Euler wa s awarde d man y honor s i n hi s life . H e wa s a membe r o f bot h th e
St. Petersbur g an d Berli n Academie s o f Sciences . H e wa s electe d a membe r o f
the Roya l Societ y o f Londo n i n 1749 an d th e Academi e de s Science s o f Pari s i n
1755. Hi s teacher , Johan n Bernoulli , no t a n eas y ma n t o ge t alon g with , calle d
him the incomparable Leonhard Euler an d mathematicorum princeps. Severa l lu -
nar feature s hav e bee n name d afte r him , an d th e lis t o f mathematica l an d othe r
scientific discoverie s name d afte r hi m i s almos t endless : Eule r lin e o f a triangle ,
Euler angle s o f a rotation , Euler-Lagrang e equation s o f th e calculu s o f variations ,
Eulerian integrals , Eule r characteristic , Eule r equation s o f motio n o f soli d bodies ,
Euler equation s o f fluid mechanics, Eule r functio n ^(n) , Euler-Maclaurin su m for -
mula, Euler's constant, Eule r products, an d so on. Th e ten franc bil l in Switzerlan d
has his picture on it. Som e of the problems he worked on are still open, and his work
links wit h a n astonishin g amoun t o f contemporar y researc h i n bot h pur e an d ap -
plied mathematics . Hi s books Introductio in Analysin Infinitorum an d Mechanica,
which wen t throug h man y editions , brough t calculu s an d mechanic s t o th e entir e
scientific world . Ther e wa s n o longe r an y necessit y o f readin g th e obscur e paper s
of Leibniz o r the work s of Newton couche d i n an opaque geometrica l languag e tha t
was unsuitable fo r mos t problems .
To survey Euler's life and work in detail in a single book is impossible. Howeve r
there ar e man y excellen t account s whic h surve y part s o f hi s lif e an d work , an d
I hav e mad e fre e us e o f these . Withou t eve n remotel y attemptin g an y sor t o f
completeness, I hav e liste d som e o f thes e i n [3] . On e o f th e mos t interestin g i s
the accoun t o f A . Wei l [3a] : i t deal s jus t wit h Euler' s wor k o n numbe r theory ,
which occupie s a bare(! ) 4 volume s o f th e 7 0 plu s volume s o f hi s Opera Omnia,
but stil l i s ove r 120 page s long . I n additio n ther e ar e als o th e introduction s t o
the variou s volume s o f Opera Omnia writte n b y expert s whic h dea l i n dept h wit h
his work containe d i n those volumes. Al l of this suggest s tha t a complete scientific
biography of Euler, treatin g all of his work with historical accuracy an d placing it i n
a modern perspective , wil l be so complex that it s length will exceed an y reasonabl e
bound.
A major par t o f Serie s I of Opera Omnia i s concerned wit h analysis . T o Eule r
this very often mean t workin g with infinit e serie s and transformation s o f series an d
integrals. H e was , withou t an y doubt , th e greates t maste r o f infinit e serie s an d
products o f his , an d indee d o f any , time . Perhap s onl y Jacob i an d Ramanuja n
from th e moder n er a ca n evok e comparabl e wonde r an d admiratio n a s formalists .
Before Euler , infinit e serie s an d product s mad e thei r appearanc e onl y i n isolate d
works (Leibniz , Gregory , Wallis , etc. ) an d alway s i n a n auxiliar y manner . Eule r
was th e first t o trea t the m systematicall y an d i n grea t depth , no t onl y fo r thei r
applications, bu t als o fo r thei r ow n intrinsi c interest . I n thi s are a h e discovere d
some of the mos t beautifu l formula e eve r t o be foun d i n mathematics. Hi s ideas o n
summing divergen t series , which he used brilliantly , fo r instance , i n discovering th e
functional equatio n o f th e zet a functio n a t th e intege r points , le d directl y t o th e
modern theor y o f summation o f divergent serie s tha t wa s a t th e cente r o f researc h
in analysi s i n the 19th centur y an d i n the earl y part o f the 20 th (se e [4] , Chs. 1,2).
In recen t year s summatio n o f divergent serie s has agai n becom e powerfu l i n a wid e
range o f problems, rangin g fro m quantu m field theor y t o dynamica l systems .
Previous Page Next Page