ALGEBRA I N ANCIEN T AN D MODER N TIME S 7

equal i n volum e t o it ; h e i s suppose d t o hav e discovere d i t whe n h e immerse d

himself i n th e bat h tub , an d wa s s o excite d tha t h e ra n ou t withou t an y clothes ,

shouting "Eureka , eureka" ( I have discovered!). Eve n today when someone discovers

something importan t i t i s ofte n th e cas e tha t th e perso n shout s "Eureka" . Fo r

example, whe n Gaus s discovere d th e proo f tha t ever y positiv e intege r i s a su m o f

three triangular number s (th e definitio n o f what i s meant b y a triangula r numbe r

need no t concer n u s here) , h e denote d i t i n a diary

1

h e kep t o f hi s discoverie s a s

follows:

EUREKA. NUMBER= A + A + A

Archimedes wa s a great geomete r a s well as an arithmetician . I n geometr y h e

proved man y proposition s abou t th e circle , sphere , an d othe r geometrica l figures.

In arithmeti c an d it s relatio n t o geometr y h e i s perhaps mos t famou s fo r hi s mea -

surement o f th e circumferenc e o f a circle . I t wa s h e wh o introduce d th e notatio n

7r (the first lette r o f the Gree k word denotin g circumference ) t o denot e the rati o of

the circumferenc e o f a circle to it s diameter , an d mad e th e famou s approximatio n

olO 2 2 1

3— T T — = 3 -

71 7 7

Everyone wh o ha s studie d mathematic s i n schoo l know s th e approximatio n 22/ 7

for7T.

DIOPHANTUS (c . 25 0 A . D )

Diophantus wa s a Gree k mathematicia n whos e work o n Algebr a an d Arith -

metic called Arithmetika exerte d an enormous influence on his successors, especially

the Frenc h Mathematicia n Pierr e d e Ferma t wh o i s generall y regarde d a s th e

father o f modern numbe r theory . Diophantus ' wor k i s supposed t o hav e containe d

13 books, bu t fo r a long time onl y 6 of these wer e available i n Greek . The n i n th e

mid 1970's Arabic translation s o f 4 more of the book s were discovered .

Diophantus was interested in solving equations involving one or more unknow n

variables, and he appears to have been the first t o ask that th e solutions be rationa l

fractions. H e was perhaps awar e that th e solutions may not b e unique although h e

does no t see m t o hav e aske d fo r th e enumeratio n o f all the solutions . Amon g ver y

simple example s o f such problem s ar e th e following .

For a commentar y an d translatio n o f Gauss' s diary , se e th e articl e b y J . J . Gray , A com-

mentary on Gauss's mathematical diary, 1796-1814, with an English translation, Expositione s

Mathematicae, 2(1984), pp . 97-130.