The Sequence of Ideas in the Discovery of Quaternions

The Sequence of Ideas in the Discovery of Quaternions Author(s): E. T. Whittaker Source: Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, Vol. 50 (1944/1945), pp. 93-98 Published by: Royal Irish Academy Stable URL: http://www.jstor.org/stable/20520633 . Accessed: 27/02/2011 09:47 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ria. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Quaternion Centenary Celebration.

THl,E SEQUENCE

OF

IN THE

IDEAS

93

OF

DISCOVERY

QIJATE INIONS. BY E. T. WHITTAKER, Edinburg THE first communiication as printed how years but

later

said3

in the Tranisactions

that

then

uniits

13,

1843,l.

gives

the multi

aiiy

inidicatioln of

' lead Novemiiber

headed

for in July,

13, 1843,"

2

1846, Hamilton

of

for the Transactions

four

appeared

wlhieh

melmoir

be of later date:

"reservinig

to the Academy,

i, j, k, withiout

extenisive

is also

formii must

lie was a more

Academy

The

obtaiined.

in its published

by Hamiltoni

for November

for the qutaterinionic

they had been

made

oni quiaternions

in the Proceedings

plicatioiu-formulae

h University.

Irish

the Royal

anid systematic accouint of his researches on the I itself lie mleintions that some slheets of it were

comiplete

subject,"5 and in the memoir

the method

of

in that paper represents the hiistorical development For information ideas in his owIn mindi. this we regardinig miust depenid

of

being

printed

exposition

in Jtunie, 1847.

sources,

otlher

There

is thus nio assuraniee

that

adopted

particularly

the Letter

in the Ptilosophical

published October

17,

1843-the

to John

1'. Graves,

Esq.,

in Decemlber, 1844, Mag(azine after the actuial discovery-amid

day

which

but was the

on was

dated lengthy

Preface to the Lectures on Quateritiots, which is dated Jtune, 1853-nearly later-togetlher

teni years

with

in a commuinication

made

different

As

memoirs.

some of

harmnonise very well with which

must

occasion

be filled

of

Hamilton's

the

first aspect

centenary,

for

an

there attemnpt

11,

of Hamilton, 1844,' and in

niot at first sight

does

rest, and as there

in the Transactions . . are

QLiaternioils. and conception,

algebraic

Couples

Academy

in Novenmber,

seventeenth

the

the material

in by conjecture,

Life

on November, are

some gaps

is perhaps

to

seeni

in the story

justification,

on tlle

to trace

the

development

tells

that

"the

of

ideas.

The memoir respecting

givenii i Graves'

indicationis

to the Academy

volume

...

of 1848

to be considered of

a continuation

wlhich 1833,

were

first

and were

as being, those

T'his

R.I.A.,

*Proc. 5Proc.

R.I.A., R.I A.,

3(1846), 3 (1846),

276.

3 (1844-45),

296. 1-16.

researches

least

in their

concerning

to the Royal

in the year

latter

iProc. 2 (1843), 423-434. R.I.A., 3Trans. 21 (1848), 199-296. R.I.A.,

at

speculations

comnmuniicated puiblished

of its Transactions."

3Proc.

us

1835,

is ani algebraic

Irish in the paper,

94

Proceedings of the Royal Irish Academy.

in which

e.g. the comnplex number

couple"

+

b /-

1

. (a,,

b2)

1848 memoir

=

a)

follows

-

(bQ a,

this up by

as a " numnber

is regarded

-

(a, b) with themultiplication-law

(b1, The

a

b

b2 a2,

initroducing

b, a2)

+

a,

in a purely

quaternionis

algebraic fashion, defining the units i, j, leby means of substitutions. But is evidence

there

that

this was

not

In 1844 Hamilton

discovered.

led " him " to conceive

the way

in which

had

they

been

told the Academy that "what originally of quiaternions

his theory

. . .was his desire

to fornm to

himself a distinct conception . . . of a fourth proportional to three rectangular of those lines were takeii into account: as lines, when the DIRECTIONS Mr. Warner

and Dr.

Peacock

had

shown

how

and

to conceive

the

express

fourth proportional to any three lines having direction buit situiated in one common plane." "The first conjecture,"he says," " respecting geometrical triplets, which

I find noted

among my

papers

(so lon)g agro as 1830) was,

that

while lines in spacemight be added according to the same rule as in the plane, they rniht be multiplied by multiplying their lengths, and adding their polar angles. if we write Now

In the method

then as that of Mr. Warren,

to me

known

. . ." the Rev.

John Warren,

A.M., was

a Fellow

and Tutor

of Jesus

who had ptublished in 1828 a book A lTreatiseon the College, Camrrbridge, qf the Square

Geomtetr-ical Representation

Boots

of Nlegative

it was

Quantities:

essentially a description and elaboration of what to-day is called the Argand diagram,

representing

the complex

a + b /-

number

1

by a vector whose

rectanigular components are (a, b). From the above extracts it is evident that Hamilton that already

had

read

this work

in 1830-three

almost

years before

as soon as it was

and

published,

the date of the memoir

oni algebra

it had suggested to him the problem ofmultiplying together two vectors in three-dimensional space. In 1834 and 1835 he devised a "general theory of triplets":

: and

"there was,"

he says,8 "a motive

which

induced me

then

to

attach a special importance to the consideration of triplets . . . This was tlle desire to correct, in some inew and useful (or at least interesting) way, calculation with geometry, through some unidiscovered extension, to space of three dimensions,

of a method

or represenitation,

of construction

which

had

been employed with success by Mr. Warren." The maticians,

and

the brothers

by The 6

was

problem

various John

final and successful

Preface

to Liotu

res

(39).

very

inuch

attempts

in

at a solutioni

and Charles attack

the minds were

Graves

and

by Hamiiilton

must

7Preface

to Lectures

of contemporary made

by Augustus

about

matlhe this

now be described.

(20),

(23).

time

De Morgan.

8 Ibid.

(31).

95

Qu*aternionCentenary Celebration.

In the Argand representation of a vector in a plane by an ordinary complex quanitity, the multiplication of vectors is deterninied by the algebraical formula 2

where

(x +

iy) (x'

+

X

= xx'

- yy',

= - 1,

i y')

=

+

X

jY

= xy'? fx'y

Y

Now the formiiulafor themnultiplication of determinants9 gives x 2 ++y2

xx'-yy'

- yy'

xx'

or

xt2

-y -

+

(1)

g,'

J'2 + y

(x + y2)(X2` + y2)j

(2)

so (xA + y2) i is the nodaluts of mucltiplication, i.e. the funietion which, in the product, has the same value as the product of the corresponding fuinctioinsof the factors. In 1843 Hamilton, abandoninig his previous notatioin, proposed to repre senit a vector

in three-dimensional

are enitities such that

x + i y + jz,

space by

j2 _ j2

1

i and j

where

(3)

_

their other properties being as yet undetermiinied. A second vector wouild be

then

represented

by

+

x'

i y' + jz',

and

obvious

the

(1) is a case of (Cauchy's theorem on the multiplication namely, Ix2+y2+Z2

x

xx'-yyj-z'

I

+t

-

+y

X +y2

| Z2

2

-y t

XJ

x

analogue

of

of two arrays, 2

-z

y

2

z

,

x

or (X2 + y2

+ z2)(X'2

+ y'2

+ Z'2)

=

(X'

_ yy'

+

(ry'

+ x'y)2

-

=t)2 (X i + X'Z)

+

+

yz)2

(yZ'

T'he left-hand side of this equation is the product of the squares of the moduli of multiplication of the two vectors, but the right-hand side containis foutr, not

three, squtares.

that

the

their

description

operation vector

geometrical which

J, we

not wheni

see that

the ratio of the leingths

on

Pondering of

operatioins triplets

but

performed in order of a and

this, however, three-dimensional

qtiadruplets:

for if we a converts

to specify

betweeil

consider

e.g.

for the

it into another need

to know

them, and

the node

this operation,

j3, the angle

requir ed

space

on one vector

to see

came

Hamilton

we

The determinantal theorems here quoted had long been known, though the notation now used for determinants had been introduced only two years previously by Cayley, and Hamilton does

not

refer

to it in this

connexion.

I use

it here

because

it brings

analogies which, as I believe, determined the course of Hamilton's thought.

into

prominence

the

96

Proceedings of the Royal Irish Academy.

and

of

inclination

altogether:

the

plane

he was

thus

in which

prepared

they

lie-that

to accept

is, fouir nuimbers (4) as

equation

the equiation

giving themodulus of multiplication of the product of two vectors. we have on multiplying

Now

+

(x + jy + jz)(x

out

= (xx' - yy' - zz')

*y + yz)

+

us compare

Let

(5) with

? j

+ 4'y)

i (x'

It appears

(4).

+ x'z)

(xz'

from

+ {jyz'

+

(5)

jizy'.

(5) thiat the equiation

for

themodulus of multiplication of the two vectors, whiclh on the left-hand side of

consists

side

on the right-hand + x'y),

(x,y'

(A2 + yt

the product the

with

order

this

(4) that

to obtaini

last

the creative

lt began

which

Boole's

matrices,

whichi

process broke

for

k

i j,

(A2 +

z2)(X'2

iy

-+ jz)(x'

in

synmbolism.

Cayley

but all tihe

and Sylvester's Gibbs'

Ausdehnungslelhre,

of qualntum-mechanics.

algebra

we have

(4) and

y2 +

and

have

quaterniomis,

rules-

Grassumaiui's

logic,

= -t

ij and the equations

2

y'z)

of mathematical

not only

yielded

'dyadics, and the Heisenberg-Dirac Writinag

of

which

it appears from the

we niust clearly

from the old

away

symbolic

square

(y z' -

be

in the history

the suipremiie moinent

othler systems

anid another

. But

have

muist

the square

= - Al.

ji This was

z y',

Z' 2),

zz' )

-

y'

square must

ij y z' + ji

this from

-y

ij y Z' + 5 i Z

corresponds to the terms comparison

xx'

(x z' + x'z),

of'-

square

+ y' 2 +

+ z2) (/2

the sqtuare of

i

(6)

(5) can now be written

+ y'2

+ z2) 2

X22

+ jz')

= X

+ Y2

+ Z2

* W2

(4a)

iY

+ jZ

kW

(5a)

where (x

+

From

(3) and

ij2 =

2

+

iy'

(6) we cani deduce

42 -I

+

the

immediately

fundameintal

jkk -

j j=i

i

equations

-kj,

(7)

15 - 5 = - i/c Hamilton

now made

a fresh start, writing w

At

one time he thought

prevailed. the equations

+

ix

of calling

+ jy

the quiadriiionoial + kz.

it a grammarithm,

The multiplication-theorem (7), and the new science

but

for quaternions is founded.

the iname quaternion follows

at once

from

Qutaternion Centenary Celebration.

97

The difficulties regarding triplets, which so long baffledHamilton; 10were solved in an entirely different way by De Morgan,"' whose triple algebra is not without interest. As we have seen, Hamilton had laid down the condition that the mnodulus of multiplication of the triplet a + hi + ej be

should

(a2 + bh +

el) I .

It

is certain

'that there

be a systenm

cannot

of triple algebra with this modulus: for the problem of finding three squares in which accented and unaccented letters enter symmetrically, and of which

the sum

showni

to be

sphere,

each

is equal

equivalent

(a2 +

to

to

the

b" + c'2),

'I)(a'2+ of

problem

is antipodal

of wlhich

b2 +

to both

three

finding

the other

can

points

be

on

a

two.

It is, however, not necessary that the modulus should be a symmetric function

of a, b, c, anid De Morgan

systeims

of triple algebra

laws

are

obeyed.

showed

can be devised

Thus,

the

denoting

if this condition

that, in which triplet

all

a + bi

by

is dropped,

the ordinary

algebraic

+ aj,

if we

impose the rules p 4

J

jt

the formuila for the prodctit

then

of two triplets

of multiplication

moduluis

1,

jJJ=

is

= (bh' + cb' + aa') + (ah' + ba' - cc') i + (ac' + ca' - bb')j.

(a + bi + cj) (a' + b'i + cj)

The

,

=

(a? +

is

b2 + c2 + ab

+ ac

- bc)t

if

for A

=

we have

+c?ba

bc'

B

taa'a

=

+

ab'

- CaC'

ba'

C

=

ac'

+

ca'

hb',

the identity

(a2 + b6 + a2+ ab + ac - bc) (a"' + b2 + a'2 + a'b' + a'c' - Y'a') = A2+B2+

For

C2+AB+AC-BC.

this system the associative, commutative, anld distributive laws are

all valid.

Anothier

of De Morgan's

proposal

was

to retain

and distributive, but to surrenider the associative law; instance with

the triplet

a + i2

10 William

=

b i + Cj j2

=

(aged 9) and Archibald Henry " Whereto be was obliged you multiply triplets can only add and subtract them."?Graves' life. Edwin

?

11Trans.

Camb. Phil

Soc,

* (1844),

241.

=j

if we

impose

the commutative

this happens for

the rules

I

" Well, (8) used to ask at breakfast, Papa, can to reply with a sad shake of the head, " No, I

98

Proceedings of the Royal Irish Academy.

Since

is not equal

(ii)j

to

i (ij),

the associative

law does not hold.

We

should now have for the product of two triplets (a +

bi +

cj)(a'

=

+ <'j)

+ b'i

Iaa'

-

(b + c)(b'

The modulus of imultiplication is i (b + c)2

at

a!2 +

t 07)

(6'

-

+

+ c')

+ ba')i

(ab'

4

(ac'

+

ca')].

{(a2+ (b + c)'J : for we have identically -

'aa'

+ c'))t

(b + c)(b'

+ (ab'

+ ba'

+ ac' +

ra'.

De Morgani discussed the geometrical initerpretation of his algebras; but it

is not

as

that

in principle,

as

so simple

advantages

long afterwards)

(discovered

of quaternions.

that

that the only

have

quaternions

Moreover,

instance

fol

inidicated

by

linear associative

the

algebras

theorem over

the

field of real numbers in which division is uniquely possible are the field of real numbers, the field of ordiinary complex numbers, and real quaternions. In the competition of systems, Hamilton's survived as the fittest; and its rivals are now forgotten.

QUATERNIONS AND MATRICES. By A. W.

CONWAY,

University College, Dublini. Introduction. Tim

term matrices

was

first

used

in a paper

by Cayley

in French

in

Crelle's Journal (1855). A formal presentation of some of the principal mnatrix properties was given by him later in the Philosophical Transactions of the Royal Society (1855). Many properties were previously known to Hamilton. In his lectures on Quaternions (1853) he published the fact

that a matrix

a cubic.

Cayley

inferred

that

quaternion

satisfied

proved

it must

as a matrix

a certain symbolical in this case equation, this for the case of a quadratic and a cubie and

be true

in general.

IVM+

1 .e.

w

V

O0

0t

1 )

V O

y

i .t

e-0 y

C. S. Peirce

ax +

w +

thus

-

J

yz

i

t

expressed

is equivalent

1

+ t

1

0 )

i) z

a to

-iz

iz0 O0/

(1881)

/3 y +

z 0

i\

i

0)