings

two

planned

algebra

he

algebra and had actually done research in this regard. In

he

influential

some later writers w e r e not, to v i e w vector analysis in

the broad perspective deeply

increasing interest in vector analysis was most

the

electromagnetic theory of light

G i b b s m a d e use of vectorial methods.

H o w e v e r he d i d not in these

p a p e r s stress t h e i m p o r t a n c e o f v e c t o r m e t h o d s ; had

read

Maxwell's

methods

used.

Treatise

Gibbs

did

would

publish

have

one

no

in fact a n y o n e w h o difficulty

with

highly mathematical

the

paper

a i m e d at demonstrating the usefulness of vector methods to a group of scientists —the of Elliptic

Orbits

in

in

1889

later

translated

aim

from

the

Klinkerfuss' the

astronomers.

Memoirs

into

This

Three of

German

Theoretische

is

his

Complete the and

" O n the

Determination

Observations,"

National

Academy

of

published

Sciences26

and

included in the third edition

Astronomie.

(1;

137)

Gibbs

of

discussed

of this rather long paper in a letter to H u g o Buchholz, w h o

edited the third edition of this The

book:

o b j e c t o f m y p a p e r was t o s h o w t o a s t r o n o m e r s , w h o are rather

c o n s e r v a t i v e . . . t h e a d v a n t a g e in t h e use of v e c t o r notations, w h i c h I had

learned

in

Physics

from

M a x w e l l . T h i s object c o u l d be best ob-

t a i n e d , not by s h o w i n g , as I m i g h t have d o n e , that m u c h in the classic methods

could

be

conveniently

and

perspicuously

represented

by

vector notations, but rather by s h o w i n g that these notations so simplify the subject, that it is easy to construct a m e t h o d for the c o m p l e t e solution o f t h e p r o b l e m . (2; In his

this

p a p e r G i b b s gave a sketch of the v e r y e l e m e n t a r y parts of

vector analysis

and

with

the

determination

Shortly Beebe

these

after

the

137)

a

Swift's

Gibbs

of

he

an

d e v e l o p e d a n i m p r o v e d t e c h n i q u e for

orbit

paper

published

method on

(little m o r e than the direct a n d skew products),

methods

m o r e w o r k on this (1;

149)

was

paper

in

Comet.27

from

three

published, which

they

It has b e e n

out more

reprints

tested

of w h i c h 276 copies

the

by

Gibbs

were distributed.

which

four polemical

bear on

papers

These will be discussed in

160

the

published detail

observations.

Phillips

and

W.

(successfully)

his

d i d not p u b l i s h his results. (199)

any other article, w i t h the exception of his

papers

W.

established that Gibbs did

problem, though he

sent

complete A.

of this

article than of

" O n Multiple Algebra,"

(1; 2 4 7 ) T h e o n l y r e m a i n i n g history in

of vector analysis

Nature

from

1891

to

in the following chapter.

are

1893.

D e v e l o p m e n t of the

Gibbs,

as

mentioned

before,

mann's works. This led h i m in son,

Modern

was

System of Vector Analysis

an

ardent

admirer

of Grass-

1887 to write a letter to Grassmann's

H e r m a n n Jr., c o n t a i n i n g t h e f o l l o w i n g : Permit me to take this o p p o r t u n i t y to express my hope that the p u b l i -

cation of another e d i t i o n of the A u s d e h n u n g s l e h r e of 1862, w i l l not be long delayed.

U s i n g that treatise in o n e of my classes t w o years ago, I

had great difficulty in c o l l e c t i n g 3 or 4 copies by b o r r o w i n g , & c . A n o t h e r m a t t e r has l o n g b e e n o n m y m i n d . the

preface

tides

in

to

wh

he

the

first

Ausdehnungslehre,

used the

principles

of the

Your honored Father, in mentions

a

work

Ausdehnungslehre

on

the

prior to

its p u b l i c a t i o n . I f t h e m a n u s c r i p t o f t h a t w o r k i s i n e x i s t e n c e , i t s e e m s t o m e t h a t its p u b l i c a t i o n w o u l d b e a n i m p o r t a n t c o n t r i b u t i o n t o t h e h i s t o r y o f t h e d e v e l o p m e n t o f m a t h e m a t i c a l ideas i n t h i s c e n t u r y . (1;

114)

H e r m a n n G r a s s m a n n Jr. r e p l i e d i n

1888 that the thesis on the tides

had

be

been

lisher. of

(1;

the

sider

but

115) T h i s

American

there

might

difficulty

in

finding

a pub-

led Gibbs to write to Thomas Craig, the editor

Journal

publishing

from this I

located

of

the

Mathematics,

work.

The

to

ask

following

if

he

would

interesting

con-

passage

is

letter:

believe that a K a m p f ums

different

methods

and

Dasein is just c o m m e n c i n g b e t w e e n the

notations

of

multiple

algebra,

especially

be-

t w e e n the ideas of G r a s s m a n n & of H a m i l t o n . T h e most i m p o r t a n t quest i o n i s o f course that o f m e r i t , b u t w i t h this q u e s t i o n s o f p r i o r i t y are inextricably entangled, & w i l l be certain to be the more discussed, since t h e r e are s o m a n y persons w h o can j u d g e o f p r i o r i t y t o o n e w h o c a n j u d g e of merit. T h o s e w h o are to discuss these subjects o u g h t to h a v e t h e d o c u m e n t s b e f o r e t h e m , & not be h a n d i c a p p e d as I was t w o years ago, for w a n t o f t h e m . (1;

115)

Gibbs'

that

prophecy,

different methods Craig

consented

when

it was

published. in

was to

a

"struggle

imminent,

publish

was

the

for

work,

decided that Grassmann's

In this

regard

Hermann

existence"

between

certainly fulfilled. this

became

unnecessary

collected works

G r a s s m a n n Jr.

the

Although

should be

wrote to Gibbs

1893: For the physical

completion

works

of a collected edition of the

mathematical

and

of my father we have to thank primarily the energetic

labors of Professor F. K l e i n of G o t t i n g e n , w h o t h r o u g h his w i d e connections

has

supplied

the

necessary

driving

force

for

the

undertaking.

H o w e v e r the credit for stirring up interest in the matter is exclusively y o u r s . (1; 1 1 5 - 1 1 6 ) As (one Gibbs

a

final

of the to

note

it may be

advocates

write

an

m e n t i o n e d that in

of vector

article

on

analysis vector

in

1899 M a x A b r a h a m

Germany)

analysis

for

the

requested Encyklopadie

161

A

History

der

of V e c t o r Analysis

mathematischen

Wissenschaften,

G i b b s declined to fulfill. By

the

time

important

only because great

that

Gibbs

contributions of his

ability

to

mathematics

died

to

request

in

vector

which

1903

he

analysis.

had

made

This

was

seize

upon

the

most

science

in

the

pendently number

in harmony with the

1880's.

and

of

numerous

possible

not

significant traditions

In the

1840's

simultaneously

similarities.

In

in

the

of the time, and thence to condemands

of the time.

It is striking that a historical p h e n o m e n o n of the again

evidently

creative p o w e r s i n m a t h e m a t i c s b u t also b y his

and physical

struct a system

a

(1; 2 3 1 )

1840's occurred

Hamilton and Grassmann inde-

created

the

vectorial

1880's

systems

another

with

scientist,

a

geo-

graphically an ocean away from Gibbs, was covering m u c h the same mathematical terrain as Gibbs. This was Oliver Heaviside.

VI.

Oliver

Heaviside

Oliver Heaviside was born with to

little

an

end

probably stone,

May

18,

1850, in

financial

security.28

Heaviside's

in

hence

lacked

1866;

through

the

Heaviside

he

influence

took

a job

of

as

a

1874

numerous to

live

research.

electrical papers.

with

Among

published

his the

parents books

(1873)

Treatise

education

university training.

his

uncle

telegraph

problems began to interest him, and in of his

L o n d o n to a family

formal

Sir

Charles

operator.

came

In

1868,

Wheat-

Electrical

1872 he published the first

He retired from this position in

and

pursue

he

studied

on

Electricity

independent study and was and

Maxwell's

recently

Magnetism.

T h e above narrative may be extended by the following statement included

by

Heaviside

began

the He

word

Heaviside by

in

telling

"Quaternion,"

a the

review story

Wilson's who,

Vector

Analysis.

enchanted by

set h i m s e l f t o r e a d i n g H a m i l t o n ' s books.

took those books h o m e and tried to

some

of

of a boy

find

out.

H e s u c c e e d e d after

trouble, b u t f o u n d some of the properties of vectors professedly

proved

were

vector be

wholly

incomprehensible.

negative? A n d H a m i l t o n was

How

could

the

square

of a

s o p o s i t i v e a b o u t it. A f t e r t h e

deepest research, the youth gave it up, a n d returned the books. He then d i e d , a n d was never seen again. He h a d b e g u n the study of Quaternions too soon.

My

own

introduction

ent manner.

Maxwell

in his treatise. learn

h o w to

to

quaternionics

e x h i b i t e d his

main

took

place

results

in quite

a differ-

in quaternionic form

I w e n t to Prof Tait's treatise to get i n f o r m a t i o n , a n d to

work

them.

I

had the same difficulties as the deceased

y o u t h , b u t b y s k i p p i n g t h e m , was a b l e t o see that q u a t e r n i o n i c s c o u l d

162

D e v e l o p m e n t of the

Modern

System of Vector Analysis

b e e m p l o y e d c o n s i s t e n t l y i n vectorial w o r k . B u t o n p r o c e e d i n g t o apply quaternionics

to

the

d e v e l o p m e n t of electrical

theory,

I f o u n d it

v e r y i n c o n v e n i e n t . Q u a t e r n i o n i c s w a s i n its v e c t o r i a l aspects a n t i p h y s i cal a n d u n n a t u r a l , a n d d i d not h a r m o n i s e w i t h c o m m o n scalar m a t h e matics.

So I

d r o p p e d out the quaternion altogether, and kept to pure

scalars a n d v e c t o r s , u s i n g a v e r y s i m p l e v e c t o r i a l a l g e b r a i n m y p a p e r s from

1883 o n w a r d .

Up to 1888 I i m a g i n e d that I was the o n l y one d o i n g v e c t o r i a l w o r k on positive physical principles; b u t t h e n I r e c e i v e d a c o p y of Prof. Gibbs's Vector Analysis

(unpublished,

1881-4). This

was

a sort of c o n d e n s e d

synopsis of a treatise. T h o u g h different in appearance, it was essentially the same vectorial algebra and analysis to w h i c h I h a d b e e n led.

as t i m e w e n t on, a n d after a p e r i o d d u r i n g w h i c h the diffusion of p u r e vectorial analysis

made m u c h progress, in spite of the disparagement

of the E d i n b u r g h school of scorners (one of w h o m said some of my w o r k was "a disgrace to the Royal Society," to my great delight), it was most gratifying to f i n d that Prof. T a i t softened in his harsh j u d g m e n t s , a n d came to recognise the

existence of rich

fields of pure vector analysis,

and to tolerate the workers therein.

I

appeased Tait considerably ( d u r i n g a little correspondence we had)

by

disclaiming

any

idea of discovering a

derived my system from

new

system.

I

professedly

Hamilton and Tait by elimination and simpli-

fication, b u t all the same c l a i m e d to have diffused a w o r k i n g k n o w l e d g e of vectors, and to have d e v i s e d a t h o r o u g h l y practical system.29 F r o m this q u o t a t i o n a n u m b e r of i m p o r t a n t facts b e c o m e e v i d e n t . First, in

it

is

striking that Heaviside,

quaternions

through

rally turned to Tait. his side

system

plification." complete

Second,

of vector

explicitly

Third,

that

it is

Tait's

from the

he

did

side

made

no

believing

mention that

he

ceived Gibbs' booklet in

VII.

Heaviside's

The first paper was and

his

1882-1883

Electric

even

natu-

developed

system.

elimination

Heavi-

and

sim-

1888 Heaviside w o r k e d in

Fourth, of

Heaviside's the

value

of

remark vector

Finally, it is significant that Heavi-

heard

indeed

there

of Grassmann

is

no

until

reason he

re-

1888.

Electrical in

interested then

had Gibbs,

"by

recognition

of Grassmann;

had

Heaviside

quaternion

this

Gibbs.30

of

eventual

analysis is certainly of interest.

for

as

clear that until

independence

concerning

Gibbs, became

Treatise.

Heaviside,

analysis

stated

like

Maxwell's

Papers

which

Heaviside

introduced vector methods

paper " T h e Relations between Magnetic Force

Current,"

published

in

the

Electrician.31

The

way

in

163

A

History

which

of V e c t o r Analysis

vectors

prising.

He

volved

in

were

its

the curl.

introduced

began

by

mathematical

line-integral

of

any

C

curve,

through

the

(5,1;

B

199)

treating a

round the

the

discussion

Cartesian

commented

sur-

which

in-

equivalent of

" W h e n one vector

closed

vector

C

curve

is

equals

called

extended

this

the

the

curl

discussion

integral

of the of the

scalar a n d vector fields. T h i s

was

of

vector curl

by

followed

of quaternions.

pointing

representing

for

somewhat

theory

related to another vector, C, so that the

Heaviside

potentials

After

reason

is

electrical

He then gave a quasi definition of curl: is

by

Heaviside

an

treatment

or directed quantity, B,

B."

by

discussing

out

the

the

great

numerous

that

quaternions

quaternions

had

Against the above

need

for

vectorial were

not been

a

vectorial

quantities

such

used.

a

method,

He

stated great advantages

in

method

for

physics,

he

but

for

suggested the

some

reason:

o f Q u a t e r n i o n s has t o b e

set t h e fact t h a t t h e o p e r a t i o n s m e t w i t h are m u c h m o r e d i f f i c u l t t h a n the

corresponding ones

in

the

ordinary system, so that the saving of

l a b o u r is, i n a great m e a s u r e , i m a g i n a r y . T h e r e i s m u c h m o r e t h i n k i n g t o b e d o n e , for t h e m i n d has t o d o w h a t i n scalar a l g e b r a i s d o n e a l m o s t mechanically. scalar

At

system,

bearing

the

there

same is

time,

great

when

working with

advantage

in m i n d the fundamental

to

be

vectors

found

in

by the

continually

ideas of the vector system.

Make a

c o m p r o m i s e ; l o o k b e h i n d t h e e a s i l y - m a n a g e d b u t c o m p l e x scalar equat i o n s , a n d see t h e s i n g l e v e c t o r o n e b e h i n d t h e m , e x p r e s s i n g t h e real t h i n g . (5,1; 2 0 7 ) He gave an illustration of the advantage of t h i n k i n g in terms of vectors,

and

Stokes.

stated

This

and

proved the

theorems

ascribed to

Gauss

and

was followed by a fuller discussion of vector potential

and curl, and finally by physical applications. It

is

noteworthy

middle and

d d — + j — +

i

that

Heaviside

of vector analysis.

J

ax

Such

d k —, n e v e r a p p e a r e d .

dy

vector

were

thus

in

a

paper

violated be

Maxwell

had arises

Heaviside

164

presentation

in

the

s y m b o l s a s i , j , /c, V ,

Heaviside

had

in

fact m a n -

its

as

The

to

usage,

absence to

Heaviside's done.

of

published

next in

his

proofs

and

or in a Cartesian form.

quaternion by

paper w i t h o u t having said a w o r d

Many

except

the

discussions

N o t h i n g in the

material

a quaternionist.

omitted

At this

stage

presentation was scarcely b e y o n d what interesting

but

probably

unresolvable

w h e t h e r at the t i m e of the w r i t i n g this paper

had b e g u n to develop his

Heaviside's was

qualitative

striking

writings

question

in this

multiplication.

would in his

his

dz

aged to introduce vectors about

began

fundamental

paper 1883

was in

the

nonquaternionic system.

entitled Electrician.

"Current (5,1;

Energy"

231-255)

In

and this

D e v e l o p m e n t of the

paper a discussion composed

of the

multiplied

by

Modern

System of Vector Analysis

of certain electrical questions

product of the

the

cosine

lengths

of the

of two

angle

l e d to a q u a n t i t y

vectorial

between

quantities

them.

Heaviside

stated:

Let

ds'

be

due

to

it

a stroke product

an

element of length

alone over

i s — AC'

the

of their

ds'

product

cos

of the

current

C\

of t w o vector quantities

magnitudes

The

( A C ) , or s i m p l y — A C '

(disregarding

portion

ds', ( i f w e

of M place

to signify that the

directions)

is

to

be

multi-

p l i e d b y t h e c o s i n e o f t h e a n g l e b e t w e e n t h e i r d i r e c t i o n s , i.e., A m u l tiplied by component of C'

along A, or C'

m u l t i p l i e d by component of

A along C ' ; as w i l l be d o n e hereafter).32 The

remainder

electrical scalar

of the

questions

product.

paper was

in

The

which

symbols

devoted to

Heaviside i, j,

k

further discussion

made

of

frequent use of the

never appeared

in

this

paper,

w h i c h as a w h o l e was ideally suited for an introduction and illustration

of the

scalar product.

Heaviside's was

entitled

side's and

next

discussion

its

paper,

"Some

published

in

the

Electrician

for

Electrostatic and Magnetic Relations."

began

1883, Heavi-

w i t h matters relating to electrostatic force

convergence:

Let

R

denote

the

electrostatic

force,

and p

the

volume

density of

e l e c t r i c i t y t h e n w e m a y say 47rp = — c o n v . R, or 47rp = d i v . R, w h e r e we use conv a n d d i v as abbreviations to be u n d e r s t o o d as follows: In terms of the components X, Y, Z of the force, we have

T h e expression on the right h a n d side of this equation ( w i t h the — sign prefixed) M a x w e l l called the " c o n v e r g e n c e " of the force; it is really the integral

amount of force,

but since + convergence

taken

algebraically,

indicates

entering t h e

unit volume;

negative electrification, we m a y as

w e l l use the t e r m " d i v e r g e n c e " for the same q u a n t i t y w i t h + sign prefixed

(5,1; 2 5 6 - 2 5 7 )

Heaviside here used the t e r m " d i v e r g e n c e " for the first t i m e in his writings.

Though

Clifford

had

introduced

this

term

five

years

earlier, it by no means necessarily follows that Heaviside h a d read Clifford, gence

physical name.

since

Heaviside

did not mention Clifford and since diver-

is an obvious n a m e for the negative of the convergence. T h e representation

Maxwell,

in terms of lines of force also suggests this

it may be

recalled,

introduced the

term "conver-

gence" to harmonize w i t h the first part of the quaternion equation

165

A

History

of V e c t o r Analysis

\dx

dy

dz)

\dy

dz)

J

Thus

one

significance

of Heaviside's

\dz

use

dz)

\dx

d y )

of "divergence,"

instead

of " c o n v e r g e n c e , " is that this t e r m harmonizes w i t h his v i e w of the scalar p r o d u c t as a positive quantity. the s y m b o l V to a scalar function;

H e a v i s i d e p r o c e e d e d to apply

he h a d used this s y m b o l before,

but sparingly and w i t h no introductory c o m m e n t on it as a vectorial operation. Later in this paper Heaviside introduced a section entitled " T h e Operator

V

and

divergence,

and

scalar

function

Its

Applications."

"space-variation" of position)

3 3

After

mention

(Maxwell's

Heaviside

wrote:

of

term

for

" N o w

it

the

curl,

VP for P a is

v e r y re-

m a r k a b l e t h a t (as w a s d i s c o v e r e d b y T a i t ) t h e s e t h r e e o p e r a t i o n s o f curl,

d i v e r g e n c e , a n d space-variation are really o n l y three different

forms

of the

nature

of

followed

some

usage

if we

operation,

function

(after

quaternion For,

same

the

the

under

effect varying

examination."

explanations)

by

according

(5,1;

the

268)

to

This

presentation

the was

of

the

of V.

denote

by

i, j,

k three

rectangular vectors

o f unit

lengths

parallel

to x, y, z, t h e n Xi w i l l d e n o t e a v e c t o r of l e n g t h X p a r a l l e l to

x,

similarly

and

for

Yj

and

Z/c,

consequently

we

may

write

R —

Xi + Yj + Zk w i t h t h e c o n v e n t i o n t h a t t h e s i g n of a d d i t i o n signifies c o m p o u n d i n g as velocities. N o w the full expression of V is

dx

J

dy

dz'

hence

E x p a n d this expression, w i t h the further conventions i 2 = j2 = k2 — —1, a n d i j — k,jk = i , ki = j , and we obtain, wo

(dX ^dY

. (dZ

dY\

i

- (dX

d Z \ ,

h

( d Y

dX\

i.e., VR = c o n v R + c u r l R. (5,1; 2 7 1 ) Referring

to

the

above

usage

of

quaternions,

Heaviside

wrote:

" H o w e v e r , this is m e r e l y parenthetical, and we shall have no more concern

w i t h quaternion expressions.

.

.

."

34

W h e n this paper was

republished, Heaviside added a footnote further to "emphasize the

166

Development of the

Modern

System of Vector Analysis

fact t h a t t h e u s e [ o f q u a t e r n i o n s ] w a s p a r e n t h e t i c a l . " (5,1; 2 7 1 - 2 7 2 ) Heaviside c o n c l u d e d this The

operator

V

section w i t h the

contains

the

statement:

whole theory of potentials, whether of

scalars o r vectors. B u t o w i n g t o t h e r e m a r k a b l y d i f f e r e n t n a t u r e o f t h e effects of V on d i f f e r e n t f u n c t i o n s , it c o n d u c e s to clearness to d i s t i n c t l y separate the space-variation of a scalar, w h i c h is easily grasped, f r o m that of a vector, a n d to instead speak of the curl or the d i v e r g e n c e of the latter, a s t h e case m a y b e , a n d a s w e h a v e d o n e h i t h e r t o . (5,1; 2 7 2 ) The paper was This is

concluded by a section

significant since the

study of

ideal area of physics by w h i c h to and

divergence

plained

the

of

a

vector

meaning

fluid

motion analogies.

motion

is

probably the

illustrate the m e a n i n g of the curl

function.

of curl,

on

fluid

In

summary

divergence,

and

Heaviside

ex-

space-variation;

or,

in m o d e r n f o r m , VP for P, a scalar f u n c t i o n of position, a n d V • R a n d V X R for R, a v e c t o r f u n c t i o n of p o s i t i o n .

Since Heaviside had not

yet defined the vector product, he could not write V V R

(V X R

in

modern

notation),

nor

(as h e d i d l a t e r )

d i d he w r i t e S VR

(V • R

in

m o d e r n notation), as he could have done, since he had defined the scalar product. In

He avoided these forms by writing div R and curl R.

Heaviside's

vectors,

but

tion of his and

its

no

later

new

papers

of

principles

1883

were

and

1884

use

introduced.35

was

In the

made

of

first sec-

first p a p e r o f 1885, e n t i t l e d " E l e c t r o m a g n e t i c I n d u c t i o n

Propagation"

and

published

in

the

Electrician,

Heaviside

i n t r o d u c e d the vector product.36 T h e vector p r o d u c t , like the scalar product,

was

questions, Hence

introduced

and the

he

wrote

definition

is

definition.

In the

C = kE

C

k

for

isotropic

C

=

VeE

equivalent same

This

medium,

the

course

of a

discussion

symbol

V was

certain

vectors,

for

to

the

modern

of physical

u s e d t o d e s i g n a t e it.

and

C,

to

€,

the

and

E.

His

quaternionic

section Heaviside gave the vector equation

(conduction

(a constant).

in

quaternion

current-density),

equation,

where

C

he

and

not isotropic, then k becomes

pointed E

are

E

(electric

out,

is

parallel;

force),

true

and

only for an

if the

medium

is

a " l i n e a r v e c t o r o p e r a t o r . " (5,1; 4 2 9 -

430) Soon after p u b l i s h i n g this was

published

completed), magnetic

over

Heaviside Wave

1-23)

Herein

vector

system.

Owing

to

published

Surface"

Heaviside He

and led up to the the

section of the paper (the

a three-year

began

gave the

in the

span,

and

(1885) the

even

whole paper

then

it

a paper " O n the Philosophical

was

Magazine.

first u n i f i e d p r e s e n t a t i o n

paper by

stating

a

not

Electro(5,11; of his

physical problem

statement: extraordinary

complexity

of

the

investigation

when

w r i t t e n out in Cartesian f o r m ( w h i c h I b e g a n d o i n g , b u t gave up aghast),

167

A

History

of V e c t o r Analysis

s o m e a b b r e v i a t e d m e t h o d o f e x p r e s s i o n b e c o m e s d e s i r a b l e . I m a y also add nearly the

indispensible,

meaning and mutual

owing

to the

connections

great difficulty in m a k i n g out

of very complex formulae. . . . I

therefore adopt, w i t h some simplification, the m e t h o d of vectors, w h i c h s e e m s t h e o n l y p r o p e r m e t h o d . (5,11; 3 ) He

pointed

method,

out

but

that

his

simplified.

method His

was

similar

to

of the

quaternion

criticisms

the

quaternion system

w e r e at that t i m e m i l d ; he p r e s e n t e d his system as a sort of half-way house

between

abbreviated

Cartesian

methods

and

quaternion

methods. The vector i, j,

presentation

of his

addition,

gave

and

rules

k.

for

he

After

defining

applying

Heaviside

system the the

these

defined

the

was

very

expression scalar

products operator

and

to

V

brief.

for

After

a vector

vector

equal

to

defining terms

products

combinations

as

in

of

and the

of i, j,

i ^ + j ^ +

and

k,

k ^ and

s h o w e d its effects w h e n it w a s a p p l i e d to a scalar as w e l l as w h e n applied to a vector by means

of the

scalar a n d vector products. He

then gave Stokes' and Gauss' theorems w i t h o u t proof. His presentation was c o n f i n e d to t w o pages of text, a n d the m e t h o d s presented correspond exactly to the m o d e r n methods. treatment

of

the

linear

vector

operator

(5,11; 4 - 5 ) H e a v i s i d e ' s

(presented

in

connection

w i t h /x, t h e m a g n e t i c p e r m e a b i l i t y ) w a s b a s e d o n T a i t ' s w o r k , a n d a proof from Tait was two

pages,

(5,11;

introduced

in the period from of Gibbs'

work.37

tinued to

use

nature

Heaviside

and

and

in

vector analysis

written d u r i n g this period he conthough

subjects

sympathy he

innovations

with,

particularly

less

frequently, because of

treated.

Gibbs'

Heaviside

work

praised

was

(except in

Gibbs'

im-

regard

development

of

w e n t on to p u b l i s h t w o other summaries of his vector

These of

the

the in

his

a paper of 1892 system:

in

the

points

Electrician,

Transactions

require

polemical

important of

first

the

papers

some

discussion

168

papers

physical

second

1892.

since

In

the

in journals,

the

ception

his

his

further

1888, at w h i c h t i m e he received a copy

linear vector operator.

method

for

In

and in

notation);

the

no

1885 to

vector formulae,

of

pressed by, to

T h i s presentation also e n c o m p a s s e d

6-7)

Heaviside

the

included.

a n d from it he proceeded to his physical investigation.

only

parts

Heaviside

to

will

be

them

will

of

1893.

Theory gave

beginning Royal

Society

brief mention

which

relevant

Electromagnetic

o f the

the

in

(with

the

discussed be

1891,

o f London

made

ex-

later) in the

following description

of

Development of the

Modern

System of Vector Analysis

I t rests e n t i r e l y u p o n a f e w d e f i n i t i o n s , a n d m a y b e r e g a r d e d ( f r o m o n e point of v i e w ) as a systematically abbreviated Cartesian m e t h o d of investigation,

and

customed to

be

understood

Cartesians,

Quaternions.

It

is

and

practically

without any

simply

the

study

elements

used by any

one

ac-

of the difficult science of

of Quaternions

without

the

quaternions, w i t h the notation simplified to the uttermost, and w i t h the very

inconvenient

minus

sign

before

scalar products

done

away

with.

(5,11; 5 2 9 ) This

quotation from Heaviside may be taken as a partial s u m m a r y

of the

nature

vector

analysis.

of his

origin;

indeed

mained

in

His

system

and

system

was

numerous

his

system.

of his

remnants

But

there

from

was

a

term

in a different sense) of his system;

cern

which

which not

altered,

in

general

side, ber

elements

unlike of

create

and

as

a

in

magnetic

Theory

VIII.

first long

articles

the

the

extensive

on

Electrician)

in

Electromagnetic

1912.

re-

(using

the

this was his ability to dis-

the

should

should be eliminated, unlike

Gibbs,

with

did

Heavi-

a large n u m -

ever-expanding

receive

credit

of vector

chapter

field

one

of

for

of

having

analysis;

(compiled from

volume

a

this

ap-

series

Heaviside's

of

Electro-

Theory

was in

published 1893, the

in

three

second in

1899,

Of these the first volume is the most imporanalysis,

treatment

Heaviside's

(1) its c o n t e n t s ,

Theory

appeared

of vector

published of

tradition

origin

1893.

history

discussion

of in

vector analysis, b u t

treatment

pages)

the first volume

and the third in

the

the lines of development b e g u n by Max-

Heaviside

(173

in

history

quaternionic

older

Heaviside,

in

Electromagnetic

Heaviside's

in

in

the

vector analysis

ideas

detailed

of

Heaviside's

volumes;

tant

did a s s o c i a t e

Furthermore the

retained.

important

in

second

analysis

new theorems

particularly

published peared

of quaternion which

Gibbs,

new

electricity, well.

and

position

unquestionably

of

chapter

(2) t h e

since

modern on

it

contained

vector

vector

the first

analysis.38

analysis

will

The

center

question of Heaviside's relation to Tait

a n d t o G i b b s , (3) t h e r e l a t i o n o f H e a v i s i d e ' s p r e s e n t a t i o n o f v e c t o r analysis

to

his

electrical

Heaviside's polemics.

writings,

Heaviside's

and

(4)

the

forcefulness

of

style can be placed s o m e w h e r e

b e t w e e n brilliant and obnoxious d e p e n d i n g on one's point of view. At

least

he

Heaviside of

the

was

as

richness

polemics

never

"the

liberal

of

dull;

indeed,

Walt Whitman Heaviside's

use w i l l

G.

M,

Minchin

of English

style

and

referred

Physics."39

the

to

Because

importance

of

his

be made of quotations.

169

A

History

In

of V e c t o r Analysis

the

preface

to

Heaviside

stated

electricity,

"for the

volume

that

he

one

would

use

of

his

vectors

his

treatment

(4;

iii)

Heaviside, like Maxwell before him,

v i e w e d this as a very important reason for using vectors. come

to

developers

be

of

sufficient reason that vectors are the m a i n sub-

ject of investigation."

has

Theory40

Electromagnetic in

known

as

of Maxwell's developed

one

of the

electrical theories;

that he

also

preface

Heaviside described his

Heaviside

great, perhaps the it is

less

greatest,

well

known

M a x w e l l ' s v i e w s on vector analysis.

In his

chapter on vector analysis:

R e g a r d e d as a t r e a t i s e on v e c t o r i a l a l g e b r a , t h i s c h a p t e r has m a n i f e s t s h o r t c o m i n g s . It is o n l y t h e first r u d i m e n t s of the subject. Nevertheless, as t h e r e a d e r m a y see f r o m t h e a p p l i c a t i o n s m a d e , it is f u l l y s u f f i c i e n t for o r d i n a r y use in the mathematical sciences w h e r e the Cartesian mathematics

is

usually employed, and we

n e e d n o t t r o u b l e a b o u t m o r e ad-

v a n c e d d e v e l o p m e n t s b e f o r e t h e e l e m e n t s are t a k e n u p . N o w , t h e r e are no treatises on vector algebra in existence yet, suitable for mathematical physics, and in harmony w i t h the Cartesian which

I

mathematics (a matter to

attach the greatest importance). I b e l i e v e , therefore, that this

c h a p t e r m a y be u s e f u l as a s t o p g a p . (4; iv) Chapter I side

of the book was entitled "Introduction." Herein Heavi-

summarized

stated: sent

"The

the

history

of electrical

crowning achievement was

Maxwell,

a

man

whose

fame,

great

p a r a t i v e l y s p e a k i n g , y e t t o c o m e . " (4; In chapter II magnetic peared.

set t h e

by

the

second

dealt

Since

stressed the

with

ordinary

quantities

compared

the

in

paragraph

now,

has,

vectors

com-

had

vector properties

which extended to

ap-

were

contains

approach

of expression,

i n p h y s i c a l science are no

to the

the

comprehension

173 pages.

statement that the majority

a vector algebra is

Cartesian

intuitive

his

geometry and

algebra

directly,

economy

ceptibility to

is

stage for chapter I I I , e n t i t l e d " T h e E l e m e n t s

polemics began with

entities

vectors.

He

and

Algebra and Analysis,"

Heaviside's

it

14)

Later in the chapter miscellaneous

introduced to

these

as

point

for the heaven-

H e a v i s i d e began the formal d e v e l o p m e n t of electro-

theory,

of Vectorial

of the

theory and at one reserved

methods needed.

for treating (4;

132-133)

vector approach

naturalness,

and

a n d t h e sus-

permitted by the

latter.

(4;

133-134) To this point Heaviside had written n o t h i n g that w o u l d displease even

the

changed

staunchest

rapidly

when

quaternion Heaviside

advocate;

this

changed

and

wrote:

B u t supposing, as is generally supposed, vector algebra is something "awfully

difficult,"

involving

metaphysical

considerations

of

an

ab-

struse

nature, o n l y to be t h o r o u g h l y u n d e r s t o o d by c o n s u m m a t e l y pro-

found

metaphysicomathematicians,

170

such

as

Prof.

Tait,

for

example.

D e v e l o p m e n t of the

Well,

M o d e r n System

of Vector Analysis

i f so, t h e r e w o u l d n o t b e t h e s l i g h t e s t c h a n c e f o r v e c t o r a l g e b r a

and analysis to ever b e c o m e generally useful.

T h e r e was a time, i n d e e d , w h e n I, a l t h o u g h r e c o g n i s i n g the appropriateness o f v e c t o r a n a l y s i s

in electromagnetic theory (and in mathematical

physics generally), d i d think

it was harder to understand and to w o r k

than the Cartesian analysis.

But that was before

I

had t h r o w n off the

quaternionic old-man-of-the-sea w h o fastened himself on my shoulders w h e n reading the only accessible treatise on the subject—Prof. Tait's Q u a t e r n i o n s . B u t I c a m e later to see that, so far as t h e v e c t o r analysis I r e q u i r e d was concerned, the q u a t e r n i o n was not o n l y not r e q u i r e d , b u t was a positive e v i l of no i n c o n s i d e r a b l e m a g n i t u d e ; . . . .

T h e r e is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). T h e vector analysis I use m a y be d e s c r i b e d e i t h e r as

a c o n v e n i e n t a n d systematic a b b r e v i a t i o n of Cartesian analysis;

else,

as

Quaternions

without

the

quaternions,

n o t a t i o n h a r m o n i s i n g w i t h Cartesians. (4; Heaviside

then

mentioned

that

and with

134-135)

Maxwell

had

pointed

out

"the

suitability of vectorial m e t h o d s to the treatment of his subject . (4;

or

a simplified

.

."

135) w i t h o u t a d v o c a t i n g t h e q u a t e r n i o n system. I t i s clear w h o m

Heaviside wished to v i e w as Heaviside

went on

to

his intellectual ancestor.

argue

that

if one

attempted to

determine

what processes occur most frequently in Cartesian mathematics, he would

find

analysis;

just

those

nowhere

processes

that

however w o u l d the

are

systematized

in

quaternion appear.

vector Humor,

history, and mathematical insight w e r e m i x e d in his next statement: " Q u a t e r n i o n " was, I think, defined by an A m e r i c a n schoolgirl to be " a n ancient religious ceremony." This

was,

however, a complete mistake.

T h e ancients — u n l i k e Prof. Tait— k n e w not, a n d d i d not w o r s h i p Quatern i o n s . T h e q u a t e r n i o n a n d its l a w s w e r e d i s c o v e r e d b y t h a t e x t r a o r d i n a r y g e n i u s Sir W.

H a m i l t o n . A q u a t e r n i o n is n e i t h e r a scalar, n o r a v e c t o r ,

b u t a sort o f c o m b i n a t i o n o f b o t h . I t has n o p h y s i c a l r e p r e s e n t a t i v e s , b u t is a h i g h l y a b s t r a c t m a t h e m a t i c a l c o n c e p t . (4; 136) Heaviside sidering spite that

turned to

obligations

of that lamentable I

should

followed

by

treatment

a

Tait's,

list

his

(or better, am

and

on)

personally

second chapter), (4;

137)

of complaints,

Heaviside

expressed

notation.

I

complain."

of Gibbs

137-138) and

then

the

his

Tait by under

it does

Naturally one

Gibbs'

agreement with

Gibbs'

writing: Prof.

seem

this

of w h i c h

"hermaphrodite

praised

to

statement

ideas,

of

(in

ungrateful

concerned

monster"

Elements

"Con-

Tait

system. Vector

but not

was

Tait's (4;

Analysis

with

his

" A s regards his n o t a t i o n , h o w e v e r , I do n o t l i k e it. M i n e is

but

simplified,

and

made

to

harmonise

with

Cartesians."

171

A

History

(4;

of V e c t o r Analysis

138)

In

Heaviside's

advocating

the

next

remark he c o m p a r e d his

"ex-quaternionic"

reasons

system to those of Gibbs:

for

"Prof.

G i b b s w o u l d , I t h i n k , go f u r t h e r , a n d m a i n t a i n that t h e anti- or exq u a t e r n i o n i c vectorial analysis was far superior to the quaternionic, which

is

uniquely

adapted

to

three-dimensions,

whilst

the other

a d m i t s o f a p p r o p r i a t e e x t e n s i o n t o m o r e g e n e r a l i z e d cases. I , h o w ever,

find

it sufficient to take my stand u p o n the superior simplicity

and practical utility of the ex-quaternionic system." Heaviside his

then

objections

letters

discussed

to

the

problems

practice

of representing

(Hamilton, Tait, and Gibbs)

Heaviside

stated:

" . . . I

In

rather

found

salvation .

in

.

.

by

leisurely

fashion

for

a

Greek

(Maxwell),

Clarendons,

for vectors

mathematics

text

and

(Phil.

Mag.,

141)

Heaviside

introduced vector addition, subtraction, and multiplication by

using, as tion

vectors

1886), a n d h a v e f o u n d i t t h o r o u g h l y s u i t a b l e . " (4;

a

then

139)

or by G o t h i c letters

i n t r o d u c e d t h e u s e of t h [ i s ] k i n d of t y p e August,

(4;

of notation. After explaining

he

of the

vector

had earlier, AB scalar

and

differentiation

and Tait,

and the

a n d VAB

vector in

a

for the

product.

manner

differences

(4;

different

were

symbolic representa-

143-163) from

discussed in

He

presented

that of H a m i l t o n great detail.

The

implication of this is indirect b u t important; it is that Heaviside was so conscious of the fact that the f o r m of his w o r k was taken f r o m the q u a t e r n i o n tradition that he felt obliged, in one sense, to justify any further deviations from this tradition. Heaviside's

lengthy

treatment

(4;

of V

163-178)

included a

large

n u m b e r of

e x a m p l e s a n d i l l u s t r a t i o n s o f its use, p a r t i c u l a r l y i n p o t e n t i a l t h e o r y and various side

part of electricity.

introduced

a

dot

or

(4;

period

178-256) as

a

In this section Heavi-

separator

(like

the

quater-

nionists) rather than as a s y m b o l for the dot product (like Gibbs).41 Heaviside entation he

then

turned

to

based

mainly

on

the

linear

Gibbs

vector

and

operator

Tait.42

Here

with as

a

pres-

elsewhere

supplied m a n y illustrations, chosen chiefly from electricity. The

chapter

was

vector

analysis

method

and

summary

by

He

began

the

a treatment physical of the

concluded a

(4;

a

polemical

attack

on

stating that he

of vector analysis

mathematics."

with last

"in

297)

the

form

Heaviside

summary

quaternion

of the

methods.

had limited himself to it assumes in ordinary then

stated

on

behalf

idea of a vector:

A n d it is a n o t e w o r t h y fact that ignorant m e n have l o n g b e e n in advance of the

learned about vectors.

Ignorant people, like

Faraday, naturally

t h i n k i n vectors. T h e y m a y k n o w n o t h i n g o f their formal manipulation, b u t i f t h e y t h i n k a b o u t v e c t o r s , t h e y t h i n k o f t h e m a s v e c t o r s , t h a t is, directed magnitudes.

172

No ignorant m a n could or w o u l d think about the

Development of the

Modern

System of Vector Analysis

t h r e e c o m p o n e n t s of a v e c t o r separately, a n d d i s c o n n e c t e d f r o m o n e another.

That is a device of learned mathematicians, to enable t h e m to

evade

vectors.

purposes, the

The

device

b u t for general

scalar

components

is

often

useful,

purposes

instead

especially for calculating

of reasoning the

of the

vector

manipulation

itself is

entirely

of

wrong.

(4; 298) This

statement

discussed in earlier

represented

the

Heaviside's

next chapter of this

position

on

an

argument

history and foreshadowed in

sections.

Heaviside then ternion

system;

analysts,

launched into his final remarks against the qua-

the

following

essentially

quotation

inadvertently,

illustrates h o w the vector

blackened

Hamilton's

reputa-

tion: Now,

a few

words

regarding

Quaternions.

It is

known

that Sir W.

Rowan H a m i l t o n discovered or invented a remarkable system of mathematics, a n d that since his the of

death the quaternionic mantle had adorned

s h o u l d e r s o f Prof. T a i t , w h o has Quaternions.

Prof.

Tait

in

repeatedly advocated the claims

particular

emphasises

its

great

power,

simplicity, and perfect naturalness, on the one hand; and on the other tells the physicist that it is exactly w h a t he wants for his physical purposes. I t i s also k n o w n that physicists, w i t h great o b s t i n a c y , h a v e b e e n careful and,

(generally

what is

speaking)

to have nothing to do w i t h Quaternions;

equally remarkable,

writers

who

take

up the subject of

Vectors are ( g e n e r a l l y speaking) possessed of t h e i d e a that Q u a t e r n i o n s is not exactly w h a t t h e y w a n t , a n d so t h e y go t i n k e r i n g at it, t r y i n g to m a k e it a little m o r e i n t e l l i g i b l e , v e r y m u c h to the disgust of Prof. Tait, who

would

preserve

the quaternionic stream pure and undefiled.

(4;

301) This

mildly

mild:

of treating Having tive

sarcastic

"Quaternions

statement

furnishes

quaternions.

Observe

granted that the

(full

science

and

that

(under

scalar

the

is

he

stated

done, the

subject is

it

of a

"should be much

by

another

simple

emphasis."

and

(4;

not

natural

so way

301)

preservation

almost

quaternions

multiplication)

"Vectors"

followed

of the associa-

an advantage, Heaviside argued that

quaternions) in

was

uniquely

quaternion

l a w for m u l t i p l i c a t i o n

quaternions

a

is

never

vector to

treated

simplified,

occur

unnatural be

physical

the

negative.

vectorially.

and we

in

for

are

square (4;

W h e n

303)

this

is

p e r m i t t e d t o ar-

r a n g e o u r n o t a t i o n t o suit p h y s i c a l r e q u i r e m e n t s . " (4; 3 0 3 ) H e c o n cluded by and by the to

the

a brief discussion final

cartesian

sentence: system

of

of Gibbs' and Macfarlane's

notations

" M y system, s o far f r o m b e i n g i n i m i c a l mathematics,

is

its

very

essence."43

T h r o u g h o u t the remainder of the book a n d in the t w o later v o l u m e s vector methods were used extensively and fruitfully to help Heaviside to m a n y of his

i m p o r t a n t results.

173

A H i s t o r y of V e c t o r Analysis

In conclusion the following statements seem justified: 1. Heaviside's

system

was

derived

from

the

q u a t e r n i o n i c sys-

tem; this statement is a m p l y p r o v e d by both his explicit statements a n d by the f o r m a n d c o n t e n t of his presentation of vector analysis. 2. Heaviside's tem

by

means

system

was

derived

from

the

q u a t e r n i o n i c sys-

of his o w n mathematical originality c o m b i n e d w i t h

u n f a i l i n g sensitivity for the needs of physical science. 3. Heaviside's whose

system

was

booklet he received in

created

independently

of

Gibbs,

1888, a n d of Grassmann, w h o m he

n e v e r m e n t i o n e d a n d p r o b a b l y n e v e r read.

Heaviside did however

m a k e use of some of the m o r e advanced sections of Gibbs' presentation.

IX.

The

As

Reception

previous

Given

to

considerations

Heaviside*s amply

Writings

indicate,

it is

not enough to

discuss a scientists' ideas; the r e c e p t i o n a c c o r d e d those ideas also demands attention. T h e singular aspect of Heaviside's writings on vector

analysis

books

on

will

be

is

that

electrical

they

were

theory.

evident in

the

embedded

Important

within

papers

and

consequences

of this

fact

f o l l o w i n g analysis

of the

reception

of his

writings. Heaviside's in

his

first

early

a change

statements

electrical

in the

paper

on

vector analysis

published

in

the

were

contained

Electrician.

In

1887

e d i t o r s h i p of this j o u r n a l b r o u g h t H e a v i s i d e a re-

quest to d i s c o n t i n u e his series of papers.

Heaviside wrote that the

n e w editor had i n f o r m e d h i m that "although he had made particular enquiries amongst students w h o w o u l d be likely to read my papers, to

find

i f a n y o n e d i d so, h e h a d b e e n u n a b l e t o d i s c o v e r a s i n g l e

one."44

This

was

not

stated, five journals x)

When

turned later,

the

to

an

Electrician

the

isolated p h e n o m e n o n , for as

refused

Philosophical

however,

Heaviside

h a d t u r n e d d o w n p a p e r s w r i t t e n b y h i m . (5,1;

the

to

Magazine

editor of the

print with

his

papers,

better

Electrician

Heaviside

results;

had

a

one

change

year

of m i n d

a n d f r o m that t i m e regularly p u b l i s h e d Heaviside's papers. F r o m t h e s e facts i t m a y b e i n f e r r e d that d u r i n g m o s t o f t h e 1880's Heaviside's partly

due

Perhaps 1887 the

writings to

the

Hertz

the

by

turning

point

p e r f o r m e d his

consequent increase

because

of confirmations

tide began to turn.

174

were

slowness

no

for Maxwell's classic

in

means

well

received.

This

was

of the acceptance of M a x w e l l ' s ideas.

interest in

of some

doctrines

experiments.

came when in

Partly because of

Maxwell's

of Heaviside's

ideas a n d partly own

results the

L o d g e at this t i m e n o m i n a t e d H e a v i s i d e for the

D e v e l o p m e n t of the

Royal the

Modern

System of Vector Analysis

Society, and he became a m e m b e r in

editors

collected trical

of the

and

Papers

Papers

Electrician

published,

appeared.

consisted

of

750

suggested and

hence

The

in

359

Heaviside's

1892

original

copies;

1891. Also in that year

that

his

two

printing

were

still

papers

be

v o l u m e Elec-

of

the

unsold

Electrical

five

years

later w h e n t h e y w e r e p u t on sale at r e d u c e d prices.45 George Electrical

Francis

Papers

FitzGerald was at

the

FitzGerald

in

the

published

August

11,

Professor of Natural

University

of D u b l i n

and

a

review

1893, and

of

the

Electrician.

Experimental Philosophy

a highly

was experienced in quaternion methods

of Heaviside's

issue

4 6

respected

scientist w h o

and w h o was, like Heavi-

side, an early a n d i m p o r t a n t advocate of M a x w e l l ' s electrical ideas. FitzGerald's

favorable

Heaviside's

Electrical

FitzGerald began papers,

which

he

practically,

and

stated

they

that

cause

of the

many

of

(6;

were

a

has

of both

of

1893.

as

"much

to

be

valuable and

"record

important

the

scientifically,

buried. . . ."

scientifically

listed, of

of nature,

too valuable

left

and

and

292)

He

practically

(6;

be-

discoveries

made

in

moreover

were

valuable

development,

of

them,

methods,

and

in an extraordinarily acute and brilliant

292-293)

Concerning side

described

FitzGerald

scientific views mind."

Theory

Electromagnetic

praising the publishers for b r i n g i n g out these

numerous

as

reception

and

historically

which

historically

by

review surely helped the Papers

the

Heaviside's faults

of

style

FitzGerald

extreme

wrote;

condensation

of

"Oliver Heavithought,

and

a

peculiar facility for c o i n i n g technical terms a n d expressions that are extremely puzzling to a reader of his Papers. seems

very

writers, Gerald's

little

hope

and write statement

like

will

So m u c h so that there

ever attain

the

clarity of some

a b o o k t h a t w i l l b e easy t o r e a d . " (6; 293) F i t z in

w o u l d be acceptable to Maxwell,

that he

regard

to

Heaviside's

relation

to

Maxwell

present historians.

every other pioneer

who

does

not

live

to

explore

the

country he opened out, had not had time to investigate the most direct m e a n s o f access t o t h e c o u n t r y , o r t h e m o s t s y s t e m a t i c w a y o f e x p l o r i n g it. T h i s has b e e n r e s e r v e d f o r O l i v e r H e a v i s i d e t o do. M a x w e l l ' s t r e a t i s e is

c u m b e r e d w i t h t h e debris o f h i s b r i l l i a n t l i n e s

of assault, of his en-

t r e n c h e d c a m p s , o f his battles. O l i v e r H e a v i s i d e has c l e a r e d t h o s e a w a y , has o p e n e d up a d i r e c t r o u t e , has m a d e a b r o a d r o a d , a n d has e x p l o r e d a c o n s i d e r a b l e tract o f c o u n t r y . (6; 2 9 4 ) Dublinite tem be

FitzGerald

of vector analysis put off by

such

discussion of the

was (6;

somewhat

critical of Heaviside's

sys-

295) but potential readers w o u l d hardly

criticism,

numerous

especially since it was followed by a discoveries

presented in these papers.

175

A H i s t o r y of V e c t o r Analysis

FitzGerald's

concluding

has

the

written,

whole

statement

was:

"Since

Oliver Heaviside

subject of electromagnetism

has

b e e n re-

m o d e l l e d by his work. No future introduction to the subject w i l l be at all final that does not attack t h e p r o b l e m f r o m at least a s o m e w h a t s i m i l a r s t a n d p o i n t t o t h e o n e h e p u t s f o r w a r d . " (6; 2 9 9 - 3 0 0 ) T h u s the

importance of Heaviside's writings for electromagnetic theory

and practice was being recognized in Britain in the German portance

scientists of

were

Maxwell's

simultaneously

work

August F o p p l , for example,

side

hence

turning

published an

to

the

im-

Heaviside.

exposition of Maxwell's

1894 w h i c h was w i d e l y read.47 I n this b o o k F o p p l i n c l u d e d

ideas in much

and

1890's.

discovering

from in

Heaviside

his

and wrote

preface:

"The

the following concerning Heavi-

works

of this

author

have

in

general

influenced my presentation more than those of any other physicist w i t h the side

to

obvious exception of M a x w e l l himself. be

the

theoretical quickly cessor

most

eminent

developments, just as

snatched from in

regard

it was

I consider Heavi-

Maxwell

in regard to

H e r t z — w h o alas w a s

so

u s — w h o w a s M a x w e l l ' s m o s t e m i n e n t suc-

to

developments."48

experimental

influenced

not

choice

mathematical

of

successor to

only Foppl's

presentation

methods

for

Heaviside

o f M a x w e l l b u t also his

the

book.

Foppl

adopted

Heaviside's vector analysis a n d d e v o t e d the first chapter of his book to an explanation of the n e w methods. This is very important since it was the first presentation of m o d e r n vectorial methods in a German

book.

cessor

to

Foppl's Maxwell

view in

of Heaviside as

regard

to

" t h e m o s t e m i n e n t suc-

theoretical

developments"

was

shared by Felix Klein and E. T. Whittaker, both of w h o m pointed out time

that the in

n o w classic M a x w e l l

modern

form

equations

Heaviside.49

in

appeared for the first

Heaviside's

influence

on

F o p p l was by no means an isolated instance; it w i l l be s h o w n later that

Heaviside's

electrical

theory

association was

the

widespread acceptance After papers;

1893

Heaviside

though

vectors

of

most

vector

analysis

with

Maxwellian

influential factor in leading to the

of vector analysis. continued were

further major contributions

to

publish important scientific

constantly

used

in

these papers, no

to vector analysis came from his pen.

I n c r e a s i n g p o v e r t y , deafness, a n d isolation m a r k e d his later years, and in

1925 the crisp w i t a n d scientific creativity of this self-edu-

cated genius w e r e stilled by death.

176

D e v e l o p m e n t of the M o d e r n System of Vector Analysis

X.

Conclusion This section w i l l be devoted to a comparison of the achievements

of Gibbs and Heaviside. T h e origins of their vectorial investigations will be considered first. From

an

interest

in electricity to the

reading of Maxwell, from

Maxwell to the

quaternionists, from the Quaternionists to m o d e r n

vector analysis,

this

circumstance

story

probably

less

has

been

than

an

twice-told accident.

in this

By the

chapter,

a

1880's alge-

braic sophistication was at a high level, and the needs of physical science w e r e sharply outlined. T h e times w e r e ready for t w o m e n of mathematical

ingenuity and a wealth

of physical experience. A n d

what Gibbs and Heaviside saw in the

1880's, others w e r e ready to

see i n s u b s e q u e n t decades.50 The two not wish

great vectorists

of the

1880's s h o u l d not b e a n d w o u l d

to be c o m p a r e d in regard to originality to the t w o great

vectorists of the

1840's. T h e task of t h e 1880's w a s o n e of s e l e c t i o n

and alteration, not chiefly of creation. Gibbs' originality should not go unnoticed however; some

as

H e a v i s i d e stressed, G i b b s h a d created

i m p o r t a n t n e w ideas.

Four vectorists and four m e n w h o l e d isolated lives; did extensive teaching of vectorial methods. however

was

the

greater;

he

associated

only Gibbs

Heaviside's influence

vector analysis

w i t h the

ever-expanding field of electricity. T h e m i d - c e n t u r y saw t w o great vectorists w o r k i n g separately; the 1890's saw t w o great vectorists

an ocean apart, u n i t e d in a c o m m o n

cause — i n w h i c h they w e r e successful.

177

Notes 1 2

Lynde

Phelps

Josiah

Willard

York,

W h e e l e r , Josiah Gibbs,

The

3

Peter Oliver

Heaviside,

Electromagnetic

5

Oliver

Heaviside,

Electrical

6

George

Francis

Scientific

7

Guthrie

Writings

Peter

Haven,

of J.

Willard

1962). Gibbs,

vol.

II

(New

Tait,

of

Elementary

Late

on

Theory,

Papers,

FitzGerald, the

Treatise

2

Quaternions,

vol.

vols.

Francis

2nd

(New

(Oxford,

1873).

1925).

1892).

Heaviside's

FitzGerald,

ed.

York,

(London,

" [ R e v i e w of]

George

I

Electrical

ed.

Joseph

P a p e r s " i n The Larmor

(Dub-

Guthrie

Gibbs:

are

by

former

American

Genius

Library from

Treatise

on

Quaternions,

3rd

ed.

(Cambridge,

I

in

who

1942])

was

was

also

a

written

scientist; for

the

a general

other

(Willard

audience

by

Rukeyser.

paper. (1;

have

Gibbs

of G i b b s City,

o f t h e b o o k c i t e d i n n o t e (1) a b o v e f o r a n e x t e n s i v e d i s c u s s i o n o f t h e

of this

given

student

[Garden

Muriel

See ch. VI

reception As

Elementary

t w o v a l u a b l e biographies of G i b b s ; o n e (cited in note [1] above) was

a

the poetess

10

Tait,

1890), vi.

There

written

9

(New

Papers

1902).

England, 8

Gibbs

1961).

4

lin,

Willard Scientific

been

107-109). able to

Through

the

examine the

c o n c e r n i n g vector analysis.

kindness

of t h e staff of Yale U n i v e r s i t y

majority of the preserved letters to a n d

Some of these were published in the book

c i t e d i n n o t e (1) a b o v e . A l l l e t t e r s q u o t e d f r o m t h i s b o o k h a v e b e e n c h e c k e d a g a i n s t the

originals.

11

From

12

(1;

lectures public

the

62). on

use

unpublished Gibbs'

It should be mechanics

correspondence.

n o t e d h o w e v e r that W h e e l e r also

at Johns

of vector methods."

Hopkins (1;

90)

University were

stated that Gibbs' noteworthy as

1880

"his first

Presumably the first of these contradictory

statements is the correct one. 13

T h i s was p r i v a t e l y p r i n t e d at N e w H a v e n in 1881 a n d 1884. G i b b s r e f e r r e d to it

a s " p r i n t e d , n o t p u b l i s h e d . " I t w a s r e p r i n t e d i n (2; this 14

17-90), a n d my references are to

printing. A d e t a i l e d analysis of the precise o r i g i n of each of the e l e m e n t s in this a n d later

chapters of Gibbs' booklet w o u l d be significant but b e y o n d the scope of the present u n d e r t a k i n g . To m a k e such an analysis a vast k n o w l e d g e of the history of calculus, electricity, mechanics, potential theory, algebra (in particular matrix theory), and of course

the

quaternionic

developments

w o u l d be needed. As almost no supporting

materials exist, this w o u l d be a task of great difficulty, p a r t i c u l a r l y since the size of Gibbs'

booklet

is

misleading:

when

E.

B.

Wilson

later rewrote

Gibbs' system in

textbook f o r m a n d retained essentially the same content, W i l s o n was r e q u i r e d to use 4 3 6 pages t o c o v e r w h a t G i b b s h a d c o v e r e d i n 73. N e v e r t h e l e s s t h e m a j o r source o f G i b b s ' w o r k is clear: he translated a n d a d d e d to the results a n d systematizations developed by 15

C.

Gibbs.

178

G. By

the

quaternionists.

Knott, E.

B.

"Review

Wilson"

in

of Vector Philosophical

Analysis

Founded

Magazine,

6th

upon Ser.,

the 4

Lectures (1902),

of J. 622.

Willard

D e v e l o p m e n t of the M o d e r n System of Vector Analysis 16

See,

for

Shaw gave used by

example,

(pp.

James

248-252)

Byrnie

a table

Hamilton, Tait,

Gibbs,

Shaw

in

his

of comparative

Vector Calculus

notations

a n d later students

by

(New

York,

listing the

1922).

notations

of linear vector functions.

This

table is helpful in "translating" results from one system to another. 17

Heaviside

and

others

expressed this

view.

For

Heaviside's

s t a t e m e n t see

(1;

117). 18

A bivector is a vector in w h i c h s o m e of the scalar coefficients assume i m a g i n a r y

values. H a m i l t o n a n d Tait b o t h w o r k e d w i t h biquaternions, w h i c h are directly analogous. F o r T a i t ' s t r e a t m e n t see (3; 6 4 - 6 6 ) . 1 9

Edwin

Bidwell

Multiple

Wilson,

Algebra"

in

vol. II ( N e w H a v e n ,

"The

A

Contributions

Commentary

on

of Gibbs

the

Scientific

to

Vector

Writings

Analysis

of J.

Willard

and Gibbsy

1936), 160. W i l s o n ' s p a p e r r a t h e r i n a d e q u a t e l y f u l f i l l e d its title;

it mainly stated w h a t G i b b s i n c l u d e d in his lectures w i t h little attention to specific questions

of originality.

made

statements

no

Despite Wilson's

thirty

years

i m p l i c a t i o n in the quotation that he had

previously

concerning

Gibbs'

statements may be f o u n d in writings by W i l s o n p u b l i s h e d in Wilson,

"On

Mathematiker "The

Products

Kongresses

Unification

Society,

16

in

im of

(1909-1910),

Additive

Heidelberg, Vectorial

418.

Fields"

1904

in

Notations"

These

Verhandlungen

(Leipzig,

statements

Bulletin

are

des

1905),

in

originality,

1905 a n d in III.

203,

of

the

such

1910. See lnternationalen

and

Wilson,

American

Mathematical

sufficiently indirect that W i l s o n

m a y be e x c u l p a t e d f r o m a charge of forgetfulness, b u t t h e y are sufficiently direct that the

historian

can conclude that Wilson's v i e w of Gibbs' originality was

different thirty years before his 20

As

quoted

raphy

Miss

Analysis.

Yale."

in

Rukeyser,

Rukeyser Yale

lists

record of such a w o r k . 21

tific

E.

B.

Wilson,

Monthly,

22

32

Wilson,

Gibbs, 2 6 8 .

This

one

by

University

work

Library

is

"Reminiscences

(1931),

has

but

Leigh.

kindly written to

of G i b b s

by

of Gibbs,"

in

the

bibliog-

M s — Gibbs'

Vector

me that they have

no

a Student and Colleague"

in

Scien-

159. T h i s article s u m m a r i z e s t h e c o n t e n t o f

1902-1903.

2 3

On this

see W i l s o n , " P r o d u c t s

in Additive

24

Josiah

Willard

Multiple

the

"Page,

219-220.

"Contributions

for

not footnoted,

Page:

I see h o w e v e r no r e a s o n to q u e s t i o n its a u t h e n t i c i t y .

the course as it was taught in

Association

not widely

1936 statement.

Gibbs,

"On

Advancement

of

Science,

Fields," 205-215.

Algebra"

35

(1886),

b e t o t h e p r i n t i n g o f t h e essay i n (2; 9 1 - 1 1 7 ) .

in

Proceedings

37-66.

My

of

the

American

references

will

M y e s t e e m for, a n d i n d e b t e d n e s s to,

this paper are b o t h large. 25

Wilson,

2 6

Josiah

"Contributions of Gibbs," Willard

Complete

Gibbs,

"On

Observations"

in

the

129.

Determination

Memoirs

of

the

of

National

Elliptic Academy

pt. I I , 7 9 - 1 0 4 . R e f e r e n c e s w i l l b e t o t h e r e p u b l i c a t i o n i n (2; 27

J.

H.

A.

Willard

"The

Orbit

Gould's

Bumstead,

Gibbs,

I

"Josiah (New

of Swift's

third

9

Willard

York,

Comet,

Astronomical Journal,

Klinkerfuss' 28

vol.

Gibbs"

1961),

1880 V,

(1889),

xxi.

The

in

Determined

114-117,

by

The

It

Three

vol.

Scientific

Beebe

Gibbs'

121-124.

from

IV,

118-148).

Gibbs,

p a p e r by

Orbits

o f Sciences,

was

and

Papers

of

Phillips

is

Vector

Method"

in

also

included

in

edition.

Biographical details on H e a v i s i d e ' s life are g i v e n in m a n y sources, i n c l u d i n g the

following: trical

(1)

Rollo

Communication

ume ( L o n d o n ,

Appleyard,

(London,

1950).

"Oliver

1930),

See especially G.

Heaviside"

211-260;

(2)

in

Appleyard,

The

Heaviside

(1930),

"Oliver

Heaviside"

o f Elec-

Centenary

Vol-

F. C. Searle, " O l i v e r H e a v i s i d e : A Personal

Sketch," 93-96, a n d Sir George Lee, " O l i v e r H e a v i s i d e - t h e M a n , " Whittaker,

Pioneers

in

Calcutta

Mathematical

1 0 - 1 7 ; (3) E . T .

Society

Bulletin,

20

199-220.

179

A H i s t o r y of V e c t o r Analysis 29

Oliver

Heaviside,

Electromagnetic

Theory,

vol.

Ill

(New

York,

1925),

135,

136,

137. 30

A m o r e exact date m a y be given: June, 1888.

31

Heaviside's papers on electricity up to 1892 w e r e collected a n d p u b l i s h e d as the

S e e (5,11; 5 2 9 ) .

b o o k c i t e d i n n o t e (5) a b o v e . T h e p a p e r r e f e r r e d t o a b o v e m a y b e f o u n d i n (5,1; 1 9 5 231).

Despite

Heaviside's

statement

in

the

preface

of his

Electrical

Papers

(5,1;

xi)

to the effect that these r e p r i n t e d papers are essentially identical to t h e original p u b l i s h e d papers, a careful check has s h o w n that this m u s t not be taken as c o m p l e t e l y true.

C o m p a r e for e x a m p l e

trician,, 1 0 ( J a n . 2 0 ,

his

definition of the

s c a l a r p r o d u c t a s g i v e n i n t h e Elec-

1 8 8 3 ) , 2 2 4 , w i t h t h e d e f i n i t i o n g i v e n i n (5,1; 2 3 6 ) . I h a v e c h e c k e d

all quotations against the originals a n d n o t e d any important deviations. 32

Heaviside,

noted

that

"Current

the

Energy"

statement

given

in

Electrician,

above

10

differs

(Jan.

20,1883), 224.

substantially from

the

It s h o u l d be statement as

g i v e n i n (5,1; 2 3 6 ) . 33

This title does

not appear in the original paper.

34

(5,1;

The

271-272).

word "expressions"

effect that one more i t e m ( w h i c h 35

was

followed by a qualification to the

is not of importance) w o u l d be discussed.

s h o u l d p e r h a p s b e q u a l i f i e d b y t h e s t a t e m e n t t h a t V 2 m a d e its first a p p e a r -

This

a n c e i n a n 1 8 8 4 p a p e r . S e e (5,1; 3 3 8 ) . N o c o m m e n t w a s g i v e n b y H e a v i s i d e a b o u t its

/ #

^2

m e a n i n g . H e s e e m s t o u s e i t a s -I- ^ ^ 4 -

^2 \ 4- -j-^J rather than, as M a x w e l l a n d Tait

had done, as the negative of the above. 36

This

p a p e r i s i n (5,1; 4 2 9 - 5 6 0 ) a n d (5,11; 3 9 - 1 5 5 ) . T h e p a p e r w a s p u b l i s h e d i n

sections,

the

last o f w h i c h

appeared in

1887.

T h e vector product is

exception

this

is that in

1886 he

cating vectors by boldface type.

S e e (5,11;

172).

introduced in

(5,1; 4 3 1 ) . 37

38

The

only

to

introduced the practice of indi-

Since.Heaviside's treatment of vector analysis in this book was highly polemical

(whereas G i b b s ' earlier presentation was not), there w o u l d be g o o d reasons for postp o n i n g t h e d i s c u s s i o n o f i t t o t h e n e x t c h a p t e r . T h e r e are b e t t e r reasons for its i n c l u s i o n at this p o i n t ; h o w e v e r , that it has r e l e v a n c i e s for t h e f o l l o w i n g chapter s h o u l d now be 39

made

George

side" 40

the

in

clear. M.

Minchin,

Philosophical

Oliver

"[Review

Magazine,

Heaviside,

5th

of]

Ser.,

Electromagnetic

38

Theory,

Electromagnetic (1894), vol.

I

Theory.

By

Oliver

Heavi-

146. (London,

1893).

I

have

used

1 9 2 5 r e p r i n t c i t e d i n n o t e (4) a b o v e , a n d a l l r e f e r e n c e s are g i v e n i n t e r m s o f it.

41

(4;

42

For

179).

F o r a n e x a m p l e o f T a i t ' s u s e o f t h e d o t see (3; 27).

Heaviside's

statements by

treatment

of the

linear vector

operator see

(4;

256-297).

For

H e a v i s i d e i n d i c a t i n g his u s e o f G i b b s ' t r e a t m e n t see f o r e x a m p l e (4;

263, 295, 300). 43

(4;

305).

By

"cartesian

system"

m e t h o d for treating p r o b l e m s 44

20

E.

T.

(1930),

See

Francis 47

August

Heaviside"

example ed.

Foppl,

papers

Joseph Einfiihrung

8

and

Larmor in

22

in

(Dublin,

die

48

Ibid., Felix

in

Calcutta

The

nonvectorial

Scientific

Mathematical

Society

Bulletin,

Writings

o f the

Late

George

1902).

Maxwellsche

Foppl's book went through four editions

49

180

"Oliver

traditional

of Cartesian co-ordinates.

211. for

FitzGerald,

1894).

meant the

209.

"Ibid., 46

Whittaker,

Heaviside

in space by means

Theorie

der

Elektricitat

(Leipzig,

i n e i g h t e e n years.

VII. Klein,

Vorlesungen

uber

die

Entwicklung

der

Mathematik

im

19

Jahr-

D e v e l o p m e n t of the M o d e r n System of Vector Analysis

hundert, p t . taker, 50

II, ed.

Mention

and

Systemes

used the Traite

de

for the This

made

Memoire

de

Droites

modern Cinematique

Mecanique

C o u r a n t a n d St.

( N e w York,

1956), 47, a n d W h i t -

Pure

VUniversite

vector

product.

of a work by Alphonse D e m o u l i n published in

sur

VApplication

(Bruxelles,

scalar

de

information

Cohn-Vossen

202-203.

should be

entitled

Divers

de

R.

"Heaviside,"

and

vector

(Paris, de

Paris),

Gand

for

(1st

ed.,

Methode

and

the

118

and

3rd

Vectorielle

pp.

cited

scalar

1879;

knew both

has b e e n d e r i v e d f r o m

section, essentially a preface.

vi

products

1862)

Demoulin

d'une

In as

this his

product ed.,

Grassmann's

a

and

Demoulin's book, pp.

de

work the author

sources

a n d J.

Paris

1894

VEtude

and

M.

Gand,

Hamilton's V - V I

Resal,

Massau,

of an

Cours 1891)

works. untitled

No further m e n t i o n of this w o r k w i l l be m a d e because

o f its b r e v i t y , b e c a u s e n o f u r t h e r p u b l i c a t i o n s s e e m t o h a v e c o m e a s a r e s u l t o f it, a n d b e c a u s e o f t h e c o m p l e x i t y i n v o l v e d i n m a k i n g a n y b r i e f i n t e r p r e t a t i o n s a s t o its ancestry.

I have seen only the

1891

edition of Massau's book, w h i c h must have had a

v e r y l i m i t e d c i r c u l a t i o n . T h i s i s i n d i c a t e d b y its t i t l e a n d b y t h e fact t h a t t h e text w a s published in handwritten form. and Hamilton,

Massau's

1891

edition mentioned both Grassmann

and, like Demoulin's book, had little or no influence.

Demoulin's books

are

significant as

reflective

of the

interests

Massau's and

of that t i m e a n d are

notable in respect to the small interest in any form of vectorial analysis in France at that

time.

181

CHAPTER

A

Struggle for in

I.

SIX

Existence the

1890's

Introduction T h e f o l l o w i n g prediction was m a d e by G i b b s in an 1888 letter to

Thomas Craig:

" I b e l i e v e that a K a m p f urns D a s e i n [struggle for ex-

istence] is just c o m m e n c i n g b e t w e e n the different m e t h o d s a n d notations mann

of m u l t i p l e algebra, especially b e t w e e n the ideas of Grass& of H a m i l t o n . "

37

Gibbs' prediction was fulfilled, for in the

early 1890's a w i d e s p r e a d a n d vigorous debate on vectorial methods took place. six

N o less t h a n e i g h t j o u r n a l s , t w e l v e scientists, a n d thirty-

publications

were

involved.

The

spirit of the

debate

characterized by L o r d Rayleigh's paraphrase of Tertullian: h o w these vectorists This

aptly

one another."38

love

debate played an

is

"Behold

i m p o r t a n t role i n t h e history o f vector an-

alysis. T h a t interest in the questions d e b a t e d was at a h i g h level is i n d i c a t e d by the fact that eight l e a d i n g scientific journals p e r m i t t e d publication general

of these

and

the

creased during the forceful,

articles.

That interest in vectorial methods in

Gibbs-Heaviside

timely,

system

in

particular greatly in-

1890's m a y in large measure be attributed to the

and

stimulating

presentations

in

these

articles.

F r o m the arguments advanced in the debate m u c h can be learned about h o w these tems

early vectorists

v i e w e d t h e i r s y s t e m s a n d t h e sys-

of their opponents.

A l l publications f r o m the p e r i o d 1890 to 1894 w h i c h contain arguments relevant to vectorial methods w i l l be discussed in this chapter.39 T h i s

time limitation is by no means arbitrary; the n u m b e r of

polemical

articles

published in

the

years

immediately before

and

after this p e r i o d is very small. M o r e o v e r the discussion is unified by the fact that nearly e v e r y p u b l i c a t i o n f r o m this g r o u p refers to one or more

other publications

from

the

group.

Though

slightly more

than half of the writings appeared in the important British scientific

182

A Struggle for E x i s t e n c e in the 1890's

weekly

Nature, t h e

d i s c u s s i o n c e r t a i n l y w e n t b e y o n d its

pages

and

t o u c h e d o n i m p o r t a n t q u e s t i o n s n o t d e a l t w i t h i n its pages.40

II.

The

"Struggle

for

Existence"

T h e b e g i n n i n g of this

"struggle for existence"

may be dated as

1890, a year i n w h i c h Peter G u t h r i e T a i t w r o t e t w o s t r o n g l y w o r d e d pleas

for recognition of the fitness of the

occasion Much

was

the

publication

Enlarged"

January,

1890,

of

his

issue

of

in

Elementary the

quaternion system.

that year of the Treatise

Philosophical

on

Magazine

"Third

The

Edition,

Quaternions. carried

an

The arti-

cle entitled " O n the Importance of Quaternions in Physics," w h i c h was an abstract of Tait's address to the Physical Society of the U n i v e r s i t y o f E d i n b u r g h , g i v e n N o v e m b e r 14,

1889.

Tait began indirectly by posing the question " W h e t h e r is experiment or mathematics the more ics?" "to

(1;

84)

their

must,

combined

some

important to the progress

He c o n c l u d e d that the

day,

or

give

alternate u p its

of phys-

question was absurd because

assaults

secrets."

(1;

everything

penetrable

84) T h e i r " i n s e p a r a b l e

c o n n e x i o n , " he stated, leads to the i m p o r t a n t c o n c l u s i o n "that e v e r y f o r m u l a w e e m p l o y s h o u l d a s o p e n l y a s p o s s i b l e p r o c l a i m its p h y s i cal m e a n i n g . " (1; 85) F o r t h i s r e a s o n t h e p r i m a r y c h a r a c t e r i s t i c t o b e sought in selecting mathematical methods for physical p r o b l e m s is neither

compactness

nor

elegance,

but

"expressiveness."

In

this

way Tait led the discussion to quaternions, w h i c h he described as "transcendently

expressive,"

compact,

and

elegant.

Tait

here

as

elsewhere invoked the name and reputation of Hamilton on behalf of the since

quaternion Graves'

cause.

immense

This Life

was

not w i t h o u t effect,

o f Hamilton

had just

been

especially published,

and this two-thousand-page tribute to the originator of quaternions must

have

brought

the

memory

of Hamilton before the

minds

of

many m e n of the time. Tait proceeded to

an

attack

on

the

artificiality of Cartesian co-

ordinates, " o n e of the w h o l l y avoidable encumbrances w h i c h n o w r e t a r d t h e p r o g r e s s o f m a t h e m a t i c a l p h y s i c s . " (1; 86) T h i s e n c u m brance, Tait argued, adopted,

and he

could be avoided if quaternions were widely

c i t e d as one reason for this that q u a t e r n i o n s are

" u n i q u e l y a d a p t e d to E u c l i d i a n space. . . ." (1; 87) T a i t t h e n g a v e a n u m b e r of illustrations of the simplicity and brevity of the quaternion sions.

expressions

as

compared with

equivalent Cartesian

expres-

He stressed these advantages above all in relation to the dif-

ferential calculus of quaternions and particularly in applications of V. At one point Tait made a statement that he w o u l d often repeat in

183

A

History

of V e c t o r Analysis

similar forms.

Concerning V he wrote:

" N o doubt, it was originally

d d d d e f i n e d i n t h e c u m b r o u s a n d u n n a t u r a l f o r m i - j - + j ~t~ + k - 7 - . B u t dx

dy

dz

t h a t w a s i n t h e v e r y i n f a n c y o f t h e n e w c a l c u l u s , b e f o r e its i n v e n t o r h a d s u c c e e d e d i n c o m p l e t e l y r e m o v i n g f r o m its f o r m u l a e t h e fragments

of the Cartesian

about

them."

(1;

91)

shell, w h i c h w e r e still persistently clinging Essentially

Tait

was

stating that

Hamilton's

frequent recourse to Cartesian equivalents was a blot on the system a n d that this tages.

had

Then

simplicity The

point

Tait

would

have

methods teresting

a

hindered number

recognition

o f its

of illustrations

many advan-

of the

power and

of V.

final

Horace;

greatly

followed

in

the

paper

wrote:

"The

it,

concealment)

the

w e r e artificial,

centered

highest

art

is

around

the

o f artifice."

absence

(1;

86)

a paraphrase (not,

as

Thus

of

Horace

Cartesian

quaternions natural. This led Tait to an in-

characterization

of nineteenth-century

physical

science:

T h e magnificent artificers of the earlier part of the century were, in m a n y cases, b l i n d e d b y t h e e x q u i s i t e p r o d u c t s o f t h e i r o w n art. T o F o u r i e r , a n d more

e s p e c i a l l y t o Poinsot, w e are i n d e b t e d for the practical t e a c h i n g

that a m a t h e m a t i c a l f o r m u l a , h o w e v e r b r i e f a n d elegant, is m e r e l y a step towards k n o w l e d g e , a n d an all b u t useless one, u n t i l we can t h o r o u g h l y r e a d its m e a n i n g . I t m a y i n fact b e s a i d w i t h t r u t h t h a t w e are a l r e a d y i n possession of mathematical methods, of the artificial kind, fully sufficient for all

our present,

and

at

least o u r i m m e d i a t e l y prospective, wants.

W h a t is r e q u i r e d for physics is that we s h o u l d be e n a b l e d at every step t o feel i n t u i t i v e l y w h a t w e are doing. T i l l w e have b a n i s h e d artifice w e are n o t e n t i t l e d t o h o p e f o r f u l l success i n s u c h a n u n d e r t a k i n g . (1; 9 6 - 9 7 ) In

conclusion the

it m a y be noted that Tait's arguments w e r e mainly

directed

to

survival

methods

a n d that h e n c e t h e m a j o r i t y o f the a r g u m e n t s c o u l d b e ap-

plied equally well

The

second

for the on

Quaternions.

on behalf of the

writing

debate

struggle between

was

Tait's

the

book

was

2

and Cartesian

Gibbs-Heaviside

of Tait from the preface

vectorial

to

year

the

directed

system.

1890 that was

third

edition

mainly

seminal

of his

toward

the

Treatise physi-

cal a p p l i c a t i o n s of q u a t e r n i o n s ; to r e c t i f y this " b i a s , " as he c a l l e d it, he a d d e d for this pect

of

edition

Quaternions"

" a n entire Chapter, on the Analytical As-

through

"the

unsolicited

kindness

of Prof.

C a y l e y . " (2; v ) S i n c e C a y l e y f o u r years later m a d e a n attack o n quaternions, Tait m a y have c o m e to regret his decision to include this chapter. T a i t stated that " l i t t l e progress has

recently been made with the

d e v e l o p m e n t o f Q u a t e r n i o n s . " (2; v i ) I n a p a r a g r a p h r i c h w i t h literary quotations he a s c r i b e d this partly to excessive efforts to m o d i f y

184

A Struggle for E x i s t e n c e in t h e

notations,

especially in

1890's

France, and concluded with the statement:

" E v e n Prof. W i l l a r d G i b b s m u s t be r a n k e d as o n e of the retarders of Quaternion

progress,

in

a sort of h e r m a p h r o d i t e

virtue

Hamilton and of Grassmann." argued that Grassmann Hamilton rect,

(correct),

but

lished

misleading),

7

4 1

pamphlet

compounded

and

Vector Analysis; notations

of

T u r n i n g to priority questions, Tait

certainly

that

internal

on

of the

did

not have

quaternions

that H a m i l t o n p u b l i s h e d his

Grassmann's

refuted by Gibbs

of his

monster,

in

and

the

1830's

external

before

s y s t e m first (cor-

Hamilton

products

had

pub-

(incorrect and

).

Tait proceeded to argue for the superiority of quaternion methods over Cartesian methods and to suggest that in quaternion w o r k even to have "recourse to quasi-Cartesian processes is fatal to progress." (2;

vii)

He c o n c l u d e d by a quotation from a letter he h a d received

long ago f r o m H a m i l t o n : factory?

Don't be

y o u feel, thanked

" C o u l d a n y t h i n g b e s i m p l e r o r m o r e satis-

as

w e l l as

hereafter?

t h i n k , that we are

Never

mind

when."

o n a right track,

and

shall

(2;

was

of course to d r a w u p o n Hamilton's great reputation to advance

viii)

This

his cause. As t i m e passed, h o w e v e r , statements s u c h as this l e d to a diminishing In

of Hamilton's

the April

2,

1891,

stature.

i s s u e o f Nature t h e r e a p p e a r e d a n a r t i c l e 3 b y

Gibbs written in response to Tait's references to Gibbs as one of the "retarders

of quaternion

progress"

and

to the

Gibbs'

system as a

"hermaphrodite monster." Gibbs began by quoting Tait's statement a n d f o l l o w e d this by a paragraph w h i c h w e l l illustrates G i b b s ' tactfulness and his

staid but forceful

style.

T h e merits or demerits of a p a m p h l e t p r i n t e d for private distribution a g o o d m a n y years ago do n o t c o n s t i t u t e a s u b j e c t of a n y great i m p o r t a n c e , b u t the assumptions i m p l i e d i n the sentence q u o t e d are suggestive o f certain reflections a n d i n q u i r i e s w h i c h are o f b r o a d e r interest, a n d s e e m not u n t i m e l y at a period w h e n the methods a n d results of the various forms of m u l t i p l e algebra are attracting so m u c h attention. It seems to be assumed that a departure f r o m quaternionic usage in the treatment of vectors is an enormity. If this assumption is true, it is an i m p o r t a n t truth; if not, it w o u l d be unfortunate if it s h o u l d r e m a i n unchallenged, especially w h e n s u p p o r t e d by so h i g h an authority. T h e criticism relates particularly to notations, b u t I b e l i e v e that there is a d e e p e r q u e s t i o n of notions

underlying

that of notations.

Indeed, if my offence had been

s o l e l y i n t h e m a t t e r o f n o t a t i o n , i t w o u l d h a v e b e e n less a c c u r a t e t o d e scribe my p r o d u c t i o n as a m o n s t r o s i t y , t h a n to c h a r a c t e r i z e its dress as u n c o u t h . (3; 511) The first "notions" vector since

products. they

He

represent

with

which

argued the

that

most

Gibbs these

dealt were the products

important

are

relations

in

scalar a n d

fundamental physics

and

185

A H i s t o r y of V e c t o r Analysis

g e o m e t r y , w h e r e a s f e w , if any, correlates are f o u n d for the quatern i o n p r o d u c t or for the q u a t e r n i o n itself. He suggested that this conclusion the

was

evident even from an examination of the practices of

quaternionists

in

dealing w i t h spatial relations. A n d he a d d e d

that vector analysis, u n l i k e quaternion analysis, c o u l d be extended to apply to space of four or m o r e dimensions. i n t r o d u c e d his

(3;

511-512) Gibbs

next p o i n t by a d m i t t i n g that the " q u a t e r n i o n affords

a c o n v e n i e n t n o t a t i o n f o r r o t a t i o n s " (3; 512), b u t h e a d d e d t h a t t h e representation Gibbsian

of

system

rotations

by

the

" s e e m s to leave

512) After a comparison wrote in summary:

of the

linear

vector

function

in

the

n o t h i n g to be d e s i r e d . . . ." (3;

use of V in the t w o systems Gibbs

" T h e s e considerations are sufficient, I t h i n k , to

s h o w that the position of the quaternionist is not the only one from which

the

subject

of vector

analysis

may be

viewed,

and that a

method which w o u l d be monstrous from one point of view, may be n o r m a l a n d i n e v i t a b l e f r o m a n o t h e r . " (3; 5 1 2 ) Gibbs

turned

statement

that

to

his

the

question

system

made

of notation

by

correcting

use of Grassmann's

Tait's

notation.

He

t h e n s u g g e s t e d t h a t h i s n o t a t i o n w a s s i m p l e r , clearer, a n d m o r e expressive than the quaternionic. exhibited

a noteworthy

In his statements in this regard he

openness

of m i n d and

in

general

placed

n o t a t i o n q u e s t i o n s in a s u b o r d i n a t e p o s i t i o n . (3; 5 1 2 - 5 1 3 )

reply4 to Gibbs'

Tait's

article was p u b l i s h e d w i t h i n the month.

It is perhaps ironic that one of the aspects of G i b b s ' system attacked strongly in Tait's reply was Gibbs' dyad, or what Gibbs called the i n d e t e r m i n a t e product of t w o vectors. Tait said this was confusing and " u n d o u b t e d l y artificial in the highest degree.

. . . " (4; 6 0 8 ) T h e

irony is that nearly half a century prior to this t i m e mathematicians attacked

the

quaternion

(noncommutative)

product

on

nearly the

same grounds; hence it is in one w a y surprising that a quaternionist w o u l d balk at a further extension in the meaning of "product." Tait wrote in response to Gibbs: Gibbs'

objections

always

considered

viz.

that

they

fore

specially

to

" I t is singular that one of Prof.

Quaternions should be precisely what I have

(after perfect inartificiality) their c h i e f merit: —

are

4

useful

uniquely

adapted

in

of the

some

to

Euclidian

space,

and

there-

most important branches

of

physical science.' W h a t have students of physics, as such, to do w i t h s p a c e o f m o r e t h a n t h r e e d i m e n s i o n s ? " (4; 6 0 8 ) F a t e s e e m s t o h a v e b e e n against T a i t , at least in r e g a r d to t h e last p o i n t . T a i t c o n c l u d e d by a discussion the t w o systems; for quaternion

186

of the

comparative

compactness

of expressions in

in this the quaternionists had a slight advantage,

products retain associativity, whereas the Gibbsian

A Struggle for E x i s t e n c e in the 1890's

vector p r o d u c t does not. (Thus in the G i b b s i a n system i X (j X j ) = X j ) X j =

Gibbs'

i.)

second article5

Tait's article.

was

published

i n Nature f o u r w e e k s a f t e r

Entitled "Quaternions and the 'Ausdehnungslehre,' "

it was written in response to historical statements made by Tait in his

Encyclopaedia

Britannica

in the preface to his

article

"Quaternions"

q u a t e r n i o n Treatise o f 1 8 9 0 .

(of

1886)

and

Gibbs' motivation

in writing this article m a y be inferred from a letter he wrote in 1888 to T h o m a s Craig to request the latter to p u b l i s h Grassmann's 1840 Theorie that

a

der Ebbe

und Flut.

"struggle

for

It w a s

in

existence"

this

letter that

among

the

Gibbs

vectorial

about to begin, a n d to this statement he added:

predicted

systems

was

" T h e most impor-

tant question is of course that of merit, b u t w i t h this questions of p r i o r i t y are i n e x t r i c a b l y e n t a n g l e d , & w i l l be c e r t a i n to be t h e m o r e discussed, since t h e r e are so m a n y persons w h o can j u d g e of pri37

ority to one w h o can judge of merit."

T h u s though Gibbs dealt pri-

marily w i t h priority questions, he was w e l l aware that m u c h more was at stake:

by correcting Tait's excessive priority statements he

added prestige to Grassmann, and in c o m p a r i n g Grassmann's ideas to Hamilton's (overtly in regard to historical questions) he put forth arguments

for

praise

and

to

recommend

Grassmann's

praise

and

to

recommend

his

marily those

the

superiority

aspects

of

Grassmann's

own

methods.

system

was

system, for he

of Grassmann's

system

And

of course

discussed

that were

to to pri-

also to be

found in the Gibbs-Heaviside system. Gibbs began by a d m i t t i n g that H a m i l t o n was the first to announce his discovery, whereas Grassmann "seems to have b e e n in no haste to place himself on record, and published nothing until he was able to give the w o r l d the most characteristic a n d fundamental part of his system 300

with considerable

pages,

which

d e v e l o p m e n t in a treatise of m o r e than

appeared

in August

1844."

(5;

79)

Both

were,

wrote Gibbs, memorable discoveries, and hence "Historical justice, and the interests of mathematical science," require that Tait's historical

statements

concerning

the t w o systems

" s h o u l d n o t b e al-

l o w e d t o pass w i t h o u t p r o t e s t . " (5; 79) G i b b s ' a p p r o a c h t o t h e p r i ority q u e s t i o n w a s t o state that t h e systems s h o u l d firs t b e c o m p a r e d in terms of what they have in c o m m o n and then in terms of what is peculiar to each. B o t h systems h a v e vector a d d i t i o n a n d the scalar a n d vector p r o d ucts, G i b b s stated, b u t the q u a t e r n i o n p r o d u c t is f o u n d o n l y in the quaternion system, w h i l e Grassmann had the linear vector function first. Gibbs then added:

187

A

History

To

of V e c t o r Analysis

w h a t extent are t h e g e o m e t r i c a l m e t h o d s w h i c h are u s u a l l y called

q u a t e r n i o n i c p e c u l i a r t o H a m i l t o n , a n d t o w h a t extent are t h e y c o m m o n to Grassmann? T h i s is a question w h i c h anyone can easily decide for himself. It is o n l y necessary to r u n one's eye over the equations used by quaternionic writers

in the

discussion of geometrical or physical sub-

jects, a n d see h o w far t h e y necessarily i n v o l v e t h e i d e a o f t h e quatern i o n , a n d h o w far t h e y w o u l d b e i n t e l l i g i b l e t o o n e u n d e r s t a n d i n g t h e functions afi,

Sap

and

Va/3,

but

having

no

conception

of the

quaternion

or at least c o u l d be m a d e so by t r i f l i n g changes of notation, as by

w r i t i n g S o r V i n p l a c e s w h e r e t h e y w o u l d n o t affect t h e v a l u e o f t h e expressions.

F o r s u c h a test the e x a m p l e s a n d illustrations in treatises on

q u a t e r n i o n s w o u l d be m a n i f e s t l y i n a p p r o p r i a t e , so far as t h e y are c h o s e n to illustrate quaternionic principles, since the objecttmay influence the form of presentation.

B u t we m a y use any discussion of geometrical or

p h y s i c a l subjects, w h e r e the w r i t e r is free to choose the f o r m most suita b l e t o t h e s u b j e c t . (5; 8 0 ) Gibbs

wrote

that

if

for

example

pages

160-371

of Tait's

Treatise

w e r e to be examined, it w o u l d be evident that "for the most part the methods of representing spatial relations used by quaternionic writers a r e c o m m o n t o t h e s y s t e m s o f H a m i l t o n a n d G r a s s m a n n . " (5; 80) After posing the

question

where the quaternion that

these were

o f t h e i m p o r t a n c e o f t h e r e m a i n i n g cases

p l a y e d a f u n d a m e n t a l part, G i b b s suggested

"very exceptional." Thus

effective w a y w h a t he h a d argued earlier: are

more

useful

Gibbs

restated in a very

scalar a n d vector products

and more fundamental than

the quaternion prod-

uct. A n d the e v i d e n c e he cited was the practice of the quaternionist Tait! G i b b s w e n t on to discuss Grassmann's point analysis, his " w e a l t h o f m u l t i p l i c a t i v e r e l a t i o n s , " a n d i n g e n e r a l t h e vast s c o p e o f his system as compared to Hamilton's. covery of matrices

came in

H e a t t e m p t e d t o s h o w that the dis-

Grassmann's

1844 work, a full fourteen

years b e f o r e C a y l e y ' s f a m o u s p u b l i c a t i o n of 1858. G i b b s p r o c e e d e d to give t w o quotations from Tait w h i c h stated in part that H a m i l t o n had

discovered the

scalar a n d vector products in the

1830's in his

" T h e o r y o f C o n j u g a t e F u n c t i o n s o r A l g e b r a i c C o u p l e s . " T h i s statement

Gibbs

simply

rejected, a n d w i t h g o o d reason.

Tait had gone

too far on too shaky a f o u n d a t i o n , for his k n o w l e d g e of Grassmann's system

was

vantage ton's

very limited. he was well

carefully

openness

acquainted with

G r a s s m a n n ' s and H a m i l -

sympathetic

Hamilton's

reasoned and well-documented paper exhibited

and flexibility w h i c h must have made it appealing to readers

other persuasion.

188

I n this r e g a r d G i b b s h a d t h e distinct ad-

systems.

Gibbs' an

that

without

He

system

being

antagonistic

to

readers

argued forcefully that w h a t was

was

also

in

Grassmann's

and

of an-

important in

that

moreover

A Struggle for E x i s t e n c e in the 1890's

G r a s s m a n n ' s s y s t e m c o n t a i n e d a w e a l t h o f a p p l i c a t i o n s u n i q u e t o it. very brief r e p l y 6 to Gibbs' article was published just one

Tait's week

l a t e r i n Nature.

In it Tait reasserted some of the views

had attacked but essentially gave tions

of,

those

views.

Tait

no n e w reasons

admitted

his

lack

Gibbs

for, or clarifica-

of familiarity

with

Grassmann's writings and in general wrote as one might write w h o was

surprised

that

anyone

would

attack

quaternions

on

such

grounds. T h e q u a t e r n i o n system was at that t i m e far better k n o w n , a n d T a i t m u s t h a v e felt little m o t i v a t i o n for g i v i n g a d e t a i l e d , tactful, and comprehensive rejoinder. Tait had long experience in defending quaternions, b u t never on such grounds as these. T h i s i s t h e last article o f 1891 that w i l l b e discussed; i t s h o u l d b e pointed out h o w e v e r that in the Electrician

Heaviside

began

a

November

series

of

13,

1891,

papers

on

issue of the

vector

analysis

w h i c h later f o r m e d the p o l e m i c a l t h i r d chapter of v o l u m e one of his Electromagnetic

Theory

(1893).

The

arguments

given

by

Heaviside

in this chapter (and hence in those papers) w e r e discussed in the previous

chapter.

In June, 1892, the debate resumed. T h e article that

month's

issue

of

the

Philosophical

Magazine

7

that appeared in

was

by

Alexander

M c A u l a y and was entitled "Quaternions as a practical Instrument of Physical

Research."

graduate

of Cambridge

Alexander M c A u l a y (1863-1931) was an (forty-ninth

Wrangler),

who

in

1886

1892

was

tutor and lecturer in mathematics and physics at O r m o n d College in Melbourne, Australia. T h e paper had b e e n given at the m e e t i n g of the Australian Association for the A d v a n c e m e n t of Science in January, 1892. T h e article is i n t e r e s t i n g a b o v e all b e c a u s e of t h e historical j u d g m e n t s m a d e b y M c A u l a y . McAulay began

by

stating that although

mathematician and although Hamilton asked:

"has

"Can

quaternions

left scarcely a successor."

any cause

arrested d e v e l o p m e n t ? " cists w e r e a s k e d this

be

assigned

(7;

Hamilton

were (7;

fairly 477)

was

a

well

great

known,

McAulay then

for this e x t r a o r d i n a r y case of

477) M c A u l a y suggested that if physi-

question, they w o u l d reply that quaternions

are n o t u s e d b e c a u s e t h e y h a v e n o t b e e n f r u i t f u l i n l e a d i n g t o scientific discoveries. to

shake

He then the

stated:

"The

paper

is

much

t o h o p e t o overturn t h a t b e l i e f . "

chief object of the present

b e l i e f of mathematica l

physicists.

further on the apathy in regard to quaternions: more

I

It

is

too

(7; 4 7 8 ) M c A u l a y c o m m e n t e d "I confess that the

think of this apathy the m o r e extraordinary does it appear,

and, as already hinted, it w i l l probably prove an insoluble p r o b l e m

189

A

History

of V e c t o r Analysis

to the future historian of Mathematics."

(7; 4 7 8 ) I t w a s a r g u e d that

most physicists w h o had rejected quaternions had studied them at a time

after

"their mathematical ideas and m e t h o d s " had "nearly or

completely

crystallized."

and

they

before

have

cease their study.

(7;

seen

479)

After

a

limited

period of study

the powerfulness of the methods, they

A major factor in this too early cessation of their

labors has b e e n M a x w e l l , w h o "is responsible to a large extent for the discredit into w h i c h quaternions have fallen among physicists." (7;

478-479)

too

early,

Young

physicists

abandon

"consoling themselves

experience

than

themselves"

limited in their usefulness.

that

and

their

study of quaternions

Maxwell . . . had had more

yet had

found quaternions very

(7; 4 7 9 )

M c A u l a y stated that physics w o u l d advance w i t h great rapidity if quaternions plete

were

exclusion

"introduced to

of Cartesian

serious study to the almost com-

Geometry,

except in

an

insignificant

w a y , a s a p a r t i c u l a r case o f t h e f o r m e r . " (7; 4 8 0 ) M c A u l a y t h e n g a v e an elaborate series of q u a t e r n i o n applications in an attempt to illustrate

his

The ciples

point.

next paper

8

for discussion is Alexander Macfarlane's "Prin-

of Algebra of Physics,"

1891, m e e t i n g Science

and

w h i c h was delivered at the August,

of the A m e r i c a n Association for the A d v a n c e m e n t of published

in

the

Proceedings

of that

society

in

July,

1892. M a c f a r l a n e (1851-1913) h a d s t u d i e d u n d e r T a i t at E d i n b u r g h and in year

1891 was teaching at the University of Texas.

secretary

and was debate

until

on

Tait's

progress reason

his

in

his

on

of the

paper by

Quaternions

quaternion

for hope.

one

section

of the

He was in that

American

Association

most active participants in the

of vector analysis.

began

Treatise

physics

death

systems

Macfarlane of

of the

t w o quotations from the preface

and

development

then has

stated been

in

slow,

effect but

that

there

is

Macfarlane r e m a r k e d that in his o p i n i o n the qua-

ternion system was on the right track, w h i l e he proceeded i m m e d i ately to

qualify his

statement rather drastically.

He wrote:

B u t at the same t i m e I am c o n v i n c e d that the notation can be i m p r o v e d ; that the principles require to be corrected and extended; that there is a more complete algebra w h i c h unifies Quaternions, Grassmann's m e t h o d a n d D e t e r m i n a n t s , a n d a p p l i e s t o p h y s i c a l q u a n t i t i e s i n space. T h e g u i d i n g idea of this paper is generalization. W h a t is sought for is an algebra w h i c h w i l l apply directly to physical quantities, w i l l include and unify the several branches of analysis, and w h e n specialized w i l l become ordin a r y a l g e b r a . (8; 65) After gave

190

mentioning

a series

the

of criticisms

Tait-Gibbs of the

debate

in

Nature,

Macfarlane

q u a t e r n i o n system. A key point in

A Struggle for E x i s t e n c e in the

Macfarlane's discussion is the following.

1890's

He insisted that the qua-

t e r n i o n s y m b o l s i, j, a n d k h a v e not one, b u t t w o m e a n i n g s . F o r example,

he

argued

v e c t o r or as "versors,"

that

i

may

be

v i e w e d either as

a " v e r s o r , " or t u r n e r .

then

ij = k m e a n s

quadrant in the

Hence

a certain

unit

if i a n d j are v i e w e d as

a right-handed rotation through

one

plane perpendicular to j c o m p o u n d e d w i t h a right-

handed rotation through one

quadrant in the plane perpendicular

to i, w h i c h is e q u i v a l e n t in result to a r i g h t - h a n d e d q u a d r a n t rotation about k.

But, said Macfarlane, Tait was too hasty in a l l o w i n g

such an interpretation since i, j, and k symbolize vectors, not vers o r s . T h e c r u c i a l p o i n t i s t h e m e a n i n g o f ii. I f t h e i ' s i n t h i s p r o d u c t are v i e w e d as versors, t h e n ii m e a n s a r o t a t i o n of 180° a r o u n d t h e i axis, w h i c h

when

applied to j

w o u l d p r o d u c e —j.

B u t if t h e i's be

v i e w e d as vectors, then Macfarlane stated that it was better to define t h e

product

as

positive,

for

example,

{ai){bi) = +ab, s o t h a t t h e

the product w o u l d be in harmony w i t h such things as i m v

2

(kinetic

energy). As a result of such considerations

Macfarlane constructed a new

system of vector analysis more in h a r m o n y w i t h the Gibbs-Heaviside system than w i t h the quaternion system. As part of this project he introduced n e w notations and defined a full product of t w o vectors

which

was

comparable to the full quaternion

product except

that the scalar part w a s positive, not n e g a t i v e as in t h e o l d e r system. This system apparently never became popular or w i d e l y employed, though

expositions

of it

were

published

rather frequently.

Mac-

farlane explained his system in a chapter in M a n s f i e l d M e r r i a m a n d Robert

S.

Woodward

(editors),

Higher

Mathematics,

of

which

edi-

tions appeared in 1896, 1898, a n d 1900, a n d in a d d i t i o n this chapter was p u b l i s h e d as a b o o k in 1906.42 Macfarlane's

article

was

intelligently written and in many ways

was impressive; nonetheless it further complicated an already complex situation. T h e i n t r o d u c t i o n in 1892 of another system of vector analysis, could

even

scarcely

a sort of c o m p r o m i s e system such as be

well

received by the

advocates

Macfarlane's, of the

already

existing systems and moreover probably acted to broaden the question beyond the comprehension of the

In

November,

1892,

the

debate

as-yet u n i n i t i a t e d reader.43

returned to the

pages

o f Nature

with a review9 probably written by Alfred Lodge44 of Macfarlane's "Principles of the Algebra of Physics." Lodge began by writing:

"This is a very suggestive contribution

to the foundations of the Algebra of Vectors as recently so strongly advocated i n A m e r i c a b y Prof. W i l l a r d G i b b s , a n d i n this c o u n t r y b y

191

A H i s t o r y of V e c t o r Analysis

M r . O l i v e r H e a v i s i d e . " (9; 3 ) L o d g e t h e n s u m m a r i z e d M a c f a r l a n e ' s work

and

made

numerous

comparisons

w i t h the Gibbs-Heaviside

and the quaternion systems. T h e r e v i e w was more descriptive than critical;

it

was

in

general

worthy of consideration.

favorable

to

It concluded:

Macfarlane's

system

as

"A text b o o k of vector alge-

b r a . . . i s m u c h n e e d e d , a s m a n y physicists are b e c o m i n g interested in the n e w algebra, o w i n g in great measure to M r . O. Heaviside's able

exposition

of its

principles

and

applications

in

t h e Electrician

a n d e l s e w h e r e . " (9; 5 )

Oliver Heaviside's paper of Energy

10

" O n t h e Forces, Stresses, a n d F l u x e s

i n the E l e c t r o m a g n e t i c F i e l d " m a y b e considered next.

It was read to the L o n d o n Royal Society in 1891 and published in the Transactions of that society in

1893, a l t h o u g h it h a d b e e n p u b -

lished

Electrical

in

1892

in

Heaviside's

Papers.

E a r l y i n t h e p a p e r H e a v i s i d e p r e s e n t e d a n e x p o s i t i o n o f h i s system

of vector analysis w h i c h was prefaced by an attack on quater-

nions. After p o i n t i n g out that vectors w o u l d be used in his paper because the physical quantities w e r e vectors, Heaviside (in mockery of Tait's preface)

discussed the "retardation" of vector analysis be-

cause of the lack of an adequate treatise on the subject, "Professor T A I T ' S w e l l - k n o w n p r o f o u n d treatise b e i n g , a s its n a m e i n d i c a t e s , a treatise on Q u a t e r n i o n s . " (10; 427) He r e f e r r e d to t h e antiquaternionic arguments this dispute as a

retarder

physics."

of

recently given by Gibbs

i n Nature a n d d e s c r i b e d

"rather one-sided." Gibbs was called "anything but progress

in

vector

analysis

and

its

application

to

(10; 428) In this instance, as in others, H e a v i s i d e spoke

v e r y f a v o r a b l y o f G i b b s ' b o o k l e t . T h e fact that H e a v i s i d e a n d G i b b s w e r e in a g r e e m e n t on all aspects of vector analysis except notation must have

strengthened their position.

H e a v i s i d e t h e n b r i e f l y p r e s e n t e d h i s m a i n a r g u m e n t s f o r h i s syst e m as against the quaternion system.

Statements such as the one

b e l o w ( w h i c h i s i n o n e sense v e r y t r u e a n d i n a n o t h e r v e r y false) illustrate

the

He wrote: must be

directness,

" . . .

humor, and humility of Heaviside's

style.

I o u g h t to also a d d that the i n v e n t i o n of quaternions

regarded

as

a m o s t r e m a r k a b l e feat of h u m a n ingenuity.

Vector analysis, w i t h o u t quaternions, could have been f o u n d by any mathematician

by

carefully

examining the

mechanics

of the Car-

tesian mathematics; but to find out quaternions required a genius." (10; In

461) summary, this paper is most appropriately v i e w e d as another

publication

by

were

associated

192

found

Heaviside

in

with

which

polemics

important

new

for

vector

physical

analysis

results.

One

A Struggle for Existence in the

1890's

might wonder what better rhetoric could be used than advocacy of a

method

followed

by

the

testimony

of actual

success

with

that

method. The

paper,11

next

side's

remarks

McAulay,

the

published

in

his

author,

in

Nature,

Philosophical

began

by

status o f t h e v e c t o r q u e s t i o n :

was

occasioned by

Transactions

giving his

paper.

opinion

Heavi-

Alexander

of the

present

" T h e r e are t w o w i d e l y - k n o w n systems

of vector analysis before the public —Quaternions a n d the A u s d e h n ungslehre—and Prof.

Gibbs's

quite

seems

a

multitude

of

less

known

ones,

of w h i c h

to be one of the least o p e n to objection, a n d of

which, in my opinion, Mr.

H e a v i s i d e ' s i s b y n o m e a n s so." (11;

McAulay proceeded to make

151)

what must be v i e w e d as a hopelessly

idealistic appeal to Gibbs and Heaviside " o n grounds i n d e p e n d e n t of the merits suggested

or demerits of their particular systems."

that

the

"woefully

small"

band

of

(11;

vector

151) He

analysists

should concentrate on making vectorial methods better known, and to do this they s h o u l d l i m i t the debate to quaternions versus Grassmann's system. Prof.

Gibbs

" T h e day for Prof. Gibbs's i m p r o v e m e n t s is not yet.

and

the small band. the c o m m o n

Mr. .

.

Heaviside

.

cause,

a n d content themselves w i t h the faith that pos-

terity w i l l do t h e m justice." McAulay biguous,

concluded his

certainly

have not yet c o n v i n c e d the rest of

L e t me i m p l o r e t h e m to sink the i n d i v i d u a l in

(11;

151)

paper with

inflammatory,

the

following

somewhat am-

statement:

To vary the metaphor, Maxwell, Clifford, Gibbs, Fitzgerald, Heaviside prescribe a course of spoon-feeding the physical public. H a m i l t o n and Tait r e c o m m e n d and provide strong meat. I do not think that harm, but rather good, w i l l come f r o m this d o u b l e treatment, as one course w i l l suit some patients

a n d t h e o t h e r o t h e r s . But l e t t h e s p o o n - f e e d e r s p r o -

vide spoon-meat of the same

kind a s t h e o t h e r p h y s i c i a n s .

Is not Max-

w e l l , Clifford, a n d Fitzgerald's food as digestible as Prof. Gibbs's a n d M r . H e a v i s i d e ' s ? (11; Needless failed.

to

Both

continuance

It was side's ical

McAulay's

statements

of the

attempt

and his

to

brash

pour

oil

attitude

on

the

waters

encouraged the

debate.

remarked previously that one major significance of Heavi1892

sections

physical

say,

his

151)

Philosophical were

results.

Transactions

embedded

in

paper a

volume

paper

was

that

the

containing

In

the

same

Alexander McAulay

was

published under the title

of this journal

paper,

polem-

important

a paper

12

by

" O n the Mathe-

matical

Theory of Electromagnetism." This

long as

H e a v i s i d e ' s , w a s i n w a y s s i m i l a r t o it. I t w a s d i r e c t e d a t ex-

nearly twice as

193

A

History

tending tained

of V e c t o r Analysis

electrical theory w i t h i n the polemics

McAulay though than

in

his

his

paper

paper

As

made

seems

Heaviside's,

McAulay

M a x w e l l tradition, and it con-

for a system of vectorial analysis —for quaternions.

to

extensive have

use

of quaternions,

b e e n scientifically less

it was an impressive display.

a n d al-

important

Early in the paper

stated:

m i g h t be expected, the mathematical m a c h i n e r y that appears to be

most c o n v e n i e n t for investigating as f u l l y as possible the consequences of these novel.

assumptions,

And I

and

others

intimately

connected

with them, is

m a y remark in passing that w h a t Professor T A I T persist-

e n t l y a n d w i t h c o m p l e t e justice emphasizes as one of the greatest boons that

Quaternions

uralness, s e e m s described.

to

grant

to

me to

McAulay's

presentation deviated

he

by

introducing

noted that though paper,

sophical

much

it

150 pages

was

1892 taken

helped

from

notation.

it perhaps

it

issue

the

viz.,

the

pure

their

perfect

quaternion

quaternion

Concerning

balanced paper

quaternions)

Moreover

of the

have

Heaviside's

vs.

Transactions.

of that time

new

with

(vectors

must

somewhat

be

side's

physicists,

nat-

illustration in the methods about to be

(12; 6 8 6 - 6 8 7 )

though

question

ungrateful

receive

for

is

this

out the

paper

sharpened the

readers

noteworthy

it may

effect of Heavi-

must have many

cause,

tradition

that

of the

Philo-

more

than

of the leading British scientific journal

up by these two papers

written in vectorial

language.

In ical his

1893

M c A u l a y published a short book

preface preface

of the

that is

book.

played

a

part

in

this

13

with a highly polem-

debate.

Tait

referred

to the preface as

McAulay (who by at the

style

1893 was

in 15

"extremely interesting

as t h e p e r f e r v i d o u t b u r s t of an enthusiast." (15;

that

McAulay's

indicated by a c o m m e n t m a d e by Tait in a r e v i e w

193)

Lecturer in Mathematics and Physics

University of Tasmania) stated at the b e g i n n i n g of his preface

the

Smith's

book

"mournful include McAulay

was

originally an

essay

submitted

Prize Competition at Cambridge.45 thing"

that

quaternions.

Cambridge As

to

mathematics

w h y Cambridge

(in

1887)

for the

M c A u l a y bemoaned the did

not in general

h a d t u r n e d a d e a f ear,

stated that he could not " b e l i e v e that she is in her dotage

a n d has lost h e r h e a r i n g . " (13; v) T h e n f o l l o w e d s o m e passages enlightening Cambridge,

for

the

the

picture

they

center of British

give

of interest

in

quaternions

at

mathematics.

W h e n I sent in t h e essay I h a d a faint m i s g i v i n g that perchance there was not a single labor. . . .

194

man in Cambridge w h o could understand it without much

A Struggle for Existence in the

1890's

T h e r e is no lack in C a m b r i d g e of the c u l t i v a t i o n of Q u a t e r n i o n s as an algebra,

but

this

cultivation

upon Quaternions he

has

as

as

is

not

Hamiltonian . . . Hamilton

a geometrical m e t h o d ,

yet failed to

find

looked

a n d it is in this respect that

w o r t h y followers resident in Cambridge.

(13; vi)

T h e only w a y to convince the nurses [the C a m b r i d g e tutors] that Quaternions

form

a healthy diet for the

young

mathematician

is

to

p r o v e to

t h e m that t h e y w i l l " p a y " in the first part of the Tripos. Of course this is an i m p o s s i b l e task w h i l e t h e o n l y q u e s t i o n s . . . are in t h e s e c o n d part a n d average o n e i n t w o years. (13; v i i ) McAulay

then

pleading

that

quaternions

addressed he

himself

"steep"

to

himself in

and promising:

the the

Cambridge "delirious

student

by

pleasures"

of

" W h e n you w a k e you w i l l have forgot-

ten the Tripos and in the fulness of time w i l l develop into a financial

wreck,

but

in

dream you will millionaire."

possession

of the

memory

of that

heaven-sent

b e a far h a p p i e r a n d r i c h e r m a n t h a n t h e m i l l i o n e s t

(13;

v i i - v i i i ) O t h e r passages

n o less

strongly w o r d e d

could be cited from this preface, as w e l l as m i x e d references to Tait. The

content of the

reader;

that even n o w I place

c o n c l u d i n g passage

nevertheless

is probably obvious to the

it may be quoted:

" L e t m e i n c o n c l u s i o n say

scarcely dare state w h a t I b e l i e v e to be t h e p r o p e r

of Quaternions

in

a

Physical

education,

for

fear

my

state-

ments be regarded as the uninspired babblings of a misdirected enthusiast,

but

I

cannot

refrain

from

saying.

. . . "

(13;

xi) T h e n fol-

l o w e d the essay itself, w h i c h was also rich in p o l e m i c a l statements, surrounded

by

numerous

highly

technical

applications

of quater-

nions to elastic solid theory, electrical theory, h y d r o d y n a m i c s , a n d the

vortex-atom

theory.

The

majority

of the

polemical

statements

f r o m the essay itself h a v e already b e e n c o n s i d e r e d in the discussion of

McAulay's

Society In

papers

in

the

Philosophical

Magazine

and

the

Royal

Transactions. conclusion

it may be

stated that

McAulay's

preface qualifies

h i m for consideration as one of the most vociferous mathematicians of the

century.

function of the wondered

However,

since

success

expressed enthusiasm

whether

McAulay's

in

debate

is

not always

a

of the participants, it may be

writing

produced

much

more

than

controversy.

T w o reviews farlane,

the

of McAulay's

book

other by Tait.

Alexander Macfarlane's review the

Physical

merit discussion,46 one by Mac-

Review

parative study.

(1893),

and

is

1 4

appeared in the first volume of

above

all

interesting

as

a

com-

Macfarlane, like Gibbs and Heaviside, was an advo-

195

A

History

cate

of V e c t o r Analysis

of a particular vector system w h i c h had been derived from the

quaternion a

system.

quaternion

had been

H e , u n l i k e G i b b s a n d H e a v i s i d e , h a d long b e e n

advocate

before

a student of Tait.

his

departure from

that system and

Macfarlane's r e m a r k s in this r e v i e w are

especially interesting w h e n they are c o m p a r e d w i t h the statements of Gibbs In with

and Heaviside

the

earlier

parts

McAulay's

t h o u gh he

on

Hamilton and on

of his

review

quaternions.

Macfarlane

showed

sympathy

enthusiastic c h a m p i o n i n g of the q u a t e r n i o n cause,

m a i n t a i n e d that a large part of the

tion into quaternion

essay was

notation of k n o w n results.

. .

"a transla-

." (14; 388) T h e

transitional passage is the following: I agree w i t h the author in his estimate of the value of H a m i l t o n ' s quat e r n i o n researches: t h e y constitute, in my o p i n i o n , the greatest mathematical w o r k of the century. T h e y contain w h a t was long sought after—a v e r i t a b l e e x t e n s i o n o f a l g e b r a t o s p a c e : I d o n o t s a y the, f o r I b e l i e v e t h a t t h e r e i s m o r e t h a n o n e . T h e C a r t e s i a n analysis i s also a n e x t e n s i o n o f algebra to space, b u t it is fragmentary a n d i n c o m p l e t e ; whereas the quat e r n i o n a n a l y s i s i s t h e t r u e s p h e r i c a l t r i g o n o m e t r y i n w h i c h t h e axis o f an a n g l e as w e l l as its m a g n i t u d e is c o n s i d e r e d . (14; 389) Macfarlane

then

disagreed

neglect of quaternions to

a small

group of defects

remedied

in

with

McAulay's

explanation

of the

a n d argued that the neglect was mainly due

Macfarlane's

in

the

own

quaternion system (which were

system).

Neither Gibbs nor Heavi-

side ever made such laudatory statements about Hamilton's w o r k as those g i v e n in the quotation f r o m Macfarlane. G i b b s d i d not do this, partly because system;

he

was convinced of the superiority of Grassmann's

Heaviside

partly because his

did

not,

partly

for reasons of t e m p e r a m e n t a n d

strategy, like that of G i b b s , was a i m e d at disso-

ciating their system from

that of Hamilton.

Macfarlane could make

a n d d i d m a k e such a statement, it w o u l d seem, because his strategy was different; he w i s h e d to present his system as w i t h i n the quaternion

tradition but w i t h the f e w quaternion defects

removed.

Thus

he felt no c o m p u l s i o n t o w a r d m a k i n g a vigorous attack on either the quaternion be

system

or on H a m i l t o n and Tait.

Thus the review may

described as favorable to McAulay's book but more favorable to

his

own

system.

Tait's review cember 28, McAulay's unnamed

1 5

1893,

of McAulay's book was the lead article in the Deissue

o f Nature.

book,

particularly

author

(Macfarlane)

the

wastes

196

pages

of a

presented

as

book

like

compared

to

another work by an

and as compared to the works men-

t i o n e d in the f o l l o w i n g passage: into

Tait began his r e v i e w by praising

as

"It is positively exhilarating to dip this

wholesome

after

pasture

toiling in

the

through writings

the

arid

of Prof.

A Struggle for E x i s t e n c e in the

Willard

Gibbs, 4 7

plexion." able;

Dr.

Oliver

Heaviside, and others

1890's

of a similar com-

Tait's remarks on M c A u l a y ' s preface w e r e not so favor-

i n fact, h e passionately a t t a c k e d M c A u l a y ' s t e n d e n c y t o w r i t e

passionately. I t i s m u c h t o b e r e g r e t t e d that M r . M c A u l a y has n o t d e t e r m i n e d s i m p l y to let his Essay speak for itself. H i s Preface, t h o u g h e x t r e m e l y interesting as the p e r f e r v i d outburst of an enthusiast, assumes here and there a character of u n d i g n i f i e d querulousness or of dark insinuation, w h i c h is not

calculated

to

win

sympathy.

I t has

too

much

of the

"Rends-toi,

c o q u i n " t o m a k e w i l l i n g converts; a n d i n s o m e passages i t runs a - m u c k a t I n s t i t u t i o n s , C u s t o m s a n d D i g n i t i e s . N o t h i n g s e e m s safe. I t i s a s t u d y in

monochrome:—the

lights

dazzingly vivid,

and the

shades

dark

as

E r e b u s ! (15; 193) Tait

described

nality,"

whose

McAulay

as

a man

reaction to the

b e e n gall a n d bitterness." (15; Intuitively

recognising

weapon which

its

of " g e n u i n e p o w e r a n d origi-

Cambridge

restrictions

"must have

193)

power,

he

snatches

up

the

H a m i l t o n tenders to all, a n d at once

magnificent

dashes off to the

jungle on the quest of b i g game. Others, more cautious or perhaps more captious, m e a n w h i l e sit p o n d e r i n g g r a v e l y o n t h e f a n c i e d i m p e r f e c t i o n s of the

arm;

and

endeavour to

convince

a bewildered

p u b l i c (if they

cannot c o n v i n c e t h e m s e l v e s ) that, l i k e t h e H i g h l a n d e r ' s m u s k e t , i t req u i r e s t o b e t r e a t e d t o a b r a n d - n e w s t o c k , l o c k a n d b a r r e l , o f their own devising,

before

it

can be

safely r e g a r d e d as

fit

for service.

(15;

Tait then commented, more benevolently than favorably, on of McAulay's

some

innovations.

Such was Tait's ternion

193)

fold.

m a n n e r of w e l c o m i n g a n e w zealot into the qua-

McAulay,

like

Tait

before

him,

had

discovered

the

quaternion system at C a m b r i d g e despite C a m b r i d g e . A l l in all Tait must have

been

quite elated about this n e w convert.

mark of Heaviside on M c A u l a y , written in may be

given:

F i n a l l y a re-

1894 in a letter to Gibbs,

" H e seems to be a very clever fellow, and he knows

it and shows that he k n o w s it a little too m u c h sometimes."

Readers

of the

brief article that his essary torial this

1 6

January

by Tait.

1893,

issue

o f Nature

found therein

a

stating that he had assumed

1891 replies to Gibbs had been sufficient to show the "necimpotence"

and

system that lacks illusion

Of this

5,

Tait began by

48

was

"inevitable the

dispelled

fifty-seven-page

unwieldiness"

of

every

vec-

quaternionic product.

But, wrote Tait,

by

Society

paper

Heaviside's Tait

read

Royal

four

paper.10

pages —then:

" . . .

I

m e t the check-taker as it w e r e : — a n d f o u n d that I m u s t pay before I could

go

further.

Quaternions

(in

I

found

whose

that

I

should

disfavour m u c h

is

not

only

said)

have to unlearn

b u t also to learn a

197

A

History

of V e c t o r Analysis

n e w and most uncouth parody of notations long familiar to me; had

to

relinguish

Tait

then

Heaviside

the

two

short

cle.

Here

as

the

Electrician.

these

from

the

of quaternions

Tait

his

questionable

so I

225)

criticisms To

passages

(16,

responded

1890

strategy

published by only by

quot-

Magazine

arti-

Philosophical

elsewhere Tait either underestimated

followed

statements

attempt."

mentioned in

ing

or

the

of treating

his his

opponents opponents'

as though they did not merit a detailed reply. Tait then

declared that the m a i n object of his present note was to call attention to a paper by Knott recently given before the Edinburgh Royal Society Tait

(the

stated

discussion that

the

of this

paper

"is

paper

is

best

a complete

delayed

exposure

until

of the

later). preten-

sions a n d defects of t h e (so-called) V e c t o r S y s t e m s . " (16; 226) F r o m reading this paper Tait claimed that he had come to understand the vectorial ideas of his opponents; final sentence:

"I

its

have

revelations

ment, In bate

the

March

with

the

tion;

16,

to

1893, 1 7

article

issue

entitled

15,

1892,

issue

much

Gibbs

wisely

commented

system. was

since

226)

re-entered the deand

the

Algebra of

o f Nature

(and

indirectly

in

re-

Gibbs began by discussing M c A u l a y (and Tait

b l a m e on the lack of u n i f o r m i t y in notathat

this

cause

could

almost no comparable mathematical

not

be

ac-

system had pre-

uniformity of notation for a longer time than the quaternion H a v i n g rejected this explanation, G i b b s suggested another: that

"simplicity,

Hamilton's

method

perspicuity,

moreover put the (17;

and

of presentation

brevity"

"geometrical

ondary

position.

Aulay's

poorly chosen metaphors,

tion

Gibbs

of the acceptance of quaternions. placed

served

o f Nature

(16;

"Quaternions

other of McAulay's writings).

had

cepted,

had

left on me is that of m e r e a m u s e d disappoint-

It was written directly in response to McAulay's paper11

slowness

earlier)

it

an

December

sponse the

his reaction was expressed in his

it difficult to decide whether the impression

or of mingled astonishment and pity."

Vectors." in

find

for mathematical

463)

After

of the

relations

making Gibbs

the

obscured

the

vectorial a p p r o a c h

had

and

of vectors" most

in

of one

a secof Mc-

suggested a law of evolu-

systems.

Whatever is special, accidental, and individual, w i l l die, as it should; but that w h i c h is universal a n d essential s h o u l d r e m a i n as an organic part of the w h o l e intellectual acquisition. If that w h i c h is essential dies w i t h the a c c i d e n t a l , i t m u s t b e b e c a u s e t h e a c c i d e n t a l has b e e n g i v e n t h e p r o m i nence w h i c h belongs to the essential.

In m e c h a n i c s , k i n e m a t i c s , a s t r o n o m y , physics, all study leads to the cons i d e r a t i o n o f c e r t a i n r e l a t i o n s a n d o p e r a t i o n s . T h e s e are t h e c a p i t a l no-

198

A Struggle for Existence in the

1890's

tions; these s h o u l d have the l e a d i n g parts in any analysis s u i t e d to the subject. (17; 464) Gibbs' those

"capital

which

are

notions" "essential

(in

the

second

and universal"

are

clearly

(first q u o t a t i o n ) .

quotation)

In this

w a y he led up to an argument central to this and to the preceding papers. If I w i s h e d to attract the student of any of these sciences to an algebra for vectors, I s h o u l d tell h i m that t h e f u n d a m e n t a l notions of this algebra were exactly those w i t h w h i c h he was daily conversant. I s h o u l d tell h i m that a v e c t o r a l g e b r a is so far f r o m b e i n g a n y o n e m a n ' s p r o d u c t i o n t h a t half a c e n t u r y ago several w e r e already w o r k i n g t o w a r d an algebra w h i c h should be primarily geometrical and not arithmetical, and that there is a remarkable

similarity

in

the

results

to

which

these

efforts

led. . . . I

s h o u l d call his attention to the fact that L a g r a n g e a n d Gauss u s e d the n o t a t i o n (afiy) t o d e n o t e p r e c i s e l y t h e s a m e a s H a m i l t o n b y h i s S ( a / 3 y ) , except that Lagrange l i m i t e d the expression to unit vectors, and Gauss to vectors of w h i c h the length is the secant of the latitude, a n d I s h o u l d show h i m that we have o n l y to give up these limitations, a n d the expression (in connection w i t h the notion of geometrical addition) is e n d o w e d w i t h an i m m e n s e w e a l t h of transformations. I s h o u l d call his attention to the fact that the n o t a t i o n [r,r2], u n i v e r s a l in the t h e o r y of orbits, is i d e n tical w i t h H a m i l t o n ' s V(p,p2), except that H a m i l t o n takes the area as a v e c t o r , i.e. i n c l u d e s t h e n o t i o n o f t h e d i r e c t i o n o f t h e n o r m a l t o t h e p l a n e of the triangle, and that w i t h this simple modification (and w i t h the notion of geometrical a d d i t i o n of surfaces as w e l l as of lines) this expression becomes closely connected w i t h the first-mentioned, and is not only end o w e d w i t h a similar c a p a b i l i t y for transformation, b u t enriches the first w i t h n e w capabilities.

I n fact, I s h o u l d t e l l h i m that t h e n o t i o n s w h i c h

we use in vector analysis

are those w h i c h h e w h o reads b e t w e e n t h e

lines w i l l m e e t on every page of the great masters of analysis, or of those w h o have p r o b e d d e e p e s t t h e secrets o f nature. . . . (17; 464) The

above q u o t e d passage is typical of G i b b s ' ability to present

powerful sage the

and

quaternion

cally true. ress

sensible

arguments.

The

remark

f o l l o w i n g this

pas-

referred to McAulay's lamentations at the poor acceptance of

of any

adopted

system;

the

remark

is both appealing and histori-

" T h e r e are t w o w a y s i n w h i c h w e m a y m e a s u r e the progreform.

the

The

shibboleth

one

of the

consists

in

reformers;

counting those the

other

who

measure

is

have the

degree in w h i c h the c o m m u n i t y is i m b u e d w i t h the essential principles

of the

reform.

I

should

apply

the

broader

measure

to

the

p r e s e n t case, a n d d o n o t f i n d i t q u i t e s o b a d a s M r . M c A u l a y d o e s . " (17; In

464) summary,

traditions

Gibbs

leading

up

maintained that to

manifested itself in the

the

present

there

were two

situation.

The

discernible

first

tradition

great physical treatises of the past w h e r e i n

stress g r a d u a l l y c a m e t o b e p l a c e d o n c e r t a i n f u n d a m e n t a l n o t i o n s

199

A

History

of V e c t o r Analysis

and operations.

The

second tradition,

running parallel

to the first

a n d i n p a r t s t e m m i n g f r o m it, c o n s i s t e d i n t h e c r e a t i o n o f f o r m a l vectorial

systems.

G i b b s argued that these t w o traditions w e r e at that

t i m e (the 1890's) c o n v e r g i n g a n d that the vectorial approach was bec o m i n g c o m m o n , e v e n t h o u g h dispute was rampant as to w h i c h vectorial

system

was

preferable.

To

decide

between

the

various

sys-

tems emerging from the second tradition, one need only analyze the content of the first tradition. Gibbs concluded w i t h a tactful and true remark which, though it criticized

H a m i l t o n , praised Tait and other second-generation qua-

ternionists able

for presenting the quaternion system in a m o r e accept-

form. N o w I appreciate and admire the generous loyalty toward one w h o m

h e r e g a r d s a s his m a s t e r , w h i c h has a l w a y s l e d Prof. T a i t t o m i n i m i s e t h e o r i g i n a l i t y of his o w n w o r k in regard to quaternions, a n d w r i t e as if eve r y t h i n g was c o n t a i n e d i n the ideas w h i c h f l a s h e d into the m i n d o f H a m i l t o n at the classic B r o u g h a m Bridge. B u t not to speak of other claims of historical justice, we o w e duties to our scholars as w e l l as to our teachers, and the

w o r l d is too large, a n d the current of m o d e r n thought is too

b r o a d , t o b e c o n f i n e d b y t h e ipse dixit e v e n o f a H a m i l t o n .

Gibbs'

paper was

(17; 464)

soon

followed by another which, though fully

in agreement with Gibbs'

a r g u m e n t s , differed m a r k e d l y i n style. I t

was written by Oliver Heaviside, whose h u m o r and brashness complemented sponse

Gibbs'

to

cerning

seriousness

McAulay

McAulay's

1 1

1 2

'

and

and Tait

Royal

1 6

Society

tact. and

Heaviside

in

paper

wrote

in

support of Gibbs.

Heaviside

wrote:

re-

Con-

"As the

h e a r t k n o w e t h its o w n w i c k e d n e s s , h e w i l l n o t b e s u r p r i s e d w h e n I say that I s e e m to see in his m a t h e m a t i c a l p o w e r s the ' p r o m i s e a n d p o t e n c y ' o f m u c h f u t u r e v a l u a b l e w o r k o f a h a r d - h e a d e d k i n d . " (18; 534)

Heaviside

ternions!

then

suggested

got u s e d to

quaternions.

I

After making criticisms

the

turned

(18;

534)

of some

of McAulay's arguments, Heavi-

to Tait and indirectly to Knott.

irrelevance

of

nionic

notations,

wrong

anyway

clearness

M c A u l a y simply give up qua-

k n o w w h a t it is, as I w a s in t h e q u a t e r -

nionic slough myself once."

side

that

" A difficulty i n t h e w a y , " w r o t e H e a v i s i d e "is that h e has

their

and

and

arguments

then

that

of expression.

his

he

on

attempted

system

Heaviside

was went

peace

nionic

have

citadel;

been alarms

pouring of boiling

200

superiority

to

show

superior on

lieved was the significance of Tait's paper. and

He chided them about

the

to

that in

state

of

quater-

they

were

brevity what

he

and be-

" T h e quaternionic calm

disturbed. T h e r e is confusion in the quaterand

excursions,

water upon

and

hurling

of stones

and

the i n v a d i n g host. W h a t else is the

A Struggle for E x i s t e n c e in the 1890's

m e a n i n g of his letter, a n d m o r e especially of the c o n c l u d i n g paragraph? B u t t h e w o r m m a y t u r n ; a n d t u r n t h e tables." (18; 534) H e a v iside c o n c l u d e d the article w i t h a discussion of some specific considerations on t h e q u a t e r n i o n n o t a t i o n a n d m e t h o d s of treating rotations. T h e m a j o r s i g n i f i c a n c e o f t h e a r t i c l e i s t h a t t h r o u g h its h u m o r , frankness, and partial refutation of arguments of the quaternionists it gave support to Gibbs' m o r e closely reasoned article. Heaviside's brashness

must

have

antagonized not a few people;

nevertheless

his brashness was not that of the y o u n g a n d little k n o w n A l e x a n d e r McAulay.

One of the was in

Cargill Vector

Society

most important papers published during this debate Gilston

Theory"

o f Edinburgh

1892,

Knott's

was

lengthy

published

for

published

1892. in

in

It

1893,

paper the

had and

was

"Recent Innovations

Proceedings

been

i n 1893. T h e f i r s t abstract w a s v e r y b r i e f ; than

19

read

twice 49

of

the

Royal

December

abstracted

19,

i n Nature

the second20 was longer

a n y o t h e r p a p e r i n t h i s d e b a t e i n Nature w i t h t h e e x c e p t i o n o f

G i b b s ' response to it, w h i c h was o n l y s l i g h t l y longer. K n o t t h a d b e e n Tait's assistant at E d i n b u r g h f r o m 1879 to 1883, then

Professor of Physics at the I m p e r i a l U n i v e r s i t y of Japan, a n d

from

1892,

versity.

Lecturer on

Applied

Mathematics

in Edinburgh Uni-

He was to become Tait's biographer and remained through-

out his life a staunch advocate of quaternions.50 T h e f o l l o w i n g discussion the

will

concentrate on the full-length paper as published in

Edinburgh

This Knott

paper, stated,

Royal as

Society

Proceedings.

nearly all others

"wholly

from

the

in the debate, was written, as point

of

view

of

mathematical

physics. . . ." (19; 212) T h e p a p e r w a s a i m e d at c r i t i c i z i n g t h e w o r k o f Macfarlane, H e a v i s i d e , and, above all, G i b b s , w h o m K n o t t c a l l e d the " h i g h - p r i e s t " of t h e " c l i q u e of vector analysis." (19; 212) A f t e r mentioning some of the earlier papers in the debate, Knott briefly discussed "the

vanced 1852.

the

work of Rev.

anti-quaternionic beyond Knott's

the

aim

stage in

M a t t h e w O ' B r i e n and c o n c l u d e d that

vector

analysts

reached

m a k i n g this

by

of

today

have

O ' B r i e n . . ."

barely (19;

s t a t e m e n t (or better,

ad-

212)

in

overstate-

ment) was to portray the Gibbs-Heaviside system as nothing new, rather as an old system that had not survived. Knott then attacked

Gibbs'

statement that the

scalar a n d v e c t o r

products were fundamental, while the quaternion product was not Knott argued that w i t h o u t the quaternion the division of one vec by

another

is

not

possible,

that

this

operation

is

fundamental

and that hence the quaternion is fundamental. To Gibbs' argument

201

A

History

that his

of V e c t o r Analysis

vectorial

system could be extended to higher dimensions,

while

quaternions

trying

"to

life

Mars. . .

in

solve

could

the

not,

Irish (19;

Knott

responded

question

217)

After

by a

a

that

this

discussion

discussion

was

like

of the

social

of n o t a t i o n

Knott

turned to the question of w h y the square of a unit vector should be e q u a l t o —1. that "*(*

ij

=

+ j ) j

k

He =

(*2

=

+

i j 2 = —j +

ik + must

equal

one.

i j

—1.

associative

s h o w e d that if we assume the associative law and

—ji, j k = ij)j 2

"

i = —kj, i2j

= But

(19;

+

and

k i = j = —ik,

kj = + i

2

j-i"

and

then

we

can

write:

i ( i j + j2) =

"»(i + j ) j =

s i n c e -\~i2j — i m u s t e q u a l — j + i j 2 , i 2 a n d j 2

221-222)

Thus

a necessary condition for the

law is that the square of a unit vector be equal to minus

Knott m a d e an issue of this, a n d it is not a small point.

Repeatedly in the paper Knott took an algebraically narrow point of view.

He

was

tinct products was

unable

to

see the l e g i t i m a c y of d e f i n i n g t w o dis-

of t w o vectors;

there was only one product, and that

the heaven-sent and Hamilton-discovered quaternion product.

Thus

Knott

criticized

the

Gibbs-Heaviside

system

because

it

l a c k e d a n y t h i n g c o r r e s p o n d i n g t o t h e q u a t e r n i o n V c o ( f o r co, a v e c tor)

(19;

223-224)

and

because

V •
and

V X pn

Gibbs' notation) had to be defined separately. Gibbs and

had

" M a x "

theory.51 series

introduced as

These

of V's.

troduction Gibbs'

Knott's

lowing

had been

humorous

Knott then tion.

expressions

turned

certain

"Pot,"

"New,"

theorems

in

"Lap,"

potential

this jokingly by suggesting the

and a "Ham."

(19;

225) This

in-

evoked one of

responses. to

position

in

in

introduced primarily to avoid writing a

Knott dealt w i t h

of a " T a i "

few

the

abbreviations

(expressed

(19; 2 2 3 - 2 2 4 )

in

Gibbs'

treatment of the linear vector func-

this r e g a r d is w e l l s u m m a r i z e d in the fol-

quotation:

In the course of the development of the theory of the dyadic, Gibbs, w i t h h i s u s u a l p r o n e n e s s t o l e x i c o n p r o d u c t s , i n v e n t s a f e w n a m e s (or n e w meanings to o l d ones), such as

Idemfactor, Right Tensor, Tonic,

C y c l o t o n i c , Shearer, a n d so on. T h e s e are all special f o r m s of t h e l i n e a r a n d vector function; and, excepting possibly the names, Professor G i b b s does

not

seem

beautiful theory.

to

have

contributed

anything of value to

Hamilton's

In no case, so far as I h a v e b e e n a b l e to see, do his

methods compare, for conciseness a n d clearness, at all favourably w i t h H a m i l t o n ' s a n d Tait's. (19; 229) One

line

was

forced

with

of attack used by Knott was to attempt to s h o w that G i b b s to

bring

in

the

a summary and by an

quaternion.

(19;

invocation of the

235) name

Knott concluded Hamilton.

Knott's paper is important in that it was the first detailed criticism of the Gibbs-Heaviside system written by a quaternionist. h i g h l y probable that it m a d e

202

It seems

impressive reading for the interested

A Struggle for Existence in the 1890's

reader. T h o u g h Knott h a d read G i b b s ' booklet w i t h care a n d w r i t t e n with

a certain

characterized

Stoic the

control,

earlier

nonetheless

papers

by

Tait

the

bitterness

and

McAulay

that had was

not

absent. Alexander Macfarlane was the first to reply to Knott's attack on paper21

"Recent

Innovations."

His

i n Nature,

slightly

than

more

a

Knott's paper had appeared.

appeared in

month

late

after the

M a y of 1893

longer

abstract

of

It was also w r i t t e n partly in r e p l y to

review.9

Lodge's Much

of

Macfarlane's

paper

was

simply

a

restatement

of his

views and hence need not be discussed. T h e major point of interest is

Macfarlane's

discussion

of Knott's

views

concerning

the square of a vector should be positive or negative. the

e n d of his

paper made

the

statement:

"The

whether

Knott h a d at

assumption t h a t t h e

square of a unit vector is positive u n i t y leads to an algebra w h o s e characteristic quantities

are

non-associative, b u t in no sense m o r e

general than the corresponding but associative quaternion quantities,

and

(19;

236-237)

whose

V

is

not the

Knott's

real

Macfarlane contention that the tive was but

e f f i c i e n t Nabla

disagreement

with

the

of q u a t e r n i o n s . " Heaviside-Gibbs-

square of a vector should be posi-

e v i d e n t not o n l y in his w r i t i n g of " a s s u m p t i o n " in italics

also

ran

arguing

through

(correctly)

his

that

whole both

paper.

views

Macfarlane

were

responded by

assumptions

(or

better

definitions) and were hence completely arbitrary algebraically. cited

as

evidence

to

Quaternions.

in

the

a

passage

from

Again

history

had

had

argued

that

1840's

Kelland

come there

full was

and

Tait's

cycle,

for

nothing

He

Introduction Hamilton

algebraically

w r o n g in defining a product that violates the commutative law, and in the 1890's K n o t t h a d a r g u e d in defense of the q u a t e r n i o n cause that there

was

associative

something

unnatural

in a system that violated the

law.

Knott (in close relation w i t h the previous 2

that V u should equal (the "real") pointing

out

Natural probandi

+ ^Jf)'

N a b l a gave this result. that

Tait

Philosophy,' " lies

+

on

the

used (21;

minus

question) had argued

"the

To this

76)

a n d by

men."

(21;

*

n c e

the quaternion

Macfarlane replied by

V2w

unreal

s

in

his

'Treatise

suggesting that

"The

on onus

76)

In conclusion it may be noted that the controversy on the important plus by

the

plicity,

or minus question was here as elsewhere being argued quaternionists

and

naturalness,

on

grounds

whereas

the

of

algebraic

vector

elegance,

analysts

sim-

argued

on

203

A

History

of V e c t o r Analysis

pragmatic

grounds —the

corresponded geometry.

to

It is

the

test

for

them

was

most frequent relations

what

most

conveniently

found in physics and

important to note that Macfarlane's arguments (un-

intentionally of course) w e r e h e l p f u l to the G i b b s - H e a v i s i d e cause, for

what

Macfarlane

equally to the Knott's

reply

l a t e r i n Nature. question

is

Knott's

written

2 2

It

behalf of his

do

system

applied

to Macfarlane's paper was published three weeks said

little

that was

little

Typical

more

of

this,

admission that the

cluding

on

system.

new.

Since the minus or plus

mathematically unanswerable,

Knott could rhetoric.

had

Gibbs-Heaviside

than

restate

and

interesting

answer was

either being legitimate,

old views couched in new because

arbitrary,

was

it

contained

Knott's

con-

paragraph:

I n c o n c l u s i o n , l e t m e say t h a t n o reasonable m a n can p o s s i b l y object to investigators using any innovations in analysis they m a y find useful. B u t in t h e p r e s e n t case t h e r e is a v e r y serious o b j e c t i o n to t h e i n n o v a t o r s c o n d e m n i n g the system, for w h i c h they have one and all d r a w n inspiration, as

"unnatural" and "weak," without in any way showing it so to

be. T h a t t h e y can re-cast m a n y q u a t e r n i o n investigations into their o w n m o u l d does not p r o v e their m o u l d to be superior or even comparable to t h e o r i g i n a l . Y e t , i n s o far a s t h e y possess m u c h i n c o m m o n w i t h quaternions, the m o d i f i e d systems used by Gibbs, Heaviside, and Macfarlane cannot fail to have m a n y virtues. " H i s f o r m h a d not yet lost A l l her original brightness, nor appeared Less than Archangel ruined."52 T w o

weeks

after

Knott's

response

to

Macfarlane, Alfred

Lodge

p u b l i s h e d an unpolemical paper23 suggesting a solution to the plus or

minus

it

question.

provoked

accepted. the

Its

sole

give

suggestion

was

not accepted;

a b e v y of reasons

importance

is

that it was

at most

w h y it should

not be

another article w i t h i n

debate.

Gibbs' 1893, had

Lodge's

Knott to

reply

nearly

been

criticisms

2 4

to

Knott

four months

published. of Knott;

was

published

after the

In it Gibbs

in

Nature

on

August

longer abstract of Knott's replied in

detail to the

17,

paper

detailed

since such criticisms a n d replies are of interest

o n l y w h e n t h e y deal w i t h major aspects of the system, the emphasis in the following discussion w i l l be on illustration rather than summarization. Gibbs

began

by

stating

that

he

felt

bound

to

reply

to

Knott's

paper because

Knott h a d attacked the w h o l e system of vector analy-

sis

on

primarily

Vector Analysis.

204

Gibbs

the

basis

replied

of first

faults to

found

Knott's

in

Gibbs'

assertion

Elements

that

he

of had

A Struggle for Existence in the

been

forced

charge

to

was

in

describing

introduce one

case

Knott's

the

due

quaternion

to

an

mathematical

into

his

inadvertence

error)

and

in

1890's

booklet.

(a tactful

two

other

This

w a y of

cases

to

o v e r s t a t e m e n t — " M y critic is so anxious to p r o v e that I use quaternions that he uses

arguments

w h i c h w o u l d prove that quaternions

Hamilton was born.',

were in c o m m o n use before

To Knott's criticism of Gibbs'

use of V,

Gibbs

(24;

the reader to consider the following statement made H e l m h o l t z first s o l v e d the p r o b l e m — G i v e n t h e liquid motion, to potentials densities the

find

of three spin;

by

ideal

distributions

equal

to

1/tt

of gravitational

of the

Kelvin:

s p i n i n a n y case

the motion. H i s solution consists in

respectively

given

365)

replied by asking

finding

matter

of the

having

rectangular components

of

and, regarding for a m o m e n t these potentials as rec-

tangular components

o f v e l o c i t y i n a case o f l i q u i d m o t i o n , t a k i n g t h e

spin in this m o t i o n as the v e l o c i t y in t h e r e q u i r e d m o t i o n . (27; 365) Gibbs then translated Kelvin's statement as he thought Knott might p r e f e r it:

"Helmholtz first solved the problem —Given the Nabla of

the velocity in solution

any case

of Nabla of the inverse

Nable

argued

that

Nabla of the velocity. of the

his

" N e w = VPot," to remember, quaternion

his

"Lap = V X as

find

the velocity.

that

Pot,"

His

square

Or, that the velocity was the

velocity."

of the

inverse

(24;

symbolic

365) Gibbs then

abbreviations

a n d " M a x = V. P o t "

"Pot,"

w e r e as

easy

expressive, and more conducive to rigor than the

equivalents.

Elements

plaining

Nabla of the

introduction

Knott that there of

of liquid motion, to

was that the velocity was the N a b l a of the

Gibbs

proceeded

to

answer

a charge

was a "tangle" and "jangle" in sections 91 o f Vector

this

was

Analysis due

to

by

thoroughly

a time

lapse

agreeing

of a " c o u p l e

by

to

104

and

ex-

of years"

b e t w e e n w r i t i n g parts o f t h e sections. S e c t i o n 101 w a s t h e last section

of the

Knott's

1881

part of Gibbs' booklet.

reference to this

finer a r g u m e n t

[against

"tangle"

Gibbs'

Gibbs then capitalized on

and "jangle"

system]

.

.

.

in the

can

be

terms:

" N o

found."

(24;

366) G i b b s gave a d e t a i l e d discussion of Knott's attack on his treatment of the could gain

linear vector function. some

knowledge

F r o m this discussion the reader

of the superiority of Gibbs' treatment.

T h i s p a p e r a l o n e , b e c a u s e o f its m a n y d e t a i l s a n d b e c a u s e i t w a s written is

as

a defense,

best viewed

serving to remove the decided between, the

The

issue

of

could not have persuaded the uninitiated.

It

as a capable and thorough response a n d h e n c e as

Nature

doubts rival

that

of someone acquainted with, but un-

systems.

appeared

paper contained a short article

25

the

week

following

Gibbs'

by Robert S. Ball, w h o h a d written

205

A

History

a

work

entitled

relation mer

of V e c t o r Analysis

royal

Theory

of Ireland

tronomy and Ball's It

The

to vector analysis.

and was

(Dublin,

1876)

that

bore

a

(1840-1913) h a d b e e n the astronoin

1893

Lowndean

Professor of As-

Geometry at Cambridge.

major point is

has

o f Screws Ball

always

summarized in the following quotation:

appeared to

me

that

the

student of physical science

w o u l d better e m p l o y his t i m e b y s t u d y i n g the " A u s d e h n u n g s l e h r e " t o which

some

of your

correspondents

have

referred

than

by

studying

quaternions. T h e w o n d e r f u l work of Grassmann is contained in a moderate-sized book

in

remarkable

contrast

to the t w o terrific volumes of Hamilton,

w h i c h e v e n Prof. T a i t a d m i t s t h a t h e has n o t r e a d e n t i r e l y . T h e fact that the ausdehnungslehre could be mastered in a mere fraction of the time that w o u l d have to be devoted to the mastery of quaternions, is not howe v e r t h e i m p o r t a n t p o i n t . (25; 391) Ball

concluded

and

by

a

by

plea

giving

for

the

an

illustration

translation

of

of Grassmann's

Grassmann's

methods

1862

Ausdehn-

ungslehre.

The

September

28,

1893,

issue

o f Nature

contained two

articles,

a short one by Knott and a shorter one by R. W. Genese. Genese's paper

2 6

will be considered first.

R o b e r t W i l l i a m G e n e s e (1867-1928) h a d r e c e i v e d his B.A. (1871, eighth Wrangler) and M.A. Professor

of

(1874) f r o m C a m b r i d g e and was in 1893

Mathematics

at

University

College,

Aberystwyth.

Genese's p a p e r constituted a s e c o n d i n g of Ball's m o t i o n that Grassmann's

book

pounds

t o w a r d t h e a c c o m p l i s h m e n t o f that task.

fact

that

be

Grassmann's

and Clifford was

Knott's in

paper27

reply to

on

a

translated;

Gibbs

number

ideas

he

even

were

pledged

not

as

yet

to

subscribe

ten

L a m e n t e d was the taught

in

England,

cited on behalf of the value of Grassmann's work.

in

the

and to

of Gibbs'

September 28 Lodge. replies

Herein but gave

issue

o f Nature w a s

written

Knott briefly commented no

n e w arguments.

This

part of the paper is best v i e w e d as no m o r e than an attempt to prevent the belief that he had acquiesced to Gibbs' arguments. Knott's last p a r a g r a p h w a s his a n s w e r to L o d g e ' s suggestion for an innovation, against w h i c h Knott gave

Knott's was issue

reply

2 2

to

five

Macfarlane's

arguments.

reply

2 1

discussed by Macfarlane in an article of

to 28

Knott's

original paper

in the O c t o b e r 5, 1893,

Nature.

Macfarlane began

by stating that he w i s h e d to aid the discussion

by restating his fundamental position. This was in brief that quater-

206

A Struggle for E x i s t e n c e in the 1890's

nion

notations

as

well

as

some

of the

fundamental

quaternion

principles needed revision, including the principles that the square (d2

of a v e c t o r is n e g a t i v e a n d that V2 =

d2

+

d2 \

+

He

c o n

*

eluded first by calling attention to t w o papers he had recently delivered at the

International Mathematical Congress at Chicago in

w h i c h he h a d e x t e n d e d his system a n d t h e n by p r e s e n t i n g the following statement:

" A s regards Prof.

'Paradise

feel

Lost,'

I

like the

Knott's closing quotation from

Senior Wrangler who, having read

through the p o e m , r e m a r k e d that it was all very pretty, b u t he d i d n ' t q u i t e see w h a t it p r o v e d . I close w i t h a q u o t a t i o n w h i c h is f r o m as g o o d a b o o k , a n d possesses m o r e logical force: 'Ye shall k n o w t h e m by their fruits.

Do m e n gather grapes of thorns, or figs of thistles?' "

(28; 541) In conclusion asked

it m a y be noted that the important question to be

concerning

Macfarlane's

this

Gibbs-Heaviside indirectly

but

Macfarlane's

paper

is

not as

to

how much

system, but whether it helped the vector

analysis

indisputably conclusions

cause.

helped

were

the

The

answer

latter,

it advanced

quaternion or the is

for the

clear;

it

majority

of

in harmony with the Gibbs-Heavi-

side vector system.

The

last

Nature

word,

went

to

literally but not

Heaviside.

This

symbolically, paper29

was

in

the

debate

published

in

in the

J a n u a r y 11, 1894, i s s u e a n d w a s w r i t t e n m a i n l y i n r e s p o n s e t o T a i t ' s review

15

of M c A u l a y ' s book.13

Heaviside values nion

began

by

arguing

that

Tait

failed

to

recognize

the

of the systems put forward by the opponents of the quater-

system.

"He

does

not k n o w their ways, either of thinking or

of w o r k i n g , as is a b u n d a n t l y e v i d e n t in all that he has w r i t t e n adversely

to

Prof.

Willard

w e n t o n t o state that h e McAulay's

innovations.

Gibbs was

and

others."

(29;

246)

Heaviside

surprised at Tait's w a r m w e l c o m e to

Then

Heaviside

argued

that the

illustra-

tion of the value of M c A u l a y ' s w o r k selected by Tait for discussion could be and essentially had been dealt w i t h in a better manner in the vector analysis system.

Heaviside concluded by a second invi-

tation to M c A u l a y to convert to vector analysis.

T h e final three papers f r o m 1893 to be discussed w e r e read before

the

Proceedings

Edinburgh of the

Mathematical

Society

Society

sometime

and

after June

published 9,

1893

in

(the

the date

of the

last p a p e r i n v o l u m e e l e v e n ) .

T h e s e papers w e r e n e v e r re-

ferred

to

debate,

in

the

course

of the

Nature

a n d h e n c e their dis-

207

A H i s t o r y of V e c t o r Analysis

cussion has b e e n p o s t p o n e d u n t i l t h e c o m p l e t i o n o f the discussion of

the

Nature

articles.

T h e first paper of the nion

30

Society in a n d its

was by Knott (who was to assume the presidency

November,

1893) a n d was e n t i t l e d " T h e Quater-

Depreciators."

etition

(at t i m e s

slightly

earlier

verbatim) "Recent

Most of Knott's

paper was

either rep-

or elaboration of his statements in his

Innovations

in

Vector

Theory."

Hence

this long paper m a y be treated rather briefly. Knott began by classifying the predators

(Gibbs,

Heaviside,

statements of the quaternion de-

and Macfarlane)

u n d e r three heads:

first, the question of "the value of the quaternion as a fundamental geometrical

conception";

second,

"the

question of notation";

and

t h i r d , " t h e q u e s t i o n of t h e s i g n of t h e s q u a r e of a vector. . . ." (30; 62)

After repeating a n u m b e r of his

question,

Knott

that although

concluded

by

earlier arguments on the first

suggesting

an

analogy.

He

stated

sin 6 a n d cos 6 occur m o r e f r e q u e n t l y t h a n 6 itself,

we s h o u l d not c o n c l u d e that 6 plays no f u n d a m e n t a l role. Similarly we

should

Va/3

a n d Sa/3 o c c u r m o r e f r e q u e n t l y .

not

infer

that

aft

is

not

fundamental

simply because

K n o t t t h e n t u r n e d to the q u e s t i o n of notation a n d cited such arguments as w e r e available on behalf of the quaternion system. F r o m this

he

proceeded

to

the

question

priate to the square of a vector.

His

concerning

the

sign

appro-

m a i n positive argument was,

as before, that the quaternion m i n u s preserved the associative law for

multiplication,

response

to

and

his

Macfarlane's

main

charge

negative

that

arguments

essentially

were

in

unit vectors and

versors should not be identified, since the square of the first should be +1

a n d o f t h e l a t e r —1.

Knott ceedings

c o n c l u d e d b y r e f e r r i n g t h e r e a d e r t o h i s p a p e r i n t h e Proof

questions

the

Royal

Society

of Edinburgh

which

discussed

these

a n d others.

This paper, like

Knott's

other long paper, was carefully written

a n d c o n t a i n e d no special excess of scorn. N e i t h e r G i b b s nor H e a v i side r e p l i e d to it, p e r h a p s b e c a u s e t h e y felt it was unnecessary or possibly because they felt it was useless (considering the audience).

K n o t t ' s p a p e r h a d b e e n d e l i v e r e d a t t h e J a n u a r y 13, ing

of the

Edinburgh

Mathematical

Society.

1893, meet-

At the next meeting

( F e b r u a r y 10, 1893) D r . W i l l i a m P e d d i e , " A s s i s t a n t t o t h e Professor of

Natural

assistant),

Philosophy"

at

Edinburgh

delivered a paper31

After commenting

that

he

University

(hence

Tait's

on behalf of the quaternion system.

believed

that

new

and

very

general

m a t h e m a t i c a l m e t h o d s (like quaternions) w e r e difficult for students

208

A Struggle for E x i s t e n c e in the 1890's

to assimilate, reason that,

Peddie

stated:

" A n d it seems

as

if it were

for this

in recent years, attempts have b e e n m a d e , by m e n of

k n o w n m a t h e m a t i c a l ability, t o s m o o t h t h e p a t h s . " (31; 85) H e w a s of course referring to Gibbs, Heaviside, and Macfarlane. Peddie's view

as

to

their

success

was

evident

from

his

next

statement:

"Practically, all these attempts consist in using, instead of H a m i l ton's,

another system

of quaternions,

cut up

into

parts;

the parts

of that system b e i n g u s e d because t h e y are i m a g i n e d to be superior to

the

corresponding

parts

of

Hamilton's

system

in

respect

of

naturalness." (31; 85) P e d d i e t h e n d i s c u s s e d t h r e e p o i n t s i n Tait's Treatise

which

Heaviside

had

called

"sticking

points."

The

dis-

cussion of the t h i r d p o i n t t u r n e d on the fact that the quaternionists began

with

fewer definitions,

but they consequently were

forced

to go through m o r e elaborate developments to get some of the fundamental results.

In response to the charge that the quaternion is

n o t a f u n d a m e n t a l e n t i t y , P e d d i e , p r e s u m a b l y o n t h e m i s t a k e n assumption that the vector system had no m e t h o d of treating rotations, argued that the w i t h rotations. that the

quaternion

was

"notions

of quaternions

any n u m b e r of dimensions." In conclusion, that.

essential

as

a m e t h o d of dealing

Peddie concluded the article by an attempt to show

Peddie's

are

(31;

applicable to space of four or

90)

paper was

competent but no more than

He had not however read Gibbs' work, and thus a n u m b e r of

his arguments w e r e vague or missed the point. T h e papers by Knott a n d P e d d i e w e r e g i v e n at the first t w o 1893 meetings Society.

(January a n d February) This

meetings

by

sequence Peddie,

was

who

of the

Edinburgh

continued

in

the

presented at these

Mathematical

March

meetings

and

April

t w o parts

of a paper entitled "Elements of Quaternions." F r o m the published one-sentence

s u m m a r y of the first paper53 it may be inferred that

Peddie discussed therein vector addition and subtraction usefulness

and the

of these operations. A six-page abstract32 of the second

part of his paper (April meeting) was published. F r o m this abstract it is clear that Peddie's i n t e n t i o n in the paper was to intersperse a review

of the

fundamental

principles

of quaternion

analysis

with

periodic forays against the vector analysts. Since these asides w e r e in

part repetitive

reasons for his

of earlier

arguments

used by Peddie and since

statements were not in general published because

of the fact that o n l y an abstract was p r i n t e d , n o t h i n g n e e d be said o f t h i s p a p e r b e y o n d m e n t i o n o f its e x i s t e n c e a n d s p i r i t .

Two magnetic

1894 Theory

reviews merit

of

the

first

discussion,

volume since

of both

Heaviside's

Electro-

commented

on

209

A

History

of V e c t o r Analysis

Heaviside's written

vectorial

by

Review.33

ideas.

Alexander

Macfarlane

The

shorter

Macfarlane

described

and

the

of

these

reviews

published

book

and

in

praised

cept the third chapter, w h i c h was on vector analysis;

the

all

was

Physical

parts

ex-

he wrote:

T h e t h i r d c h a p t e r e x p o u n d s t h e e l e m e n t s o f v e c t o r analysis. T h e exposition,

while

c l e a r , a n d s u i t a b l e f o r t h e class

of readers addressed,

contains principles w h i c h appear of d o u b t f u l validity to those w h o have m a d e a s p e c i a l s t u d y of t h e m a t t e r . F o r instance, t h e scalar p r o d u c t of t w o vectors is not distinguished by any prefix, though the vector product is, a p p a r e n t l y o n t h e s a m e p r i n c i p l e t h a t t h e e l d e s t d a u g h t e r i s suffic i e n t l y d i s t i n g u i s h e d b y M i s s , p r o v i d e d all t h e y o u n g e r sisters h a v e t h e i r Christian

names

p r o d u c t are

appended.

only partial

But

the

scalar

the

simpler name belongs properly

products;

product

and

the

vector

to the complete product, w h i c h is their sum. Here we have an indication of one of the p r i n c i p a l defects of the analysis; it is fragmentary, not a method This the

totus

teres

statement whole

phorical

atque

(33;

153-154)

is

illustrative of t w o practices c o m m o n throughout

debate.

T h e first is the p e n c h a n t of vectorists for meta-

expressions;

tendency

rotundus.

of

many

the

second

of the

and

writers,

the

more

particularly

important

the

is

the

quaternionists,

to treat the question of products as s o m e t h i n g m o r e than a question of arbitrary

definition.

In

Macfarlane's

case

his statement that the

scalar a n d vector products are o n l y partial products was p r e s u m a b l y not

made

an

argument

through

algebraical The

own

by

George

Magazine.

Minchin's and

M.

34

of

rather

commented upon

and

lengthy on

Heaviside's

on

the

assumed

readers. Electromagnetic

published

review

Heaviside's

style

it is best v i e w e d as

based

Heaviside's

Minchin

concentrated

rather

ideas

of Macfarlane's

review

written

ignorance;

Heaviside's

"naivete"

second

favorable

his

against

was

Theory

in

the

in

general

innovations.

was

Philosophical very

Minchin

in the following way:

A reader of M r . H e a v i s i d e ' s w r i t i n g s is at once struck by the extraordinary on

style

which

distinguishes

h i m from every other English writer

Mathematics or Physics; and the impression w h i c h is produced by

this style is often the reverse of pleasing. T h e r e is a c o m p l e t e absence of the

conventionalities

which

are

the w r i t i n g of a scientific treatise.

generally recognized as proper to Mr.

Heaviside is the Walt W h i t m a n

of E n g l i s h Physics; a n d , l i k e t h e so-called " p o e t , " he is certain to raise a v e r s i o n to his p e c u l i a r i t i e s . (34; 146) Minchin

however was

not

in all ways averse to Heaviside's style;

he w e n t on to praise h i m for his clearness in exposition. After an

extended

troduced by vector

210

discussion

Heaviside,

analysis.

He

Minchin

criticized

of some

of the electrical terms in-

discussed Heaviside's chapter on

first

of all

Heaviside's notation, for

A Struggle for E x i s t e n c e in the

example, his

use of Clarendon type

tors.

Minchin

was

very

argued, as

inconvenient

others

when

for the

representation of vec-

had argued, that this

one

1890's

wished

to

write

convention

vectors

in

a

manuscript, b u t it was at least better t h a n M a x w e l l ' s use of G o t h i c capitals. T u r n i n g to the ideas in w h a t he c a l l e d H e a v i s i d e ' s cal

Vector

Analysis,"

Minchin

pointed

out

that

he

used

"Hereti-

this

term

" w i t h o u t necessarily i m p l y i n g a n y censure. . . ." (34; 154) M i n c h i n presented no real arguments against Heaviside's vector system; his tone h o w e v e r revealed that he this

innovation,

praised

most

favorably pared

of

this

Heaviside's

inclined to

with

his

of

review

In

summary,

ideas,

vector system.

FitzGerald's

but

he

Minchin

was

not

so

His review may be com-

of

Heaviside's

last chapter.

Electrical

Both reviews

Papers

must have

Heaviside to become better k n o w n , and their discussions

Heaviside's

many

not favorably inclined toward

system.

electrical

w h i c h was discussed in the helped

was

"Heretical"

vectorial

important

ideas

results)

must

(which have

system (which was little k n o w n in

had

at

helped

least

Heaviside

served

to

make

to

that

1890) m u c h better k n o w n .

T h e final t w o papers of the debate are a m o n g the most interesting. T h e disputants w e r e C a y l e y a n d Tait, a n d the q u e s t i o n was o n e that h a d n o w h e r e else b e e n discussed. Arthur Cayley (1821-1895), whose extremely

prolific

and

very

paper appeared first, was an

well-known

mathematician.

He

held

the Sadlerian Chair for Mathematics at C a m b r i d g e University; the remarks

in

his

paper

are

probably represent the

thus

point

especially

of view

interesting,

that

he

passed

since on

they

to many

C a m b r i d g e students. It is probable that M c A u l a y ' s statements about quaternions view

at

Cambridge

of quaternions.

were

Cayley

directed

had

against Cayley

long been

interested

and

in

his

certain

aspects of q u a t e r n i o n s a n d was in fact the first p e r s o n (after H a m i l ton)

to

publish

mentioned

a paper on quaternions.

previously,

written

a

M o r e o v e r Cayley had, as

chapter

entitled

"A

Sketch

of

the Analytical T h e o r y of Q u a t e r n i o n s " for the

1890 edition of Tait's

Treatise

as

on

Quaternions.

Tait

described

this

being

included

t h r o u g h Cayley's " u n s o l i c i t e d k i n d n e s s . " C a y l e y a n d T a i t h a d frequently

corresponded,

and

this

correspondence

more

or

less

di-

rectly led to the t w o papers to be discussed.54 M o s t of the arguments that w e r e

eventually

these letters. their

different

presented

As early as views

s h a l l r e m a i n so. . . . "

in

these

of quaternions: 55

papers

may be found in

1888 C a y l e y h a d c o m m e n t e d c o n c e r n i n g " w e are

irreconcileable

and

Nevertheless they continued the discussion

in later letters, a n d in June,

1894, C a y l e y sent T a i t a c o p y of a p a p e r

211

A

History

entitled to

of V e c t o r Analysis

"Coordinates

publish

Cayley Royal and

in

allow

his

Society

Tait's

Quaternions"

Messenger

paper

to

of

be

were

read

published

After

the

Cayley planned

Tait

before

and

Cayley

at the July 2,

in

which

Mathematics.

read

Edinburgh.56

of

reply

sequently

versus

the

suggested published

agreed,

by

that the

paper35

his

1894, m e e t i n g a n d sub-

Proceedings

of

the

quotation

from

Royal

Society

of

Edinburgh. Cayley's

paper

first edition

began

of Tait's

later editions.

with

Treatise,

a

a preface

which

the

was

preface to the

reprinted in

the

Tait had written:

It must always be r e m e m b e r e d that Cartesian methods

are m e r e par-

t i c u l a r cases o f q u a t e r n i o n s w h e r e m o s t o f t h e d i s t i n c t i v e f e a t u r e s h a v e disappeared; and that w h e n , in the treatment of any particular question, scalars

have to be adopted, the quaternion solution becomes identical

w i t h the Cartesian one. is

generally

methods.

gained,

N o t h i n g , t h e r e f o r e , i s ever lost, t h o u g h m u c h

by

employing

quaternions

in

place

of ordinary

I n fact, e v e n w h e n q u a t e r n i o n s d e g r a d e t o scalars, t h e y g i v e

t h e s o l u t i o n o f t h e m o s t g e n e r a l s t a t e m e n t o f t h e p r o b l e m t h e y are app l i e d to, q u i t e i n d e p e n d e n t of any limitations as to choice of particular c o o r d i n a t e axes. (35; 2 7 1 ) Cayley s u m m a r i z e d his

own views

in the following paragraph:

M y o w n v i e w i s t h a t q u a t e r n i o n s are m e r e l y a p a r t i c u l a r m e t h o d , o r say a t h e o r y , in coordinates. I have the highest a d m i r a t i o n for the n o t i o n of a

quaternion;

but . . . as

I

consider the

full

m o o n far m o r e

beautiful

t h a n a n y m o o n l i t v i e w , so I r e g a r d t h e n o t i o n of a q u a t e r n i o n as far m o r e b e a u t i f u l t h a n a n y o f its a p p l i c a t i o n s . A s a n o t h e r i l l u s t r a t i o n . . . I c o m pare

a

quaternion

f o r m u l a to a p o c k e t - m a p —a capital t h i n g to put in

one's pocket, b u t w h i c h for use m u s t be u n f o l d e d : the formula, to be u n d e r s t o o d , m u s t b e t r a n s l a t e d i n t o coordinates. (35; 2 7 1 - 2 7 2 ) In

Cayley argued that35 the

short,

pure

mathematician

is

quaternion as

interesting

and

an

important,

entity for the

but the

quater-

n i o n m e t h o d as a m e t h o d for the applied mathematician (consideri n g the g e o m e t e r in this case as an a p p l i e d m a t h e m a t i c i a n ) is nothing

more

than

a

shorthand

u n d e r this aspect it is Cayley The

gave

lem:

Given

to the

a

scalar.

Cartesian lated

212

quaternions

He

solution,

into the

of his

point of view.

a comparison of the solutions,

and by Cartesian methods, of the prob-

two lines OA and O B ,

w o u l d be

of abbreviation, and even

illustrations

find

plane containing OA and OB.

solution is

elementary

second illustration consisted in

worked out by

m

three

or a m e t h o d

not a useful method.

He

the line OC perpendicular stated that the quaternion

my = Va/3 w h e r e y = O C , a = O A , commented but

was

that this

was

unintelligible

Cartesian equivalent.

much

and

= OB, and

briefer than the

useless

u n t i l trans-

A Struggle for E x i s t e n c e in the

Cayley's would

final

say

statement

that

while

was

the

following:

coordinates

are

"In

applicable

1890's

conclusion, to

the

I

whole

science of geometry, a n d are the natural a n d appropriate basis a n d method

in

the

science,

quaternions

seem

to

me

a

particular

and

very artificial m e t h o d for treating such parts of the science of threedimensional the

g e o m e t r y as are m o s t naturally discussed by m e a n s of

rectangular coordinates

x,

t / , z."

(35;

275)

Cayley's

arguments

could of course be translated directly into arguments against almost any vectorial

Tait's of

the

system.

reply

3 6

to

historical views Tait

Cayley

Quaternion

began

was

Method."

entitled

It

is

" O n

the

especially

Intrinsic

interesting

Nature for

the

were

not

presented therein by Tait.

the

paper

by

admitting

that

quaternions

applicable to spaces of h i g h e r d i m e n s i o n than three; t h e n he a r g u e d that H a m i l t o n was viations, to

and

yet

quaternions.

of all people most able to dispense w i t h abbrehe

In

had devoted the

the

last t w e n t y years

following quotation

Tait

of his

life

stated his primary

argument: It will be

gathered from

what precedes

that, in my o p i n i o n , the t e r m

Quaternions means one t h i n g to Prof. C a y l e y a n d q u i t e another t h i n g to myself: To

thus

Prof.

Analytical

Cayley

Quaternions

Geometry;

and,

as

are

mainly

such,

a

essentially

Calculus,

a

made

of those

up

species

of co-

ordinates w h i c h he regards as " t h e natural a n d a p p r o p r i a t e basis of the science." T h e y artfully conceal

their humble

origin, by an

admirable

species of p a c k i n g or f o l d i n g : — b u t , to be of a n y use, t h e y . . . doubly dying, must go down To the vile dust from w h i c h they sprung! To

me

Quaternions

are

primarily

a

Mode

of Representation:— im-

m e n s e l y s u p e r i o r t o , b u t e s s e n t i a l l y o f t h e s a m e k i n d o f u s e f u l n e s s as, a d i a g r a m o r a m o d e l . T h e y are, v i r t u a l l y , t h e t h i n g r e p r e s e n t e d : a n d a r e thus a n t e c e d e n t to, a n d i n d e p e n d e n t of, co-ordinates: g i v i n g , in general, all the m a i n relations, i n t h e p r o b l e m t o w h i c h t h e y are a p p l i e d , w i t h o u t t h e n e c e s s i t y o f a p p e a l i n g t o c o - o r d i n a t e s a t all. C o - o r d i n a t e s m a y , h o w ever,

easily

be

read

into

them: —when

anything

(such

as

metrical

or

n u m e r i c a l d e t a i l ) i s t o b e g a i n e d t h e r e b y . Q u a t e r n i o n s , i n a w o r d , exist i n s p a c e , a n d w e h a v e o n l y t o r e c o g n i z e t h e m : — b u t w e h a v e t o invent o r imagine c o - o r d i n a t e s Tait

then

of all

kinds.

(36;

277-278)

to

discuss

the

history

proceeded

of quaternions.

He

first stated that the quaternion of the latter half of the century m u s t be

viewed

"as

having,

from

at

least

one

point

of view, but

little

r e l a t i o n to that of t h e s e v e n last years of t h e e a r l i e r h a l f . " (36; 278) Tait view,

argued and

that

he

Cayley's

suggested

view

that

of the

quaternion

Hamilton's

was

the

earlier

greatest contribution

was

213

A

History

that

of V e c t o r Analysis

through

arose,

as

if

him

by

meaning is

"From

magic,

an

the

most

absolutely

intensely

natural

one!"

artificial (36;

of

279)

systems Tait's

clarified by the following quotation:

M o s t u n f o r t u n a t e l y . . . H a m i l t o n ' s nerve failed h i m in the composit i o n of his first great V o l u m e . H a d he t h e n r e n o u n c e d , for ever, all deali n g s w i t h i , j , /c, h i s t r i u m p h w o u l d h a v e b e e n c o m p l e t e . H e s p a r e d A g a g , and the best of the sheep, and d i d not utterly destroy them! He had a p a t e r n a l f o n d n e s s for i, j, k. . . . He h a d a f u l l y r e c o g n i z e d , a n d p r o v e d to others, that his

i, j, k w e r e m e r e excrescences a n d blots on his im-

proved method:—but he

unfortunately considered that their continued

(if o n l y partial) recognition

was

i n d i s p e n s a b l e to the r e c e p t i o n of his

m e t h o d by a w o r l d steeped in Cartesianism! Through the whole compass o f e a c h o f his t r e m e n d o u s v o l u m e s o n e c a n f i n d traces o f his d e s i r e to a v o i d e v e n an allusion to i, j, k; and, along w i t h t h e m , his sorrowful conviction reader.

.

that, should he .

. And

do

so,

he

should be

left w i t h o u t a single

I f u r t h e r b e l i e v e t h a t , to this cause alone, Q u a t e r n i o n s

o w e t h e scant f a v o u r w i t h w h i c h t h e y h a v e h i t h e r t o b e e n r e g a r d e d . (36; 279-280) Tait own

then

admitted

book,

that

a defect he

c a l l e d for.

the

same

planned

to

defect

remedy

Tait stated that from his

could

be

found

if a fourth

in his

edition was

abundant correspondence with

Hamilton he had learned of Hamilton's changed view of quaternion analysis.

Thus,

criticisms which

in

summary,

Tait's

argument

was

that

Cayley's

simply did not apply to the modern quaternion methods,

avoid

quaternion

use

is

of i, j,

k,

"altogether

and hence

of coordinates, of w h i c h the

independent"

and to w h i c h

it is

"ante-

cedent." Among

Tait's

analogy to

concluding

replace

Cayley's

statements

is

"pocket-map"

the

suggestion

of

an

analogy.

A m u c h m o r e natural a n d adequate comparison w o u l d , it seems to me, liken may

Co-ordinate employ

on

Geometry . . . to

any

a

steam-hammer, w h i c h an expert

d e s t r u c t i v e o r c o n s t r u c t i v e w o r k o f one general kind,

say t h e c r a c k i n g o f a n e g g - s h e l l , o r t h e w e l d i n g o f a n anchor.

But you

m u s t h a v e y o u r e x p e r t t o m a n a g e i t , f o r w i t h o u t h i m i t i s useless. H e has to toil a m i d the heat, s m o k e , g r i m e , grease, a n d p e r p e t u a l d i n of the s u f f o c a t i n g e n g i n e - r o o m . T h e w o r k has t o b e b r o u g h t t o t h e h a m m e r , for it

c a n n o t u s u a l l y be t a k e n to its

hand,

are

like

the

elephant's

w o r k . . . Q u a t e r n i o n s , on the other

trunk,

r e a d y a t any m o m e n t f o r anything,

be it to p i c k up a c r u m b or a field-gun, to strangle a tiger, or to uproot a tree. P o r t a b l e in t h e e x t r e m e , a p p l i c a b l e a n y w h e r e . . . d i r e c t e d by a little native w h o requires no special skill or training, a n d w h o can be transferred

from

one

elephant

to

another

without

much

hesitation.

S u r e l y t h i s , w h i c h a d a p t s i t s e l f t o its w o r k , i s t h e g r a n d e r i n s t r u m e n t ! B u t t h e n , it is t h e natural, t h e o t h e r t h e artificial, one. (36; 283) If Heaviside was the "Walt W h i t m a n of English Physics," the above quotation of English

214

certainly

merited

Physics."

Tait the

title

of the

"Rudyard Kipling

A Struggle for E x i s t e n c e in the 1890's

These

two

papers,

which

conclude

the

debate,

may

serve

to

r e m i n d us that though most of the papers in the debate discussed the

question

o f which v e c t o r i a l

t i o n w h e t h e r any v e c t o r i a l

method

should be

method should be

used, the

ques-

u s e d was h a r d l y for-

gotten.

III.

Conclusions

If the

materials

discussed p r e v i o u s l y in this chapter are v i e w e d

broadly, a n u m b e r of important generalizations emerge. Altogether thirty-six

publications

publications

(thirty-eight

discussed

from

1890-1894.

with

Nature

from

England,

in

Eight

carrying

the

leading

twenty

including

last chapter)

the

two

scientific journals

articles.

Heaviside

appeared in the years

Twelve

were

involved,

scientists,

writing

Scotland, Australia, a n d t h e U n i t e d States, partici-

pated. Nearly all the articles referred to other articles in the debate, so that the A high much with

should be v i e w e d as a definite historical unit.

of intensity a n d a certain

of the

debate

interest.

And

metaphors ternions meat,

debate

level

fierceness

characterized

and must have led m a n y readers to follow it the

penchant

of

the

(at t i m e s b y t h e i r a b s u r d i t y )

participants

for

striking

led t h e m to compare qua-

with, among other things, a Highlander's

archangels, a map, and an elephants

trunk!

musket, Such

strong

analogies

d i d at least lead to readability, t h o u g h o n l y i n d i r e c t l y to m e a n i n g f u l discussion. T h e ratio of heat to light was

especially high

in the writings of

the quaternionists, and it could hardly have been otherwise. Gibbs and

Heaviside

must have

appeared

to

the

quaternionists

as

un-

welcome intruders w h o had burst in upon the developing dialogue b e t w e e n the quaternionists a n d the scientists of the at a m o m e n t w h e n success s e e m e d n o t far distant.

day to arrive

Charging forth,

these t w o vectorists, t h e o n e brash a n d sarcastic, t h e o t h e r s p o u t i n g historical irrelevancies, h a d promised a bright n e w day for any w h o w o u l d accept their overtly pragmatic arguments for an algebraically c r u d e a n d h i g h l y arbitrary system. A n d w o r s t of all, the system t h e y recommended was, "Grassmannian"),

not some n e w system

but

only

(even

a perverted version

if Gibbs called it of the

quaternion

system. Heretics are always m o r e hated t h a n infidels, a n d these t w o heretics had, w i t h wrenched

little

major portions

c l a i m e d that these

parts

u n d e r s t a n d i n g a n d less a c k n o w l e d g m e n t , from

the

Hamiltonian

surpassed the

whole.

system

and

then

So at least it m u s t

have appeared to the quaternionists, and if this description of their reaction is only half correct, still it s h o u l d be sufficient to explain why

there

was

so

little

communication

between

the

contending

215

A

History

parties.

of V e c t o r Analysis

Darwin's

remark

in

the

final

c h a p t e r of his

Origin

o f Species

m a y serve to r e m i n d us that the quaternionists were not unique in their reaction.

Darwin

wrote:

A l t h o u g h I am f u l l y c o n v i n c e d of the truth of the views given in this v o l u m e u n d e r the f o r m of an abstract, I by no means expect to c o n v i n c e experienced naturalists facts

all

viewed,

directly

whose

minds

are s t o c k e d w i t h a m u l t i t u d e o f

d u r i n g a l o n g course of years, f r o m a p o i n t of v i e w

opposite

to

mine. . . . A f e w naturalists,

endowed with much

flexibility of m i n d , and w h o have already begun to doubt the immutab i l i t y of species, m a y be i n f l u e n c e d by this v o l u m e ; b u t I look w i t h confidence to the future, —to y o u n g and rising naturalists, w h o w i l l be able to v i e w b o t h sides of the question w i t h impartiality.57 Though should

all

the

not be

quaternionists

forgotten

that

wrote

with

Knott wrote

some

with

care

bitterness,

it

and thorough-

ness, T a i t w i t h the prestige that c a m e f r o m his d i s t i n g u i s h e d career science,58 a n d M c A u l a y w i t h some success in demonstrating the

in

usefulness haps

too

of quaternions

frequently

favorable

in

Edinburgh

Among as

sive

the

opponents

are

Prophets

articles.

It is

thoughtful

of

of the

perhaps

to

may

However,

per-

for the already

never

be

heard

in

quaternion

system

Gibbs

carefully reasoned,

stands

and persua-

not too m u c h to describe his papers as

mathematical

readers

wrote

s e l d o m h e a r d a n y w h e r e else.

having contributed timely,

masterpieces

application.

and Peddie

audience.

their h o m e l a n d , but patriots

out

physical

Knott, Tait,

accept

rhetoric the

and

as

capable

Gibbs-Heaviside

of leading

system.

Their

effect on t h e less c a r e f u l r e a d e r c e r t a i n l y w a s less great. H e a v i s i d e ' s short polemical

papers

of readers.

greatest contribution

tion

His

of the

vectorial

probably had a certain appeal to both types

effectiveness

methods

noteworthy

too

in

of the

his

important

that it was

in the

were becoming recognized as tists.

was

vectorial

h o w e v e r his approach

electrical

demonstra-

by

his

publications.

use

of

It

is

1890's that G i b b s a n d H e a v i s i d e

extremely important physical

scien-

There was no one in the quaternion camp w h o was to receive

such acclaim except perhaps Tait.

Gibbs and Heaviside wrote only

eight of the papers, but these papers w e r e especially effective since the authors

had the

the

advocated by their opponents,

well In

system

acquainted w i t h his 1890 the

least w i d e l y side

system.

of any

through

There

The

the

opponent's

quaternion

known;

system:

discussed.

216

advantage that they w e r e fully experienced in only

Knott was

system, though

not widely used, was at

such was not the case w i t h the G i b b s - H e a v i are

two

stages

preliminary

it must first become quaternion

first

whereas

system.

system

stage a n d was

known had

well

to

the acceptance

and then

from

into the

1844 to second.

be

tried and

1894 passed The Gibbs-

A Struggle for E x i s t e n c e in the 1890's

Heaviside

system

had

made

roughly

the

same

progress

in

the

p e r i o d f r o m 1881 t o 1894. T h e p u b l i c i t y associated w i t h t h e d e b a t e was very helpful to the Gibbs-Heaviside system but was unnecessary a n d p e r h a p s e v e n d a n g e r o u s for t h e q u a t e r n i o n system. F r o m this point of v i e w the c o n t r i b u t i o n of Macfarlane m a y be evaluated. Since

he

presented

his

system

as

resulting

from

defects

in

the

quaternion system a n d since s o m e of these defects w e r e also n o t e d by Gibbs a n d Heaviside, his five papers m u s t have acted to support the vector analysis

( G i b b s - H e a v i s i d e ) cause.

written

of

on

behalf

written came from support of his

the

full

N o t a great deal was

Grassmannian

system.

What

was

Gibbs, w h o used his discussion of it m a i n l y in

o w n system, and from

B a l l a n d G e n e s e , w h o s e ar-

ticles w e r e e x t r e m e l y brief. It is

interesting that Gibbs, Heaviside, a n d to some extent Mac-

farlane took w h a t m a y be described as a pragmatic approach to the question of w h i c h system was to be preferred.

M a n y of their argu-

ments were on grounds of expressiveness, congruity w i t h physical relationships,

and

ease

of understanding.

The

quaternionists,

on

t h e o t h e r h a n d , p u t s o m e w h a t greater stress o n m a t h e m a t i c a l elegance and algebraic simplicity.

Some quaternion advocates looked

askance at their opponents' use of a n u m b e r of products rather than the single quaternion product. T h a t history had in this regard c o m e f u l l circle has p r e v i o u s l y b e e n m e n t i o n e d . There

was

probably of

too

(which

was

natural

however

British

mathematicians

mathematics

little for

had

much

direct two

far

stress

relevance)

reasons.

of the

on in

It was

the the

the

notation debate.

common

question This

was

opinion

of

nineteenth century that continental

surpassed

English

mathematics

by

1800

in

large part because of the superiority of the L e i b n i z i a n notation for calculus as compared to the N e w t o n i a n notation.

Hence it w o u l d

b e n a t u r a l for t h e m t o p l a c e a n u n d u e stress o n n o t a t i o n . A s e c o n d and

more

important

when Gibbs dissociate way to

their

do

question

reason

is

historically

rooted in

the

fact that

and Heaviside created their systems, they w i s h e d to systems

this

was

was

from

through

further

the

quaternion

changes

complicated

in

by

parent.

symbolism.

the

fact

O n e natural T h e notation

that

Grassmann's

symbolism bore no relation to that of H a m i l t o n , Gibbs, or Heaviside.

Macfarlane

too

tion.

Thus

were

there

had suggested numerous,

many

changes

d i s t i n c t sets

on

in

symboliza-

notation

to be

considered, and since the symbols themselves became symbolic of the differences b e t w e e n the systems, the w h o l e question was hotly debated. The

notation

problem

reached

major

proportions

in

the first

217

A H i s t o r y of V e c t o r Analysis

decade

of

which

the

twentieth

resulted

in

in failure.59 It is

century;

committees

were

organized,

part in suggestions for n e w symbols but mainly

surprising that only t w o papers in the debate di-

rectly discussed the fundamental

q u e s t i o n o f w h e t h e r any v e c t o r i a l

system should be adopted, especially since vectorial systems were at that time

still

participant

to

something of an innovation. Cayley was the only

argue

that

no

vector

system

should

be

adopted,

w h e r e a s a large n u m b e r of t h e readers of the debate m u s t , at least initially,

have

been

of that persuasion.

Cayley is

also

unique

in

that he was the only mathematician w h o contributed a major article to the debate. who

were

T h e vast majority of the participants w e r e physicists

interested

primarily

in

the

applications

of vectorial

methods to physics. Some

results

Shunkichi Nature

o f this, d e b a t e

Kimura

addressed 60

nions. " entitled

and

to

now

be

Molenbroek

"Friends

and

considered. published

Fellow

Workers

In

1895

a notice in

in

Quater-

S i m u l t a n e o u s l y t h e y p u b l i s h e d a s i m i l a r n o t i c e i n Science

"To Those

of M a t h e m a t i c s . " 6 1 Yale,

may

Pieter

and the

Interested in Quaternions and A l l i e d Systems Kimura,

Dutchman

a Japanese

Molenbroek

scientist then

suggested in

residing at

these

notices

that an association of those interested in various systems of vector analysis

s h o u l d be formed. T h u s c a m e about the International As-

sociation tems

for P r o m o t i n g the

of Mathematics.

of the

International

pearing in and

Ball

In

Study of Quaternions

March,

Association

1900, the

a n d A l l i e d Sys-

f i r s t i s s u e o f t h e Bulletin

w a s p u b l i s h e d , t h e last issue ap-

1913. T a i t w a s e l e c t e d t h e first p r e s i d e n t , b u t d e c l i n e d , was

elected

with

Macfarlane as

general

secretary.

The

m e m b e r s h i p as l i s t e d in t h e first issue i n c l u d e d o v e r sixty scientists from

fifteen

countries.

One

major

result

of the

Association,

pri-

m a r i l y a t t r i b u t a b l e t o M a c f a r l a n e , w h o b e c a m e its real l e a d e r , w a s the

publication

1913)

of

Mathematics to

roughly

a which

in

1904

(with

Bibliography contained

twenty-five

supplements

of

Quaternions (with

hundred

the

articles

in and

the Bulletins until Allied

Systems

supplements)

of

references

in the vectorial tradition.

T h i s association, w h i c h m u s t have advanced the vector cause considerably, c a m e about in large part as a result of the "struggle for existence" of the early

1890,s.62

A n o t h e r major result of this

series of articles of the early 1890's

seems to have been a lessening of Hamilton's reputation. Tait and the

other

quaternionists

had,

as

noted previously,

frequently in-

voked Hamilton's name and reputation on behalf of the quaternion cause. T h u s G i b b s a n d H e a v i s i d e , w h o w e r e o f course proponents of the system that e v e n t u a l l y t r i u m p h e d , t e n d e d to take their stance

218

A Struggle for E x i s t e n c e in the 1890's

against H a m i l t o n . T h e result was that the failure of the q u a t e r n i o n cause,

which

acted

to

had become

diminish

and probably Gibbs

so

closely

Hamilton's

linked to

historical

need not have been

the

Hamilton's name,

stature.

case.

This

It is

and Heaviside had presented their system

should

possible as

not

that if

a direct an-

cestor of H a m i l t o n ' s creation, if they h a d t r i e d to capitalize on the successes

of this tradition and h a d v i e w e d themselves as

moving

w i t h i n i t rather t h a n a s b r e a k i n g f r o m it, t h e n e t r e s u l t o f t h e e v e n tual t r i u m p h of their cause w o u l d have b e e n an increase in H a m i l ton's

reputation.

Macfarlane,

it

presenting his

should

system.

contending parties

to

be

recalled,

had

But intellectual extremes,

and

taken

debates

hence

this

Gibbs

have b e e n v i e w e d as the great revolutionaries;

position

tend to and

in

push

the

Heaviside

it should be clear

h o w e v e r f r o m w h a t has b e e n w r i t t e n that s u c h a v i e w does serious injustice to H a m i l t o n and even to Tait. indeed the

Gibbs and Heaviside were

b u t their cause was 90 percent in h a r m o n y w i t h

Hamilton-Tait orthodoxy.

lectual this

heretics,

ancestors,

Hamilton

and Tait were

their intel-

but this genealogy was obscured in the heat of

debate.

Finally, it should be noted that in the course of the debate there was m u c h speculation as to w h y the quaternion system h a d b e e n as yet poorly received. the relevance

In the light of this discussion a n d because of

of this

question

to the general t h e m e of this

study

some attempts to explain this p h e n o m e n o n w i l l n o w be made. It is first of all important to note that the quaternionists painted t h e p i c t u r e far too b l e a k l y .

By

1890 considerable recognition

had

been given to the quaternion system; as it was pointed out earlier, twenty-seven quaternion books and over four h u n d r e d articles h a d been published by

1890, a n d over half of these books a n d r o u g h l y

a t h i r d of the articles w e r e p u b l i s h e d in the

1880's.

Moreover, to

consider the question from a comparative point of view, there were few

new

teenth

mathematical

century

or

that were

physical rapidly

systems

presented

accepted.

in

the

Appropriate

nine-

compari-

sons m i g h t b e n o n - E u c l i d e a n g e o m e t r y , g r o u p t h e o r y , o r M a x w e l l ' s theory.

The

quaternion

system

represented

a real innovation;

its

f u n d a m e n t a l l a w s b r o k e w i t h l o n g - e s t a b l i s h e d t r a d i t i o n s ; its d i s c o v ery was a key d e v e l o p m e n t in the birth of m o d e r n algebra; applied

system

its

only

traditional

reference

points

were

as an such

things as the parallelogram of forces or velocities. O t h e r reasons for the

s l o w acceptance are that H a m i l t o n ' s style

was unsuited to an introductory exposition and that mathematically anything that could be

done by the

application of quaternions

in

219

A H i s t o r y of V e c t o r Analysis

g e o m e t r y a n d physics c o u l d also b e d o n e w i t h the Cartesian methods,

though

thus

were

usually not

by

longer

absolutely

processes.

considered

a

The

sine

vectorial

qua

non

for

methods progress.

A l s o r e l e v a n t is t h e fact that f e w i m p o r t a n t p h y s i c a l discoveries h a d been made by quaternion methods; indeed the physicist of the first two-thirds than

his

of the

century

h a d far less

n e e d for vectorial methods

c o u n t e r p a r t of t h e last t h i r d .

It also took t i m e to p u t the

quaternion system into the most fruitful form for application to the needs

of the

physicist.

Systems

of vectorial

physical

developments

velopments; 1840

to

algebra

This

was

analysis but

the

great

progressed

also

changed

in

relation

greatly

and

achievement of Tait.

not

only to

in

relation

mathematical

matured

rapidly

to de-

from

1890 to allow a perspective w i t h i n w h i c h the quaternion

system might be judged. There were of course innate mathematical difficulties

within the

quaternion

s y s t e m t h a t h i n d e r e d its accept-

ance, for e x a m p l e , t h e scalar p r o d u c t was negative. F i n a l l y , up to 1890

there

had

been

no

widespread

published

discussion

of the

merits of the q u a t e r n i o n system. T h e r e are certainly other reasons, b u t these are a m o n g the most important. The

q u e s t i o n of the success of the q u a t e r n i o n system s h o u l d be

v i e w e d as s u b s e r v i e n t to a n o t h e r question. T h i s is the success of the

vectorial

approach

in

general.

If this question is emphasized

(as G i b b s s t r e s s e d t h a t i t s h o u l d b e ) , t h e n t h e s i t u a t i o n a p p e a r s f a r f r o m b l e a k . T h e v e c t o r i a l a p p r o a c h i n its m a n y f o r m s w a s b e c o m i n g evermore common. T h e challenge facing Gibbs and Heaviside was by no means the challenge that had faced H a m i l t o n a half century earlier. T h e G i b b s - H e a v i s i d e system came into existence in an entirely which

different climate from that of the had

earlier

longer present.

acted

against

the

1840,s.

M a n y of the forces

quaternion

system

were

no

For example, the maturation of algebra supplied a

p e r s p e c t i v e t h a t m a d e t h e n e w s y s t e m s e e m far less r e v o l u t i o n a r y . M o r e o v e r , t h e r e was a l o n g t r a d i t i o n by t h e n of w o r k in vector analysis. Physics h a d c h a n g e d ; the physicist was daily forced to deal w i t h vectorial entities. T h e G i b b s - H e a v i s i d e vector analysis system was c l o s e l y a s s o c i a t e d w i t h i m p o r t a n t n e w p h y s i c a l i d e a s i n H e a v i side's the

writings.

If one w i s h e d to read Heaviside, one had to learn

language of vector analysis.

of the

Finally, the mathematical defects

quaternion system as an a p p l i e d system w e r e in part elimi-

nated in the n e w system, a n d the merits of systems of vector analysis h a d b e e n w i d e l y d i s c u s s e d . T h e times were thus in many ways disposed to the acceptance of s o m e v e c t o r s y s t e m ; t h a t s y s t e m w a s t o b e t h e G i b b s - H e a v i s i d e system; that acceptance w i l l be discussed in the next chapter.

220

Notes 1

Peter

sophical 2

Guthrie

Magazine,

Peter

Tait,

"On

5th

Ser.,

Guthrie

Tait,

3

4

Peter

Nature, 5

43

(April

Guthrie

43

(April

Josiah

Peter

7

1891),

Alexander

search" 8

in

9

An

the

Role

Role

of Quaternions

"Quaternions

Tait,

"Quaternions

and

and

"Quaternions

Magazine,

Macfarlane,

5th

as

in

in

the

the

"Principles

for

the

a

Ser.,

the

the

Quaternions,

3rd

Algebra of Vectors"

Algebra

of Vectors"

'Ausdehnungslehre'"

Ausdehnungslehre"

practical

33

Instrument

(June,

of t h e

1892),

Algebra

Advancement

i n Nature, 4 7 ( N o v e m b e r 3 ,

in

in

Nature,

in

Nature,

44

of

of

of Physical

Physics"

Science,

Re-

477-495.

40

in

Proceedings

(1891,

of

published

1892), 3 - 5 .

Note

that the

author of t h e re-

work.

O l i v e r H e a v i s i d e , " O n the Forces, Stresses, a n d F l u x e s of E n e r g y in t h e E l e c Field"

183 A ( 1 8 9 2 , r e a d i n Heaviside, the

Electrical

Phil.

in

Philosophical

Transactions

1891, p u b l i s h e d in

Papers,

Trans,

2

(London,

1892),

Alexander

McAulay,

"Quaternions"

Alexander

McAulay,

"On

Transactions

1892, p u b l i s h e d in 13

Alexander

14

Alexander

Peter

16

Peter

17

Josiah

in

Physical

Willard

Oliver

521-574.

in

Nature,

Mathematical

Royal

Utility

Society

of

of

Quaternions

"[Review

Review,

Tait,

Nature, 4 9

Guthrie Tait,

( M a r c h 16,

the

the

Macfarlane, in

Royal

Society

of

London,

References

will

be

given

47

(December

Theory London,

15,

1892),

151.

of Electromagnetism" 183

A

(1892,

read

in in

1893), 6 8 5 - 7 7 9 .

Guthrie

McAulay"

of

McAulay,

McAulay"

the

publication.

1 2

Philosophical

of

1893), 4 2 3 - 4 8 4 . P u b l i s h e d earlier in O l i v e r

11

18

on

Lodge], " [ R e v i e w of] 'Principles of the A l g e b r a of Vectors' By A l e x a n d e r

tromagnetic

15

Treatise

of Quaternions

v i e w used an incorrect title for Macfarlane's

A.

Elementary

105-106.

Association

Macfarlane"

for

Philo-

65-117.

[Alfred

1 0

in

608.

Gibbs,

McAulay,

American

1892),

Physics"

511-513.

"The

1891),

Philosophical

Alexander

the

in

in

84-97.

1891), 7 9 - 8 2 .

Guthrie

(June 4,

"On

1891),

Tait,

30,

Willard

44 ( M a y 28, 6

2,

of Quaternions

1890),

1890), v-viii.

Josiah W i l l a r d Gibbs, Nature,

Importance

(January,

"Preface"

ed. ( C a m b r i d g e , E n g l a n d ,

in

the

29

1

(1893),

"[Review

(December

28,

of]

"Quaternions

Physics Utility

of

(London, Quaternions

1893). in

Physics.

By

387-390. Utility

1893),

"Vector Analysis"

Gibbs,

in

of]

in and

of

Quaternions

in

Physics.

By

A.

193-194. Nature, 4 7 ( J a n u a r y 5 , the

1893), 2 2 5 - 2 2 6 .

A l g e b r a of V e c t o r s "

in

Nature, 4 7

1893), 4 6 3 - 4 6 4 . Heaviside,

"Vectors

versus

Quaternions"

in

Nature,

47

(April

6,

1893),

533-534. 19

of

Cargill

the

1893), 20

Royal

Society

Knott,

"Recent

o f Edinburgh,

19

Innovations

(1892,

read

in

Vector

December

Theory" 19,

1892,

in

Proceedings published

212-237.

Cargill

Nature,

Gilston

47

Gilston

(April

20,

Knott, 1893),

"Recent Innovations

in Vector T h e o r y " (An Abstract)

in

590-593.

221

A H i s t o r y of V e c t o r Analysis 21

Alexander

1893), 22

Macfarlane,

"Vector

versus

Quaternions"

in

Nature,

48

(May

25,

75-76.

Cargill

Gilston

Knott,

"Vectors

and Quaternions"

i n Nature, 4 8 ( J u n e

15,

1893),

148-149. 23

Alfred

Lodge,

"Vectors

Josiah

Willard

Gibbs,

and

Quaternions"

in

Nature,

48

(June

29,

1893),

198-

199. 24

17,

"Quaternions

and Vector Analysis"

i n Nature, 4 8

(August

1893), 3 6 4 - 3 6 7 .

25

Robert

1893), 26

S.

Ball,

Robert

William

(September 28, 27

Cargill

1893), 28

Gilston

Nature,

48

(August

'Ausdehnungslehre'"

and Vectors"

and

"Quaternionic

Knott,

Peddie,

24,

in

Nature,

48

in

Nature, 4 8

(September 28,

Quaternions"

in

Nature,

48

(October

5,

Innovations"

in

Nature,

49

(January

11,

Alexander

in

"The

M.

of

the

Elements

its

Depredators"

January

Principles

Edinburgh

13,

in

Mathematical

Proceedings

1893),

of Quaternions

2

of]

(1894),

Society,

62-80.

and other 11

(1893,

(Abstract)

April

Electromagnetic

of]

Ser.,

(1893-1894

read

14,

Theory.

in

Proceedings

1893), Vol.

I.

of

130-136. By

Oliver

152-154.

"[Review 5th

"Coordinates

20

of Q u a t e r n i o n s "

(1893,

"[Review

Magazine,

Cayley,

11

Review,

Minchin,

o f Edinburgh,

read

(1893,

Fundamental

Society,

Physical

Philosophical

Arthur

Society

and

Proceedings

Macfarlane,

in

George

the

Quaternion 11

1893), 8 4 - 9 2 .

Mathematical

Heaviside"

side"

in

10,

Peddie,

Edinburgh

"The Society,

"On

Analyses"

William

35

"Vectors

Mathematical

read February

34

"Grassmann's

"Quaternions

Macfarlane,

Edinburgh

Vector

33

Knott,

Heaviside,

William

32

in

246.

Cargill

the

Quaternions"

540-541.

the 3 1

Genese,

Gilston

Oliver

30

on

1893), 517.

Alexander

1894),

of

Discussion

516-517.

1893), 29

"The

391.

38

Electromagnetic (July,

versus session,

Theory.

1894),

By

Quaternions"

in

read

1894,

July

2,

Oliver

Heavi-

146-156. Proceedings

o f the

published

Royal 1895),

271-275. 3 6

Peter

Guthrie

Proceedings 2,

of

the

Tait,

"On

Royal

Society

the

Intrinsic

of

Nature

Edinburgh,

20

of the

Quaternion

(1893-1894

Method"

session,

read

in

July

1894, p u b l i s h e d 1895), 2 7 6 - 2 8 4 . 37

Lynde

38

E.

tin,

T.

20 39

It is

40

W h e e l e r , Josiah

Whittaker,

(1930),

The

(New in

Haven, Calcutta

1962),

115.

Mathematical

Society

Bulle-

force.

For obvious reasons

such articles have not

reader m a y w i s h to consult the recent article by Alfred M. Bork, " 'Vectors Quaternions' —The

manuscript;

I

In

March,

had written

pleased to

find

odological

difference

discussion

to

(2;

vi).

the

Tait

wrong.

Letters 1965,

in

Nature"

in

Professor Bork

my study of the

American

Journal

of

Physics,

34

generously sent me a c o p y of his

debate

in

the s u m m e r of 1964 a n d was

that we had reached m a n y of the same conclusions. T h e chief methbetween

articles

either

neither of them;

pletely

222

Gibbs

necessary to m e n t i o n that an important article of only mathematical

a definite polemical

(1966), 202-211.

41

Willard

Heaviside"

included.

Versus

knew

"Oliver

205.

hardly

c o n t e n t has been

Phelps

did this

in

the

not is

two

studies

is

that

Professor Bork

limited

his

Nature. know

Gibbs'

evident

since

notation his

or Grassmann's

notation

or

statement about notation is com-

A Struggle for E x i s t e n c e in the 1890's 42

A.

Macfarlane,

second edition 4 3

It

may be

of Algebra" of

1892,

ciety,

Vector

Analysis

and

Quaternions

(New

York,

1906),

50

pp.

No

of the book appeared. noted that

at the

which

volume

Macfarlane

delivered another paper,

American Association

was 41,

published

33-55.

(in

This

for the

December,

paper

does

"On

the

Imaginary

Advancement of Science meeting 1892)

not

in

require

the

Proceedings

further

of that

discussion,

so-

since

it is essentially an extension of the earlier paper. 44

The

is

review

that

of

was

not

Macfarlane

Mathematics

(Dublin,

the better-known 45

(13; v).

46

1

do

signed.

did

The

so

1904),

evidence

in 51.

his

for attributing

Bibliography

Alfred

of

Lodge

it

Quaternions

should

not

to

Alfred

Lodge

and

Allied

Systems

be

confused

with

scientist Oliver Lodge.

It seems q u i t e clear that M c A u l a y ' s essay d i d not w i n the prize.

not believe

it

is

necessary to

marizing review written by A. S.

discuss

a lengthy,

favorable,

mainly

sum-

Hathaway, w h o taught at Rose Polytechnic Insti-

tute in Terre Haute, Indiana, and w h o was a quaternion advocate a n d author of an 1896

Primer

Bulletin

of

47

(15;

of

Quaternions

the

(New

American

193).

York).

Mathematical

Heaviside

Hathaway's

Association,

never attained

1st

review Ser.,

the doctoral

appeared

3

(1893),

degree

(unless

t h e n certainly after 1893) since he n e v e r a t t e n d e d a university. Tait w o u l d k n o w this.

in

the

179-185.

honorary and

It w o u l d seem that

It is doubtful then that Tait's b e s t o w i n g of a " D r . " on Heavi-

side was a simple slip of the pen. 48

F r o m a letter dated August 6,

preserved at Yale 49

1894, in the "Scientific C o r r e s p o n d e n c e " of G i b b s

University.

T h e first abstract w a s part of a r e p o r t on t h e E d i n b u r g h

of D e c e m b e r 50

and

This

19,

1892.

information

Scientific

Works

S e e Nature, 4 7 ( J a n u a r y

has

of

been

Peter

obtained

Guthrie

Tait

19,

mainly

Royal

Society meeting

1893), 287. from

(Cambridge,

Cargill

Gilston

England,

1911);

Knott, see

Life

espe-

cially the title page. 51

J.

W.

Gibbs,

The

Scientific

Papers

ofj.

Willard

Gibbs,

vol.

II

(New

York,

1961),

44-50. 52

(22;

53

William

149).

Mathematical 54

Society,

Cargill

Life

of

Knott's

Gilston

Tait,

is

from

"Elements

11

of

(1893),

Knott

Milton's

55

Ibid.,

159.

Ibid.,

164-165.

57

Charles

5 8

The

Darwin,

in

Lost,

I,

591-593.

Proceedings

of

the

Edinburgh

104.

published

Origin

vector versus

part.

selections

from

o f Species,

quaternion

Throughout

his

life

6th

ed.

dispute

he

(New

was

this

correspondence

in

his

most famous

regard frequently been accused

that

Tait

had

argued

1963),

only

444.

dispute

in

which

Tait

manifested a tendency to become e m b r o i l e d in

a n d h e has

in this

York,

not the

controversies. T h e

tioned

Paradise

Quaternions"

154-166.

56

took

quotation

Peddie,

of these concerned the history of thermodynamics,

in

his

1880

of chauvinism.

article

for

the

It may be men-

Encyclopaedia

Britannica,

entitled " H a m i l t o n , " that H a m i l t o n was actually of Scotch ancestry. 59

See

on

this

point

Standpoint:

Arithmetic,

A

o f Mathematical

History 6 0

Shunkichi

Quaternions" 6 1

Algebra,

Kimura in

Nature,

Felix

Analysis

Notations,

Klein,

(New vol.

and

Pieter

52

(1895),

York, II

Elementary n.

d.),

(Chicago,

Molenbroek,

Mathematics 65, 1952),

"Friends

from

and

an

Advanced

Florian

Cajori,

136-139.

and

Fellow

Workers

in

545-546.

Shunkichi K i m u r a and Pieter Molenbroek, " T o Those Interested in Quaternions

and Allied

Systems

of M a t h e m a t i c s "

in

Science, 2 n d

Ser., 2

(1895),

524-525.

223

A H i s t o r y of V e c t o r Analysis 62

Another

of a prize ods

event,

almost

certainly

engendered

by

this

debate,

was

the

offering

1894 by the " D u t c h Society of Sciences" for a comparison of the meth-

of Grassmann,

physical sche

in

science.

Hamilton,

This

and Cauchy with

information

Ausdehnungslehre"

in

is

given

Zeitschrift

fur

emphasis

on their applicability in

in Victor Schlegel, Mathematik

und

" D i e Grassmann'-

Physik,

41

(1896),

53.

I have not been able to determine to w h o m or even whether the prize was awarded.

224

CHAPTER

The

Emergence System

SEVEN

of the

of

Modern

Vector

Analysis:

1894-1910

I.

Introduction I n t h e last c h a p t e r t h e d e b a t e o n vectorial m e t h o d s that o c c u r r e d

in the period 1890 to 1894 was discussed, and it was suggested that as

a

result

of this

widely known.

debate

But by

the

1894

Gibbs-Heaviside

this

system

system

became

still h a d not b e e n w i d e l y

accepted by the scientific community. T h e present chapter w i l l deal with

the

developments

vectorial

analysis

from

1894 to

came to be

1910.

It w i l l be argued that

accepted widely

d u r i n g this period;

that the form of vectorial analysis that came to be accepted was that in the tradition side;

and

established by

Hamilton, Tait,

that

the

most influential

stemmed from

the

association

theory,

an

association

to

be

force

of vectorial credited to

Gibbs,

and Heavi-

in producing acceptance analysis Maxwell

T h e evidence employed to support the

with and

electrical Heaviside.

statement that vector an-

alysis c a m e to be a c c e p t e d w i d e l y d u r i n g this p e r i o d consists in the establishment of the cations

14

fact that a substantial

presenting

the

now

were published at this time.

common

n u m b e r of major publi-

system

of vector analysis

I t has b e e n a s s u m e d that t h e p u b l i c a -

tion of such a w o r k indicates a belief on the part of both author and publisher that

such

works

would be

w i d e l y read. W h e n later edi-

tions of a b o o k w e r e c a l l e d for, this is t a k e n as

evidence that the

b o o k was in fact w i d e l y read. To s u p p o r t the c o n c l u s i o n that the ancestry

of the

majority

Hamilton, Tait, the

works

has

of these

Maxwell, been

publications

Gibbs,

analyzed

in

and

extends

back

Heaviside tradition,

terms

o f its

origin

and

to

the

each

of

content.

225

A

History

of V e c t o r Analysis

M u c h i n f o r m a t i o n c o n c e r n i n g the origins o f the w o r k s has b e e n derived also

from from

There period

biographical

reviews were

1894

discussed

perhaps

to

or

information

concerning

the

authors

and

of the books.

1910

1000 journal

in

which

employed.

articles

published

during the

some form of vectorial analysis was

These

have

not been

analyzed since the

e v i d e n c e d e r i v e d f r o m the major p u b l i c a t i o n s has s e e m e d sufficient unto

the

propounded.15

conclusions

In a n u m b e r of cases t h e analysis of t w e l v e m a j o r p u b l i c a t i o n s has served

to

reveal

important

lines

of

development,

and

there

has

b e e n no hesitation to discuss these b r o a d e r aspects. In the conclusion to this chapter the fate of the

quaternionic and the Grassman-

nian traditions is briefly treated.

II.

Twelve 1894

to

August Elektricitat1

Major

Foppl's of

1894

vector analysis was

one

Publications

in

Vector

Einfuhrung

in

die

is

an

but

of the

important

also

first

in

stated:

from

book

Maxwellsche not

only

in

history of electricity. in

German

Maxwell's

ideas.

Theorie the

der

history

Foppl's

of electricity

of

book

as

pre-

In his foreword F o p p l

" T h e circle of ardent followers of Maxwell's electrical theo-

consisted

cists.

the

expositions

sented in accordance with

ries

Analysis

1910

until

recently

almost exclusively of English

physi-

Earlier considerable attention had b e e n accorded this theory

in G e r m a n y , b u t scientists w e r e still too biased by the ban on action a t a d i s t a n c e t o b e a b l e t o b e c o m e f u l l y a c c u s t o m e d t o i t . " (1; Foppl

proceeded

Heinrich an

Hertz

in

to

cite

1887

the

famous

experiment

performed

III) by

to verify M ax well's statement that light is

electromagnetic wave, as the turning point in G e r m a n hostility

to

Maxwell.

tire

system

After Hertz's

led h i m to write his book. Foppl

e x p e r i m e n t the interest in M a x w e l l ' s en-

b e c a m e widespread, and F o p p l stated that this interest

w e n t on:

(1;

III—IV)

" I n the mathematical formulation of the theories

discussed I have throughout m a d e use of the symbols and methods of v e c t o r c a l c u l u s ; t h e s e are d i s c u s s e d in t h e first c h a p t e r to t h e extent that they w i l l be used. T h e m a n n e r of presentation is very simple and, as

I dare to assume u n c o n d i t i o n a l l y , also v e r y easy to un-

derstand."

(1;

analysis, ods

Foppl

in his

V-VI)

After

discussing

the

advantages

of

vector

revealed h o w he came to introduce vectorial meth-

book.

In the presentation of vector techniques and on many other points, I f o l l o w e d t h e p a t t e r n set b y O . H e a v i s i d e i n his p a p e r s , w h i c h h a v e re-

226

Emergence of the M o d e r n

System of Vector Analysis:

1894-1910

cently been collected into book form. T h e works of this author have in general influenced my presentation more than those of any other physicist w i t h t h e o b v i o u s e x c e p t i o n of M a x w e l l h i m s e l f . I c o n s i d e r H e a v i s i d e to be the most e m i n e n t successor to M a x w e l l in regard to theoretical dev e l o p m e n t s . . . . (1; V I I ) Among

Foppl's

arguments

for

Germany was the following:

the

adoption

of vector

analysis

in

" T h e country w h i c h p r o d u c e d a Grass-

m a n n s h o u l d n o longer stand b e h i n d the c o u n t r y o f H a m i l t o n i n regard

to

the

introduction

of these

important

improvements

in

the

m a t h e m a t i c a l aids t o t h e o r e t i c a l p h y s i c s . " (1; V I I ) Thus

Foppl

d e v o t e d his first t h r e e chapters (84 pages) to an ex-

planation of vector analysis

and employed the symbolisms of Max-

well and especially Heaviside the

Gibbsian

book was

i n d o i n g so. T h e G r a s s m a n n i a n a n d not used.16 T h u s F o p p l ' s

systems of symbolism were

the first written in G e r m a n to present a detailed exposi-

tion of the m o d e r n system of vector analysis, and it was a very popular book:

a

second edition appeared in

ham,17 and a t h i r d edition in Foppl's These Die

book

additional

additions

were

Geometrie

der

of the

more

some linear

vector

theory. The

1907.

subjects

based

Wirbelfelder,

1904, edited by M a x Abra-

In the in

on Foppl's in

1904 A b r a h a m edition of

vector

analysis

were

which

Foppl

had

a d v a n c e d parts of vector analysis,

function

and

Here as before

the

vectorial

presented

including the

treatment

of

potential

F o p p l closely followed Heaviside.

importance of Foppl's

ment made in the

treated.

short publication of 1897,

w o r k w a s w e l l s u m m e d u p i n a state-

1910's by Felix K l e i n , w h o was a c o n t e m p o r a r y to

Foppl and to the developments at that time.

Klein wrote:

H e a v i s i d e i s also l i n k e d w i t h t h e f i r s t i n d e p e n d e n t p r e s e n t a t i o n w h i c h v e c t o r a n a l y s i s has f o u n d i n G e r m a n y . T h i s i s A . F o p p l ' s " G e o m e t r i e d e r Wirbelfelder" Foppl's

(1897),

which

"Einleitung in

two publications

there

die

was

a completion of the presentation in

M a x w e l l s c h e T h e o r i e " (1894).

F r o m these

later arose t h e t w o - v o l u m e " T h e o r i e der E l e k -

trizitat" w h i c h was w o r k e d on by A b r a h a m and w h i c h is n o w one of the most frequently used textbooks in electricity.18 Foppl uber

also

technische

published Mechanik

a

four-volume

(1897-1900)

in

work which

entitled he

Vorlesungen

used

vector

analysis extensively. T h i s publication m u s t have b e e n very successful, since a second edition was i m m e d i a t e l y called for a n d appeared from

1900-1903.19

A book similar to Foppl's 1894 publication was p u b l i s h e d in 1899 by

the

tecnica,20

Italian and

presentation

Galileo

Ferraris,

like

Ferraris

under

Foppl,

followed

the

title

Lezioni

Heaviside

in

di

Elettro-

both

his

of electricity and of vector analysis.

227

A H i s t o r y of V e c t o r Analysis

T h u s the first extensive published presentations of modern vector analysis to appear in England, Germany, and Italy were included in books

on

electricity

One

was

written

from

him.

Edwin Book upon

Bidwell

for

the

the

lished

Lectures

Wilson's of

of J.

that was

book

Students

Willard

Maxwellian

other two

of

of

Gibbs,2

entirely

a

the

1901,

Vector

Analysis:

and

Physics

Mathematics was

the

devoted to

point of view.

stemmed

first

directly

A

formally

presenting the

Text

Founded pub-

modern

of vector analysis.21

Wilson's best

from

Heaviside;

Use

book

system

presented

by

textbook

books

to

be

on

vector analysis was one of the longest and

published on

that subject.

The

story of h o w it

came to be written is directly relevant to the present study, since it reinforces reached

one

of the

concerning

more

surprising

and

important conclusions

the m a n n e r i n w h i c h m o d e r n vector analysis

came to be developed and widely applied. W i l s o n t o l d t h e s t o r y i n f u l l e s t d e t a i l i n a p u b l i c a t i o n o f 1931.13 As W i l s o n was c o m p l e t i n g his undergraduate w o r k in mathematics at H a r v a r d in 1899, the m a t h e m a t i c i a n W. F. O s g o o d suggested to h i m that he do graduate w o r k at Yale and study w i t h Pierpont and Percy Smith. Of his teachers only B. O. Peirce m e n t i o n e d Gibbs by referring to h i m as someone " w h o m some of us here think a rather able

fellow."

Pierpont

(13;

and

211)

Smith.

Thus

At

his

Wilson first

w e n t to Yale to

registration

study with

for courses

Wilson

p l a n n e d to register for o n l y three courses u n t i l D e a n A. W. Phillips required that he take a fourth and suggested the course in vector analysis taught by Gibbs. To this W i l s o n objected that he had had a full year course in quaternions at Harvard from J. M. Peirce and that c o n s e q u e n t l y t h e c o u r s e w o u l d in large part serve o n l y as a review. Somewhat unhappily Wilson followed the Dean's advice and j o i n e d the h a n d f u l of others w h o h a d registered for the course. W i l son f o u n d t h e c o u r s e easy, a n d soon after h e h a d c o m p l e t e d it, h e was

asked

by

Professor

Morris,

editor

of the

Yale

Bicentennial

Series, to p r e p a r e a b o o k on v e c t o r analysis for that series. W i l s o n m e t w i t h G i b b s , w h o was e n g a g e d in p r e p a r i n g a w o r k on statistical mechanics and thus " w o u l d not have time to advise on the composition of 'Vector Analysis,' to read the manuscript or the proof. . . . " (13; 214) W i l s o n stated that his contact w i t h G i b b s c o n c e r n i n g the book was

essentially l i m i t e d to that meeting.

Nevertheless Gibbs

later m e n t i o n e d t o W i l s o n that h e f o u n d t h e b o o k satisfactory. (13; 214) It is interesting to note that Gibbs and W i l s o n corresponded in 1903

228

on

the

question

of whether Wilson

should

write

a shorter

Emergence of the

Modern

System

of Vector Analysis:

treatise or an a b r i d g m e n t of the vector analysis book. However no

such book came

(13; 2 1 7 - 2 1 8 )

into print.

That the primary source of Wilson's book was established by the

1894-1910

following quotation

Gibbs'

lectures

is

from Wilson's general pref-

ace. B y far t h e greater part o f t h e m a t e r i a l u s e d i n t h e f o l l o w i n g pages has b e e n t a k e n f r o m t h e course o f lectures o n V e c t o r A n a l y s i s d e l i v e r e d annually at the

U n i v e r s i t y by Professor Gibbs.

S o m e use, h o w e v e r , has

been made of the chapters on Vector Analysis in M r . tromagnetic

Theory

lectures

on

(Electrician Die

Series,

MaxwelVsche

1893)

and

der

Electricitat

Theorie

in

H e a v i s i d e ' s ElecProfessor

Foppl's

(Teubner,

1894).

M y p r e v i o u s s t u d y o f Q u a t e r n i o n s has also b e e n o f g r e a t assistance. (2; ix) The

book

was

v e r y clear a n d t h o r o u g h i n its p r e s e n t a t i o n ;

it in-

cluded, w i t h a b u n d a n t explanation, nearly all the material in G i b b s ' earlier work. ple,

an

velopments ceived a

Society,22

Ziwet

one

Victor

barycentric calculus the

number of very

by

referred

to

in

the

the

favorable

in book

sense

review

thor was

C.

G.

the

and some

function.

reviews, of

one the

reviewer pages,23

and

Fortschritte

der

Mathematik

in

"essentially

a

of nine

the

ten

as

in

the

book

a

Mathematical des

shorter

which

working

out

Philosophical

re-

pages by

Bulletin

sci-

one

by

Schlegel

of quaternion

A u s d e h n u n g s l e h r e . . . ,"

appeared

further de-

The

American

in

to

ran

of t h e

Knott;

vector

Bulletin

anonymous

which

Schlegel

linear

favorable

in

an

mathematiques

theory

quantity of n e w material was added, for examto

concerning

Alexander

ences

A small

introduction

24

A

Magazine.

very

un-

The

au-

his criticism was that the b o o k was not qua-

ternionic.25 T h e success of the b o o k is perhaps best indicated by the fact that a s e c o n d e d i t i o n w a s p u b l i s h e d in In

summary,

tuted

the

the

publication

appearance

of the

first

m o d e r n system of vector analysis. with

this

book

than

one

would

1909.26

of W i l s o n ' s full-length

Vector book

Analysis

consti-

presenting

the

I n o n e w a y G i b b s h a d less t o d o expect,

and

some

of Heaviside's

ideas w e r e included. T h e b o o k was w e l l r e c e i v e d a n d popular. It is striking,

but in

one

sense

typical of developments that have been

previously discussed, that the author of the book was a convert from the

quaternion

Alfred Analysis book

Heinrich mit

on

camp.

the

Bucherer's

Beispielen

aus

modern

der

system

He had in

of

1903,

Elemente

Physik,3

of vector analysis

many. Bucherer, w h o was born in at Bonn.

book

theoretischen

was

published

der the in

Vektorfirst Ger-

1863, was in 1903 a Privatdozent

1897 p u b l i s h e d a book relating to electricity27

and was to publish another in

1904.28 It is t h u s p r o b a b l e that his in-

229

A H i s t o r y of V e c t o r Analysis

terest in

vector analysis

electrical

theory.

s t e m m e d from his interest in M a x w e l l i a n

Further

support for this

speculation

is

derived

f r o m t h e f o l l o w i n g s t a t e m e n t f r o m t h e f o r e w o r d o f his b o o k : " I n regard to the form of presentation I have on the w h o l e followed Heaviside a n d used the same symbols as d i d A. F o p p l in his excellent w o r k ' E i n f i i h r u n g in die M a x w e l l s c h e Theorie.' . .

(3; I V ) T h a t

Bucherer d i d not f o l l o w F o p p l in all things is s h o w n by the following statement:

"I have nevertheless believed it to be expedient to

make

of G r a s s m a n n ' s association of vectors w i t h surfaces.

full

This

use

s e r v e s t o s i m p l i f y t h e d e r i v a t i o n s o f m a n y t h e o r e m s . " (3;

IV)

A t o n e p o i n t h e also m e n t i o n e d W i l s o n ' s r e c e n t b o o k . (3;

12) B u c h -

erer's

(addition,

book

covered

the

major topics

in vector analysis

multiplication, a n d differentiation of vectors, potential theory, and the transformation theorems) w i t h the exclusion of the linear vector function.

It offered a n u m b e r of applications chosen especially from

mechanics,

hydrodynamics, and electricity.

T h e r e c e p t i o n g i v e n h i s b o o k i s i n d i c a t e d b y t h e fact t h a t a seco n d e d i t i o n a p p e a r e d t w o years later.29 I n t h e f o r e w o r d t o this edition

B u c h e r e r stated:

" W h e n I w r o t e the first edition of this small

work, the discussions and deliberations concerning a uniform symb o l i s m for vector analysis w e r e still in flux. Since that t i m e t h r o u g h the adoption of a suitable m e t h o d of designation by those w o r k i n g on

the

Encyklopadie

an

put forward."30 The cyklopadie

der

important

system

of

symbolism

encyclopedia referred to

mathematischen

Wissenschaften

has

been

w a s t h e f a m o u s En-

which

was

appear-

i n g i n those years a n d w h i c h w i l l b e discussed later. O t h e r than the change in notation, no major alterations w e r e m a d e for the second edition. T h u s Bucherer produced the first separately published treatment of m o d e r n vector analysis written in German;

h o w e v e r in the year

(1905) of the p u b l i c a t i o n of his second edition t w o other such books w e r e published, all three c o m i n g f r o m the same p u b l i s h i n g house, that of B. G. Teubner.

Dr. ungen

Richard auf

Gans'

die

Einfuhrung

mathematische

Gans was born in

in

Physik4

die was

Vekt or analysis published

mitAnwendin

1905.

1880 and by 1905 h a d b e c o m e Privatdozent at the

University of T u b i n g e n . Since his inaugural dissertation dealt w i t h electricity31

and

theory in 1906

32

since

he

published

major writings

on

electrical

a n d in 1908,33 it is p r o b a b l e that G a n s also c a m e to

vector analysis through electricity.

F u r t h e r support for this conjec-

ture as w e l l as other interesting information is given in the followi n g quotation f r o m his foreword:

230

Emergence of the M o d e r n

Since

however

through

System of Vector Analysis:

the

development

of the

1894-1910

electrodynamics

of

m o v e d b o d i e s a n d o f the t h e o r y o f e l e c t r o n m o r e d e m a n d s are c o n s t a n t l y b e i n g placed on readers in relation to their c o m m a n d of the methods of vector analysis, it seemed to me not out of place to write a book w h i c h w o u l d be a i m e d at fullfilling the needs of these branches of science, for it cannot be questioned that the important results attained in the abovementioned fields will

not safely be surveyed by m a n y persons, since

t h e y are n o t s u f f i c i e n t l y f a m i l i a r w i t h t h e m a t h e m a t i c a l t e c h n i q u e s i n v o l v e d . It is also certain that those w h o have t h e i n t e n t i o n of a s s i m i l a t i n g t h o r o u g h l y a n d w i t h t h e greatest p o s s i b l e ease t h e n e w e r l i t e r a t u r e a n d that w h i c h is yet to appear, or of w o r k i n g i n d e p e n d e n t l y in the field of t h e o r e t i c a l e l e c t r i c i t y , m u s t a b o v e all p r o v i d e for t h e p r o p e r tools, i.e., for a k n o w l e d g e of v e c t o r analysis. (4; v) T h e notation used by Gans was based mainly on the system used by

Lorentz

and

Wissenschaften.34

Abraham

Gans

treated

multiplication, the

in

the

such

differential

Encyklopadie

topics

as

and integral

der

vector

mathematischen

addition

calculus

and

of vectors, the

transformation theorems, and curvilinear co-ordinates;

and he gave

n u m e r o u s applications to mechanics, h y d r o d y n a m i c s , a n d above all electrodynamics. composed

Gans'

primarily

book,

like

Bucherer's,

for physicists

and

was

engineers,

a

short

with

no

work

discus-

sion of such topics as Mobius', Grassmann's, or H a m i l t o n ' s system. Like

Bucherer's

edition the

book,

second edition

on tensors

The

and the

second

Eugen

Gans'

was

the

vectorial

book

success,

major change

for a second introduced

in

of a seventeen-page chapter

published

(1863-1921)

indicated

his

title

in

page)

Bergakademie

zu

in

uber

Geometrie,

Physik.5 J a h n k e

Konigl.

inclusion

Vorlesungen

auf

on

met with

only

linear vector function.

Jahnke's

Anwendungen

book

1909.35 T h e

appeared in

Germany

die

Mechanik 1905 of

und

held the

Berlin."

Unlike

1905

was mit

mathematische

academic position

"Etatsmassiger

Gans, but like Wilson, Jahnke was

in

Vektorenrechnung

Professor

Foppl,

(as

an

der

Bucherer,

and

primarily a mathematician; this

is indicated by the nature of his other publications a n d f r o m the approach taken in his In

one

sense

vector analysis book.

Jahnke's

book should not,

despite

its

title, be

in-

c l u d e d in this study of the first books p r e s e n t i n g m o d e r n vector analysis, books

for

its

approach

published

Grassmann

in

tradition

this

was

primarily

period

have

been

and

Grassmannian. written

excluded

on

directly the

The

other

within

grounds

the that

they differ from m o d e r n books on vector analysis in regard to a n u m b e r of f u n d a m e n t a l ideas (for e x a m p l e , a s l i g h t l y d i f f e r e n t d e f i n i t i o n of the vector product), and that their contents include m a n y mathe-

231

A H i s t o r y of V e c t o r Analysis

matical most

ideas

(for e x a m p l e ,

modern

vector

p o i n t analysis) w h i c h are not f o u n d in

analysis

books. Jahnke's book merits discus-

s i o n s i n c e its o r i g i n s l i e i n b o t h t h e G i b b s - H e a v i s i d e t r a d i t i o n a n d the

Grassmannian

tradition.

Jahnke,

though

primarily

influenced

by Grassmann's ideas, was also i n f l u e n c e d b y , a n d took advantage of, t h e ideas of t h e H a m i l t o n - T a i t - M a x w e l l - H e a v i s i d e tradition. Jahnke

began

his

foreword by quoting from a letter written by

G a u s s i n 1843 i n w h i c h Gauss, r e f e r r i n g t o M o b i u s ' b a r y c e n t r i c calculus, suggested that all n e w calculi are accepted o n l y slowly, a n d accepted o n l y w h e n it is seen that they p r o v i d e d m e t h o d s for dealing

with

problems

referring Jahnke

to

the

stated:

recently

of

complicated

"However

achieved

question

too

slowness

an

in

of the the

the

major cause

results

theory

electromagnetic

b r o u g h t a b o u t a c h a n g e . " (5;

for traditional methods. acceptance

of vector

After

analysis,

w h i c h vector methods have

of electrons and in regard to the foundation

for

mechanics

have

I I I - I V ) Jahnke then suggested that a

of the hesitation of the mathematical w o r l d was to be

ascribed to the division in the historical development between the traditions

stemming

from

Hamilton

and from Grassmann. Jahnke

proceeded to compare these traditions. After stating that some have argued that the Hamilton-Heaviside line of development is suited for physical applications b u t of l i m i t e d value for geometrical applications, Jahnke revealed his o w n feelings by the statement: " O n the other h a n d the direction established by Grassmann allows applications to geometry in the broadest sense of the w o r d as w e l l as to m a t h e m a t i c a l p h y s i c s . " (5; After

describing the

IV)

contents

of his

book, Jahnke

briefly men-

t i o n e d his system of s y m b o l i s m . H i s s y m b o l i s m was of course symbolic

of his

approach and was

derived primarily from

Grassmann

b u t also partly f r o m the tradition that l e d to F o p p l ' s book. Jahnke mentioned

sixteen

heading this

men

list was

whose

works

had

influenced

his

ideas;

Grassmann; m e n t i o n e d in the list w e r e such

m e n as Schlegel, Kelland, and Tait, H y d e , Peano, Bucherer, Abraham,

Foppl, and Gans;

were

the

names

of

n o t e w o r t h y b y their absence f r o m the list

Hamilton,

Gibbs,

Heaviside,

and

Wilson.36

J a h n k e ' s b o o k was d i v i d e d i n t o t w o sections, t h e first w a s e n t i t l e d "Vectors

in

the

Plane,"

latter section took up

and the

second

"Vectors in Space." T h e

s l i g h t l y less t h a n t w o - t h i r d s o f t h e text. T h e

first section began w i t h a chapter on the addition and subtraction of point,

i.e.,

the

Mobius-Grassmann

point analysis

f r o m the Grass-

m a n n i a n point of view. In the next t w o chapters Jahnke introduced free

vectors

done,

232

and

through

line-bound

vectors

(defined, as

Grassmann had

the subtraction a n d m u l t i p l i c a t i o n of points respec-

E m e r g e n c e of the M o d e r n System of Vector Analysis:

tively).

1894-1910

In these first three chapters, as w e l l as later, Jahnke pre-

sented numerous applications to physics and geometry; his intent was

clearly to

interest scientists a n d engineers as w e l l as mathe-

maticians in the contents of his book. T h e next three chapters dealt with vector multiplication and applications thereof to physics; the p r o d u c t s d e f i n e d w e r e (1) t h e i n n e r o r s c a l a r ( d o t ) p r o d u c t c o m m o n to

modern

vector

analysis

and to

Grassmannian

analysis,

(2)

the

G r a s s m a n n i a n o u t e r p r o d u c t w h i c h i s (as d i s c u s s e d e a r l i e r ) s i m i l a r t o t h e m o d e r n v e c t o r p r o d u c t , a n d (3) t h e G r a s s m a n n i a n r e g r e s s i v e p r o d u c t w h i c h has n o correlate i n m o d e r n vector analysis. Thus same

e n d e d the first section; the second section dealt w i t h the

topics

as

the

first

section

but

developed

for three-dimen-

sional space a n d w i t h special attention to p o i n t relations. In addition t w o chapters on vector differentiation, the differential operator V, and a short section on tensors w e r e included. It was especially in these t w o chapters that the influence of the H a m i l t o n - M a x w e l l Tait-Heaviside tradition was evident, for terms (and ideas) such as "Gradient,"

"Curl,"

"Divergenz,"

and

"lineare

Vektorfunktion"

appeared, as w e l l as applications to topics in physics such as Maxwell's equations and electron theory. To sum up, Jahnke's book was primarily, but not entirely, w i t h i n the Mobius-Grassmann tradition. Numerous physical applications were i n c l u d e d in the hope of interesting those w h o were in the process of adopting a system of vector analysis for physical application. I k n o w of four major reviews of Jahnke's book that shed light on t h e n a t u r e o f t h e r e c e p t i o n a c c o r d e d it. T h e v i e w o f t h e b o o k t a k e n by

E.

matical

B.

Wilson

Society

lectures

are

is

writing

summed

really

up

lectures

in in on

the the

Bulletin following

multiple

of

the

American

statement:

algebra and form

Mathe-

"These an

ex-

cellent introduction to the subject —[but] no more than an introduction. . . ."

37

In short, W i l s o n v i e w e d it as a w o r t h w h i l e b o o k for t h e

m a t h e m a t i c i a n , b u t m i s l e a d i n g i n its title. J u l e s T a n n e r y 3 8 w r i t i n g from France, and O.

Staude

39

writing in German, praised the book

a n d d e s c r i b e d its c o n t e n t s i n s o m e d e t a i l . E m i l M i i l l e r , w h o w a s a Grassmann enthusiast, criticized the w o r k as g i v i n g a poor picture o f G r a s s m a n n ' s ideas.40 M u c h l i g h t i s also s h e d o n its r e c e p t i o n b y t h e fact that w h e r e a s B u c h e r e r ' s b o o k a t t a i n e d to a s e c o n d e d i t i o n , Gans' book to a seventh (1950) a n d to translations into E n g l i s h a n d Spanish, and Wilson's book had both a second edition and a major paperback

reprinting,

Jahnke's

book

was

never reprinted,

repub-

lished, or translated.41 T h u s Jahnke's b o o k m a y be v i e w e d as a carefully conceived but unsuccessful attempt to bring about the adoption of a somewhat revised Grassmannian system.

233

A

History

In

of V e c t o r Analysis

1906 Gibbs' original treatise on vector analysis was p u b l i s h e d

part of his collected works.6 Published in the same v o l u m e were

as his

six

this

papers

barely

relating

to

constitutes

vector analysis.

the

publication

If narrowly considered,

of a

vector

analysis

book;

h o w e v e r if broadly considered, it constitutes an important publication for a n u m b e r of reasons. cles,

his

"Multiple

First, since G i b b s ' four p o l e m i c a l arti-

Algebra"

paper,

and

his presentation of a vec-

torial m e t h o d for c o m p u t i n g orbits w e r e i n c l u d e d in the volume, it offered

the

Second, famous before ers

reader

by

1906

than

any

1910.

short

provided

Pavel

ern

father,

and

quite

valuable

materials.

famous, certainly more book

book and were influenced thereby.

public

supplemented with

a

published

Wilson's

longer

In one sense treatise

and

c o m p a c t treatment of vector analysis.

S o m o f f (or S o m o v ) p u b l i s h e d i n

of vector analysis

interest

Joseph

his

important

1907 one of the

early books on vector analysis;7 w i t h his book the mod-

system

from

his

Osipovich

Somoff's

of

had become

other writer of a vector analysis

treatise

the

best of the

array

T h u s it is probable that a substantial n u m b e r of read-

turned to

Gibbs'

an

Gibbs

own

in

made

vectorial

Somoff (Ossip interests

in

its f i r s t a p p e a r a n c e

analysis

Ivanovich

mechanics

in

Russia.

probably s t e m m e d from his Somoff,

and

or

Somov)42

mathematics,

and

which

is

witnessed by his publication of roughly t w e n t y papers on these t w o subjects

in

Russian

and

German journals

from the

1880's to 1907.

T h e m o r e direct sources o f Somoff's w o r k o n vector analysis are clear from

the

following

quotation

taken

from

the f o r e w o r d of his

book: In the study of this subject, it is possible to discern t w o directions, an older one connected w i t h the names Grassmann ( " D i e lineale Ausdehnungslehre,"

1844)

and

Hamilton

("Lectures

on Quaternions,"

1853).

T h e other direction, w h i c h is m o r e recent, appeared at the same time as the study of vector fields and developed through the work of Maxwell, H e a v e s i d e [sic], G i b b s , F o p p l , a n d o t h e r s . . . . W e w i l l h o l d t o t h i s seco n d a p p r o a c h i n o u r p r e s e n t a t i o n o f v e c t o r a n a l y s i s . (7; i i i - i v ) Somoff's parts

book

corporated well as

dealt

with

of vector analysis,

as

many

both

examples

brief explications

those

of Mobius

and

the

elementary

including the selected

primarily from

of other systems

Hamilton.

and

the

advanced

linear vector function.

It in-

mechanics,

as

of space analysis, such

S o m o f f s e e m s t o h a v e b e e n ac-

q u a i n t e d w i t h the majority of the existing works on vector analysis, and

in

Tait, Foppl,

234

the

course

Heaviside, Bucherer,

of his

b o o k he m e n t i o n e d such names as Resal,

Henrici

and

Turner,

Gans, and Jahnke;

Gibbs,

Wilson,

Fischer,

correspondingly his presenta-

E m e r g e n c e of the M o d e r n System of Vector Analysis:

1894-1910

tion was eclectic, w i t h the Gibbs-Wilson s y m b o l i s m and techniques being the

most frequently used. T h u s w i t h Somoff vector analysis

entered Russia—and entered in a very respectable manner.

Siegfried

Vekt o r analysis8

Valentiner's

was

published

in

1907.

Valentiner was b o r n in 1876 a n d h a d by 1907 b e c o m e Privatdozent for physics at the U n i v e r s i t y of Berlin. his book p r i m a r i l y for physicists, as earlier. T h u s

B e i n g a physicist, he wrote

Bucherer and Gans had done

nearly half of the book was devoted to the discussion

of applications of vector analysis to electricity a n d mechanics. H o w ever Valentiner, unlike Bucherer and Gans, i n c l u d e d in his book a treatment of the linear vector function presented from the Gibbsian point of view. Valentiner was in general m o r e eclectic in his presentation

than

Bucherer or Gans,

cidedly influenced by the to

the

Gibbs-Wilson

and

his

presentation,

Heaviside-Foppl

tradition.

tradition,

Valentiner's

while

owed

de-

much

book must have been 1912.43

w e l l received, for a second edition appeared in

Cesare Burali-Forti (1861-1931) and Roberto Marcolongo (18621943)

published

numerose

Fisica-Matematica. this

in

applicazioni In

1909

their

alia

geometria,

the

following

Elementi

di

calcolo

alia

year

a

vettoriale

meccanica French

con

e

alia

translation

of

was published.9 At this time Burali-Forti, w h o was pri-

book

m a r i l y a m a t h e m a t i c i a n , was a professor at the M i l i t a r y A c a d e m y of Turin.

Marcolongo

was

a physicist and professor of rational

me-

chanics at the University of Naples. Burali-Forti's first book relating to vectorial analysis was his 1897 Introduction H.

a

la

Grassmann.

Grassmann

It

Peano

highly

stemmed

early advocate infinitesimal

geometrie

is

differentielle

probable

from

another

of Grassmann's

calculus

(1858-1932).

Peano

had

la

methode

Burali-Forti's

citizen

system.

at the Royal

suivant

that

of Turin

This

was

who

the

de

interest was

in an

professor of

University of Turin, Giuseppe

received

his

doctorate

in

mathe-

matics in 1880, a n d in 1887 he p u b l i s h e d a b o o k that i n c l u d e d m u c h discussion metriche colo

of

del

Grassmann's

calcolo

geometrica

ideas;

this

was

This

was

followed

infinitesimale. secundo

VAusdehnungslehre

his

di

Applicazioni by H.

his

geoCal-

Grassmann

(1888) a n d by a n u m b e r of other w o r k s , i n c l u d i n g his f a m o u s w o r k on

the

foundations

of

mathematics,

Formulaire

de

Mathematiques,

one part of w h i c h was d e v o t e d to vectors. Peano thus was an i m p o r tant proponent of the Grassmannian system, a n d in addition was a man

who

had

the

perspective

provided

by a knowledge of other

systems, such as that of Hamilton.44

235

A

History

of V e c t o r Analysis

After his first Grassmannian publication of 1897 Burali-Forti continued

to

of view.

publish

on

O n e of his

vector analysis from the Grassmannian point

articles that

the

controversy

Marcolongo) deserves

mention

typical

of the times and influential on the presentation and recep-

tion

in

(published with

special

provoked

by

it was both

of their books of 1909-1910. There was during the first decade

of this century a heated debate of considerable magnitude concerning

the

best

for vector analysis.45 A major cause of this

symbolism

d e b a t e a n d its i n t e n s i t y w a s t h e fact t h a t t h e s y m b o l s u s e d s y m b o l ized the

essential

tending

systems —the

Hamiltonian,

mathematical

differences b e t w e e n the still-con-

Gibbs-Heaviside,

the

Grassmannian,

the

others.46

and

At the Naturforscherversammlung at Kassel in 1903 a commission was

set u p t o

Klein

the

"about

deal with

only

three

result

new

the

notation

of the

notations

activity came

question. of the

into

According to Felix

commission

existence!"47

was

The

that

Fourth

International Congress of M a t h e m a t i c i a n s was s c h e d u l e d for R o m e in

1908, a n d in preparation for this m e e t i n g Burali-Forti and Marco-

longo p u b l i s h e d an extensive study of the origins of the various notations.

T h e y also r e c o m m e n d e d the establishment of a n e w system

of symbolism,48 but the seems

to

have

been

only that

major result of their recommendation mathematicians

(which used the proposed symbolism) more bolism In the

than

the

reviewed

their

books

in terms of their sym-

their content.

preface

following

to

their book

statement

Burali-Forti

which

above concerning their interest in

supports

and the

Grassmann's

Marcolongo conclusions

made given

approach:

K n o w l e d g e of vectorial methods is already being forced not only upon physicists a n d electricians b u t also o n those w h o w o r k i n p u r e mathematics.

M a y this book, p r e p a r e d in the year w h e n G e r m a n y celebrates

the first centenary of the birth of Grassmann, contribute to spread and m a k e k n o w n the methods of this great mathematician, an e n d on behalf o f w h i c h w e have l a b o r e d w i t h faith a n d patience for m a n y years; m a y i t c r o w n the w o r k b e g u n among us w i t h so m u c h talent by Mr. P E A N O ! (9; v) T h e authors d i v i d e d their b o o k into t w o parts: the f i r s t p r e s e n t i n g vector methods two

chapters

calculus. is

devoted

to

vector addition and the barycentric

T h e title of their first chapter is s y m b o l i c of m u c h else; it

"Prodotto

was

and the second dealing w i t h applications. T h e first

were

vettoriale

e

intorno." Thus

their scalar (dot)

product

n a m e d in accordance w i t h the Grassmannian tradition, where-

a s t h e i r v e c t o r (cross) p r o d u c t w a s n a m e d a n d d e f i n e d i n a c c o r d a n c e with

236

the

Hamilton-Heaviside-Gibbs

tradition,

the

product

being

Emergence of the M o d e r n System of Vector Analysis:

1894-1910

another v e c t o r rather t h a n a d i r e c t e d area. T h e final t h r e e c h a p t e r s in the first part dealt w i t h rotations in a p l a n e (for w h i c h t h e y introduced a n e w and controversial operator) and the differential calculus of vectors, i n c l u d i n g t h e o p e r a t o r V (the s y m b o l a n d u s u a l presentation of w h i c h they rejected). In the second part, applications to geometry,

mechanics,

trodynamics

were

hydrodynamics, elasticity theory, and elec-

given

along with

formation theorems.49 In the

the d e v e l o p m e n t of the trans-

Italian edition a section

of historical

notes was a p p e n d e d at the end, a n d in the F r e n c h e d i t i o n the historical well

section

as

was

expanded,

presentations

while

some

philosophical

ideas

as

of both quaternions a n d the original Grass-

mannian system were added. Since m a n y o f t h e b o o k r e v i e w s w e r e u n f a v o r a b l e 5 0 a n d n o seco n d editions appeared,41 not w e l l like

received.

Jahnke's

it may be concluded that their books were

T h e books

book,

o f B u r a l i - F o r t i a n d M a r c o l o n g o are,

especially interesting as

representing partial

departures from the Grassmannian system toward the system finally accepted.

As compared to Jahnke's book, their books were

some-

what more r e m o v e d from the Grassmannian tradition than his; b u t because of their tendency toward the inclusion of original symbolisms a n d ideas, t h e a u t h o r s c a n n o t b e v i e w e d a s n e a r e r t o t h e system that was eventually accepted than Jahnke.

Joseph Vector

George

Methods

Mathematics

10

Coffin's

and

was

Their

book

Vector

Various

published

in

Analysis:

Applications

1909.

Coffin

An

Introduction

to was

Physics born

in

to and

1877,

and, after s p e n d i n g four years in E u r o p e ( S w i t z e r l a n d a n d France), he entered Massachusetts Institute of T e c h n o l o g y in 1894 a n d gradu a t e d w i t h a B.S. in 1898. C o f f i n c o m p l e t e d his P h . D . in p h y s i c s at Clark University in 1903, a n d by 1909 he h a d taught physics at b o t h Clark and the City College of N e w York. His book requires little discussion; it may be described by recalling that W i l s o n corresponded w i t h Gibbs about the advisability of publishing a shorter a n d more elementary w o r k f o l l o w i n g the same format

as

Wilson's

book.

Wilson

never

published

such

a book;

Coffin did. Coffin followed the Gibbs-Wilson tradition in both symbolism

and in

methods, t h o u g h his explanations a n d proofs w e r e

less f u l l a n d r i g o r o u s a n d h i s c o n t e n t s s o m e w h a t less i n c l u s i v e . H e d i d include all the major topics in vector analysis up to the linear vector f u n c t i o n , w h i c h h e treated rather briefly. T h e last t w o chapters, w h i c h c o m p o s e w e l l over a t h i r d of t h e b o o k , w e r e d e v o t e d to applications to electrical theory and to mechanics; applications

were

also

given

in the

and numerous

earlier chapters.

Coffin,

like

237

A H i s t o r y of V e c t o r Analysis

nearly all the writers of w o r k s on vector analysis f r o m the latter half of the first decade of this century, was acquainted w i t h the majority of the

earlier books. T h e reception accorded his book is indicated

by t h e fact that a s e c o n d e d i t i o n w a s p u b l i s h e d in 1911 ( w h i c h has b e e n reprinted a large

n u m b e r of times a n d is still in print) and a 1914.10

French translation appeared in

W.

V.

Ignatowsky

Vektoranalysis

und

published

ihre

his

book

Anwendung

in

on

vector

der

analysis,

Die

Physik,n

theoretischen

i n t w o parts i n 1909 a n d 1910. D r . I g n a t o w s k y h a d w r i t t e n a n earlier book,

entitled

Electrodynamics

Solution

with

the

of

of

1902 a n d w r i t t e n in Russian. very

probable

interest in

that

his

electricity,

n u m b e r of papers Ignatowsky's

in

Some

Help

of

Electrostatics

Analysis,

and

published

in

F r o m the title of this book it seems

interest on

Problems Vector

which

in

vector analysis

subject he

came from

h a d also

his

published a

G e r m a n journals.

vector

analysis

book

was

in

the

tradition

estab-

lished by Bucherer, Gans, and Valentiner, although his book was somewhat

longer than

Lorentz, and Abraham; ham,

theirs.

His

notation

stemmed

from Foppl,

and his bibliography listed works by Abra-

Foppl, Gans, " G i b b s - W i l s o n , " Jahnke, Jaumann,51

Bucherer,

and Valentiner. As he stated in his foreword, the book was intended f o r p h y s i c i s t s (11,1; iii), a n d h e n c e t h e e n t i r e s e c o n d p a r t d e a l t w i t h applications.

In his first part he presented the algebra and calculus

of vectors, i n c l u d i n g special discussions of types of fields and curvilinear co-ordinates, a n d he c o n c l u d e d w i t h a section on tensors. A n u m b e r of his methods, vectors

methods

of presentation

differed from the ordinary

for example, he m a d e every effort to avoid d e c o m p o s i n g

into

their i, j, k components in proving theorems, and he

d e f i n e d t h e v e c t o r (cross) p r o d u c t t h r o u g h a n i n t e g r a t i o n . T h e applications in the second part w e r e to nearly all the branches of physics w h e r e v e c t o r analysis w a s a p p l i c a b l e . T h e b o o k m u s t h a v e b e e n w e l l received, since by

1926 a third edition had appeared.

T h e f i n a l w o r k t o b e d i s c u s s e d is, n o t a b o o k o n v e c t o r analysis, but

the

famous

and

mit

Einschluss

Wissenschaften and

extent

of

influenced the century.

The

the

influential ihrer

vectorial

developments publication

of

methods in

Encyklopadie

Anwendungen.

der

The

included

in

mathematischen fact,

the

form,

Encyklopadie

the first decade of the twentieth

the

Encyklopadie

began

around

1898

a n d e x t e n d e d to 1935; a F r e n c h translation a n d revision was b e g u n about

1904.

Included

238

in

the

part

of

the

Encyklopadie

devoted

to

geometry

E m e r g e n c e of the M o d e r n System of Vector Analysis:

1894-1910

were extensive treatments of both the Grassmannian and the H a m iltonian systems, as w e l l as shorter treatments of such systems as that o f M o b i u s , b u t these sections w e r e p u b l i s h e d o n l y after 1915.12 For

the

part

Timerding

of

the

wrote

a

Encyklopadie section

dealing

entitled

with

mechanics

"Geometrical

Mechanics of a Rigid Body," w h i c h he

finished

Basis

H.

E.

of

the

in F e b r u a r y , 1902,

and in w h i c h he gave an elementary presentation of the Grassmansystem.12

nian

chanics

Max

section

Abraham

dealing with

also

wrote

for

the

part

of the

me-

deformable bodies a section entitled

"Fundamental Geometrical Concepts," which he finished in April, 1901.12 A b r a h a m as

i n c l u d e d in this a presentation of vector analysis

applied to various types of fields. His

presentation partly fol-

lowed Grassmann and partly Gibbs and Heaviside. It is interesting to note that in 1899 A b r a h a m wrote to G i b b s asking h i m to write a presentation Encyklopadie,52

of vector Gibbs

analysis

did

not

for

the

fulfill

mechanics

section

of the

request.

Others

made

his

l i m i t e d use o f v e c t o r m e t h o d s i n t h e m e c h a n i c s sections; m o r e extensive

use

entitled

"Physik."

One

however

of the

is

major parts

"Electricity and

Optics."

found

of this

in

the

section

section

The first two

on

of the

physics

parts of this

Encyklopadie

was

entitled

section were

m a i n l y explanations of M a x w e l l ' s electrical ideas. T h e s e w e r e written

by the

1928),

and

great the

Dutch

physicist Hendrik Antoon

Lorentz

Heaviside-Gibbs form of vector analysis

(1853-

was

used

t h r o u g h o u t . L o r e n t z h a d f i n i s h e d these sections b y late 1903.12 H e had used vector analysis at least as early as Versuch ungen

einer in

bewegten

received 1884.53

Theorie

a

The

Korpern

copy

had

of Gibbs'

section

"Electrostatics

der

elektrischen appeared. original

following

on

It

is

optischen also

writing on

Lorentz'

and Magnetostatics";

w h o finished it in

1895, w h e n his b o o k

und

its

Erschein-

known

that

he

vector analysis

in

sections

was

entitled

author was Richard Gans,

1906.12 H e r e a n d i n t h e later sections o n electri-

city by Lorentz, A b r a h a m , and others, vector methods w e r e extensively used. T h i s fact m u s t have l e d m a n y scientists to an interest in vector analysis, for it seems the

Encyklopadie

more

probable that there w e r e f e w parts of

carefully

and

more

widely

read

than

this

section on electricity.

III.

Summary

and

Conclusion

Since the detailed study of the reception of the m o d e r n system of vector analysis from 1894 to 1910 is n o w concluded, it w i l l be profita b l e t o set s o m e o f t h e facts w i t h i n s h a r p e r p e r s p e c t i v e a n d t o sub-

239

A H i s t o r y of V e c t o r Analysis

sume

them

where

possible

into

generalizations

reflective

of the

times. First of all,

it

should

not be assumed that these w e r e the only

books presenting vectorial systems that w e r e p u b l i s h e d at this time. Books

o n t h e q u a t e r n i o n s y s t e m c o n t i n u e d t o appear; thus, for ex-

ample,

Hamilton's

annotations 1899

and

by

the

as

in

1904

sition

Introduction

of

such

to

books

Grassmann Hyde's

system

in

was

two

republished

volumes

edited

Quaternions

by

and

with

appearing in

1 9 0 5 e n t i t l e d Manual published

Kelland

and

the

Tait.

third Simi-

Grassmannian tradition was represented to some degree

as

Wyllys

Quaternion

Knott

book of Burali-Forti

well

and

and

of

larly the in

of

Jasper Joly

1901. Joly also p u b l i s h e d a b o o k in

o f Quaternions, edition

Elements

Charles

was

s

Grassmann

None

Joseph

V.

C o l l i n s ' A n Elementary Expo-

Ausdehnungslehre

presented

Quaternions.

and M a r c o l o n g o a n d in that of Jahnke,

as

s

in

of

Space his

the

(1901)

Analysis

1906

above

and

(1906).

publication

however

Edward

Macfarlane's o f Vector Analysis

seems

to

have

re-

q u i r e d a second edition.54 Thus

i n t h e first d e c a d e o f this c e n t u r y o t h e r systems o f vectorial

analysis besides the m o d e r n system w e r e published

on

them.

Grassmannian interval

and

Yet the

the

rate

in

use, a n d books w e r e

of publication

Hamiltonian

systems

of books

was

less

on the

during the

1 9 0 1 - 1 9 1 0 t h a n d u r i n g the i n t e r v a l 1 8 9 1 - 1 9 0 0 , a n d the rate

w a s to decrease f u r t h e r after 1910. T h a t n e i t h e r t r a d i t i o n has passed away even n o w is indicated by the publication of books w i t h i n each tradition in the

1940's a n d 1950's.

Whereas interest in books presenting the Grassmannian forms of vectorial

analysis

decreased,

Heaviside tradition table

in

which

increased.

some

of the

interest in

the books

in the

Gibbs-

This is evidenced by the following publication

history

of books

in

the

Gibbs-Heaviside tradition is presented. Author of Book and Year of First Publication Foppl,

1894

Some Translations

2 n d ed.,

1904

3 r d ed.,

1907

4 t h ed.,

1912

1st E n g l i s h e d . ,

1932

16th G e r m a n ed., Wilson,

1901

and

Republications

2 n d ed.,

1957

1900

8th reprinting,

1943

Paperback reprinting,

240

1960

Emergence of the M o d e r n System of Vector Analysis:

Author of Book and Year of First Bucherer, Gans,

Some Translations

Publication

2 n d ed.,

1903

2 n d ed.,

1905

1905 1909 1913

1st E n g l i s h e d . ,

1931

2 n d Spanish ed., 7th G e r m a n ed.,

1940 1950

Paperback reprinting,

1906

Valentiner,

and

Republications

3 r d ed.,

Gibbs,

2 n d ed.,

1907

7th ed.,

2 n d ed.,

1909

1961

1912 1950

R e p r i n t of 7th ed.,

Coffin,

1894-1910

1954

1911

1st F r e n c h e d . ,

1914

2 n d ed., 6th impression, 9th reprinting,

Ignatowsky,

3 r d ed.,

1909-1910

1923

1959

1926

F r o m this table three generalizations m a y be made: first, the books listed were indeed in the tradition that survives and flourishes at present; second, m a n y of these books w e r e to play a part in the history of vector analysis after 1910, a n d i n d e e d most are still in print; third, the books w e r e w i d e l y read f r o m 1895 to 1915, w h i c h is indicated by the fact that n e w editions The

books

in

the

Grassmannian

were

so frequently necessary.

tradition

differed

from

these

books in two ways: first, the Grassmannian books contained some ideas that varied f r o m c o r r e s p o n d i n g ideas in vector analysis books — for example, the

Grassmannian outer product —and, second, the

Grassmannian books contained m a n y ideas and methods for w h i c h there was no correlate in the vector analysis books, for example, the regressive product and the

system of point analysis. T h e s e differ-

ences seem to have b e e n sufficient to deter m a n y readers, for n o n e of the

books

using

the

Grassmannian

approach

or

the

modified

G r a s s m a n n i a n a p p r o a c h (Jahnke, B u r a l i - F o r t i a n d M a r c o l o n g o ) required second editions. T h u s it m a y be c o n c l u d e d that the Grassmannian tradition p l a y e d no

major role

alysis. played

It no

was

in

the

acceptance o f t h e m o d e r n

shown

major

role

previously in

the

that

the

development

system

of vector an-

Grassmannian of the

tradition

Gibbs-Heaviside

241

A H i s t o r y of V e c t o r Analysis

system,

that

conclusions other major

is, are

the

conclusions: role

in

modern

system

of vector analysis. T h e s e t w o

p a r t i c u l a r l y s t r i k i n g w h e n t h e y are set b e s i d e t w o the

both

Grassmannian the

development

system and

could

the

have

played

acceptance

of

a

vector

analysis. T h e traditions

t h a t did p l a y a m a j o r r o l e i n t h e a c c e p t a n c e o f t h e

m o d e r n f o r m of vector analysis are the f o l l o w i n g . T h e tradition that began with Gibbs and led to the books of Wilson and Coffin played a large part.

It should be noted h o w e v e r that Wilson's interest had

roots directly in the q u a t e r n i o n system a n d that Coffin as a physicist probably

attained

some

Heaviside tradition.

of his

The

interest and information

same remark applies

in

from

the

stronger form to

P . O . S o m o f f . T h e m o s t i n f l u e n t i a l t r a d i t i o n w a s t h a t w h i c h h a d its beginning

in

Heaviside and in

Heaviside's

development of Max-

w e l l ' s ideas. T h e source of the interest a n d i n f o r m a t i o n attained by Lorentz,

Foppl,

n a t o w s k y has

Abraham,

Bucherer,

Gans,

Valentiner,

and

Ig-

been traced to Heaviside and located specifically in

his association of m o d e r n vector analysis w i t h Maxwell's electrical ideas. T h i s fact is s u r p r i s i n g in at least one w a y , for G i b b s ' presentation was

fuller than

Heaviside's

a n d i n c l u d e d ideas that w e r e original

w i t h G i b b s a n d m o r e o r less u n i q u e i n his system. F u r t h e r m o r e , o n t h e c r i t e r i o n that w h a t has s u r v i v e d is w h a t was fittest, G i b b s m u s t be given credit for h a v i n g d e v e l o p e d a better notation than Heaviside. ential

T h e conclusion that the than

statement

the that

Gibbsian many

Heaviside tradition was more influ-

tradition

who

became

m u s t also

be

interested

qualified by the

in

vector

analysis

w i t h i n t h e H e a v i s i d e t r a d i t i o n w e r e h e n c e l e d t o t h e G i b b s i a n tradition and profited from the greater riches therein. F i n a l l y it m a y be n o t e d that the vast majority of the authors of the books

presenting the modern

form

of vector analysis

were physi-

cists. T h i s i s a p p r o p r i a t e i n that t h e great f u t u r e for v e c t o r analysis lay

in

physical

science;

at present nearly all books on electricity

and mechanics use vector analysis, a n d it appears not infrequently in b o o k s on optics a n d heat c o n d u c t i o n . It is also u s e d in m a n y parts o f m o d e r n p h y s i c s , a n d its a p p l i c a t i o n s for t h e e n g i n e e r are legion. V e c t o r analysis has b e e n o f great v a l u e t o t h e g e o m e t e r , b u t g e o m e ters are f e w i n n u m b e r a m o n g m o d e r n mathematicians. Such mathe m a t i c a l creations as matrices, v e c t o r spaces, groups, a n d fields are associated sense.

In

only

indirectly

m a n y cases

with

vector

analysis

in

the

traditional

h o w e v e r their roots e x t e n d back historically

to the b r o a d stream of d e v e l o p m e n t that c u l m i n a t e d in the first decade of this

242

century.

Notes 1

August

Foppl,

Einfiihrung

Bidwell

Wilson,

in

die

Maxwell'sche

Theorie

der

Elektricitat

(Leipzig,

1894). 2

Edwin

Mathematics York,

and

1901).

Alfred

Heinrich Physik

Richard

5

6

7

Text

Book for

Lectures

reprint

of the

the

of J.

Use

of Students

Willard

slightly

Gibbs

revised

of

(New

second

edition

Willard in

der

Vektor-Analysis

mit

Beispielen

aus

der

in

die

Vektor analysis

mit

Anwendungen

auf

die

1905).

Vorlesungen und

Physics

Elemente

Einfiihrung

Jahnke,

in

A

the

1903).

(Leipzig,

Mechanik

Josiah

1906),

Bucherer,

Physik

Eugen

dents

to the

(Leipzig,

Gans,

mathematische

Geometrie,

are

Analysis:

upon

1960).

theoretischen 4

Vector

Founded

References

(New York, 3

Physics

uber

mathematische

die

Vektorenrechnung

Physik

Gibbs,

Elements

Gibbs,

Scientific

of

mit

(Leipzig,

Vector

Papers

Analysis

of J.

Anwendungen

auf

1905). Arranged

Willard

for

Gibbs,

the

Use

vol.

of

Stu-

II

(London,

(in

Russian),

17-90.

Pavel

Osipovich

(Saint Petersberg, 8

Siegfried

9

Cesare

Somoff,

Valentiner,

applicazioni

(Bologna, breuses

Analysis

Vektor analysis

Burali-Forti

numerose

Vector

and

Its

Applications

1907).

and alia

1909). a

la

alia

translation:

geometrie,

1907).

Marcolongo,

geometria,

French

applications

(Leipzig,

Roberto

a

Elementi

meccanica

Elements

de

di

e calcul

la

mecanique

et

Analysis:

An

Introduction

calcolo

con

Fisica-Matematica

vectoriel

a

vettoriale

alia

avec

la

physique

to

Vector

de

nom-

mathematique,

trans. S. Lattes (Paris, 1910). 10

Joseph

Their

George

Various

translation: 11

W.

12

Calcul

v.

Physik, T e i l

to

Vectoriel,

Vector

Physics trans.

Ignatowsky, I

Die

and Alex

der

Mathematics Veronnet

Vektor analysis

(Leipzig and Berlin,

Encyklopadie

Anwendungen

Coffin,

Applications

1909);

Teil

mathematischen

(Leipzig,

1898-1935).

und II

(New (Paris, ihre

York,

in

der

(Leipzig and Berlin,

following

and

French

1914). Anwendung

Wissenschaften

The

Methods

1909).

mit articles

theoretischen

1910).

Einschluss in

the

ihrer

Encyklopadie

are

e s p e c i a l l y i m p o r t a n t f o r t h e h i s t o r y o f v e c t o r a n a l y s i s : (1) H e r m a n n R o t h e , " S y s t e m e g e o m e t r i s c h e r A n a l y s e , E r s t e r T e i l , " vol. I l l , pt. I ( L e i p z i g , 1 9 1 4 - 1 9 3 1 ) , 1 2 7 7 - 1 4 2 3 ; (2) A l f r e d L o t z e a n d C h r .

Betsch, "Systeme geometrischer Analyse, Zweiter Teil,"

v o l . I l l , pt. I ( L e i p z i g , 1 9 1 4 - 1 9 3 1 ) , 1 4 2 5 - 1 5 9 5 ; (3) H . E . T i m e r d i n g , " G e o m e t r i s c h e Grundlegung der 1908),

125-189;

(Leipzig,

Mechanik (4)

Max

1901-1908),

netische

Theorie"

theorie,"

vol.

V,

pt.

3-47;

and II

eines

starren

Abraham, (5)

Hendrik

"Weiterbildung (Leipzig,

13

E d w i n Bidwell Wilson, Scientific

14

Monthly,

32

der

Lorentz,

I

(Leipzig, vol.

"Maxwells

Maxwellschen

II (Leipzig,

"Reminiscences

(1931),

By a " m a j o r p u b l i c a t i o n

Antoon

I V , pt.

Grundbegriffe,"

1904-1922), 63-280;

statik u n d M a g n e t o s t a t i k , " vol. V , pt.

in

Korpers," vol.

"Geometrische

Theorie.

IV,

1901pt.

3

elektromagElektronen-

(6) R i c h a r d G a n s ,

"Elektro-

1904-1922), 289-349.

of Gibbs by a Student and Colleague"

211-227.

in vector analysis" I

m e a n a relatively long w o r k pre-

senting a system of vectorial analysis similar or identical to the n o w - c o m m o n system.

243

A H i s t o r y of V e c t o r Analysis

T h e presentation

h a d to i n c l u d e exposition of some of the a d v a n c e d parts of vector

analysis a n d be p u b l i s h e d as part or all of a book. T h u s all expositions p r e s e n t e d in j o u r n a l s have b e e n e x c l u d e d . T h e m a j o r i t y o f the w o r k s t o b e discussed are books o n v e c t o r a n a l y s i s . T h e t w o e x c e p t i o n s are t h e w o r k s l i s t e d i n n o t e s (1) a n d (12) a b o v e . These

two

works

included

pecially influential. ture

of the

(1)

North

Whitehead,

published),

(Cambridge,

and

(London,

Rotors

Vector

(Leipzig,

Diagrams

des

vecteurs

1903?);

of vector

discussed have

Treatise

on

1898);

(3)

(4)

(London,

(Paris,

A

England,

1904);

analysis

and

were

es-

(2) O .

Victor

William

1909);

Universal

Henrici

Fischer,

Cramp

(5)

not b e e n treated. Algebra,

C.

and

vol.

and G.

T h e s e are

(only

C.

Theorie

F. et

volume

T u r n e r , Vectors

Vektordifferentiation

Charles

Fortin,

I

und

Smith,

Vektor-

Vectors

applications

and

elementaires

1910).

Whitehead's book was systems

presentations

period might expect to be

Alfred

integration

extensive

F i v e books that a reader k n o w l e d g e a b l e in the vectorial litera-

of Symbolic

a i m e d at presenting "a thorough investigation of the various

Reasoning allied to ordinary Algebra." Hence the majority of

the w o r k dealt w i t h material extraneous to vector analysis, and that part dealing w i t h vector analysis mentary

work

was

primarily based on Grassmann.

written

mainly

Henrici's book was a very ele-

under the influence of Heaviside.

also v e r y e l e m e n t a r y a n d was

in the

tradition

begun by

Fortin's book was

Bellavitis.

Fischer's book

was a specialized m o n o g r a p h and, according to the reviews, one of very poor quality.

The

Cramp-Smith

alternating currents 15

1

have

book

of course

an

exposition

of complex

of Steinmetz'

numbers

read many of these journal

support the conclusions 16

was

by means

m e t h o d s for treating

a n d e l e m e n t a r y vector analysis.

p u b l i c a t i o n s , a n d i n all cases t h e y

arrived at t h r o u g h the m o r e systematic m o d e of analysis.

F o r a table w h i c h gives the systems of s y m b o l i s m used by nearly all the authors

discussed in this chapter see James B y r n i e S h a w , " C o m p a r a t i v e N o t a t i o n for Vector Expressions" of

in

Quaternions 17

and

August

Bulletin Allied

Foppl,

Theorie

der

third edition appeared in 18

Felix

Klein,

hundert, p t .

2.,

ed.

of

the

Systems

Elektricitat,

uber

19

A f o u r t h e d i t i o n a p p e a r e d in A

Wissenschaftliche 21

Gibbs

translation Grundlagen

repeatedly

ed.

die

C o u r a n t a n d St.

20

German

Association

Mathematics

for

(1912),

Max

Promoting

Abraham

of

der

Entwicklung

1911, a n d a

Ferraris'

(Leipzig,

that

his

short

1904).

The

im

19.

Jahr-

1956), 47.

in 1951.

appeared

trans.

Study

1957.

Mathematik

( N e w York,

fifteenth

book

Elektrotechnik,

insisted

der

Cohn-Vossen

the

18-29.

1907 a n d a sixteenth edition appeared in

Vorlesungen R.

International

of

in

Leo

printed

1901;

Finzi work

Galileo

Ferraris,

(Leipzig,

of

1881-1884

1901). was

not

" f o r m a l l y p u b l i s h e d . " It h a d in any case a v e r y small printing. 22

Alexander

Bulletin 23

of

24

des

die 25

Cargill

son"

in

Alfred

Alfred

(Leipzig,

244

Gilston

Knott,

6th

E.

of]

Ser.,

26

B.

(1902),

"[Review

Magazine,

Ser.,

of]

33

By

8

Edwin

4

Bidwell

(1901-1902),

(I.-W.) et W i l s o n

2nd

"[Review

Mathematik,

Vector Analysis. Association,

Wilson,

Wilson"

in

207-215.

(E.-B.).— Vector analysis" in

(1902),

21-30.

Vector

Analysis"

i n Jahrbuch

uber

96-97. Vector Analysis. (1902),

By

Dr.

Edwin

Bidwell Wil-

614-622.

T h e s e c o n d e d i t i o n w a s i n its e i g h t h r e p r i n t i n g i n

trochemischer 28

der

Philosophical

appeared in 27

mathematiques,

Schlegel,

Fortschritte

of]

Mathematical

" [ R e v i e w of] Gibbs

sciences

Victor

"[Review

American

Anonymous,

Bulletin

26

Ziwet,

the

1943, a n d a paperback r e p r i n t

1960. Heinrich Krafte

Heinrich 1904).

Bucherer,

(Freiberg,

Grundziige

einer

thermodynamischen

Theorie

elec-

1897).

Bucherer,

Mathematische

Einfiihrung

in

die

Elektronentheorie

E m e r g e n c e of the M o d e r n System of Vector Analysis: 1 8 9 4 - 1 9 1 0 29

Alfred

Heinrich

theoretischen

Physik,

30

Ibid.,

31

Richard

32

Richard

ed.

Elemente

(Leipzig,

der

Vektor-Analysis

mit

Beispielen

aus

der

1905).

V.

matischen 33

Bucherer,

2nd

Gans,

Uber

Gans,

Wissenschaften,

Richard

Induction

in

"Elektrostatik vol.

Gans,

IV,

pt.

Einfiihrung

rotierenden

und 2

in

Leitern

(Leipzig,

die

(Leipzig,

Magnetostatik"

Theorie

in

finished

des

in

1902),

1906),

Magnetismus

33

Encyklopadie

pp.

der

mathe-

289-249.

(Leipzig

and

Berlin,

1908). 34

F o r a c o m p a r i s o n see S h a w , " C o m p a r a t i v e N o t a t i o n , "

35

Richard

mathematische

Gans, Physik,

Einfiihrung

2nd

ed.

in

die

(Leipzig

and

t h i r d e d i t i o n c a m e in 1913, a seventh in

18-29.

Vektor analysis Berlin,

mit

1909).

Anwendungen

It

may

be

auf

noted

die

that

a

1950, a n d translations into E n g l i s h a n d into

Spanish in the interim. 36

H a m i l t o n , G i b b s , a n d H e a v i s i d e are m e n t i o n e d in the b o o k itself, b u t in such a

w a y that there is no indication that Jahnke h a d read their works. T h e ideas of H a m i l ton and Heaviside of course 37

Edwin

Von

Bidwell

E.

Jahnke"

(1905-1906), 38

Jules

O.

40

I

J.

Ser.,

Vorlesungen

American

their followers).

uber

Mathematical

die

Society,

Vektorenrechnung.

2nd

Ser.,

12

O.

mechanics; Mechanik,

E.

Physik,

11

Jahnke,

Vorlesungen

(1907),

iiber

Physik,

of Congress,

in

die

Vektoranalysis

in

Bulletin

tiber

die

.

des

.

.

sciences

318-322.

of]

und

( R . ) — Einfiihrung Vektorenrechnung"

die

(1906),

in

Von

Dr.

E.

Jahnke"

in

56-57.

of such w o r k s as the

Nationale.

discussed previously in connection w i t h H e n r i Resal.

I h a v e as-

the

son

Museum,

and

p u b l i s h e d cata-

Bibliotheque

Somoff is

British

Vektorenrechnung"

268-275.

Vektorenrechnung.

17

statement on searches

Library

Somoff was

Gans die

"Vorlesungen

based this

of the

of]

(1905),

und

(1) O s i p o v i c h m e a n s

retische

29

Mathematik

serted that P.

in

the

uber

"[Review

Miiller, fiir

have

logues 42

"[Review

Mathematik

Emil

Monatshefte 41

2nd

der

of

of]

354.

Staude,

Archive

"[Review

Bulletin

( E . ) — Vorlesungen

mathematiques, 39

in

Tannery,

Jahnke

influenced him (indirectly through

Wilson,

of J. S o m o f f on t h e basis of t h e f o l l o w i n g facts:

" s o n o f J o s e p h " a n d (2) J . S o m o f f h a d a s o n P a v e l w h o w o r k e d

concerning this vol.

I

see

(Leipzig,

Alexander 1878),

Ziwet's

f o r e w o r d to J.

Somoff,

Theo-

v-vi.

43

In

1954 the seventh e d i t i o n of Valentiner's b o o k was p r i n t e d for a s e c o n d time.

44

In

the

preface

to

his

Calcolo

infinitesmale

(Turin,

1887),

v-vi,

Peano

mentioned,

besides H a m i l t o n and Grassmann, M o b i u s , Bellavitis, Resal, a n d J. Somoff. 45

1 w o u l d estimate that over t w e n t y papers were devoted entirely to the notation

question.

If however papers that were only partly or indirectly concerned w i t h the

question were to be included, the n u m b e r w o u l d probably triple.

M a n y of the book

r e v i e w s o f t h e v e c t o r a n a l y s i s b o o k s p u b l i s h e d a t t h a t t i m e w e r e l i t t l e m o r e t h a n attacks 46

on

notation.

For example, Alexander

Macfarlane continued to

defend

his

system

and sym-

bols. Also w i t h i n each tradition listed above there w e r e different s y m b o l i s m s ; thus Heaviside's

symbolism

differed

from

Gibbs',

and

Grassmann's

earlier

symbols

d i f f e r e d i n s o m e cases f r o m his later ones. 47

Felix

metic. 48

Klein,

Algebra. Cesare

zioni

Analysis,

Elementary trans.

Burali-Forti

vettoriale"

in

E. and

Mathematics R.

Hedrick

Roberto

Rendiconti

del

from and

an

C.

A.

Marcolongo, Circolo

Matematico

Advanced

Noble

"Per

(New

Standpoint: York,

l'unificazione

di

Palermo,

23

Arith-

n.d.),

65.

delle

nota-

(1907),

324-

328; 24 (1907), 6 5 - 8 0 a n d 3 1 8 - 3 3 2 ; 25 (1908), 3 5 2 - 3 7 5 ; 26 (1908), 3 6 9 - 3 7 7 . See also Roberto

Marcolongo,

"Per

l'unificazione

delle

notazioni

vettoriali"

in

Atti

del

IV

245

A H i s t o r y of V e c t o r Analysis Congresso 197.

Internazionale

The

Mathematical 49

In

with

some

matica

of the

See

for

(1908),

greater

(Chicago,

vol.

Ill

detail

1952),

of]

G.

of

Edwin C.

alle

in

(Rome, Florian

1909),

191-

C a j o r i , A History o f

136-139.

relevant to vector analysis; derivate

Elements in

dargestellt

Bidwell

Burali-Forti

applicazioni

and

Society,

de

Calcul

Nature,

86

Jaumann,

aus

advanced topics applicazioni

Mathematical

Marcolongo" 51

in

rispetto

ad

un

this

was

their

e

alia

mate-

punto

1909).

example

numerose

punkt

more

[Review

American

view

II

con

(Turin,

con

Roma

discussed

vol.

vettoriali

tions,"

the

Notations,

matematici.

is

1909 Burali-Forti a n d Marcolongo p u b l i s h e d a short book in w h i c h they dealt

Omografie

5 0

dei

controversy

"The

vettoriale

. . .

con

C.

di

applicazioni

415-436,

Prof.

of Vectorial

Elementi

and

calcolo in

G.

Bulletin

B.

Burali-Forti

Notavettoriale

M.,

and

of

"[Re-

Prof.

R.

75.

Grundlagen 1905).

By

Unification

Marcolongo],

(1909-1910),

vectoriel.

(Leipzig,

R.

Omografie 16

(1911),

Die

Wilson, and

In

der

Bewegungslehre

this

book

von

Jaumann

einen made

modernen extensive

Standuse

of

vector m e t h o d s a n d f o l l o w e d especially t h e G i b b s - W i l s o n tradition at least in notation. 52

Lynde

53

Ibid.,

54

This

Phelps

W h e e l e r , Josiah

Willard

Gibbs

(New

Haven,

1962),

231.

222. statement

requires

a

partial

qualification.

The

books

of H y d e

and

Mac-

farlane had originally appeared as chapters in Mansfield M e r r i a m and Robert Simpson

Woodward's

Higher

Mathematics,

first

published

in

1896.

This

book

consisted

of e l e v e n chapters s u r v e y i n g parts of h i g h e r mathematics a n d m u s t have b e e n quite popular, since it was in a third edition by

1902. A b o u t 1905 it was d e c i d e d to p r i n t

the chapters as separate books, a n d thus the title pages of Hyde's and Macfarlane's books

contain

the

words

"Fourth

Edition."

It seems

that these t w o chapters were

not a m o n g the most popular, for no "fifth e d i t i o n " of t h e m appeared.

246

CHAPTER

Summary

and

EIGHT

Conclusions

This chapter focuses on a s u m m a r y v i e w of the history of the idea of a vectorial system. As in g e o g r a p h y , so in history, a v i e w f r o m afar is at t i m e s a clearer v i e w . Concerning

the

early

history

of vectorial

concepts

it

was

sug-

gested that the idea of a parallelogram of physical entities was indirectly influential.

A l t h o u g h this concept does not necessarily in-

volve the idea of a vector and d i d not lead directly to any vectorial systems, it d i d provide the early vectorists w i t h an area in w h i c h the usefulness

of elementary vectorial methods

could convincingly be

illustrated. To L e i b n i z belongs credit for h a v i n g seen the desirability, a n d to a l i m i t e d extent the nature, of a system for the analysis of t h r e e - d i m e n s i o n a l space. L e i b n i z h a d n o direct a n d m a j o r effect o n the

later

history

of vectorial

analysis,

but

his

indirect

effect

was

noteworthy. A tradition nions

that

led

directly

to

Hamilton's

discovery of quater-

( a n d to t h e less successful searches of others for a s y s t e m of

space analysis) w a s t h e c o m p l e x n u m b e r tradition. W i t h i n this tradition

two

branches

are discernible:

the first is that of the repre-

sentation of c o m p l e x n u m b e r s as ordered pairs of real n u m b e r s a n d the second is that of the representation of complex n u m b e r s through g e o m e t r i c a l lines. T h e significance o f t h e f i r s t b r a n c h w a s less t h a n that of the

second;

indeed the first branch was only significant in

that it p r o v i d e d H a m i l t o n w i t h an epistemological justification for his

"four-dimensional"

quaternions.

Within

the

second

tradition,

Wessel, Argand, Frangais, Servois, M o u r e y , J. T. Graves, De M o r gan, Bellavitis, H a m i l t o n , a n d perhaps others sought a t h r e e - d i m e n sional vectorial system. Wessel was the first to add vectors in threed i m e n s i o n a l space, a n d H a m i l t o n was the first to p u b l i s h an i m p o r tant

type of multiplication

Within tradition,

another authors

tradition, focused

of entities which on

in three-dimensional

may

be

geometrical

called the entities

space.

geometrical and

devised

247

A H i s t o r y of V e c t o r Analysis

methods stricted

for

operating

sense

directly

Bellavitis,

Giinther Grassmann,

with

Justus

them.

Mobius

Giinther

and

Grassmann,

in

a re-

Hermann

a n d Saint-Venant m a y be located w i t h i n this

tradition. An algebraic tradition is discernible w i t h i n w h i c h authors focused forms

on

relations

and

hence

between

mathematical

devised more

general

entities

or algebraic

meanings for such opera-

tions as multiplication. T h e spirit of this tradition stimulated many authors, above all Grassmann. It was n o t e d that Grassmann's major w o r k was done w h i l e he had no k n o w l e d g e of the geometrical repr e s e n t a t i o n o f c o m p l e x n u m b e r s . A l s o i n t e r m i n g l e d w i t h t h e s e traditions

was

the traditional

quest for more natural, more compact,

and m o r e p o w e r f u l mathematical methods for physical science. T h e s e m a n y traditions led to the creation of m a n y systems of vectorial men

character.

It was a r g u e d that the fact that a large n u m b e r of

sought for (and some found)

such systems indicates that this

quest should be v i e w e d as a definite m o v e m e n t w i t h i n early nineteenth-century mathematical t h o u g h t a n d that the existence of this m o v e m e n t augered w e l l for the eventual w i d e s p r e a d acceptance of such a system. tems

O n t h e basis o f c o m p a r i s o n s w i t h o t h e r s i m i l a r sys-

a n d their acceptance it was suggested that the acceptance of

vectorial methods d i d not take an inordinately long time. Of the m a n y systems created, t w o w e r e predominant; these were the systems of H a m i l t o n a n d of Grassmann. If these t w o systems are compared, a n d m o d e r n vector analysis is taken as the standard of c o m p a r i s o n , it is r e a d i l y apparent that the H a m i l t o n i a n a n d Grassmannian

systems

have

many

similarities.

Both

were

vectorial

in

character; both contained vector addition and subtraction and opera t i o n s s i m i l a r t o t h e m o d e r n scalar (dot) a n d v e c t o r (cross) p r o d u c t . Hamilton's

system h o w e v e r i n c l u d e d these t w o products as parts

o f t h e q u a t e r n i o n p r o d u c t ; t h e p r o d u c t o f t w o v e c t o r s a/3 w a s f o r h i m c o m p o s e d of a scalar p a r t Sa/3 a n d a v e c t o r p a r t Va/3.

The former

of these is the n e g a t i v e of t h e m o d e r n scalar p r o d u c t ; the latter is the modern vector product. A m o n g Grassmann's products the inner p r o d u c t is e q u i v a l e n t to t h e m o d e r n scalar p r o d u c t , w h i l e his outer p r o d u c t yields a directed a n d o r i e n t e d area rather than another vector. Both

H a m i l t o n and Grassmann dealt w i t h vector differentiation,

and both created full-blown systems rich in content and application. Hamilton's by certain

system

was

deletions,

transformed into the system there

248

a vectorial

modern

however

two

and redefinitions

vector analysis

was also in m a n y w a y s were

system of such a character that

simplifications,

it could be

system. Grassmann's

similar to m o d e r n vector analysis;

important

types

of

differences.

Grass-

Summary and Conclusions

mann's system contained some elements w h i c h were definitely not equivalent to their correlates his outer product.

in the m o d e r n system, for example,

Moreover there were many elements

in

Grass-

mann's system for w h i c h no correlates can be f o u n d in the m o d e r n system,

for

example,

his

Broadly considered, the

point

system

and

his

other

products.

structure of his system was very different

from the structure of the m o d e r n system, for example, vectors w e r e derived from

points.

T h u s it was c o n c l u d e d that Grassmann's system, like Hamilton's, could h a v e letion

l e d to m o d e r n vector analysis t h r o u g h a process

and

alteration.

Concerning

priority

in

relation

of de-

to the

ele-

ments c o m m o n to these systems H a m i l t o n deserves credit for priority of publication, whereas Grassmann deserves credit for b e i n g the first to create his system and the first to publish an extensive e x p o s i t i o n of it. I t i s w o r t h w h i l e n o w t o c o m p a r e these t w o m e n , especially i n regard to factors influential on the history of their systems. H a m i l t o n was born in

1805 a n d d i e d in

and died in

1877;

1865;

Grassmann was b o r n in 1809

they w e r e thus close contemporaries.

Hamilton

was a prodigy w h o before he was thirty h a d m a d e a n u m b e r of very important

discoveries

which

had brought h i m a large measure of

fame; by comparison Grassmann's youth was undistinguished, and he

published

Hamilton

his

first

scientific

paper

only

when

he

was

thirty.

attained a university professorship w h i l e he was still an

undergraduate;

Grassmann

never attained this

distinction despite

n u m e r o u s efforts. T h u s w h e n i n t h e 1840's t h e i r systems w e r e b e ing offered to the public, H a m i l t o n had as he said a "capital" of personal

fame

that aided h i m

in

gaining recognition

for his system,

whereas Grassmann was then and long remained almost unknown. Both in

m e n had strong interests outside mathematics, particularly

regard

to

languages,

philosophy,

and

religion.

Within

mathe-

matics both had a similar style that l e d t h e m to w r i t e long, scarcely readable w o r k s e m b o d y i n g philosophica l ideas. B o t h w e r e isolated geographically

from

their

important

contemporaries:

Hamilton

lived in D u b l i n at a distance from Trinity College, and Grassmann lived at Stettin, f r o m w h i c h he h a d to travel to B e r l i n to read the Comptes

rendus

of the

Paris A c a d e m y .

W h e n each

m a d e his

discov-

ery, he expressed a readiness a n d a desire to s p e n d a m a j o r part of his life d e v e l o p i n g his system. B o t h also h a d o n e e s p e c i a l l y i m p o r t a n t f o l l o w e r , n e i t h e r o f w h o m h a d b e e n a student of his master. Grassmann w a i t e d nearly until his death for a measure of fame; H a m i l t o n ' s fame came early b u t probably d i d not increase d u r i n g his later years or in the decades i m m e -

249

A H i s t o r y of V e c t o r Analysis

diately after his death. B o t h w e r e geniuses a n d alike in that the fate of their systems was to d e p e n d on the development and acceptance of another system, that of m o d e r n vector analysis. Correspondingly, j u d g m e n t s on the significance of these t w o m e n as creators of vectorial

systems

are c o m p l i c a t e d by the fact that their systems have

n o t b y a n d large s u r v i v e d i n tact. T h e i r s y s t e m s w e r e ancestors o f the

modern

have

system,

but this

ancestry they themselves might well

disclaimed.

Concerning study

was

the

made

reception

of the

accorded

publications

their

systems

a

statistical

relating to each system from

1841 t o 1900. T h i s s t u d y r e v e a l e d that r o u g h l y 594 p u b l i c a t i o n s relating to

quaternions

appeared

from

1841 to

1900, whereas there

were roughly 217 publications relating to Grassmann's system duri n g this t i m e span. T h u s there w e r e s o m e t h i n g on the order of 2.73 q u a t e r n i o n publications for each Grassmannian publication. T h e r e were 38 quaternion books as compared to 16 w i t h i n the Grassmann i a n tradition, or 2.37 q u a t e r n i o n b o o k s for each b o o k in the Grassmannian tradition.

Thus

interest in

the quaternion system was on

the order of t w o a n d one-half times as great as that in the Grassmannian

system.

This

that during the tions

within

generalization

last d e c a d e

the

was qualified by the observation

of the

Grassmannian

number published within the

century the number of publica-

tradition was

nearly

equal

to the

Hamiltonian tradition.

In relation to geographic distribution of interest it was observed that a m o n g the British, American, French, a n d G e r m a n peoples interest in

each system was

s t r o n g e s t i n t h e l a n d o f its o r i g i n , n e x t

strongest (for b o t h systems) in A m e r i c a , a n d substantial in the other t w o countries. T h e r e was sufficiently widespread interest in quaternions

outside these four countries that ten quaternion books were

p u b l i s h e d i n s e v e n o f t h e less i n t e l l e c t u a l l y p r o d u c t i v e c o u n t r i e s o f the

late n i n e t e e n t h - c e n t u r y w o r l d . A n o t h e r major conclusions f r o m

the

statistical

study was that interest in vectorial systems was in-

creasing rapidly After this the

during the

latter t h i r d of the nineteenth century.

q u a n t i t a t i v e study e s t a b l i s h e d the degree of success of

quaternion

system,

the relation of Tait, Peirce, Maxwell, and

Clifford to the quaternion tradition was analyzed. Tait was Hamilton's

leading

better known.

disciple

and

did

much

to

make

Hamilton's

ideas

In a d d i t i o n to this he d e v e l o p e d m a n y parts of qua-

t e r n i o n analysis, a n d s o m e o f these d e v e l o p m e n t s w e r e later transferred into modern

vector analysis.

M o s t important was Tait's de-

v e l o p m e n t o f t h e o p e r a t o r V ; a s M a x w e l l p u t it, T a i t was t h e " C h i e f Musician

upon

theorems

associated

250

Nabla." with

Thus V

the

first

very

important

appeared

in

transformation

vectorial

form

in

Summary and Conclusions

Tait's writings.

M o r e o v e r T a i t r e p e a t e d l y s t r e s s e d (as H a m i l t o n h a d

not) the applications of q u a t e r n i o n m e t h o d s in physical science. D u r i n g the nineteenth century, physical science (above all electricity) d e v e l o p e d in such a w a y that the n e e d for a vectorial system increased. T h u s for example emphasis was p l a c e d on the field concept, and potential theory was developed. M a x w e l l , partly t h r o u g h Tait,

became interested in quaternion analysis

1870.

His

enthusiasm

for the

"ideas,"

some time around

but not the

"methods," of

q u a t e r n i o n analysis was great, a n d his critical analysis of the quaternion

system

system.

M o s t of these defects w e r e e l i m i n a t e d in the G i b b s - H e a v -

iside

system.

led h i m to definite views as to the defects

T h e chief significance of M a x w e l l

in this

is that he associ-

ated certain parts of quaternion analysis w i t h i m p o r t a n t ideas presented

in

his

Treatise

on

Electricity

and

Magnetism.

Though

the

use of q u a t e r n i o n m e t h o d s in this influential b o o k was v e r y l i m i t e d , it was

sufficient to

lead other scientists,

most notably Gibbs and

Heaviside, to the study of quaternions. Certainly not all, p r o b a b l y not e v e n a majority, of the physicists of the

1870's a n d 1880's w e r e favorably d i s p o s e d t o w a r d vectorial

methods.

Lord

Kelvin

was

not

only

the

most

influential

British

physicist at this t i m e ; he was also an o u t s p o k e n critic of q u a t e r n i o n methods as w e l l as vectorial methods in general. Tait was not the only

vigorous

advocate

of quaternions.

Benjamin

Peirce

was

an

early and influential p r o p o n e n t of Hamilton's ideas, a n d m u c h of the interest in quaternions

i n t h e U n i t e d States can b e t r a c e d t o h i m .

Peirce was also discussed as illustrative of the not s m a l l n u m b e r of m e n w h o w e r e led f r o m quaternions to mathematical ideas of great importance, Kingdon

but

ideas

Clifford

was

outside seen

of the

quaternion

system.

William

as a very capable mathematician w h o

was k n o w l e d g e a b l e i n b o t h t h e H a m i l t o n i a n a n d G r a s s m a n n i a n traditions. o f Dynamic

Clifford's was

presentation

viewed

as

o w e d the presentations from

1865 to

o f a v e c t o r i a l s y s t e m i n h i s Elements

prophetic

in

the

sense

of Gibbs and Heaviside.

that it foreshadThus

the

period

1880 was characterized as a t i m e of " r e a l i z a t i o n s " as

opposed to "discoveries." D u r i n g this p e r i o d m a n y scientists came to realize that vectorial

systems were important and useful, and a

f e w scientists b e g a n t o d i s c e r n w h i c h e l e m e n t s i n t h e e x i s t i n g systems were most significant.

T h u s the stage was

set for G i b b s a n d

Heaviside. T h e m o d e r n system of vector analysis originated w i t h Josiah W i l lard G i b b s and O l i v e r Heaviside. T h e s e t w o great physicists, w o r k ing independently of each other, constructed their systems (which w e r e essentially identical) d u r i n g the late

1870's a n d early 1880's.

251

A H i s t o r y of V e c t o r Analysis

T h e position of both in relation to previous traditions was identical. It was tem

d e m o n s t r a t e d t h a t e a c h b e g a n t o s t u d y t h e q u a t e r n i o n sys-

under

writings. Treatise

the

Both

on

stimulus were

Electricity

supplied

exposed to and

by

Maxwell's

Maxwell's

Magnetism

and

to

electromagnetic

critical remarks in his the

practice

followed

b y M a x w e l l i n t h a t w o r k . T h u s G i b b s a n d H e a v i s i d e c r e a t e d a syst e m that successfully avoided the defects ascribed to the quaternion system by M a x w e l l a n d s i m u l t a n e o u s l y salvaged those parts of the quaternion approach that M a x w e l l praised. T h o u g h neither was influenced by Grassmann,

Gibbs

d i d become an advocate of some of

Grassmann's ideas in m u l t i p l e algebra. D u r i n g the

1880's the G i b b s - H e a v i s i d e system started to b e c o m e

known through Gibbs' his

lectures

at Yale,

selectively distributed booklet and through

and through

Heaviside's

and

elsewhere.

Gibbs'

analysis

was

greater than

Heaviside's, though Heaviside played a

greater

role

in

gaining

forth vector analysis

creative

publications in the

Electrician

acceptance

in his

contribution

for their system,

to

since

very important electrical

vector

he

set

publications.

I t w a s p o i n t e d o u t that b o t h G i b b s a n d H e a v i s i d e m a d e efforts f r o m a n u m b e r of directions to m a k e their systems better k n o w n and that one factor that m u s t have influenced m a n y readers was the rapidly increasing fame of each as a physical scientist. T h u s by

1890 the

m o d e r n system of vector analysis had b e e n created and offered, as it were, to the public.

It was n o t as yet w i d e l y k n o w n , b u t this situa-

tion was soon to change, for a "struggle for existence" was about to begin. T h i s " s t r u g g l e for e x i s t e n c e " took place f r o m 1890 to 1894; thirtyeight

publications

written

during

that

time

by

more

than twelve

scientists w e r e discussed. It was stressed that the debate was widespread,

conducted

physicists,

and

with

presented

vehemence, in

a

very

participated readable

in

manner.

mainly The

by

main

q u e s t i o n d e b a t e d was, W h i c h vectorial system is best? A l l the participants was

except Cayley w e r e c o n v i n c e d that a vectorial

desirable.

The

main

contesting

systems

were

the

approach Hamilton-

Tait and the Gibbs-Heaviside systems; the merits of the systems of G r a s s m a n n a n d Macfarlane w e r e debated, b u t to a lesser extent. A major result of the

debate

was

that the

quaternion

system, w h i c h

was w e l l k n o w n , was forcefully attacked; and the Gibbs-Heaviside system, w h i c h until then was not well k n o w n , received m u c h publicity, though certainly not always of the most favorable kind. It was the

suggested that the Gibbs-Heaviside system emerged from

debate w i t h i m p r o v e d chances for w i d e s p r e a d acceptance; this

w a s p a r t l y d u e t o t h e fact t h a t t h o s e w h o w r o t e o n its b e h a l f p r e -

252

Summary

and Conclusions

sented a unified front, w e r e more tactful in their presentation, w e r e more

knowledgeable

of the

opponents'

system,

and

(judging by

later history) w e r e right in their arguments. F i n a l l y it was suggested that an unnecessary, but nonetheless real, result of the debate was that H a m i l t o n ' s fame suffered. T h i s was unnecessary in that, rhetorical

considerations

justice

aside,

Gibbs

presented themselves

as

and

Heaviside

descendents

could

from

have

Hamilton

with and

Tait rather than as opponents of them. That they d i d not do this is certainly no cause for censure; that they chose not to do this was a result of the natural tendency of the quaternionists to d r a w on the capital

of fame

that had accrued to

Hamilton.

Thus

by

1894

the

Gibbs-Heaviside system had become k n o w n to a substantial n u m ber of scientists; it was not as yet w i d e l y used. T h e m e t h o d o l o g y selected for the discussion of the acceptance of the

Gibbs-Heaviside

system

was

to

analyze

the books published

f r o m 1894 to 1910 that presented the n o w - c o m m o n system of vector analysis

in order to

determine their origin, their nature, and their

success in attracting readers. It was c o n c l u d e d that it w a s p r i m a r i l y through Heaviside and his association of vector analysis w i t h Maxwell's electrical ideas that the m o d e r n system entered into the German-speaking lands, especially through the writings of such m e n as Foppl,

Lorentz,

Abraham,

Bucherer,

Gans, Valentiner,

and Igna-

towsky. T h r o u g h Ferraris it became available in Italy, and through Heaviside Through

himself

it

had

Gibbs a n d his

already

become

available

in

Britain.

pupil Wilson an excellent presentation of

the m o d e r n system was published in America. Coffin's shorter book further h e l p e d to m a k e this system available. T h r o u g h P. O. Somoff, w h o was acquainted w i t h both the Gibbs and the Heaviside traditions, the m o d e r n system was presented in Russia. The

Grassmannian

Burali-Forti,

tradition

was

kept

alive

in

Marcolongo, Timerding, and Jahnke.

the

writings

of

In the writings

of these m e n Grassmann's ideas w e r e dominant, t h o u g h the methods a n d style of the G i b b s - H e a v i s i d e tradition w e r e not absent. To a far lesser d e g r e e works

as

cluded

on

those the

bases

that the books the

ones

the

Grassmannian

of Wilson

and

tradition entered into such

Valentiner.

of republication

and

However, translation

it was

con-

information

directly w i t h i n the Gibbs-Heaviside tradition w e r e

that w e r e

widely

used by the

scientific world.

Thus,

in

short, the H e a v i s i d e tradition was the most important, the G i b b s i a n tradition second in importance, and the distant third. pear,

but

Books

their

Grassmannian tradition a

w i t h i n other traditions h a d not ceased to ap-

number

had

decreased

greatly.

The

pure

Hamil-

tonian a n d the p u r e G r a s s m a n n i a n systems are not yet m a t h e m a t i -

253

A

History

of V e c t o r Analysis

cal antiques; thus, for example, H e n r y G e o r g e F o r d e r p u b l i s h e d in 1941

a

book

entitled

Grassmannian 1950's t w o

The

tradition,

long books

Calculus

and

Otto

within the

1

o f Extension F.

which

Fischer

is

in

the

published

in

the

H a m i l t o n i a n tradition.2

Heroes the history of science must have, though w i t h i n the scope of the

present

study

so

many

scientists

have

contributed

signifi-

cantly that the discernment of an order of heroic accomplishment is far f r o m ing

easy.

o f last

Each

made

quoted,

Nonetheless, if t w o of the m e n discussed are deserv-

mention, a

but

they

prophetic

prefaced

are

certainly

statement,

by

the

Grassmann

and

remark

these

that

the

and

Hamilton.

statements present

will

almost

be

uni-

versal a d o p t i o n of vector analysis has fulfilled their prophecies, not to the

letter of the

Writing ond

in

statements, but to the spirit thereof.

1861,

Ausdehnungslehre For

I

Grassmann

concluded the

with

following

the

f o r e w o r d o f his sec-

statement:

remain c o m p l e t e l y confident that the

labor w h i c h

I h a v e ex-

p a n d e d o n t h e s c i e n c e p r e s e n t e d h e r e a n d w h i c h has d e m a n d e d a significant part of my life as w e l l as the most strenuous application of my p o w e r s w i l l n o t be lost. It is t r u e that I am a w a r e that t h e f o r m w h i c h I have given the science is imperfect and must be imperfect. But I k n o w a n d f e e l o b l i g e d to state ( t h o u g h I r u n t h e risk of s e e m i n g arrogant) that even

if this

years

or even longer, w i t h o u t entering into the actual d e v e l o p m e n t of

work

s h o u l d again r e m a i n u n u s e d for another seventeen

science, still that t i m e w i l l come w h e n it w i l l be brought forth from the dust of o b l i v i o n , a n d w h e n ideas n o w d o r m a n t w i l l b r i n g forth fruit. I k n o w that if I also fail to gather a r o u n d me in a p o s i t i o n ( w h i c h I have up to n o w desired in vain) a circle of scholars, w h o m I c o u l d fructify w i t h these ideas, a n d w h o m I c o u l d stimulate to develop and enrich further these ideas, nevertheless there w i l l c o m e a t i m e w h e n these ideas, perhaps i n a n e w f o r m , w i l l arise a n e w a n d w i l l enter into l i v i n g c o m m u n i cation w i t h contemporary developments. For truth is eternal and divine, and

no

phase in the development of truth, however small may be the

r e g i o n e n c o m p a s s e d , c a n pass o n w i t h o u t l e a v i n g a trace; t r u t h r e m a i n s , even

though

the

garment in w h i c h poor mortals clothe it may fall to

dust.3 Tait nions

in

the

quoted

Hamilton

preface from

had

a

think,

that

we

Never m i n d when.4

254

the

third

written

edition to

him

of his in

Treatise

1859

by

on

Quater-

Hamilton.

asked Tait:

Could a n y t h i n g b e as

to letter

s i m p l e r o r m o r e s a t i s f a c t o r y ? D o n ' t you feel, a s w e l l are

on

a

right

track,

and

shall

be

thanked h e r e a f t e r .

Notes 1

P u b l i s h e d at C a m b r i d g e , E n g l a n d , xvi + 4 9 0 pp.

2

See

alcade Natural 3

Otto

Philosophy

Hermann

Werke, v o l . 4

As

F.

Fischer,

(Stockholm,

I, pt.

quoted

1951), with

in

Peter

England,

Gilston 1911),

Mechanics

Fischer, Physical

Grassmann,

(Leipzig,

(Cambridge, England, Cargill

Technical

Giinther II

Universal and

1896),

Guthrie

and Five

Hamilton's Mathematical

Quaternions, Structural

Quaternions

(Stockholm,

Gesammelte

mathematische

A

Cav-

Models

in

1957). und

physikalische

10. Tait,

An

Elementary

Treatise

on

Quaternions,

3rd

ed.

1890), viii. T h e date of the letter (April 12,1859) was g i v e n by

Knott,

Life

and

Scientific

Work

of

Peter

Guthrie

Tait

(Cambridge,

134.

255

Chronology

1673

Wallis'

1679

Leibniz'

treatment

1799

Wessel

of complex

letter to

numbers

H u y g e n s on a g e o m e t r y of situa-

tion publishes

3-dimensional

the first 2-dimensional vectorial

and first

system

by 1799

Gauss discovers geometrical representation of com-

by 1 8 0 0

Parallelogram

plex

numbers

common

of

forces,

in physical

velocities . . . becomes

science books

1805

Birth

1806

Argand publishes the geometrical representation of

of Hamilton

1806

Buee

publishes

1809

Birth

of Grassmann

1814-1815

Argand,

complex

numbers

Mobius

complex

Frangais, and Servois

dimensional 1823

paper on

vectorial

publishes

centric

his

numbers

publish ideas on 3-

systems first

treatment

of his

bary-

calculus

1824

G r a s s m a n n ' s f a t h e r (J. G . G r a s s m a n n ) p u b l i s h e s h i s

1826

Hamilton

idea of a geometrical

omy

becomes

at Trinity

tronomer 1827

Mobius'

product

Andrews'

College,

Professor

Dublin,

and

of AstronRoyal

As-

of Ireland Der

barycentrische

Calcul

Warren publishes the geometrical representation of

1828

complex Mourey

1828

of complex Four

by

1830

numbers

publishes

men

the

geometrical

representation

numbers

had

published

geometrical

representa-

t i o n o f c o m p l e x n u m b e r s , b u t all are n e g l e c t e d H a m i l t o n begins searching for a 3-dimensional vec-

1830

torial Gauss

1831

256

system publishes the geometrical representation of

complex

numbers

Chronology

1831

Birth

1832

G r a s s m a n n gets first ideas for his calculus of exten-

of Tait and Maxwell

1835

Bellavitis

sion publishes

calculus 1837

his

first

major

paper

on

his

of equipollences

Hamilton

publishes

representation

of

complex

numbers as "couples" of numbers 1839

Birth

of Gibbs

by 1840

At least t e n m e n h a v e searched for a " t r i p l e " alge-

1840

Grassmann submits his dissertation on tidal theory

bra

containing the first presentation of his calculus of extension 1843

Hamilton

1844

Grassmann

discovers

quaternions

publishes

his

system

in

his

Ausdehn-

ungslehre 1845

Saint-Venant publishes his vectorial system

1846

H a m i l t o n i n t r o d u c e ? Sa/3 a n d Va/3

1850 1852

Birth of Heaviside O'Brien

publishes

his

most complete treatment of

his vectorial system 1853 1853 1855

Hamilton's

Lectures

Peirce

of

1858

Tait-Hamilton

1862

Grassmann's

1865

Quaternions

strongly recommends

System

by 1865

on

C a u c h y ' s " S u r les clefs a l g e b r i q u e s "

Death

Analytical

quaternions

in

his A

Mechanics

correspondence second

begins

Ausdehnungslehre

of Hamilton

Grassmann's publications still totally neglected

1866

Hamilton's

1867

Tait's

1867

Thomson's

Elements

Elementary

of

Treatise

and

Tait's

Quaternions on

Quaternions

Treatise

on

Natural

Philos-

ophy 1867

Hankel

praises

der 1871

Grassmann's

complexen

Maxwell's

ideas

in

his

Theorie

Zahlensysteme

"Mathematical Classification of Physical

Quantities" 1872-1875 1873

Schlegel's Tait's

System

Elementary

der

Treatise

Raumlehre on

Quaternions

(2nd

ed.) 1873

Kelland's

1873

Maxwell's

and

Tait's

Treatise

on

Introduction Electricity

to and

Quaternions Magnetism

257

A H i s t o r y of V e c t o r Analysis

1877

Death

1878

Second

of Grassmann edition

of

Grassmann's

1844

Ausdehnungs-

lehre

by

1878

Schlegel's

1878--1887

Clifford's

1879

Death of Maxwell

Hermann

Grassmann

Elements

of

Dynamic

and Clifford

1880

Death

1880

G i b b s begins teaching vector analysis courses

of Peirce

1880

German

translation

of

Tait's

Treatise

on

Quater-

nions 1881

Peirce's " L i n e a r Associative Algebra" is published

1881--1884

Gibbs'

1881--1884

German

Elements

of

Vector

translation

of

Analysis

Hamilton's

Elements

of

Quaternions 1882--1884

French

translation

of

Tait's

Treatise

on

Quater-

nions 1882

Kelland's (2nd

and

Tait's

Introduction

to

Quaternions

ed.)

1 8 8 2 - • 1 8 89

Grave's

1883

H e a v i s i d e begins using his system in his papers in

1886

Gibbs'

paper " O n

1890

Tait's

Elementary

1890--1894

Thirty-eight

1891

Maxwell's

Life

the

of

Sir

William

Rowan

Hamilton

Electrician Multiple Algebra"

Treatise

on

publications

Quaternions

discussing

(3rd

ed.)

the merits of

the various vectorial systems Treatise

on

Electricity

and

Magnetism

(3rd ed.) 1891

Macfarlane's first presentation of his system

1892

Heaviside's

1893

Heaviside

Electrical

Paper

publishes

vectorial

a

system

lengthy

in

his

exposition

of his

Electromagnetic

Theory

1894--1911

Grassmann's

1894

F o p p l publishes first presentation of modern vector

1895

K i m u r a a n d M o l e n b r o e k propose that advocates of

1898--1900

Tait's

1899

Ferraris publishes

analysis

in

collected scientific

papers published

German

vectorial methods form a society Scientific

vector analysis

by

first l o n g e x p o s i t i o n in

of modern

Italian

1899--1901

Hamilton's

1900

O v e r 1000 vectorial publications: 594 quaternionic; 217

258

Papers

Elements

Grassmannian

of

Quaternions

(2nd

ed.)

Chronology

1900

First

issue

sociation

of

the

for

nions

Bulletin

Promoting

and

Allied

of

the

the

International

Study

Systems

of

of

As-

Quater-

Mathematics

1901

Death

of Tait

1901-1909

M o d e r n vector system used in important articles in the

Encyklopadie

der

mathematischen

Wissen-

schaften 1901

Wilson

publishes first book-length presentation of

m o d e r n vector analysis 1903

in

English

Bucherer publishes first German vector

book

on modern

analysis

1903

Death

1904

Second edition

of Gibbs

1905

Gans'

1905

Jahnke's

1905

Bucherer's

1906

Gibbs'

of Foppl's

Einfiihrung

in

Vorlesungen

Elements

of

Vektor analysis

uber

Elemente

der

collected

book

die

Scientific

Vector

(2nd

ed.)

containing

his

Analysis

1907

Valentiner's

Somoff publishes first book in

Vektorenrechnung

Papers

1907

alysis

die

Vektor-Analysis

Vektor analysis on

modern

vector an-

Russian

1907

Third edition

1909

Coffin's

of Foppl's

Vector

book

Analysis

1909-1910

Ignatowsky's

Die

1909

Wilson's

Analysis

1909

Gans'

1909

Burali-Forti

1910

French

Vector Einfiihrung

Vektor analysis

in

and

(2nd die

ed.)

Vektor analysis

Marcolongo's

(2nd

ed.)

di

calcolo

Elementi

vettoriale translation

longo's by

1910

Interest

in

1913

Last

nions 1925

Death

and

Marco-

interest

and

in

Grassmann's

system

Gibbs-Heaviside

system

increasing

issue

sociation

Burali-Forti

quaternions

declining; rapidly

of

b o o k ' o f 1909

for and

of

the

Bulletin

Promoting Allied

of

the

Systems

the

International

Study of

of

As-

Quater-

Mathematics

of Heaviside

259

Index

A

B

Abraham,

Max,

161, 227, 2 3 1 - 2 3 2 , 2 3 8 -

239, 242-244, 253

Addition of vectorial entities: to Bellavitis, 50, 52-54; 140;

to

Gibbs, 61,

199;

to

according

Baltzer,

to Clifford,

83,

Grassmann,

67-69,

73-74,

86-87;

Hamilton,

28-30,

86-87;

to

Heavi-

side,

172;

to M a x w e l l ,

138; to

168,

M o b i u s , 50; die,

209;

Tait,

to

to O ' B r i e n , 98; to Ped-

to

Saint-Venant,

to

Wessel,

122;

parallelogram quaternions,

of

147

3,

force,

81;

247;

2,

Heinrich

Adhemar.

58,

73-74,

Mobius, Ceva, Barycentrische

28-30

See

Calcul

160,

Algebraic keys

247,248

Ampere, Andre Marie, Andrews, Thomas,

the

118,

Advance-

190

124,

137, 147

128

Apollonius,

147

16, 34, 82, 85, 2 4 7

origin Atled.

See

origin

of term,

of

6 6 ff., 9 3 ,

idea,

15-16,

69;

15-16

Betsch,

Chr., 243 of

Quaternions

144-145,218 M., 84

157,

B l a n c h a r d , C.

(Grassmann):

18, 4 3

179 H., x 103

B o c h n e r , Saloman, 127, Bolyai, Johann, 26, 44

of

1844,

138, 1 5 3 - 1 5 4 , 206, 234; of

1 8 6 2 , 7 6 , 8 9 ff., 1 5 3 - 1 5 4 , 1 6 1 , 2 5 4

260

Friedrich Wilhelm, 33

Bobillier, Etienne,

D e l , origin of t e r m atled, 146

Ausdehnungslehre

14,

Bessel,

Bivector,

A r g a n d , J e a n R o b e r t , 9 - 1 0 ; 5 , 8 , 11, 1 3 -

Associative:

102-103,

Birkhoff, George D.,

13-14, 37, 49

Bolyai, Wolfgang, 44 Bolzani,

48-52

14

B i q u a t e r n i o n , 36, 123, 149, 179

Archibald, R a y m o n d Clare, Archimedes,

94,

Binet, Jacques P. 179

(Mobius),

Beman, Wooster Woodruff,

lane), ix,

127

Appleyard, Rollo,

87-89,

Bibliography

Apelt, Ernst Friedrich, 79

of and

Giusto, 5 2 - 5 4 ; 34, 39, 4 6 - 4 7 ,

50,

for

236;

179

Bellavitis,

158,

229,

232, 234, 239;

Battaglini, Giuseppe,

Allardice, R. E.,

Association

188,

48-52,

Allegret, Alexandre, 39, 41, 46

m e n t of Science,

of Grassmann, 57-

159,

B e l l , E. T., 18, 43

of Cauchy, 83-85

80,

102

Beebe, W.,

102

69,

Saint-Venant

Barycentric calculus:

to

in

50,

Barker, G. F., 148 Barre,

and

247;

Richard,

102-103

Airy, George Biddell, 21, 35

American

149, 2 1 7 - 2 1 8 ,

222

Adams, John Couch, 22

56-57,

Bacharach, Max,

B a l l , R o b e r t S., 2 0 5 , 2 0 6 ;

Professor, 39

Bond, James W., x Bork, Alfred M., 222

147

244-245,

104-105

(Macfar-

Index

Boyer, C a r l B., 102, 103

Cayley,

British Association for the A d v a n c e m e n t of Science, 29, 33-35, Brun, Viggo,

Hugo,

Heinrich,

229-230;

C h r y s t a l , G e o r g e , 119,

William

109-110,

179

Cesare, 235-237; 240-241,

243,245-246,

253

126

155;

and

149,

206;

Calculus of extension (Grassmann analysis): d i s c o v e r y of, 5 5 ff.; e a r l y r e c e p of, 6 5 - 6 6 , 7 7 ff.;

interest in as

countries,

114-117;

later

Coffin,

239;

by

Apelt, 79;

by Ball, 206;

by

Joseph

Complex

Forti, 235-237;

quater142,

237-238;

241-

104, 2 4 0

Christopher, 96 origin of idea,

Bellavitis'

15-16, 2 8 -

15-16

(two-dimensional): calculus

of equipol-

53-54

Clebsch, 91-92;

DISCOVERY

of

by Clifford, 93, 140; by C o l l i n s , 240;

sentation

of:

by

Argand, 9-10; by Buee, 9; by Eu-

Cremona,

by

and

and Tait,

George,

numbers

lences,

Burali-

149;

31, 69; origin of term,

by

by

141-142;

Collins, Joseph V.,

and

88;

139-140, 142-143;

Coleridge, Samuel Taylor, 22

Baltzer, 69, 80; by Bellavitis, 87-88; Bretschneider,

93,

243,253

Commutative:

on, 96,

250-

142-143, 153-

149

250;

view of taken by Abraham,

193,

Heaviside,

139-140,

Columbus,

number of publications

and

139-143;

165,

Grassmann,

r e c e p t i o n of, 9 0 ff., 1 1 0 ff., 2 4 0 - 2 4 1 ,

111-117;

148,

Pearson,

nions,

Cajori, F l o r i a n , 1 4 , 1 6 , 127, 1 4 7 , 2 2 3 , 2 4 6

153-154

Kingdon,

144,

251, 259; and Gibbs,

and C

tion

145

14

Clebsch, Rudolf, 91-92, 94 Clifford,

85

related to

154,

147

Clausius, R u d o l p h , 80, 94,

160

Bumstead, H. A.,

E.,

132,

102

Clagett, Marshall,

Buee, Abbe, 9; 5,8,10-11, 15-16,54,82,

Byerly, W.

35,

C h r i s t e n s e n , S. D., 13

Alfred

Burali-Forti,

211-213;

188,214,218, 252

Chase, A r n o l d B., 126,

231-235, 238, 241-245, 253 Buchholz,

184,

Ceva, Giovanni,

130

13

Buchenau, A., 13 Bucherer,

Arthur,

158,

87-88;

by

Drobisch,

geometrical 5-10,

247,

repre248;

by

79; b y E n g e l , 92, 96; b y F o r d e r , 105,

ler,

254;

G a u s s , 8 - 9 , 1 0 - 1 1 ; b y M o u r e y , 11;

by

by Gauss, 78; by Genese, 206;

Gibbs,

62,

153,

159,

161,

187-

189; b y G r a s s m a n n , 5 7 ff.; b y H a m i l ton,

69,

85-87;

H y d e , 240; Klein,

92,

McAulay, by

by

91;

96;

by

Kummer,

193;

by

Macfarlane,

Marcolongo,

bius, 69, 78-80; Peano,

Hankel,

by

by Jahnke, 231-233; by

235-236;

235-237;

81;

by

190;

by

Mo-

by Muller, 233; by by

Preyer,

93;

by

Schlegel, 91-93; by Staude, 233; by

by

14;

by

Truell,

Frangais,

14;

by

9-10;

Wallis,

6;

by

by

W a l m e s l e y , 14; b y W a r r e n , 11; b y Wessel, DISCOVERY couples

6-8 of representation of

numbers:

6,

of as

247;

by

Bolyai, 26,41; by Hamilton, 23-26 Concentration:

origin of term,

Conrad, Carl L u d w i g , 56, Convergence,

165-166;

131

104

origin

of term,

132

Stern, 92; by Sylvester, 93; by Tait,

Conway, A. W., 46

185; by J. T a n n e r y , 233; by T i m e r d -

Coolidge, Julian Lowell, 8-9,14-15,102

ing,

Copernicus,

239;

by Whitehead,

105,

244;

by Wilson, 62 Cardan, Girolamo, 6

Nicholas, 95

Coulomb, Charles,

129

C o u r a n t , R., 108, 181, 2 4 4

Carmichael, Robert, 38

C o u t u r a t , L o u i s , 5, 14, 107

Cassirer, Ernst, 13

Craig, T h o m a s , 161, 182, 187

C a u c h y , A u g u s t i n , 14, 4 7 , 5 0 , 8 1 - 8 5 , 8 8 89, 106-107, 158, 224

Cramp, William, 244 Cremona, Luigi, 87-88,

107

261

Index

Cross

product.

vectorial Curl,

164,

See

Multiplication

of

Edinburgh

Mathematical

166-167, origin of term,

132

Eichhorn,

207-

Minister of Culture, 81

Einfuhrung

in

der

D

die

Maxwellsche

Elektricitdt

(Foppl),

Theorie 226-227,

229, 230, 232, 240

Darboux, Gaston,

103

Einfuhrung

D a r w i n , Charles, 95, 216, 223 Darwin, G.

H.,

Broglie,

Del

154

Nabla

origin

V, < ):

Burali-Forti, 237; 152,

Papers

175,

of term

and Gibbs,

die

Vektoranalysis

o r i g i n of, 32,

(del),

146;

and

and Clifford,

142;

156,

(Heaviside),

Electromagnetic

Theory

168-177, Elementary

186, 2 0 2 - 2 0 3 ,

163-169,

192 (Heaviside),

189, 2 0 9 - 2 1 1 , 229 Treatise

(Tait),

41,

on

118-125,

Quaternions

132,

205; a n d Grassmann, 61; a n d H a m i l -

146,

155-156,

162-163,

ton, 33, 41; a n d Heaviside, 164-168,

185,

187-188,

190,

172;

212,

and Jahnke,

233;

and

Macfar-

lane, 207; and Marcolongo, 237; and Maxwell,

131-133,

a n d O'Brien, 99; 123-124,

135-136,

184,

118,

202,

Alphonse,

15, 21, 2 6 - 2 7 , 31,

131, 36,

vectors,

122-125,

Diophantus,

134 41,

Dirichlet,

156

31, 69;

152,

155-156,

198-199, 205

of idea,

165-167;

vectorial 28-32,

(Hamilton),

Analysis 192,

(Gibbs),

204-205,

234,

der

15-16, 28-

15-16

origin

entities:

201;

term,

quater-

according

to

Wis-

230-231,

102-104,

108

E q u i p o l l e n c e s , calculus of, 5 2 - 5 4 Essai

of

mathematischen 159-160,

Friedrich, 75, 78-79, 82, 85, 90,

sur quantites 10,

of

nions,

Vector 185,

92-94, 96,

142,148 Division

140-

238-239 Engel,

Lejeune, 50

origin of term,

Divergence,

of

senschaften,

of Gibbs,

origin

(Clifford),

241

Peter Gustav

Distributive:

(Burali235-237,

Quaternions

150-158,

Encyklopadie

185-187,

vettoriale

Marcolongo),

Dynamic

of

Elements 61,

6

product

160,

of

Elements

D i r a c , P. A. M . , 18, 43 Direct

calcolo and

39-42, 46,118, 240

Differentiation 9 8 ff.,

(Buch-

143,251

181

Descartes, Rene, 37-38, 86, of

211,

241

Elements

34, 40, 4 4 - 4 5 , 69, 8 5 - 8 6 , 108, 247

183-

209,

Vektor-Analysis

di

Forti 240,

DeMorgan, Augustus,

Demoulin,

der

Elementi

203,

(Gauss), 8

145-

171,

erer), 229, 232, 233, 241

250-251 Nova"

192,

143,

214

Elemente

139;

and Tait, 41,

131-133,

"Demonstratio

(Gans),

E i n s t e i n , A l b e r t , 95, 128, 147, 151 Electrical

Louis, 21

(Atled, 61;

in

230-231, 233, 241

D a v i d , R. W . , 145 De

Society,

209

entities

Euclid,

une

maniere imaginaires

de

representer (Argand),

les 9-

13 16,

127

Euler, Leonard,

14

G r a s s m a n n , 68 ff.; to W e s s e l , 7 Dot

product. torial

Drobisch,

See

Moritz Wilhelm, 79

Dugas, Rene, Dutch

Multiplication

14, 2 1 , 4 4

Society of Sciences, 224

Dyadic,

of vec-

F

entities

123,

156-157,

186, 202

Faraday, M i c h a e l , 129, 134, 151, 172 Favaro, Anton,

102-103

Ferraris, Galileo, 227, 244, 253 Feuerbach, K. W.,

103

F i e l d concept, importance in the history of vector analysis, E Edgeworth,

262

Finzi, Maria, 22

127-131

Leo, 244

Fischer, O t t o F.,

18, 2 3 4 , 2 5 4 - 2 5 5

Index

Fischer, Victor, 244 FitzGerald,

178,

George

Francis,

120,

154,

232,

234-235,

176, 180, 2 2 9 -

sis:

240,

1911,

238,

228-229,

237,

242-244,

253

number

242;

early

151

Gibbs-Heaviside system

1 7 5 - 1 7 6 , 178, 180, 193, 2 1 1 Foppl, August, 226-228;

179,

life a n d f a m e of,

of

of vector analy-

books

on,

before

225-226

R E L A T I O N

to Abraham, 227, 239; to

Footnote system explained, ix

Bucherer, 229-230,241; to Burali-

Forder, Henry George,

Forti,

105, 254

235-237;

F o r s y t h , A. R., 148

142-143, 251;

Fortin, C., 244

241;

Foucault, Leon,

124

to

to

Clifford,

Ferraris,

227;

to

176, 2 2 6 - 2 2 8 , 234, 241;

F o u r i e r , Joseph, 130, 184

139,

to Coffin, 237-238, Foppl,

to Gans,

230-231,239,241; to Grassmann's

Frangais, Jacques-Frederic, 9 - 1 0 ;

11,13,

15, 3 4 , 2 4 7

calculus

of extension,

90,

107,

94,

174,

Fresnel, A u g u s t i n , 22, 36, 118, 124

47, 54, 77,

151-155,

158-163,

217, 227, 230-233, 241-242,

248-250, 252-254; to Ignatowsky, 238, G

241;

to Jahnke, 231-233;

to

L o r e n t z , 2 3 9 ; t o M a c f a r l a n e ' s system, 190-192,207; to Marcolongo,

Galileo, 37, 95 Gans, Richard, 230-231;

232-235, 238-

239, 241-242, 245, 253 Gauss, Carl Friedrich, 8 - 9 ; 5 , 1 1 , 1 5 , 2 5 26,

34,

50,

58,

77-78,

81-82,

89,

1 0 4 - 1 0 6 , 129, 151, 199, 2 3 2 Gauss' T h e o r e m ,

124-125,135,164,168;

early history of,

235-237;

to

152-154,

162-166,

251-252;

tem,

96-101,

system,

186-189, 220,

H.

G.

Grass-

254;

60;

152-158,

162-163,

ant,

80-81,

Analyse"

(Grassmann),

plained, mann's

situation 3-5;

(Leibniz):

relation

to

ex-

Grass-

236-237;

238;

by

Somoff, 143,

187-188, 217-

G e r h a r d t , C. I., 1 3 - 1 4

by

Willard,

187-189,

150-162,

198-200,

185-

204-205,

1 6 2 ff.,

33,102,104,105,108,110,146,

237;

149,

178-182,

Knott,

221,

223,

236-

228; by

by

237Fitz-

Foppl,

176,

1 5 2 ff.,

185-186,

187-

198-200, 205; by Heaviside,

234;

195,

by Burali-

Coffin,

by Gans, 230-231, 239;

Gibbs,

189,

by

175-176;

226-228;

186,

O.

122-125,

Ferraris,

Gerald,

system, 4-5, 80

Josiah

P.

Tait,

Bucherer, 229-230;

Forti,

G e r g o n n e , J o s e p h D i a z , 9 - 1 0 , 13, 1 5 - 1 6

Gibbs,

152-

170-173,

V I E W of, t a k e n b y A b r a h a m , 2 2 7 - 2 2 9 ; by

83 of

151,

220, 254; to Valentiner, 234, 241

81-82

Geometry

sys-

quaternion

168,

to

to

"Geometrische

to

192, 1 9 8 - 2 0 0 , 215, 2 1 7 -

229,

234-235;

of Saint-Ven-

O'Brien's

166,

m a n n , 5 7 ff.; o f J . G . G r a s s m a n n , 5 7 of Mobius, 50-52;

143, 174-

19, 32, 4 2 , 4 7 ,

Genese, Robert William, 206; 217, 222 of

to 201;

162-163,

Geometrical

product

137,

169-171,

177, 219, 2 2 5 - 2 2 8 , 230, 234, 239, 242,

158,

146-147

Maxwell,

192-193; by

by

Kelvin,

119-120,

Ignatowsky, 119-120;

201-203,

by 204,

246,251-253; and Clifford, 142-143,

206, 208, 229; by Lodge, 191-192;

153-155;

by

151,

and

153-155,

217, 229;

Grassmann, 158-162,

and Heaviside,

163,

168,

177,

216,

242,

251-253;

192-193,

137,

139,

143,

219,

225;

and Tait,

150,

150,

152-158,

and

62,

187-189, 154,

157,

Lorentz,

193;

Marcolongo,

200-201,

chin,

Peddie,

137,

239;

210-211;

by

208-209;

Page, by

by

by

Min-

157;

by

Schlegel,

160,

154, 229;

143,

Tait, 163, 1 8 4 - 1 8 5 , 1 8 6 - 1 8 7 , 1 9 6 -

by Somoff, 234-235; by

196-

197;

by Valentiner,

son,

157, 2 2 8 - 2 2 9 ;

185-189,

McAulay,

191, 210;

236-237;

200, 2 1 9 - 2 2 0 , 222; a n d W i l s o n , 157,

177,

by

by Macfarlane,

Maxwell,

152-155, 122,

76,

235;

by Wil-

by Ziwet, 229

263

Index Ginsburg, Jekuthiel, 45

110,

112,

Grassmann

132,

134-137,

142,

145,

146,

148-

158,

161-163,

171-173,

179,

181-

202-203,

205-

analysis.

See C a l c u l u s o f e x -

tension Grassmann, 9,

Hermann

13-15,

110-114,

34, 142,

53,

145-150,

182,

185-186,

224,

227,

190,

240,

244-245,

54-96;

182,

184-185,

193,

206,

209,

220,

231,

233-236,

152,

193,

230-231,

181-

196,

222,

mann,

252-255;

85-87,

tion,

35-36;

discovery

of

his

system,

early life of, 5 5 - 5 6 ; a n d elec-

trodynamics, 1862,

80,

93-94;

and

a n d Clifford, 93, Gauss,

58,

94;

78;

life

of,

Cauchy,

after

83-85;

139-140, 206; and and

Gibbs,

62,

76,

245,

227,

247-254;

121,

232, 248-250;

99-101,

108;

and

and Salmon,

a n d Tait, 36, 3 9 , 4 1 , 1 1 8 - 1 1 9 ,

183,

223;

and Warren,

10-11,

2 5 , 8 6 ; a n d a l g e b r a , 2 3 ff.; b e c o m e s Royal and

Astronomer

calculus

of

Ireland,

of variations,

complex numbers, 9-11,

20;

17;

and

15, 2 3 - 2 7 ;

187-189,

discovery of quaternions, 26-33, 44,

217, 229; and J. G. Grassmann, 55,

45; e a r l y life of, 2 0 - 2 1 ; f a m e of, 1 7 -

151,

153-155,

158-162,

5 7 - 6 0 ; a n d H a m i l t o n , 19-20, 23, 60,

23,

183,

187-188,

64, 69, 77, 81, 8 5 - 8 7 , 93,

213-214,

218-219,

232,

248-250;

174, 217;

and

187-189,

Heaviside,

163,

a n d L e i b n i z , 4 - 5 , 80, 88;

and Maxwell,

138; a n d M o b i u s , 5 0 -

52, 57-58, 73-74, 78-80, 83-84; a n d Saint-Venant, 56, 81-84; gel, 54, 91-94,

and Schle-

154 161

Grassmann, Justus, 92 Grassmann,

Justus

Giinther,

55, 57-60,

13, 15, 2 1 , 27,

34, 4 3 - 4 4 , 102, 108, 183 124-125,

Green's

124-125;

147 early

his-

the

history

K N O W L E D G E

plex numbers: 10,

Hankel,

105-

H Hamilton, Archibald H., 29

Edinburgh),

138

Sir

12-13,

91, 93-95,

Edwin, 29

William

Rowan,

17-42;

16, 4 6 - 4 7 , 5 1 - 5 5 , 6 6 , 8 8 , 102-103,

10,

15;

of Wallis,

10,

14-15,

39,

46,

91,

158

H a t h a w a y , A . S., 2 2 3 147

Heath, Thomas,

See

14

106-107,

109-

system of vector analy-

Gibbs-Heaviside

vector

analysis

Heaviside,

Oliver,

200-201,

Hamilton, James, 20

Hamilton,

10,

Hermann,

94,139-140,

sis.

Hamilton, William

15;

of Frangais,

H e a t h , A . E . , 14, 79, 1 0 3 - 1 0 5

79-80,

(of

15;

com-

10,

of Warren, 10-11, 86; of Wes-

Heaviside-Gibbs

William

of

of Argand,

15; o f G a u s s , 9 - 1 0 ; o f M o u r e y , 15;

Hawkes, H. E.,

Sir

17-

o f early ideas o n geo-

H a y w a r d , R. B., 119

108

August,

of science,

representation

F.,

107

5-6,

in

19, 23

146-147

Johann

253;

and me-

H a r t , A . S., 4 0

G r e s w e l l , Rev. Richard, 34, 35

Hamilton,

1843, 3 0 - 4 2 ;

H a r d y , A . S., 1 3

Green, George, 77, Theorem,

195-200,

sel, 10, 15

Graves, John T., 31, 34, 44-45, 87, 247 Graves, Robert Perceval,

192,

249-250,

c h a n i c s a n d optics, 17, 2 1 - 2 2 ; position

15;

Graves, Charles, 31, 39, 42, 4 5 - 4 6

Grunert,

after

of Servois,

104, 248

Gregory, D.

of,

of Buee,

Grassmann, Robert, 89, 92

tory of,

life

metrical

G r a s s m a n n , H e r m a n n , Jr., 9 2 , 108,

264

224-225,

240,

187-189,

O'Brien,

81;

130-

1 9 - 2 0 , 23, 60, 64, 69, 77, 81,

attempts to secure a u n i v e r s i t y posi-

5 5 ff.;

217,

128,

a n d C a y l e y , 3 5 , 2 1 1 - 2 1 4 ; a n d Grass-

239-

233-236,

248-250,

123-126,

102-107,

Giinther,

47-49,

117,

207;

system

162-177, 19,

33,

96,

of

192-193, 99-101,

120,

125,

150,

154,

157-158,

182,

184,

187,

189,

191,

178-

194-198,

202-204,

208-211,

214-215,

218,

220-223,

229-230,

232-236,

239-

241,244-245,251-253; and Clifford, 142-143,

165; a n d G i b b s ,

154, 157,

Index

163,

168,

177,

192-193,

216, 242, 251-253; 163,

174,

217;

139,

143,

150,

174-177,

and

Maxwell,

162-166,

131,

225-228;

and

Tait,

137,

143,

162-166,

168,

170-

173,

177,

192,

196-198,

219;

early

162-163;

174-176; ings,

given

his

writ-

R., 245

Hegel, Georg W. Helmholtz,

Kant, I m m a n u e l , 24-25, 44 Kelland,

133-134,

Lord

Friedrich, 79

119-120,

Thomson),

129-131,

147-149,

133,

Shunkichi, 218, 223

Kipling,

Rudyard, 214

176, 13

Knott,

Herschel, Sir John, 33, 3 5 - 3 7 , 40, 119,

145,

Hiebert, E r w i n N., x,

145,

129,

154

Penyngton, 38

K l e i n , F e l i x , 9, 15, 92, 96, 107, 155, 161, 180, 223, 227, 236, 2 4 4 - 2 4 5

Klinkerfuss, E.

H e r o (of Alexandria), 6,

vii,

135,

154, 205, 2 5 1

Kimura,

Kirkman, Thomas

Henrici, O., 234, 244

146,

146

(William

Kirchhoff, Gustav Robert,

Hermann von, 118-119,151,

Hertz, Heinrich,

120-121,

Kellogg, Oliver Dimon, Kelvin,

154, 205

119

174, 1 7 6 , 2 2 6

148

F. W . , 160,

179

C a r g i l l G i l s t o n , 9 , 15, 9 6 - 9 7 , 9 9 -

100,

108,

178,

198,

120-121, 200,

144,

205,

148,

157,

209, 216, 2 2 1 -

223, 229, 240, 244, 255

H i l l , T h o m a s , vii, 3 7 - 3 8 , 45, 126, 147 Horace,

Philip,

203, 232, 240

life of, after 1892,

reception

174-177

Hedrick, E.

K

169-171,

122,

life of,

219,

200-201,

and Grassmann,

K o r m e s , J. P., 102

184

Kramar, F. D., 14

H o u e l , Jules,

13

Kummer,

Ernst Eduard,

81

Hutton, James, 95 Huygens, Christian, 3, Hyde,

Edward

13, 8 0

Wyllys,

54,

104-105,

232, 240, 246

L Lacroix, Sylvestre-Frangois, Lagrange, Joseph Louis, 119, 149,

I

Imaginary

numbers.

241-243

See

complex

Lame, num-

bers 128,

147

83,

Laplace, Pierre-Simon, 20, 45, 60, 119

86 Association

103

for

Promoting

the Study of Quaternions and Allied of Mathematics,

origin

of,

218

178

L a t t e s , S., 2 4 3 Lavoisier,

International

Systems

M.,

Larmor, Joseph,

product of Grassmann, 70, 80-81,

Antoine-Laurent,

Lectures

on

Quaternions

121,

Leibniz, Gesellschaft

der

Wis-

179

Jahnke,

Eugen, 231-233;

234, 237-238,

240-241, 243, 245, 253 Jammer, Max,

14

Jarrett, T h o m a s , 3 4 - 3 5 J a u m a n n , G., 238, 246

Marie,

Gottfried

13-14,

37,

geometry

senschaft, 5, 80 Jacobi, Carl Gustav, 21, 33, 50

(Hamilton),

234

Legendre, Adrien

Jablonowskische

95

19-20, 25, 35-39, 40, 46, 77, 85, 87,

L e e , Sir George,

J

103

Gabriel, 84

Laquiere,

Infeld, Leopold, Inner

13

199

Laisant, Charles Ange, 53, I g n a t o w s k y , W . v., 238;

10,

14, 21, 33, 57,

80, of

10

Wilhelm, 88,

104,

situation,

3-5; 247; 3-5;

2, and and

Grassmann, 4 - 5 , 80, 88 Leverrier,

Urbain, 21

Lie, Sophus,

13

Linear product of Grassmann, 63, 70, 83 Linear

vector

function

and

Foppl, 227;

a n d Gans, 231; a n d G i b b s , 152, 1 5 6 -

Joly, Charles Jasper, 46, 240

157,

J o n e s , P. S., 14

63,

Juel, C., 13

202;

168, 76,

187, 202; a n d G r a s s m a n n ,

187;

and

and

H a m i l t o n , 36, 41,

Heaviside,

167,

172;

and

265

Index

Maxwell,

134-136,

139,

152;

and

Tait, 123, 134, 1 3 5 , 2 0 2 ; a n d W i l s o n ,

222

229 Lloyd,

Milton, John, 223 M i n c h i n , G e o r g e M . , 2 0 9 - 2 1 1 ; 169, 180,

Humphrey,

21-22

Mobius,

Lobachevski, Nicolas, 95 Lodge,

Alfred,

191-192, 204;

174, 203,

206,221-223

Ferdinand,

48-52;

4,

106-107,

127,

151,

158,

231-234,

239, 245, 248; his barycentric calcu-

Lodge, Oliver, 223

lus,

L o e m k e r , L e r o y E., Lorentz,

August

34, 47, 69, 82, 8 5 - 8 8 , 94, 102-103,

13-14

48-52;

relation

to

50-52, 57-58, 73-74,

H e n d r i k Antoon, 43,

155, 231,

238-239, 242-243, 253

Grassmann,

78-80, 83-84,

89 M o l e n b r o e k , Pieter, 218, 223

Lotze, Alfred, 243

Moritz, Robert Edouard, 45

Lowell, A.

Morris,

L a w r e n c e , 126, 147

Professor, 228

M o u r e y , C. V.,

M

11; 5 , 8 ,

15-16, 34, 82,

247

McAulay, Alexander, 189-190,

193-195,

M c C o r m a c k , T h o m a s J., MacCullagh, James, MacDuffee, C. Macfarlane, 196,

139,

nions, 28-32; to Wessel, 7 - 8 190-191,

206-207,

144-146,

192,

201,

208,

240,

245-246,

121, 1 9 0 - 1 9 1 ,

252,

209-210;

148-149,

218-219, 259;

195-

173,

221-223, and

Tait,

D O T P R O D U C T (modern): 45;

and

lar

28-29,44,

Burali-Forti's

product,

236;

and

product,

internal

Clifford's

140-143;

and

scaDe-

m o u l i n , 181; a n d Gauss, 105; a n d Gibbs'

196, 203, 2 1 7

156,

Mach, Ernst, 14

direct

160,

product,

185-186,

152,

187,

155-

198-199,

205; and Grassmann's inner prod-

M a d d o x , J. R., 14, 44 Marcolongo,

entities:

G r a s s m a n n , 5 7 ff., 6 8 ff.; t o q u a t e r -

34-35

Alexander,

ix, 96,

of vectorial

A C C O R D I N G to Bellavitis, 52-54; to

14

C., 4 4 - 4 5

203-204;

Miiller, E m i l , 233, 245 Multiplication

196-201, 203, 207, 211, 221, 223

Roberto,

235-237;

240-

uct, 70, 8 0 - 8 1 , 86, 248; a n d Grassmann's

241, 243, 245-246, 253

linear

product,

63;

and 32,

Hamilton's

Matrices, 76,

248;

a n d H e a v i s i d e ' s scalar p r o d -

Maxwell,

188

James

33,

uct,

163-172;

124,

142,

144,

and

Knott,

159-160,

172,

180,

Lodge, 204; and Macfarlane, 190-

232-234,

239,

Clerk,

109-110,

119-120,

146-149,

151,

190, 242,

193-194, 250-253;

ternions,

129,

139,

150,

138;

and

150, 219,

127-139;

211, began

study

of qua-

132; a n d G i b b s ,

137,

152-155; and Grassmann, Heaviside,

162-166, 225-228;

129,132-133, Mehmke, Rudolf,

137,

169-171, and Tait,

139,

143,

174-177, 117,

128-

sur

ant),

les

sommes

geometriques"

Michelson, Albert,

les

diffe-

(Saint-Ven-

projective O'Brien,

191, 2 4 6

154

208;

233; and

product,

181;

and

product, 9 7 ff.;

236;

Mobius'

51-52;

and

and Peddie, 209;

a n d q u a t e r n i o n scalar p r o d u c t , 32, 1 2 2 - 1 2 3 , 1 5 2 , 1 5 5 - 1 5 6 ; a n d Resal,

44,

and Tait,

122-123,

P R O D U C T 45;

and

236-237; product, 181;

81-83

Merriam, Mansfield,

266

et

Jahnke,

internal

Massau,

and

155-156

(modern):

Bucherer,

Burali-Forti's

(Buee), 9, 15

rences

and

CROSS

102

and

201-202,

191, 2 0 3 - 2 0 4 , 207, 210; a n d M a r colongo's

106;

137-138

" M e m o i r e sur les q u a n t i t e s i m a g i n a i r e s "

"Memoire

scalar

product,

M a s s a u , J., 1 8 1

vectorial

and

28-29,

230;

and

product,

Clifford's

vector

140-143; and Demoulin, Gibbs'

152,

155-156,

205;

and H.

skew

product,

185-188,

198-199,

G. Grassmann's geo-

Index

metrical product, 57 ff.; and H. G.

O'Sullivan,

Mortimer, 37

Grassmann's outer product, 70 ff.,

Osgood, W.

F., 228

86-87, 248; and J. G. Grassmann's

Ostrogradsky,

geometrical

product,

Ostwald, Wilhelm,

Hamilton's

vector

248;

57-60;

and

product,

32,

Outer

uct,

163-172; and

and

Jahnke,

Knott, 201-202, 208; lane,

and

and Macfar-

190-191; 203-204, 207, 210;

and

Marcolongo's

uct,

236-237;

vectorial

and

Massau,

prod-

multiplication

87,

Ignatowsky, 231-233;

146

152

points, 74-75;

and Heaviside's vector prod-

238;

Michel,

of Grassmann:

248

P Page, L e i g h , 157, Parallelogram

179

of

velocities

and

and Mobius' geometrical product,

Parish, Charles,

Peacock, George, 14-15, 34,

O'Brien,

Peddie, 209; tor

product,

155-156;

97ff.;

and

a n d quaternion vec32,

122-123,

152,

a n d Saint-Venant's geo-

metrical product, 81-82; and Tait, 122-123, SCALAR

See

Dot

prod-

VECTOR PRODUCT.

Pearson, Karl, 141, 149

Peirce,

Benjamin,

125-127;

147,

38,

Peirce,

Benjamin Osgood, 228 Charles

44, 126,

44-46,

158, 2 5 0 - 2 5 1

Peirce, See C r o s s p r o d -

Santiago

Saunders,

28,

147

Peirce, James M i l l ,

uct

108

Peddie, W i l l i a m , 208-209; 216, 222-223

109-110,

PRODUCT.

108

Peano, Giuseppe, 232, 235-236, 245

155-156

uct

forces,

2, 34-35, 58, 6 1 - 6 2 , 219, 247

181;

57-60;

and

of

of vectors, 70 ff, 8 6 -

126,

147, 2 2 8

P h i l l i p s , A. W . , 160, 179, 2 2 8 N

Pierpont, James, 228

N a b l a . See D e l , o r i g i n o f t e r m n a b l a , 1 4 6

Plarr, Gustave,

Nagel, Ernest, 79,

Plucker, Julius, 22, 50, 91,

104-105

N a t i o n a l A c a d e m y o f S c i e n c e ( U . S.), 2 2

Poinsot, Louis,

Neander, J. A. W., 55

Point

Neugebauer, O.,

108

Newcomb, Simon,

Poisson,

159,

103

184

analysis

73ff,

154

121

of

Grassmann,

57-58,

188, 229, 2 3 6

Simeon-Denis,

129-130

N e w t o n , H. A., 126, 147, 151

Preyer, W., 93

N e w t o n , Isaac, v i i , 18, 3 7 , 5 5 , 1 2 7 - 1 2 8

Price, D e r e k J. de Solla, x, 148

N i c h o l , J. P., 38, 46

Projective product of Mobius, 51-52

N o b l e , C. A., 245

Pythagoras,

37

Norgaard, M a r t i n A., ix, 13 N o t a t i o n : c o m p a r a t i v e table of, b y S h a w , 179,

244;

debate

on,

in

early 20th

century, 218, 236

Q Quaternions: Gauss'

early reception supposed

Note system explained, ix

Hamilton's

Noth, Hermann, 93

terest

in,

O

201 Directionens

Multiple

tion

of,

193, analytiske

i n g " (Wessel), 6 - 8 , "On

9; in-

related

interest

to

in, in

countries, U.S.,

114-

117, 1 2 5 - 1 2 7 , 147, 251; later recep-

O ' B r i e n , M a t t h e w , 9 6 - 1 0 1 ; 47, 102, 108,

" O m

of,

d i s c o v e r y of, 2 6 - 3 3 ; as

114-117;

of, 3 3 - 4 2 ;

discovery

Algebra"

Betegn-

ff,

184-185,

198-200,

220, 240, 250-251;

189-190,

213-214,

219-

number of publi-

cations on, 4 1 , 1 1 0 - 1 1 7 , 2 5 0 ; proper-

13 (Gibbs),

110

196,

158-

160

ties

of,

taken

by

Ball,

28-33, Airy, 206;

122-124; 35;

by

view

by Allegret, Bellavitis,

of 39;

" O n Symbolic Forms" (O'Brien), 97-101

by

Open product of Grassmann, 76

Bolzani, 39; by Cayley, 211-213; by

39;

by

267

Index D e M o r g a n , 34; by Fischer, by

Gibbs,

187-189, mann,

94;

by

side,

162-164, by

Hill,

vii,

Joly,

240;

251;

by

208,

by

by

166-173,

by

by

193-195;

vii,

34;

by

119-120, 204,

McAulay,

by

Macfarlane,

by

views,

190,

251;

18-19;

by

Peddie, 208-209;

by

B.

34;

125-127,

251;

17;

by

Somoff, 234;

P.

O.

by

38;

by

Schlesinger,

by Tait,

183-185,

186-187,

196-198,

211-214;

by

189,

Ludwig,

229,

15 D., 55

102

104

Schrodinger, Erwin,

Servois,

17, 21, 4 3

Frangois-Joseph,

10;

11,

13,

15-16, 34, 82, 85, 247 Shaw, James Skew

Byrnie,

product

160,

179, 2 4 4 - 2 4 5

of Gibbs,

185-187,

Smith, Charles

152,

155-156,

198-199, 205

F., 244

S m i t h , D a v i d E u g e n e , 1 3 - 1 4 , 16, 43, 45,

Whittaker,

18-19

224,

Searle, G. F. C., 179

Schrodinger,

117-125,

102,104,

154-155,

244

Schlomilch, Oskar,

Peirce, 3 8 -

39,

152,

Schliermacher, F. E.

modern

Nichol, by

107-108, 232,

of

121

Schlegel, Victor, 54, 85, 91-94,

Schroder, Ernst,

195-196,

multiplication

Friedrich, 79

Scherff, von, G.,

206,

203-204, 206-207,210; by Maxwell, 130-138,

Schelling,

See

entities

189-

MacCullagh,

190-191,

product.

vectorial

200-

119;

Jarrett,

201-203,

240;

Heavi-

192,

35-37,

Kelvin,

Knott,

by

Hamilton, 30-

223;

Scalar, o r i g i n of t e r m , 3 1 - 3 2 Scalar

Grass-

Graves, 34;

by

37-38;

207, 190,

T.

Herschel,

by

185-186,

205;

Hathaway,

201;

by

by J. 34-35;

18, 254;

172,

198-200,

Greswell; 31;

152-153,

102 Smith, Percy, 228 Smith, Robertson, Sohncke, L.,

R

146

104

Somoff, Joseph, 106, 234, 245, 260 Rayleigh,

Lord

154,

(John

William

Strutt),

182

243, 245, 253

Reference system explained, ix Regressive

product

of

Spottiswoode, William,

Grassmann,

75-

76, 86, 233 102

Staude, O., 233, 245

Resal, H e n r i , 106, 181, 234, 245 Riemann,

Bernhard,

St. C o h n - V o s s e n , 108, 181, 2 4 4

91

S t e e l e , W . J.,

R o g e r s , S t e p h e n J., x Romorino, Angelo,

14

Rothe, H e r m a n n , 102,

Steinmetz, Charles 146, 243

Stokes,

Edinburgh,

35,

197, 200

Rukeyser,

Gabriel,

119,

124-125,

Theorem,

124-125,

168; early history of,

135,

164,

146-147

S t r u i k , D i r k J., 125, 1 4 7

Society of London, 9,22,163,175,

192,

Stokes'

198,

212-214 Royal

George

135, 1 4 6 - 1 4 7 , 154, 164, 168

Royal Irish A c a d e m y , 22, 2 9 - 3 0 , 31, 35 of

Proteus, 244

Stern, M. A., 92

A c a d e m y of D e n m a r k , 6

Society

118

Steiner, Jacob, 50, 55

Rosse, L o r d ( W i l l i a m Parsons), 22

Royal

138

Stackel, Paul, 44 Stahlman, William D., x

Reinhardt, Curt, 50,

Royal

Somoff, Pavel Osipovich, 234-235; 242-

Sturm, Rudolf, "Sur

Muriel,

178-179

les

clefs

104 algebriques"

(Cauchy),

83-85 S y l v e s t e r , J. J., 9 3 , 154, 1 5 8

S

S y n g e , J. L . , 17, 4 3 , 46

Salmon,

George,

System

35-36

der

Raumlehre

(Schlegel),

91-93

Saint-Venant, C o m t e de (Ademar Barre), 47, 56, 81-85, 87-89,

102, 106-107,

T Tait,

158,248

Peter

Guthrie,

186-187, Sarton, George, 77,

268

104-105,

107

189,

117-125,

183-185,

196-198,213-214; 9,

Index

15,

32-33,

136,

109-110, 151,

193-195,

206-209,

188, 221,

45,

144-148,

225,

131,

169,

232-233,

235,

134-

178-180,

Varignon,

216,

218,

Vector

240,

250-

Vector

255; a n d del, 41, 1 1 8 , 1 2 3 - 1 2 5 , 1 3 1 133,

184,

Cayley,

202,

203,

184,

250-251;

211-214;

and

and

150,

152-158,

196-198, and

177,

199-200,

219-220,

H a m i l t o n , 36, 39, 41,

121,

183, 223;

137,

143,

and

168,

Analysis

.

237-238,

its

.

of

.

J.

Founded

Willard

product.

118-119,

Vector types:

122,

74,

See

Vectorial

flux,

131, 147; l i n e b o u n d ,

systems,

and Knott,

N - D I M E N S I O N A L :

202, 204; a n d M a c f a r l a n e , 121, 1 9 0 196,

203,

217;

and

132-133,

202;

Maxwell,

137-138

83-85;

Forti,

Taylor, Richard,

143;

of Grassmann

182

235-237; of

Teubner, B. G., 230

of Gibbs,

Thales,

56 ff.;

Theorie

127 der

Ebbe

und

55-57, 60-63, "Theory

of

Flut

2 7 ff.;

187

Conjugate

Algebraic Couples" 27,

Graves,

(Grassmann),

Functions,

or

of

(Hamilton), 23-

235-237;

T h o m s o n , J . J., 1 5 4

Mobius,

William

(Lord

Kelvin).

101;

See

T h o m p s o n , S i l v a n u s P., Timerding, H. Treatise

Electricity

Treatise tion

160,

128,

Quantities

the the

Geometrical Square

(Warren),

Truel, Dominique, T u c k e r , R.,

Roots

Argand,

143, 1 5 2 -

by

Buee

10;

14

of

106;

of 97-

of Saint-Ven-

Somoff,

106;

of

of Wessel, 7-8

10;

(?),

181;

O'Brien,

by

9;

by

Mourey,

sought

for

Bellavitis, 53;

11;

Frangais, by

9-

Servois,

144

of Argand, 9-

of Bellavitis, 52-54;

9;

of Frangais, 9-10;

9;

of Mourey,

11;

11;

of Buee,

of Gauss, 8-

of Warren,

10-

of Wessel, 6 - 7

Vektor analysis Veronnet,

(Valentiner),

235,

241

Alex, 243

Victoria, Queen, vii, 37

36

Uylenbroek,

of

Macfarlane,

of Marcolongo,

T W O - D I M E N S I O N A L :

11

U

T.

10

Representaof Negative

by

of J.

162-177;

of

of J.

122-124;

10;

9;

Hamilton,

Massau,

48-52;

by

Turner, G. C., 234, 244

Ulysses,

31;

of

T H R E E - D I M E N S I O N A L :

162-163, 251, 252

on of

134-139,

of

Magnetism

and

of

(?),

195-196, 203-204, 206-

81-83;

Tait,

145

E., 239, 243, 253

on

(Maxwell), 153,

31;

of Resal,

ant,

Kelvin

140-

31;

of Grassmann,

Graves,

207, 209-210,240;

Thiele, T. N., 13

Thomson,

Gauss

231-233;

190-191,

6 9 ff.

Burali-

27,

Heaviside,

Jahnke,

188

of

150-162;

27,

of

of Clifford,

181;

of C.

186,

of Cauchy, 6 3 ff.,

DeMorgan,

Demoulin,

31

Gibbs,

and Peddie, 209;

Tannery, Jules, 233, 245

Tertullian,

26,

and

T H R E E - D I M E N S I O N A L :

146

of

232

119-120;

191,

the (Wil-

multiplication

F O U R - D I M E N S I O N A L :

117,128-129,

upon

entities

196-198, 200; and Kelvin, 198, 2 0 1 -

(in

Gibbs

177, 192,

121,

241

Applications

son), 2 2 8 - 2 2 9 , 230, 2 3 3 , 2 3 4 , 237 Vector

vectorial

170-173,

and

Lectures

222;

Heaviside,

162-166,

(Coffin),

Analysis

Vector

Clif-

185-189,

Analysis

Russian, by P. O. Somoff), 234-235

ford, 142, 149; a n d G i b b s , 122, 137, 143,

Pierre, 72

Vector, origin of term, 31-32, 240

Vorlesungen

13-14

uber

die

Vektorenrechnung

(Jahnke), 231-233, 237; 240, 241 V

Voss, A., 14

Valentiner, H., Valentiner,

Vrai

13

Siegfried,

243, 245, 253

235;

238,

241-

Theorie des ginaires

des

quantites

quantites (Mourey),

negatives

pretendues 11,

et ima-

16

269

Index

w W a l l i s , J o h n , 6, 11, 14 Walmesley, Charles,

Wills, A

P., 138,

Wilson,

Edwin

104,

14

Windred, G ,

82, 86, 106

Wood,

155

W e s s e l , C a s p a r , 6 - 8 , ix, 5 , 9 , 11, 1 3 - 1 5 , 34,

109, 2 4 7 162

Wheeler,

107-108,

222,

Phelps,

White, J

Whittaker, 45,

270

62,

178-179,

14

De Volson, 39, 46

Woodward, Robert Simpson,

191, 2 4 6

Y Young, Thomas, 95

21 Z

S , 112, 145 Alfred North,

Whitman, Walt,

228-229, 162,

178,

246

Whewell, William,

Whitehead,

157,

Wordsworth, William, 22

Wheatstone, Sir Charles, Lynde

146,

230-235, 237, 240, 242-246, 253

Warren, John, 10-11, 5, 8, 15-16, 25, 34,

Weber, Wilhelm,

125,

149 Bidwell,

104-105, 244

169, 210, 214

Edmund

Taylor,

Z e u t h e n , H. G , ix, 13 Ziwet,

18-19,

43,

129, 131, 1 4 8 , 1 7 6 , 1 7 9 - 1 8 1 , 2 2 2

245

Alexander,

104,

106,

229,

244,