The Story of Mathematics

STORY OF

EMATICS

.......... STORY OF

EMATICS

From creating the pyramids to exploring infinity

Anne Rooney

fIl

ARCTURUS

Aclmowledgements Wim thllnk, rv th= of my Flluhook f.und!;rho hllre htlped in i'llriOU! "".ry<, parriro/Ilriy .\fi,hlltl A ,ui {ZiJIlO Jilltd (lIllrrllrd FaCIIlrylCllmbridj(e FIlCIIlry/BMron lItA), Gordon Joly (London), John Nllllj(hrvll (Camhridge Aillm '68, The Open Univ/'rriry FIl,ulry),Jlldi SchofteM (London/GlIllrdilln Nru'!llnd .Iledill) Ilnd Bill Tb01f'P50n (LonJonlCllmhridge Fllro/rylCiry UK Fllmlfy).

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ARCTURUS

Arcturus Publishin g Limited 26/ 27 Bickels Yard 151-153 Bermondsey Su'eet London SE 1 3HA Published in association with

foulsham w,

Foulsham & Co , Ltd, Th e Publi shing House, Benne tL.. Close, Cippen ham , Slough, Berkshire SLl 5AP, England ISBN: 9i8-0-5i2-O:l41:1-9

This edition primed in 2008 Cop)Tight © 2008 Ar(!urus Puhlishing limited /Anne Rooney (http://l-mw.annerooney.(o,uk) All

right~

rese rved

Covcr design and al't dirc(tion: Heatriz Waller Design: Zoe Mellors The Copyright A(! prohibits (sul~e( t to «:rtain \'Cry limited excc ptions) the making of (opies of any ropyriglll \\'ork o r of a su bstantial part ofsu(h a \\'ork, induding tllC making of (opies by photo(opying or similar pro«:ss. Written pe rmission to make a (OpY or (opics must therefore normaU)' be obtained from the publishe r in fl(km(e, It is advisable also to (ons.uh the publishe r if in any doubt as to the le~rality of any (opying which is to be ullt!t;rtaken. British Library Cataloguing-in-Publi<:ation Data: a (ataloguc re(ord for tllis book is available from tlH~ British LiIH'fII')' I'rimed in China

Contents Introduction: The Magic of Numbers 6 Chapter 1 Starting with numbers 8 " ' here do numbers come from? • Numbers and bases • More numbers, big and small

Chapter 2 Numbers putto work 34 Putting two and

twO tO~,'ether

• Special numbers and sequences · Unspeakable numheN

Chapter 3 The shape ofthings 60 The measure of everything • Early breometry • Trigonometry

Chapter 4 In the round 92 Curves, circles and conics • Solid geometry • Seeing the world · Other worlds

Chapter 5 The magic formula 120 Algebra in the ancient world · The birth of albrebra • \Vriting equations • Algebra comes into its own • The world is

1l(.'Vcr

enough

Chapter 6 Grasping the infinite 144 Coming to terms with infinity . The emeq,rence of calculus • Calculus and beyond

Chapter 7 Numbers at work and play 166 Cheer up, it may never happen · Samples and statistics · Statistical mathematics

Chapter 8 The death of numbers 186 Set theory • Getting fuzzy

Cha pter 9 Provi ng it 194 Problems and proofs • Being logical • Mat were we talking about?

Glossary 204 Index 206

I NTR O D UCTIO N

THE MAGIC OF NUMBERS Think of 111111111herfrom 1 to 9. Multiply it by 9.

If y ou have a two-digit 1lI1111ber, add the digits together. Take away 5. Multiply the 111tl1lber by itself

everything from th e hehaviour of subatomic particles to the expansion of the universe i~ Lased on mathematics. MATHS FROM THE START

The earliest records of mathematical activity - beyond counting - date from The allswt!r is 16. How do es it- work ? 2,000 years ago. They come from the ferti le It all depelUls 011 a crucial bit of 1l11mher ddtas of the Nile (Egypt) amI the pbins 'IIIagic: adding tog ether the digits of between the nVQ river:;:, the Tigris ~nd 1111tltiples of 9 always gives 9: Euphrates (Mesopotamia, now lraq). \Ve know little of the individual m:lthematicians 9: 0+9=9 of these l"arly l"Ulrures. 18: 1+8=9 Around 400sc the Ancient Greeks 27: 2 + 7 = 9 a.lld so 011. developed an interest in mathematics. They Th erellfter, it's all pltlin stliling: went heyond their predecessors in that they were interested in finding I"Ules that could 9- 5 = 4,4 X 4=16. he applied to any problem of a similar type. ThL')' worked on concepts in mathematics here is plenty more magic in which und~rlie all th:!.t has come since. numbers. Long ago, some of Some of the greatest mathcmaticI:!.IlS of :!.Il the earliest human ci~i l izations time lived in Greece and the Hell enic discovered the strange and fascinating centre of Alexandria in Egypt. quality of some numbers and wo\'t~ them A~ the Greek civi lization came ro an end, into their superstitions and religions. mathematics in the \Vcst entered a dead Numbers have entr:mced people ever since. zone. Several hundred years btcr, lslamic and still hold the pOwer to unlock the scholars in th(· Nliddle East picked up the universe for us, by providing a key to thr hOlton. Baghdad, built around 750, became a secrets of science. Our understanding of dazzling intellectual centre where Arab

T 6

I NT ftOD UcnON

Europe was struck by th e cataclysm of th e Black Death (1347-50). Between a quarter and a half of the populat ion died in many European countrics. l t was the 16th cenOiry before much new progress was made, bur then there was a flurry of intellectual acrh~ry, in mathematic~ as in science, art, philosophy and music. The invention in Europe of the printing prcss accelerated the spread of new learning. European Toledo ill Spaill htYonn rhe jfareilwy rbroujfh which mathematicians and scientistS began to J Irh cmmr)'- shape modern mathematics and to find Arah leamiT/jf fIIte11'd Europe ill rh,' myriad applications for it. i\-luslim scholars pulled tOgether the legacy \Vhile this has been the path of of both Greek and Indian mathematicians development of present-day mathematics, and forged something new and dynamic. many cultures have developed in parallel, Their progress was b'l"eatly aided by their often making identical or comparable adoption of the Hindu-Arabic number discoveries but nOt feeding into the main system which we now use, and given .~tory centred on North Africa, the Middl e impetus by their interests in astronomy and East and Europe. China kept itSelf separate optics, as well as the requirementS of the from the rest of the world for thousands of Islamic calendar and thl! need to find the years, and Chin(;!Se mathematics flouri shl!d din'eoon of A-tecca. Howl-'Ver, the demands independently. Th e meso-American societies of Islam wh ich were once a spur to in South America developed their own dl!velopment eventually stifled further mathematical :.-yStems tOO, but they were growth. Jv[uslim theol o~,'y ruled against wiped out by European invaders and intellectual activity that was considered colonists who arrived in the 16th century. spiritually dangerous - in that it might Early Ll(lian mathematics did feed into th e uncover truths that should stay hidden, or Arab tl-adition from arowld the 9th centul1', and in recent years India has bt'comc a rich challenge the central mysteries of religion. Luckily, the Arab presence in Spain source of world-class mathematicians. !lude the transfer of mathematical At the very end of our story, a single knowlt'dgc to Europt' quite straightforward. number l>ystcm and mathematical ethos From the late 11 th ct'nrury, Arab and Greek has sprL'ad around t he globL', and texIS were translated into Latin and spread mathematicians from all culmrcs including rapidly around Europe . Japan, [ndia, Russia and the US work There was little new development in alongside those of Europe and thL' Middle mathematics in Europe during the M:iddle East towards similar goals. Though Ages. At the point where a few people were mathematics is now a global enterprise, it equipped to carry mathematics forward, has only recently become so.

"Iff

7

CHAPTER 1

STARTING with numbers

Before we could have mathematics, we needed numbers. Philosophers have argued for years about the status of numbers, about whether they have any real existence outside human culture, just as they argue about whether mathematics is invented or discovered. For example, is there a sense in which the area of a rectangle 'is' the multiple of two sides, which is true independent of the activity of mathematicians? Or is the whole a construct, useful in making sense of the world as we experience it, but not ' true' in any wider sense? The German mathematician Leopold Kronecker (1823-91) made many enemies when he wrote, 'God made the integers; all else is the work of man.' \Vhichever op.inion we incline towards individually, it is with the positive integers - the whole numbers above zero - that humankind's mathematical journey began.

III

tb~ b~gillllillg

..

CIW~II/m C01i1d

pllillt, bllt colild tbey

COli lit?

~ ....n", wn"

""M""

CAN AN IMALS COUNT?

Could

the

mammoths

count

thei r

attackers? Some animals can apparently count small numbers. Pigeons, magpies,

rats and monkeys have all been shown to

be able to count small quantities and distinguish approximately between larger

quantities. Many animals can recognize if one of their young is

missin ~

too.

FOUR M AM MOTH S OR

M O RE MA MMOTHS?

Imagine an early human looking at a herd of potcntiallullch - buffalo, perhaps, or woolly mammoths. There arc a lot; the hunter has no number system and can't count them. He IV/" ngll"lte fill a~cts of0111' lift by III/mbns. bllt (bat

or she has a sense of whether it is a laq,rc herd or a small herd, recognizt!S that a

bas I/Of ak,ays hem rbe crISe. Tbe w;'lIIfe halld <1}flS

added ro clixJ:s ill 1-1-7'), tbe Sl'Colld halld arOlllld 1560.

Where do numbers

come from? Numbers are so much a part of our everyday lives that we take them for granted. They're probably the first thing you see in the mor ning as you glance at the clock, and we all face a barrage of numbcrs throughout the day. But there was a rime before number systems and counting. The discovery - or invention - of numbers was one of the crucial stcps in the cultural and civil development of humankind. It enahled ownership, trade, science and art, as well as the dL'vclopmellt of social Structures and hierarchies - and, of course, brames, puzzles, sports, gambling, insurance and even birthday parries! 10

Mlllly agaillst ollr is ilion likdy ro mSiIIl' a safr OlltrOlllf alld a mM! for blllltrrs t''lllippt'd ollly with prilllitive WMpollS.

W HER£ 00 NUMB ERS COME FROM?

HOW TO COUNT SHEEP WITHOUT COUNTING

As each sheep leaves the pen, make a notch

on a bone or put a pebble in a pile. When it's sheep bedtime, cherk a notch or a pebble for each sheep that comes in. • If there are pebbles or notches unaccounted for, go and look for the lost sheep. • If a sheep dies, lose a pebble or scratch out a notch. • If a sheep gives birth, add a pebble or a notch.

single mammoth makes easier prey, and knows that if there arc morc hunters the task of hunting is hath easier and safer. There is a clear difference between one and 'more-than-one', and between many and few. But this is not counting. At some point, it becom~ useful to quantify thc extra mammoths in some way or the extra people needed to hunt them. Precise numbers are still not absolutely essential, unless the hunters want to compare their prowess. TAllY-HO! Moving on, and the mammoth hwlters settle to herdin g their own animals. As soon as people star ted to keep animals, they needed a way to keep track of them, to cht..'Ck whether all the sheep/goatslyaks/pigs were safely in the pen. The easiest way to do this is to match each animal to a mark or a stone, using a tal/y.

It isn't nec~sary to count to know whether ~ set of objects is complete. \Ve c~n glance at a tahle with 100 places set and see instantly whether there ~re any places without diners. One-to-one correspondence I S learned early by children, who play games matching pegs to holes, toy Dears to beds, and so on, and was learned earlr br humankind. This is the basis of set tht..'Ory - th~t one group of objects can be compared with anothcr. We can deal simply with sets like this without a concept of number. So the early farmer can move pebbles from o ne pile to another without counting them. The Ilecd to record numbers of objects led to thc first mark-m~king, the precursor of writing. A wolf hone found in the Czech Repub lic carved with notches more than 30,000 years ago apparently rcprt..'Senrs a tally and is the oldest known mathematical object. 11

STARTIN G WITH NUMBlRS

FROM TWO TO TWO·NESS

ONE, TWO, A LOT

A tally stick (or pile of A tribe in Brazil, the Piraha, have words for only 'one', 'two' and 'many'. Scientists have found that not having words pebbles) that h as been developed for counting for numbers limits the tribe's concept of numbers. In an experiment, they discovered that the Pirah;i could copy sheep can bi.' pur to other patterns of one, two or three objects, but made mistakes uscs. If there arc thirty when asked to deal with four or more objects. Some sheep-rokens, they can also philosophers consider it the strongest evidence yet fo r be used for tallying thirty linguistic determinism - the theory that understanding is gOatS or thirty fish or thirty days. It's likely that ring.fenced by language and that, in some areas at least, we can't think about things we don't have words for. tallies were used early on to count time - moons or days until the birth of a baby, for example, or from planting to cropping. The concrete objects counted heralds a concept realization that 'thirty' is a transferable idea of numher. Besides seeing: that four apples and has some kind of independence of the can be shared out as two apples for each of two people, pL'ople discovered that four of anything can always be divided into two b'TOUPS of two and, indeed, four 'is' twO twos. Ar this point, counting became mort: than mllying: and numbers nl.:!eded names. BODY COUNTING

Many cultures developed methods of counting: by using parts of the body. They indicated different numbers by pointing at body parts or distances on the body following an established sequence . Eventually, th!;' names of the body p:lrts probably came to stand for the numbers and 'from nose to big toe' would mean (say) 34. The body part could be used to d!;'note 34 sheep, or 34 trees, or 34 of allY thing else. TOWARDS A NUMBER SYSTEM

How nllllly

hln~

71'e gotl A Porrugllese villryrmi

work" lIotcbes II Ulllystick ro "ecord Mcb bnrkt1 of grllpn rhllr passer by.

12

Makin g a single mark for a .single counted object on a stick, slate or cave wall is all very well for a small number of objects, but it

WHER£ DO NUMBERS {OM£ FROM?

quickly becomes unmanagl.!able. BeJore humankind ("(mid use numbers in any more complex way than simply tallying or counting, we needed methods of recording them that were easier to apprehend at a glance than a row of strokes or dotS. \Vhile we tan only surmise from observing nonindustrialized people as to how verba l counting systems may have developed, there is physical evidence in the form of artefactS and records for tht:' development of written number ~yswms. The earliest number systems were related to tallies in that they began with a series of marks corresponding one-tn-one to counted objects, so 'lIT' or ' .. .' might represent 3. By HOOne, the Ancient Egyptians had developed a system of symbols (or hieroglyphs) for powers of ten, so that they used a stroke for each unit and a symhol for 10, then a different symbol for 100, another for 1,000 and so on up to 1,000,000. \Vithin t':!ch group, the symbol was repeated up to nine times, grouped in a consistent pattern to make the number easy to recognize.

In l\Jlesopotamia (current-day [raq), a simibr system existed from at least 3000Be. A still-familiar simple grouping syStem is Roman numerals. Numbers 1 to 4 are represented by vertica l str okes :

I, II, III, 1111 The Romans gave up at Ill, switching a symbol for five, V. Later, they sometimes used rv for illl. [n this case the position of tbe vertit':!l stroke determines its meaning - five minus one. In the same way, lX is used for nine (ten minus one). Different symbols are used to denote multiples of five and ten: to

V

5

X", 10

L

50 C:: 100

D

500

M '" 1,000

Numbers are llUilt up hy grouping unilS, tens and so on. So 2008 is represented by MMVlll. The characters for 5, 50 and 500 can't be lL~ed more than once in a number, since VV is represented by X, anrl so on. Some numbers are quite laborious to write. For example. 38 is written XXA'VU]. The system doesn't allow subtraction from :Inything except the next symbol in the 1111111, numeric:!] sequence, so 4Y can't be written IL (50 minus I); it has to be written XLLX (50 minus 10; \0 minus 1). The next Step is a system 10,00() 1,000 100,000 1,000,000 which instead of repeating the :.ymbols for a number Em'~Y Egl'ptiml hhroglypbs repn!Si'lIIt d IlIIlIIbl'rs I~illg POW"" of tw, (A..,"\.,"'( for 30, for instance) (lml cOllid sb{J'J) JIIlmben lip to 9.999,999. uses a ~ymbo l for each of the

I, II, III, II· 1111, III'

1111' I!! '

I u ,.

~:~

~

"

'{.sl

",n,", W'ffi "'M" "

digits 1 to 9, and thell this is used with the symbols for 10, 100 and so un to show how many lOs, IOOs and 1,000s arc intended. Th e current Chinese system \rnrks on this principle . So:

shown by three digits. Roman llullu;,rals, on the other hand, need between ant' :lIld four digits for the numbers 1 to 10 and hetwc(;!11 one and eight digits for numbers up to 100. CIPHERED SYSTEMS

11]-r- 4 x 10",40 but;-G: andlZll-rlZ!l

The hicroglnJhic ..,ystem described above

10+4",14 4 X 10+4 = 44

This is kn()wn as a multiplicative grouping system. The number of characters needed to represent numbers is more regular with this typl! of ~ys [em. Numbers 1 to 10 are shown by one digit; numbers 11 to 20 are shown by twO digit~; thereaher, multiples of 10 up to 90 :lrc shown by two digits (:20, 30 ctc.) :md the orn er numbers up to 99 are

(see page 13) was only one ofrh ree systems uscd in Ancient Egypt. There were twO cip hered systems, demotic and hi erati c. A ciphered system nOt on ly has different symho ls for the numerals I to 9, but distinct symbols for each of the. multiples of 10, 100 anti 1,000. H.ieratic is th e old est known ciphered system . It could e..'\: pre.~s numhers in a very eompact form, hut ro use it people mU St learn a large number of different symbols. This may have served a soeia J purpose, keeping numbers 'specia l' and so endowing those wl1l) knew them

HOW OLD 15 THE COW AND HAVE YOU BEEN PAID? In Babylon (from southern Iraq to the Persi an Gulf) two systems of writing numbers were used. One, cuneiform, consists of wedge.shaped marks made by a stylus in damp clay which was then baked. A different system, curvilinear, was made using the other end of the stylus, which was round. The two scripts were used to represent numbers for different purposes. Cuneiform was used to show the number of the year, the age of an animal and wages dUE. Curvilinear was used to show wages that had already been paid.

(SO . 1) (60) (60)] .. 11 (60)'+ 47(60)+111 _ 2S11.~

40 · 2

~

W HER { DO NUMB ER S COME FRO M ?

UN ITS

TENS

HUNDREDS

THOUSANDS

TEN S OF THOUSANDS

H UND REDS OF THOUSANDS

,

,

U 11\ ll.<j

"\ '"

~

1\ >r 7J

A

A

~

)l ?

~

!

"i

-

.::.I-

,

,

t.

=?

rt

51

llll

-

~

? ?," /3~.3 ~

1 ?

~

!!l; ~

~

~

Egyptlllll burnt/(' mflflt'rflir qfrbe New Killgdllm (l600-JOOOsc) /lsed /f101T' symbols rbrlll ""foil', 1I1r/!.:illl!, IIIfIIlbny lIIore call/pllet bur barrier ro lellrl/ W /lse.

with extra power, forming a mathematical elite. In many cultures, numbers have been closely allied with divinity and magic, and preserving the mystery of numbers helped to maint:lin the authority of the priesthood. Even the Catholic Church was to indulge in this 10,000 jealous b'l.lardianship 54,321 == 5 X 10,000 of numbers in the 10,070 == 1 >< 10,000 European .M.iddle Ages. Other cil>hcrcd systems include Coptic, Hindu Brahmin, Hebrew, Syrian and early Arabic. Ciphered systems often use letters of the alphabet to represent numerals .

position of the numerals to show their meaning. This ean only work when there is a symbol for zero, as otherwise there is no way of distinguishing between num bers such as 14, 204 and 240, a problem encountered by the Babylonians.

GETTING INTO POSITION

dated was developed by the Sumerians from 3000 to 1OO0BC, but it was a complicated system that used both 10 and 60 as its bases. It had no zero until the 3rd century Br:, leading to ambiguity and probably confusion. Even after zero was introduced, it was never used at the end of numbers, SO it was only possible to distinguish between, say, 2 and

1,000 4 X 1,000

100 3 X 100

10 2 X 10

1 1 Xl

a x 1,000 a x 100

7 X 10

1 XO

A positional system loan show very large numbers as it does nOt need new names or symbols each time a new power of lOis reached . l1lt~ Mrli~t

Positional number syStems, such as au r own modern SYStem, depend on the position of a digit to sho\v its meaning. A positional system de\'elops from a multiplicative grouping system such as Chinese by omitting the characters that represent 10, 100 and SO on and depending only on the

positional

~y;rem

that em he

IS

STAItTlN G WITH

NU MBllt~

SUMERIANS AND BABYLONIANS The fertile area of the two

Me~opotamia,

river ~ ngri~

between

and Eu p hrates, has

been called the cradle of civilization. Now in Iraq, it was settled by the Sumerians, who

by

the

middle

of

millennium BC had established

the

fourth

perhap~

the

earliest civilization in the world. Invading Akkadians in the 23rd century BC largely adopted Sumerian culture. The period from around 2000BC to 600BC is generally called Babylonian. After this, Persian invaders took over, but again continued rather than replaced the culture of the area.

200 from the cOntext. This was sometimes easy and sometimes not. The statement '1 have 7 sons' was unlikely to be interpreted as '1 have 70 SOilS' - but a statement such as 'An army of 3 is approaching' contains dangerous ambiguity. An army of 300? No problem. An army of 3,000, or 30,000 or even 300,000 is a very different Illa tter. One of the two number systems in use in Ancient Greece, that most popular in Athens, used letters of the Greek alphabet to represent numbers, bCb>lnning with alpha for I, DNIl tor 1 and SO on up ro 9. Next, individual letters were used for multiples of ten and then for muh:iples of lOO, so that any three-digit number could be represented by three letters, any four-dib>lt numher by four letters, and so on. They didn't haw enough letters in their alphabet to make it up to 900 with this system, so some of the numerals were reprL'Sentcd by 16

archai c letters rhey no longer used for writing. For numbers over 999 they added a tick mark to the right of a letter to show rhat it must be multiplied by a factor of 1,000 (like our comma as a separator) or the letter 11111 as a subscript to show multiplication by 10,000. To distinguish numbcrs from words, they drew a bar over numbers. GrL'ek philosophers larer came up with methods of writing very large numbers, nor because they especially needed them, but to counter claims rhat larger numbcrs could nOt exist since t here waS no way of representing them. The Mayans used a complete positional system, with a zero, used thoroughly. The earliest known use of zero in a Mayan inscription is 36nc- Mayan culture was discovered - and consequently wiped out, along with the Mayan civilisation - by Spanish invaders who came to Yucatan in

W HER£ DO NUMBUS COM E nOM ?

the early 16th century. The Mayan number system was based on 5 and 20 rather than 10, and again had limitations. The first perfect positiona l system was the work of the Hindu s, who used a dot to represent a vacant position .

M~()p otami:l

THE BIRTH OF

Adding a diagonal line hetween the horizontal strokes of the Brahmi '2' and a verti ca l lin e to the right of the strokes of the Brahmi '3' m:lkes recognizable versions of our numeral~. The Brahmi numerals were part of a ciphered loystcm, with separate ~ymb()ls for 10,20,30 and SO 011.

H I NDU-ARABIC NUMBERS

ahout ;\0650 refers to nin e Hindu numbers. 2

3

= -

4

5

+

" "

6

7

1

B

...,

9

I

The numhers we use today in the \Vest have a long histOry and originated with the In dus valley civilizations more than 2,000 years ago. They are first found in early Buddhist inseri ption s. The use of a single stroke t() stand tor 'one' is intuitive and, nOt surprisingly, many cultures came up with the idea. The MOVING WESTWARDS orientation of the stroke vari es - while in The Arah writer Ibn al- Qifti (\ 172- 1242) the -\;Vest we still use the Hindu-Arabic records in his OJTOIIO/O&'Y of tbe SeiJo/tlTS how vertica l stroke. (I), the Chinese use a an Indian scholar hrought a hook to the horizontal stroke (-). But what about the .~t'.c()ll d Ahisid C:lliph Abu Ja'far Abdallah other numbt:rs? The squiggles we now use ihn Muhammad al-Mamllr (7 12-75) in to represent 2,3,4 and .~o on? Baghdad, Ira q. in 766 . The hook W:lS The earliest, 1, 4 and 6, date from at least the 3rd century Be :md are found BRAHMAGUPTA (589-668) 111 the In dia n Ashoka The Indian mathematician and astronomer Brahmagupta inscriptions (these record was born in Bhinmal in Rajasthan, northern India. He thoughts and deeds of the headed the astronomical observatory at Ujjain and Buddhist Mauryan ruler of published two texts on mathematics and astronomy. Hi s India, Ashoka the G reat. work introduced zero and rules for its use in arithmetic, 304-2328C). Th e Nana and provided a way of solving quadrati c equations Ghat inscriptions of the equivalent to the formula still used today: second century Ile added 2, 7 .\'. _h:i:..)4t1(+'? and t) to the li st, and 3 and 5 2" are found in the N asik eaves of the 1st or 2n d century AD . Brahmagupta'5 text Brohmasphufaliddhanta was used to A text written by the explain the Indian arithmetic nef'ded fo r astronomy at the Christian NestOrian bishop House of Wisdom. Severns Sebokht livin g in 17

STARTING WITH NUMBlRS

probOlbly the Bmhmaspbllfasiddbantn (The Opening of tbr U7Iiverse) written by the IndiOln mathematician Brahmagupta in 618. The caliph had founded the 1-1ouse of Wisdom, an edueJtional institute that led intellectual development in the Middle East at the time, translating Hindi and Classical Greek texts into Arabic. H ere, the BmlmltlsplJllftlsiddbtlllftl was translated into Arabic and Hindu numbers tOok their first step tOwards the \;Vcst. The diffusion of the Indian numerals throughout the 1\'liddl e East was assured by two very important texts produced at the Housc of \Visdom: 011 tbe CaJC1IJat;ofl with Hindll Numem/s by the Persian

mathematician al-Khwarizmi (c. 815), and 011 the Use of tbr Indian Numerals by the Arab Abu Yusuf Yaqllb ibn [shaq al-Kindi (830). A system of counting angles was adopted for depicting the numerals 1 to 9. It's easy to see how the Hindu numerab could he converted by the addition of joining lincs to fit this system - try counting the angles in the straight-lin e forms of the numerals we use nOw:

1Z~~Sb lB~

MUHAMMAD IBN MUSA Al·KHWARIZMI, c7B0-8S0 The Persian mathematician and astronomer al·Khwarizmi was born in Khwarizm, now Khiva in Uzbekistan, and worked at the House of Wisdom in Baghdad. He translated Hindu texts into Arabic and was responsible for the introouction of Hindu numerals into Arab mathematics. His work was later translated into latin, giving

--1"1

Europe not just the numerals and arithmetic methocls but also the word ' algorithm' derived from his name. When al·Khwarizmi's work was translated, people assumed that he had originated the new number system he promoted and it became known as 'algorism'. The algorists were those who used the Hindu·Arabic positional system. They were in conflict with the abacists, who used the system based on Roman with an abacus.

18

W H ER { 00 NUMBERS COME nOM?

A FU SS ABOUT NOTHIN G

Zero was adopted around the same time;

zero, of course, has no angles. The Arab scholars devised th e full positional system we lISC now, abandoning th e ciphers for multipl es of ten used by the Indian math ematicians. Not long after, the new fu sion o f Hindu-Arabi c number systems made il5 way to Europe through Spain, whi ch was un der Arab rul e. The earli est European tt;'xt to show the Hindu-Arabic numeral s was produced in Spain in 97 6. ROMA NS OUT!

Of course, Europe was already using a number system when the Hin du-Arabi c nOtation arrived in j\'loori sh Spain. Mter the fall of the Roman Empire in th e \Vest, tradition ally dated A04 76, Roman culture was only slowly eroded. Th e Roman num ber system was un chall cnbTCd for over 500 years. Alth ough th e Hindu -Arabi c numerals crop up in ,\ fLow works produced or copied in th e 10th century, they did not enter th e main stream for a long time. 1

I

5

V

10

X

50

L

100

C

500

0

1,000

M

5,000

(I)

10,000

(I)

50,000

(I)

100, 000

(I>

The conce pt of ze ro might seem the antithesis of counting. Wh ile zero was only an absence of items counted, it didn't need its own symbol. But it did need a symbol when positional number systems emerged. Initially, a space or a dot was used to indicate that no figu re occupied a place; the earliest preserved use of this is from the mid·2nd millennium Be in Babylon. The Mayans had a zero, represented by the shell glyph:

~ This was used from at least 368e, but had no influence on mathematics in the Old World. It may be that Meso-Americans were the first people to use a form of zero. Zero Glme to the modern world from India. The oldest known t ext to use zero is the Jain Lokavibhaaga, dated AD458. Brahmagupta wrote rules for working with zero in arith metic in his Brahmasphutasiddhanta, setting out, for instance, that a number multiplied by zero gives zero. This is the earliest known text to treat zero as a number in its own right. AI·Khwarizmi introduced zero to the Arab world. The modern name, 'zero', comes from the Arab word zephirum by way of Venetian (the language spoken in Venice, Italy). The Venetian mathematician luca Pacioli ( 1 445~1514 or 1517) produced the first European text to use zero properly. While historians do not count a 'year zero' between the years 1 Be and ADT, astronomers generally do.

\9

STARTING WITH NUM BlRS

LETTERS FROM ABROAD The Romans used written numerals before they could read

and write language. They adopted numbers from the Etruscans, who ruled Rome for around 150 years. When the Romans later conquered the Greek·speaking city of (umae, they learned to read and write. They then adapted the numerals they had taken from the Etruscans to make Roman letters.

As the Empire grew m extent and sophistication, the Romans needed larbrer and larger numbers. They developed a system of enclosing figures in a box, or three sides of a box, to show that they should be multiplied by 1,000 or 100,000. The system wasn't used consistently, though, so

fV1 could me:ln either 5,000 or 500,000. Arithmetic is virtua lly impossible with R oman numerals and this was to lead to its eventual replacement. XXXVIII +

XIX LVI!

(38 + 19 = 57)

For the purposes of accounting, taxation, census taking and so on, Roman account;lIlL~ always used
FibOlltlcci, the It,iliall matbeTlTaticiall, /efll1led abollt Hilldll-Al"flmc I/lfTlTemir ar a boy "/:.rbile tl"lwellillg ill .Nonb Afiiro -;;;itl1 hir rrader/a/ber.

Hindu-Arabic system, particularly amongst tbe mercbants and accountants. Even so, it took many centuriC!:> and considerable struggle before Europe moved over completely to the use of the Hindu -Arabic system (see Unspeakable numbers, pa~,'c 56). Roman numerals continued to he used for many things long after they were replaced in mathematit~Jl functions. They

'The nine Indian figures are:

987654321 With these nine figures, and with the sign

o ..

any number may be written. '

Fibonacci, Liber Abaci, 1202

W HER£ 00 NUMBERS (OM£ FROM?

NOT OVER YET

It would be a mistake to think that our numbers have Phrases that incorporate a number in Roman numeralsstopped evolving. tn the last chronograms -were often used on tombstones and books. century we have seen the By picking out certain letters and rearranging them, a date is revealed. For example, My Day Is Closed In Immortality is development and :.ubscquent declin e of the zero with a a chronogram commemorating the death of Queen slash through it, 0, to Elizabeth I of England in 1603. The capitals read MOO!! when put together, which corresponds to 1603. A coin distinguish it from capital struck by Gustavus Adolphus in 1627 includes the latin ' 0' in computer printouts, and the reprl!Senration of inscription Ch,lstVs DuX ergo tflVMphVs ('Christ the leader, digits as a collection of therefore triumphant') which is a chrono gram for MDCXWVII o r 1627. straight lines so that they can be shown by illuminating bars on an LED display. are still often used on elock faCl!S, fur Computer-readable character sets, too, have example, and to show the copyright date of been developed for usc on cheques and movil!S and some TV programmes. other financial documenL~, taking our numerals fur from their cursive origins. In addition, we have developed new types of notation for writing numbers so unimaginably large that our anCl!Stors could have had no conceivable usc for them (sec pab'es 26- 33). CHRONOGRAMS

lilies I)f differe1l1 tbickllesses

Lllm Pacioli 7L"IlS f1 Frallciscall frial: III rbis pmTrair

Bar rodes

lISe

by Jm:I)/IO de Barb"'7 (I;. 1495), be if dmTl)llstratillg

/"!'p/"~mt

/llimbers: tbeY( all' reml by CIJmplltfl7zrd

I)/I~

I)f Eudid's rbel)/"ems.

sca mnrs 'Il·bicb

~ce'

(I)

tbem ar lilimbers.

21

STAItTlNG WITH

NUM Bl lt~

IlL'Ople may be required mking hold of or pulling on A new s~tem for writing numbers, and particularly for the fingers, tor example. entering them on a calculator or other machine, was A highly developed system, more complicated developed in 1993 by Jaime Redin. Called 'verbal numbers' it than aims to be more intuitive and quicker to use than the ordinary finger standard positional system. For example, the number counting, was used in both Europe and the Middle East. 4,060,087 would be written 4M60T87 and would be entered into a calculator by It was more like a sign pre~ing 4 - M (for million) ... 60 - T ce:lffiGJeB language, and enab led ~GJfI3m (for thou!>
FIN GERS AND THUMBS

\-Vt;' have probably developcd our decimal system because most people have ten fingers. Although it seems vel)' obvious that we ca.n count on our fingers, different cultures through the ages have de\,e]opeJ different ways of doing it. Fingers Imy be extended or filldeJ down to indicate a number; joints may be counted as well as fingers; one hand may be used to show tens and the other units, Dr interaction het\vcen 22

Pl'opll' dOIl'/ jlar rollin 011 fillfl:l·,r. Thir hWiy-rolll/tilll{ rysrml if IlSl'd by rill' Fartl Of Ptlpl1il NrJ) GllillM. BA CK TO BAS E

D espite the obvious recourse to tingcrs as counting aid, not all cultures have used a decimal system of counting. Indeed, we owe many of our strange weights and 3

NU M 8ERS AND BASU

ORIENTAL FINGER BARGAINING

Secret systems of bargaining with the fingers were widespread between Algeria and China for centuries. The two participants needed to know the approximate price they were negotiating ~ whether it would be in units, tens, hundreds or thousands. One negotiator would hold the index finger of the other to indicate 1 (or 10, or 100), the index and middle fingers to indicate 2 and so on. Clasping the whole hand meant 5. In different places, different methods were used for the numbers above 5. In some places, for instance, 6 was indicated by twice gripping the fingers fo r 3, in others by grasping the thum b and little finger.

measures to cultures that have used different counting systems. Binary, or hase 2, is used by computers as it (.~Jn designate onc of twO states, TRUEIFALSE, or hold a negative or positive electrical charge. But there have heen human users of hinary systems. Some of the oldest trihes in Australia usc a counting system in which the names of the numbers arc defined in relation to twO and one. The Gapapaiwa of Milne Bay have sligo 'one', 1"/111 'twO', then 1"1111 11/11 Jago for 'three', which is literally, 'nl'o and one', and rlltllJ1a rlll1 or 'n\'o and [\.vo' for 'four', 1"IIt11lla 17111 l/1ll sago ([\,1'0 and [\.\'0 and one) for 'five'. Although it differs &om computer binary in thatit uses one. and twO rather than zero and one, it sti ll has on ly twO distinct numhcrs.

The mTerllftrlQIllt! filllglltlg~ Qfrom1l1erce: 11 tm/I"lst btlrgflills 7:litb (/

local nufi vmdQr

/let/I"

tbe

fin

if" rbe

Te17·tlCQttl1 Al""llty, Xinll, Cbi/lli.

Often, the negotiators hid their hands inside a sleeve or concealed them in a robe so that onlookers could not see the price agreed.

The indigenous peoples of Ti erra del Fuego and partS of South America have used number systems with bases three and four. Base-4 systems may have emerged hcc
HH

II

== 5 + 5 + 5 + 2 '" 17

Thi s 'rule of four' li cs hehind many cu ltural oddities. In ancient Rome, for

~ m.n", wn"

Thr" an

Tbt

"'M""

IIM/ly

ways of fomning

I11(Jrr ((IIIII/IOII/Ise

011

rbe fillgny.

I",SI' 5 01" bllSe 10.

in~[ance, the first four sons were given 'proper' names, such as Marcus or Julius, hut after four they were given numbernames: Quintus (fifth), Sixrus (sixth), Septimus (sL'Vcmh) and so on. A f(;!w culn\l"t~s lise a quinary (base-5) system, including the speakers of Saravcca, a South American Arawakan language, the IiOllb'Ot, a head-hunting tribe trom the Philippines and some [ndOllcsi;m ~ocietics. The hll-JS had a quinary system

widely Q'iCd dozen and gross (144 = 12 X 12), the 11 inches in a foot and 12 months in a year. The ancient Sumerians had a sexagcsima I syStem - one tha t used hase-60. ft is clearly difficult to rememher 60 different names for the digits, So for lower numhers they used base 10. From the Babylonians, we still haw 60 seconds in a minute and 60 minutes in an hour; when we write 2 hrs 14 mim 3H secS we arc using their hase-60 ~1's[em. It is not entirely clear why they USL-d 60, but sixty has many factors (numbers .i t can be dividL-d by), making it a useful base. Arab mat hematicians

with names for numbers up to at least

10,000. It is ca:.y to sec how a quinary system could have evolVl'd frOIll cOllnting on the fingers of one ham!. Other common systems arc base 6, duodecimal (base 12) and vigesimal (hase 20). Bases 12 and 20 have often been used with other bases in a complex ~1'stem where a small base number is used for low numhers (up ro 5 or 10) and a large one for numbers O\'er a certain limit. Remnants of a base-l0 SYStem linger on in the French ' quatre-\,ingrs' for 80, for example. Ve~tigcs of a hase-12 ~ystem art' all arowld us, in the

calculations and then mO\~llg back to 60 to express their final result. How MANY FINGERS DOES A COMPUTER HAVE ?

For bases higher than 10, we need to rope in other ~1'mbols to stand for digits we don't have in our decimal system . Computers count either in binary or in hexadecimal (hase 16 = 2"). TIl represent the numbers hen\'een 9 and 10 (= \6) in hexadecimal notation we use letters of the alphabet. 3

4

5

6

7

10

11

100

101

110

111

2

10

11

12

20

21

1

2

3

10

11

12

13

Base 5

I

2

3

4

10

11

12

Base 6

1

2

3

4

5

10

11

Decimal equivalent

1

2

Base 2 (binary)

1

Base :3

1

"'''''

N U M 8E R S A N O BAS ES

Decimal Hexadecimal

9

10

11

•

12

14

13

15

9

A

C

D

E

F

Decimal

I.

Tl

1.

19

20

21

22

Hexadecimal

10

11

12

13

14

15

I.

Decimal

23

24

25

2.

27

2.

29

Hexadecimal

17

1.

19

lA

1.

lC

10

Decimal

30

31

32

33

34

35

3.

Hexadecimal

1E

1F

20

21

22

23

24

Clearly all numbers above 10 mean different things depending on the hase system used, so there is plenty of scope fur confusion: '11' in hexadecimal means 17 in decimal. Computer hooks often use the hash Si~,'11, #, hefore a hexadecimal number, so '#11' = 17 (and 23 would be represented

like

by'll).

perhaps less yellow, slimmer and

Because computers count m hexadecimal, some strange numbers are beginning to creep back into L·veryday life, too. "Vhile we sti ll buy eggs by the dozen, we might also buy a memory card that will hold 512 MBofdata, oran iPodwith 8GB of storage. The decimal ~"yStem has by no means taken over completely.

with more hair), we might now use a ,;..oy.,

CARTOON COUNTING

Cartoon characters are most often drawn with three fingers and a thumb. Had we all evolved to look Homer

base·S

Simpson

(though

counting

system in which '10' only eight doughnuts.

1\11111] irrolf are n'lIdt11

ill '1l1l11nities rbtlr dOIl'r reilire

fa

rysTi'1Il -

tbe decimal fIIcb as

eggs

wid by rbe doul/.

25

STARTING WITH NUM BlRS

W HEN IS 1,000 NOT 1,000?

Although for convenience we think of a kilobyte as 1,000 bytes and a megabyte as 1,000 kilobytes, the terms are ambiguous and mean different numbers in different contexts. Because computers work with binary, a kilobyte is often used to mean 1,024 (= 2 1 ~ bytes and a megabyte to mean 1,024 kilobytes (= 1,048,576 bytes). But they can also be used to mean 1,000 bytes and 1,000,000 bytes respfftively. To try to end the confusion, new units of mebibytes and kibibytes were introduced in 1998. A kibibyte is now officially 1,024 bytes and a kilobyte is 1,000 bytes; a mebibyte is 1,048,576 bytes and a megabyte is 1,000,000. But confusion still reigns. In common usage, a megabyte is 10" bytes (1,000,000) when measuring hard disk size, is 231 (1,024 x 1,024) bytes when measuring computer memory and often file size, and 1,024,000 (1,000 x 1,024) when measuring USB flash drives or the old 1.44 MB flop py disks. In either system, a byte = 8 bits.

More numbers, big and small Natural numbers have a special status Th e first numbers humankind used related hecause they can be related om~-ro-one to directly to object s in the real world; indivisible objects in the real world. For they were the positive integers. _ ......-~_ convcnience, intinity is thc namc But th (;'se are not th c on ly givcn to th e largcst possible numbers we recob'llize now. A~ number, th c cnd of counting time has passed, we have though clearly this could never developed ways of quantifying exist since. howt..'Ver large a an absence with negative number is we could always add numbers, showing fragrnenrs or one more, and allother, and so portions that are less than one, on. lntinity is represented by th(;' and representing numbers so ~ymbol 00, first used by J ohn \Vallis in his book D e scctiQflibus large they tax our normal systems for writing numbers. fOil/CIS (Of collical sectiolls), We have (;'ven develoJled a way publish(;'d in \655. of talking about imaginary (complex) numbers. LESS THAN ZERO Ncgative numbers don't relate directly to the physical world in INTEGERS Integers are the positivc and that we can't co unt a negative negativc whole numb ers, Tbc scale all a Ib~ro/Omcrer spallf extcnding infinitely in both directions from (ami including) IIrgarive alld pofirive mnpU"Iltlires, zero. The positive intc~,'ers arc alld is II follliliar applirarioll of natural numhers. lIegarivt J/Il1nlll'/"s. ca ll ed 26

MOllE NUM8US, 81G AND SMAll

'God created the integers. All the rest is the work of Man.; leopold Kronecker,

1823~91

number of objectS - we can't sct' 'minu!S twO cows', for example. But as soon as concepts of ownership emerge, negative numbers have a meaning. They were used early on m indicate a debt (money or goods owed). They are also used in some types of scaled measurement, such as rcmperature. Negative numbers are first mentioned in a Chinese text called ]illZhtlllg SIIaflsbll (Tbe Nille Chapters on the A1athrJJlaticai Art). It was compiled by several authors during: the period from the 2nd century Be to the 1St century AD . The Bakshali Manuscript, an Indian text of uncertain date but no later than the 7th century, also uses negative numbers, though they are confusingly indicated by a '+' sign. Th e minus sign was first used to show a negative number by Johannes WIdmann in 1489. It is from the Indian texts that negative numbers entered western mathematics.

sand. But some things - and most nH..':ls""Ures - can be divided into portions smaller than a unit. A loaf can be broken in half, or a person can drink a third of a hottle of wine, or a stick may be half a metre long. Fractions are a useful way of expressing the !Size of a portion . Fractions may be expressed as one number (the numerator) divided by another (the denominator), such as Y, (one di\~ded into four parts). A fraction is also called a rational number as it expresses the ratio between numbers - so).':i shows the ratio 1:4. In a decimal fraction, numbers a her the decimal point indicate tenths, hundredths, thousandths and so on according to their position. Decimal fractions can express irrational numbers - those which are not a ratio of twO whole numbers and have an infinite number of digits after the decimal point. hntional numbers caused problems for early mathematicians (see page 56). The Persian mathematician and poet Omar Khayyam (1048-1131) accepted all positive numbers, rational and irrational, so had a much broader number system than th e Grecks, who refused to acknowledge irrational numbers.

PARTS AN 0 WHOLES

Some thinb'S arc nOt divisible we can't speak of two and a h:Ilf people or three quarters of a grain of AIallY rbillgr {OIl be bmkm don'lI iura slIIallcr (/lid s1I1allCI"

portimls - but a l(}tlf Inv('m illto {"/"{/"Illbs is of lilllr IISC. 27

STARTING WITH NUM BlRS

al-Samaw'al al-Maghribi systematic
GETTING TO THE POINT

Dccimal fractions were recorded in Indian units of measurement around 1800Bc. "Veights found III th e archaeological remains of :l settlement l':l11cd Lothal in the Indus Valk'Y area (modern-day Gujarat) weigh 0.05, 0.1, 0.1, 0.5,1,1,5,10,20,50, 100, l{Xl and 500 units (a unit being around 2H grams or one ounce). Abu'l Hasan Ahmad ibn Ibrah.im al-Uqlidisi ([.920-80) wrOte the first known Arab text on the lL~e of decimal fractions. 1n the 12th century, the franian ma t hematician Ihn Yah)'a 28

MOllE NUM8ERS, B IG AND SMAll

a vertical bar instead of a decimal point. Th e first European trt::ltiseon dceimals was produced by Simon Stc'~n in 1585 , and he is gencrally creditcd with introducing decimal fractions into Europc. Stcvin used a diffcrent notation from that we use now, writing 5.912 in the foml: 5 @ 9 @ 1 ® 2 G)

The French mathematician Fram,:ois Viete (1540- 1603 ) cxperim cntcd with several ways of writing dccimal fractions. H e tried raisin g and undcrlinin g the fractional part (627 ,115 ~), and showing me dccimal as a fraction (617,125 I,~~:;;;;), using a vertical stroke to scparate thc integral and decimal partS (627,12515 12,44) and showing the intcb'Tal part in hold type (627,125,5 12,44). But he was nOt to comc up with the mcthod which has stuck. Th e carli est printcd usc of the decimal point was by Giovanni j\1agini (1555- 161 7), an Italian map-maker,

Pl"lllIfOis Viru l1;orhd for tb~ ClJIII7 of fhlll] of Nfw(lny.

By Cl"flckillg a dplJn' IfJd by t};~ Spflllish,

b~ mflbhd tbe Prmrb 10 r~ad clle-my despatcJJfs.

EGYPTIAN FRACTIONS The Egyptians had a strange way of working with fractions. They had special characters for half, ~ , and two.thirds, en"" . Thereafter, a fraction was shown by the character c:> written above the denominator, which was shown using the usual Egyptian symbols for i'iit 1; numbers. SO III means 7. However, with the exception of )1" the Egyptians only used unitary fractions (those with a num erator of 1); there was no way to show a numerator, so it was impossi ble to write % or Yt. To complicate matters further, it was not allowed to repeat a fraction - so %could not be written as ~ + K Instead, it was necessary to fin d a way of making %from unique frac t ions:

29

STAItTIN(; WITH NUM8lllS

GOOGOL AND GOOGOLPLEX

The terms 'goog ol' and 'googolplex' were invented by Milton 5irotta (1911-1981), the 9-year-old nephew of American mathematician Edward Kasner (1878-1955). A googol is 1 followed by 100 zeroes; a googolplex is 1

followed by a googol zeroes. These numbers are inconceivably large. There are fewer than a googol fundamental particles in the known universe (fundamental particles are sub-atomic; there may be 10'" in the universe). If you could write down a googolplex in standa rd 1 0 point type, it would be 5 x 1O~ times longer than the diameter of the known universe, and writing at two digits per second would take I Ou times the age of the universe to complete.

astronomer and friend of Kepler, who used it in 1592. Even so, it did not I..-arch on until J ohn Na pier used it in his t;lblcs of logarithms over t\VCnty years later. Napier sugb"Csred in 1617 that tht: fidJ stop or COlllma could be used, and settled on the full stop in 16 19, though many European commies have adopted the I..'Olllilla as their decimal separator.

17 being zeroes and the last I. This can be extended to show other numbers as IlItLltipk'S of a power of ten. For instance, 10 J = 1,000 and so 6.93 x 10) = 6,930 and 6.93 x IO-J '" 0.00693. Scientific notation is much easier to understand, and more compact to write, than a long string of digitS. The first use of scientific notation is not known, but it was already current in 1863 when an encyclopaedia included the followin g text:

'a om'tllt jor« (q ual 10 10,000,000,000 timrs thr /''IIIII( givell hy the qlllJtimt of J 7IIet1'e by I secol/d 0frill/e, rhar is, I d (J metre/secol/ds'.

BI GG ER AND BIGGER

\-Vhilc fractions and decimals provide a W'3y of writing very sma ll numbers, developments in science have led to a need for W:J.ys of representin g and r:llking about increasingly large numocrs. Scientifi c notanon uses powers of ten to show both vc ry large and vcry sma ll numbers. A power of ten shows how many fi gu res come befor<.' or after the decimal point. For example, 10 'ft is 1 followed by 18 zeroes. in the other dirL'Ction, 10-1• is a decimal point followed by 18 digits, the first JO

Joh1l Napi!'l; Ihl'illt'C7II0" Of oflogarithms, bdirvl'tilbllllhl' world 1l'01i1ti (01111' ro alll'llJ l'ilhl'I'

1688 0/' 1700.

ill

MO RE NU MBER S, B IG AND SMAll

THE SA ND RECKONER In what was effectively one of the world's

first

research

papers,

Archimedes boasted in the 3rd century Be that he could write iI number larger than the number of grains of sand it would take to fill the universe. He was able to do this using the new Ionian num ber system and his own notation, which in effect used powers and was based on the 'myriad', or 10,000. He worked with powers of a myriad my riad, or 100,000,000. Archimedes' estimate of the size of the universe, while far larger than previous figures, was nowhere near modem estimates. His number of grains of sand was 8 x 10" .

SClENT1FIC NOTATION

US NAME

EUROPEAN NAME

10'

Thousand

Thousand

10'

Million

Million

10'

Billion

1000 million (milliard)

10"

Trillion

Billion

10"

Quadrillion

1000 billion

10"

Quintillion

Trillion

10"

Sextillion

1000 t rillion

10'00

Googol

Googol

10'01

Centillion

10= 10-«

Centillion Googolplex

Googolplex

10 to power 10'00, or 1 followed by googol zeroes

31

STARTING WITH NUM BlRS

The usc of scientific notHion also gets arowld the confusion over the different names used tor large numbers in the US and elsewhere. Although names arc the same up to a million, they then diverge. Th e American billion is only a thousand million (10''), while a European hillion is a million milli On (lO l!) and 109 is JUSt called a thousand million. The pattern continues with even larger numbers. As science and mathematical proofs demand writing ever larger numbers, even scientific notation becomes unwieldy and finally unmanageable. Solutions to the problem include using" or - to indicate powen; of powers, and even using polygonal shapes to indicate powers. In 1976 Donald Knuth proposed a notation using" to indicate powers. The expression n"m means 'rai se n to the power of III (n X n

III

n"2- n'

3"2 is 3' - 3 x 3 - 9

n"3 - n' n"4", n'

3"3 is 3' - 3 x 3 x 3 - 27

Doubling the " symbol to n""m means 'calculate n"n III times' and n"""m means 'n""m' 111 times'. So while 3"3 is 3' "" 27 3"""3 is 3"(3"3) '" 3" "" 7,625,597,484,987 And tripling tbe " symbol to """ rapidly leads to very large numbers: 3"""3 is 3""(3"""3) "" 3".." ,'9" ,s.o,\II!'''3,,6' \ '9',...,,'W

times)'

THE LARGEST NUMBER EVER The largest number that has been cited in any theoretical mathematical problem is called Graham's Number, named after American mathematician Ronald Graham (right). It was devised by Graham as the upper bound of a possible solution to a problem. The num ber is so large that it is impossible to write it in any of the notational forms covered here. It is said that if all the matter in the universe were turned into ink it would not be enough to wri te the number out in full. Ironically, experts suspect that the real answer to the o riginal problem is '6'.

32

3"4 is 3' '" 3 x 3 x 3 x 3 "" 81

MO R( NU M 8{ RS, B IG AN O lMAll

As the number of " characters incrt":lses, the numbers get harder ro read (as well as unimaginably large). John Conway (b. t 937) suggestS condl'nSing the numbers by using right arrows-to indicate the number of" characters. So n"""4 would be written n ..... 4 ..... 3.

Another way, called tetration, expresses

n

nn

n

So 42 is

a~ ' n.

2

2

22

•2

24

•2

16

• 65,536

all nested' . At each ~tage, the number is ev:.tluated and lL'>Cd fur the next stage, so 1 in a square is 1 in twO nested triangles. The first nested triangle is 21 '" 4, so the next nested triangle is 4~ = 256. ® (a number /I in a pentagon) is equivalent to 'the number n inside n squares, which are all nested' . Originally, this was the limit ofStcinhaus's system and he used a circh:, for this: @.

Cd) starts from 156 Jj~ and evaluates this in the same way 256 times. Steinhaus gave the numher @ the llame a tJlCgll, and @ the name lIIogistolJ. Most'r~~ number is 1 inside a polygon with mega sides. MOVING ON

Another system, Steinh:ms-i\1oser notation, uses polygona l sh:tpe.~ to show how many times a numher muSt be raised to a power.

@] (a numher 'T/ in a square) is equi'~Jlent to 'the numher n inside n triangles. which are

Now that we are equipped with l1rgt' enough numbers, we em begin to put them to work. \iVhat numbers can (k) on their own is the suhject of pure mathematics; wh:lt they can do when they are recruited into the service of other discipline.~ is applied mathematics. A culture must develop :It iL'ast a littl e pure mathematics before it can start appl~'ing numbers Il) real-world problems such as building, economics and :lStronomy, SO we will start with number theory.

CHAPTER 2

•

".'

..... I

...-

. ,~~

,

'to

NUMBERS put to work

•

Counting is a good start, but any more sophisticated application of numbers requires calculations. The basics of arithmetic addition, subtraction, multiplicanon and division - came in to usc early on through practical applications. As soon as people started to work with numbers in this way, they began to notice patterns emerging. Numbers seem to play tricks, to h.'1VC a life of their own and to be able to surprise us with their strange properties. Some arc simple bur elegant - like the way we can muJtiply a two-digit number by II simply by adding the digits together and putting the result in the middle: 63 x 11 :: 693 (6 + 3 '" 9, put 9 between 6 and 3). Some are breathtaking in their sophistication. Number theory, which includes arithmetic, is concerned with the properties of numbers. Ancient people imbued numbers with special powers, making them the centre of mystical beliefs and magical rituals. Modem mathematicians talk of the beauty of numbers.

A

11/1/1/

/Ires

1/1/

IIbflCIIS ill fI Jllpllllere >I1:m·d shop, c.1 890.

~

/ ' NUM8ERS PUT TO WOR k

/~ Putting two and two together Thl'_ rules of arithmetic provided the ancients with methods for working out fairly simple sums, bur as the numbers involved grew larger, tools to help with and eventually to mechanize - calculation become increasingly important. Tools ro simplifY addition, subtraction, multiplication and division emerged very early on. Over the last few centuries these simple aids have nOt been sufficient and our tools for working with Ilumbers have hecome increasingly complex and technically sophisticated, until we now have computers that carry out in a fraction of a

second calculations that would have seemed quire inconceiv:Jble to the earlieSt mathcmatici:ms. STRIN GS, SHELLS AND STICKS

The earliest mathematical tOols wcrc counting aids such as tallies and beads, shelL~ or stones. The Yoruba in west Africa used cowry sbells to represent objects, always reckoning them in b'TOUPS of 5,20 or 200, for !!Xample. Other civilizations have used different objects. In lvleso-America, the Inca ci,~lization had no written number system but used khipll (or flllipll) - groups of knotted srrinbr:; - to record numbers. A khiplI consist~ of coloured strands of alpaca or llama wool, or sometimes COtton, hanging from a cord or Shdls

hac~ hel'lI

{/lid {/S C1111"fllry.

36

lISell liS colilltillg aids

A Sooin fCboolboy IISes a Rllssiall ab,wlf - fhr schocy - during tl11lfIth,·/eSSOIl still widely I/wd

ill

ill

1920. AbacllSes {IIY

Rllssifl, bur

110

umgl'r

ill

schools.

rope. 1t could be used to record ownership of goods, to calcubte and record taxt!S and census data, and to Store dates. The strings could be read by Inca acc()untanl5 called flllipuralJlayors, or 'keepers of the knots'. Different-coloured strands were apparently used to record differi ng types of information, such as details relating ro wal; taxes, land and so on.

PUTTING TWO ANO TWO TOGETHER

KNOTTY PROBLEMS

The position ot a group of knots on a khipu shows whether that group represents units, tens, hundreds, etc. Zero is indicated by a lack of knots in a particular position. Tens and powers of ten are represented by simple knots in dusters, so 30 would be shown by three simple knots in the 'tens' position. Units are represented by a long knot with a number of tums that represents the number, so a knot with seven turns shows a seve n. It's impossible to tie a long knot with one turn, so one is represented by a figure-ot-eight knot. Khipus recorded information such as population censuses or details of crops harvested and stored. Alrhollgh it lookr lif f a dfforativr jrillgr, tbr

khipu 1:.'ar a sopbisticaud new llllting aid. T bis ollr was n/(ulr ill Prru c.1.J 30- 15 32.

N orth American tribes also used knotted strings, called W01JlP01Jl, and knots in leather straps have been used in less sophisricated arrangement<; by the Persians, Romans, Indians, Arabs and Chinese. In Papua New Guinea tally ropes were IlScd to record the trade in gold lip pearl shells. In Germany, bakers l.L~ed knotted ropes to tally bakery orders until the late

_. ..... .•. .....

19th cenmry. Herdsmen in Peru, Bolivia and &."uador used a furm of kbipll, with !,'TOUpS of white strings for sheep and !,'Oats and green strings for cattle, unril the 19th century. The practice has proved remarkably enduring. In Tibet, knotted prayer strings still help Buddhists to keep track of their prayers; the same function is performed by Muslim prayer beads and Catholic rosaries.

-

~

~~ .:-. "~

...,. po

-

:.

, .

, 37

T IMES TABLES

Tables of numbe rs for looking up the results of calculations, partkularly multiplication, have been used for thousands of years. Clay tablets dating from around 1800BC preserve ancient multiplication tables used in Mesopotamia . Tbe idea of compiling tables of tbe resu lts of common arithmetic operations is as old as written mathematics. Th, mathematicians of ancient Babylon inscribed their work on clay tablets; many of these present mathematical tables for multiplication, squares and cubes and their rOOL<;, and reciprocals. BEADS AND BOARDS

A dlly pIau flum c.2500BC, follllll ar Lagash

Some cultures developed quite in~:,''C nious tOols and systems to help with calculations. One of the more familiar is the abacus, developed around 3000BC in Mesopotamia and still in use in some eastern cultures. It began life as a boar d or slab covered in sand, used in ancient Babylon for aligning numbers or writing; it later developed into a

;11 Ira,!, cumaills fI record of IlIIlIlberr ofgOilts fllld shup.

board with lines or grooves for counters. Th e modern abacus with counters threaded on to rods or wires requires more technolobrical acl\':lncement to produce, but is used in much the same way. The position

.-

SHOW ME THE MONEY

The name Chancellor of the EXChequer for the minister in charge of the country's finance in the UK comes from the use of the 'exchequer' board ~ a counting board similar in design to a chess board used as an abacus. Tb~ F.xch~'!tur ch"rg~d

38

wirh

was

rb~ 1I/~di~vfll

COIl~clillg

English il/Sfitlltioll

rOYfll ,TIJflllles.

~1I/~

-

I' lIfTlNG lWO AN D lWO TOGHHU

LOGARITH MS logarithms offer a quick way of carrying out long division and multiplication. They work on the principle that to multiply powers we can add them together. 10 ' _10 10' "" 100 10 ' x 10'"" 1,000= TO ] looking at Ule powers: 1 + 2", 3 The logarithm of a number n is the power to which the base number (in this case, 10) is raised to give n. So the logarithm of lOis 1 because 1 0' '" 10; the logarithm of 1 00 is 2, because 10' "" 100. The logarithm of 2 is 0.30103 because 2'1.10,0. "" 1O. Any two numbers can be multiplied together by adding

century and is still used there as well as in the Middle East and China. Earlier Chinese mathcmaticians used rods of different lengths which they laid out in matrix on a special table or board. Th t principl e was similar to the abacus in that the position of the rods indicated their value. [n Europe, merchantS continued to use the aiJa(.'Us until at least th e 17th century, when it wa s replaced by arithmetic algorithms foll owi ng the ascendance of Hindu-Arabic

their logarithms. So log0010 + 10910100 "" logoo 1,000.

num eral~ .

The subscript shows that we are using logarithms to base

Certain early Arab mathematicians took ovcr the basic algorithms for calculation from India, and around 950 Abu' l Hasa n Ahmad ibn Ibrahim 011Uqlidisi adapted them for use with pen and paper rather than the traditional Indian dU5tooard.

10 - i.e., working with powers of 10. The same p rinciple of working with powers obviously holds wiUl other numbers besides 10:

(16

x

1,024 '" 16,384)

So using logarithms with a base of 2, the logarithm of 16 would be 4. logarithms can be constructed to any base.

TRI CKIER CALCULATIONS

of a bead or cowner denotes whether it Stands for a unit, a t CIl , a hundrcd and so on. A practised, proficient user can move the beads or counters at gn~at speed, carrying out calculations as quickly as many latcr mechanical calculators. As late as the 1920s, accountantS training in the City of London had to be able to usc an abacus as well as arithmetical methods. The abacus spread to Japan in the 16th

As both science and commerce bt.."Clme more advanced and sophi sticated, the need to work with large numbers, fractions and decimals increased . Calculations hecamc hard and time-consum in g and pcople .~earched for ways to make them more manageable. Tb e mOSt ingenious and enduring solution was the development of loga rithms by the Scottish mathematician John Napier in th e early 17th cClltury. 39

JOHN NAPIER ( 1550-16 17)

John

Napier

Scottish and

was

a

mathematician

eighth

laird

of

Merchiston. He entered the

University

of

St

Andrews at the age of 13, but left without a degree. He is best known as the inventor of logarithms and another calculating device called bones'.

He

'Na pi er's began

working on logarithms around 1594 and published his treatise, Description of the

The TO/if ill Napicrf BQIlt's mrri,'d T11l1lriplimriQII

Marvelous Canon of Logarithms, in 1614.

rabies ~bicb mlltlr CtI/CIIlati(lIIs 1nllch sin/pIn; bllr

Na pier's bones comprised a system of small

fbI'] did lIor work

ill [be Slime /:Jay

lIS

logtlr-irhms.

rods used for calculating; they were the forerunner of the slide rule. Napier was also an inventor of artillery, and suggest ed to lam es VI of Scotland

to use the dot as a decimal point separating the parts of a decimal number -

his

logarithmic tables are the first document to

something like a tank - a metal chariot with

use the decimal point in the modem style.

holes from which small bore shot could be

He was ardently anti·Catholic and beli eved

fired. He is known, too, as the first person

the Pope to be the anti-Christ.

Tables of logarithms were published first in 1620 by the Swiss mathematician JOOSt Burgi whn discovered logarithms independently of N apier henveen 1603 and 1611. To usc logs it was necessary first to look up the logarithms of the numbers to he multiplied, add them together and finally to look up the antilogarithm of the answer. For di\'ision, it was necessary to subtract one logarithm from the other and then look up the antilog.

Logarithms also offer an easy way of finding powers and rOOts. To find a square, multiply the logarithm by twO and look up the antilog; to tind a square rOOt, divide the log by mo and look up the antilog. To find a cube, the logarithm is tripled; to find a cube root, it is di\~ded by three, and SO on. Children in western schools were taught to usc rabIes of log'arithms until the latc 20th ccntury when electronic calculatOrs finally took OWr the rol e of comple..'o: calculations.

P UTTING TWO AN D TWO TOGHH U

The development of logari thms made much else possible. For scientists, the comple-x ealcularions required, particularly for astronomy, hecame much easier and so progress in this field speeded up. It didn't take long heton:~ logarithms moved from printed tables to physical calculating devices. The first was the Gunter scale, developed by Englishman Edmund Gunter in 1620. It was a large plane scale with logarithms printed on it. A10ngside a pair of compasses, sailors used it to multiply and divide distances . A1though hase-iO logarithms are hardy used any more (their function has been taken over by cakulators and computers), base-e logarithms (natural logarithms; sec pand) are still widely used in sciellce. An enduring mechanical cakularor that used logarithms was the slide rule. The first slide rule was circular and designed by \Villiam Oughtred around 1632; he made a rectangular version in 1633. The slide rule ha.~ decimal numbers Oil one .scale and their logarithms on another. By lining up the scales in the right way, it's possihle ro read off the product of two numbers .

ALL ABOUT e

e

is

a very

significant

number

in

mathematics. It is defined (among other methods) as the sum Df all numbers in the series

1+1+1+1+1+ O!

1!

2!

3!

4!

where n! means n·factorial (n multiplied by each digit smaller than itself

~

so

4! '" 4 x 3 x 2 xl", 24). By convention, O! == 1. SD

As it is an unending series, it has an infinite number of digits and is an example of an infinite series.

MACHINES FOR MATHS

Charts and tables, and then the slide rule, offered a grt.-"":lt advantage over carrying out calculations with paper and pen, hut the enormous burden of cakulation required hy emerging science, especially astronomy, by commerce, finance and navig'ation cried out for better mechanical aids. The first commercial attempt at a cakulating machine was made by Blaise Pascal (see page 43) in 1642- 3 ro help his father, an adminisrra tor in Rouen, France,

SmiTing ill tbe If/te 17tb crimi/). tbe slide rille rl'iglledfor300yefmasrbrkillgofcalelllf/tl)/"S.1r ll"IJr

slIpt'rseded by ,be pocket caleulf/w,' ill 'he 1970s-.

who h:ld ro dea l with 'It is unworthy of excellent men to lose hours like slaves in the complicated tax figures. The labour of calculation which would safely be relegated to Pascaline, as it was called, anyone else if machines were u~ed.' consisted of a OOX containing a series of notched whed~ or Gottfried Leibniz gears. A complete rotation of unc wheel advanccd the adjaccnt wheel one tenth of a rotation. It Leibniz died in poverty. The prnrorypt': of could only mrryout addition and subtraction, his machine was ignored and lay hidden in and was not hugdy useful as French currency a.n attic in thl' University of Gottinb'l'n, at the rime was not dtcimal: there were 12 Germany, until 1879. dl:lliers to a wI and 20 sols to a livre. ([his is the The 18th Ct':ntul), S:lW a flurry of same as the system in. use in the UK wltil calculating machines based on the same 1970, which had twelve pt::Ilce to a shilling principles as those of Pascal and Leilmiz, and twenty shillings to the pound.) but nont': really took offconUllCrcially since A slightly earlier machine that also used mechanical limitations meant that they rotating cogs was the Calculating Clock were never qukk and easy to usc. designed by vVilhclm Schicl...ard (1592-163 5) The first successful calculating machine in 1623. He made a prowtype, which was was made by Charles Xavier Thomas de destroyed in a fire, and possibly a second Colmar (1785-1870), in France. Hi s copy which h:ls never heen found. Schiekarrl Arithmomt':tt' r worked on tht': same died ofbuhonic plab'1H~ and hi.~ invention was principle as Leibniz's machine and could lost to histOry. However, he deS(:ribed his c;lrry Out all four arithmetic operation.~ invention in papen; (including letters to the easily. Between 1820 and 1930, 1,500 wert' aStronomer Johannes Kcpll'r) which were sold and similar devices appe;lred from discovered in the 10th century, enabling his other manunlcrurers. machine to be recreated in 1960. Gottfried Leibniz (sec page 154) TOWARDS A COMPUTER developed the principle of Pasca l's machine The precursor of the modem computer is intO a fully functional calculating machine generally considered to be the Analytical th:\[ could handle addition, subtraction, Engine designed by Charles Babbage multiplication and division . Like Pascal, (1791-1871). At the time Babbage was L eihniz W:lS a child prodigy. H e had learned working, complex ealeubtions were carried Latin by the age of 8 :lnd gained his second Out using tables of figollres, including doctoral degrel' at 19. Hi s first 'Stcppl'd logarithms, compil ed by people called Rl'ckoner' prototype was built in Paris in 'computers'. The tabk"S tended to have a lot 1674. It used a central cylinder with a setof of errors - Babbage's aim was to makl' a rod-shaped teeth of different ll'ngths that machine that could perform calcul :l[ions extended alon g thl' drum. This turned a without making mist3kes. He began series of toothed whi:'l'ls. Dl'spite his genius, designing his first such machine, a

PUT11NG TWO AND TWO TOGHH (R -.I

BLAISE PASCAL (1623-62)

The French mathematician, physicist and philosopher Blaise Pascal laid the foundation of probability theory and invented the first digital calculator. His mother died when he was a small child and the family moved to Paris, where his father took on his son's education. Pascal was something of a prodigy, publishing his first paper on mathematics at the age of 18. As well as designing his calculating machine, he worked on pressu re and hydraulics, formulating Pascal's law of pressure and making a mercury-filled barom eter. In his thirties, he underwent an intense religious experience, adopted the strict moral code of Jansenism and entered the convent of Port-Royal in 1655, giving up his interest in mathematics. Tb~ ParCfllill~

1II/"lI/b~rY

could d~al witb

lip ra 9,999,999,

bllr C01l1d ollly b~ llsed for additioll aud !illbrmctlOIl. Ir war operared by movillg

tbe dia/r.

rb~

solurioll ro

rb~

plVblelll tlppettrillg ill rbe

wil/MUls above.

Difference Engine, in 1822 to work Out the values of polynomial functions (functions that contain morc than onc tcnn, such as 4x! + 5y). The first Differencc Enginc hc designcd necded around 25,000 parts weighing a total offiftecn tOlLS (13,600 kg). It would have stood 2.4 metres (8 fcct) tall. !-Ic nevcr built it, hur designcd an improved version, Differencc Engine No.2 . Again,

COW CATCHER

Babbage also invented the cow-catcher - the metal frame attached to the front of trains to clear the track of obstacles.

.P

~ NUMBERS P UT TO W O RI(

/~

'The Analytical Engine weaves algebraic

patterns, just as rhe Jacquard loom weaves flowers and leaves. '

Ada lovelace

Difference Ellgille No.2

1J'{1.f

[OIlm-IKud by tbe

Scimce /r.1I1felll!l ill Lolldoll ro celebrate rbl' 200rh

TH E JACQUARD LOOM

The mechanical lacquard loom, invented by Joseph Marie Jacquard in France in 1801, uses punched

cards to store a woven pattern and control the loom to reproduce the

pattern. It was the first piece of machinery to be controlled by punched cards and, although it was entirely mechanical rather than computerized, it is considered an important step towards computer programming.

flllllwl'mny of Bflbbugl'f bin/;. Ir worked jlrrwlersly.

Bahbage did nOt 1ll00ke it, but it was built to his design in the Science Museum in London in 1989- 1991, to the engineering tolerances of Babhage's time. It produced a solution accurate ro 31 dibrits at its first triaL Babbage abandoned plans for the Difference Enbrine and emba r ked on a more ambitious project - to design an Analytical Engine which could accept programmed instructions on punched cards. Abrain he did nOt actually build it, but refilled the design repeatedly. The mathematician Ada Lovelace read of his design and constructed

P UTTING TWO AN D TWO TOGHH U

BERNOULU NUMBERS

a B

1

-I

2

4 -~

6

,

"

a program to calculate Bernoulli numbers using the Analytical Engine. Bernoulli numbers arc a sequence of positive and negative rational numbers important in number meoly and analysis (see tahle above).

10

a

14

!

'rVe have come a long way from Babbage's plans for machines that filled a room and performed only arithmetic, though there is some dehate about who created the very first truc computer.

AUGUSTA ADA KING, COUNTESS OF LOVELACE ( 181 S-S2) - 'PRINCESS OF PARALLELOGRAMS'

Augusta Ada King, often known as Ada Lovelace, was the daughter of the poet Lord Byron and Annabella Milbanke; the couple separated two months after her birth. Her mother hoped that instruction in mathematics might root out any madness Ada could have inherited from her father and engaged Augustus De Morgan, first professor of mathematics at the University of london, to teach her. Ada fi rst took an interest in Babbage's work around 1833. During a nine·month period in 1842-3, she translated the work of Italian mathematician Luigi Menabrea on Babbage's Analytical Engine. Her instructions for calculating Bemoulli numbers using the Engine are widely considered to be the world's first computer program, even though they were never actually implemented. Ada worked with Babbage until her prematu re death, which

was caused by over·enthusiastic bloodletting on the part of doctors trying to treat her for cancer of the uterus.

~

/ ' NUM8ERS PUT TO WOR k

/~ The Gcrm:m engineer Konrad Zuse (1910-95) CHIPS WITH EVERVTI-HNG made the first binary The microchip was first invented in 1952 by a Ministry of computer, the Z3, in 1941 Defence work er in the UK, Geoffrey Dummer. However, bur it was only partially the MOD refused to fund development and a patent was programmable. The first filed in the US by Jack Kilby seven ye ars later. completely programmable computer was the Colossus, designed by Tommy Flowers (J905-9S) for they had ever been built. Stepping into the the UK Secret Service during the Second gap left by the disowned Colossus, the US \"'orld War and used to crack the high-level claimed the tlrst computer with the codes of the German army. The first E1\T[AC designed by John j\lrauchly and J Colossus went into service in early 1944 and Presper Eckert and completed in \ 946. ten had been lJUilt by the end of the war. After the war, enormous mainframe HowL'Ver, most were dcsrroyed at the end of computers were produced for use in the war and for many years the British industry and hy governmentS and other government refuscd to acknowledge that large organizations ami the commercial computer industry was born. These early computers COH hundreds of thousands of dollars and worked with punched tapes or cards. They had no screen or kt.:yboard and were used largely for scicntific, military and fin311ci3l applications. A COMPUTER IN THE HAND .••

Dt'rpitt' fillillg a room, Colossus bad less pnxemllg ptr,;;tT

rball all iPod. Afrer rbe

il'tlIi

rbt' Bririsb

gfTvenmlfnr dt"llied ir brld evt'l·l'xifted.

Computer5 for everyonc became 3 re3lity with the development of the microchip in 1958 by Jack Kilby, who worked for Texas Instruments in the US . A microchip, or imegr3ted circuit, p3cks 311 the circuitry required for a complex electrical system on to a tiny wafer of si licon using etching tcchnolob'Y' It enabled the miniaturization of the prc\'iously huge machines used for calculations. The microchip led to the introduction of the first h3ndheld c3lculators in 1970 and then person3l computers. The lntel 4004, developed in 1971, W3S the first microchip to put 311 the functions of 3 computer on a

PUTTING TWO AN D TWO TOGETHU

single chip, which heralded the revolution in computer design. The number o f instructions that could be tltted on to a microchip doubled every year as manufacturing advanced. A modern microchip can hold features .smaller than mierometre across (one millionth of a merre, or a thousandth of a millimetre). M.icrochips arc everywhere, controlling our planes, caN and household appliances they arc even cheap Tbe cilrllitry QII a lIIiCHXbip is {JJQsTllall (Q see ;]. ,i tb tbe lIaked rye. enough to put into J\1i17wbips are fVl'IJwbere, CQl/(I"Qllillg lIear!y nil QII/' tecblw!Qgier. birthday cards that playa towards quantum computing, making use of tune when opened. The speed and powcr of computers the sub-atomic properties of matter to store continues to in crease at an astonishing rate. and manipulate data. The L'Omputers used to put a man on the Early calculating aids intended to speed moon in 1969 could be outwitted by a up arithmetic. Computers first did this, mobile phone today. facilitating bulk calculations, sa\'ing time Our fastest supercomputers carry out and giving accurate results. ComputeN can hundreds of trillions of operations a second, deal with exo'cmely complicated tasks as which is around a million times faster than a long as the tasks can be specified in logical standa rd desktop computer. A steps. This has provided an impetus for supercomputer capable of 10 quadrillion advances in IOb';C and its notation, and even (10 1') operations is expected by 2010. for computers to handle these tasks. Now, The deman ds we pbce on computers computers are used to manipulate are ever-increasing, too. Decoding D NA, mathematical expressions in symholic form, analy"t:ing radiation from outer space for working directly \vith algebraic equations tell-tall' signs of a deliberate m(..'Ssage and rather than calculating \\.;th numbers fed rendering digital movies at the highest into equations. For example, the resolutions still demand hours and days of Schoonschip program, developed by Nobel dedicated computer tim e. The ncxt physicist Martinus Veltman (born 1931) generation of L"Omputcrs may leavc silicon handles the alcgebra required for highcircuitry behind altOgether and move energy physics.

"

SLICE OF PI The fi rst calculation of n by an electronic computer was made on the ENIAC in 1949. 11 produced 2,037 digits in 70 hou rs. A simila r program run on a mid.range PC in 2000 took one second to achieve the iame result, so ran

defective, even, having no real facto~. But there are duste~ of interesting phenomena around them and they have become central to number theory.

approximately 250,000 times as fast as ENIAC.

FIND ING PRIMES

Special numbers and sequences People have long been fascinated by the apparently magical abilities of numhe~ to fall into patterns and to throw up surpri~ing mil'S. Some of these be'came apparent to very early mathematicians. These numbers were often incorporated into mystical or religious rituals, buildings and artefacts. Th e strange properties of numbers arc now the domain of numher theoristS.

It's easy to find small prime numbers; we can all do it in our heads. But tlnding larger primes becomes increasingly diffic-ult. Prim e numher theory attempts to predict the frequency of primes. The French maril(:'matician Adrien-Marie Legendre (1752-1833) showed in 1798 that

PRIME NUMBER S

Primes are a special class of intebTCrs: they are numbers which have no factors (cannot be di\'ided by anything) except themselves and 1. The primes under 20 are 2, 3, 5, 7, 11, 13, 17 and [9 (1 is usually nOt included). A~ numhers get largel~ primes become less frequent, but remain surprisingly common. Even with numhers around 1,000,000 about 1 in 14 is prime. People have studied prime numhcn: tor millennia, originally ascribing some mystical or religious significance to them. The Greek mathematician Euclid was the fi~t to prove that there is an unending sequence of primes around 300ne. Sti ll, more than 2,000 years later, we have no fornmla for predicting primes. Prime numbers sound as though they are nothing special - perhaps rather

Elldid prrU1nillg his lJ·ork ro Killg ProlelllJ I Sorl'1·

ill Ale:mlldl"ia. Tile ilJlI#mlioll is by Lollis Figllifl" alld dnrfs jivm 1866.

SP(CIAl NUMBERS ANO SEQU ENCU

THE SIEVE OF ERATOSTHENES

The

Ancient

Greek

mathematician

Eratosthenes (276- 194BC) developed a algorithm

simple

for

finding

How to sieve p rim es: 1.Begin by drawing

prime

up

a square

grid

containing all the numbers from 1 to your

numbers, called the sieve of Eratosthenes.

top limit for primes. Cross out 1 - it's not a prime.

2

1

4

5

6

7

8

9

10

PRIME NUMBER

2. The first prime is 2; write this at the top of your list. Cross out all multiples of 2.

15 16 17 18 19 10

11

12 11 14

21

22 21 24 2.S 26 27 28 29 10

3. The next remaining number is the next

11 12 11 14 1S ]6 17 ]8 ]9 40

prim e (3), so write this in the list of prim es.

41

42 41 44 45 46 47 48 49 SO

Cross out all multiples of 3.

51 S2 51 54 S5 56 S7 58 59 60

4. The next remaining number is the next

61 62 6J 64 6S 66 67 68 69 70 71

prime (5); write this in the list of primes and

72 7J 74 7S 76 77 78 79 80

81 82 8] 84

as

cross out all multiples of 5.

86 87 88 89 90

91 92 9] 94 95 96 97 98 99 100

5.Continue to the end of the square. The

101 102101104 lOS 106 107 1011109 110

numbers in the list (and not crossed out in

111 112111114 llS 116117118119110

the square) are the prim es . J

]

11

11 11 14 15 16 17 18 19 '20

11

12

n

21

22 21 24 2.S 26 27 28 29 10

21

22 21 24 2S 26 U

•

2145678910

PRIME NUM8ER

4

5

6

7

8

9

10

14 IS 16 17 18 19 20 28 29 10

] 1 11 11 14 ]S 16 ]7 18 ]9 40

11 12 11 14 U

41 42 4] 44 4S 46 47 48 49 SO

41 42 41 44 4S 46 47 48 49

SI S2 5] S4 S5 S6 57 58 S9 60

51 52 S] 54 SS 56 S7 S8 59 60

61 62 61 6 4 6S 66 67 68 69 70

61 62 6] 64 M

71

71 72 7) 74 7S 76 77 78 79 80

72 7J 74 75 76 77 78 79 80

81 82 81 84

as

86 87 88 89 90

]6

17 18 19 40

so

66 67 6B 69 70

91 92 9] 94 'liS 96 97 98 99 100

101 10210] 104 lOS 106 107 1111109 110

101 102 10] 104 lOS 106 107 1011 109 110

111 112111114115116117118119110

111112111114 llS 116 117118 119 110

x/ln(x)

where In(x) is the namrallogarirhm of x. As the size of x increases, the inaccuracy of the

11 11 17 19 21 29 ]1 17 41 41 47 51 59 61 67 71 71 79 8] 89 97 101 10] 107 109 111

81 82 81 84 8S 86 87 88 89 \lO

91 92 91 94 9S 96 97 98 99 100

the number of primes below or including a number x is approximately

PRIME NUM8ER

approximation becomes vanishingly small. The chances of a number, x, being a prime arc l/ln(x).

This means, for instance, that a number

~

/ ' NUM8ERS PUT TO WOR k

/~ ERATOSTHENES ( 276--194Bc)

Eratosthenes was born in libya, but worked and died in Alexandria (Egypt). A friend of Archimedes, he was in charge of the library

at Alexandria. Around 250sc he invented the armillary sphere, a spherical model with

intersecting bands that is used to demonstrate and predict the movement of the stars. It was used as an astronomical instrument until the 18th century.

Eratosthenes developed a measuring

also system for

longitude

latitude, drev.> a map of the

whole known world and made the first recorded calculation

of the Earth's circumference (see panel, page 87: Measuring the Earth). The later writer Eusebius of

Caesarea

(d.A0339-40)

attributes to Eratosthenes a calculation of

the distance from the Earth to the sun which is accurate to within one per cent of the figure now accepted.

around 1,000,000 ha s a chance of about 1 in 13.8 of being prime since the natural logarithm of 1,000,000 is U.s. TWIN PRIMES

Twin primcs arc pairs of prilll~ numbers separated by only 2. Obvious examples are 3 and 5, 5 and 7, II and 13, or 17 and 19. The twin primes conjecture states that there IS 50

The Eanb is n>pl"esmred by rbe baH at rbe cmTIT of tbe flrmillary

~bf"n,

rbe apparellt O1vits of

o/bel" bodies by tbe l"illgr aI"QIlIll/ ir.

all infinite number of twin primes. That s~cms r~asona ble, as it on Iy means they don't have to run om at some point. Bm it hasn't heen prown to be true. Therc is also a 'wcak' twin primes conjecture, which has been dcmonstratcd. This states that the number of twin primes below a number x is approximately given by this horribly complicated expression:

SP{C IAL NU M SER S ANO UQU ENCU

THE GOLDBACH CONJECTURE

In 1742, the Prussian mathematician Christian Goldbach wrote a letter to the Swiss mathematician and physicist Leonhard Euler in which he set out his belief that every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime, which mathematicians no longer accept. The conjecture has since been refi~ed and now states that every even number greater than 2 can be written as the sum of two p rimes. Goldbach could not prove his belief (which is why it is a conjecture and not a theorem), and no one has been able to prove it since. it has been verified by CDmputer for all numbers up to 1,01 8 (to April 2007), but a theoretical proof is still needed.

[~

J(logx)' ~ 1.320323631

J~

(Iogxi

Th~ll! '1:. 'I1SII" {/ IVCf/( J~{/IIJj'slI/lIl1

ill (be ferrer GQId/!(lch 'i:J1'1)rr

(Q

rtlll.:

Elllet;

Inn {blll'r hUiJ.1 ir is wir" 1ffillbofloricillllS,

propl'r di\>isors. Thi~ means that if you add together all the numhers that the numher can be divided by, the answer is the number itself. For t!Xample 6", 1 +2+3=lx2x3 28", T + 2 + 4 + 7 + 14 '" 1 X 2 X 14", 1 X 4 X 7

Don't worry about the e,x pression - it doesn't matter. VYhat is interesting to think aOOut is why it exists at all. \\!hat is it about numbers that makes it possihle ro find an Euclid first proved that the formula 1",1(2"_ 1) expression like this? Th e number in the gives an even perfect number whenever middle, 1.310313631, is called the prime conStant. It has no other known M1HA1LESC U'S THEOREM relevance except in this in 1844, Belgian mathematician Eugene Charles Catalan prediction of twin primes. (1814- 94) conjectured that 2' '" 8 and 3' '" 9 form the only example of consecutive powers (Le., 2 and 3, with cube PERFECT NUMBERS and square, 8 and 9), It was finally proven to be the case Perfect numhers are those by the Romanian mathematician Preda Mih~ilescu in 2002. which an:' the "''lllll of all their

51

2n _ I is prime . There arc currently 44 perfect numben; known, the hight:st of which is 2 J!.,tIl.b5h x (2 11·SIIl .657 -1). It has 19,616,7 I 4 digits .

o o 0 000

AMICABLE NUMBERS

Amicable numher.~ comc in p3irs. Thc proper divisors of om.' of the pair, added together, produce tht: other. The numbers 220 3nd 284 ~re an amicable pair. Tht: proper divisors of 220 arc 1,2,4,5,10, II, 20,22,44,55 and 110, which 3dd('d tOgether make 28+; and the proper divisors of 284 arc 1,2,4, 71 and 142, which tOb'Cther make 220. Pythab'Oras' followers studied amieJble numbers, from around 5OO8C, believing them to have 1l13ny mystical properties. Thabit ibn Qurrah (836-901), 3 tr3nsbt()r of Greek mnhenutlC31 tc.xt.~, discovered 3 rule for nnding amic3ble numbers. Arab m3them~ricians continued to study them, Kamal aI-Din Abu'I-H asan Muh3mmad 31-Farisi (c.1260-1310) discovering thc pair 17,926 and IH,416 and Muhammad B3qir Y.1Zdi nnJing 9,363,584 and 9,437,056 in the 17th century. POLYGONAL NUMBERS

Some numhers of dot.~. Stones, seeds or other objcct.~ can be. arr3ngeJ into regular polygons. For example, six stonl!.~ can be 3rranged into 3 perfectly regular triangle.

Six i.~ therefore :I. triangular number. If we 3dd ~n cxtra row of stoneS at t he bottom, we get the next triangular numher, ten:

o o

0

000

o

000

Nine StOnes can be arranged intO

o

0

:J.

s(]uare:

0

000 000

The nl!.xt square number has four side, giving 4! = 16.

011

e3ch

Some numbers, such as 36, 3re both triangular 3nd squ3rl': 000000 000000 000000 000000 000000 000000

o 'Six i5 a number petfeet in itself, and not because God created all things in six days; rather, the converse is true. God created all rhing5 in six da~ becau5e the number is perfect.' StAugustine (AD354 -430), The City of God

52

00 000 0000 00000 000000 0000000 00000000 P()lygon~1

numbers 3rc increased by incremcnting each side by onl' extra unit.

SP(CIAl NUMBERS AN O UQU(NCU

TRIAN GU LAR NUMBERS

[

3

6

•

o

o

••

o

[0 o

o

0

•••

o

0 00

••••

SQUARE NUMBERS

•

0

[6

9

4

[

•

••

0 0

0 0

• •

•••

0

0

0

0

0

0

0

0

0

• • •

••••

PolYb"Onal numbers have been studied since the time of Pyt hagoras and were often used as the basis of arranb>ementS for talismans. Notice how the prc\'ious triangubr or square number is incremented to form the next in the series. TRIANGULAR

SQUARE

NUMBERS

NUMBERS

1

1

3 (= 1+2)

4 (= 1+3)

6 (= 3+3) 10 (: 6+4)

9(=4+5 )

15 (= 10+5 )

25 (= 16+9)

AI/Ionio Galldl iliaR-pmlTl'li fJ1!IIlgic SqIllD? ill th~ Ciltlxt/ral rf dx Sllgmdll Fllmlfi" iI/Btl/TrlOI/a. TIx 1II11gir IIImm- is 33,

it) has three squares on each side and the magic constant is 15:

16(=9+7)

21 (= 15+6)

36 (= 25+11)

28 ('" 21 +7)

49 (= 36+13)

MAGI C SQUARES

A magic square is an arrangement of numbers in a square grid so that each horizontal, vertical and diagonal line of numbers adds lip to the same total, ca lled the mabric constant. The small est magic square (apart from a box with the figure 1 in

th~ YIIPfX1.w ~~ rfChrist at his deatll.

15

2

7

6

15

9

5

1

15

4

3

8

15

15

15

15

15

This IS known as the Lo Shu sguare after a Chinese legend recorded as early as 650BC This tells how \,illagcrs tried to appease the spirit of the flooding rivcr Lo and a turtle came our of the watcr with markings on its back that depicted the magic square. The people were able to use the pattern to control the river.

Magic squares have been known for arowld 4,000 years. Th!..')' are recorded in ancient Eb'YPt and Lldia and have. been attributed with special powers by cultures around the world. The first known magic squares with fin' and six numbers on L'11ch side arc described in an Arab [ext, the RflJa 'il Ib~"Wml fll-S(/fo (E1lt),dopedia of tbe Brefbrt'll of Pllrity), written in Baghdad around 983. The tirst European to write about magic squares was the Greek Byzantine scholar Manuel .M oschopoulos, in 1300. The Italian mathematician LUL-a Pacinli, who recorded the system of double-cuny book-keeping in 1494, collected and smdied Illabrie squares. (He also compiled a treatise on numher puzzles and magic that lay undiscovered in the archives of the University of Bologna until it was published in 2008.)

PI A~

well as numbers that foml series or pattems, there are several strange and significant single numbers. The fir.~t to he discm'ercd W;lS pi, n . This ddines the ratio of a cirde's diameter to its cireum ference, ~o that the circumference is nd

where d is the diame.ter. Th e. vJlue of rr is a decimal number with an infinite number of digits after the decimal poinL It IH.'gins 3.14159 (which is a good enough approximation for most purposes). That the ratio ofthe diameter of a circle to irs circumference is always the same has been known for SO long that iL~ oribrins can't be traced. The Eb'YPtian Ames Papyrus,

c.l650BC, uses a value. of 4 X (819f = 3.16 for 11". In the Bible, measurements relating to the building and cLJuipping of the temple of Solomon, c. t)50nc, use a value of 3 for 1L The first theoretical calculation seems to h:tve been carried Out by Archimedes of Syracuse (287-212nc). He obtained the approximation

He knew thar he did not ha vc an. accurate value, but the average of his twO bounJs is 3.1418, an error of a bout 0.0001. Later mathematicians have refined the approximation by discovering: more decimal places.

e Another strange and very significant number is e. The value of c was first discovered by Jakoh Bernoull i, who tried to discover the value of the expression lilll

•

""

(I . . 1)' II

while working on calculating compound interest. \-Vhen evaluated, t he. expression gives th e series that defines e.. The tI.rst known use of the eon Stant, represented by the letter Il, is in letters from Gottfried Leibniz to Christi:lan Huygcns written in 1690 and 1691. Leonhard Euler was the first to use the letter e for it in 1727, ~nd the fir.') t published use of e was in 1736. He possibly chose I.' as it is the first letter of the. word 'exponential'. I.' has an infinite. numher of digiL~ after the decimal place, as it is defined (among

SP{ CIAl NU M 8[RS ANO UQUENCU

othcr methods) as the sum of allnumbcrs in an infinite series - see panel. page 41, UNREAl!

The imaginary number, i, is detined as the square root of minus 1. The term imaginary numher was used by the French philosopher and mathematician Rent! Descartes (1596-1650) as a derogatory term, but now mcans a numher that involves the imaginary square rOOt of -I :

en ,+ T '" O. This, known as Euler's identity, is a special case of a rule which rdates complex numbers and trigonometri c functions.

(A negative number t::J.n't 'really' have a square rOOt as when a number is squared, whether it is positive or negative to Start with, it always gives a positive result.) A complex numher z is defined as z'" x + iy

where x and y afe ordinary numben:. Imaginary and complex numbers were encountered first in the 16th cenrury by Gerolamo Cardano and Niccolo Tartaglia while investigating the roots of cubic and quartic equations, and were first described by Rafael Bombelli in 1572. H owever, even negative numbers were distrusted at the time, so people had littl e time for imaginary numbers. It was in the 18th century that it began to he taken more seriously. It was brought to thc attention of mathematicians properly by Carl Friedrich Gauss in 1832. Strangely, the special numhers come together in the expression which has been call ed the most startling in the whole of mathem:uics:

Tb .. Grnk ffltllbcmoridllll Pyrhogol'llrdl'1noll#rarrs

bir rb..o/'~m 1)1/

1)1/

rbt grolmd,

dgbr'llIIg'.-d "'71l11ghs by dl'tr;1'illg

,;S-/ '1 NUM 8 US

PUT TO WO Rt(

1'/ Unspeakable numbers The t'oncept of banning a number may seem bizarre, but it has happened for millennia and still happens even tOday. Some numbers havl' been considered just too difficult or dangerous to countenance and havl' been outlawed by rulers or mathematicians. But a lwnned number dOCiin't go away, it just goes underground for a while.

'It is rightly disputed whether irrational numbers are true numbers or false. Because

in studying geometrical figures, where rational numbetJ desert us, irrationals take their place, and show predsefy what rational numbers are unable to show.. we Gre moved and compel/ed to admit that they are correct .. Michael Stifel, German mathemat ician

(1487-1 567) PYTHAGORAS' NUMBER PURGE

The ancient Greek mathematician Pythagoras did not recognize irrational numhers and banned consideration of negative numhers in his School. (An irrational numher: is one that eannot be expressed as a ratio of whole numbers; so 0.75 is a rational numher as it is ;/. hut j"( is irrational.) Pythagoras had to aclmowledb't! that his ban caused problems. His theorem, which finds the 1t~nbrt.h of a side in a rightangled triangle from the lengths of the other two sides, insrantly runs into problems if only rational numbers are recognized. The length of the hypotenu~ (longt!st side) of a right-angled triangle with two sides one unit long is the square rOOt of two - an irrational number ("" 1.414).

Pythagoras was una hIe to prove by logic that irrational numbers did not exist, hut when H ippasus of Metapontum (born (.500Be) demonstrated that the square root of 2 is irrational and argued for their existence, it is sai d that Pythagoras had hi m 56

drowned. According to legend, Hippasus demonstrated his discovery on board ship, which turned out to have been unwise and the Pythagoreans thre<.v him overboard. P),thagoras' ban was b ased nn his aesthetic and philosophical objection to the existence of irrational numhers. Later censors have had political, economic and social reasons for trying to outlaw certlin numbers or catebrories of number. ARABS

v. ROMANS

There was conside.rable resistance to the introduction of Hindu-Arabic numerals in Europe in the Middle Ages . The ease with which arithmetic could he carried out with the new number system made. it attractive. As Hindu-Arabic numbers threatened to democrati ze numeracy they were demonized by those who had an interest in restricting numerncy and retaining it as a special tool of the dite. If mathematics were opl'ned up to everyone, a source of power would be lost. The Catholic Church wanted to kl'ep control of l'ducation by maintaining its holJ on numbers, and in addition opposed the. sysw m from the Islamic world

UNS P [AKA6 l£

on religious grounds. MathemOlticians who practised the arcane systems of mathematics using an Olba(:us were protected by the Church. So strong was the opposition to the popularizatio n of Hindu-Arab numerals that, it is said, some poor souls were even burned at the smke as heretics for usin g them. However, merchant~ and accountant~ wanted to usc the new system as it made their tasks easier. The battle between the algoristS - those who u.scd algorithms, or calculating methods, with Hindu-Arabic numerals - and the abacists - who used an abacus and Roman numerals - raged for ccnmries. The emergence of printing in Europe eventually contributed to the domin:mce of the Hindu-Arabie syStem. Dissemination o f Olrithmetical methods became easy and it was no longer possible to runtain the flow of numbers. Ewnmall y, of course, the cs tabli~hment buckled under pressure and the number system we usc now triumphed. But Roman numerals and the abacus continued to be used in some areas of life for many yea rs. The French were the first to release themselves from the tyranny of the abacus. After th e French Revolution (1 789), there was a complete reversal and use of the abacus wa~, in mrn, banned in schools and government offices .

,

NU M 8t R~

LIIlly Arithmenc TI")eet:; rbe flb'Klts fol"

Nil/till-Am" IIl1mr/"a!s ill rbe MargJri[J

Philosophic:! by Gl7f,OrillS ReisTb, J503. THE SECRET OF ZERO

The names for 'zero' in use at the time when Hindu·Arabic numbers were banned in Europe were dIm, chifre, tziphm and so on. These names came to stand fo r the whole number system that included zero. As the system was u sed secretly, the name also came to mean a code or secret and developed into the word 'ci pher'.

666 - THE NUMBER OF THE BEAST

Many religions depend he3vily on number sy mbolism and me speci31 numerical methods for discoverin g or conce3ling secretS. In th e early ye3rs of Christianity, the Romans were using as a talism3n the Magic Squ3re of the Sun. [n this lmgic square of six by six numbers, the numbers 1

to 36 3re armnged so th3t the rows, columns 3nd di3gonals add up to 111. The sum of all numbers from I to 36 is 666. The Church banned the po5.'icssion of the magic square because in Christianity 666 is the N umber

57

-P /

1 NUM8ERS PUT TO WOR k

/// 666l)'lIIboliurtbemdoJ

NOTO NEGATIVES

tbe1l'{JrldforChristifills,

In Renaissance Europe, negative numbers were not recognized. Solutions to mathematical problems that included negative numbers were ofren disregarded. Evt!1l though early Chinese and Indian mathematicians had explained the U~ of nCbr;nivc numbers, u~ually by relating them to economic debt, later mathematicians in Europe strub'gled with them. Michael Scifel (sec panel, page 56) called numbers less than

bllt ill

Cbtl/ere [I/Itlll"l' it

is amsidl'lYdIlIcky.

of the Beast, thought to he the enemy of God identified in the Book of Revelation. Possession of the magic square became punishable by deat h.

ill fbI' UallflllTllell Sqlllll"e 1I1t1SfIl{11' 011 4 Jilin 1989. It if illegal to lise o rbe dnre of rbe masmcre as II PIN m code ill Cbillilo

Sevaal thol/Sf/lld people were killed

58

UNS PEAKA8 U NUM81RS

zero 'absurd numhers', for =mplc. The French mathematician Albert Girard (1595-1632) was probably the first major academic fully to accept negative numbers in solutions, but it took until the early 19th century for a proper foundation for arithmetic with negative numbers to be .set out. DANGEROUS DIGITS

666 is nOt the only specific number to have been demonized. In China it is illeg';!l to usc the date of the Tiananmen Square massacre (8964,4 June L989) as a password or PIN or in any other form that might link it with the event (rather than as the ncxt number in the natural sequence of counting). Ll the US, there is a hexadecimal (32digit) number which has acquired the status of ' ill egal number'. 1t is the key to encrypting high-definition DVDs, and its publication is technically illegal (since by using the hy with the appropriate mechanism it would be possible to unencrypt the DVDs). The AACS (Advanced Access Content System) claims that it is a copyright circumvention device possession of a copyright circumvention device I S III brcach of the Digital lvlillennium Copyright Act (US, 1998). H owever, within a short time of its being revealed, the 'secret' number was Jlublished on 300,000 web sites; attempts to remove it from the public domain were dearly b,'Uing to be fruitless. The AACS also claims to own many other numbers used for encryption but won't say what they arc (as their usefulness depends on them being secret). The only 'special' feature of these numbers is that they arc nOt at all special, but arc g'Cnerated

random numbers. There is, understandably, considerahle n:sisrance to the notion that anyone can 'own' a number and prevent others knowing or using it. Computer enthusiasts rushed to lay claim to their ovm numbers that they could tell everyone else nOt to use in retaliation and mockery of the AACS. So we can stop now. No more numbers can be used in this book as someone else owns them all! MOVING ON

Examining numbers and their properties is all very interesting, and was b'U()(] enough for the Ancient Greeks (who disdained applicati ons of mathematics), lmt for most people the value of mathematic'S li es larbrcly in its usefulness. Num hers all ow us to measure, count, make things, run economies and examine the universe around us. [n truth, they are the kc'Y to all of science and a lot of art and h3ve played a key role in every ci\'ilization.

NII'II/berr ,·Ide (/lid define rbe wmid's economies, nlld

ill

impnct all

till Ol/l·acrivitief. 59

CHAPTER 3

THE SHAPE of things

Not everything can be counted. A herd of catt1e contains a countable number; even a field contains a number of blades of grass that could, theoretically, be counted. But some things can only be measured - we can't count how much water is in a IlOnd or the distance between the hill and the sea, yet it is useful to be able to quantify them. Geometry - working with distances, areas and volumes in the real world - was one of the earliest applications of mathematics. (The word is derived from geQ, ' earth' and metro/I, ' measure' in Greek.) It is likely that some of the first calculations ever carried out related to building monwnenrs, marking out land or making artefacts for religious purposes. A first necessary step was to develop units of measure, in itself a major conceprualleap from counting. Measuring makes an artificial distinction, dividing something continuous into nominal units.

Tix

Gold RtlSb prodllCt'd gold ro b~ w~igbed. Ilor coills to b~ cOllllt~d.

TIn SH APE O f THI NGS

The measure of everything Measuring is essential as soon as a society starts to enclose and own land, or to trade in anything but the mOst basic objects, or to start building any hut the simplest stmctures. Early ci\~lizations needed to he able to lllea:mre distances, areas, volullles and time. Some thinb'"S that could be counted, such as grains of wheat, are more easily measured by volume, tOo. But units of measure didn't develop methodically. The jumhled mix of mea~ures still in use in the UK and the US is the legacy of earlier systems orib~nating in Ancient Babylon, Eb'YPt and the Roman Empire with later Scandinavian, Celtic, Germanic and Arabinfluenced systems. AT ARM 'S LENGTH

Most of the earliest measurement systems, from China to pre-Columhian America, were based on dimensions of the human hody or common objects, such as brrains of wheat. Americans (and older British people) still measure short distances in feet, and a grain is scill used as a unit of wcight. (It is the weight of a brrain of harley, and has remained constant for over 1,000 years.) The measure of ~,'old and gemstones, the carat, has irs origins in the caroh seeds used originally hy Arah jewellers to weigh

A tldly 711f1l/ ill all EJ/glish bop fold:

picJ.1'I"S';Jel"~

paid by tbe bllYbd (a me/JSI/remmt of roll/we) alld nceiVffi half" tnlly stick by

,""/ly

of naipt.

precious metals and stones. The caroh has seeds that are remarkably uniform in weight, making it ideal for measuring vcry valuable commodities . The c"Ubit, the unit of length familiar

from the Old Testament in AHEAD Of THEIR TIME

The inhabitants of the Indus Valley were amongst the first to produce a unified system of weights and measures and coul d measu re dist ance, mass and time with great precision. Their smallest measure, at 1. 7 mm, is the finest known from Bronze Age civilizations.

62

which Noah measured his ark, was an Egyptian measure equal to the distance from the elhow to the fingertips. It was subdivided into other units that also related to parts of the hody:

THE MEASU RE O f EV ERYT HI NG

cubit == 28 digits (a digit is a fingerwidth) 4 digits", 1 palm 5 digits == 1 hand

12 digits", 1 small span

~ FLEMISH ELL ~

14 digits", 1 large span

c:

~ CUBIT ~

•

•

0(

ENCLISH Ell

•

FRENCH Ell

FAT HOM

. • •

Bu t the human body comes in all shapes and sizes, so one person's 'hand' may be another's 'palm'. To overcome the obvious difficulties and potential for dispute, standard measures were Tlx biswricnlllllits as Sbl1WII ill chis [ needed. The cubit Uollll1do's Virnn~an M.an. sticks used in Egypt were all copied from a royal standard made measured in furlongs (or stade), leagues and of black gran irc and mea.~uring 524 mm miles. A furlong was an t:!ighth of a mile, a (20.62 inches). The system successfully mile was 5,000 feet and a league was 7,500 imposed uniformity. The Great Pyramid at feet. These measures, along with the Giza is built on a square hase 440 by 440 Roman measures for weight based on cubits with variation of no more than 0.05 pounds and ounces, spread through Europe per cent on any side - making it accurJte to and, hundreds of years later, were carried around the world. 115 mm in 230.5 metres.

•

•

ROMAN FEET The foot divided into twelve inches originated with the Romans, though their foot was probably equivalent to 11.65 modern inches, or 296 mm. (There is some variation in the Roman foot, which appears to be deliberate bur has never been fully explain ed.) They also had a palm, which was a quarter of a foot. Larger distances were

•

BIG FEET AND SMALL FEET During the centuries after the fall of Rome, measures developed and proliferated around Europe, bur there was no unifomlity. The length of a foot or the weight of a pound varied from place to place and sometimes according to what was being measured. So a gallon of wine contained 231 cubic inches, hut a gallon of ale was 282 63

TH£ SHAPE O f THINGS

DEFINING THE PITCH OF A PITCHER

Chinese weights and measures developed independently of those in the West and the Middle East. The system was unique in incorporating acoustics in its standards. The standard vessel for measuring volumes of wine or grain was defined by the weight it could

hold, its shape and the pitch of the sound it made when struck. Two vessels of the iame shape, material and weight will only make the same sound if they hold equal volumes. The same word in Chinese is used for 'wine bowl', 'grain measure' and 'bell'. Cbillil is going lIIen·if liS tm de illcreilSes ,rirb rhe rest

of rbe

world, bllt rbe old lJ'eigbts IIlId measures sysrn ll persists mllcb

cubic inch es. (Th e first, known as the Queen Anne brallon, is still the standard gallon in the US, though in th e UK the gallon was redefin ed in 1824.) Standardization came slowly, progressed by separJte legislative acts in different countries. In the US, the older En gli sh units survived after the UK had redefin ed

ill

I!f tbe (1XIIltl)'.

th em, resulting in the discrepancy between US custOmary and UK imperial units today. WEIGHTS AND MONEY

It's no coincidence that the pound was both the unit of weight and o f currencr in the UK for many centuries. VYhen coins were made from preci ous metal s, th eir weight

'Uniformity of weights and measures, permanent, universal uniformity, adapted to the nature of things, to the physical organization and to the moral improvement of man, would be a blessing of such transcendent magnitude, that, if there existed upon earth a combination of power and will, adequate to accomplish the result by the energy of a single act, the being who should exercise it would be among the greatest of benefactors of the human race. '

John Quincy Adams, American Secretary of State, 1821

THE MEASU RE O f EVERYT HI NG

The IWnmll5 illtrodured thr poUIld, 'which har rilll:/! hem tkdlllali~d

and /lUly

SO/I/~

day give u'ay ro the Ellrv .

was important, since weight and value were equivalent. The Hebrew shekel was perhaps the earlit!.~t mea~1.lrc used for money and weight, and the Romans introduced the pOllnd, which was then used in Europe for the next 1,000 years. Ln 1266 Henry III fixed the weight of a penny at 32 b'Tains of wheat, with

A NEAR MIS S

An

identical

eventually

metric

introduced

system in

to

that

France

was

proposed in T668 by Bishop John Wilkins, a founder of the Royal Society in England. In a long book on the p ossibility of an international language, he proposed an integrated system of measurement based on a decimal system and almost identical to the modern metric syst em. His unit of measurement was 997 millimetre.<> - almost exactly a metre. The unit of volume was the equivalent of the litre. Wilkins' proposed system was never promoted and wen t largely ignored until

rediscovered

by

Australia n researcher Pat Naughtin in 2007.

twenty pennil!S making an ounce and twelve OUllC!;'S to the pound. Eight pounds was the weight of a gallon of winc. Although the twelve and twenty switched places in thc monemry system, with twelv!;' pence to th(' shilling and twenty shillillgs to the pound, the equivalent of 240 pellct' to the pound both sterling and avoirdupois - was established. The shilling has gone and the currency has been re\'ised, bur the leg·acy of the Roman pound and penny still survives in the British monetary system (though the Euro will probably replace them soon). ' FOR ALL PEOPLE, FOR ALL TIME'

The scientific community worldwide now llses SI uni ts (SystNue Inte111ntiOlla/ d'Unitis) the seven standard metric units (gram, metre, Keh
THE SHAP E O f TH INGS

STAR QUALITY OR 'MAD AS A BADGER'?

Numbered

scales

can

be

used

fo r

qualitative comparisons. The star rating commonly used to grade hotels is a familia r and universal example of a qualitative scale. The b ad ger ra ting for deg rees of eccentricity used in parts of southern England is a localized system. Many websites invite users to rate preferences and experiences numerically. There is a whole science of evaluating the effectiveness of such rating systems .

and second). The metric system was first developed in 18t h-century France. The need for a simpler, unifi ed and standard system of measures was pointed Out by th e vicar and mathemati cian Gabri el Mouton in

1670, hut it took anoth er 120 years before anything was done to provide it. [n 1790, Charl es-Maurice de TallyrJlld set the ball rolling ag'ain and the French Academy of Sciences recommended that a team detcrmine the distance from the North Pole w the Equator, guing through Paris. The first stage was to meas ure the distance on the meridian from Dunkirk in northern France to Barcelona in Spain. The day after King Loui s }.'v r gave hi s approval , he was imprisoned by the French Revolutionary rulers. As a result, it was an oth er year before the expediti on started LOltir X VI grrve hir permirriliJl fur the expeditif»l -he

U'fIS

gllillotilled five II/(JI/thr Inter.

A TlMEliNE Of WEIGHTS' AND MEASURE 12 IS c.800 c.30008C

I-Inly Rmll:1n Em p"ror Ch" dema PI"

Egypti:Jnsd~\~ lop ~

ll)Val smn d"rd f';' t heir basic measure of I"ngth , th~ l'UlJit

(r.i68-S H ) tri~s w regulntc w"ighrs anJ rnell5ures

An English n~tin nni st~ n d"td for weightl; ami rn ~"su res is agreed ~nd ell.l'hrined in th" Magru Cm:.. , the charr~r gr~nted by King J oh n (1 199-12 16)

1352

Ed "·~rd

m or

Engh nd e ..r:ahll>hes th~ t (XI" Stone C<1" als H IXllmili. , ,·,,-Iue th ot rem"in.~tn th is Jay

1588

New standard.. issu<;>::! hy Eliz"beth I in Eng land (r. 1558--1603)

-- r.220 11C The Ii ~t ('"11 i nese Emperor, Shi I-luan g OJ (r.121-109l1 10oc) standard izes all weightl; and measu re<;, "wn spai l}'ing th~ precise axle len¥rh to be used o n carLS

66

%0 The fi rst king of all Engl~nd, King Edgar (".957J!7S), dl'Cn:.-eS tho t weif,
with a SCln
02 66 m Ii.us th~ relationship between 1110n(1' ~nd weig:ht in Eng lish curren,")" making one penny the weight o f 32 grains o f wh~~ t ~nd 240 pcnc~ t<.> the p
1496 New St,:mdllr
TH E MEA SU RE O F EV ERYT HI NG

A map of the herwfIIs

from the

H annonia

lvlarnx:osmica Atlas by

Cellmills (1660).

1824 1790

1670 GHlxiel Mouton proposes a metric ,,}Stem o f weigh.,; and meaSur"" in France

166' John \\r,lkins proposes a unive"",1 Ilwtric system o f weight., and measures in England

George \Va~hington's fi r;t mes>;agc to Congress St~teS the nee<1 for 'uni formity in ~"Ufren<.y, weigh.,; and measur",,'; lAngr""s retains the English weigh.,; and measures system

Re
1878 The yml is redcfin("(1 in the UK

1707

1799

1866

I %()

A gallon of wine is fIXed Ht 231 eubic inches. The measure had been us<:<1 sill~"c the time o f Edw~rd I (r.12 7 2- 1307), but the act of 170i fi xed the size

Standards of the metric s},;tern are defined in Paris, Fran~"C

The M!-1:ric Act allows the use o f the metric s}stem in the US

New Systol1e /"te""II;o",,1 d'U,,;tis (lnrernati ona I System of Units Or SI) fonllul.at("(1 in Paris, Fran~"C at the II th General Con feren~"C o n \\'eigh.,; and Aleasures

67

TIn SHAPE O f T HI NG S

and even then it was beset with dift-lcultics. \Var in france and Spain SO hi.ndered the project that it took six years to complete the journey. But in 1799 the metric ~ystem was formalized with twO new uniLS of measure, intended to be universal and enduring. The metrc was defined a~ 'one tenmillionth part of a meridional quadrant of the Earth'; the gram was the mass of a cubic centimetre of pure water at 4°C (the

temperarure at which iL~ density is b'n:atcst). A platinum cylinder, the Kilogram of the Archives, became the standard for the kilogram (1,000 grams). The kilogram standard is now made of platinum-iridium alloy kept in ScYres, nCar Paris; the kilogram i~ the only basc unit still defined by a physical object. AttemptS to find a better wayof defini ng the kilogram are ongoing.

SillY NUMBERS? Calculating with the 1,760 yards in a mile,

by

light

in

a vacuum

in

the 16 ounces in a pound or the 160

1/299,792,458 of a second. And a second

square rods that make an acre has

is the duration of 9,192,631,770 cycles of

been the bane of many a schoolchild's life.

the radiation associated with a specific

The metric system looks simpler, based on

change in energy level of an atom of the

multiples

isotope caesium·133.

of

relationships

ten

and

between

with measu res

clear of

different quantities. But the 51 units have

III 1793 a mml' was defilled

some

of the distallce fiwlI tbe Pole to the 0jllarOJ~

even

more

bizarre

defining

numbers. A metre is now the distance

68

travelled

(IS

11/0,000,000

IIcr.lJ its defilled by rhe speed ofligbr.

EARLY G EOMETRY

STONEHENGE

Stonehenge is a vast arrangement of concentric circles of stone and holes that were perhaps intended to hold posts or other stones near Salisbury in Will.5hire, England. The remains of the monument, built in three phases over a period of around 1,000 years between 3000 and 2oo0BC, consist of huge standing stones, some surmounted by stone lintels. The arrangement shows an ability to work with circles in space, and the curved lintel stones

demonstrate an understanding of arcs of a circle ~ when all were in place, the lintel stones would have formed a true circle, not il series of straight stones. The only tools available to the builders were picks made of deer antlers and stone hammers, yet they were able to Gllculate and measure portions of a circle and distances. The northeast axis aligns with the position of the rising sun at the summer solstice, suggesting that some form of Gllenda r had been developed.

Stonehenge if flfllrly /JS

old as fbe PJrtnllids

{ou a mil/III!/" l"role.

If, roo,

<J'lts

bllilf

;rlrb a sopbisricnted lillriersrrllldillg

of

gwml'fry alld Ibe ·/I/OVnllnll

of rbe filii .

Early geometry Geometry deals with dist:J.nces and angles, with lines, areH and volumes. In its simplest and earliest forms, it works with lines and linear shapes in a flat plane. But &om this it has been extended to dealin g with c"Urvcd lines in three-dimensional spacc and cyen to curved spaces in more dimensions tha t help US to explain the very fabric of the universe. On the way, Some of rbe ell/·lint deco1"tlted objrcts /JrlVr S')"mllen·ica/ paftf'J"lIs.

it has f,';ven us architccture, aStronom~', optics, perspective, l~Jrtography, ballistics and much more. PATTERN AND SYMMETRY: THE ESSENCE OF GEOMETRY

The earliest engagement with geometry predates written number systems. Many early peoples have left evidence of their interest in repeated patterns, symmetries and shape in the fornl of geometric patterns decorating their objects, structures and dwellings. 69

TH( SHAPE O f THINGS

Some of these date from 25,OOOnc. Early structures built or aligned with considera ble precision are further testimony to our ancestors' b'Tasp of some simple form of geometry. PROBLEMS WITH LAND

Practical problems of geometry must have been tackled in building projects long before tht..}' were recorded III written form. The Sumerians, the Babylonians and the Egyptians became quite adept at working with the geometry f/fI'odorus (c.484-425BC) has

bem dern"ibed as 'the Fflther of

Nisrory'.

Tbe flllI/llfI/ flooding of the Nile, which <'rami bOlllldfllJ nTflrJ:s,

ilXIS

olle of tbe pmmplY to tb(

droelopmmt ofnTathemfltics in Anciellt l:.gypr.

of two-dimensional shapes and threedimensional objects, calculating distances, areas and volumes. Documents from around 3100BC reveal that the Egyptians and Babylonians already had some mathematical rules for mt..'asuring storage containers, surveying land and planning buildings. The Great Pyramid at Giza \vas constructed around 1650BC, demonstrating that the Eb'Ylltians already had a good grasp of geometry. According to the Greek historian H erodotus, the Eb'YPrians needed to be able to calculate areas because the seasonal floodin g of the Nile swept away property boundaries. They needed henchmarks and surveying techniques to restore them properly. Egyptian geometers were sometimes referred to as 'rope-stretchers' after their way of measuring and marking Out distances and shapes using ropes. The

EARLY G EOMETRY

STRANGE GEOMETRIES

Enormous geometric patterns drawn in the Nazca Desert, Peru resolve into glyphs when seen from the air. They were treated by the Nazca culture between 200Be and AD6OO. There are 70 individual figures, ranging in complexity from simple lines and geometric shapes to stylized animals, plants and trees. Their significance is unknown, b ut their construction is evidence of some considerable skill with geometry among a people about whom we know little. The uny Iby climate o/tbe Nazca Desn7 bas be/prd to prem-'i'e tbe girlllt geometric parnrlls dr/r'';)11 all [be

/fwd amlllld 2,000 ),em'Y ago.

same technique no doubt served just as well to mark ground plans for building projects as to reclaim land that had been flood ed . WRITING IT DOWN

The earliest known m:uhematical document I S the Ahmes papyrus (sometimes called the Rhind papyrus) from Eb'YPr. It was written by the scribe Ahmes around 1650BC, copying from an older text written about 100 years pre,~ously which itself may have contained much older material. It is a papyrus scroll 33cm tall by over 5 metri..'S long ( 1 foot by 18 feet). It presents 84 mathematical problems, covering topi'-'S in

arithmetic, albochra, b"Cometry as well as weights and measures. The problems arc all given a strictly practical presentation; for exam pl e, one asks 'a round field has diameter 9 khcr. \Vhat is its area?' The socalled Moscow papyrus from the same date includ es instructions for working our th e volume of part of a pyramid. Because the Eb'Yptians wrOte 011 papyrus, which is fra gil e, littl e of their mathematical writing su r.~ves. The people of l\tt csoporamia, Tbis Bary/ollillli cia)' tablet jeawl'illg a problem ill geolllrny is alVl/lld 4,000 )'MI"f old.

71

THE SH APt O f T HI NCi

th e fertil e hasin drained by the Tigris an d Euphrates, wrote instead on clay uhl ets which th ey haked. These are much more enduring and over IOO,lX10 survive. A seri es o f Babylonian day tabl etS datin g from 1800- 1650BC shows methods for cal (."Ulating th e hypotenuse of a right-angled rriangle usin g what we now call Pythagora s' theorem, and for working with the areas o f rectangle ~ , triangles ~nd circles. One probl em, for instan ce, asks about the di smn ce th e foot o f a l::tdder moves if the top, leanin g against a wall, slips down. Tbey include an apprnximation for the square roOt of 1 wbich is accurate tn five decimal places. The positional number sy.;tem used hy the Babylonians was better sui red to all kinds of calmlation than the Egyptian ~y ste m th ough it is hard to say whether an intercst in numbers prompted the development of th e hener !>y~tl.' m Qr resulted fro m it. U nlike the Egyptians, th e B~bylonians .~ eem to have had a concept of general principles - that .~o m e mathemati cal st~t e m e n tS will always he tru l' in any situation of a given type. For exampl e, one clay tablet shows the ratio of the sides to th e dia gonal of a square. Babylonian mathematicians had derived th e ratio 1: J2, th e impli cati on bein g that it is possible to fin d th e dia gonal o f all} square by multiplyin g its side by J2 . H owever, bo th th e Egyptians and th e Bahylonians showed a cavali ~r disreg'afd for accuracy. In some cases, th cir c~lculations give precise answers. ill other c~sl!s, they uSt quite approximatc mt thods for findin g areas, but never co ncede th~t these areas are not ~ccuratc . The area of this sha pl', for t.'.Xam pie, 72

b

,

,

ll±£x.b±.rl 2

2

d

th ey would l'al culate ~ s a formul~ give only an ~pproxim~te n.:.~ ult.

th~t

would

THE BIRTH OF MATHEMATICS

Egyptian and Bahylonian mathemati cs always relate to particular, practical situations; it was a later civilization , the Ancient Greeks, wh o first tOok an interest in purely abstract pro blems. The anCl'stors of the An ci ent Greeks beb'TI.n to enter the Grel'k peninsula from th e north around 2000BC and were a force to be rL'C koned with by around HOOBC. Th l'y ventured in tO Egypt and MeS<Jporamia, trading with and learning from th eir hostS.

MATHEMATICS IN VERSE

The earliest Indian texts to present mathematical problems are the Sulba sutras, Sanskri t texts which p resent problem~ and solutions relating to the construction and positioning of sacrificial altars. There were 5utras - collections of aphorisms ~ on many different topiCS. The aphorisms are written in verse with prose commentaries and expositions. The sutras were originally passed on orally, the verse serving as an aid to memory. The Sulba sutras are among the oldest Hindu texts, the earliest dating from perhaps BOOse

EARLY G EOM U RY

There is no mention of Greek mathem:uics before the sixth century Be , when the figures of first Thall'S and thcn suddenly Pyth agoras appear. Thales of Miletu~ appa rently brought Babylonian mathematics to Grecct~ around 575BC. He

has been called the 'first mathematician' on account of having evolved a theorem and then demonstrated it, though whether he actually did this is impos~ibl e to say. What we know of Thales comes from later summaries of his reputation and we can't nOw tell how much of his mythic stature is deserved. The theorem which bears his name (the Theorem of Thale~) States tllat

any angl e inscribed in a semicircle is :J. right angl e (90°) . This was known to the Babylonians around 1,000 years earlier, and Thales could have learned it III Mesopotamia. His demonstration or th e theorem, if it existed, has not survived. Writing around IOU years after Thal cs' death, Produs (cAI0-485BC) credits him with severa l fundamental geometric theorems: • a circle is bisected by any diameter • the base anglt:s of an isosceles triangle are equal • the opposite angles formed by nvo intl!rsecung lin es arc equal

THAlES OF MllETUS (c.624-546BC)

Thales was one of the Seven Sages of Ancient Greece. He may have studied in Egypt as a young man and was almost certainly exposed to Egyptian mathematics and astronomy. If he wrote any works, they have not ~urvived. One story rep orted by Aristotle tells how he was able to predict a good harvest of from observations weather patterns and bought up all the olive presses in Miletus to prove of how mathematics coul d ~

make him rich. Diogenes -;;;;:::;;:;; l aertius reported that Thales " was able t o calculate the height of the pyramids by measuring their shadows, and he is

said to have used his knowledge of geometry to determine the distance of a ship from the shore. He put his mathematical ability to military use, too. He is said to have predicted an eclipse which then led to a peaceful settlement in a war, and later to have helped King Croesus to get his army across a river by telling him to dig a diversion upstream to reduce the flow of the river until it was possible to ford it. Thales is credited also with a cosmological model of the Earth as a vast disk floating in wa ter. Ironically, reportedly died of watching a

73

THE SH APE O f THINGS

• tWO triangles are congruent (of equal shape and size) if twO angles and a side arc equal.

THE TETRACTYS For the Greeks, the number 10 was the most perfect number. They called it tetractys and revered it for being a triangular number, the sum of the digits 1 to 4, having as many primes as nonprimes before it and being, in the words of Philolaus (died c.390Bc), 'great, all-powerful and all-prooucing, the beginning and the guide of the divine as 01 the terrestrial life'.

vVhile Thalcs may be ca ll ed the first mathem atician, rhe title 'father of mathematics' is often given to Pythagoras, who lived fifty years later. He is perhaps the best knowll of Greek mathematicians. No one can have come through school mathematics without learning Pythagoras' famous theorem: that in a right-angled triangl e, the square on the hypotenuse is equal to the sum of the squares on the ()ther twO sides.

•

P)'lhfl.'fprtls'rheort:m fl 1~ 1/ + r~

It is likel~', though, that the theorcm was actually dL'Veloped later by members of the Pytha gorean schoo l rather than by Pythagoras himself. As with Thales, no wri tin gs by him survive and we arc depcndent on later rcports of his work and hjs reputation. (It is also possible the theorem was based on the bn_"'akthroughs of earli er marlH::maticians in Ebry}lt or India.) Th e PythagorL'flns were a secret brotherhood and held knowledge in common, so that individual attribution of work is now impossibl e. Th ey took deli ght in th e pattcn1.S :l.lld properties of numbers and sequences and believed that numbers were at the hcart of all thi.nb>S. The group continued tor many years after the death of PythagorJs.

TH E NIN E CHAPTERS The earliest Chinese mathematical text, The Nine Chapters on the Mathematical Art, was first produced in the 15t century Be. Many commentaries were written over the ensuing centuries, the best of which was by liu Hui in AD263. The text demonstrates Pythagoras' theorem (derived independently) and shows how to calculate such dislilnces as the height of a tower seen from a hill, the breadth of an estuary, the height of a pagoda and the depth of a ravine. It also deals with finding are
EARLY G EO MU RY

PYTH AGORAS (c.SBO-SOO8c)

Pythagoras

was

an

Ionian

(Greek)

mathematician and philosopher. After travelling in the Middle East he moved to sou thern Italy around 532nc to escape the tyrannical ruler of his homeland, Samos. He is best known for the theorem which bears his na me. A contemporary of Buddha, l ao Tze and Confudus, he established the Pythagorean Brotherhood at his academy in Croton. This was

a

religious

and

philosophical

movement that influenced Aristotle and Plato and made an important contribution to the development of western philosophy. Pythagoras and his followers believed that everything was related to mathematics and everything

could

be

predicted

and

measured in rhythmiC patterns or cycl es. The Pythagoreans were

v~etarians

as they

believed in the trammigration of souls, and

them. Pythagoras is said to have been

so any animal could house a former human

slaughtered by an angry mob when he

soul. They also, rather curiously, believed

refused to run through a bean field to

beans to be special and would not eat

escape their pursuit.

THE GOLDEN ACE OF CLASSICAL GREECE

the great math ematicians of the day. Eve n so, we can deduce enough to see that math emati cs waS pursued fo r its own sake, for a delight in knowledge and bt..'c ausc th e

Athens in the 5th century IlC, between the Persian and Pdoponn esian wars, saw one of th e gre:lt~ t flow erings of intell ectual life in the hi story 'All things which can be known have number; for it is not of the world. Sad ly, no mathemati cal tex t~ survi ve pmsible that without number anything can be either conceived or known .' from the perio d and WI: have only a few scrappy accounts Philolaus, 4th century Be ofth e problems addressed by

75

TH £ SHAP E O f THINGS

'God is like a skilful geometrician.;

Sir Thomas Browne, Religio Medid, c. 1636

made a distinction between the practical arithmetic of everyday life (of which nothing of theirs is recorded) and the higher pursuit of mathemat ics and logic, which has come down to us through the wTitings of those who bendited from their legaL),. THREE PRO BLE M S

Thr Allfiem Graff Wi'I'I: n<1Jilrt fhnt rbr &'11111

is II

spbi'n mouillg ill SPflf/! IIlId belit'ut'd rbllr lIlflfbt'lflftrics is tbt kq ro I/Jldffl"Stfilldilig rJJf IlIliu~/'YI'.

Greeks believed that the working~ of nature could be understood through mathematics. To them we owe the concept of the universe as harmonious cosmos governed by laws that discovered by reason th an govern ed by all unknowable deity), the idea thar the earth sphere and moves space, and the concept of mathematical proof. Th e Greek math ematicians

,

76

Greek mathematicians defined three great classical problems in gL'()metry: squaring the circle, trisecting the angle and doubl ing: the cuhe, all to be ach ieved only with compass and straight edge. Thcse problems were to tax mathematician~ for 2,200 years until they were all proved to be impossibl e. The issue of squaring thl;' circle first appears in relation to Ana..xagoras, a natural philosopher who ""TOtc the first scientifi c best-sell er, 011 Natllre. (A copy could be bought in Athens for OJle dracJmla.) Anaxagoras was imprisoned for denying that the ~l.I n wa~ a deity. saying: rather that it was a huge red-hot Stone, bigger than the whole of the Greek pcnin sul:I and that islands, and reflected li ght from the illuminated the Allilxllgonu

(c.500-42SIlC)

.iJ}{Lf

pllt ill jail for dmyillg

( A RLY G EOMHRY

III 'The Allrimr of Days', poer (/lid painrer rVillifllll Blake depicted God flf mrbirur of tbe III/WnW, prot/lid/If{ rbe Eartb witb f{eollletric illstnlll!ellis.

moon. While 111 prison, Anaxagoras oc(."Upied himselfhy trying to discover a way of 'squaring the circle' - b';ven any circle, creating a square of exactly equal area using only straight edbTt! and compass. The problem of doubling the cube emerged at the rime of the great plague in Athens (430nc). Eratosthenes repOrts that the people consulted the oracle at Delos, and the god Apollo demanded that to stOp the plague they must double the size of his altar. They duly doubled the dimensions of the altar, but that of course increased iD; volume by a factOr of eight (1 J), nOt two. Apollo was nOt satisfied, and the plague went on to kill around a quarter of the

population. The Greek craftsmen, who could not work out how to achieve what Apollo required, asked the philosopher Plato for ad\'ice. H is answer was that the intention of the oracle was to shame the Greeks for their neglect of mathematics and geometry in particular. ([here is another version of the origins of the problem in which .M.inos, king of Crete, commissions a tOmb for his infant son, Glaucus, who died by falling into a vat of honey. Minos (k'ddes the tomb that is proposed is too small and demands that iD; size be doubled.) The Indian Vedic scriptures state that a second plea at an altar in the same place as a first pit;:;) demanded a (."Ullie altar nvice the volume of the first, and this may have sugbrcsrcd the problem to the Greeks. The German mathematician Carl Friedrich Gauss stated that it was nOt possible to douhle the cube with only straight edge and compass, and this was proved hy Pierre \Vantzel in 1837.

Tbe plaf{lIe of Atbmr is illtnpreted SUlItr
77

m£

SHAP E O f THINGS

'In proceeding in fa mechanical] way, did not one lose irredeemably the best of geometry?'

Greeks for pure or thl.'Orerical methods led them to ('Ontinue th e quest rCb":lrdlcss of the lack of a practical need .

Plato GEOMETRY RULES THE UNIVERSE

As we have seen, Greek mathematicians Trisecting the ~ngle is ~ l e~ eng~ging problem and has no e.xciting mythical histOry attached to it; it is po~ib l e that it devdoped from the Egyptians' need to divide angles between SClrs in order to tell the time at night. The problem is simply to divide any angle into three equal parts using only straight edge and compass. It is possible to trisec[ some angles (a right-angle, for example) and there arc mechanical methods for trisecting: any angle which were known to the Greeh. However, the desire of the

ACHillES AND THE TORTOISE

To demonstrate the absurdity of dealing in whole units, however small, leno proposed a race between Achilles and a tortoise. The tortoise has a head start, but even though Achilles can run very quickly he can never overtake the tortoise. In the time it takes Achilles to cover half the distance from the starting block to the tortoise, the tortoise has moved on. When Achilles covers half remaining distance, the tortoise has moved on further, though

to come level

78

were rcluct~nt to recognize irr~ti ona l numbers (those which can't be expressed as a ratio of two whole numbe rs). Geometry cannOt explain all things if it is limited to who le numbers and ratios of whole numbers - which becomes apparent as soon as we look at the diagonal of a square with a side of one unit. This alone would bave been enough to bring the Pythagorean edifice crashing down. A second problem, expressed in the paradoxes of Zeno the Eleatic (c.450nc), made matters worse.

EARLY G EOMETRY

Zeno's parOldoxes show that, however much a unit of measurement is subdivided, it never expresses the continuum that we see in real life - even a sequence of infinitesimal steps is still artificial. Confronting these twO difficulti es - the existence of irrational numbers and division into infinitesimal portions - forced a paradigm shift in Greek mathematics. At the time of Pythagoras, numbers were thought of as points, often represented concretely by pebbles (ealled fll/CII/i, gi\'ing us the word 'calculation'). But by the time of Euclid, 200 years magnitude was later, represented by line segments - atOmism had given way to continuity and the model of the basis of the universe had shifted from the discrete numbers of mathematics to th e measurements of geometry. \¥hile J2 can't be represented as a number (in the Greeks' terms) it IS very ea~y to draw it as a line segment.

DEMOCRITUS AND THE INFINITESIMAL

The

chemist

and

philosopher

Democritus

(c.460-370BC) proposed that everything is made up of infinitely small and varied particles moving around in em pty space. The creation of our own and other worlds came about, he claimed, because the particles coagulated

in

different

config urations,

giving

materials wi th certain similarities and differences. (The idea had already been suggested by leucippus.) Extending this to geometric figures, a square pyramid, for example, can be seen as a stac:k of infinitely thin squares ranging from the largest at the ba se to the infinitely small at the apex. Because the layers are infinitely thin, each square is effecti vely the same size as its neighbou rs - but of course it can't be, for then the pyramid would be a cube. Breaking down an area or volume into infinitesimally thin slices is the underlying prindple of integral calculus, but

Democritus

could

progress towards this as a method

because

he

could not get past his logical objections to the

slices

being

different sizes. The method was used successfully

by

Antiphon, Eudoxus and

later

by

Archimedes, derived the 'method of exhaustion' to find the area of a shape (see page 146). Thl' odd cOllplt : DnlllX'ritlls, fhl' ItllIgbillg pbilorphn; piert/red alollgsidl' Hmldillls, fbI' {"Iyillg philompbl'1:

79

TIn SH APE O f THIN GS

BRINGING IT ALL TOGETHER

Practically no Greek mathematical textS are extant from before the 4th century, bur we have nOt lost the work of this period. Perhaps the most famous mathematician of all time, Euclid of Alexandria, gathered togcther and recorded the inheritance of ancient geometry, codifying and extending it in his Ele1J/('//ts around 300oc. By this time, the Greeks had discovered many of the standard curves (ellipse, parabola, hyperbola and so on), a fore-runner of integral calculus in the method of exhaustion, and methods for dercmlining the volume of a cone and a sphere. Though Plato was not a mathematician himself, his academy 111 Athens was the centre of the mathematical world and had helped to crystallize the distinction between pure maths and the practical application of numbers. Euclid's Elements nOt only demonstrates the mathematics of the Ancient Greeks bur also their d".."elopment of logical method. Euclid establi shed five axioms and five

'common notions' and deduced from these several hundred theorems or proofs, exemp lifying the principle of logithJI deduction which endured for centurit:S. Although Euclid's text is famous for its treatment of planar gcometry (the geometry of flat, two-dimensional shapes), it abo deals with number theory, algebra and solid geometry. rt was intended as a textbook of elementary mathematic's and does not deal with either simple arithmetic (which would have been beneath its intended readers) or the more complex geometry of curvilinear shapes and conics investigated later by Apollonius (which would go bL)'ond what was required). The five basic axioms from which Euclid develops everything else are: I. Any twO points can be joined by a single straight line. 2. Any finite straight lin e can be extended as a stra ight line. 3. A circle can be drawn through any centre and with any radius. Tile OxyrbJl/cbl1S papynlSef. fillds frulll toWII

fill

al/rimt

dllmp. conrailled tbe oldest ,md lIIOSt

[(1II1plete difl?!YD/1f frrllll Euclid's Elements.

80

EARLY G EOMETRY

4 . All ri ght an gles are equal to each oth er. 5. If twO str.:light lin es in a plane arc crossed by another s trai ght lin e (call ed th e transversal), and the illtt:ri or an gles bern'eell the two lines and the transversal lyin g on one side of th e tr.:lnsversal add up to le5S than twO right angles, then the t\\'o strai ght lin es can he extended until th ey eventually intersect on that side of the transversal.

EU CLID OF ALE XANDR IA (c.3 25- 26SBC)

Euclid wa s a Greek mathematician who lived in Alexandria, Egypt, almost certainly during the reign of ptolemy I (323-283BC). He is often considered to be the 'father of geometry'. His most popular work, Elements, is the most successful te xtbook in the history of mathematics and was used for over two thousand years. Euclid also wrote works on perspective, conic sections and spherical geometry. Euclid would have written on papyrus scrolls and, as these decay readily, his work has come down to u~ only in copies. The oldest surviving version of Euclid's Elements is in a Byzantine manuscript written in AD888 - making it closer in time to us than to Euclid! We can't be sure that we have Euclid's own text, rather than something improved upon or altered by later scholars.

The last of th ese is also call ed th e 'pa rall el postulate'. It is not as self-evident an d self-suffici ent as the first four. Plato demanded that axioms should be simple, self-evident and so cl early trut;' that they need nOt be proven . \Vhil e th e first four meet his conditions, th e fifth does not. Thi s wa s probab ly evident even m Eucli d's lifetime. However, it took wltil the t 9th century befure anyone coul d prove the final axiom. Ll addition, Eucli d stated five 'common notions' which are less sO'ictl y related to gL'Ollletry: I . Things whieh arc equal ro the same thing are al so equal to onc another. 2.1f equals are added to equals, th(' wholes are equal. 3. If cqual s are subtracted from equals, the remainders arc equal. 4. Things whi ch coincide with one an other are equal to one another. 5. The wh ole is grcater than the part.

Eucli d W:1S writing just a fter th e end of th e Hell eni c peri od, when both Al exander th e Great and AristOtl e had died. Th t! empire of Al exander was broken up, and Athens lost its supremalY as an intell ectual centre , the intc lli gen t.~ ia convenin g in stc:1d in AJ ex:1ndria in E gypt (Eucli d included). AJexandria was th e capital of Egypt and th en fell wlder Roman rul e when Cleopatra's army lost the battl e of Actium in 31Be. Th e first to benefit from Euclid 's work were th e Rom:1n s, bur math ematics W:1S not hi ghly rega rded by R oman scholars and was taught for its practi cal usefuln ess rather than anythin g else. So an architect would nced to understand geometry, calculation and loadbearin g and a merch311 t would need to understand 3rithmetic, hut no one was parti cularly con ce rn cd to extend the boundarics of kn owl edge in mathcmatics for its own sake. B1

THt SHAPE O f THIN GS

Bllilt .·1070- 80, tbt" COIOSft"lnil

ill Rnlllt" is all

t"Uipliml alllpbitbt"fitl"t"

allli a mllftnpiue of ROJ/lUll mrbirurlll"t"

alill t"IIgillUl7l1g.

The end of the Roman Empire in the \Vcst, when G ermanic tribes under the leadership of Odoa cer overran mueh of modern Imly, saw the end of math ematical

HYPATIA Of ALEXANDRIA (c.370-41 5)

Hypatia was the daughter of Theon of Alexandria, a notable mathematician and philosopher (it is from his yersion of Euclid's Elements that all surviving texts are derived). Hy patia was a Neopla tonist, and lectured on Neoplatonism and mathematics. She is the earliest

known

mathematician.

significant Sadly,

none

female of

her

mathematical works suryives, though her commentari es

on

the

work

of

other

mathematicians may be preserved in some of the annota tions that have come down to us. She was murdered by a Christian mob in 415, indted by the patriarch of Alexandria to wipe out

pagan

scholarship.

The

library

Al exandria was destroyed at the same time.

82

at

activity in Europe for a long time. lnstead, we muSt look to Lldia and then the Midd le East for (k-vclopmellts in geometry as 111 other areas of mathematical endeavour.

T RIGONOMHRV

Trigonometry Trigonometry is the branch of mathematics concerned with calculating angles, particularly in right-angled triangles. Until the 16th century, it was really a part of geometry, but since then it has come to be considered an independent area of mathematics. As any polygon can be reduced to a number of triangles, trigonometry enables mathematicians to work with all areas or surfaces that arc hounded by straight lines. Plane trigonometry deals with areas, angles and distanct!s in one plane. Spherical trigollometry deals with angles ami distances in 3D space.

The Eb'Yptians wt!rt! not rib,{lrOUS in thdr study of rriangles, though. As in other areas of mathematir.-s, they wt!re interested in practical applications rather than pure trigonometry. Early Indian mathematicians, too, knew something of trigonometry. The Sulba slltras, in the context of describing altars, contain a calculation of the sint! of rr/4 (45°) as Y.t2 . However, it was left to tht! Greeks to develop trigonometry properly. THE

360 ° TURN

The Greeks took tht! straight line and tht! eirelt! as tht! basis of their geometry and from this devel0Jled trigonometry. Tht!

TRIANGLES INTO PYRAMIDS

The Egyptians had some knowledge of trigonometry as their building of the pyramids demonstrates. The Ahmcs papyrus includt!S a problem that finds the seiad, or slope, of a pyramid from the height and tht! bast!. ft was expresst!d as the opposite ratio to our measurt! of gradient. Hi nill rbuw gnu/ient tIS {I Ttlrio of the

BELOW. Arty polygon mil be dividrd illto tritlllgkr,

ABOVE:

wbicb 1I1f1kes rb~ atfmlnrioll of tlH'tI wry if)'01l

verrirnl me {llid

liN

rnppl/01I/n7J.

b0/7~0Ilrrt!

distnl/ce, rbollgb we've

revened rbe ordo' gll[f tbe AI/{'i~nr ElOprillllf.

83

TH E SH APE O f T HIN GS

SfKfD ..

RUN .. 180 CU81TS ~ S ~ PALMS/ CUBITS RISE 250 CU81TS "

250 CUBITS

, ,,,

, ,,, ,, ,,, ,, ~- - - - . ---

• 180 CU8 1TS

250 CUBITS

convention of 360° in a circle and 60' in a degree ori!,rinates with Hellenic maths - it seems to have been in use by the time of Hipparchus of Bjthynia (c.190-110BC). It probably derives from the Babyl onian aStronomical division of the wdiac into twelve signs or 36 decans, and the annual seasonal lyde of approximately 360 days. Th e superior system used by the Babylonian s for representing fractions

The EgyprimlY colat/tired rbe 5cket! 01" rlope of tI

pymmid by ill1agillillg a righr-allgled fliallgh

iI/ride the rtl"IICfIll?

made it more usefu l than either the E!,ryptian or Greek ~ystems, and Ptol emy (C.AD90- 168) followed their base-60 system in di\'iding de!,"Tees into 60 minutes (partes lIIil/little primtle) and each minute into 60 seconds (Ptl11es mil/little sec/mdtle). SPHERICAL AND PLANAR TRIGONOMETRY

\,yhil e a planar triangle is on a flat plane, a spherical triangle is one inscribed on the surface of a sphere. It is made up of arcs of three intersecting circles drawn around the sphere, or planes cutting through the sphere. Th e first definition of a spherical triangle is found in a work by the Zodiac dock: ir ll'as tIn Baby/olliaw; l1:ho divided rhe zodiac iuro 12 siglls 01· 36 dfCalls - njlfCIillg the;'· rearollal cycle ofappl"oxill/tlrely 360 dayr.

TR IGONOMETRY

Tbe first nmlfiOIl of fI spberiml

t/it/llgh cOllles ill

/I
by rbe Egypriflllll'!lenelt/{j,f ofAlexlllllb-ia.

, ,,

spherical surfaee and the first to treat trih"Onometry as a discrete discipline. H e ,, developed spherica l ...... trigonometry to its ,, '." ,-." ". current form .

/

:

',/

/,

" "",

// Egyptian Menelaus .' / of Alexandria , / (c.AD 100). , ,/' THE RI SE OF THE H e developed TRIAN GLE \\" , - , / , / / the equivalents of Hipparchus was /''-'. Euclid's principles of the first to compile planar trigonometry ta bles of trigonometric but appliL-d to spherical functions. His interest triangles. Spherical triangles was in imaginary triangles are clearly essential in aStronomy 'drawn' on the imagincd sphere of the nights)..), that related heavenly bodies and mapping. \-Vhile the angles of a planar triangle to one another SO that he could calculate ahvay-; add up to 180°, those of a spherical and predict the positions of planets. triangle always add up to more than nmo. Hipparchus considered L'":lch triangle to be There are other fundamental differences, inscribed within a circle and developed a too. Until around 1250, and the work of system of calculating angles from chords. Nasir ai-Din al-Tusi (1201 - 74), spherical H e drew up rabies of the chords produced trigonometry was always integrated with by drawing angles of different sizes astronomy. AI-Tusi was rhe first to li st six which relate to the modern concept of sines distinct types of right-angled triangles on a and cosines.

..

TRIGONOMETRIC FUNCTIONS

There are six trigonometric func tions that enable us to calculate the size of an angle given two sides of a right-angled triangle, or the length of a side given one side and an angle.

,

•

,

,

sin A cos A

,, ,

side o~~osite hypotenuse side adjacent hypotenuse

b

,

tan A

Ii

side o~~osite side adjacent

csc «0_) A ""

,c

h;tpotenuse side opposite

'" A

cot A

c Ei

,b

==

hypotenuse side adjacent side adjacent side opposite

8S

THE SHAPE Of THINGS

In his aStronomical text the Almagest, the Greek astronomer Claudius Ptnlemy (ADlOO-170) extended Hipparchus' work, deriving better trigonometric tables and loosely defining the inverse trigonometric functions arcsine and arccosine. He llsed a nominal radius of 60 as the basis of his table of chords and gave values in steps of Iho from 0° to 180° accurate to 1/3600 of a unit. This is equivalent tn a table of sines for L'very 1/4" from 0" to 90° . Ptolemy worked with Euclid's axioms and concentrated on planar triangles in order to d("ve]op his model of the heavenly bodies revolving around the Earth. Ptolemy lived and worked in Alexandria. Details of his life bave not been preserved it's even possible that he may have been Greek in origin. His was th e l':lrlicst work on trigonometry to be circulated in Europe in the Nli ddle Ages and W;L~ used for many ccntu ri es. His model of the heavens sun'ived intact until th e work of the Poli sh

f1ippa/rhus ill his obsrrl!(trory ar AJ~x'lIIdria lookillg ar rb~ sran. H~ bas b~1I adired wirh rbe iln'mr;oll ofthf am'olabe (IS <1.'ell lIS (be 111711illlll] sph~Jr.

astronomer Mikolaj Kopernik (Copernicus, 1473-1543), which put the sun at the centre of the solar system. SINE OFTHE TIMES

After th e Greeks, lndian and Arah mathematicians worked on trigonometry. Arab scholars translated and mastered the work of their Greek predeec..~s()rs and soon went hL'Yond it. The Indian mathematicians were largely working in their own tradition, which had drawn independently on the Egyptian and Babylonian heritage. The geIJCf'IIrrir (EaNh-cmtnd) IlIIivflYf' (H39) sb(T'.J.'s Al'istork's fo ur dell/f'IIts sl/n'Ollllded by rbe pia/lets alld

86

we abode ofGot! in [t'r alia.

T RIGONOMHRV

TRIANGLE S AND WATER

One practical application of trigonometry was to calculate the gradient of water flow. The Sinhalese inhabitants of the city of Anuradhapura, Sri l anka, used trigonometry for this purpose. Theirs was one of the greatest Asian civilizations of the ancient world. To farm the dry land around and supply water to the huge city, the Sinhalese built a highly sophisticated irrigation system, which consisted of overground and underground channels, reservoirs and ponds.

The Hindu mathem:lticians were the first to work with sines as we now define them. Early in the fourth century, or perhaps late in the fifth, the unknown author of the Indian astronomical treatise Suryn SiddJJtrllttl, calculated the sine function for intervals of 3.75° from 3.75° to 90°. The date of the text is nOt known - the

III "wderll AIIIII'adb"pIITa, bom/plnllps b,W~ rep/ared rb~

CQTl/pkx il"11gllTiQII

ryrtffll

sta/1rd 1900 j

f'tl/T

"SQ,

sun~ving

version may date from around AD400 - hut it claims to have been passed down directly by the sun god m 2,163, J OJ Be! The Al)'tlbJmtiytl by Aryabbam I (c.475- 550), which summarizes H indu mathematics as they stood in the first half of the 6th century, includes a table of sines. Brahmagupt:l also published a table of sines for any angle in 618. The first table of MEASURING THE EARTH tanbTt!nL~ aJld cotangents was Eratosthenes (276--194sc) noticed that, while the sun is conso'ucred around 860 by overhead at noon at the summer solstice in Syene (now the Persian astronomer Ahmad ihn 'Abdallah Aswan), in Alexandria, 500 miles (800 km) northwest, it is at an angle of 7" at the same date and time. He assumed Hahash al-Hasib ali\tlarwazi. the sun's rays to be nearly parallel when they hit the Earth, The Syrian since the sun is so far away. Working with trigonometry astronomer, Ahu 'abd Allah Muhammad Ibn Jabir Ibn and the known distance between the two cities, he Sinan al-Batt:llli al-Harrani calculated the circumference of the Earth. The accuracy of as-Sabi' (c.85H- 919), gav!! a his calculation can't be assessed exactly because the length of his unit of measure, the stadia, is not certain. rule for finding the elevation of the sun ahove the horizon 87

THt SHAP E O f THINGS

CALCULATING SINES The sine func ti on is the ratio of the side opposite an angle in a right. angled triangle to the hypotenuse. To calculate the sine function, draw a circle of radius 1 and draw the

The Persian mathematician and aStronomcr Abu al-\Vafa 3l-Buzjani (940- 997/8) worked principally on trigonometry, bur most of his work h3s been lost. H e intrOduced the wngent function and improved methods of ca lcllbting trigonometry tables. H e discovered the sine formub for spherical geometry:

required triangle within it, like this. sin (A)

sin (6)

sin (C)

sin (0)

sin (b)

sin (c)

--=--=--

A cr3ter on the moon is named after him in honour of his extensive studies of the motions of the moon . Arab mathematicians con tinued to refine tables and trigonometric exclusively in the service of astronomy until al-TlIsi The distance OP, the hypotenuse, is the radius of the circle, 1. The y coordinate of point P gives the sine of angle a (:AP/l). The ci rcl e, called the unit circle (because it has radius 1) is, by convention, used to derive and relate all the trigonometric functions.

hy measuring a shadow (the principle 011 which SWldials work). His 'table of shadows' is effectively a tabl (' of cotangentS for 3ngles from 1° to 90 0 , n inrcrv31s of 1°. H e 3lso calcubtcd the tilt of the E3rth's 3xis, 23 ° 35'. h W3S through al-B3ttani's work th3t sines came to Europe 3nd he m3Y h3vc discovcred them independently of the work of Aryabhata. B8

AI·Barrani mlclllrmd tbe lilt of thl' Earth:r axis at

13 0 15'. HI'; also er/r1llated (be Imgtb of a soIaryefl/· IQ be ;65 days. 5 bOllrs, 46

lIlilllltl'S

and 14 sl'fOm/s.

T RIGONOMHRY

AI- n ISi} pllpil Qmb ai-Dill al-Shira':J

11'1IS

the /n,r

penoll to come lip with II sdmrifir nplalllltioJl of tbe rIIlllb(T<1J.

established trigonometry as a separate discipline in his observatory in A-laragheh in the 13th century. One early development was the mathematical explanation of the rainbow by al-Tusi's pupil Qutb aI-Din alShirazi (1236-1311). Ulugh Beg, the grandson of th e great Mongol conqueror Timur (famhcrlaine the Great), established an observatOry at Samarkand in the early 15th century ami created tables of sines and tanbTCnts for every minute of arc, accurate to five scxagesimal places. It was one of th e greatest achievements in mathematics up to this time.

surface, with circles mapped either to circles or straight lines. This had first heen used by Apollonius and Ptolemy. From the 9th century, the Arabs perfected the astrolabe, an astronomical instrument originally designed in Ancient Greece. It consists of a series of concentric metal rings etched with the positions of th e sun, moon , Stars and planets. Simpl y moving the rings replaced ream s of tedious calculation. The astrolabe could be used for astronomy, timekeeping, surveying, n:l\~gation and triangulation. The combined Greek and Arab knowledb'C of triangles came to Europe with

FINDIN G DIRECTIONS

One spur to Arab advances in gcomerry and surveying was the need to determin e the direction of Mecca (q ib/a) from any place, so that the devout Muslim could face the holy city for prayer as demanded by the Qu'ran. \Vith this need in mind, Arab geometers adopted the sten.'Oscopic projection, which produces a planar image of a spherical

T be 1l'qllire111mr for Mllslims to pray to NlecclI several rimer II dlly

71'IIS II

spm· to imprlJl!nnmrs

illfilidillgdirertiollS.

89

TIn SHAP E O f T HI NGS

the translation of many Arab texts intO Latin from the 11 th century. The Europeans rook to the astrolabe enthusiastically remained the na\'igation instrument until the development of the sextant 18th century. INTO THE MODERN WORLD

Although medil-,val European scholars translated Arab and Greek works on trigonometry and other breometry, they added nothing new of their own. It was not until the explosion III scientific and mathematical knowledge in Europe from the Renaissance that trigonomerry prob'l"cssed again. Johannes Miiller von Konigsberg (\436-76), also known as Regiomontanus, was the author of the first hook entirely devoted to trigonomerry, all Triallgles of Eve"-J Killd, printed in 1533. It brought tobocther all the formulae required to work with planar and spherical trigonomctl1' and was greatly admired and intluentiaL Hi s work was ll~ed and adapted by the great Polish astronomer Kopcmik (Copemicus) in his new model of a solar ~"ystem centred on the SU Il. Kopernik worked with the help of Prussian math ematician Georg Rh eticus (1514-76). In his own work, Rh eticus went fuITher than Regiomontanus, finall y making trigonometry about triangles. H e diSClrded the old tradition of considering trigonometric functions with respect to the arc of a circle, freeing the triangle to stand alon e. H e calculat ed detai led tablcs for all 90

Th~ s~xtllln

17IJollitiolliud

IlIIvigatioll. It tlllowed soilon to plot their {IOyitioll by trflrillg the

YIIII

S cOline

agaillst tbe borlum.

trigonometric functions, and embarked on a set calculated w ,n even greater degrce of accuracy but died before completing them. (They were finished by one of his pupils.) These developments cam e just before trigonometry, and geometry as a whole, took a new direction, becomin g involved

III rbe 16tb cmtlllY, J\1iko/aj Kopemil.: dnllollStmted tbllt tb~ Earrh circles rbe filii a/ollg with tbe orber pla l/ets. Tbis tl/I"I/ed pn·viol/s belieft 011 tbeil" bmd.

TR IGONOM ET RY

DEADLY TRI ANGLES

Galileo Galilei (1564-1 642) discovered thllt the movement of a projectile is parabolic

Vo sin 2A

9 where 9 is the acceleration due to gravity

and could be separated into vertical and

(about 9.81 metres/second')

horizontal movement. This led to the

Vo is the muzzle velocity (velocity at which the

formula for calculating the range of a

cannonbllil leaves the cannon, or bullet leaves

cannon

the gun)

or

other

artillery

weapon,

A is the angle of elevation

disregllrding air resistance:

The mllximum range is achieved when A "" 45°. ~',----------------,c-----------------,

// , ,,/

]So

/'

( 160

mit )'

~~ ~:.t

~ m"m".~

~,

'" '"

~

1100 m

~.

/

Tlx IlIqllisirian jim:t'd Cali/eo

ro

I"fCIJm

his bdi~fthtlf fJx

Eill1h 1ll00l'd INTJlllld

fIx SIIII.

" with algebra and the slow evolution of algebraic geometry (sec Chapter 5). WIth this hmdamental shift, trigonometry became more theoretical, separated from the. realworld shapes with which it was oribrinally concerned, and later even embroiled with imaginary and complex Ilumber.;. At the same time, though, the practical applications of trigonometry were growing. The invention of accurate clocks, better navigation methods and artillery, as well as new applications for optics and advanc,-'S in astronomy all demanded the application of trigonometry and aided it.~ development in new directions.

MOVING ON

Triangles and circles are inextricably linked in the histOiY of mathematics, and with Galileo's work on projectiles another curve, the parabola, becomes involved. Circles, curve; and the effects of revoh~ng shapes in space to produ{'e solids of revolutioll lift g,-'Ometry away from straight lines on the flat page. and move it into space. \Vith the circles and curves, roo, we begin to take steps towards contemplating infinity - that great bugbear of early mathematicians which evcntually would free b'Cometry even from three dimensions to cavort through as many dimt'mions as we care to imagine.

"

L-:"-:

CHAPTER 4

In the ROUND

The world around us has provided the impetus for much of the development of mathematics. That the Earth itself is a sphere , and the sky looks like an inverted bowl above us, has put cunres, circles and spheres at the heart of geometry from early times. These features of the world have led to challenging problems with explaining, depicting and modelling the universe as we experience it. How can we represent the three-dimensional environment we see in a flat drawing? How can we map the spherical Earth on a two-dimensional chart? Grappling with these issues led to a theoretical investigation that threw up further questions about dimension and geometry. Sometimes, the world does not seem to conform to the geometry laid out by Euclid which was accepted for 2,000 years. New models for dealing with these situations have opened exciting and fruitful new avenues for mathematicians.

III tbe relll world it seems ns ifpllmlle/liller came ta!!,ffber.

I N TH£ ROUND

Curves, circles and conics The circle-lies behind all trigonometry, as it defines a full n..'Volution about a point. The triangle and the circle togcther fomH~d the basis of astronomical geometry, with astronomical problems being explored by drawing imaginary triangles on the circular dome of the sky. Galileo's model of projectiles brings another curve ro trigonometry and introduces a link between trigonometric functions and conics - curves which can he derived from slicing through a solid cone. III fact, triangles and t:urves have been inseparable since the earliest geolllt::tries. The trigonometric tables were all defined initially from triangles drawn within circles, using diameters and chords. Angles arc mea~"Ured with reference to the full r(.-volution defined by a circle, which was di\'ided at least by the time of Hipparchus into 360°.

1t From the earliest times, the circle has heen endowed with reli gious and mystical sib'llificance. It is the perfect shape, having no sides (or infinit e sides), the endless line, found everywhere in nature. People have known for thousands of years that the ratio between the diameter and circumference of a circle is always the. same. and have given thi s number sl)ecial significance. \Ve represent the ratio by the G reek letter 11 (pi), notation popularized by the Swiss mathematician Leonhard Euler (1707-83) in 1737, bur first used by William Joncs in 1706. Pi is an irrational number; it has an infinite number of decimal places (sec table on page opposite).

THE MAGIC RATIO -

Sir iratI{" Newton. Oll~ of tb~ ,",ost QlIlStdllliillg 'II/{lthc'lfIaticillllf offill time. calmlaud Jf to J 6 plilrer.

CALCULATING

1t

Th e Babylonians used an approximate value of the ratio we now call 11, 3.1 25, which they ohtained by calculating or measuring the perimeter of a hexagon drawn inside a circle.

o The Ahmes papyrus Egyptians used a value of

shows !)I'hl

that

or about

3.16049. The Chinese text Thr Nine Chaptm gives instmctions for finding the area of a circle by squaring the diameter, di\~ding by 4 and multiplying by 3, SO using a value of 3 for Jr. Archimedes dcvelopt'd more sophisticated method that involved drawing polygons both within and Hound a circle, giving upper and lower limits for a value of

CURVES, CIR CUS AND CONICS

By adding more sides to the poIYb'tlll, he could obtain increasingly precise limiting values . He settled on 96 sides, which gives a value for IT hetwecn mil! and 1.217, or an average value. of about 3.1418. lt was Archimedes, too, who discovered that the same ratio can he used to calculatc the afca of a circle when multiplied by the square of the radius (lTr). IT.

0 0 0 Chinese, Indian and Arab mathcmaticians caleulated j[ to b'Teater degrees of precision but had no better method than tha t of Afchimedcs. For example, in AD263 Liu Hui used a polygon with 3,072 side.~ and obtained a value of 3.1416. At the end of the 17th ccntury, hetter methods of calculation were developed. The English mathematician Sir Isaa c Newton WHO

WHERE

Ahmes

Egypt

Archimedes

WHEN

(1642- 171 i) USed the binomial theorem to calculate j[ to 16 places. Today, j[ has heen calculated by computer to more than 10" places and on personal computers there are ptt;,nry of programs for calculating It to a billion or more places. This degrec of accuracy is completely unnecessary for most practical purposes. If the Earth's circumference is calculated from irs radius using a value of]( accura te to only 10 decimal places the result is accurate to arOUlld a fifth of a mi.llimetre. SQUARING THE CI RCLE

Th e problem of squaring the circle, although made famous by Anaxagoras. troubled earlier mathematicians, tOo. Ahmes givcs a method for constructing a square of almost the same areu as a circle, which consists of taking away QIlC ninth of the diameter of the circle and using the remainder us me side of the square, tbough this is showll as a way of calculating the arca of a circle, nOt solving the. classical problem. WHAT

(3. 16049)

c. 16508c

!!%

Greece

c.25011C

~l';,l < n <

3.1622 (Jl0)

1111 (3.1418)

Chang Hong

China

AD130

Ptolemy

Greece

c. 150

3. 1416

liu Hui

China

263

l.911/l.l 10

Zu Chongzhi

China

480

JJS/ II ) (3.141659292)

Aryabhata

India

499

t,l,~J ~ O.wo (3.1416)

(3 .1416)

al-Khwarizmi

Iran

c.800

3. 1416

Fibonacci

Italy

1220

3.141818

fran

c. 1430

3. 14159265358979

France

1593

3.1415926536

Adriaan van Roomen Belgium

1593

3. 141592653589793

Ludolph van Ceulen

1596

3.1415926535897932384626433832795029

al-Kashi F ran~ois

Viete

Germany

I N THE ROUND

(It is from this that we dedun! th e Egyptian v:llue for rr of 3.16049.) We h:lve already seen that the Greeks tried and failed to solve the problem geometrically. La ter mathem3ticians also tried and 311 f3iled. Squaring the circle with compass and ~traight edge bl'{:ame ~uch a preoccupation of both professional 3nd 311Ulteur m:lthenmticians in 18th-century Europe th3t in L775 the AC3dcmie des SciencC$ in P3ris p3ssed a resolution s3ying that it would nOt look at any more proposed solutions. Soon aftenvards, the ROY31 Society in London did the same as they were inwldated with faulty solutions . Some mathematici3ns even tried to fudge the iS~l.Ie by assigning a different value to rr. "Vhen Carl Louis Ferdin3nd \'on Lindemann (IS52- 1939) proved in 1880 th3t n is 3 tr3nscendental number (i.e., nOt the foot of any :llg't!braic equation with rational coefficients), thi.~ demonstrated finally that

squ:lring the circle is, in fuct, impossible it is quite impossible to work with :l tr.mscendental number using straigh t edge and comp3SS. C O NI C SECTI O NS

A circle is not the on ly curve. \-VhiIe the circle 3nd circubr 3rcs wen' the first curves to be studied 3nd uscd, there 3re three other n:!guI3r curves which C3me early to the attention of geometers. These 3re the par3bola, hyperbob and ellipse. Each can he formed by cutting throubrh a cone. These are called conic sections. The first influenti3l work on conic sections was hy Apollonius of Perg3 (c.262-190BC), an Alexandrine-Greek geometer and astronomer known as 'the Gre3t Geometer'. Although Apollonius wrote other works, only his treatise on conics has survived. The first" sections dr3w on previnm writings, but the later parts are

8101'1' .--J.pcIlOlliIlS,

diJjmllf-sb(lpffl COllI'S WI'1"f' Ilsi"d to dn;''t! <'fIcb CIRCLE

ELLIPSE

rype ofwrvl'. ApoDol/illr rbowed 1111 rollld bi" dun't'd fi"(JIll tbi" rlllllr

- - - PARABOLA

conI'

by

ITtljlLftillg rbi"

(Illgil' oftbi" pllllli" rlirillg tb1VlIgb tbt: roll~.

HYPERBOLA

96

CURVES, CIR CUSANO CONICS

BECOMING USEFUL

Apollonius was proud that his theoretical work was of value for its own sake - 'They are worthy of acceptance for the sake of the demonstrations themselves' - and much of his work had little practical application in his day, but has since found uses in many areas of science. His work on the hyperbola produces a result equivalent to Boyle's law that defines the action of gases and his

completely original. Apollonius' work completely replaced all work on conics that had come before as surely as Euclid's work had replaced all pre\;ous Greek breometries. Apollonius describes the derivation and definition of the curves he names and considen; the shortest and longcst straight

study of the tangents of an ellipse (though he did not know the term) is fundamental to understanding the movement of the planets and stars as well as in planning space travel. 1\101"C rball "'-.;JO rbol/.flllld )'MI"S afrer Apolumills liued, fpau tmud bas brrome n pmcriml flpplimrioll of bis work 011 CIlrvt'S.

lines that can be drawn from a given point or pointS on the curve. rn this, he lays all the groundwork for the definition of curves by quadratic equations in the Cartesian coordinate system. Indeed, 1,800 years later, Rene Descartes tested his analytic geometry against a genera lization of 97

I N T H ( ROUNO

Apollonius' theorem relating ro a moving point and its relation ro fixed lines. Both the Arab and Renaissance mathematicians were heavily indebted ro Apollonius. Though several Arab mathematicians srudied conics, finding ways to calculate the art;:as and volumes of figures derived from conic sections, it was left for Omar Khayyam to take their study in a new direction. In using conics in his general proof of c..'Uhic equations, he anticipated Descartes ro some degree, bringing geometry ro hear on algehra (though he expre~~ed a hope that his successors would he ahle ro find algebraic solutions ro finding roots). The rediscovery of Apollonius' work in the European Renaissance provided the groundwork for many of the advances in optics, astronomy, cartography and other practical sciences. 98

The fablllolis illterior of the Hagill Sophifl ill [mmblll, forme/iy COllrrlllltillOple: tbe flltllr is lit by fill/light at 1111 hom, of tbe dilY.

BEGINNING WITH OPTICS

I n one of Apollonius' lost works he apparently discussed parabolic mirrors and demonstrated that light reflected off the inside of a sphere is not reflected ro the centre of the sphere. Optics was ro become a major area for the application and development of work on (.'Urves. lr could have most starding practical applications, too. In {.lOOBe, Diodes demonstrated geometrically that rays of light that are parallel ro the axis of a paraboloid of revolution (a solid produced by rotating a parabola) meet at the focus of the paraboloid. Archimedes is reported ro have used this ro set light ro enemy ships from

CURV(~,

the shore. The focal properties of the ellipse were used by the architl'Cts of the Hagia Sophia Cathedral in Constantinople (537) to make sure that the altar was illuminated by sun li ght at any hour of the day. Several Aral) scientist.~ inwstign ted tht: properties of mirrors mad(' from conic SL'CriOns. Ibn alHaytham found the point on a convex spherical mirror at which an observer would be ablc to see an object at a gi\'en position and showed how to design the cu rves needed for sundials. The same properties can be applied to sound - the galleri es in borh the US Capirol and in St Paul's Cathedral in London are eonstrueted so that a whisper uttered at one focus of the ellipsoid gallery can be heard at the other focus, but nowh~re clse. Even more recently, satellite dishes and solar coll cction dishes have used the reflective propertie.~ of;J parabolic surface to focus the rays that strike them on to a central receiver or coll ector. In surgery the same geometry

CIR CU S AND CONICS

is exploited to fOclis ultrasound waves on organs or stones within the body. Galileo's work on projectiles and Keplcr\ on planetary motion were among the t'a rli (.'St applications of conics to suhjecrs other than optics. Kepler found that the Earth moves around the sun in an dliprical orbit, with tht sun at one focus of the ellipse. Later work on conics and curves used infinitesimal analysis to try to detcmline the area under curves or their length, but it was the invention of analytic geomet ry by Descartes and Fermat in the 17th ct::nlliry that paved the way for the modern definition of conics. In stead of dt::riving conic secti ons by cutting through a cone, the mathematicians of th e t 7th century and later defined them with algebraic equations. as the patb traced out 0 11 a plane by pointS moving according to a second -degree equation in twO variabl es. At this point in our Story; coniL"i dis3ppcar from geometry an d re-emerge in algebra.

THE PERFECT PENDULUM The Dutch scientist and mathematician Christiaan Huygens (1629-95) developed the pendulum clOck after discovering a new curve, the cyclOid. He discovered that a pendulum released from any height within a cyclOid bowl will reach the bottom in

preci sely the ~ame time - it doe~ not matter how fa r it has to travel. Huygens went on to demonstrate properties of other curves. He use d methods from analytiC geometry and infinitesimal analysis to discover the lengths of curved lines and to become the first person to discover the surface area of part of a solid of revolution called a paraboloid, rormed by rotating a parabola.

99

I N T H E ROUNO

100

Solid geometry

BASIC SHAPES

Solid b'"Cometry - the b'eometry of solid, three-dimensional objt.-'Cts - was needed as soon as humankind began building anything more complex thaJl a simple hut (when trial and error rather than mathematics may have ~l1fficed). One of the three problems facing classical mathematicians, the doublin g of the cube, is a problem of solid geometry. Problems in solid geometry relate to measuring the dimensions or volume of a three-dimensional shape. The volume measured need not he of a solid; it is likely that early uses for solid b'eometry related to measuring capacities as well as cak-ulating the dimensions for buildings. Some of the problems in the Babylonian and Eb'Ylltian texts concerned cak-ulating the volume of cellars and pyramids.

Plato identified five rebrular polyhedral solids; he associated these with the basic clements which he helieved made up the phy.:;ical world. These PlatOnic solids are the pyramid (tetrahedron), cube (he.xahedron), octahedron, doc\(.-cahedron an d icosahedron. PlatO claimed that earth was made of cubic particles, fire of pyramids, air of octahedrons and water of icosahedrons. He claimed, the god used [the dodecahedron] for arranging the constellations on the whole heaven.' Ll his Eif'JJ1mts, Euclid gives a thorough account of the PlatOnic solids and repeats a proof briven by Plato that there are no more than five regular solids. B1are rfdl"fillg work all

d

pyl"tJlIlid. EJ!J'ptiall

bllildny bad to m!cttlate ifr volllllle ill ord.r to nrqllir£ fbe rigbr tmlOllllf of stolle.

SO UD GEOMETRY

TETRAHEDRON

HEXAHEDRON DR CU8E

OCTAHEDRON

The German astronomer Johannes Kepler (1571-1630) tricd to associate the Platonic solids with the known planets and formed a model of the solar ~"'Ystem in which the solids were nested within one another. Although he had to give up the modd, he did, in thl! process of working on it, discover twO regular stella ted polyhedra in 1619. These are formed by extending the edges, or faces, of polyhedra until they meet, forming new shapes. Louis Poinsot discovered a further two in 1809. In 1812, Augustin Cauchy proved that there were no more regular star polyhedra.

DODECAHEDRON

ICOSAHEDRON

Pilltollic solidr: rbese lUI"C rbe rusie ele1l!mrr rbat Pinto believed mllde liP rbe pbysim/7.J.1()I-id. fir ji/l"rber believed Cfmb WtlS millie of l7Ibic parricles,jire of pyra mitis, ail, ofoctllbedlVlls lind water ofirofilbedlVlls.

Although Plato is credited with first describing the Platonic solids, they are all represented on carved stone balls 4,000 years old found in Scotland. At least one of Kepler's polyhedra was known before he wrOte about it, too. A stellated polygon is depicted on the marble floor of the Basilica of San l\;larco in Venice, Italy, which dates from the 15th century.

Stdlilud rrgll",r IIIld irreglilm' polybedra ml' {"/"f:ated by eumding rbe flUes of a polybrdroll IIl1tiltbry illli'l"Sl'ct. Some polybedm prodll{/' 1111111y srellariol/f. otbers very fr.v. Tbere lin '/10 srd"uiolls for 1/ mix. 101

I N T H ( ROUND

The ftI'olllldplrm for tbe Bari/ica of S(III J\1(1rco. Vel/ice:

(I

rtell(lted

polygoll if depimd 011 rbe lIH1rVle floor dnrillgfiWI! tbe lStb eel/wry.

MEASURING VOLUME

Just as a two-dimensional polygon can be reduced to a series of triangles, so a threedimensional polyhedron can often be reduced to rebrular solids for the purposes of calculating volume. Methods for calculating the volume of a cube, square or triangular pyramid, cylinder and cone were known to the Ancient Egyptians. But the volume of sha pes that call1lOt he reduced to any of these is harder to calculate. Archimedes IS credited with realizing that the volume of an irregular shape can be found by measuring the volume of water it displaces, a discovery that reportedly led him to leap naked from his hath and run down the street shouting, 'Eureka!'

THE GOLDEN CROW N

The Roman write r Vitruvius (died c.25BC) told a story in which King Hieron commissioned a solid gold crown and asked Archimedes to determine whether the crown the jeweller made was really solid gold. Clearly, Archimedes could not damage the crown to test it. He realized while in the bath that he could measure the water dis placed by immersing the crown, then weigh the crown and calculate its density. By comparing this with the known density of gold he could work ou t whether or not it had been adulterated with a cheaper, less dense metal.

102

SO U O GEOMtTRY

IYhile 1I1OSt fa'II/OtlSfor his 'Eureka!' 1!!(]I/lmt ill the batb, Arrhill/eder afro explailled the prillcipkr of tbe lever, the dn;iCl' UPO" whirb 1I1ecballics is based.

A sphere is a special case of a rehrular solid as it has no angles, edges or faces. Archimedes proved that the volume and surface area of a sphere are nvo-thirds that of a cylinder the same height and diameter. The earliest demonstration of the volume of a sphere, J.1 nr J, was by the Chinese mathematician Zu Chongzhi (429- 500). VOLUME S AND CALCULUS

After establishing mathematical methods for discovering the volumes of regular polyhedra, or solids that can be broken intn reb'l.llar polyhedra, and :l practical method

for measuring the volume of irregular solids (by immersion), there was little left to calculate hut the volume of irrebrular solids and solid conics. These problems were not solved until the invention of calculus at the end of the 17th century.

211 Chollgzhi 'iJ'as tbe [lI"st to 1I1famre tbe vo/mne of a sphen, bur aim r?"eated a

/lew

{II/nldar syrre1l1, af

[(]Il11!lf1lHffated ill tbis statile ill Shallgbai. 103

IN THE ROUNO

Seeing the world For all the theoretical purity of the Greek mathematicians, maths comes from and impacn; on our relations with the real world. The developments that had begun with an interest in rarefied logic, distanced from real-world applications, led in Renaissance Europe to a rich cross-fertilization hetween the arn;, sciences and mathematics that resulted in new wa~ of seeing the world. These, III turn, led back intO new mathematical ideas. The way we see the world - indeed, the universe - around us interested and inspired geometers for centuries. Not only the mL'ChaniL"s of how we sec and how light hehaves, hut the diftk-ultics of representing and modelling what we sec has hoth henefited from and prompted d(.'Velopmenn;

in geometry. Perspective geometry is the study of the relationship hetween fibrures and how they are mapped or represented, and hegTIn with the study of shadows cast hy ohjects and the way items in the distance a ppear to the eye. PUTTING THINGS IN PERSPECTIVE

The Arab scientist and mathematician Abu Ali aI-H asan ihn al-Haytham (c.965-\040) worked with gL'(lmetry to formulate his ideas on optics. He developed some of Euclid's work, redefining parallel lines, and used conic sections to help III his exploration of the reflection and retraction of lighL He arrived at the accurate model of light rays emanating from an ohject rather than heing sent out by the observer's eye (which was the model adopted by some scientists). H e descrihed a pyramid of rays coming from the ohject, some of which reach the eye of the observer. H e went on to determine the point of reflection from a plan e or curved surface, using conic sections. The work of al-Haytham came to the West through Latin translation and prompted olle of the greatest revolutionary events in the histOry of art - the discovery of linear perspective in the ltalian Renaissance. It was the Florentine architect and engineer The Dead Christ byAlldn:a Manregllil (1431- 1506) is il rIIperbearJy Filippo Brunelleschi eXffnlple of rhe applicatioll of rbr prillciples ofperspective ill llIestem art. (1377-1446) who first

10<

SU ING T H( WOR LD

rediscovered the architecrural principle of linear perspective that had been known to the Greeks and Romans. Brundlescbi demonstrated the principle of perspective in (vm illustrative panels that have been lost, but in 1435 his work was ill corpora ted in Del/a pittlll"tl (011 Paintillg) by Leon Battista Alberti (1404-72). Alberti suggests that a painting is like a projection of an image on to a vertical plane L"Utting through the pyramid of rays of light at some point between the object (the apex of the pyramid) and the observer's eye. The painting includes a 'point at infiniry' (now called a vanishing point) at which parallel lines in the painting converb'C· MAPPIN G THE WORLD

Recording larger-scale images of the world required a different ryPt! of application of geometry. Surveyors used trigonometry for the new method of trianb'1.llation that made accurate maps possible for the first time. Triangulation was first suggested in Europe by the Flemish mathematician Gemma Frisius (1508-55) in 1533, though crude versions

Tbe dome of tbe Catbedml of StlIltfi !lldria del Pian ill Plorwce (1420- 36) if rbe pillllfu/e of Bnlllellercbi'r m:bievenlfllt; it is brld lip by p,YSYI/JY fI"(Jm rlx weight of tinlbfl"Y.

105

I N THE ROUN O

J\1elllbnT uf rbe 187-# grogmpbim/ S/Irut')' c(mdlliT rrial/gli/ariull wurk fir rbe tup ufSllltall l\1ulllltaill

ill Sail Jail CUIIllIy, Culurado. USA.

of triangulation were used in Ancient Egypt and Greece, and H eron of Alexandria described a primi tive theodolite in the 1st cenrury AD. From each end of a hase line, angJt..-s of sight to a distant object are mea~l..Ired using a theodolite. Trigonometric methods then give the distance to the ohject. By covering a b'eob'Taphic ~l..Irface with mea~"Ured and calculated triangles, the entire area can be mapped. Tht;' first large-scale mapping A Rmllissallce IrQ/"1d map bawd 011 rbe w,itlllgr uf Prolmry} Geography. Tbe d17 ufc1II1ugrapby develuped dralllatically during tbe age uf discuverier.

106

SEE I NG TH( WO RlO

PTOLEMY AND THE AMERICAS

Though Ptolemy's most famous work was the Almagest he also wrote a Geography which remained influential for over a thousand years. He developed two projections and introduced lines of latitude and longitude, though the inaccuracy of measurements led to considerable errors in his longitudes. He also overestimated the extent of the Earth's surface covered by the Hellenic lands and consequently his calculated size of the Earth was smaller than the real thi ng. The earliest surviving Eu ropean maps from the Middle Ages are heavily reliant on Ptolemy'S Geography. When explorers planned to sail to India by heading west they would have expected the journey to be much shorter than it actually was. Perha ps if Columbus had realized the true natu re of the undertaking he would not have attempted the voyage that led him to the Americas.

proj(."Ct was carried out by \Villehrord van Roijen Snell (1581 - 1616), who survL'Yed a stretch of 130 kill (80 mi les) in Holland with 33 triangles. The French government decided to survey the whole of France, which rook more than a hundred years to complete. The British surveyed all of India hetween 1800 and 1911, discovering Mount Everest in the process. From the mid-15th century onwards, explorers werc discovering and charting new lands, bcginning with the Portuguese

A nJlllflllriciud m1isr's impITssi(}/1 ofCo/lI'Illblls laudillg ill AlIlenm ill 1492. /-Ie l!!Im have bfl'lI reiirJt:d to srrikr laud aftn- a IOllge/~ thall-e:xpected jOilmry.

exploring the African coast. \Vhilc surveying dea ls in straight lines, the cartographers who were aiming to record the laq,rc e.\':]lanses of newly discovered lands needed a way of representing in twO dimensions terrains which arc actually draped over the surface of a sphere. The method PtOlemy had used in his Geogmpby (rediscovered in Renaissance Europe) did not work for the enlarged world. I nstead, cartographers adopted t he stereographic projection that astronomers used to portray 107

IN THE ROUNO

,

_ _ _ C.. ntral meridian

G ~at

dl ' tol1lon In high latitude,

hample, of rhumb lin .. . (dl""'tlon tru .. l>etw .... n any two points)

~'

i ==t=

Equator touche, cyWnder 11 qllnMr I. tang .. nt R... sona bly tru .. "'ope, ond dl'tan<.. wi thin 1S' 0 1 Equ .to r

the sky. But, of course, that depicts the interior of a hemispherc and the cartographers needed to represent the exterior of a sphere. (A stereographic projt!{~tion projects a sphere on to a flat plan!;! - a circle - from a projection point which is then not visibl!;! on the map. Areas nL':Jr to the projection point ar!;! distOrted). The mOst successful variant developed was the Mercator projection, made first by the Flemish map maker G!;!rardus MercatOr (1512- 94). H e drL''W the Earth as though projccred on to a cylind!;!r tangential with th !;! equator. Parallels and meridims arc drawn as straight lines spaced to produc!;! an accurat!;! ratio of latitude to longitude at any point on the map . \¥hen the cylinder is

A map oflb~ wm1d {lSillg tb~ Mt"I'Cfltm' prtjectioll, rbuwillg bull' il if dn'iwdfi'om tb~ plVj~Clioll of Ib~ globe 011 W a cylilldn:

unrolled, th!;! flat map is revealed. Although the projection was useful for navigation, it distorts areas particularly nl;!ar the poles. A .MercatOr projt-,crion of the Earth shows Greenland as approxima rely thl' sam!;! size as Africa, for !;!xample, whereas in fact the ar!;!a of Africa is around 14 times that of Greenland. AND BACK TO MATH5...

The int!;!nSt! discussion of perspL'Ctivc and projections fed back intO mathematics, stimulating discussion of the properties of

PROJECTIVE GEOMETRY

Projective geometry formalizes the principle central to linear perspective in art of showing parallel lines meeting at a point which represents infinity. It is a non-Euclidean geometry in that it rejects the fifth axiom (the parallel postulate). Desargues extended the convenient trick of perspective drawing, taking it off the artist's page and formulating a non-Euclidean space in which parallel lines actually do meet at infinity. He used this projection to study geometric figures, including conics.

108

SEEING T H( WOR LO

perspective in general. The most significant Outcome was in the work of Girard Desarb'lles (1591-1661) which eventually led to the development of a more rigorous projL'Ctive gcomerry in the 19th century. Desargues was a Frcnch mathematician, architt..,o and artist, a friend of both Rene Descartes and Pierre de Fermat (1601-65), the leading mathematicians of his day. He developed a geometric method for consrructing perspective imabTCs of objects Gn"IJrdlls lvll'1'mror holdi IIg hir !!,Iobe.

and ,wOte a highly theoretical text explaining the geometry of constructing perspectives in 1636. The ellb'Taver Abraham Bosse restated Desargues' work in 1648 in a more accessible form, presenting what is now known as Dcsargues' theorem. This St:ltes that if twO triangles are siruated in three-dimensional space so that they can be seen in perspective from one poinr, then corresponding sides of the triangles can be extended so that they intersect, with the points of inrerSL'Ction all lying on a line. This works as long as no two corresponding sides are parallel. A modification of the thL"()rem rakes account of this. Dcsargues' work was popular for around 50 years, and was read by Pascal (sec pabTC 43) and Leibniz, but was then larbTCly ignored until it was rediscovered and published again in 1864. Both Desargues and Pascal srudied the properties of figures that were preserved and those that were distOrted by different methods of projecrion. For example, a flat map of a spherical world cannOt accurately represent hoth distance and shapes.

"

'

-

....

Fmm rhe viewy,!!, pDim, V, rhe tl"Itlllgles lilY III P<""Sf!ective.

If rorrerp(Jlldil'!!, ed!!,t!S of rhe rrillng/es alT

n.1mdrd tllltil rhey 1II«t (Be IIIld B'C, ete), rhe poillts oiillten«tioll, PI, 1'2 alld 1'3, lie ill a m llighr line. 109

I N THE ROUND

1,1 fldilitiQII !{ff)7llel1J~

ro bi< <J:Q1-k (}II pnyntivi'

PQwe/er u- Ti"f{flHktlllf rhi'm(JS(

illjIlIClIlifl/ wgillt-'fI"

i/I

birrory. pnming

rhl'
Euclidea.n 6'Cometry provides liS with the tools we need for working with the geometry of planes - but perfectly flat planes exist in only small or ideal environments. We live on a spherical Earth, in a universe with at least three physical dimensions. In representin g the curved surface of the Earth, or of the sky as it appears to us, on a flat piece of paper we are necessarily distOrting it. Some of these problems are addressed by projective geometry. H owever, as we move away from the perfect, regular curvature of a sphere, more problems of geomerry relatin g to curved surfaces emer6'1:: . Though the ancients were aware of difficulties in marrying Euclid's geometry to curved sur&ces, it was not until the I Yth century that mathematicians developed new models to address them.

Th e principles of projective geometry were rediscovered by J ean-Victor Poncclct (1788-1867) in the early 19th cenrury. Poncc1ct had heen left for dead at Krasnoy, Russia, in 1812 after fighting111 Napoleon's Russian campaign . He was then ('aptured and imprisoned at Saratov, and worked on problems of perspective and conies while in prison. Hi s solution to the need for a modification to Desargues' theotCm in the case of parallel sidt;!S was ro change the narnre of Euclidean space. POllcclet postulated poinL~ at infinity, each lin e having a point at infinity and parallel lines ha\'ing a point in common at infinity. This bet-ame the basis of the new projective geometry. Poncelet ignored geometric measurements of dist3nees and SPHERICA L G EOM ETRY angles in order to find other properties of The first non-Euclidean geometry t o fit,'l.Irt:s which do nOt vary when thL)' are develop, spherical geometry, tackles projcr_ted . These invoked collinear poinL~ ­ lllt!;ISUremenL~ on the surface of a sphere. It points which fall on a line in the original is the geometry of the surf:JCc of a sphere. also fall on a line in the projection and so me 'It haJ been demonstrated by mathematic; that the surfoce of special ratios between distances. Projective the land and water is in its entirety a sphere ... and that any geometry could be used to plane which passeJ through the centre makeJ at its surface, further work on conics (since that il, at the Jurface of the Earth and of the lky; great circles.' all con ic sections can be seen ptolemy, Geography, c.AD150 as projections of a circle).

PONCELET

110

OT HER WO Rl DS

Usillg spheriC/II gl!llIl1f11-y, 11't are ablr to uteaSll/"/' distaltres 011 plrlllt'ts

alld 1111)1)/1S

with

SWllt dtgree of(Il"Cl/l"Ilry.

One anomaly of spherical geometry IS immediately apparent. A line in spherical geometry is the shortest distance benveen two p·oinrs, juSt as it is in the geometry of nat planes, but it looks very different. A line drawn across the surface of a sphere, if continued long enough, meetS its own heb,rinning, becoming a circle with its centre at the eentre of the sphere. This is called a geodesic, or great circle - so a straight line becomes a circle! All other asp!;!ct'i of planar geomctry are then adapted accordingly - so angles are defined bep,ve!;!n great circles, for instance. A line on a spherical surface is defined nOt by its length but by th e angle under which in; end pOUlts appear when '';'L'wed from the centre of the sphere. This angle is called the {f/"c angle. It is mually measured in mditlllS. The arc angle multiplied hy the

radius of the sphere gives the lcngth of the line over the surface. Some differences between planar and spherical gcometry quickly become obvious. On thc surface of a sphere, we can define a shape using only nvo lines (or grcat think of a circles) seb,'lllent of ::m orange. Clearly, we can't make a shape from only two straight IUles in a plane. Spherieal triangles have other special propcrties. The angles always add up to more than 180"; how much more than 180" is determined by the size of the triangle and is callcd the spherical excess (E). This can be used to calculate the area of the triangle: area == E x r'

where r is the radius of the sphere and E is given in radians. This is called Girard's Thcorem after the French mathematician Albert Girard (1595- 1632).

RADIAN S AND DEGRE.ES degrees; there are 2n radians in a circle, or n radians on a straight line. Radians were fi rst used as a measure of angles by the English mathematician Roger Coates in 1713. He recognized that the radian is a more natural unit of measure than degrees, though he did not u~e the name. The term first appears in print in an exam paper set at the Queens College, Belfast in 1873. One radian""

I*,,,

Tb(' OIafl' !lIrfou of all

Ol"flllgf

sl'ff'l1mt if fl sbajlf' bolmdfd

'ry Dilly rwo rtrfligbr IillfS. 111

IN T H£ ROUND

Early astronomers and surveyors were working with spheres as they looked at the sJ..-y and the Earth. They beeallle aware early on of difficulties with Euclidean geometries when applied to spheres. However, it took many centuries for the possibility of

'The hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines.' Saccheri

alternative 6>'eomerric rulC!:i to be accepted . ELLIPTIC AND HYPERBOLI C GEOMETR IES

Curved surfaces give rise to two nonEuclidean geomf:'tries. Lines drawn on curved surfaces do not behave in the same way as those drawn on a plane, as we have seen with spherica l geometry. J" l ()st importantly, Euclid's fifth posrulate does not hold . In Euclidean planar geometry, twO lines both drawn perpendicular to a given line, L, will be parallel. For a curved surhtce this is nOt true. On an elliptical surface, there are no such lines - twO lines drawn perpcndiUllar to a. third line will eventually intcrSL'C[. A perfectly elliptical .';urface is a .~pherc and spherical geometry is a special the simplest - model of elliptical geometry. On a hyperholic surface, twO lines drawn perpendicular to L will diverge. If the CUl"varure is exactly right, the hyperbolic surface will he the inside of a sphere, but

otherwise it will h~ some vast howl. Clearly, the reverse of a hyperbolic surface is elliptical - the outside of a sphere is elliptical ~nd the inside of the sphere is hyperbolic. REJECTING ALTERNATIVE GEOMETRIE S

That the behaviour of lines on a curved surface is COntrary to Euclid's rules of geometry disturbed mathematicians. For many centuries, thL}' tried to deny all nonThe Italian Euclidean geometries. mathematician Giov~nn.i Girol~mo Saccheri (1667- 1733) tried to prove that rht'}' could not exist, but ended up d(ling the

Threr rill/pie tljtl!{1Yl1IlS n"fPnrminf{ rhr behaviour oflillff wiTh 11 CQ1II1Ilon perpmdiCII/l1r 171 filCh ofrbr rhra: I)'per ofgral!lrtry. A frillnglr

tln/wll

all tbc

hyperb(J/ic'ill/fllU of II

HYP",bolk:

112

Euclldeln

EIHptk:

rllddll' '/(-II/(JII'ifrllrrr rhllt rhe III/gIn iI/ride 11 njflllglc ill hYflirbolic geomrlry ClIII I1IId up (0 Iffr rbull 180°.

OT HER WOR LDS

equivalent to hyperbolic h"Cometry and that of the obtuse angle gives elliptical gt..>Qmetry. Saccheri's work had little impact at the time he was writing, and its importance was not recognized until Eugenio Beltrami rediscovered it in the mid-19th cenrnry. DAWNING ACCEPTANCE

Jdllor Bo/yair ronlpasser ill

tb~

Bloyai MUre/I'IIl,

Marorvdrdrbely. Romallin (wbrn bf died).

opposite, demonstrating the possibility of alternative geometries and deriving some of the principles of hyperbolic geometry. His work apparently drew on writinb'S of the Iranian mathematician Omar Khayyam (1048- 1131), though he may have developed his arguments independently. Saecheri took as his starting point a parallelogram proposed by Omar Khayyam. The parJllelob'Tam is formed from a pair of parallel lines, with sides drawn between them, perpendicular to the nvo lines. (Ill normal planar gL>Qmetry this looks like a reb'1llar rectangle.) l ie then considered three possibilities: that the internal angles are 90°, less than 90° (acute) or more than 90° (ohtuse). Although it looks pretty obvious that they are 90°, his aim was to prove that they could nOt be anything else, and SO support the fifth posrnlate. It turned out that these alternate hypotheses were not as absurd as Saecheri had hoped. His reasoning for refusing the other nvo possibilities was nOt sufficiently rigorous and, though he rejected them, he did nOt disprove them. It emerged over time that the case of the acute angle gives a system

Hyperbolic geometry re-emerged with the independent work of the H ungarian J anos Bolyai (1802-60) and the Russian Nikolai lvanovieh LobaehL'Vski (1792- 1856) around 1830. Bolyai published in German and Lohachevski in Russian; it was nOt until Lohaehevski published in German, too, that his work came to wider attention. The great German mathematician Carl Friedrich Gauss claimed to Bolyai that he had already discovered most of what Bolyai rt..'Vealed A rtllrtlf ofJdllor Bolylli fllld

bir fa/bel; Fm·kllr, afro II

ilI~lI­

klllr<1J111!!fllbmlflliriflll,1:.'/;o illlTrllcud bir rOil [r(Jm all ertrly agr.

UlllJl:.1

ttl!t till'

113

I N THE ROUND

CARL FRIEDRICH GAUSS <1777- 1855) Carl Friedrich Gauss was a child prooigy, born to uneducated, impoverished parents

in Germany. He had an amazing capacity for mental arithmetic and claimed to be able to calculate logarithms in his head more quickly than he could look them up in a table. Gauss mad£> great advances in mathematics and its applications in

his ideas. In fact, Gauss's personal diaries suggest that on several occasions he came up with ideas many years and even decades before others published them, but did nothing with them himself. The GalifS- lVt-b''1·1fIOlllllm'llr ill Giittillgm, C"/"fared

by (ad Fffliilllllld /-Iarfzr,· in 1899.

astronomy, statistics, earth sciences and surveying. He produced several important theorems and proofs in many areas, and his work on curvature underpinned Einstein's theory of relativity. Working with his physics professor Wilhelm Weber, he studied the earth's magnetiC field, dev£>loping methods that were still used until the second half of the 20th century. The pair also constructed the first electromagnetic telegraph in 1833. Gauss's claim to have worked out hyperbolic geometry before Solyai placed a strain on relations between the two, since Solyai considered that Gauss was trying to steal

when he published in 1832, hut had not him!ielf publici zed it. This was possibly true, and hoth Lobachevski and Bolyai had links with Gauss that could have gi\ren them insights into thoughts conmined in his teaching and correspondence. [n this case, Gauss would haw been the first to develop a consistent, nOI1, 1<

Euclidean geometry. The work of Lobachevski and Bolyai had little impact until Gauss's ideas were puhlished after his death in IH55. Gauss suggested treating hyperbolic and elliptical A pDl·n·ait ofNiko/fli /vmlOvich Lobamwski. Ht spellt most of bir wrter rts

1/

pmfessor ar KnWIl UlI1c'ffriry.

OT HER WO RlDS

surfacf$ as 'spaces', since although thL)' c.xist in three dimensions they actually have only twO dimensions and on ly two \'ilriables are needed to specifY a point on them. H e showed that a surface could be described entirely with reference to distances and angles measured on it, and without giving information about its placcment in threedimensional space. RIEMANN AND IRREGULAR CURVES

Although Bol yai and Lobachevski had demonstrated that a set of alternative methods for working with hyperbolic sur&ces was feasible, there was no model equivalent to Euclid's planes, lin es and points for dealing with the geometry of curved sur&ces. Such a model was provided by the Italian EUb"t:nio Beltrami (1835-99) in 1868. lmponantly, he demonstrated that hyperbolic geometry was consistent if

Euclidean gL'Ollletry is consistent. Beltrami developed spatial models which are now ca ll ed the pseudospherc, Poincare disc, Klein model and the Poincare half-plane. On the Poincare disc, distances at the edges are larger than distances near the centre, though this is nOt apparent as the disc curves away from the viewer. In Escher's picture, 'Circle Limit Ill ', the figures arc the same size allover the sur&ce. The way that the Mercator map projL'Ction distorts the size of cowltries near the poles is similar - Greenland looks larger than it is, for example - but on a Poincare disc the distortion is the other way, with distances seeming smaller than they an:. The shortest distance between twO points 011 the ed!,'l: of a Poincare disc is an arc of a circle drawn at right-angles from the boundary of the disc.

Similarly, the centre of a hyperbolic circle is not in its middle:

A rumpllw' gmpbir sb(T;J ,illg rb~ rlirvatlllT of rpare [n~ared

by a b1iU"k bok. Tbe rlirvalllrc of.
was €Stablisbed by Einsteill]- rbem) of ndativity.

IN THE ROUND

German Bernhard Riemann 'The views of space and time which I wish to lay before you (1816-66) extended hyperbolic have sprung from the soil of expen"menral physics, and geometry ro work with surfaces that do nOt have uniform therein lies their strength. They are radical. Henceforth curvature. H e dcveloped a space by itself, and time by itself, are doomed to fade away system for describing the into mere shadows, and only a kind of union of the two will curvature at any point on a preserve an independent reality' surface in space using only ten Hermann Minkowski, 1908 numbers . Riemann geomctry involves postulating higher dimensions (i.e., dimensions beyond the be twisted into shapes which appear to be three familiar, physieal dimensions). He three-dimensional ohjects, producing began with a concept of II-dimensional curious anomalies that have no distinct spacf;!, and used calculus to provide inside or outside . geodesics for any cun'ed surface. His work The simplest example of this idea is the underpins much of modern physics. .M ohius srrip, easily made by talcing a strip including Einstein's theory of relativity. of paper, twisting it once and gluin g the The attempts to prove non-Euclidean ends together. The strip has (lnly <)Ill' side geometries led to greater rigour III you can run a finger over the whole . .l.lrface, scrutinizing the Elr1l1wts. The German both side.~ the original pa.per strip, in a mathematician Moritz Pasch (UH3-1930) continuous movement. The KJein bottle is an c.xtension of this saw the need for conceptS, axiom.~ and logical deductions based on these axioms to principle requiring :J further dimension. underpin the new geometries as well as old Although tht' bottle is necessarily drawn mathematics. This contributed ro the drive, int ersecting irs own surface, as it is led by David Hilbert at the start of the 10th eenrury, to axiom:uize all of mathematics and provide a firm foundation in proof for even the most seemingly obvious deductions (sec page 199).

or

INSIDE OUT?

Curved surfaces form the basis of the branch of mathematics called topology. lr became one of the most important areas of development in m:uhematics in the middle of the 10th century (1925- 1975). Although, as Gauss and Riemann showed, they c.xist in II-dimensional space they have only two dimensiolls of thejr own. Surfaces can even 116

A Miibills ship. Thr Vis/litl deception m:"tUm by SlId)

sbapa - is it F.J:O- or rhru-diJ/lC/lsiollal? - fan/IY rbe bartY of rbe

l);1»-k

of "rtisr M.e. £reb" (IS98- 19 72) .

OT HER WOR LDS

EIN STEIN, RIEMANN AND THE SPACE-TIME CONTINUUM

The general t heory of relativity postulated by Albert Einstein (1879~1955)

uses

concepts

of

geometry

and

the

Riemann an

extra

dimension, making a fourdimensional

space

called

space-time_ (Space-time was first suggested by Hermann (1864~1909),

Minkowski

following the publication of Einstein's special theory of relativity in 1905_) In general relativity,

is

space-time

curved, with the degree of curvature increasing close to massive bodies_ Curvatu re is produced by the interaction of mass-energy and momentum producing the phenomenon we know as gravity_ Thus Einstein's theory replaces the 'force' of gravity familiar from

Follawillg IHillhm:ski's lend, Albert Eillsteill added a follrlh

Newtonian mechanics with

dill!<'IIrioll tv the 7:.r,w/ds of1l1atbe1l1atics alld gro1l1etJ)' with bis

multi-dimen sional,

tbeorJ ofspaa-tillle.

non-

Euclidean geometry. This cu rvatu re, and t he principle of relativity,

was

observations

of

proved an

in

1919

ecli pse.

by

Einstein

At the moment of an eclipse, a star would appear to be in slightly the wrong place because of this distortion. Measurements (1882~ 1944)

predicted that light rays would be distorted

made by Sir Arthur Eddington

by the cu rvatu re of space produced by the

in Principe Island, Gulf of Guinea, proved

gravity of a nearby star or planet.

that this was indeed the case.

117

IN TH£ ROU ND

flATLAND

The novelill Flatland: A Romance of Many

to

Dimensions w ritten and illustrated by Edwin

dimensions might exist, but he won't be

Abbott Abbott in 1884 slltirized the social

persuaded. It becomes a criminal offence

hierarchy

of

Victorian

Britain

in

a

mathematical tale. The narrlltor, a square, occupies

a

two-dimensional

world,

Flatland. He dreams that he visits a one-

the

lineland, but cannot convirn:e the ruler that life in two dimensions is possible. The square 1I

sphere,

but can't conceive of a three- di men,>i ona I worl d until he visits it. The squMe then tries

sphere

that

dimensional world is possibl e. In

another

introduced

dream,

to

square

the

Pointland

and no

- 0 . . _ ... .......... . _ _ ... ... •

__

_.t-

I\ROM~ci~ 0 ' M~NV DuiilN~IOfoI"y

moddled mathematically it is curVt'd through four dimen sions and ha s no intersectio n. The in side becomes th e outside seam l l!.~sl y. A Klein bottl e can he dissected to give twO N16bius strips. The Dutch artist M. C. Escher drew several scenes whjch played with the ideas

of altemative worlds.

:.~ -\-

-..;~ ~'i1'~

--....9_

success

there of the existence

~...c..::;¥.2Z"A.;;;yzj

....

i~

again

persullding the ruler

~~-:::~

~~

more

to suggest in Flatland that a three-

world,

dimensional

is visited by

convince

Tirll'-pagf i11l1rtmtioll

.....:.

.",,";';..

-,

--

-

_~_

froll! Flatland, sbOllrilig 1111 llll' plllt:,·s rbe fq llill"f visits ill bis dl"ffllll.

of impossible surfuces and structures. The Penrose triangle, first drawn by the Swedish artist Oscar Reurc.rs\·ard in 1934, was popularized by th e mathematician Roger Penrose in the 1950s. He called it 'impossibility in its purest form'. MOVING ON

Impossihle geometri es, it TRIVIA The name 'Kl ein bottle' is a misinterpretation of the German Kleinsche Fldche ('Klein surface'), taken as Kleinsche Flasche ('Klein bottle'). The name has stuck (even in

German) and severlll glass-blowers have made literal 'Klein bottles', though necessarily with an intersection. There is display 01 these in the Science Museum in london.

11 8

1I

turned o ut, are nOt so impossible after all , and thc fact that wc can't visuali ze .-.omcthing doesn 't mean that it can't exist". As with mapping three-dimensional space to flat planes, all that is needed is a l.."'Onsistent and

OT HE R WOR LDS

Tb~

impossible mallgle ill fur( I'errb, IVesrem

Australifl. Tbe

frrtl((lIn' if

fieri/filly disjointed (It rbe

A mathematician named Klein

rap, fllIIl btlrbem pboragmpbedfrom one of rbe rwo

Thought the MObius band was divine.

rpotr fivm wbieb it IT Mfigned ra be rUII.

Said he: 'If you glue The edges of two,

thorough method. The way these representations work, espL'Cially tor spaces in more than three dimensions, is by way of a coordinate system. This can be explored and manipulated mathematically using algebra. Alb'Chra and geometry dL'vcloped in parallel, with considerable cross-fertilization, until the 17th century_ Then the

You'll get a weird bottle like mine.'

Anonymous limerick

remarkable work of twO Frenchmen in that century brought them tOgether and provided the tools needed for Riemann and other non-Euclidean geometries to be formulated.

119

CHAPTERS

The MAGIC FORMULA Algebra is familiar to most people in the fonn of equations that must be solved, either equations set as exercises at school or equations fonned to model problems in economics, science or some other discipline. The representation of unknown quantities by symbols, which is fundamental to algebra, evolved slowly. Although Ancient Egyptian and Sumerian mathematicians dealt with problems that involved unknown quantities, they did not express them in the fonn of equations as we do now. Indeed, not until the late 16th century did the familiar form of an equation evolve. We now have many ways of solving equations, including the use of graphs. This has been made possible by the crowning achievement of Rene Descartes, who brought together geometry and algebra in the system of Cartesian coordinates which allows an equation to be plotted as a graph.

Ar rbis al/If/e, rbe Tawn' if Babel defier rbe Tides ofGod alld KerJlIletry,

TH( MAG IC fOR M ULA

Algebra in the ancient world I t is impossible to disentangle simple algebra from geometry, for it was in problems relating to twO- and threedimensional gt..>Qmetry that algebraic questions first surfaced. Early on, specific, practical problems in algebra were neither systematized nor represented in a way which we would now recognize as algebra yet they provide th e origins of algebra as it was later fOnlmlated. FIELDS AND CELLARS

Babylonian clay tablets III the British Museum include a number of problems which would now be formulated as quadratic or cubic equations. These rdate to building projects and involve working with areas and \'Olumes. Some problt:ms related IU dividing up an area in partS with different proportions. 1t is easy to see how a problem in area can lead to a quadratic equation.

, , 2

Here the area of the larger (enclosing) rectangle is (a + 2)(a + 1) '" a' + 3a + 2

Similarly, cubic equations can be derived from Babylonian problems relating to digging cellars. The earliest known attempt 122

The lIlnhod for solvillg rrmllltfllll'oflS

N11/an011S

IIlnlled aftl'l' Carl Friedrich Gauss bad bel'll tlsed

;11 rhl' East 2.000 JtYlrs ulrlin: to write and tackle cubi c equations is in the form of 36 problems about construction in a clay tablet nearly 4,000 years old. Such problems were expressed in words by both the Babylonians and Egyptians, and by mathematicians for many centuries afterwards - for example 'the len gth of a room is the sa me as it.~ width plus 1 cubit; it.~ height is the same as its length le~"S 1 cubit.' The Babylonians did not attempt any general rul~ or methods of treatment for problems of these types. They dealt only with the specifies of each problem and seem to have had no grasp of a general algorithm that could help them solve all problems of a simi lar type. The Ancient Egyptians, tOO,

ALGEBRA I N TH( ANCI£NT WOR lO

solved practical problems that would now be exprcssed as linear or quadratic equations, but again without rcC()urse to any formal notation and without recognizing them as equations. The Chinese text The Nille OJllpfers (2nd to bt cenmry Be) includes a chapter on soh~ng simulcmeous linear equations for two to seven unknowns. They wen' solved using a counting board or :;l.Irnce and could include neg'ative coefficients. The description of c'luations with negative coefficients is the earli e.~t known use of negative numbers. The method used is now known in the \Ve~t as Gmssiall elimination after Carl Friedrich G:I\.l~S who used it 2,000 years later.

yielded more than one solution he sropped after arriving at the first - (..'Ven if there were an infinite llumber of solutions (as for an equation of the type x - y '" 3). H e developed a method for representing equations which was less cumbersome than writing them um in wurds, bur was ~rill nOt comparable with modern methods. A~ the Greck~ used thi:' letters of their alphab~t for numbers, thert: were no recognizable ,"ymbuL~ immediately ava.ilahle to represent varia bles. \Ve ean usc x, y, a, b, m, n and so 011. to stand for variables and mmtants because we have separate symbo ls for numbers, and SO an e:"llression such as 2x is unamhiguous. Diophanrus adopted some variants on Greek letters, and used symhols FROM GEOMf.TRYTOWARDS ALGEBRA to indicate squaring and cubing . His ~YS[CIl] 1"n the middle of thc 3rd century AD, the of ablm..,'Viations was:m intermediary st:lge Hellenistic mathematician Diophanrus of between the purely discursive explanation Alexandria developed new methods for uf pro hi ems and the purely symholic in usc so h~n g proLk'IJl.~ that would now be sho ....'T\ now. It also g"Jve him the opportunity, not as linear and 'luadtatic equations. H.is work, .~een ur exploited before, of dealing in ArifbllletiCtl (of which only part has higher power.~ than mbl'.s. Some of his survived), cOIlt'Jins a number of algebraic problems include a notation that means cquations and methods for solving them. 'square-square' or 'cube-cube', indicating Diophantus applied his methods ro the powers of 4 and 9 respl.'Ctively. problems in hand, but did not e.xtend them 1n addition, Diophanrus had no concept to general solutions. Like the earlier of an equality - of twO balanced expressions Greeks, hc dismissed any solutions that between which parts could be moved or on were less than zero, ami whcn an equation which idcntic-al operations could be carried (JUt. Nor did Di ophantu s deal with more than one INDIAN QUADRATI CS unknown at a time. H i:' An ancient Indian text, one of the Sul ba sutras written by always sought a way to convert a sC(:(md unknown Baudhayana around the 8th century Be, fi rst cites and then into a.n expression built solves quadra tic equations of the form ax' '" c and axl + bx = c. Tht>..'ie occurred in the context of building altars, and around the first. So, for 50 relate to a practi[al problem in three dimensions. example, in a problem that ca ll s for two numbers whose

n'"'"" " ~OR M UtA

sum is 10 and the sum of whose squares is 108. D iophantlls would not write, as we may. x + y = 10; x1 + y.' '" 108, but might tCflll them (x + IU) and (x - 10), the second equation then becoming (x + lOy +(x - 1W :20S.

ORDERS OF EQUATION

Polynomial equations are those that contain a series of terms, each of which has a variable raised to any power, multiplied by a constant (ordinary number). For example in the following equation

D I OPHANTINE EQUATION S

Diophantine equations arc those III which all the numh~rs involved, including those in the solutions, are whole numbers (which can be positive Of negative) . They fall into three categories: rhost': with no solution, rhose with a fixed numher of solutions and those with infinitely many solutions. For e.x:lmple. the equation 2x+2y:l

has no solutions, because ml'n"! arc no values for x and y that are whole numbers

the first term consists of "II! X 1, the second 01 Xl X 2 and the last the constant B (or XO X .8). Mathematicians refer to polynomial equations as bei ng of the first order, se(ond order and so on depending on the highest power they contain. So a quadratic equation such as that above is called a second-arder equation; an equation including a cubed term (x~ is a third· order equation.

4x + 6y '" 24

that can give the amwer t (the sum of two

even numbers is always even). The equation x - }' = 7 has illfinitel~' many solutions as we can continue to pick larger and larger values of x and y. The equation 4x '" S has only one solution: x = 2. Diophantine equations ;Ire useful for dealing with qu:mtities of obje('t~ that cannot be divided - such as numbers of p(.'Ople. So, for instance, if there is a choice of (~ ars to rake 24 people on ~ trip, some of whieh c~rry four and some of which carry six passengers, and ~ll must be full, we could write ~ Dioph~ntine equation, since the only useful solutions assib'll whol", numbers of people to whole numbers of cars: ,2<

(This has the additional requirement that the values of x and y muSt both be positive.) Maths problems of the toll owing type USt' Diopbantine equations: ';I hoy has spent 96 cents on ~"Weets and bought 4 chocolate mice,2 lollipops and a chocolate bar. \Vh~t is the cost of each item?' Diophantine equations of the form ax + by '" c ~re

lin(.":lr equ~tions (a graph dr~wn of the would he a straight line). Another Diophantine equation,

equ~tion

A LGEBRA I N THE AN CI £ N T WO RlO

relates to Pythagoras' Theorem and produces Pythagorean triplets (e.g., 3,4,5: Y+16=25).

AJthough Diophantine equations arc named after Diophanm~, he was not the tlrst to work on them. The lndi~n Sulb~ surras deal with several Diophantine cqu~tions. H owcvl'r, Dioph~nrus differed m~rkcdly from e~rlicr Indian and B~hylonian m~them~ricians in th~t his probll'ms were purely thcoreric~l - he was nOt concerned with building altars, digging cellars or taxi ng grain, and his numbers do nOt relate to qU~J]riries in the re~ l world. He was also concerned only with precise answers using whole numbers. It is probably for this laSt rcason that there arc few cubic equations 111 D i()ph~nrus' A1"itlJ1llfficn . Although t he questions that Dioph~ntuS deals with m~y nOt look unu:,·ually difficult, his ~ppro~ch w~s gcnuint::ly innov~rive and has h~d ~ lasting effect on la ter m~thematici ans. Indeed, it was while trying to generalize ~ problem r~ised by DiophanUls, to divide a square into twO squares, that Fermat arrived ~t his famous Last Theorem (see page 140).

GOING BEYOND THE CUBE

VYhile Diophanrus had a form of notation for powers gn:ater than three he did nOt make any great use of it. Another AJcx~ndrine, P~ppus of Al ex~ndria, appro~ched the issue, hut ~gain did not come to grips with it. He was rhe first to st~te cle~rl y that l ine~r, or first-order, ~ l gebr~ic problems rcl~te to a single line or one dimension; secon d-order problems relate to two dimensions or areas, so arc planar, and third-order prohlems rd~tc to three dimensions or volu mes, so arc solid. Investigating the properties of eurves defined by IUles in planes and voluml's, he came up a!,rainSt the possibility of equations of a higher order. However, he dismissed it since 'there is nOt anything contained by more th~n three dimensions'. Dioph~nrus w~s toO much wedded to algehra and Pappus to !,'"l'omerry for either of them to make th~ conceptu~l J e~p into algebr~ic geometry, though they both approached the jumpulg-off point. It was one of Pappus' geometric problems of lines and loci that eventually led Descartes to invent algebraic gt.."Ometry in the 17th century.

OIlSERVAT IO DOMINI PETRI DE FER ,\ IAT ubum autcIIl in duos mbos, aut' lJuldratoquadratum in duos quadr.Jtoqulldrgto!i & gCllcr-.llitcr null am in infinitum ult'r-J quaur.JtunJ potest,1tcm in duos eiusdcm nominis f~ s ~Sl diuidcrc mius rei dClllonstrationcm mir.Jbilcm "'ln~ dctexl. Hanc marginis cxiguit':ls non ClpcrcL

C

Trnllslnt;oll of F tTm nt 's Las t Tbeol"e'm : It is illlpom'ble for (/ ruile to be the SIIW of mlo wiles, n follrrh power to br tbe srn" of r.1'O fOlln/; PIY'':I(TY, or;/1 g meral for n/~Y III11flbt r Ibn! is n ptr':II:I· gf"Cnul" dlflll tbe

seco"d ro be tbe SIWI of r.:·o like PO".l'I'1Y. I bttVt dirtov/' I"(tI n n·lliy ultl/7.'fllo/ls demO/lSm/rioll oftbis propositioll Ihar tbis 'lIl1lrgill is too 1!mTIF.IJ to fOllfnill. 125

11'"'." " ~OR M UlA

Al-MAMUN'S DREAM

The caliph al-Mamun (786- 833) is klid to have had a dream in which Aristotle appeared to

him. As a consequence, the caliph ordered translations to be made of all the Greek texts that could be found. The Arabs had an uneasy peace with the Byzantine empire and negotiated the acquisition of texts through a series of treaties. Under al-Mamun's caliphate and at his House of Wisdom, complete versions

of Euclid's Elemenfs and Ptolemy's Almagest

were translated, among others. A 15th-eel/wry palmillg ojAriswrlc. AI-M'Uf/lmr uigll 'was /loted for his bilge effurlY ill rbe tn/llslarioll

ofGn'l'k pbilosopby ami scimu.

126

The birth of algebra

Al JABR WA-l-MuQABALA

With the development of the Indo-Arabic number system and the adoption of zero, something approaching modern algebra hecIme possible. The Arab mathematicians, in drOlwing tOgcther the best of Indian and Greek mOlthematics and extending it, laid th e foundations of a proper algebrOlic system and even gave us the term 'algebra'. They found algehra more Olppealing than the Greeks had done and there were also spurs to its development wi thin their own society. The incrcdibly complex hws of inheritance, for example, made the cakuhtion of proportions and fractions a tedious necessity. On tOp of that, the constant need to find the direction of Mecca made algebra, like geometry, a tOol worth developing.

Th e word 'algehra' is derived from the title of ~ tre~tise written by the Persi~n m~thematician ~nd member o f the House of Wisdom, Muhammad ibn Mus~ alKhwarizmi, (""ailed AI-Kifab al-Jabr wa'l!Wllf/aba/a CThc lA)mpendious Book on Calcuhtion by Completion ~lld Balancing'). This presented systematic methods for solving lineOlr and quadratic equations. The modern word 'algorithm' comes from the name '~I-Khwarizmi', too. In his hook he gives methods for solving equations of the types ax" = bx, ~x' = c, bx = c, ~x' + hx = c, ~x" + c = bx, ~nd hx + c = ax' (in modem notation). Like D ioph~ntus, he only considered whole numbers in equations ~nd their solutions; he had the addition~1 requirement that the numbers must also be

"T HE BIRT11 O F A LGEB RA

Olllfir Kbayya1l1 ,HIS also rt>spollsibll' fm· tbl' n'lm711 of fbI' Pl'lYiall colmdar. His Jilla/i (almdflr is tbl' bamof rbflt still ill IISC ruM)' ill Im ll fllld Ajgballisrflll.

positive, while Diophanrus allowed neg'Jtiw numbers. Al-Khwarizmi wrOte out all problems and solutions in words and had no symbolic nOtation. Ironically, since his work is credited with introducing Hindu-Arabic numeral s to Europe, he even wrOte th e numbers our in full. Afi:l;'r showing how to tackle equations, al-Khwarizmi went on to usc Euclid's work to provide demonstrations using b'l!ometry. Euclid's propositions were entirely geometric, and al-Khwarizmi was th l;' first to apply them to quadratic equations. The method he developed, of systematizing the

GHIYAS AD-DIN ABU Al-FATH OMAR IBN IBRAHIM KHAYYAM NISHABURI (1048- 11 3 1) Omar Khayyam was a mathematician,

empi re. His Treatise on Demonstration of

astronomer and poet born in Iran, probably

Problems of Algebra (1070) set out the basic

to a family of tEfit-makers. He lived most of his

principles of algebra and was responsible for

life on a modest pension provided by a friend

the transmission of the Arab work on algebra

who became grand vizier to the Seljukid

to Europe. He worked on the tri angular arrangement of numbers known as Pascal's triangle and is sometimes considered

the

originator

of

algebraic geometry, which uses geometry

to find

solutions to

algebraic equations. A 19rb-cmrmy ElIglisb tmlls/arioll of 01l1ar KbIlJJII'I11's collectioll offolir-lilll' pol'lllS, rbl' Rubaya!:. A11111J PI'Tsiall scbo/ars wt?·e alro f!OI'ts.

127

II '"'"" "

fOR M U L A

(a+ b)" == a"+na""b + n(n-l )a""'b' + n(n-l )(n_2)a1>lb' + n(n-l )(n.2)(n_3)a""'b 4 +

I

1x2

lx2x3

cases and then applying a geometrical solution, was adopred by later Arah math ematicians and perfected by Omar

Khayyam

(5(,C

Tbe njlltltion s/J(T'':'S hlr.lJ 10 jill/I 1be cOI'ffidm1:i tllIIl vfII7t1bks for tilly iXlMlld.." billominl c:tpn:ssioll of tbe /01"111 (a -+ b)".

below). Al-Khwarizmi's work

stands for algchra as Euclid's Elcmwts did for gl.'Olllcrry, and remained the clearest and best dCllll'ntary n"Camlcnt until modern Omar Khayyam followed a similar procedure TO al-Khwarizmi, using Greek geometric work on conic sections to demonstrate his solutions to cubic (thirdorder) equations. Omar Khayyam produecd general solutions for cubit: equations where the Indian mathematicians had worked only with speci fi c L':lSCS. In 13th-century China, Zhu Shijie developed .~o lutjon s for cubic eq uations without reference to Omar Khayyam's work.

had been studied in fndia by Pin gala (5th-3rd century Be), though I'lll ly fragments of his work survive in a bter commentary. Another Arah mathematician, Abu Bakr ibn Ahhammad ibn OIl Husayn alKaraji (c. 953-1029), had also worked on it and is credi ted with being the first to derive the binomial theorem (sec above): The Indian mathematician Bhattotpala (c. 1068) wrOte Out the triangle up to row 16. The triangle provides a quick '\"Jy of c.\:panding expressions .~uch as (x -+ y)\ since all that i.~ needed is to take the coefficienL~ from (in this ca.5e) lin e 3 (since it is a thirdorder equation), giving the result:

SHAPES, NUMBERS AND EQUATIONS

lx' -+ 3x'y + 3xy + l y'.

times.

In Pascal's triangle, each numIH.:,r is the sum of the twO numbers above it. The pattern forms the binomial coefficient series . In [ran, it is call ed Khayyam's triangle and in China Yang Hui '~ trianglc after the Chine!iC mathematician Yang Hui (1238-98) who also worked on it.

2 3

4 6

3

4

MOVING AWAY FROM AREAS

Although geometry provided good methods of prm'ing OIlb'Cbraic solutions, it was as algebra moved away from the restrictions of rcal-world gL'Ometry that the. idea of an abstract equati on, relating to numbers rather than measures or quantities, heCllllc

'Whoever thinks algebra iJ a trick in obtaining unknawns ha! thought it in vain. No attention !hould be paid to the fact

1

1

Before Omar Khayyam wrOte on P:Jscal's triangle, it '28

+ nab,," 1+ b"

1x2x3x4

that algebra and geometry are different in appearance. Algebra! are geometric facts which are proved. ' Omar Khayyam

THt HI.TIl Of ALeUtR""

.-/11 iUIIJtr'llliOll jivlII lkSClll1tS' "Ibe \"or!{l ill "i!·hicb hi R1rknLfI his Ibrorit'f QlI light. /br fl'llS/'S,

biQlogy IIIIl/mlllly otlll'l'lrJpia.

possible. The Arab were mathc nmticians willing to treat (:ommensurah le a nd incommensurable n Ul1lbc~ alongside Ollt' anothe r, and to 1l1lX ma gnitudes 111 different dimensi()ns, both of which the Grech were unwilling to do. Combined with the J-I illd u~AT"dbic number sysrem and the acccptance

of

zero,

this

:l llowed

algebra to move fo rwards away from its root.. in pr:I(.:tical b'Comctry. When Om:!r Khayyam and :ll ~ Khwa rizmi had recourse to geo metry to demonstrate

;lIld

their algebraic results they

D " 1,,

illllmmiOl/ sllowiug tbe

PrilKiplcs of Philosophy.

III OVfllftlll

of ob)rcu, from

D crC1II11!J'

were not im3gining their algebraic prob le ms III Ic rm.<; of lengths, areas and volumes but using geometry theoretically as a tool to represent algellraic problems . This relation ship between the two, d eveloped O\'e r the next 500 years, resulted cvcntually in the :l nalytic geometry of Dcscartes and FcrmaL 129

THE M AG IC ~ORMU t A

Writing equations Omar Khayyam died in 1131 ,lIld already Arab mathematics was in decline. Scholars from the Arab world were to make few further contributions in the field. Luckily, at the .'lame rime that political and religious groups were fracturing the Arab cultural world, the in tellectual spi rit was reawakening in Europe. Durin g the 12th cen tury Gerard of Cremona translated 87 work<; of Greek aJld Arab scholarship into Latin, working at Toledo. These included Ptolemy's Aimogrsf, Euclid's EirJllwfs and al-Khwarizmi's Aigrbm. In England, Robert of Chester translated al-Khwarizmi in 1145 and Adclard of Bath translated Euclid's Ehmf'llts in 1142. After centuries spent recovering :md cQllso lidating earlier learning, EutQpean mathematicians hegan to make their own cOlltribution to the dL'Velopment of algebra. Gennany wa.~ the foms of th ese new developmcntS in the 16th century. Perhaps th e most important of the Ill'W German works on algebra was Ari,IJlJletica illfegm by Michael Stifd (L".1487- 1567). He allowed the use of neg-ative coefficient<; in quadratic equations and as a consequence reduced the various types of quadratic to a single form. H e introduced negative powers, tOO, b';\1ng

and SO on. Even so, he did not allow negative roots in c(Juarions and referred to negative numhers as f1//"/J/rri absllrdi. l-re was similarly distrustful of irrational numbers, which he said are 'hidden under some SOrt of cloud of infinitude' . He proposed using a single Ictter to denote an unknown quantity, 130

repeating th e letter for Ilowers of the numher - so if c is the unknown, cc is C and ccc is c' . TOWARDS A NOTATION FOR EQUATIONS

Algebra without the ~l'mbols we use now was {:umbcrsome and long-\\1nded. Yet the modem notation is a Inte arrival on the scene. In Italy, the symbols li and iiI came to be used for plus and minus as abbreviations for the words phi (more) and mmo (less). But Latin was full of abbreviations for words and groups of letters that are wrirren repeatedly and this was not partieularly original. The introduction of arithmetical operatOrs - ~ymhoL~ showing [he type of eomputation to carry (Jut - did not begin until the hue 15th century. The first symhols tQ be used were + ~nd - , though originally they were to show a surp lus and a de6cit 1Il warehouse quantities. They soon t()()k on their mmll'm role as arithmetic operators. They werl' first printed in a book by Johan \iVidmann (bom c.1460), one of severa l German mathematicians who published on algehr:l in the late 15th and early 16th centuries. Even after the dc'Vclopment of symbols, many mathcmaticians continued to follow the rhetorieal model, writing our the prob lems th ey were posing and solvin g as discursive text with little or no re<:ourse to ~"ymboli c abbreviation (synmpation). Although vVcstem maths did nOt have a thorough and consistent symholic algebra until the 17th century, the wcstem part of the l~lamic world used symbolic notation in 14th-century comm entaries intended for teaching.

WRITI N G EQUATIONS

ROBERT RECORDE (1510-58)

Robert Recorde was born in Wales and taught mathematics at the Universities of Oxford and Cam bridge. He trained in

mathematic~

wanted to make

-:;";;;;:-1 r

as accessible

as possible. Most of his works were written in the form of dialogues between a master and a student. In

medicine and was private

1551 he published an abridged

physician to Edward VI and

version of Euclid's Elements, making

then Mary I. He was also

the text available in English for the

Controller of the Royal Mint.

first time. He first used the equals

Recorde

re-established

mathematics

in

when the country had not seen a for 200 mathematidan years.

sign, though using much longer

,..r._..

lines than we do now. It took 100

England,

years before

the sign

:~,:~~~~~~~~~J~universaIlY accepted alternative notations.

was

above

He explained /11 1558 R~rolrk "i:.'ar

everything in careful detail, in steps that

iJ//pris/.m~d flr/ailillg ro PIlY

were easy to follow and in English, as he

£J ,000 libel cbm ges. He died

'CC.D...__~

ill prisull rbe Sllllle yo·m:

BOL

DAn

SOURC

+ (plus) • (minus)

1489

Johan Widmann, Germany,

.J (square root)

1525

Christoff Rudolff, Germany, Die Coss .

'" (equals)

1557

Robert Recorde, England,

x (multiply)

1618

William Oughtred, England, in an appendix

S

Rechnung auf allen Kauffmanschaften .

The Whetstone of Witte. to Edward Wright's t ranslation of John Napier's Descriptio. a, b, c for known

1637

Rene Descartes, France,

quantities (constants)

Discours de la methode pour bien conduire sa

x, y, z for unknown

raison et chercher la verite dans les sciences.

quantities (variables)

+ (divide)

1659

Johann Rahn (or Rhonius), Germany, Teutsche Algebra.

TH E MAG IC fORMU LA

can be imagined, was nOt well pleased and the twO or Gemowe lines of one /engthe, thus: ==, bicause noe, 2. hattled for ten years owr thynges, can be moore equal/e, ' Cardano's disclosure. Robert Recorde Tirtaglia had hoped to rcmin the revelation of the cubic to publish as rhe crowning THE BEGINNIN G O F MODE RN MATHS? achievement of his career. (T.1rtaglia had, For all the importance of Stifel's Aritbmetiftl previously, published the findings of others illfegm, it was to be superseded within the without acknowledging his own debt, which year. In 1545 a work appeared that was so may reduce our sympathy for him a little.) Cardano was a little more open-minded revolutionary in ItS eentral concept that some people have taken it to mark the start with rebrard ro negative numbers than most of the modern period in mathematics. In of his predecesson>. Althoubrb be JUSt about An magllfl, Gerolamo Cardano (see panel entertained the possibility of a negative opposite) explained how to solve cubic rOOt, he dismissed it as being 'as subtle as it (third- order) and even. quartic (fourth- is useless'. order) e(luations . However, it was not a Cardano's hook represented the greateSt straightforward triumph of individual advance in algebra since the Babylonians genius. The solution to mbiL"S had probably hlld discovered how to so lve quadratic been discovered by Scipione del Ferro equations by completin g the square. Although it was of no pracrical use ~ (/:".1465- 1526), a professor of m:lthcmatics at the University of indeed, the solution of cubic~ by Bologna. On his death, he appro.ximation by passed the information to :t Jamshid al-Kashi (1380-1429) was more useful than Cardano's student, AntOnio Maria Fior. method - it stim ulated further Either from Fior or working development of algehra and toOk independently, Niccolo Tartaglia (c.I 500-57) the subject beyond the realm of the discovered the physical world. If quartil'S could disclosed it to Carda no, on he solved, then why not tifrborder equatiOJlS, sixth-order condition that he did not reveal it. Cardano promptly published it. H e did, admittedly, own that he Ir sums likdy tbar had a clue from Tarmglia. H e CO'o/llmo Cm'ilflllo /;e!lefited fi"flm fbI' w/)rk also acknowledbTCd that the (lJld ideas of olbl'J' solution of the quartic had 'II1arbmfflricillllf;1I bis heen by his amanuensis, Ludovico Ferrari grolllldbrraitillg book (1522---65). Tartaglia, as An m3gLl3 (/5-15). '/ will sette as J doe often in waorke use, a paire of parol/e/es,

132

WIIITI NC EQUATIONS

GEROlAMO (ARDANO <150 1-76) Bom in Pavia, Italy, Gerolamo Cardano was the

ill~itimate

child of Fazio Cardano, a -~

friend of leonardo da Vinci. His mother tried to abort him, and his three siblings died of plague. After

~ome

......,,-.,

. .. ~

IJ~"''''''''''''''' . . ·e:", ....... ,,"""

,....J.'P/III ~ •• _".f<~.w

difficulties in

. . ,., Q

__

I"

...... ~ ...... If."

being accepted, he trained as a doctor and

~". I'N

was the first to describe ty phoid. He became professor of medicine at Pavia in 1543 and at Bologna in 1562.

As well as being a physician, Cardano was one of the foremost mathematicians of

C(mllll/o'y honJSropr oIJfYlIY Cb,.llr rhm got him 11110

his day. His publication of solutions to cubic

fO 1Il11ch NUl/bit'. A c01l1n ill rbt' ll.fCWdllllf Lib}"a ellll

and quartic equations in Ars mogno secured

be mm

his place in history, but he also published

'whik the Uti}" GaSIor 11/ Grmilli pr.·ilim'd v/olmer

the first syst ematic work on probabili ty a

7:.·irhill Chrirts lifo.

hundred yeal"5 before

Pa~cal

(/f /111

inraprl'flltloll of the ftll}" of Ilnblehelll.

and Fermat.

Cardano's private life was colourful, and

poisoning his wife. In 1570, Cardano was

certainly fed into his interest in probability. He

accused of heresy and imprisoned for several

was

months for calculating the horoscope of lesus

always

short

of

money

and

supplemented his income by gambling and

Chri st.

As

a

consequence

he

lost

his

playing chess. His treatment of probability,

professo rship at Bologna and the right to

which he applied to gaming, includes a

publish books. He died on the day he had

Sfftion on how to cheat effectively.

previously predicted, but he might have aided

life wasn't

easy for

Cardano.

His

favourite son was executed in 1560 for

equati ons and even high er? Suddenly, algehraic prohJcm.~ no longer needed to relate to real-wo rld problems in th e dimensions we recognize. Furth er dimensions, for the ~"ake of mathematics, could he postulated, at lea st in theory. VVhile further dimensions were dearly absurd to Cardano's contemporarie~, of interest only in the arena of fantastic

the fulfilment of his prophecy by committing suicide .

math ematit~Jl l'xplor:ltioll, they would come into their own several centuries later. By opening up the possihility of al!,Tt'bra and algebraic geometry c.xtending into more than three dimensions, Cardano laid the foundations for Riemann b'Comerries and the four-dimensional space-time continuum with which Einstein would remodel the universe (sec pages 115- 9).

IB

THE MAGIC

~OftMULA

Algebra comes into its own The gulden age o f European algehr:t which began with C:trdano's publication of the solution of cubics and quarrics, encompassed the legitimization of negative and complex numbers, the dL'Vclopment of the Cartesian coordinate system, the marriage of algchra and geomerry in analytic geometry as wdl as considerabk steps towards the development of integral calculus. British mathematieians t:ame into their own again after a long ahscnce from the scene. but did n ot displace thc Italian. German and Polish mathematician.~. Some of these men wcre now writing in their OW11 languages rather than Latin. TO WARDS COMPLEX NUMBERS

Soon after the Cardano-l artaglia s() luti ~)!l of cubics ami quarties appeared, the ftali:m mathematician Ra£1d Bomhdli (c.l526-72) hcc:tme the first to introduce complex numbers on t o the scene. (Complex numhers arc those that involvc the .~quare roOt of -I , i.) Wor king wi th tube roots, he developed equations whieh used im,lginary roOts as a Stn6'C in deriving final soluti ons that are real numbers. H e descrihed it as 'a wi ld thought' and it did nOt in f:tct help in his computations, but it did signa l the impo]"[an ce that ('omplex num bers were to have fur algebra in the furure. DEALING WITH NUMBERS AND NOTATION

D espite all their advances, the algebraists and rrigunometcrs of the 16th century still did not have a widely used nOtation for deci mal fractions. \Vhen Rhetieus beg:lll his ,3<>

most ambitious u·igonometric tables. he lL~cd triangles with sides of length 10') units ro attn in the degree of accllracy he wanted without having to use fractions of any kind. (lr doesn't matter which unit.~ as he didn't actually construct the triangles, just suggc~ted them.) Fran~"Oi s Viete (see box opposite) was on ly a part-time mathematician, but made progress in various fields - arithmetic, trignnomerry, geometry and, most impormntiy. aI6'Cbra . He w:ts instmmcntal in bringing about changes in notation that mad e furth er progress possible and promoted the lise of decima l rather than sexagesimal fractions. Viete's most important contrihution was in brin6ring consistent n(lt;lti()n to algebra . TillS enabled him to develop a syStematic w:ty of thinking and a new method of working with !,'Cneral forms of equations. H e adoprcd vowels to represent unknown quantitie.~ and consonants to represent known quantities. H e also showed how to change the form of equations by multiplyi.ng or di\'irling each side by th e same maglllrude.. For example, he showed how to transform the equation x' + OK= b'x

into K+OX=b'.

Vifte sti ll did not recognize negative or zero terms, so he could nOt reduce the number of possible equations to :t single form in each ordn. (\Ve have the foml ax1 + Lx + c '" 0 as the standard foml which can descrihe any quadratic equation be('""3use, by allo\\~ng a, b

ALG£BR A COMB INTO ITS OWN

FRANC;:Of S VIETE <1540- 160 3) Fr an~ois Viete was a French mathematician and Huguenot 5ympathizer. Trained in law, he became a member of the Breton

Viete made great advances in several fields of mathematics, but always working in his spare time. Being weal thy, he printed

parliament, then of the King's Council serving Henri III and Henri IV. He was proficient at deciphering secret messages intercepted by the French. Indeed, he was so successful that the Spanish accused him of

numerous of his papers at his own expense. For a period of nearly six years in the second half of the 15805, he was out of favour at court and concentrated almost eXClusively on mathematics. In the 12th

being in league with the devil, complaining to the Pope that the French were using black magic to help them win the war.

century, al-Tusi had found the same method of approximating roots of equations as that discovered by Viete.

r)r c to be negative or zero, it covers such possibi lities a$ x' - 7 '" 0, where b is 0 and c is negative.) It is impos.~iblc to c;Jvcrst:1tc the importance of good, L"Qnsistcnt notation for the prob'TC,5S o f algebra. Yet this was not View's only achievement. H e arrived at formulae for multiple angles, was the first person to usc the law of tangents (although he did nOt publish it) and the first to sce that trigonometry ~ould he used to solve cubic equations that could nOt be reduced . Hc also produced the first thL'Orctical precisc numerical expression for IT:

', r I,. x

~. \

I

I

". j I

!"-rJ1 lo;

I I !'~

g o I

I

""! ""!

I

! '"

Although the method is nOt new, it w~s the first time the infinite scries had been cxprl."ised ana lytically. Algebra and trigonometry were mO\~ng morc and more

tcl\vards a concern with the infinite. - both the infinitely large md the infinitely s m~ 1 1. Progress :lccelerated as a clutch of talented mathem~ticiallS applied themselves ro developing algehr~ in irs new directions. French mathematician Albert Girard recognizcci that the number of roots an equation has depends QIl the Qrder Qf the equation - SO a second-order equation has twO rOOts, a third-order equation has three rOOtS, and so on. The hreakthrough came because he was sufficiently open-minded to allow negative and imaginary numhers in roots. Englishman Thomas Harriot (1560-1611) introduced the symbols> and < for greater than and less than . H e was also the first proper mathemiltician to set fQot on American soil, having been sent in 1585 by Sir Walter Raleigh as a surveyor. More influential than Viete in promQting the adoption of decimal fractions was the Flemish mathem:ltician Simon SW\'in (154H-1620). He :llso urged the :ldQption of

m

THE MAG IC

~ORMU t A

a decimOlI system of wcights 'There are enough legitimate things to work on withow the and meOlsures, though this was nOt ro happen for need to get busy on uncertain matter.' another 2UU years. Stevin Simon Stevin, T585 adopted a nOtation for powers which is similar to that in usc now, llSing a number in a circle matter of choice wh ether one solved a raised ahow the lin e to show the power - so problem by geometric or 3lgebr~ic 5° means 5!. H e even used fr3ctional methods. B}' bri nging trigonometry to bea r powers to show rnOL~, so 51!! means ~5. But on algrhra he was widening the scope of the Stevin was primarily a practical subject and promoting its alliancc wi th mOlthematitian, and he dismissed any geometry. Vien: was in fact one. of rhe first considerOltion of complex numbers. people to view mathematics as a wlified The eon6dence with which the best whole ratht!r than di fferent branche.s to he mOlrnematicians now approaehcd algebra - considered separately. fn 15 72, Bombelli 's Aigebm h::td and the distance ir had travelled from its roolS in real-world gL'Ometrie problems in presentcd many geometric problems which up TO three dimensions - is dear in the he solved algebrairally. For eX3mple, he ga\'C public challenge set in 1593 by the Bcl gi:m :dgebraic so lutions of cubics and then lllahematician Adri3en nn Roomen showed gL'Oll1 erric demon strations of his (1561-1615) to solve a 45 th-order equation: solutions. (H owL'Ver, this part of hi s trc3tise was nor included in the printed edition and - 3795xl + 45x = K didn't appear llntil \929.) Seventy-five yl'Olrs later, D escartes would t3.kc gcomctric No CI) llcept of 45 dimensions was need&!. problcms, convcrt them to an albrebraic form Viete. rose to the ch::tllen gc and solved the to simpli!)' them as far 3S possible, then when all , - - - - -- - -- - - , rerum to geometry for a final equation amh3ssador to the court of L ' A L G E BRA solution. bl this, his analytic o JI ERA g~ometrv., he comil leted a H enri rv said that there was Ili""." I . . ... ""..,.... no Frenchman capable of it. J.~~;-~..:.;,rljOW"llcy begun by Apollonius c:.o=~:'-='" when he showed th::tt conic ~.j,.60 ... ')J,.,..""IWiIf. sec tio n.~ cou ld rCllreSent TH E APPROACH TO ALGEBRAIC GEO METRY quadratic equations. VietC'S so lution related to sines and he used hi s Tbe rith paKf to II / 579 edition of multipl e-a ngle fo nnula e to BOll1belli's Algebra. 11JI'fim derive it. In pro~'i din g a rb,·u vclllmrr of all i//tmded jiill ll'f71: published in 1572. Bombrll; consistent symbo li c systrm I N BOLOGNA. IoII)IX1Wf, for representin g- algeb rai c rbilt J'I'I/I" befor.: be could ilird ... 100. .. .. "'T-. filla/i'U rb" lim r-.JJO vollflmr. equations. he also made ir a

(..

_. --.

......

,.,.a...-; ~ _ .

136

AlGURA COMf.S INTO ITS OWN

GIANTS OF THE 17TH CENTURY

From the first half of the 17th century there was communication hdween mathem:ltician.~ than there had bcen at any time since Plaro'.~ Academy. In many countries, mathematical societies grew up alongside the other learned societies then appearing. I..n Britain, the mathematical socicty had the enticing name of the 'lnvisible College'. in Francc, communicati on was further facilitated by Father Marin J'1ersenne, wh() corresponded with hundreds of mathematicians, scit::IItists and other learned men, acting as ~ conduit for knowledge and a sort of early networking guru. Thi s meant that there were fewer incidences of mathematicians privately developing work that was then lost and had no impact on others. Mersenne facilitated disagreement as much as anything, but at least no olle was in any doubt ahout what everyone else was doing. By a process of steady accretion, the fowldatiolls of modern mathematics were laid. Two men, hoth French, were to playa leading role in that process. Neither of th e two toweri ng figure.~ of the age was a profession~l mathematician. Rene Descartes (see page 138) was a minor scion of the French nobility who is more ramou~ as ,1 phi losopher than as a mathematician. Hi s explan ati on of his system of analytic geometry is provided in a.n appendix to hi s philosophical text, Discollrse 011 M ftbod, as a demonstration of

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how he u~d reason to arnve at hi s resu lt... Pi erre de Femm (set page 139) was ~ bwyer 3nd then 3 councillor who pursued his interest in mathemHics in his spare time. Yet his ability rivalled that of Descartes. MARRYIN G ALGEBRA AND GEOMETRY

D esca rtes fOlllld neitber geometry nor algebra entirely satisfactory 3nd set abour taking the best of both. By seeing the quan titi es in his equations as line !iCgments, Descartes avoided 3ny conceptual diffi culty in working with higher-order equ~tions and de3ling with equatio ns that did nOt h3ve cxpressions of the sam!.' order on L'ach side. F or cxample, the Greeks could not 3llow 3n equ3tion such 3$ / + bx = a bec3use the two parts on the left-h3nd side are considered areas and that on the right is considered a lin e; an area and a line cannOt be considered equal. Deseartes refin ed Viete's notation, using letters nC3r th e Start of the alphabet for known quantities (a, b, c) and letters near the end of the alph3bet for unknowns (x, )', z) . H e u~ed r3ised numbers to indicatl:~ powers 3nd used the .~ymbo l s for the arithmetical opcr3tors which we sti ll use. Only his symbol for equ3lity W3S different as he had nOt 3dopted Rohert Recorde's p3ir of pa rallcllin!.'s. 137

TH( MAGIC fORMULA

RENE DESCARTES ( 1596-1650)

propounded in his Discourse on Method, that knowledge must be acquired through reasoning. He maintained that sensory perceptions are not a reliable guide to the world around us and Glnnot be depended upon to yield true information. His famous dictum, ' I think, therefore I am', is part of his demonstration of the few things which can be relied upon - the exist ence of the thinking mind, of God and of the material world. The dichotomy between mind and body was another of his preoccupations. His belief in free will was paramount; he adopted

the

anti-Calvinist

view

that

salvation can be earned through the operation of free will and does not depend Rene Descartes was born in Tourraine,

only on God's grace.

France. His mother died when he was only

Descartes was always Sickly and, when

a year old. His father remarried and moved

Queen Christina of Sweden invited him to

away, leaving the infant Descartes in the

her court to teach her philosophy, and

care of relatives. He trained in law, taking

demanded that he get up at Sam each day,

his degree in 1616, and then travelled. It

he quickly succumbed to the Scandinavian

was while he was in Bohemia in 1619 that

winter and died.

he developed analytic geometry. Descartes practices of

shared the

some

views

and

All mgmvillg

mys tical

group

th e

QjQIlf'm

Rosicrucians. like their full followers, he

Gor;ltilltl of

moved around a good deal, always lived

Swedm
alone and practised medicine without

Iln1'f'aS07ltlble

charge, but he rejected their mystical

df"Tlltllllir fIJI"

beliefs. He promoted religious tolerance

;llsh'lIctioll by

and championed the use of reason in his

rbe gnat

scientific and philosophical writings.

pbilofOpber

Descartes has been Gliled the father of modern philosophy for his contention,

13'

precipitmed D(!sCllrtes'demb.

Al GEB RA COM U INTO ITS OWN

PIERRE D E FERMAT ( 160 1-65)

Born in the Basque region, Fermat studied

law and later mathematics. He developed independently of Descartes the principles of using a coordinate system to define the positions of points. Fermat worked extensively on curves, developing a method for measuring the area under a curve that is similar to integral calculus, and to generalized definitions of common parabolas. He worked extensively, too, on the theory of numbers and corresponded with Blaise Pascal on this subject. This was his only contact with other mathematicians. He was a senetive recluse, who generally communicated only with Marin Mersenne (see page 137). Fermat

was

the

most

productive

reluctant to publish that he gained little

All t'IIW·{roll1g of Pierre dr Ft1"'lIlfir t(ltt'll from UJ/lis Figllier's Vies {Its Savants UlusO"e<;

credit for his work during his lifetime.

(,Lives of me GrCJt Scientists), of 1870.

mathematician of his day, but was so

Descartes proposed that the position of a point in a plane could be identified by reference to twO intersecting axes, used as measuring !,ruides, so developing the coordinate system whieh is now known as the Cartesian system. For ~ll the familiarity of his ~Igebraic notation, Descartes' graphical representations of equations do nOt all resemble ours, for he never used negative vnlues of x in his graphs. The familiar form of a grapb divided into quadrants by axes that cross at (0,0) was introduced later by Isaac NewtOn. In addition, his axes were not always set at

right angles to each other. Dcsc~rtcS believed that any polynomial t!Xpression in x and y could he expressed as a curve and studied using analytic geometry. At the same time as Descartes was formulating his analytic geometry, anomer Frenchman, Pierre de Fermat, was doing much the same thing. Both arrived at comparable results independently. Fermat stressed that any relationship between x and y defined ~ curve. He recast Apollonius' work in ~lgebraic terms, ~iming to restOre some of Apollonius' lost work. Both Descartes and Fermat proposed using a

TH( MAGIC

~ORMU l A

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discoveries, the rather obscure rectification of the scmicubal p~r~bol~ (a method for discovering the length of a curved lin e). FERMAT'S LAST THEO REM

third axis to model three-dimensional curves, but this was not advanced until later in the 17th century. Neither Descartes nor Fermat sought to publicize their work widely. Descartes did publish his, writing in French so that more people could under.~tand it, but he did not explain in great detail and much ofthe work was impenetrable to many readers. It is not entirely clear whether Descartes wanted to exclude pl."ople whom he fclt weren't sufficiently serious or whether he wanted to give his readers the plt;'asure of discovery by making some of the intellectual leaps and hounds themselves, hut either way it did little to help the dissemination of his ideas. Soon, an anonymous introduction was added to his work to help explain it. In 1649 Frans van Schooten published a Latin edi tion with explallat01Y commentary. Fermat was little better at promoting his work than Descartes, bcing a confirmed rccluse who refused to puhli sh . Di ssemination of his ideas during his lifetime was almost exclusively through the mediation of Marin Mersenne; indeed, F ermat puhlished only one of his

,.,

Fermat is most famous now for his so-called 'last' or 'great' theorem. H e noted in the margin of his copy of Diophantus' Aritbnutial that there arc no solutions to the equ~tion

for values of n greater th~n 2. He added, 'I have discovered a truly marvellous proof of this, whi ch, however, the m ~rgin is not large enough to cont~in' - and so the proof was lost ~nd the subsequent search for it taxed mathematici~ns tor more than 300 years. Because the problem is so easy to understand, many people tried to solve it hefore it was finally mastered by the Engli sh mathematician Andrew \Viles in 1993. Wiles proved Fermat's theorem "'~th a method that uses elli ptical curves . H e had tried to solve it as a chi ld, as soon as he h'-'fIrd about it, ~nd continued through his degree course in mathematics. l ong after he h~d given up he re~lized that it was rcl~teJ to his work on curves and remrneJ to tht: problem again . His proof is highly t:omplc.x ~nd m~y well nOt be the s~ m e as that Fennat cl~imed to have found.

T H { WORLD I S N{V[R ENOUG H

The world is never enough Descartes brought al6rcbra and geometry together by defining a point by coordinates and using this to draw graphs from t'quarions. Ll doing rhis, he pro\~ded rhe means for a later development of algebraic geometry into untold new dimensions. Any two-dimension~l sh~pe c~n be representcd by giving the coordinates of irs vertices (corners), e~ch as nl'O numbers. The principle c~n be extended to three dimensions easi ly - by gi vin g three coordinates WI: define a point in threedimensional sp~ce. It is easy to work Out the differences between points, too. Ll a rn·odimensional system, with points (a,b) and (c,d) wt:' can use Pythagoras' theorem to work Out the distance betwet:'n the points. \·Ve im~gine a tri~ngle, with the nvo poinrs defining the ends of the hypotenuse. The Icn!,.'1:h of thi s line - the distance henvcen the points - is then RCc -yaf + (d - b)} We can e..xtend the s~me formula to thrct:' dimcnsions: the distance benveen rhe poinL~ (~,b,c) ~nd (d,e,f) is J((d - af + Ce - b)1 + (fc»). \Vh~t is there to stop us taking this further and dealing with distances in four dimensions, defined by four coordinates? Or 16 dimcnsions? Or 4,;19 dimcnsions? We may have a conceptual objection because \\·c can't visu~lize tour-dimensional

space. stilll css 4'; 1I)-dimensional space, but mathcmatics is not concerned with whcthcr we are comtortablt:' with the concepL \Vhat use is multi-dimensional sp~cc? If we c~n step back from the problems of trying to vi~l.Ialize it as a re~ l -wo rld space, the theoretical space with many dimensions is actually quite useful. \Vc often dr~w graphs that plot nvo variables - specd against time, for e.."(ample, or temper~turc ~gainst growth rate. There arc many situations in the real world in which fur more than nvu variables are involved. Ifwt:' track weather conditions. the 0' performancc of com panics in a stOck market, or the mortality rateS III a population, therc are many variahlt::.~ to take into account. By allocating values for perhaps seven, eight or rune variables to each data point we can envisage, if nt)t visualize, a map in sev!;!n, t:ight or nine dimt:nsions from which we can make measurements and predictions could be made. 1r isn't necessary to draw thc map algebra can take L"'are of the calculations without that - hut the conceptual sp~ce has been sug6'Csred in which rhe 6'1"aph t:'xis~. II

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n'"' "" " ~ORMUtA

THE KOCH SNOWFLAKE

It is even possible to conceive of geometry in fractional dimensions. A fumolL~ model of this is the Koch snowflake, dcvdopecl by the Swedish mathem:nician Niels vOn Koch (1870-1924). The Koch snowflake is an

example of a fr~ctal, ddincd.

Olle

of [he earliest

Draw an equilateral rrianglt!; divide each side inll) three equal ponions. Remove the middle portiun from each side, replacing it with twO sides of another equilateral triangle the same size as the removed section . Keep doin g this. The result is a shape like a snowflake.

The nIT'll' has infinite length. The tOml length increases byonc third at each step and so the length after n Steps is (:Yo)' . It is not a one-dimensiona l lin e, as any portion is unmeasurablc - it is infinitely long. Yet it is nOt enclosing an area, so it is nOt twOdimensional either. It is said to have a fract al dimcn~ion oflog4/log 3", 1.26, brreater than the dimension of a line, but less than the dimension of a t-'l1nre. (A fractal dimension is also called a Hausdorff dimension after one of the founders of modem. topolob'Y') OTHER FRACTALS

lr's po~siblc to carryon doing this an intinitc number of times. The result is a shape m:lt has an area defined by rhe formuhl

when s is the me:l.sure of one side of the original triangle. H owever, the perimeter is infinite - an infinite perimeter enclo~es a finite area. Carrying nut the same operation with a singlc line segment instead of a triangle, the resulting lin e approaches a curve as the line seb'lnents get smaller and small er. The curve is called a Koch cunrC. ,<2

A fractal is a structure in which a pattern is repeated from the large scale to the small scale, so that lookin g more closely at the structure re'lt'als the same or simil:lr figun"s. There arc 1113ny ne3r fractals in Il3turC, including snowfbkes, trees, gJ laxi t!S and blood-vessel networks. Fractals are tOO irregubr to be described using standard Euclidean geometry and generally ha'le a Hausdorff dimension which Jiffers from their normal topological dimension. rrJctals 3re often produced by space-filling algorithms. Th e Sierpin ski triangle is an example. Starting wi th a simple triangle. make three copies of it at one half the size o f the original, and place the copies in the corners of the origin31. Carry on repeating this step ad illjinitum. The resulting pattern is identical at any magnification. It was first described by the Polish m3thematician Waclaw Sicrpin ski (1881-1%9) in 1915 in

T H( WOR LD I S N{V( R ENOUG H

the form of a mathematically defined curve rather than a geometric shape. It has a Hausdorff dimension of log 3/log 2 "" 1.585.

'Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.' Benoit Mandel brot

The best known example of a fractal is the Mandclbrot set, described by the Polish mathematician Benoit Mandelbrot (born I 924). This is the result of drawing a geometric figure of a set of quadratic equations that involve complex numbers. Mandelbrot drew togcther earlier examples of fractals, brave them the name 'fractal' and defined their conditions . He explored their prevalence, both in the natural world and in artificial systems such as economics, and determined that they are a very common model, more frequently found than the simple structures of Euclidean geometry. Fracta ls can often express the 'r ough' quality of the real universe, whereas Euclidean geometry deaL~ with smoothness, which is rarely found in nature. Mandelbrot sugbrcsted a model of the universe in which stars arc fract:llly distributed. This would solve Olbers' paradox without the need for a Big Bang, though it doe... not prL"Clude a Big Bang. (Olber.;' paradox states that the night sJ..Y' is dark when it should he bright, since looking in any direction we should see a star. Although it was described by the Gernlan astronomer H einrich Gibers in 1823, itwas tlrst noted by Kepler.) Although fracta ls generally begin as equations, th ey are best realized as geometric shapes.

MOVING ON

\-\lith fractals, linL'S expand into infinity. V\!orking with the graphs produced by Dt:Scartes and Fermat, even without the added complexity of infinite line lengths, soon produced a need to calculate the areas under L"Urves and the Icngths of (:lITVed line segmcnt.~ . The way of dealing with this and, later, of dealing with fractals - involved looking ru the infinitely small. In the late 17th century, mathematicians finally came to grips with the idea of infinity. Bmoit A/llllddbrol, 'the fiaber offrllctlll g eometry·.

,<3

CHAPTER 6

Grasping the INFINITE

Geometric methods for fmding areas and volumes are easy enough when dealing with polygons and solids with straight edges, but they fall down when confronted with curved areas and volumes as well as with the even more challenging spaces and surfaces of Riemann geometry and fractals. Early approaches to the problems of working with irregular shapes and volumes tried to divide the area or volume into small, regular parts then add together the parts. The essential elements of this method were described by Eudoxus and Archimedes more than 2,000 years ago, but rigorous development and application were not possible while mathematicians still baulkcd at the idea of infinity. The late 17th century finally saw a systematic method for dealing with these problems. As happened with analytic geometry, the new method - calculus - was developed simultaneously and independently by two of the greatest mathematicians of the time.

T he ceaslemnss of the sea if all eiffthly rymbol of the illfillite.

GRASP ING T HE INfiNITE

Coming to terms with infinity Irrational numben;, like it , e and J2, are infinite series. \Ve can go on refining them to ever more decimal places, but the task will never he completed. Both the infinitely large and the infinitely ~mall (the infinitesimal) had worried mathematicians for two millennia. The Greeks disliked irrational numbers to the point, perhaps, of murdering l-lippasus for pro\'ing their existence. Bur III the 17th century mathematicians nude moves to approach and eventually embrace the infinite and the infinitesimal. These concepts and numbers were finally to hecome useful rather than JUSt confounding to eXllL'Ctations and beliefs that were held dear. AN EARLY PRECUR SOR

A/"rbim~des, pictl/red ben' ill nil ollacbnmistir po'Trail

Tbe method Archimedes adopted for ofJ 620,rollfnmt~dp/Ublelllsofillfi"iry(lIIdlilllils calculating the area of a circl e (and so ;l;birh wolild b~ addn:lYI'd I/etlrly 2,000 yetm·lnrer. obtaining a value for J't) depended on drawing polygons inside and outside a circle would converge at that point. As the and calculating their respective areas. These number of sides tends tOwards infinity, the gave upper and lower limits for the area of difference between thc area of the polygons the circle. A greater degree of accuracy \\':lS and the area of the circle tends tOwards zero achieved by using L'Ver-larger numbers of and the limits coincide. sides for the bounding and inscribed polygons. Here Archimedes encountered THE WEIGHT OF PAPER two concepts which would Just as Archimedes had found the volum e of an irregular become hugely important shape by measu ring the volume of water it displaced, so later - that of limit~ and that Galileo discovered a practical solution to th.e problem of of infinity, for the perfect finding the area under a curve. In the absence of geometric area would be given by a and algebraic tools to calculate the area, he would plot his polygon with infinitely many curve, then cut it out and weigh the paper. By comparing sides. A circle may indeed be the weight with the weigh t of a pie(e of paper of known called a polygon with infinite area, he could work out the area of his curve. sides, SO the twO polygons

'"

COM ING TO fiRMS WITH INF IN ITY

The possibility of working out :m area or volume by dividing a fib'1lre into a very large number of very thin slices was not Ilt.-'W to Archimedes. Democritus had rejected it 200 years earlier as he could not work around his objection to the 10brical difficulty that, if the slices are infinitely thin, there is no difference henveen them, so every pyramid becomes a cube. Antiphon developed the technique into the 'method of exhaustion' (though that tenn was nOt used until 1647) and Eudoxus made it rigorous. The principle was to rdate the

area to bediscovered to another area, which was easier to c-ah::ulate, and prove tiNt that the unknown area is not b'Teater than the known area and then that it is nOt smaller than the known area (so the)' arc equal}. It is a non-constructive method of proof, since the answer must be known before the proof can be used. In the 17th c:cnrury, when mathematicians finally became more comfortable thinking about the infinite and the infinitesimal, the method finally came into its own with proper albTCbraic formulation and emerged as integral calculus. This could not happen until analytic bTCometry had been dt.-'Vdoped and a rigorous understanding of limits had emerged. STEPS IN THE RI GHT DIRECTION

A 19rb-cmrmy wgravillg of Delllocrillls wirb straigbr edgl'. a!lI1pflfS alld globe.

1n the second half of the 16th century, the great rush of development In science and meehanit's brought new incentives for calculation with areas, volumes and properties such as velocity. The German scientist Johannes Kepler (1571 - 1630) and the Flemish engineer Simon Stc'~n (sec page 18) both worked on calculating the areas of irregular shapes by dividing them into very thin slices and both approached the problem from a practical point of view with a specific problem in mind. Stcvin used the technique to address the problem of t-alculating the centre of b'Tavity

GRASPING TH E INfiNITE

of a solid object. He inscribed parallelograms inside a triangle to find the median on which the centre of gr:l\~ty would lie. Kepler had an altogether more interesting question . "''hen paying for wine by the barrel, the price was calculated according ro hoW" full the barrel was. BIlt this was measured with 3 dipstick and rook no 3ccount of the vcrtic31 curvature of the barrel. Only 3t the point when the barrel W3S e.X3Ctly full or h3lf-full did the dipstick give 3n accurate measure of the volume, since the barrel is wider in the middle than at the top or bottom. If a barrel was a qU:lrter full (in depth), it cont:lined less than a quarter of a full h:lrrcl and Kepler would be cheated if he p:lid :l quarter the price of a full barreL He proposed cutting the barrel into infinitely thin circular slices :lnd adding

'[The infinite and infinitesimal] transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; imagine what they are when combined. ' Galileo, 1638

'"

up thcir :lreas as a method of calcuhting the true volume. In fact, he also needed to measure the :lreas under curved paths for his work on astronomy - but the wine barrel presents 3 more compelling problem. Galileo stated his intention of writing 3 treatise on the infinite, but if he ever did sO it has not survived. Instead there are passages that relate to calculating are:lS and volumes by reference to infinity and infinitesimals, but Galileo still grappled with the str:lnge logic of these concepts. interesting Perhaps his mOst observation, and one that looked forward to the dt.-'Vclopment of set theory in the 19th century, W:lS tha eaeh integer can be squ:lred, so, since there is an infinite number of integers, there is an infinite number of squares: 'we muSt say that there arc 3S many squares as there arc numbers.' But can the infinite number of squares be brger than the inflllitc lllrrnberof integers? He had come dose to recognizing a feature of infinite sets - that a part of the set can be equal to the whole set. However, he backed away from this condusion, saying inste:ld tha 'the .lttributes "equ:lI", "h"Tearer" and "less" are nOt applicable to infinite, but only to finite quantities.' It W:lS another Julian, Bon3ventura Calvieri (1598--1647), who drew tob>ether the work on infinite division from Archimedes to G:llilw. In 3 text published in 1635 (though conceived six ye3TS earlier), he explained his method of 'indi\~sibles'. He used laborious geometric methods that were soon superseded, but: achie\'ed an impressive re~"Ult. He managed sometlling that was equivalent to the calculus that would be invented only 50 ye:lrs later.

Tln EMUIG£N(( O f CA LCULUS

Calculus IIsed

(Q

be [filled 'fbe CfI/m/lis'

IlIId today ir is if"rm /lsed 10 sowe c01l/plicared problems 'iI'bieb algrbra a/oll e [fI//

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dell101:irb.

BACK TO ACHILLES AND THE TORTOI SE

The emergence of calculus The invention of cait'ulus was one of the great turning points in the histOry of math ematics. It tackled problems that h:ld taxed mathematician.s for 1,000 years and opened doors that no one t.-'\'en knew cxistl;'d bcfore.

The paradox of Zeno, in which Achi ll es em never catch up \\~th thc tOrtoise if the tOrtoise is given a head start (scI;' page 78), can bc expn.:!sscd (bur not solvcd) using cakulus. Using d to represcnt the distance from the starting point that the tOrtoise has travelled, and t to represent the amount of time that has passed, we have a SL'{luence of times :lnd corresponding distanct:S, tI, 11, tJ. . and dl, d!, dl .. .. Th e spced at \cvhich the tOrtoise moves is a function of time and distance, and gives the r:lte of chan ge in the tortoise's position. I-li s speed over an interval between twO times, tI and 11, is given hy:

A BIT ABOUT CALCULUS

Cait'ulus provides a way of measuring rates of change and th e effects of change. ('Calculus' is me Latin namc for a small Stonc used for counting.) It is di \~ded into twO parts which are the inverse of each other: differentiation and integration. The fundamental (hL'Orem of calculus is that applyingdiffcrentiation to an integral returns thc original expression, and \~ce vcrS:l. Both are essentially methods of approximation, bur aim to use li mits that make the error involved (the inaccuracy of the approximation) tend towards zero. The principle is easicr to understand when illustrated by an example.

dl - d l

12 - 11 If after 15 seconds thc tortoise is 3 metres from the starting block and after 10 SL'{:onds he is 4 Illetres from the startin g block, his speed is

4- 3 20-15 or

~

metre per second . A graph of the tOrtoise's movement would be a straight line, as the relationship between disCincc and time is constant:

'"

GRASP ING T HE INfiNITE

If the tortoise moves a t a steady speed, a graph of speed against time would he a straight horizontal line d = 0.2L

, o. ,

0"

E 0.1

,

I• o.,, 0.0

0

,

,

TIme (1) In .e
EARLY DYNAMICS The French bishop Nicholas Oresme discovered c.1361 that the area

under a graph of speed against time is equal to the distance travelled. In his conversion of a p roblem in dynamics to geometry, he was probably the first to use a coordinate system outside cartography.

The distance covered is the area under the graph, speed x time, which is easy to calculate in this instance. The rate of change of speed (acceleration) is given by the slope. In this case, the line is flat, as there is no acceleration - the tortoise goes at uniform speed . Now assume the tortoise has been given an electric scooter. Lnstead of a uniform speed, he now accelerates until the scooter reaches its tOp speed. The first part of the speed graph looks like this:

/

i ~-------------~ / ~~

. ~----------~.~~

J ~----~yC~----~

----TIme (t) In serond.

The situ3tion is altogether more complex. 10 find the distance the ronoise has rovered, we need the are3 under the 6'Taph, but this is not easy to calculate. To find the accelerntion at any pa.rticular inst:Inr, we need to mea~urc the slope of the curve at that point. The first is solved by integral caklllus ;md the semnd by differential caklllus. INTEGRATION

lntegration finds the area under the curve by drawing a series of infinitesim311y thin rectangles under the curve 3nd adding tOgether their areas. h's very similar to Kepler's slices of wine barrel (sec page 148) or the slices of pyramid that troubled Democrirus (see pa6TC 79). \Ve can make a rough approximation of 150

Tln EM Ul G£NCE Of CA LCULUS

the area under the curve by drawing reetangles so that th e t:urve passes through the midpoint of the top of (;!:teh rectangle:

,..--"

---

/ /'

The line cutS off part of the top uf eaeh rectangle to the left, hut there is a space under the line tu the right. If the spare bit of rectangle were flipped over, it wuuld fit the space pretty well. The smaller the rectangles, the better the fit to the curve.

DIFFERENTIATION

The average acceleration over an interval of time (on the graph of rime against speed) is given by the slope of a straight line drawn between the start and e nd pointS of the intetv'JI. This line is call ed a secant. The acceleration at an instant is given by the slope of the curve at thftt instant (or of a tangent to the curve). Differential calculus ptO\'ides ft way of approximating the slopc of the curve by assuming a very short time interv:l.land calculating the slope of the secant for that interv:l.l. (The very short interval is called L'l.t, 'ddm-t', the Greek capital delta bein g used to show a small quantity.)

/ secant

~

./

~

7'"

-'

------

III rbis ("XI/mph, rhe Stralll

/KIll/rf (!II rbe

is d,.ITi1 ·11 bawl'rli tiliO

lilll!.

Th e calculated area (the sum of all th e re("tangles) approaches the true area under th e L"Urve as the number of rectangles approaehes infinity. This area is the integral of the function f (t). Th e expression for th e integral is written as hi tbk t":wmplc, rllt fflllgwl flJl/cbfS rbe l illt 1/

where ft and h are the limits we arc working within (the upper and lower values of t that bound the area) and 'dt' means a very small change in time.

III

poml.

The time 1n is a vcry short interval. Making 1'n sma lkr and s maller producc .~ a more and more accurate result, though it 151

GRASP IN G THE IN fttHTE

will never bc quite the same as the slope of the curve because we can't set .6t to zero. However, as .6.t approaches zero, the line appr().1ches a perfect match. This introduces the concept of the limit: the limit of the function "pprotlcbcJ the required va lue (the acceleration at an imtant) as L'1t approaches zero. This is the process of differcntiation. NEAR MISSES

SEEING THE WAY FORWARD

With the development of analytic geometry, it bet:ame possible to describe movement algebraically. The Ancient Greek~ had introduccd the idea of a curve as the path (locus) of a moving point. Algclmic geometry pro \~ded a tool for de~ribing that locus in the form of~n equation, gener.:dizing about the shapes of the cw·ves produced by different types of motion and identifYing patterns that had predictive value. For example, Calvieri noticed that the area tmder the parabola defined by y '" x\ between 0 to " on the x-axis is oJ/ 3. Similarly, for the CUITC y '" Xl, the corresponding- area is d!/4. [t was nOt then difficult to guess that the general formula for the area under a CUlve y '" Xfi is an-'/(II + I).

Fermat's work Oil analytic geometry includes a rl'latiollship which is fundamental to the theory of calculus. Fermat dealt with finding tanb't!lltS to CUlVes and areas under CUlves. The expressions he derived had an ulVerse relationship, yet it seems to have escaped his attention, for there i~ no evidence that he pur.~ u ed it or tried to e'\llbin it. l EIBNIZ AND NEWTON Blaise PaSt'al is another who could easily The fundamentals of calculus - both have taken a final .~tep and di.~covered differentiation ~nd integration - were calculus. Pascal's interestS in mathemati cs discovered around 1670 independently by were varied and he flitted from topic_to topic. H e also gave up mathematics after he SEKI KOWA OR SEKI TAKAKAZU (1637/42-1708) underwent a religious Seki Kowa was born in Japan in either 1637 or 1642. He developed a new notation for expressing equati ons up to ecStasy, and he died youngtwo further factors that the fifth degree, using kon;; characters for variables and unknowns. He discovered discriminants, which led him to contributed to rob him of the prize that lllight have some re.\ults in differential calculus at around the same time been his. Pascal came so as Newton and l eibnitz discovered them in Europe. There close ro discovering calculus is no known communication between the European and while working on an Japanese mathematidans. (A discriminant is an expression that shows a relationship between the coeffidents of a integration of the sine polynomial equation. For example, lor the quadrati c ax 2+ function that L eibniz later wrote that it was reading bx + C, the determinant is b2 - 4ac. Whether the discriminant is positive, negative or zero give information Pascal\ work that signpostcd calculus for him. about the nature of the equation and its roots.)

152

TH E EMUlG£Nn Of CA LCULUS

SIR ISAAC NEWTON <1642-1 727) Isaac Newton was born prematurely on

at Cambridge and could dedicate himself to

Christmas Day in 1642 and was so Sickly

his scientific and mathematical work. He

that he was administered the last rites. His

discovered that white light is made up of a

father had d ied before his birth and, when

spectrum of coloured light, l ormulated his

he was three, his mother left him to be

laws of motion which underpin classical

cared for by his grandmother while she

mechanics and defined a fo rce which directs

went to live with her new husband.

the motion of falling bodies, gravity. His of

Cambridge, where he studied

his

discoveries,

Phi!osophioe Natllmlis Principia

the classical science required

is

by the curriculum but also

perhaps

most important

read the new works of

scientific publication of

Descartes

,nd th' chemist Robert Boyle.

all time.

When the university was

mystical

closed

alchemy. He also had a

for

because

two of

Newton was attracted to

years plague,

psychotic

He developed his ideas of which

and

intolerance of

other scholars disagreeing

Newton worked at calculus,

matters

he

with him. He spent years

called

isolation,

'fluxions', and a good deal of scientific work, but at this point none of his work was published. After the plague, Newton became a professor

any Nr.NOIi

-was rlU' firsr

/0

Tt'lIliu rbar whitt' light could

I", split iura

rbe CQIQlIl"f

the sperfTtnl,.

the Engli~h ~t'ientist and mathematician Isaac: Newron and the German polymath Gottfried Leibniz (sec page 154). VVhat both m~n did was to discover a method for cak-ulating the tangent of a curve ata sp("cified point on the curve , briven only the equation defining the L-urve. The slope of the tan g"Cnt (which defines the line geometrically) shows the rate of change of

of

possible

shunning source

of

conflict. Professional disputes sometimes prompted him to burst into tantrums that took the scientific world by storm.

the function (such as the speed of a moving hody at a particular instlnt). Borh men also realized that integration is thl' inverse of this process of differentiation - that integrating the rcsult of differentiation leads back to the original function and \~cc versa . This revealed a surprising relationship between total values and rates of chang·l.'.

GRASP ING THE INfttHTE

one of the benefitS of his (~SC()vcri~ in calculus

The notation

was the ability m tackle power series - infinite sums of multiple power.-; of x, such as

for the expression giving the area under m.. curve y = x1 was adopted by Leihniz hecau~c hI! Saw it as the sum (indicated by thr clollbratccl 5, J) of the c.\vn:ssioll, in this c.1.~C

He de\'Clopcd a calculus of power series,

the x-axis (d,,). Newton and Lcibniz stressed

showing how to differentiate, integrate and invert them. Lcibniz was morc intere;tcd in the properties of changing systems and

diffcrcnt aspects of calculus and had quite

in summing infinircsimals. His work [reared

separate intentions in lLSing it. For Nl'Wtol1,

continuOllS quantities as though they were

x\ divided into infinitely small segmenlS along

GOTTFRIED WILHELM lEIBNIZ (1646-171 6)

Gottfried leibniz was largely self taught as a child, then entered the university of

branch of science called dynamics which is concemed with the movement of object.'i

leipzig to study law. The university refused him a doc torate because he was too young and he consequently left the city, never to return. He was awarded his immediately at NLirnberg. leibniz moved to Paris,

and the forces acting upon them, then worked in the 1670s in practical mechanics and engineering, designing and many kinds of machinery. He is considered the originator of geology after the observations he

and most of his writings are in French or latin. He worked in the service of several noble families during his lifetime, pursuing his interests in

made at the mines in the Harz mountains. He was the fi rst to propose that the Earth had initially perfected binary notation

which is at the heart of computer science. HiS' philosophy

mathematics, philosophy and many branches of science ~me time. He developed a calculating machine which he presented to the Royal Society on a visit to l ondon. He developed the

IS'

Ldblliz. a'rotl" le1IS oflhollSdllds of dXIIJ//l7ItS 011

a 7.l .idl" rdllgr of

SIIljocts {llId IIlw:h ofllis work rmlflills IlI/pllMishul to this day.

was optimistic he believed that the world represented the best of all possible worlds that God could have created.

THE EM ER G ENCE O f CA LCULUS

Zt'IIQ} PlIl¥ldl).Xl'f fellt/'/' (lI'IHlIId dividillg up COl/nll/lOflS U/m~1I/1'IIr

inrl)

dispute over priority and the merits of each mans methods had long-la.~ting repercussions, isolating British maths until the 19th cenrury.

lill] 1Il01lll'llts - a pmhlrm glossed liver by mlt:-lIllIs.

THE IMPACT OF CALCULUS

In srudying falling bodjcs, Galileo (who died the year Newtoll was horn) needed to calculate the speed of an object at a particular instant in time. For this type of problem, differential calculus is the perfect tool. Since the time of NewtOn and Leibniz, tbe heart of Zeno's paradoxes. calculus has been applied to countless NewtOn fuiled to publish any of his problems in lllLochanics, ph~ics, astronomy, findings rdating to calculus until \693, nine economics, social science and many other years after l.eihniz published. The emlJing fields, revolutionizing ph ysics and briving new impetus to the fiuther development of mathematical tt'Chniques. CalculllS bas spawned a whole branch of mathematics, called analysis, which deal~ with continuous change. In summing a la r!,'t! set of small quantities, integration is useful in as problems such determining the distance travelled by a hody moving at 'l:Irying speed or calculating the total fuel consumptioll of :l vehicle. Differentiation can be used in such varied problems as modelling disease epidemics and determining the path an aircraft needs to take CalclllllS has widl'-nll/gillg pmuiCilI appliCflriolis fIJI" fbI' modrm world. Ir to avoid co ll iding with another. rom bl'lpr tv I'ttablirb safr jligbtpathr fin· ain:raft ill 0/11' bllsy skin discrete, a logical flaw tbat be and otbers overlooked tbough it was an issue so old tbar it was tbe difficulty at 1..J..........w

155

GRASP IN G TH E INfiNITE

Calculus and beyond A-lathematicians had never been comfortable with the concept of infinity and calculus highlighted this concern. The Anglican bishop George Berkel ey (1685- 1753) mad!;! a well-argued rcfutation of calculus and th.is prompted a productive debate which led t o the rigorous definition of limitS and infinity. This ultimately benefited the development of calculus and enabled analysis to grow our of it. Berkeley's objection was not fully answered for morc than a hundred years. A further century la ter, the 10brician Abraham Robinson (i 918-7 4) finally showed that the idea of the infinitesimal is lo gicall y consistent and that infinitcsimals can be considered a kind of number. Analysis deals with continuous change and with processes that have emerged from .~tudying it, such as limitS, differentiation and integration. In particular, differentiation is one of the principal t001.~ of analysis. Relating rates of change to present values, it is possible - at least in thL'O'1' - to predict future behaviours. This puts analysis at th!;! heart of many modelling and predictive activities, from weather forecasting to epidemiology, from astronomy to fluid mechanks. USING INFINITY A~

BirbQP Berkeley T<'spandffi ro -;rblll be SiTlJ.lliS IIx

lIIull"l7//illillg Qfl"f/igio// by /If/tural pbilaropby (jJf wbirb co/mitIS ,~'ns II key p1l11) ill

The Analyst (1734).

irrational lin e segment.'), the procedure never terminated, becoming an infinite process. Euclid used this property to tCSt irrationality. A similar willingness to ignore th e logical diffinllries of infinity and infinitcsimals (or lack of rigour in applying such logic) led NewtOn and Leibniz to fud ge calculus. They treated reality as both discrete and continuous at the same time, depending on quantities that were so tiny they could wink out of exiStence when convenient, or could be u:.;ed to put a Stop to

long ago as th!;! time of Ancient Greece, infinity had been employed in mathematics, despite the 'And what are these Fluxions? .. They are neither finite difficulty of such a concept. Euclid used an algorithm to quantities, nor quantities infinitely small, nor yet nothing. find the greatest common May we not call them the ghosts of departed quantities?' divisor of IT pair 0 f numbers. George Berkeley, 1734 If he applied it to a pair of

1S6

CALCULUS AND BEYON~

MADHAVA OF SANGAMAGRAMA

( 1350-1 425) The Indian mathematician Madhava

infinity. However, the results achieved by calculus, in SO many fidd s, were so valuabl e and impressive that the inconsistencies at the heart of calculus wcre nOt immediately addrc5Scd.

of Sangamagrama is considered by many to be the earliest originator of

AFTER NEWTON AND LEIBNIZ

analysis as a method. He founded

The disagrecment bctwccn Newton and Leibniz over priority had the result that subscqucnt work with calculus was polarizcd. As Lcibni z had come up with the usable notation, his was carried forward and Continental Europl! bccame the arena in which developments in calculus were played out. J\Tewton's interest III geometry obscured the cak"Ulus hl! was using and his aversion to his rival's notation mcant that few people in Britain followed him, despite admiring the results he achit.-'ved. In thl! decades after Lcibniz's publication, thl! Swiss brothers Jakob (1654-170H) and Johann Bcrnoulli (1667-1748) dominatcd calculus, along with Leibniz himself Thc Bernoullis dl!veloped the rules for differentiation, the intc~,'Tation of rational functions, the theory of

the Kerala school of mathematics and astronomy, which flourished between

the

14th

and

16th

centuries. He was the first to accept limits tending to infinity and to define infinite series. He discovered the

infinite

series

trigonometric developed

of

functions

several

the and

methods for

calculating the circumference of a circle and two methods for finding n. He also made steps towards the development of both integral and differential calculus. His writings have

not

survived,

achievements reputation only.

are

so

his

known

by

Jllkob (lift) 111/{1 Jobfllll/ (rigbt) Hemal/iii Clnne fivl/J II

f,dellfed fllmily of frllde!;mell IIIIlI !;dJoIa/"s (/ml11."fr?

Ollly two ofeigbr 1I1f1rbemaricilll/f ill tb eJrtmily! 157

GRASP ING TH E INfiNITE

elementary functions. applications to mechanics. and the geometry of lllrVes - in fact, most of the fundamentals of classical calculus with the exception of the power series that most interested Newton. Thl' Bernoulli brothl'r.; even used call'ulus to demonstrate NewtOn's mvn inverse square rule applied to gm\~ty in an elliptical orbit, which Nl'wton had nOt explained well. []l the mid-19th century, Riemann refined the method for calculating an integral, suggesring comparing two sets of thin slices. one inscribed and one circumscribcd. As the two values approarn each other (with thinner and thinner slices) the o·ue imeb'Tal is found.

In,crIb..d ,lie",

DEALING WITH DILEMMAS

H owever useful calculus was, the inconsistencies at its heart would not go away and sooner or later thL"}' had to he dealt with. Dealing with these resulted in the development of analysis - not itsclf a computational technique, hut a sound logical basis for usin g cakulus. Two dilemmas, highlighted by the (Titics of early apillications of calculus, arise as soon as we Start to think more rigurously about differentiation than did NewtOn and Leilmiz. One is L'VoeativeJy eaptured in Berkeley's 'ghosts of departed quantities'; the other might be ca lled the 'brhost of a moment' . The rea l world is better 158

characterized hy the model o f a con tinuum than by a set of discrete parts. (This r('Calls the disrinction we made at th e very start between counting and mea.'>uring, between arithmetic and gl'ometry, as wl'll as the problems of the continuous and thl' discrete at the heart of Zcno's paradoxe.'i.) Think about any system that involvcs continuou~ change - water flowin g over a dam or air over an aircraft wing, for example. As local conditions vary, the rate of flow is not conStant. l\ 'l easuring it at any moment involves some kind of approxim:nion or averaging as the time interval could always he made small er. Only by freezing time could we take an accurate measurement. But fl ow depends on time. so if we freeze time the flow is zero. 1t is not only rime that can he endlessly subdivided. For c.xample, as tl'mperature chanf,'"Cs from 1" to 3", it must go through aJl infinite numher of imermnliate stages; even 1" and 3° thcmselvcs are infinite decimals, with an infinin.· numher of zeroes aftcr the decimal point. In modelling continuous change, we must deal with these fleering values - and th ey arc necessarily infinite decimals. The concepts and deductive stl1.lctures hehind infinite quantities came to preoccupy mathematicians working with calculus as they struggled to develop ri gour. For analysis to become a rigorous and dependable tool, mathematicians first needed to tind some way of dealing with thc vagueness of these ghosts of quantities and moments. The Gernlan mathematician Karl Weierstrass (1815- 97) was the first to produce a complct~ly satisfactory definition of the limit o f a seri es. H e became known as

CAlCUlUS ANO BHONO

the father of modern analysis for devising a test fur the convergence of series and fur his work on functions. Using the sequence

INFINITE SERIE S

An infinite series is a series with an infinite number of terms. For example,

\Veitmtrass would say that all we need to do is pick the level of error (or approximation) that is acceptable (~) and then continue with the series until we reach a term lIn which is smaller than the error, then we C3 n say th3t the series has reached its limit. Thi s the need for nebulous removes infinircsimals and gives a real numher which satisfies the requirements. Also, although the series 3pproaches its limit, it docs not have to reach it for Weierstrass's condition to be met. Now, the margin of approximation could he stHed 3nd the

is an infinite series, with each term being half of the last. The limit of this series - t he number that would be reached if we could get to the end of the infinite number of terms - is 1. Because the series reaches a definite limit it is said to converge. Other series do not converge, such as

This series diverges as it never settles to a limit. Some convergent series can be ambiguous: 0+1-1+1-1+

oscillates between 1 and O.

degree of accur3CY quantified . There was no need to worry about quantities that h3d to disappear from existence - analysis was put on to a logical footing.

Gt'1"mIl1l1l1arhemllticillll KRrllf/(ierstll1J:f ,ray

rom:lTlled 7.:.'itb elimil/tltillg illCl)/lsisrmries ill edirt/IIIS (111£1 defillil/g rbe limir of II series.

CALCULUS BECOMES ALGE.BRAIC Durin g the 18th century, calculus moved 3W3Y fi-om itS geometric roots in the work of NewtOn and Leib niz and became increasin gly algcbr3ic. Geometric curves became less important 3nd algebraic functions moved to centre sta ge. Soon, complex numbers moved in on the scene. Differentiation offers a useful tool for 159

GRASP ING T HE INf IN ITE

finding local m3ximum and minimum values benvecn upper and lower limits. If we draw a curve of a function, the slope approaching a maximum point flattens out; the curve is momentarily tlat (has a slope of 0) at the maximum point, then it curves downward again, its slope reversing. As rate of change is equivalent to a tangent draWll to the curvc, it is ea!>y to Spot maximulll or minimum points - it is those pinel's at which the curve has a slope of zero and tbcn reverses its slope. This knowlcdb'C makes it possible to find the changes of direction all local maXlllllllll and mlllllllUlll points benvecll bowldarie5 without drawing the graph. Where the function differentiates to zero, the tangent to the curve is parallel tc) the axi~. Differentiation is also useful for working with :dl of the many phenomena which cxhibit exponenti:tl growtb or decay - such as population b'TUwth, or radioactive decay. By examining the rate of change at given moments, it's po~iblc to extrapolatc to find valucs for the furure (or past). WAVE FUNCTION S

The ability of calculus detcrmine ma.Xllna nlllllllla has made especially valuable for working with all kinds of waveform, from acoustics to optics, from clectromagnetism to seismic activity. The earlie;t work in this field was carried our by the English mathematician Bl"Ook T.1ylor (1685-1731) who produced 160

a mathematical description of the vibrational frcquent), of a violin string in 17 14. The French mathematician Jean Le Rond d'AJcmhcn (1717-83) refined the model in 1746 to take account of more conditions and limit'i, and of v,;uiation in some pmpenies alon g the length of the stl"ing. His dClllonStl"ation had twO wavcfonllS travelling in different dircctions. The Scottish physicistJamcs ClerkMa.nveil (1831-79) found the same thrccdimensional wave when exploring electromagnctism. It enabled him to predict the cxisrence of radio waves. Radio, television and radar are all dL·vdopments dependent on the early analyric work on the waveforms of musical inso·unlenl5. Further work on thc propagation of sound hy th e Swiss mathematician Leonhard Eulcr found a trigonomctric scries:lt the heart of the problem (1748), In 1822 the French mathematician J()seph Fouricr (1768-1 ~30) also found a trigonometric series ddining the way heat spreads along a mctal rod. From this hc developed Fourier analysis, which enabled him ro find the values necdcd to model heat spread for any initial temperature distribution. Fo uricr analysis is used to analyse complex, composite waveforms, brcaking them do wn into their component..:; and values. For instance, an audio signal can be amlyzed into Nidmrrmnl 'Drrfty' at «hool ill Edillburgb. .1f111l1'f Nffl,l"U.'tll p/'odllml l];/JI ·k

to rivrr/lllly

gt"f.'ill

pbyridsr.

CAlCUlUS AN O BUON O

LEONHARD EULER ( 170 7-83)

The Swiss mathematician and physicist

than any other mathematician has ever done,

Leonhard Euler spent most of his life in

his work filling 60 to 80 volu mes. He worked in

Germany and Russia. He published more

many fields, making significant breakthroughs not just in analysis, but in graph theory, number theory, calculus, logic and several branches of physics.blished much of the notation used now, including {(x) for a function of x, the notation fo r the trigonometric functions, the use of the symbols e (e is sometimes i and

called

Euler's

number)

and

L (for summations). He also popularized

(but did not originate) the use of the Greek letter

Jr.

His most startling discovery was

Euler's identity. 1n e + 1 '" O.

iL~ different frequenci es and amplitudes. Although his methods were not rigo rous, they were later refined and are, in essence, used today - ror compre~ing sound into downloadable At1P3s, tor example.

Too HARD

Some problems proved intractabl(! even with th e usc of calculus. The movement of the planeL~ in the solar syStem, for t;!XOlmple, is toO complex to be accountcd for by straigh tforward series. The field of dynamic system th eory has developed to ta ckle such problems. Esscntially, local data drawn from particular sit(!s within a much larger tleld arc analyzed and reslllts from these arc applied to known global properties of

Tbl' 'Kdlligwcrg bridges' problem

';l}//S

solved by

Elller ill 1736. Ir ash ;J.· babn· il is porsible 10 crors ellcb ofrbl'si'lJt7I bridges ill Kdl/igsbrrg Ollly OIlU, 1l'lIIl7Iillg to

(be Hllrlillg paim. Ettl,.r pl"Ovl'd rbllr ir

is 11(1(, dl'fillillg rbl' Elflniall prllb ill rbe PI"f)Ct'ss (II pillb (bar follU"J.JS etUb edgr, bllt ollly Ollce). His proof is rbf jirst Ibeol"e," ofgl"flpb Ibl'o'1. 161

GRASP IN G T HE INfiNITE

Kil/g Orefl/" /1 wbo affired a

GREENHOUSE GASES

Fourier was the first person to suggest, in 1827, that gases in the atmosphere may lead to increasing temperature on a planet - the greenhouse effect.

the whol e system . Today, computers analyze, approximate and ass(;!Ss solutions created in this way. Dynamic systems theory was fi rst developed by Henri Poincare (1854-1911) for a competition. [n 1885, King Oscar II of Sweden and Nonvay offered a prize to detcmlin(;! the stability of the solar ~ystem - saying whether it would continue in much the same state or whether, for e.'l:ample, a planet could fly off Oil a rob'lIt! journey of its own , perh~ps colliding with the sun. Hl'llri Pail/cOli v.}as blessM witb a formidable

1f1171/oI)

f/lld was ablf to master mfllly disaplillfs.

162

prize fol· detfl""II/illillg

b(r.l.1

Stftble fbe mla/" rySWII 'I:.'as.

NewtOn had used the inverse square law of gravitation to demonstrate the elliptical orbit of planets th~t Kepler had noticed, but he also fOlUld that the syStem was too complex to calculate if more thall two bodies were involved. The king now wanted a solution involving nine bodies - the sun and the eight planets known at the time. Poincare's solution did not, in fact, deal with nine bodies. H e restricted himself to three and even then assumed that one had negligible mass (and so negligibl e gravitation~l effect). H e modelled ~ sample of what may happen in a limited arl'a - where the path of 3 planet intersected with this are3 - and extr3polared the rate of change to come up with a prediction for the stability of the whole system. Although Poinc3f(! WOll the prize for his partial solution, he noticed a mistake in his solution and spent more than the prize money III reprinting his solution. From the end of the 1Hth century, mathematicians were more willing to accept

CALCULUS ANO BUONO

, Polnu.r~

.octlon

The hllle dire rl'[ln'fi'llts

(11/

tina

ill <:..birb dnrfl

abollt p/flllftflry 'II1ovmlelllS fin' rol/uud. Tb.>

poil/ty of imnwcrioll wit/; rbe

trajectory

discovered that many functions cou ld not he integrated, or hehaved in a The blind Belgian physicist Joseph Plateau (1801-83) studied the fi lms and bubbles created by soap solution. bizarre way if integrated. A~ Soap solution forms minimal surfaces- the minimal surface a consequence, integration area that can cover a space. Minimal surface mathematics was redefined by the French mathematician H enri-Leon is a productive area of research. The West German pavilion Lebesgue (1875- 1941 ) around at the Expo 1967 World's Fair in Montreal, designed by Frei Otto, was based on minimal surfa ce studies of soap films. 19CX1. Instead of taking thin slices of the b'nph vertically beneath the curve, Lebesgue complex numbers and Gauss began suggested taking thin slices horizontally. applying the principles of analysis to them This gn.'arly increased the usefulness of in 1811. Analysis using cumplex numbers - inreb'Tal calculus as it could now be used complex analysis - is possible because with discontinuous functions. It expanded com pie-x numb ers arc deemed to follow the possible applications of Fourier analysis. many of the same laws as real numbers. There are very many different brJncht.'s Modern analysis differs in many regards and applications of analysis and they spi ll from ea rly analysis . Mathematicians over into all areas of scienc£'. SOAP BUBBLES AND ARCHITECTURE

TradiUnoollntegrollon. taking """ I
.II<~ '

163

GRASP I NG T H( INFI N IT(

The Lorl'llZ attraclm;

(J

model of the 1I1iJVe'1llmtolit'l·

CHAOS THEORY

Poincan:'s method IS the foundation of challges ffi'lll rnmr,m, t};e01.lfmll ryrrelll jrA"1llf a paTteI7/. chaos theory, which developed greatly during the 20th century. It is a method which enab les useful data to be drawn from apparently LOST GLORY The Japanese researcher Yoshisuke Ueda discovered a random systems. Computers chaotic system in the same year as lorenz (1961), but have made the study of chaos and chaotic systems possible. The his supervisor did not believe in chaos and would not let him publish his findings until 1970. work involves carrying out the same calculations again and ag'ain time of (I chaotic syrrml. (here, wCf/thn). Even tbollgh

,6<

CA LCULUS ANO BHOND

of the selJuence. He found the resulting weather A popular encapsulation of chaos theory is the idea that the prediction was radically diftcrent from the one he movement of a butterfly's wings may cause or prevent a had obtained the first time. tornado, as the small local effect is amplififfi as it triggers or The reason, he found, was prevents other changes in the atmosphere. It is likely that the idea come~ trom a science fie lion story by Ray Bradbury, that his printout rounded fi!,'llres to thrce digiL~ (from 'A Sound of Thunder' (1952), in which a time traveller six) and this Slmll error was causes subtle changes in human history by inadvertently killing a butterfly during a visit to Jurassic limes. enough to produce a hugely different result. Chaos theory is applied with different valucs - this would be to many areas of science, including physics, virtually impossible without computers. medicine, tectonics, computing, mldics of A system which appears to be chaotic (in lasers and electricity. It also has applications the usual sense of the word) in faCt follows oUL~ide science in areas as diverse as strict rules. HowL'Ver, the system is SO econom.il"S. psychology and soci()lob'Y' scnsitive to tiny changes in the starcing point of its variables tha t ie; hehaviour is, to MOVING ON all intt:llL~ and purposes, unpredictable. The tl;!chniques of calculus and analysis arc \¥eather forecasting is notoriously difficult useful in examining trends in dara of many beyond the very short range because a vcry kinds. Yet before dara can be examined, large number of factors can affect the they mUSt hI;! collected and processed. weather and the outcome is very sensitive to Surprising as it .~eems, the idea of collecting starting condicions. It IS effectively data on which to base deci.~ions - and impossible to produce :lll accurate forecast making that data collection rigorous and beyond a few days . fair - is a relatively rl;!cent dt..'Veiopmem. It was while working on weather The branch of mathematit..os which has forecasting in 1961 that Edward Lorenz come to be known as statistics has grown up (b.I917) made a significant discovery in over only the last 400 years. Intercstingl~', chaos theory. He wanted to repeat a weather its emergence coincided with the modelling operatioll, but to save time he development of calt-ulus, which has become input figures from a printout he had run an important tool m statistics and earlier, starting his model from the middle probability studies. THE BUTTERflY EFFECT

'"

CHAPTER 7

NUMBERS

at work and play

Calculus and analysis arc a long way from the everyday encounters with numbers that many of us have. Even though most of the science we come into contact with, most of the products we usc and much of the world around us depends on activities in higher mathematics, our everyday encounters arc more likely to lie with statistics and probability. In finance, gambling, games, the economy and many other spheres, numbers as predictors and risk assessors help us to make decisions - whether about buying a lottery ticket, taking out life insurance or flying on a plane.

- -- - - -

Nllmbers, alld fh( possibilirier they o./frr, are with liS 1/11 rh( time.

NUMBERS AT WOR K AN D PlAY

Cheer up, it may never ha p pen Humankind has played gamt!S of chance for millennia. This is playing with numbers; the fall of the dice or roll of the roulette wheel arc effectively random, and winning at these games demands either large slices of luck or great proficiency in calculating probabilities and risks. Vcry simple probabilities are ea~y to see if we toss a coin, there is a 1 in 2 chance that it will land hcads and the same chance that it will land tails. If we toss a coin a large number of times, we will probably get about as many heads as t::J.ils. This was first noted by the Swiss mathematicianJakoh Bernoulli in a treatise published posthumously in 1713. H e did acknowledge that the result is so patently ohvious that l.'ven a very stupid person would notice it, but he is sti ll given credit for it as he spent 20 ycars developing a rigorous demonstration of why it is true. He called it his Golden Theorem, but it is generally known now as the Law of Large Numbers. Casinos depend on it; although an individual gambler may have a I1.m of good luck, over time a casino can expect to keep ;.3 per cent of all the money bet on a roulette wheel.

Altbollgb j{ is pruYible to 'bear rbe !milk' ovn' a rb01"{ pn'iod of rime, rbe cilSilio is fai,.,y cent/ill to 11,ill ill tbeiollg1'lUi.

Between the obvious probabilities and the Law of Large Numbers, problems of probability become more complex. "Vhat are the chances of getting tails exactly five times in a row? If we throw three dice, what is the chance of getting three sixes? "Ve need to do a littl e work with probability to be abl c to L-alculate tht!.~e; the chance of getting tails fi ve timt!S in a row is 1 in 2' = 1 in 31; the chance of throwing three sixes is 1 in 6) =1 in 116. DICE AND CHAOS For most of tlle many Although the fall of dice or spin of a roulette wheel are th ousands of years th:n people have been playing effectively random, they are actually determined events. games of chance, they had The starting position and all prevailing conditions, no way of working Out thc including the direction and force of the throw, the surface probabilities of different of the table and the exact featu res of the dice will determine the outcome. However, there are too many Outcomes beyond the few that are very obvious or for conditions, and their measurement is too difficult, for the which it is easy to enumerate outcome to be modelled or calculated. the possibilitit!s.

168

CHHR UP. IT MAV NtV[R H APP£N

A GAME

OF CHANCE

Probability - the chance or likelihood of an evcnt happening - entered mathematit..os in the 17th ccnrury and it was in the context of a game of chance. Although Gcrolamo Cartlano had wriUt'n on games of chance in the 1520s (sec page 132- 3), his work was not published until 1633 50 he lost Out m Fermat and Pascal. In a series of letters, the pair discussed a problem proposed by a gambler, the Chevalier de Mere: Two players are playillg a gll1lle ofpllre cballa Oil 1vbich Mcb bas bet 32 coillS. 77)('. first to will three times ill a row claims tbe pot. HfToJ)r·i.'l'1; the;,. game is illffnwptfd after Ollly three gaJJus. Phlyrr A hm WOII t7J.lia Illid player B has 001/ once. How C/1/I they divide the pot fairly? The two mathematicians both came up with a 3:1 distribution in favuur of player A, though they arrived at the .solution hy different methods. Fermat gave his answer in terms of probabilities. Two more games is the mOSt that would bt' needed Il) decide the match, and there are four possible outcomes AA, AE, BA, BB. Only the last would make B the overall winne]'. so he has a one in four chance and should receive a quarter of the winnings . Pascal propo~d a solution based on expectation. Assuming B wins the next round, each player would have ~n equal claim to 32 coins. Player A should receive 32 coins anyw~y as he definitely has twO wins. The chance of B winning this next game is 50 per cent so he should h~ve half of the remaining 32 coins. Pl~yer A also has a 50 per cent chance of winning and should

have the laSt 16 coins. Again, pla}rer A receives 48 and player B receives 16 coins. Pascal's strategy w~s the one which won approv~l among mathematicians de~ling with chance evcnts. All'S FAIR.•. Although gaml!S of ch,mce continued to interest mathcm~ticians, another impetus was the lebTfiI idea of a fair contr~ct. 1n a f~ir cOntr~ct, the parties have equal expectations. This was an important concept bCc:IUSI! fair expl!ctations were at the heart ()f the justification for mOlll!y-lending. Christian doctrine b~ns lL~Ul"y - profiting fn)m lending monl!y. To get around the difficulty, lenders were considered to he investors who put in money at their own risk and could fairly cxpect to share in the profits. Until the 17th century, the rates for loans and annuities were fixed with no regard for any m~themati('""al concept of risk or how it might be c~ Il"Ulated. The first trc~tise on calculating risk ~ppl!ared in thl! Netherlands in 1671, produced by Jan de \Vit after consulting Christia~n H uYb't'ns. At the time, anllu iti es were sold by th e state to r~i.,e money, uften to finance wars. The retum Jilll dr /fItt ITt//iull

tbat 11rk sbollld govt"1"II 1"afnofn711I"rI.

'69

NUMBERS AT WOR K AN D PUY

was Ollways a seventh of the vOllue of the annuity, paid eOlch year unril the holder's death. The age or health of the holder was nOt taken into aeL'()UnL Clearly, without 3ny assessment of how long the state m3Y h3\'e to pay the 3111luity holder, this could be cxpcnsive. Even though de \Vir could sec the fhws in the system, there were 3t the time no dat3 on mortality 3t different ages, so little could be done to improve the syStem - ami little W3S done. rt was nOt until 1762 that 3n insurance company III London, Equitable, began to price iL~ policies on the basis of calculated risk, or probability. GOD EXISTS - PROBABLY

Prob3bility did nOt become 3n exact mathematiC31 concept until the 18th cenrury, and was sti ll gcnerally considered an indisrinct ide3 based on common sensc into the 19th century. The French mathematici3n Pierre-Simon de Laplace (sec p3ge 174) refcrred to probability as 'good sense reduced to calculation'. Interestingly, a link between chance and religion hecame a central interestofn3tural theology in the 18th century. John Arbuthnot (1667- 1735) produced evidcnce that God defmitdy exists from a study of christening statistil's in London hetween 1629 and 1710. He showed tim there were slightly more boys born th311 girls - 14 boys christened for every 13 girls - yet hy the 3ge of marri3ge the b3lance of thc sexes was equal. If wc 3ssu me th3t the chancc of 3 chi ld heing born a boy is 0.5, the ch3nce of more boys th3n girls being born every year for H2 years is 0.5~! . Th e same p3ttern of more male births is found throughout the 170

PASCAL'S WAGER In 1657·58 Blaise Pascal wrote a philosophical essay in which he described the 'wager a sceptic should make. The penalty for not believing in God (the Christian God, for Pascal) could be eternal damnation; however, the cost of believing in God if He turns out not to exist is slight. At most, the person who chooses to believe may relinquish a few fleeting pleasures and spend a few fruitless hours in church. Although the sceptiC may feel that the chance of God's existing is very small, the cost of lOSing the wager is so high and the price of belief so comparatively low, that it is a better bet to believe than not believe.

Arbuthnot took this as world . incontrovertible evidence of Divine Providence Olt work, setring up society with the perfect habnce. (it doesn't seem to h3ve occurred to him th3t Divinc Providence could equally wdl h3\'e killed fewer boys on the p3th to adulthood, mus 3voiding the suffering of bere3vcd Inrenrs at the S3me time as achieving the required balance.) The argument was generally adopted and

CHHR U P, IT MAY NtV[R H APP£N

refined. HowC--'Ver, Nicolas Bernoulli, the more rat ional Swiss mathem:nician, sUb'gestcd that perhaps the probability of a male birth was not 0.5 at all but 0.5169, which would producc exactly the required re~ult with no need for divine intervention. MAKIN G DEC ISION S

As with Pascal's wager, many dl'cisions that may he influenced by a knowledge of probability are also affccted by a more subjective perception of desirable outcomes and the concept known as 'marginal utility'. Imagine a national lottery, in which tickt:r.~ COSt one ducat (a coin in IL~e in much of Europe in the 18th century) and the prize is a million ducats. For a poor man, a ducat is very valuable, and the payout immensely so. For a rich man, a ducat is of little consequence, though the payout is still valuable to bim. Th(' rich man can better afford to bet a Jucar tllan the poor man, bur a.~ he has less need of the prize he might not bother. A1though the probability of winning is equal for hoth, the decision about whether to buy a ticket is very different for each. Ll the 17 50s and I 760s, inQ{:ulation against smallpox was a topical subject of deb'lte. The inoculation used live smallpox virus and in a small number eJses produced smallpox Oenner's vaccine produce- such as blindnc.~s or brain damage. Someone who did not have the vaccine stood :I high chance of contracting smallpox at some time in the future, and a I in 7 chanceof dying from it. Someone who chose to have the vaccine stood a ~lllall

chance (not measured) of dying immediately of smallpox brought on by the inO<.lllation, Imt othenvise virtually no chance of dying of smallpox III the future. The purely mathematical calculatiun , canied out b)' Daniel Bernoulli, suggested that there was only one .'icnsiblc choice - inoculation. But the Frrnch mathematidan Jean Le Rond d'Alembert, among others, argued that many pcople may prefer the better chance of sun'iving the next week or two to the aS~llrance of safety in the furore. (TI)day, plenty of people prefer the immediate adv:l1lmg\! of long-haul flighn; m the longteml henefit of still having a planet [() live on.) INDEPENDEN CE

People are nOt only affected by marginal utility and the preference d'Alembert noted for short-term benefit. They may also be swayed by .~llperstition that has no grounds in statistical probahility at all. Imabrine flipping a coin ten times; the prohahility of getting heads each time is 1 in 210. Suppose the first time it is heads. Now the probability of all ten flips being heads is I in 29. If the first nine come up heads, the probahility of ten heads, by the last time, is I in 2. Now suppose you want to fly on a plane. You know that the chances of dying in a plane crash are, say, I in a million on any particular flight (this is not the real prohability). You have already made 1,000 flights

•

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-with t}nir/ollg-w'm fllrtln:

if it

1m'tlns gnills ill tbe shon tfl711.

171

NUM BER S AT WO RK AN D PlAY

HERD IMMUNITY

Some diseases have been completely or nearly eradicated by national inoculation programmes. An example is measles, once endemic in the westem world but now rare in countries with inoculation programmes. However, worries about the safety of the vaccine in the T9905 led to a reduced take-up of childhood vaccimtion in the UK and measles began to take hold again. While the vast majority of a population has immunity, a few unprotected individuals benefit from the 'herd immunity' as the disease can't get a foothold amongst the inoculated population. However, as the number of unprotected in dividuals rises, the presence of the disease increases to the

safely. Your chances of dying this time are still one in a mill ion - the previous flights do nOt Olffcct this one. In this case the events are independent; even if you had made 99!J,999 tlights safely - or tell million - the chances of dying in the next flight would still be only 1 in a million. But it doesn't feel like that to many people. The perception is often that if we hav~ been 'lucky' up to now, our luck is du~ to run out. it can work the other way, too. People may pick the same lottcry number each week bccause they believe their Ilumber 'must come up sooner 172

point where it can spread amongst the uninoculated population. The dilemma facing parents who were unconvinced about the safety of the vaccine mirrored that of the people making a choice about the early smallpox vaccine. For SOCiety as a whole, there was a moral dimension was it right that a few individuals should avoid the (pOSSible) risk posed by the vaccine and depend on benefiting from the herd immunity acqUired at the cost of everyone else taking that risk? For mathematiciam and medics, there was a different question: what proportion 01 the population could remain unvaccinated before their safety was compromised?

or later' . few people. pick numbers 1, 2, 3, 4, 5 and 6 because thL'Y believe (irrationally) that this combination is less likely to be drawn than any other. This tendency is nOt so far removed from the Ancients who beliL'Vcd the number 3 had special properties, or who wore a magic square for protection. INTERDEPENDENCE

\"'hen choosing whether to hoard a plane, people are dealing with random eventS they have no control oyer whether the plane will crash. A situation that is harder for

CH HR U P, IT MAV NtV [R H A PP£N

the benefi t for themselves. They may also try to minimize the detriment to others - or they may pay no atten tion to th e impact on others, or evcn act to spite them. Game theory trics to rake accOunt the motives and insights of people acting in thc situation that is moddled, as well as many other rel evant a~pccL~. For example, players - which may be individuals, groups, nations or corporations, for example - may he in direct competition or may cooperate to a greater or lesscr dcgree. They may be competing for a finite resource or infinite resources. They may he in full possession of all relevant information, including the actions of other players, or have only partial acce~s to information. There are different brame theory modds to cover these and other possibilities. Game theory often produces a matrix of out comes which can then bc analyzed.

or

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mathematicians [Q modd i~ that in which one person's actions are dependent on or linked with those of another person (such a~ the decision aoout whether [Q vaccinate a child). This is addrL'Ssed by game theory, developcd in the 1~s by the Hungari an\'on American mathematici:m John Neumann and the German-American Oskar M orgenstern. Dcsp irc its name, game theory is concerned with the serious pursuits of econom ics ra ther th an th e frivolity of games. Nlorgenstern and von Neumann saw that the mathematicalmodds developed for systems in physics and other areas of science werc poor tools for working with economics and other ~rudics that involve human behaviour lK'Cau~e they werc ba~ed on the actions of disinterested parties. \Vh en people make choices, they try m maximize

BACK W ARD REASONIN G

Proofs such as that of Arbuthnot that God exists work backwards from effects to C::lIl.ses - there arc equal numbers of marriagcahle men and women, th erefore God exists. J akoh Bernoulli demonst rated that, if th e probability of an event is nOt known, it can be inferred from looking at the results of experiment or observation as long as th e observer has sufficient kno wl edge and experiencc. He gave as an example the fun that if a coi n is tOssed enough times, the ratio of heads to tails approaches ever more closely the ideal 1: 1. A formal demonstration of prolmhiliry in this way was made independently by Thomas Bayes and by Laplace and is now known as Bayes' theorem. Laplace famously used it to argue 173

NUM BER S AT WO RK AN D PlAY

the probability of the sun rising tomorrow, given our knowledge that it has risen every day for the last 6,000 year.; (which in 1744 was considered to be the age of the Earth). laplace and his contemporaries tried to put probability at the heart of the moral .~cicncc5, though their attempt was somewhat dubious. Enlightenment philosophers and reformers were concerned with the value of the judgements made by electorates and juries - \\"ould they reach the right decision or deet the hest t':lndidate? They addressed this as a problem in probability. As.'mming that each juror aeted independently (French juries did not

delih erate) and had a grea ter than 0.5 chance of reaching the right verdict, they worked Out the optimum size of jury and the majority needed to reach a safe conviction. The practice of deciding jury S17..t.' and majority using probability continued until the 18305. By then the system was coming into disrepute and a pupil of l aplace, Simeon-Denis Poisson, used ncw statistics to produce a better model. Before probability could be used effectively in any area, though, reliable information was necessary. Statistics and probability go hand in hand.

PIERRE·S IMON , MARQUIS DE LAPLACE (1 749- 1827)

The French sdentist and mathematician Pierre·Simon de laplace was most famous for his work on astronomy and his application of probability to scientific problems. He was the son of a peasant farmer, who revealed mathematical ability while at a military academy in Beaumont. In 1766 he went for one year to the UniverSity of eaen, but left for Paris, where lean d' Alembert helped him to secure a professorship at the Ecole Militai re. He taught there until 1776. laplace applied Newton's theory of gravitation to the movement 01 the planets. He perfected the contemporary model of the sol ar system and

demonstrated that apparent changes are not cumulative, but occur and correct themselves in predictable cycles. (Isaac Newton had suggested that divine intervention was sometimes needed to put the solar system right!) laplace was the first to suggest that the solar system was formed by the cooling of a van cloud of gases. His explanation of planetary motions made him a celebrity. laplace was president of th e Board of longitude, helped to organize the development and introduction of the metric system, and for six weeks was minister of the interior under Napoleon.

SAMPLE S AND STAT1STI CS

Samples and statistics \Vithout infor mation on which to base decisions, it is lX)ssibl c to cakulate only the most basic prohahilitiC$. A~tonishingly, it was not until the latt! 1ith century that peoplt! beg'a n to recognize the true value of collectin g n u meric informatiun ahout populations and economies. Suddenly, statistics were everywhere and computing with them gave new insights into how societies might: work. For the first time, the gUL"is\\'ork was takert out of planning and the burgeoning science of statistical analysis had material to work with and aims to work towards. PEOPLE COUNTING

Collecting information about the number of people living in an area by t:lking a census has been practised intermittently for thous,mcls o f years. The Babylonian s, Ancic.nt Chinese, Egyptians, Greeb and Romans all held population cem-uses. in Cbristian trad ition, the p:Jrents of J esus travell ed to Be thlehem immediately before His birth bccau$t' the five-yearly census required everyone in the Roman Empire to rerurn to their place of birth to be counted . The very basic information collected in thcse early censuses allowcd rulers to work Out how much money could he collected in taxes, how many people could be recruired for an army o r building project and how much food could be produced or would be needed. In Et,'Y]lt, it was also used to redistribute land after the annual flooding of the Ni le. But no additional analysis of population dat,1was carried Out and on~' the most hasic details were collected. Often, the census dat"J were not reliable. If p(..>(Jple

exp(..>(:tcd to be taxed on the hasis of how many lived in a house, a few might he missed out, for example. In 1066, arrer the conquest of Britain by Norman invaders, \Villiam the Conqueror ht"ld a thorough audit of hi s new lands. Thi s included a Cenl>l1S and a listing of every item of prnperry in the land . It was written up in the Domesda), Book - a m,1ssivl' undertaking for the 11 th century and one which still providcs valuable statistics for historians. Thereafter, there '\~JS no enthusiasm for regular census-taking. Although bishops in many parts of Europe were supposed to keep count of the families- in their dioct::scs, there was littl e information about popu lati on levels. Some people even believed that taking a cemus wa s sacrilegious, citing a Story !Tom the Bible in which King D~vid attempted a census which was interrupted hy a terrible plague and never completed. The fir.~ t regular census in modem times was carried Out in Quehec, Canada in 1666. In EurQpe, fccland was the first in 1703 , followed hy Sweden in 1i49. The US held its firH ten-yearly ccnsus in 1790 ~nd the UK in 1801; the US had jlL~t under 4 million inhabimnts and the UK JO million (previous estimates had put the UK population at between 8 and 11 millio n). THE RISE OF STATISTICS

In 1662, the English Statistician J ohn Graunt published a set of .~ tatistics drawn from mortality records in London, and in the 1680s the political economist \"lilliam Petty publi shed a series of essays on 'politica l arithmetic' whieh provided smtistical records with calculations - some 175

NUMBERS AT WORK AN D PlAY

THE CENS US AND COMP UTERS

The demands of census.taking were a

considerable

spur

to

the

development of technological aids to calculating. The first machine for working with census data was used in

1870.

Census

data

were

transcribed on to a rolling paper tape displayed through a small window. In 1884 Herman Hollerith (1860- 1929)

acquired

the

first

patent for storing data on punched cards and organized the health records for Baltimore, Maryland, New York City and New lersey, which won him the contract to tabulate the 1890 census. The huge success of this census opened other markets

to

Hollerith

and

his

machines wen> used in Europe and Russia.

He

incorporated

his

Tabulating Machine Company in 1896, which later became IBM.

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SOC IETY IS TO BLAME

QII Tbr: idf"ll

The Belgi~n m~them~tician Adolphe Quetclet (1796-1874) was :1 champion of st~tisti cs as the basis of the social study which hi:' tcmled 'social physics'. He examined d~t~ of all kinds, using the techniques common in som .. scientific di~cipline.~ of amassing a vaSt collection of dam ~nd looking for emergent p~trems. To his surprise, he found them everywhere, not JUSt in the ar(.'as where Divine Providenl'e

rbllr 1111 pnl"fJ1111/ dllrt! C01l1d be cMrd

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176

quite bizarre, such as the monetary value of all people in Ireland. On the whole, governments encouraged or financed statistical survey.~ ~nd gu~rded the rL'Sults jc~ lous ly, using them to increase the power of the st~te. They were still inextricably tied up with ~uperstition and followed vcry unscientific methods. One of th .. most f~mous 'p(Jlitic~1 ~rithmeticiaIlS' w~s the Prussiall J()h~nn SlSsmi1ch, who published three volumes over mOre than twenty ye~rs, ending: in 1765, proving again the existenee of God revealed in the harmony of socia l stati stics. Other SL1tistics were collected by scientist.~, profession~ls of different types and hum~nitarians. Lldeed, there was a growing enthusiasm for statistics, which became something of a mania during the early 19th century. Suddenly, everything was studied, counted, audited - the weather, ~6'Tieulture, population muveml'nts, tht.: tides, the land, the Earth's magnetism ... The European countries that h~d l'mpircs surveyed their new ~l'quisitions and took cenSUSL'S in their colonies. As Americans moved westward, claiming morc land. they charted it and logSrcd its resources.

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mi ght be expected to operate. In particular, he was impressed to find th,u crime figure s follow ed a predictable pattern. He conjectured that they are a product of society rather than indi,~duals and that, while an individual criminal may be able to rt'sist the urge to commit a crime, the overall pattern of crime rates is altered little by individual actions. He felt that the proper study was of crime ratt's rather than criminals and that the proper remedy to c rim e lay in social action, including education and an improved judicial ~ystem. Careful use o f statistics to examine the effects of changes and suggest directions for future change would, he felt sure, produce the desired results. Quete1et's thesis promptt'd some debate on the apparent contlict bet\vcen statistics and the doctrint' of frce will - if crime rates can bc deternlined by statistical methods and arc unchanging over time, how much freedom do individual s really havc over their action s?

_21 ' Sept 14 _9 ' Sept. 9

0 ' Aug. 18

0111' if the mtJSr l/{'cowplished gmphim/ J'tpl'l!wlltariml:f ofrtlltirticr nm"ntndl' is (vurles .~1illaJ'd 'r gmph of Napole(JII} disasN'o/ts campaigll ill RI/Ssill ill 1812, Ir shlTiJ.'s /l/olTality

011

the 1l-'
l\1OSW..v

//lId Wll'1.'liltfl/l1:ith telllpn11t1ll'l!, The width of the

grull {lilt! ortillge lilies I'I.'PJ'fSl'IIlS the size of the anNY, shlT,.JJilllf, hlT.JI it tT<1Jilldws, Ollly 4 PC!' cellt n'tt1l11ed

frum tbl' (iflllpaigll, STATISTICS MEET SCIENCE

Perhaps surprisingly, it was nOt wuil th c middl e of th e 19th century that statistics beg'Om to be applied to science with th e same enthusiasm and ri gour :::IS th l.)' had been applied to social science. In the 1870s th e Scottish physicist J :lIlH!-S Clerk Maxwell often explained his theory of gases with reference to social statistics. From the very large numher of random movcmenL~ of molecules hc derived thernlOdynamic laws order from chaos. He argued that, JUSt as statistics relating to crime or suicide can yield consistent rL'Sul1!; from the unordert'd acts of individuals, so predictahle outcoml."s 177

NUM BER S AT WO RK AN D PlA'

FLOREN CE NI GHTI NGALE ( 1820- 19 10)

Florence Nightingale enjoyed a privileged childhood in England, where her father taught her languages, philosophy, history and mathematics. She claimed to have had a message from God telling her she had a vocation and later wanted to train as a nurse. Her family resisted and she became instead an expert on public health. She did later train as a nurse and, du ring the Crimean War (1854--56), was put in charge of the hospital at Scutari, in Turkey, where she revolutionized healthcare for wounded soldiers. She kept copious notes and after the war put together an extensive report from the statistics she had gathered. She used innovative ways of presenting information in graphs, such as the 'coxcomb' graph (above right). Nightingale worked tirelessly to improve conditions in the British army. She founded the first training school for nurses anywhere in the world, the Nightingale School for Nurses in london, and established the professional footing of nursing.

on a large scale could be extracted from acts that are unpredictable on the small scale. Bur before statistics could be applied, it had to develop as a mathematical discipline. Mathematical methods specifically applying to statistics beg:m to emerge from the end of the 18th cenrury and proliferated rapidly. 178

C.u",.ol mortaHty In the army In the East. April 1854 to M.",h 1855

D D

noo.ballle

battle

Nigbtillfl,flle 11'/1$ 1I pi(lIIe,~' ill

rbe

flllalyris 111111 prtSelltlltioll of stllrmicf, 'CQxromb'

grllpbr

'wert derigllfll to be

.. ""d""ood by et'I'1)'W
'{S tatist/dans] have already overrun every branch of science with a rapidity of conquest rivalled only by Afti/a, Mohammed and the Colorado beetle. '

Maurice Kendall, 1942

STATISTI CAl MAT11£MATICS

Statistical mathematics

Abraballl de !\1I)wre 11'flS

tI

pil)neel"

;11 flllalyric

grolllfny aNd Ibe lImn] 1)[ prwabi/if)'. beillgfirst WHAT ' S NORMAL?

II) 1I1)Iiu Ibe /lI)l7Ilt1l distribllti(JIl CIIrve.

Abraham de MoivTl' (1667-1754) was the first pe~n to notice the characteristic bell curvc of the nQmlal distribution (~ below} The curve plots the &cquency or probability of valucs a!:,'ninst the vnlues themselvcs. The mOst frequently occUlTing result s are at the top, representing the mean value; the results that d,,'Viate most from this noml and occur least frequently are on the lower anns of thc cUive. The slope of the curve is determined by the deb'Tee of ,~ariation within the sample. Approximately 6g per cent of the values in the nomlal

" diStribution are said to fall within one standard

ITom phr>ieal attributes such as height to characteristics of psychological profiling ~llch a~ propensity to get married or commit suicide. WORKING WITH ERROR

The early 19th cennuy saw a rapid fiSC III mathematical method~ involving statistics. \"'ork on mcasuring the Earth's longitudinal cin:umferenee in order to detcmlim: the lenbl'(h of a metre (to be 1140,000,000 of the cin'umference) needed statistical methoos to deal with errors and inconsistenlies in geodetic measurements. In 1805, the Frencb mathematician Adrien-Marie Legendre (1751- 1833) propoSl'd a tedlllique which has come to be known as the 'least squares' method. H e took values minimized the sums of the squares of deviations in a set of observations from a point, line or l"ul"'iC drawn through

deviation of the norm.

them. GjUSS became

The normal distribution curve and the concept of standard deviation ITom the noml were widely used to assess statistics in many different fields. L1place used the model, too, in his probability stu(~cs, JXlnlcularly in applying probability to very large numbers of evenn;. Quetelet argued that virtually all human train; confonned to the normal distributioo <.1.lIve,

interested 111 the method and showed 111 1809 that it Adrim-Milr;e ugmdre hilS a emul"

1)/1

(be Mooll

1IIImed tlftn' bim. 179

NUM8ERS AT WORK AN Il PUY

METHOD OF LEAST SQUARES

The method of least squares calculates the best line through a set of points by working out the smallest possible sum of the squares of deviations from the line of ilil the points. Squares are used to remove the difficulty of dealing with both positive and negative deviations, since when squared they will both give a positive result.

gave the best possible estimate if we assume that the errors in mea~l.Irement follow the Ilomlal distribution. The method of least squan:s was applied to statistics ill all fields and became the principal tool of statisticians in the 19th century. 1t was often used to estimate whole populations from a study of a small sample. PERFECTING HUMANITY

Francis lfollton, a wusin of Ch~rles Darwin, took an interest in the \'~riation highlighted by normal distribution and standard de,~utions.

o

02

H e used a model, known as the C:illiton board, to show how a normal distribu tion is achieved (see below). A set of peb'S is arranb't'd in a triangle alxwe a row of cups. Ball bearillb'S dropped at the top of the triangle bounce down through the pegs to fall into u cup. A few full into outlying mps but m(l)t full into the mps in the middle of the board, forming ~ normal distribution curve. Galton applied st~tistic~ l ideas to heredity to show how v~riation tends to be bred out, and generations of an organism tend to revert to similar levds of variance.

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Ball bwrillgs dropped all ro the Gil/tOil board at the top art! dif/«tn/ illfo tbe ClIpS at (be boltom. Tbe distriblltioll Of bill/ betlrillgs ill ClipS dmlOllsn'atrs tlllorma/ distributioll CIIrve.

180

STATISTI CAl MAT11£MATICS

So although the chi ldren of ext·eptional parents may he exceptiona l memselvt!S, at least in some ways, on me whole, they tend to regress towards me general population as a whole. Galton took his research in an alamling direction, becoming the founder of the eugenic; movement which aimed to guide human evolution tOwards perfection. H e wanted to breed in 'good gencs' in the way that breeders select me best genes in farm animals and crops. Although originally he was in ten.:!sted primarily in genetics and heredity, Galton recognized the application of his statistical methods to other areas and Stressed the adaptabili ty of me tools he developed.

the individuals included should bechosell at random. The first triumph of this tf..>chnique of stratifi ed sampli ng came in 1936 when George Gallup'S poll predicted the reelection of Franklin D. Roosevelt in the US, whi le a larger, unstratified sample, confidently (and wrongly) predicted the opposite result. Gallup drew on a sample of on ly 3,000 voters, while Literary Digest, me opposing pollsters, polled 10 million. Roosevelt won with the largest landslide in American history. A large sample is no b'llarantee of a representative sample or an accurate result. Experimental design went hand in hand with the development of statistical tools. The use of a control group to compare with

COURTING RANDOMNESS

Developments in statistics aimed to enable information rrom a sma ll sample of data to be extrapolated or applied to a larger populati on. By deciding the rate of crime or marriage or an inherited disease in a sample of the human population, for example, researchers hoped to reach conclusions ahout th e rate in the whole population. The result.~ of any stati stical ~L1 rv t.'Y depend, of course, on th e quality of the sample measured. The head of the Nonvegian Central Bureau of Statistics, A. N. Kiaer, aimed to draw samples that covered the full ran ge of representative varia bl c.~ in the population, such as old and young, rich and poor. The English statistician Arthur Bowley was one of the first to try to introduce randomness into sampling. Th e Polish statistician J erzy Neyman hrought these two concerns tOgether in 1934, trying to ensure that a samp le included representatives of major variables but that

Th~ lnlllislidr I"(-rim/Oll of Fmllklill

D. Roos.7Jdr ill

1936(f1I11CllSIIIJSII,.p,.isrtoGallllpwhlJhlldll.r~d

sn"fllijlrd sffmplillg to pl'edict stich {t /Twit.

181

NUM 8 ERS AT WORK A NIl PlAY

'Nothing is more dangerous than to live in the temperamental atmosphere of a Gallup Poll, always taking one's

.:

I,. • •

th e experimental group, and the random all ocation of individuals to the control or experim ental group, em e r~,''C d as stan dard procedure during the early years of the 20th cenlliry. [n parriL'Uiar, th e Briti sh geneticist and statistician Sir Ronald Aylmer Fisher (1890- 1962) reshaped experiment design in many fields , including p~ych o l o gy, medicine and ecology in th e years after the Second Wo rld War. I-I e began hi s research in geneti cs, where h e used statistical analysis t o reconcil e incon sisten cies in Darwin 's

theory of evolution that ha d been thrown up by th e e.xperimental work on inheritance of the Austrian hotanist G regor M endel. H e devel oped the method - which now seems ridiculously obviou s - of varying only One condition in an exp eriment at a tim !! and comparing re~lllts with a control group. Although earli er exp erimenters had donc thj s to som e degree, it was fclt to be immoral where human subjects were concerned, and so rigorous use of control groups and random allocation of individuals t o the control or experimental group had not been pra cti sed previously. Fisher also advocated repeatin g experiments and T be IrrJara lld rill/dum IIl1mber geuerrlrm' developed

by Bob l\1 w de ill 1996 produced n m dmll II l1mbel'S Jlsillg a ro mpllrer program seetkd wirb digiral pborograpbs of tbe parterlls produced by !twa lumpr.

THE DIFFICULTY OF BEING RANDOM

It is not only in sampling that randomness is actively sought. In g ames of chance for high stakes, there is an im perative to make sure that events that are supposed to be random are in fact just that. Cryptography also demands randomly generated numbers. This is much harder than it at first appears. As chaos theory demonstrates, many events that look random are actually not, but are governed by complex laws and a la rge num ber of variables. The systems used to pick numbers fo r large-scale gambling ventures such as national lotteries are very carefully designed and engineered to remove, as far as pOSSible, all bias in the selection methods. It is very difficult to produce com puter algorithms fo r picking random numbers, so most lotteries use mechanical methods instead. (These also have the advantage of looking more spectacular than computers.) Computers that can generate genuinely random numbers do so using a physic.:!1 source such as atmospheric noise (e .g., www.random.org),

182

STATl s n CAl MAT11(M ATICS

looking at the variation in results to determine the margin of error. The most influential statistician of the 20th century, Fisher summed up his findings in the highly innuenrial text Statistimi Il1rthods fllld Scientific JIIJrrcnce (1956). One of his most important developments was in the analysis of variance (called A.l'\fOVA) which looks at the points in a sample which vary from the norm. It is used to assess whether or not results are stati stically significant - that is, whether they are likely to reflect a real trend, change or cause, or whether they could have come about by chance. COMPUTERIZATION

The hurden of calculating with very large sets of dam has becn made easier by the widespread use of compurcrs . Mile earlier statisticians had the laborious task of carrying Out calculations for each data point by hand, their modern counterparts can feed all their data directly into a computer and lcave it to apply the necessary statistical tools and provide the analysis and graphs. Often, the data are even collected by computers directly from sensors. \Vc can now handle inUllense data sets, so large that thcy could not have bcen handled in a whole lifetime without computcrs. It means that statistical analysis can be applied in all areas of life, detcrmi ning patterns and projecting outcomes in areas as diverse as the effect of

Bririsb mtlrhelflilticiall Johll C(}/rwny's 'Cmlle of ufr' quickly gaill~d II mit folkr
ill (I

'society oflivillg orgllllisms '.

early education on crime rates, the likely spread of epidemic disease and the effl-'ctS of global warming. A famous illustration of the importance of initial conditions is J ohn Conway's 'Game of Life' (1970). This is a cel lular automaton - a computer simulation of an evolving population or universe in which an initia l organism or automaton makes copies of itself which succeed or f:I il accordi ng to various conditions (~l.Ich as overcrowding, lack of resources, etc.). Conway created it in response to a problem presented by John von Neumann in the 1940s relating to constructing a machine that could make copies of itself. T he 'Gamc of Life' is not a game in the usual sense of the word, in that there are no active players. After the 183

NUMBERS AT WOR K AND PUY

SETI@HOME The SEn project - Search for Extra· Terrestrial Intelligence - collects radio data from space on a continual basis, and is starting to look also for pulses of laser light. Its stated aim is 'to explore, understand and explain the origin, nature and prevalence of life in the universe'. SETI's task is to examine the constantly growing data set for patterns that might indicate a deliberate radio transmission. To do this, it asks volunteers around the world to install a screensaver which imports chunks of data from SETI over the Internet and processes them on the computer while it is not being used. In this way, SETI makes use of millions of hours

,8<

LookillgfOl' riglls ofliif": nulio IlIIteimile 'which

of free computer time on personal computers around the world. Each PC reports its results back to SET! and any possible patterns are flagged fo r further investigation. An unimaginably large task in statistical analysis is being carried out at very little cost and much more quickly than

form plll1 of the

it could be managed using dedicated computers. The SETI equation The Drake equation (1961) is suggested as a way of calculating the likely number of ptanets

ne '" the number of planets in each solar system with an envi ronment suitable for life. fl '" the fraction of suitable planets on which life actually appears. fi .. the fraction of life.bearing planets on

that have intelligent life in the Milky Way: N '" R* X fp x ne X fl X fi X fc x l

which intelligent life emerges. fc '" the fraction of civilizations that

where N '" the number of civilizations in the Milky Way whose electromagnetic emissions are detectable.

develop a technology that releases detectable signs of their existence into space. l '" the length of time these civilizations release detectable Signals.

Vt-I)'

ulrge An"lly /lSfrollomicuJ

ohmvll(01)';1I New Mexico, USA. R~ "" the rate of formation of stars suitable for the development of intelligent life. fp '" the fraction of those stars with planetary systems.

STATISTI CA L MATHE MATICS

'Nothing in the universe is unique and a/one, and therefore in other regions there must be other Earths inhabited by different tribes of men and breeds of beasts,' lucretius, SOse

instigatOr sets up initial conditions, the game runs, producing generations that flourish or perish according to the consequences of the starting conditions. The original game used populations of coloured squares III a grid, but it spawned a whole industry of computer simulation games, some of them immensely complicated, that produce populations of creatures or other entities. The interest in cellular automata that grew out of Conway's game has found applications in many fields, including research into human, animal and \'iral populations, growth of crystals,

economic problems and many other areas in which complex patterns develop organically. MOVING ON

Much of the work on statisric'5 in the last hundred years or SO has led to analysis of groups or sets of data in quite complex ways. The hehaviour of sets - whether of numbers or anything else - is the suhject of set theory, fi~t developed in the second half of the 19th century. The appearance of set theory has heen one of the most important developments in the history of mathematics.

185

CHAPTER 8

The death of NUMBERS

AnalY.l.ing data gathered from populations, experiments and other sources leads to a search for patterns that can be used to categorize and group items. The natural result of this is to divide items into sets and compare these. Everything in the universe can be defined by its membership of sets. Relationships between these sets gives further infonnation about the objects. \¥hen mathematicians turned to set theory at the end of the 19th century, they fOlmd a rich lode that would provide methods for dealing with everything. As set theory developed, it built a logical language of its own that was eventually turned back on itself. Set theory could be used, it appeared, to explore mathematics, to prove mathematical theorems and even to dissect and analyze set theory. It allowed mathematics to encompass everything and yet to become so abstrdct and abstruse that it apparently dealt with nothing.

SEF! (page 184) aims to idemify I' milsn ofphllldS 'U'bicb bost lift.

TH1 D EATH O f NUM8£RS

Set theory At the very beginning of the history of numbers heforc humankind began counting, it is likely that people compared set~ of object~ - are there enough spe~rs for hunters to have one each? Arc there more or fewer sheep in the pen than pebblc.~ in the tally pile? A~ the requirementS of hum:mkind bl'c~me sophisriC:Jted, m ~thcmatics moved away fi-om these setS of objects, ut'vc1oping a concept of numher that could be applied universally. Now, thousands of years l~tcr, mathematicians have returned to sets, hut with a new insight - the possihiliry of an infinite set. OR I GINS O F SET THEORY

Set rhL'Ory was developed by Georg l..antor between 1S74 ~nd 1879. He defined a set as a collection of definite, Jistinguisb~ble 'objects of perception or thought' th~t were conceived as a whellt' entiry. So there is a scr of positive imegers which can be thought of ~ s ~n object in its own right. But there is also a set of people who arc employed ~s firemen, a set of molecular structures that form hydrocarbolls, and so on. Although

INSULTS FLY

Amongst the insults hurled at Cantor and set theory were that it was a 'grave disease' infecting mathematics (Poincare), that it was 'utter nonsense', 'laughable' and 'wrong' and that mathematics was 'ridden through and through with the pernicious idioms of

set theory' (Wittgenstein), and that Cantor was a 'scientific charlatan', 'renegade' and

'corrupter of youth' (Kronecker).

188

the basic principle is e.nrcmcly simple, logical thought about sets soon leads into complicated co ncepts that blur the boundary between math ematics and philosophy. Early critics ~rgued th~t set thl'Ory de~1t only with fictions, not with ~llything th~t reflected re~Jity, th~t it violated the principles or religion and th~t it was not mathematic~1. [t is true that set theory is a hr3llcb of pure mathematics that has little ~pplic~tion to ordi ll ~ry e:\llericnce aod the everyday world. It has proved immensely valuable, however, in enabling: manipulation ()f complieated mathematical concept~. Set theory is itself eapahle of heing defined, analyzed and refined by applying the logic of setS to its concepts . SETS FOR BEGINNER S

The fund~mental concept of sct.~ is very simple. Any group of ohjITt5 or numhers, whetber or not they h~\'c real or enduring existence, forms a set. Any indi\,idual member of a set may be a mt'mher of m~ny sets. Sets overlap, and some contain other sets (subsets). A set may have all infinite number of members, in which case it is an infinite set. Arithmctic with sets is nOt quite equivalent to arithmetic with numbers. If two set~ arc ~dded togethcr (called the union of set.~), the new set eo mprises all the members of hoth sct~ but with no duplicate entries. The intersection of two set~ comprises the members that are in hoth sets. A set with nu members is a null set, designated by 0. In general, the order of clements in a sd is not rcle\~Jnt. So whi le the coordin~te pairs (x, y) and (y, x) are not the same,

UTTHfORY

GEORG FERDINAND LUDWIG PHILIPP CANTOR (1845- 1919) Georg

,

Cantor

German

friends. Their exchange of letters brought

mathematician who is best known as the

Cantor to his most important work, on sets

creator of set theory. His parents were

and the concept of transfinite numbers. He

Danish, but settled in Frankfurt in

faced considerable opposition from

1856. His mathematical ability

l eopold Kronecker (1823- 91),

became obvious during his

who blocked publication of his

early teens. Cantor went to

work and his advancement

university in Berlin and

within

Zurich

Kronecker

to

stu dy

the

univerSity.

had

previously

mathematics, philosophy

been Cantor's mentor, but

and

now believed Cantor's work to

physics.

taught

by

He

was

meaningless

Weierstrass,

transfinite

whose influential work on analysis impressed Cantor. His

numbers

and

the

he

was

writing about to have. no existence. Opposition came from the ChurCh, too,

PhD thesis, submitted after a single term, had the title In mathematics the art of

which fel t his work challenged the unique

asking questions is more valuable than

infinity of God. This opposition made

solving problems. He became a professor at

Cantor particularly sympathetic to young

Halle in 1879. Cantor worked

scholars later in life, whom he helped and first

on

theory

01

numbers, then tumed to the theory of trigonomctric series and extended the work

encouraged in the face of an entrenched and resistant system. Cantor's work established set theory as a

of Bernhard Riemann. He met Richard

branch

Dedekin

groundwork for much of the development

( 1831-1916)

while

on

his

honeymoon and the pair became lifelong

rh e sets {x, yJ :Ind [y, x} :Ire id enrit~JI. Cantor's definition of a set was that it is a grouping into on e entity of objects of any type, though

of mathematics

and

laid

the

in mathematics in the 20th cen t ury.

The condition x E A is true only if x meets the conditions of the formula

it.~

'Nowadays it il known 10 be possible, logically speaking, to

own identity. For3nyobjcctx

derive practically the whole of known mathematin from a

in; relation to a set A i.~ 3lw3Ys that it is or is nor 3 member ~ hown by x E A or x f£A

single source, The Theory of Sell. '

t":3ch ObjCl·t 3i.<;O rer:tins

Nicolas Bourbaki, 1939

"9

TH1 D EATH O f NUM8£RS

BERTRAND RUSSEll <1872- 1970)

Bertrand Russell was a British mathematician and philosopher; bom into an aristocratic family. He was orphaned at six and brought up by his grandmother, hometutored and isolated from other children. He went to Trinity College, Cambridge to read mathematics, but changed after a short time to philosophy. Much of his philosophical work was on the philosophy of mathematics. He was influenced by Weierstrass, Dedekir1d and Cantor who all wanted a formal, logical basis for mathematics. Russell aimed to prove with Principia Mathematica that mathematics was nothing but logic.

himself. Who shaves the barber? logicians

However, he discovered a paradox which he simply could not resolve except by redefining the basis of logic a barber says he shaves everyone who does not shave

have found different ways of adapting set theory to deal with the paradox, which attempt to make the definition of a set more restricted and precise.

Sex) which defines members of the set. This is the principle of abstraction . That a set is defined by its members is the principle of extension. The number of elements in a set i~ called its cardinal number - the set {4, 5, 6) has a c-ardinal number of 3. The cardinal numhel· of a Set A is wrinen A. Any subset of a finite set has a smaller cardinal number than the original set. If we imagine a set of 'all cars', there are clearly fewer members in thc subset 'red cars'. WORk i NG WITH INFINITY AGAIN

The concept of equivalence - that hI'o sets are equivalent if they have the .~ame number 190

of members - does not depend on the. sets being finite. So the sct of positive integers and the set of negative integers are infinite but equivalcnt setS, since there is a negative inwger to match every positive integer. A~ Cantor quickl), realized (and others, including Galileo, had realized before him), each natural number can be squared, so there is an infinite set of natural numbers and an infinite set of their squares. Yet thc squarcs are a subset of the set of natural numbers. Galileo's conclusion in 163H had been that the conceptS of 'equal to', 'grt':ltcr than' and 'less than' did not apply to infinity. But Cantor developed instead a

G£TT1NG FUUY

concept of transfinite numhers which ret'ognized different sizes of infinity. AXIOMS AGAIN

Attempts to deal with paradoxes at the heart of Sl't theory led to the devdopmem of axiomatic set rheory. Thi~ aims to develop axioms to FornI thl' ground rules for set thl'ory, much as Eudid'.5 axioms form the grOlmd rules for trigonometry. Severa l conOicting sets of axioms have heen prOJ»:ied, tOO complex to go into here. The b:lsic (yiteria for :lxiom~ are that they should be • consistent: it should not be possihk to prove a statement and its opposite • plausible: they should accord with gt!ner:ll beliefs about sets • c:lp:lblt' of producing re~ults of Cantorian .~et theory. Axiomatic set theory is further divorced from the real world than CantOrian (or ' nai've') set theory since it requires no knowledb"C of what the sets discussed arc. It concentrate:; only on rcl:nions between sets and their properties in a rather ncbu l o\L~ way that gives fuel to the few mathematicians who still claim set theo ry deals with nonexistent fictions. Set theory has influenced many areas of :!Oth century mathematics, but rem:lins in runllOil. Th e search for acceptable axioms rec-ollls the difficulties faced by b'eollleters trying to find n cw models and rules for non Euclidean geometries, hut so far is a long way off resolutiOn.

(;etting fuzzy At the heart of set theory is the apparently simple rule that an object either is or is nOt

a member of a set. Aesop's fahle of the tOrtoise and the hare, like Zeno's paradox of Achilles and the tortoise, pits a fast and slow contestant in a race against each other. Both the han~ and thl' tortoise arc memhers of the set 'animals'. The hare is a member of the suhst't 'mammals'; tht tortoise is a memher of the subset 'reptilc.~'. Now suppose we had a set of'fust animals'. \Ve might say the hare is a member and the tortoise is not. But th is is a subjective judgement and proves more difficult with some other animals. \Vh:lt about a dog - is it a fJSt animal? Or a sn;]ke? Or a girJffe? \Ve are likely to say that thL1' arc somewhar fast, or quite fJSL H ow~"er, set membership is binary - either an animal is fast or it is not. Thi s is dC:lrly problematic :IS it requires an absolute L·ut-off point that is not necessarily satisfactory. If we say that an animal which can run;]t 15 mile~ an hour is fa.5t, then an animal which can run at 14.95 miles an hour is nOt fast; the distinction begins to look silly. CATERING FOR IMPRECISION

AristOtle identified tht' problem of the 'excluded middle' - the objects that can be cbssified as ncither o ne thing nor another (a slightly fast animal, for example). But mathematics had no way of dealing \\~th th e in dcreml in :nc, :md rhe middl e ground remained excluded until the 20th centulY Bertrand Russell, in his paradoxL'S of tht' barber who mayor may not shave himself and the set that contains aJI sets, highliglned the problem again, showing it as a contradiction in set thL'Ory. In the 1nOs, the PQli~h logician Jan Lukasiewicz worked out the principl es of 191

TH{ D EATH O f NUM8£RS

multivalued logic, in which st;}remenL~ (:',1Il t;}ke a fractional truth v;}lue between I (wholly true) and 0 (wholly false). In 1937, philosopher Max Black applied llluitivalued logic to setS of objects and drew the first 'fuzzy' set cUJ"\les; he called thc.se see; 'vague'. From these outlines, the American mathematician Lotfi Zadeh developed the concept of fuzzy logic and fuzzy sets in 1965. Thtse pro,~(k a way of working with imprecise values and categories. There is some disagreement ahout the validity and nature of fuzzy theory. Some mathematicians see it as a variation Oll probability theory. which can. be called possibility theory; others see proba hility as a specia 1 ease of possibility in which certainty can be applied. Fuzzy COUNTIN G

The di.~rinction we saw early on betweCIl counting and measuring addresses the problem of things which do not full wholly into one set or another. InstL'll.d of an clement belonging to ;} set or not belonging to it - a binary distinction, with values of either 0 (nor a member) or 1 (a member) fuzzy seL~ can support dcgrc/;:s of membership. Membership of a SCt can have a value between (amI including) 0 and I. So in a set of fast animals, a cheet:lh may have a membership value of I, Achilles a membership v:1lue of 0.5, and a tortoise a membership ,'aim' of 0.1 . Something which does not move at ~ II, like a m~ture barn~clc, would h~ve ~ value of 0 ~nd not be a member of the set. Fuzzy thL'Ory makes use of linguistic c:ltegories, such :IS '!iOmewhat', 'quite' and 192

'very'. So an animal might be very fast, or quite fast. If 0.6 membe.rship of the set of fast ~nimals i~ called 'quite fast', 0.8 membership might be 'very bst'. Fuzziness is nOt ~bout uncertainty, but about the vagllc b{)und~ries between categorit:S. Fuzzy sct~ may overlap . So an animal might h~ve 0.2 membership of the set 'fust ~nimals' and 0.8 membership of the set 'slow anim~ls'. By combinin g values from more than one set, useful inform~tion can he gained that gives ::t better description of ;\ situation or object th~n the straightforward binary membership/nollmembership of a conventional set. Some. but nOt all, of the mathematics of conventional sctS apply to fuzzy sets. In fuzzy seL~. ;}n object may be a member of two complementary sets (such ~s slow animals ~nd f~st ~nim~L~), wherc;}s in conventional sct theory this :is not possible. The only restricti~m is t hat its tutal membership v::tlue for the two sets ~dds up to 1 (such as 0.1 fast and 0.8 slow). USING FUZZINESS

Fuzzy logic is the ::tpplication of fuzzy SCL~ to decision making and computer programs. It is used in m~ny enb~neering control ~y~tems to approximate human judgement ~nd m~ke the operation of a device ad~pt to prevailing t'ondi[ions. It is commonly used in consumer eleerronics, household ~ppliances ~nd vehicles. A di6rit~1 c~mera, for ex~mplc, uses sensors to dcternline the light lcvc.ls ~l1d the {)hjecL~ in the \~ew which the phot(Jgr~phcr is likely to want to fiH-"Us on (from detecting the edges of object.~), then adjusts focus ~nd t':."lJOsure ~ppropri~tcly. A washing maehine determines the best

G£TT1NG FUUY

fL>atures for the wash L'),cle from the quantity of washing SENDAI SUBWAY and how dirty it is, for In 1988 Hitachi produced a fuzzy logic system to run example. It will calculate subway trains in Sendai, Japan. The trains need only iI optimum amounts of snap conductor and no driver. The fuzzy logic .system controls and water, the best acceleration, cruise speed and braking, taking into account temperature and the Itngth SiI(ety, comfort, fuel efficiency and the need to stop of wa.~h required. The first accurately at target positions (station platforms). system controlled by fuzzy by logic was created Ebrahim A1all1dani and Seto Assilian at refuted diagnosis can thell be fed back into Queen .Mary College, London in tbe early the s},stt'm to improve irs future t 970s. They wrote a set of heuristic rules performance. for eomrolling tbe opm-ation of a small Sets - fuzzy and classical - have steam t:Ilgine and boiler, then used fuzzy redefined mathematics fO I" the 10th and 11 St setS to c{)nvert the rules intO an algorithm to centuries. In SOllle ways, they have allowed control the system. The first commercial mathematics to be divorced from the real use of a fuzz.y ~)'stCIl] was to control a world. Higher set tbeory deals not with cement factOry in Copenhagen, Denlll:lI"k in numhers or OhjecL~ in the wurld, but with 1980. Exploration and uses of fuzzy logic concepts and relations bt'tween cuncepts. increased massively in the 19)30s, especially YCt in accommodating the imprecision and in Japan. contingenL), of the real world set theory, Fuzzy logic is not only used in control like &accds, it acknowlcdbFt'S the 'roughne.'iS' but also III expert systems, artificial of the rcal world and provide.~ a more intelligence and ~lpplicatio ns such as voice accurate (if messier) model of reality than recognition and image processing softwarc. earlier mathematics. It tries to minimize the human intervention needed in a systt'm by approximating MOVIN G ON human judgel1lenL For this it needs an Set theory works '.\';th mathematil-s far expert human to set up the rules on which divorced from numbers. A~ it docs so, it judgements are uased. uut intelligent IlL-tomes increasingly dependent upon systems t':111 then improve themselves logic. Although it may seem that logic has learning from adjustments an operatOr hcen at the heart of mathematics from the makes to the settings that the system start - after all, Euclid attL'mprcd to deri,,(chooses. [n diagnostic medicine, for all geumetry througb a sequence of logical example, 3 fuzzy systcm can look at all the steps - the applicatiun of logic was neither symptoms reportcd or monitOred in a rigorous nor closely examined until the 19tb patient and assess the likdihood of different century. Set thL'Ory, it tunled out, could he diagnoses based on tht' deb'Tt'e to which applied to devc.loping the logic needed to t:'aeh symptom is preSt'nt. A confirmt'd or give mathematics firm foundations.

oy

193

CHAPTER 9

PROVING IT

As in law, evetything in mathematics must be proven before it is accepted as true. Even the most blatantly obvious ' facts ' arc not accepted as facts unless a mathematician can provide a rigorous proof. It is not enough to put one apple with another apple to show that one plus one equals two: it must be proven that one plus one always equals two, that there are no cases in which one and one might make one , or zero, or three, or 1.7453. Often, it is much harder to prove something than to discover it and decide that it is almost certainly true. Sometimes, it takes many centuries for a theorem to be proved, as in the case of Fermat's Last Theorem. But it is the proof that defines a theorem - it must be possible to demonstrate its truth through a line of logical reasoning from axioms and other established theorems.

Tbar

rb~ filII

bar ah~}ayf

,·isell

ir

I/O

proofir ,rill rire tomorl"/JW.

P ROV I NG IT

Problems and proofs It took Jakob Bernoulli 20 years to prove that tossing a coin a large number of times will give close to a 50:50 split between hl':Jds and tails - yet as he pointed Out, th e result is olwious to anyone. \Vhy did he hother? And why did it rake So long? Although the Ancient Egyptians and Babylonians were content to work with specific examples :lnd problems, tht Greeks moved towards theorems ~nd axioms that muld be aJlplied uni\'crsally - they demanded proof. Proving th~t ~n idea holds true requires some kind of logical theoretical treamu:nt, since it is [JOt possible to try Out all Jlossible eases - to test Pythab'oras' theorem for all possihle rightangled triangles, for e.x~mp le. Proof~ aim to find fruitful relations hips between mathematical statements ~nd objects. For this reason, even theorems wbich h~vc been adequately proven in the paSt - slich as Pytha gor~s' tbeorem - may be proven anew, opening up fresh avenue,>:; for exploration. Over time, simpler proo(~ ~re di~covercd and the earli er, often cumbersome, proo f can be replaced. Many developments in mathematics came about as the result of people tcsting and Dying to prove theorems and axioms and even doubting long-h eld beliefu. The dispute over Euclid's fifth poStulate, for in~tance. was th e spur to much of the

'Ead! problem that I solved became a rule which served afterwards to solve other problems.'

Rene Descartes

progress made in geometry and ultimately the emergence of new, non-Euclidean geometries in the 19th century. Rigour in m~thematical proof increased at the end of tbe 19th ct'ntury when mathematics and logic t~me together. A sy~1:elll~tic nOtation for logic came to be used by marhematicians and some philosophers. The development of se t theory required a method of representing: logical relationships ~nd a way of dealing with concepts which did not necessarily involve any numbers ~t alL Set thec.lry even becaml' a useful ml'aIlS of demon.strating matJlL"m~ticaJ theorems. to

UN BELIEVA BLE PR O OFS

A famous problem th~t produces a proof which many people find hard to accept is the Monty Hall paradox. Named after the host of a US game show, it goes like this: Suppose (/ gtJme s};ow host shows yOIl thn:c doors. Behil/d 11UO of thrill tho·r is II golft;

behilld the /ost there is illvites

yOIl

to pick

a etll".

11 dQQI:

Tbe horl

He will tbfIJ

open nl/other door, rt'vrnlil1g (/ gont , ([lid 'Nobody blames a mathematician if the first proof of a new theorem ;! dum!y. '

Paul Erdos

giVl' you the chnllc/'. to change YOl/r cboice. IYill YOllr dUll/ITS of7ll illlli1lg be impmved ifYOII S1J,·itc/J doors? (rbe problem nSSllmes tlmt YOII would rntber hove tI {tn· tl}(ln n gont)

196

PR O BLEMS ANO PR OOfS

Most people say their chances of getting a car are unaffected if they switch doors. Mathematicians say that the chance of getting the car is increased if you switch doors: you had a 1 in 3 chance of choosing a car, 3nd this is unaffeCted by the opt'ning of another door; the chance that you chosc COrrL-ctly is still 1 in 3. If you switch, you 3rc making a new choice, where the chances arc I in 2. Switching will get you a C3f 50 per cent of the time, bur st3ying with the first door will yidd a C3r only 33 per cent of the time. The logic is easier IP follow if you think of 1,000 duors with goats behind 999 of them. Your chances of picking the door with the car the first rime are 1 in 1,000. After 998 b'03ts have llCen released IP run amok, there is a 1 in 2 chance that the other door hides the car. The obvious objection here is that there must 31so be 3 1 in 2 chance that the original door hides the. car, since prohabilitit!S must add IIp t(J one. The trick is that the problem is nOt as it appears. Your choice i~ r3ndom, hut the host IWOWJ where the C3r is. If the host randomly opened doors, coincidentally picking those that conce31ed goats, the ch3llce of finding a car 3t the end would be the same as the chance of finding a goat, whether or nOt you switched doors. The proof of this problem uses mathematical notation to show probabilities and break~ it down into small , logical Steps which naturally follow one from 3nother. This is how mathCIll3tici:ln.~ now demonstrate truth. But it has not always been the case.

Tradition maintains that Thales pro\'ed th3t the angles 3t the base of an isosceles triangle are equal, that 3 diameter CutS a circle inll) two equal parts, that opposite angles formed h}' two intersecting line.5 arc equal 3nd tha two triangles arc identical if any twO angles and one ~ide arc t:qual Sint:t' nonc of Thall'S' writings survives, it is impossihle to S3Y whether he really produced rigorous proofs of these theorems. Around fifty years latcr, Pyrh3goras proved his theorem for right-angled triangles. Since the time of Thales and Pythagoras, the basis of proof In mathematics has been to derive more complicated Statements from faCtS which are apparently simpler (though they may not actually be simpler). Genemlly, anything 1ll ge(.lllu~ try that C3n be demonstrated in clear, logic31 steps from Euclid '.~ postulates counL5 3S proven, for instance. Bllt this does not mean that a new idea is deduced first from the existing f.1Cts. Mathematicians commonly h3vc the idea

DEDUCTIVE PROOF THAT 1 '" 2

let a '" b. So it follows that

a' '" ab a1 +a' =a' +ab 2a' =a ' + ab 2a' - 2ab=a ' + ab _ 2ab 2a' _ 2ab", a'_ ab This ciln be rewritten as 2(a' _ ab) '" 1 (a ' _ ab)

EARLY PROOFS

Dividing bot h sides by a' - ab gives

The earliest known mathem3tical proofS arc said to have been provided by Thalcs.

2=1

197

P ROVIN G IT

first - perhaps as an intuition, or as something suggt'srcd by the re~1Jlts of an experimcnt or an !;!xploration - and then turn to the known fact.~ to prove it. Sometimes, an attempt to find proof refutes the new theory and it must be rejected .

Sometimes, finding a proof appears an intractahlt' problem and the theorem rt'mai ns unproven - for hundreds of years in somt' cascs. PROOF BY DEDUCTION

the Greeks out was refined and defined more rigorously much later is indirect proof. There are several types of indirect proof, including proof by contradiction and proof by reductio ad absurdulJI. Proof by contradiction aims to prove a statement is true by showing that itS opposite is nOt trut'. Proof by redllctio fld abSlmillni aims to prove a statement is true by using it to prove untrue something that is known to be true (so producing an absurd result). Hipassus' proof of the existence of irrational num hers was all indirect proof and is the earli est known.

Proof hy deduction works in small steps to deduce new truths from known truth~. For example, if we say. 'Humans are mammals' and ' Peter is a human' we can then say. PROOF BY INDUCTION 'Peter is a mammal' . Deduction is nl)[ The Greek model of proof was followed by wholly reliahle, even if the initial StatementS the Arab mathematicians and taken over are genuinely true, as the reasoning may nOt from them in the Middle Ages by early be valid. So we might say, 'Humans arc European scholars. But in [575 a new mammals' and 'Peter is a mammal', therefore 'Perer is :::t human' but the fir.~t All HORSES ARE THE SAME COLOUR The Hungarian mathematician George Pa[ya (1887-1985) statementS would al.~o he true if Peter were a dog or a used proof by induction to show that all horses are the hamstt'r or any other same colour. The case fo rn '" 1 (one horse) is dear - a horse mammal. Proof by deduction can only be the same colour as itself. Now assume the is nOt accepted as sufficiently theory is correct for n "" m horses. We have a set of m horses, all the sameco[ou r (1, 2, 3, ... m). There isa second rigormL~ by modern mathematicians, though it set of (m + 1) horses (1, 2, 3, .. . m + 1). We take out one was used extensively by the horse from this last set, so that it contains horses (2, 3, . Ancient Greek..; :mcl by m + 1). The two sets overlap; this second set is a set of m medieval mathematicians. horses, which we know is a set of horses the same colour. Parmenides is credited with By the principle of induction we can continue this for all the first proof by deduction further horses, therefore all horses are the same colour. in the 5th cenrury Be The argument is, of course, invalid as the statement is not true. The crucial point is that when n '" 2 the stat£'ment does INDIRECT PROOF not hold true: fo r this value, the sets do not overlap (the first Another meth od of proof contains only horse 1, the second contains only horse 2). which also originated with 198

PROB U MS ANO PR OOfS

DAVID HILBERT (1862- 1943)

David Hilbert is considered one of the most influential mathematicians of the 19th

He began as a pure mathematician and, when he turned his attention to

and 20th centuries. He was born

1912,

was

what

he

in East Prussi a in lin area that is now

part of Russia. As a

student, he met Hermann

considered

the

approach to

sloppy

math~

most

taken

physicists.

Minkowski and the two

by

stayed lifelong friends,

Hilbert also devised a

each

conceptual space that

mathematical

had infinite dimensions

cross-fertilizing other's ideas.

Hilbert worked in many

(called a Hilbert space). and

his

students

to the maths for his contributions to the

behind Einste.in's Theory of

axiomatization of mathematics.

Relativity and quantum mechanics.

modd emcrg~d; in Ar;lbmeticorlfnl Libri Dllo Francesco Maurolico (1494-15 75) b'llVC thc fi n;;t known description of mathematical induction, though him.~ of this method can be found earlier in worh by Bhaskara :md :ll-Karaji (C.AD lOOO). Proof by inductio n was also developed independen tly by Jako b Bernoulli, Blai se Pascal and Pi erre de Fermat. Proof by induction works by showin g firstly tim a hypothesis holds true for a first value (often n = I), then that it holds true for a later value (.~'ay n '" Ill) and also for the following value (n '" m + I). From the demonstration that it holds trw: for n '" III and n '" m + I it can be interred that this process could he repeated indefinitely to prove that it holds true for all further values. It's a bit like a row of dominoes, arranged on

end and equally spaced SO that if one falls It will knock the next over. If knocking th e fir.~ t do mino ovcr causcs til e ncxt to fall, it will in evit:lbly follow that they will all fall. A1aurolico used proof by induction to demonstrate that the sum of the first 11 odd imeb't'rs is n ~ : 1 + 3 + .5 + 7 + 9 + ... n -'" n

1

ASKING QUESTIONS

VVith the advcnt of calculus, complex numbers and later non-Euclidean bTL'Omctri es, more and morc W:lS denllUlded of proof. Berkel ey's o bjection to calcu lus as dealing with th" 'ghosts of quantitiL'S' was :I spur to greater rigour, nOt o nly in definin g the quantities and concepts with which mathematicians were working but in providing proofs. 199

P ROVIN G IT

The earliest writer Oil logic, in 1945, a boy in eighth grade taking part in a maths Pbru, died ill 347 or 348m:. Plato presents his Olympiad in Russia won first prize even though he did not philosophical works in the attempt to solve even one problem. The prize was foml of dialogues, o r awarded on the basis of a remark he submitted with an unfinished proof: con\'ers~tions, between ' I spent much time trying to prove that a straight line philosophers. They read as argum~nrs, with each can't intersect three sides of a triangle in their interior participant putting fiJn\'ard points but failed for, to my consternation, I realized that I his case in a series of have no notion 01 what a straight line is.' Statements which his opponent tht!l1. r~futes and Bur it '\~JS the 19th century which s:).w he thell defends. The argument uecomt::s the b'Teat revolution in mathematic:).l proof int:rea~ing:ly l"Olllple.x as the subject is as new methods of logic were developed and tackled rigorously. This m ethod, called people for the first time tried to apply dialectic, fonned the model for logical formal logic to mathematics. This requi.red debate until the ."'liddle Ages. Although a reassessment of the very basis of logic was a major conCl~m of these medieval mathematics and brought mathematics and scho lars, they did not think to apply it to philosophy rugNher. Mathematici:).ns, mathematics. It took more than 2,000 years unsettled Ly rL'Cem discoveries that threw for logic and mathematics to come together long-aL'Cepted truths into doubt, sought properly. new proof~ and questioned L"Vcn the most fundamental ideas underpinning their MATHEMATICS BECOMES LOGI CAL discipline. Suddenly, nothing could be taken One of the first ru tadde the issue was the for granted . Italian mathematician Giuseppe Peano (1858- 1932). H e wanted to develop the Being logical whole of mathematics from fundamental At the end of the 19th century and starr of propositions using formal log:ic . He the 20th century there was a flurry of developed a lo&,;c notation, butal~o a hybrid interest in the applieation of logi e to intern~tiollal lan6ruage which he hoped mathematics or, more precisely, the would be used for scho larship. Called derivation of mathematics frOm logic. It lnt~rlin~,'ua, it was based un the vocabulary came about largdy as a rcsult of rapid of Latin, French, German and English, but changes in mathematics and irs applications, used a very simple grammar. His use of it and critici sms of its rigour and \':llidity. hindered the acceptance of his Proof in mathematics is only parr of thc mathematical work. larger tOpic of logic which has developed The breakthrough III relating and grown since the time of the Ancient mathematic.~ to logi c came wlth the work of Greeks.

IGNORANCE EQUALS W ISDOM

'00

rigorolL~

B U NG LOG ICAL

the German logician and mathematician Gotdob Frege (1848- 1925), who has sometimes been called the gre:lt~t IObrici:ln since Aristotle. He set out to prove that all arithmetic could he derived logically from a set of hasic axioms and he is essentially the founder of mathematical logic. H i:' devised a way of represenring IObric using varia hIes and functions. A SEARCH FOR NEW AXIOMS

The German mathematician Davit! I·Elbert laid the foundations for the formalist movement that grew upin the 20th century by requiring that all mathematics should depend on fundamental axioms from whieh everything else ean be proven. He required any syStem to he both complete :lnd consistent, incapable of throwing up any contradictions fi-om the application of its axioms. He reformulated Euclid'.~ axioms himself as the fir.~t step in trying to find this faultless axiomatic basis for maths. Hilhert ramously proposed 23 problems which were still to be solved in 1900. Th~e effectively set Ollt the agenda for 20th century mathematicians. The mOSt important of Hilbert's problems for the dL'Velopment of logie in mathematics is the second . Hc proposes that it is neeessary to ser up a systcm of axio ms 'wh ich contains all exact and complete descrip tion' of the relations between basic ideas and requires ' that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory resultS'. In parricular, this was seen as a call for axioms to prove the basics of Peano arithmetic.

An~...vering Hilbert's call for an axiomatic basis for all mathematics. Bertrand Russell and Alfred North Whitehead published the three-volume PriIJfipia A1atbrmflfica in 1910- 13. Ambiriously named after NeWtOn's seminal work with the same title, the book aims to deriw all of mathematics from a sct of hasic a.xioms using the ."ymbolic logic set forth by Frege. It CO\'Crs only set theory, cardinal numbers, ordinal numbers and real numbers. A planned volume to cover breomt:'-try w"as abandoned as the authors were tired of the work. After getting a good way into the work. Russell discovered that a lor of the ~';J.me ground had been covered by Frcge and he added an appendix pointing Out the differences and acknowled bring Fregc's prior publiL';J.tion. The test of the Prif/cipil/ re.~ted on whether it was complete and consistent in H ilhert's tl'l"ms - L"1)uld a mathematica l statement he found that could not be proven or Jisproven by Prillcipil/'s methods, and cCluld any contradictions be produced using itS axioms?

MOVING THE GOALPOSTS

Before Prillcipia had a chance to stand the test of time, the key questions were taken away by German mathematician Kurt COdel. He produced twO 'incompleteness theorems' ( \ !J31) which dealt wi th Hilbert's proposal for the
P ROVIN G IT

and the system is inconsistent. If G is not prm':l.ble, then it is true and the systcm is incomplete. The second theorem states that basic arithmetic cannot be used to prove it~ own statements and, by extension, can't be used to prove a.nything more complex, either. lOGIC AND COMPUTERS

During the 20th centu ry, the Jt'vcl opnwnt of computers has given logic and mathematics a field of their own . Computer programs use logica l sequences to carry Out calculations. Thi s is the basis of all computer applications, evt::Il those that look nothing like mathematics to the user, such as animation, music production and imab't! processing. Computers can also be used to test the orems. They can produce a proof by exha ustion - which involves trying all possihle va l ue.~ - which a human c(JUlJ not manage. Th ere ~re, toO, cQlllputer programs to construct proof~ by other methods. The program V~mpire, developed at Manchester University hy Andrei Voronkov, h,lS won the 'world cup for theorem prm'crs' six timcs (1999, 2001 - 05). Perhaps th e time has come when computers, with their impeccahle logit, will take over from human mathem:uici~ns as the experts at applying logic to mathematics, or tracing itin mathematics.

What were we talking about? At no point in this book haw we stopped to ask what mathematics is or if, indeed, it 'is' anything. This might seem lik.. a considerable oversight. But on the whokmathematics crept up on humanit)" made 202

itself at home with no introduction and encouraged IL~ to build our cultural edificc

'For any consi!tent forma'- computably enumerable theory that prove! basic arithmetical truth J, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both comistem and complete.' Kur t GOdel, 1931

arouud it. There was no point at which it was scnsibl e to ask what it was all about. At the Stan of the 20th ct'nrury, Illathematic.~ turn ed to fundamental questions ahout its very nature. A central question e:1ll be hridly .~ ummarizcd as 'h Illathematic.~ discovered or invented?' There are three principal positiQns. A Pl~tonist realist vic\\', such ;H that of Kurt G6del, says that the laws of mathemati cs are evt'rywhere true and immumble, like the laws of nature. Ahthematicians discover them; they are pre-exi~ting. A formalist view, such as that taken by D~vid Hilht'rt, says that mathematics is a codification, a I:m!,'1lage or even a game in which theorems are built on axioms throu gh lo gil~J l demonstration . There is no particular re~son to prefer onl! set of axioms over another if both sets seem to hold true. This \'iL'w was dealt a near-futal blow by Giidcl's incomplett'ness thwrems which showed that no set of axioms could be enti rely com plett' and consistent. Finall y, the intuitionist view holds that mathematics is

WHAT W[ R£ WE TAlKING AUOUT1

entirely a fabrication of the human mind, construcred to explain the world we find around us but having no existcnce or v~lidiry outside hum~n culture. This \'itw was propounded by the Dutch mathcm~ticirul L. E.]. Brom\"cr (1881-1966) and he was TcmoT~cl~sly ridiculed and persecuted for it (not lc~st by Hilbcrt). Ovcr the l~st hundred years, the qucstion of the fuulllbtion of mathem~tics has nOt becn ~lllswered, but has slipped Out of vit-,'v. Hilbert's form~list Stance ~"Uffered from the assault of the incompleteness theory, yet logic ~lld axioms still lie at the hean of m~thelll~tics as it is practised. A more mode,f[J vicw is the empiricist one promoted by W. V Quine (i 908-1(00) ~nd Hilary Putn~m (born 1916). They m~intain th~t the existence of numbers ~nd other mathem~tic~l entities c~n he deduced from observation of the n:::d world. It is rc\~ted to realism, hut is morc grounded in re~liry and human culturc. Quine's view is that mathematics seems tQ be 'true' bccause all our c~:pcrience and science is wovC'n around it :tnd aJlpe~rs to endorse it. It would be vcry difficult to rebuild our model of the universe without m~them~tics.

If the last sentence sounds like a challenge, it is onc th:tt was taken up by American philosopher Hartry Field (born 1946). 01 the 1980s he proposed that mathematic~l sto1tcmentS arc all fictiun~l and th~t science can he creatcd without mathcm:ttics. According to his fictional ism dnco-inc, mathem~[ical statementS arC' uscFuI structuring devices, but should nOt be accepted as literally true. And why would we

'make up' mathematics? One an~·wer is that the structure of the hum~n mind makes it inevitable. The embodied minds theory is based in cognitive psychulogy; it was dtveloped for m~thcmatics by American cogmtlve linguistic George Lakoff and psychologist Rafael NllllCZ. Their argument, expressed in their book Wlm·e Mllfbt'lllllfics Comes From (2000), is th~t the Structure of the human brain and the w~}' our bodies operate in the world has dict:lted the way we have developt:d mathematics. A~ we can't divorce ourselves from our brains to examine the universe without our cognitive process!;!s getting in th!;! way, we will nOt be able to tell whether mathematics

'Mathematics may or may not be out there ill the \IVOrld, but there'! no way that we Kielltifically could possibly tell. '

George l akoff, 2001

has any existence outside hum~n culture. Plenty of m~thematil'ians disab'Tl:'.e v,ith Lakoff and NWlez, and with the proponentS of all the oth!;!r ideas outlined here and they will no doubt arb'lle the case for many decades or cenruries to come. The question of its foundations has little impact on our d:ty-to-d:ty use of mathematics . We will carryon playing the lottcry, building aircraft, looking for life in outcr spac!;! ~nd insuring against catastrophes without knowing whethcr lll~them~tics is in all}' sC'nse 'real' and 'out there', just as the Egyptians built their pyramids and th~ Incas counted their llam~s without giving the matter a second thought. 203

G LOSSA RY

GLOSSARY algebraic geometry - geometry using algebraic eguntions and e"~pressions

conics - the faIlljl~' of curves produced by cutting through a cone, or Ule srudy of these curve.'i conjecture - all unproven theorem

algorist - someone who ealcuhtes using the IIindu-Ar~bic number system to earry out arithmetic ratlier than using an ah:1CUS algorithm - a mle for carrying out:1 calculation an3.I}'tlc geometry - geometry using coordinates axiom - a basic h1w which is self evident and reguires no proof

cot3.ngent- tlle ratio of Ule side adjacent to an angle to the side opposite the ~ngle in a right...angled triangle decimal fraction - a number in which frnctiOllal p~rtS ~re expressed :IS ~ decimal (showing tenths, hundredtlls, thOllSnndtJls and so on )

base - ule basis or 01 L"Qunting system; the bnse number is ulnt to which numbers are counted before shifting the pbce value (to tens. hundreds, etc)

differential calculus (differentiation)method for calcubting the slope of a <"lJ1ve a p3rticuhr point

binary - counting system th3l has only t:\vo digil5, I and 0 (base. 2)

Diophantine equation - :In eq uation in which all tlle numbers illl"oh-ed are whole numhers

binomial coefficients - the sequence of coefficients used with vm;:lbles when :I hinomi:ll expre.~sion is exp3nded

fractal - a curve or OUler figure which repe~["~ it'i overJll pattern or sbape ill portions of cOIlSI:lntly reducing sire, so thilt:J portion of the figure when m:l/,'llified looks the same as the \vhole fig-me

calculus - hranch of mathematics concerned \\'i th C3lcllhting the sum of infinitesimal quantitie.~ to approximate the ~rea wIder ~ cun'e or ule rnte of cllange of a curve chord - a straigh t line joining the ends of :In arc (a IlOrtion of the eircumference of ~ circle) coefficient - ~ constl.llt or number hy which a variable is multiplied in all :llgebraic expression commensumble - (or more than one quantity) able to be measured or mmp3red to a conic section - a curve produced by ctJIting a section through a cone

2""

cosine - the ratio of the side ~djacelll. to all angle to the hypotenuse in a right-:mgle(! triangle

~t

function - a !1lal:hem3ticai eXl>ression WiUI one or more variables hyperbola - n cun'e produced hy $licing tltrough a cone with a plane \\~ I:h a small e.r angle nt it.~ :lxis than the side of the CQlle h yperbolic geometry - geometry Ul3 t deals with Sh3pe.'i drnwn on curved surfaces imaginary number - a number which involves the square root of ~ 1

GLOSS ARV

incommensumble - (of more than one quantity) not able to be direcuycompared or measured by tile same standnrd

quadratic equation - an equarion of tlle fonnax2+hx+c",O quinary - bnse 5

infini tesimal towani~ zero

~

very sln:J1l guantity, rending' ntional number ~ a number which c:m he e.xpre!!osed :lS:l r:Jtio of t\\'o whole numbers

integral - tile pronuct of intef:,'l'Jtlon int egral c:llculus (integratio n) - metllOd of calculating Ule area under ~ cun'e by approximating the sum of J large number of infinitely thin slices of tile ::Irea irncio nru number- a number which C':lnnOl be expr~ed :J5 the nUl!) or two wht;lle numbers limit - the lowest or higheSt function will be calculated

\~Jlue to

which a

log:lrithm - the power [Q which ::I b:lSe figure (usually 10 or e) must be r:Jised to gille a specified nu mber

real number ~ any !)()Sitive or neg-drive m unher that does not involve the square mot of -I Riemann geometries - geometries of surtnces which do not follo\\.\ the standard rules of tr:lditiollnl pbnar geometry set' - relnted gl1)up of entities sexagesimal- b:lse 60 sine - tile r:Jtio of tile side opposite an nngJe to the hYJXItemL~e in a righ r~angled trinngle spherical trigonometry - the study of triangles {!rawn on the surface of::l sphere

optics - tile study of lenses, lIision and light parabola - curve produced by slicing through a cone with a plane par:Jllel to tile side of tile cone I)anlle! postulate - Euclid's fiftll posrubte, \.\'hich stntes the condition which must be met for lines not to be P:Jf.ll1el (::Ind so by reversing the postulate gives tile condi tion for lines to be pnrallel) perpendicular -

ilt

polynomial equation - an equation w hich powers of a vnrinble (e.g.

x2+4x+ I =0)

theorem - st:ltement of n rule tllat is not self-evident but which can be proven by logical steps tOf)()logy - the study of geometric properrie~ which are not affected hy ch:mgcs of shape or sIze

righ t :l.IIgles to

perspectille geomeny - tile study of how three - dimensiollalfigure~ appear and em be represented in two dimensions

in\'olve~ non~zero

tangent- tile fJtiO of the side opposite Oln :Ingle to the side :ldjacent to the nngle in n right-:lngled tri:lngle

transfinite numbers- !lumbers which relnte infinities of different m;\brnitudes; so the infinite number of whole num bers is smaller ulnn the infinite number of renl numhers to

triangulation - procedure for measuring or mapping n surt~ce by dividing it into trbngles and Cllcubting dist:lnce~ and nngles

205

INDEX

INDEX

[".,.'S

A

.I.. cus l809, 57 Ahhon, ~:'Iwin AI~)On II~ Ad.. m.. John Q ~incy 61 Ahmes 1'"1')' '''' 71, 94 105 All",,.;, Leon B.. .Igeh ... in Ancient Greece I ll · 5 .rulcoleul .. 159-60 in Chino l!l "'lLUtion. Il6 ·9, 1l0· 1, IH-5 in Europe 110 .rulgeometty In · l,ll6-4 l .",1 Gero!.m" C'II'(UI>O Ill· l in,\[<sopo"mi. Il l · l in,\[i<1,lIeE2lytical Engine 4 5 An.>x:ogo"" 76· 7,95 Ancient Eg)l't froction. in 1 ~ , 19 8""metryin 70).1 n~mhersin 1l, 14_15 trigonometry in ~ l Ancient Greece .Igeh ... in Ill · 5 8""metry in 71·81 n~ml",rs in 16 trigonometry in ~ l · 6 Ancient Rome 8""metry in 6l, ~ 1 · 1 n~mhersin 1l,19· 11 .ngle, trisecting me 7 ~ Antiphon 147 Apollon ... ~O, 96-7 Arbuml>O.. John 170 Archim«les l I, 54, 94- 5, 9 8 , 101, 146-7 Ari.totle 7l .rimmetic .",I.[".cus l809 .rulcoleu!'tingn",chines 41 · 7 .rulcomputers 4 1· 7 .rul co~nting •• Is l6·9 in I"", Empire l6 .",II<>g2I'ithm. 19 in,\[<sopo"mi. l ~ .00 ~ .. ~'e Americons l7 .",1 n~n~",r "hies l ~ Arl'\/ugnu(C",Uno) I ll· J A'pblu" [ 87 AshobmeG ... t 17 Assili>n, Seto 19l ••tro!.he ~9 , 10 ••trononl)' calculus in 16:> trigonometry Ul ~ 5 · 1i, ~ 7 _ 8 .. ion"tic..,t theory 191

ni."

•

Il.>hl"Se, Ch..rles 4 1· 5 Il.>hyloni:1n n",then",tics '" JI/".p.,,,,,,iu Il.>rl",rl" ... ,lox 190

206

n ·5

B.. yes'theo",m 17l·4 B.. yes, Thrn",. 17J· 4 Il«le n heU cu,\'< 179 Beltr::lm~ Eugenio I ll, 115 Berkeley, Bi.hopGeorge IS6 B"mouU~ D .. niel 171 B"mouU~J.l:nh 54, 157·8, 1 6 ~ B"mouU~Jolunn 157·8 B"mouU~ ~icol,", 171 B"mouUi n~ml",rs 4 5 B!.ck,,\hx In Bolpi,J'''''' I ll · 14 Bomhel~, Rof:oel 55, IH, 116 Bo.se, Ah...h"", 109 Bowl"y,Arthur I ~I a",lhu,)" K.y 165 Bourl .. k~ ~icob. 1~ 9 B... hn",gup" 1 7 · 1 ~ , 19, 87, Il~ Bn1bmuKYPtuiiMlklntu (B...hn"gup") 17· 18,

"

Brou"-er, L. E.J. 10l B runeUesch~ Fi~ppo

104- 5 butterflyeff<"" 16S .1. Buli.ni, Ahu .I· \\,.f:o ~~

,

, .. lcubting C lock 4 1 calcubting m""hines 4 1· 7 calculus 149·65 , .. ~ieri, Bono,-en,"", 148 ,IS1 ' .. nlOr,Georg I ~~ , 1 ~9, 110· 1 , .. "Uno, Gerol.mo H, Ill · l, 169 e>rIog"",hy 105-9 , .."bn, Eugene 51 CEllS..,.,. 175 cluos meory 16+5 Chino .I.....,h", in I II .rimmebc in J9 geometry in 64, H numhersin 14 weights in 64 Chongmi,;r.... IOl chmr>og",n" 11 Clmmol"KI oftb,Stb. "".,(.I .Q ifti) 1 7 · 1 ~ circle .",1 conic section. 96-9 .",1 cu"-es 9809 .",1 Pi 9+6 .",1 optics 98-9 "' .... nngof 76· 7,95·6 '.<>.>tes, Roger III C"Iosoos(computer) 46 coml'lex numhers 16l coml'uters binory ""Se !l •• coleu!'tingn",chines 4 1· 7 .",1""",.. 176 computer. re:ouhle numhers 11 .",1 counting.ystem l6 .",1 hex:o,lecin"'II ....,. 1+5 .",1 logic 101 .",1 microchip 46-7 .",1 ... ",Iom nun~"'rs l~l · J, I~ S conic section. 96-9

,1.."

' .on""y,.Iohn I ~ l ' .opemicus, ,I[il:n!.j 90 cui"" d"uhling of 77 cuhits M · l cu"-es n ·9 cydo.1 'Ii D

,I'Alemhert,Je:rn LeRon,1 100,171 De section ih .. conici. (\ \ '.I~.) 16 ,leein,,1 fr.:rctions 17 _JO ,1",lucti,-e proof In ,Iegrees III Demoerit .. 79, 147 D"",rgues, Gi ... ,,1 IW Des=-tes, Rene 55,97 ·8, 99, IW, 115, 1l7·-1O Diff"",,,,,,, Engines 4 1·4 ,Iiff"",nti:rtion 151 · 1 , IH, 159·60 Diodes n ·9 Dioplunti"" "'lLUtion. 11+ 5 DioplunrusofAle"""lri:r Ill · 5 D ... I:e"'l .... tion 1 ~4 Dummer, Geoffrey 46 ,Iyrun~c.ystemmeory 160· 1

~:.rth, circumfereoce of So, 87, 95 e 41 ,5+ 5 Edhert,J.l'resper 4<\ ~:'I,lington, Sir Arth u r 117 Ei"'tein, An",n 117 ElemenlS(Eudi,l) ~ 1, 100 elliptic8""metty II ] · IS E.",AIC 46, 4 7 encrypnrn on DVD. 59 "'l .... tions 116 ·9, IlO· I, 1l+5, ISl "'lui,,,k,,,,,, 190· 1 Er::Ito>menes 49,50, 77 E",her,,\I.C II~

numhers in 10 Eud.1 of Alex:on,lri:r 48 , Sl, ~· I, 100 Eud,,-,us 147 Euler, Leonlu,,1 51, 160, 161 Euler'. i,lentiry H Europe .!geb", in IlO fr.:rctions in l~ · JO ..,,1 HinOmetry in ~ 9-91 "",I "",of zero 57 E"",biusofC.>es'" 50 ,\[ulumn",1 Sl

F.. hnogi D~lungiri II Fern",,, [ ~erre ,Ie 109, 115, I l7, I l9-40, 151, 169 Fern",'-' [--2ining II Fi.her, Ron'!,1 Arlm"r I K'

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114

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