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EUCLID-ARCHIMEDES-DIOPHANTUS

In th e fron t o f th e boo k ther e ar e tw o tablet s containin g th e name s o f som e

mathematicians an d their life spans. Th e first on e consists of some mathematician s

of ancient times , from Centra l Asia, China, Greece , and India. Th e second contain s

some o f th e principa l character s i n th e creatio n o f moder n algebra , mostl y fro m

Italy.

What i s mos t strikin g abou t thes e mathematician s i s th e originalit y o f thei r

ideas an d th e fac t tha t afte r al l thes e centurie s thes e idea s ar e stil l aliv e an d in -

spiring moder n research . I n th e developmen t o f mathematic s ther e i s therefor e a

continuity that goe s back all the way to the pre-Christian era . I t is therefore a good

idea to star t b y lookin g briefl y int o som e o f the idea s an d problem s tha t occupie d

mathematicians fro m thes e epoch s an d tr y t o dra w a line fro m the m t o wha t i s of

interest t o u s i n the present .

EUCLID

Euclid is one of the most famous mathematicians of all time. Almos t every edu-

cated person knows about hi m and his work in geometry. No t much is known abou t

his life except tha t h e lived in Alexandria, Greec e around 30 0 BC an d ha d a schoo l

of disciples . Hi s achievement , monumenta l b y th e standard s o f an y epoch , wa s t o

establish Geometr y a s a deductive science based o n a small number o f fundamenta l

principles called axioms. H e did this in a work called the Elements, consistin g of 13

Books [H-E] , which systematically develope d th e geometrical facts abou t triangles ,

circles, an d othe r plana r an d spatia l figures. I t i s not a n exaggeratio n t o sa y tha t

no mathematical treatise has been more influential tha n Euclid' s Elements. Amon g

the mos t famou s o f the theorem s i n the Element s ar e the following .

The su m o f the thre e angle s of a triangle i s equal t o tw o right angles .

The are a o f the squar e o n th e hypotenus e o f a righ t angle d triangl e i s equal to th e

sum o f the area s o f the square s o n the othe r tw o sides .

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http://dx.doi.org/10.1090/mawrld/012/01