Preface

The origins of this book go back to a course on the history o f mathematics tha t

I gav e a t UCL A i n th e winte r quarte r o f 2001. I n mos t universitie s suc h a cours e

follows a standar d curriculu m tha t start s wit h th e Babylonian s an d work s it s wa y

to more recent topics . I decided to do it differently an d focus attentio n o n the work

of a singl e grea t figure. I chos e Euler , attracte d bot h b y hi s universalit y an d th e

great relevanc e o f hi s wor k t o wha t shoul d be , i f no t wha t is , th e undergraduat e

program i n mathematic s today . I discusse d mostl y Euler' s wor k o n infinit e serie s

and products , an d wa s guided b y the chapte r o n Euler i n A . Weil's beautifu l boo k

Number Theory : An Approach through History from Hammurapi to Legendre tha t

had bee n published i n 1984. Th e result, a s far a s I was concerned, wa s predictable .

I fel l unde r th e magica l spel l o f Euler' s personalit y an d mathematics . I t wa s a

huge personal discover y fo r m e to learn how alive Euler's theme s stil l are. I starte d

writing thi s boo k afte r th e course , havin g bee n encourage d b y m y friend s tha t i t

would b e a good thin g t o tr y t o do .

Before I get t o Eule r I really wan t t o emphasiz e a n importan t point . I believ e

that writin g historically abou t mathematic s shoul d no t b e limited t o questions lik e

who did wha t whe n an d t o who m he/sh e wrot e abou t it . I a m o f the opinio n tha t

the histor y o f mathematics an d mathematician s shoul d g o beyond thes e concerns ,

however legitimat e the y are . I fee l tha t a complet e mathematica l histor y shoul d

pay grea t attentio n t o th e historica l evolutio n o f idea s an d ho w the y mes h wit h

what w e know and ar e interested i n today. T o modify a famous remark , th e histor y

of mathematics i s too importan t t o b e lef t entirel y t o historians . N o one ha s don e

this typ e o f historical writin g mor e brilliantl y tha n A . Weil, a s on e ca n tel l b y hi s

historical memoir s tha t ar e scattere d throughou t hi s Oeuvres Scientiftques. I hav e

tried t o follo w hi s example ; ho w fa r I hav e succeede d ca n b e judge d onl y b y th e

readers.

No singl e perso n o r boo k ca n describ e completel y th e many-side d geniu s o f

Euler o r hi s sunn y an d equabl e temperamen t tha t inform s it . I hav e thu s trie d

to limi t m y focus . I hav e bee n concerne d onl y wit h th e tas k o f tellin g wha t th e

themes of Euler were and how they can be connected to current interests . Moreover ,

I hav e limite d mysel f mostl y t o hi s wor k o n infinit e serie s an d product s an d it s

repercussions i n moder n times , namel y th e theor y o f zet a values , an d divergen t

series and integral s (Chapter s 3 and 5) . I n Chapte r 2 I have give n a brief overvie w

of some other part s of his work, for instanc e i n elliptic integrals and numbe r theory .

His work on elliptic integrals is the forerunner o f the modern theory of elliptic curves

and abelia n varieties , an d hi s work o n number theor y raise d question s whic h coul d

only b e understoo d full y afte r th e developmen t o f clas s field theory . I n Chapte r 6

I have sketched a brief accoun t o f the theory o f Euler product s which he started bu t

which reall y starte d t o unfol d wit h th e wor k o f Dirichle t an d which , i n th e cours e

vii