Documenting Riemann'SCOMPLEX

8/11/2019 Documenting Riemann'SCOMPLEX

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E N E U E N S C H W N D E R

Docum enting R iem ann s

Im pact on tho Theory of

C om plex Functions

long w~th Euler Gauss and HzIbert Ber nhard R~emann ~s one of the mos t ~llus-

tmous m ath ema tic ian s of all t~me H~s pro min enc e ~n the fie ld of complex analy-

s~s may be appreciated s~mply by noting that ~n the

athemat~sches Worterbuch

by Naas-Schmld [1961, Vol 2,510-524], the ent nes relate d

to Rmmann's function-theoretical work (Rmmann map-

ping theorem, R mmann differential equation, Rmmann sur-

face, Rlemann-Roch theorem, Rmmann theta-functlon,

Rmmann sphere, Rmmann zeta-function, etc ) take up al-

most as much space as those related to all the work of

Euler or Gauss m total Rmmann's contnbu tlon to com-

plex analysis rests, on the one hand, on his publications,

especially his inaugural dissertation on the foundations of

complex analysis (1851) and his articles on the theory of

hypergeometnc and Abehan functions (1856-57), but his

several lecture courses in this field are also very impor-

tant

These were given in the yea rs fr om 1855 to 1862, and

regularly began with an introductor y general part Rmmann

then tur ned either to the theor y of elliptic and Abehan func-

tions or to hypergeometnc series and related transcen-

dental functions Much of the more advanced parts of

Rmmann's courses were published m his Collected Papers

[Rmmann 1990, 599-692] and m a boo k by H Stahl

[Rlemann 1899], but not his intro ducto ry lectures o n gen-

eral compl ex analysis Thus the latter gradually fell into

obhwon, despite their lntnnslc interest, and despite their

declsl've influence on later developments through the

closely related wntmgs of Dur~ge, Hankel, Koemgsberger,

Neumann, Prym, Roch, and Thomae It there fore seemed

appropnate to publish them m a cntmal edition

[Neuenschwander 1996], m order to make them accessible

to a broader circle of readers For this edition, I also pre-

pared an extensive bibliography of the history of the m~-

pact and influence of Rmmann's function theory

This newly assembled blbhography is mamly intended to

close---at least m the field of complex analysis--the con-

slderable gap existing between the blb hograplues of Purkert

and Neuenschwander, which were appended to the reprint

of Bernh ard R~eman n s Gesam melte Mathemat~sche Werke

[Rmmann 1990] The penod from 1892 to 1944, not system-

atically co~ered there, was scrutnuzed, usmg the indexes of

names m B~blwtheca Mathematzca (1887-1914), and the

Revue semestr~elle des Publ wat wns mathdmat~ques (1893-

1934/35), and ex ammmg the sections on Gescluchte und

Plulosoptue and Funktlonentheone (or Analysis ) m the

abstractmg journal Jahrbuch uber dze Fortschmtte der

Mathe~nahk (1868/71-1942/44) Additionally, I consulted

the holdings of older books in the libraries of the Institutes

of Mathematics at the Umversmes o f Gottmgen and Zurich

Generally, I included only publl cano ns which co ntmned ex-

plw~t reference to Rmmann's work (e g, quotations with

9 1 9 9 8 S P R I N G E R V E R L A G N E W Y O R K V O L U M E 2 0 N U M B E R 3 1 9 9 8 1 9


2/8

e x a c t i n d i c a t i o n o f l o c a t i o n ) , b u t e v e n t h i s c r l t e n o n l e f t

m o r e t h a n 1 ,0 00 o u t o f t h o s e 8 ,0 0 0 ti t l e s p r o v i s i o n a l l y s e -

l e c t e d o n t h e b a s i s o f t h e r e w e w s a n d t h e s y s t e m a t i c h -

b r a r y i n s p e c ti o n , t h e o n g m a l s w e r e t h e n l o o k e d u p , to

m a k e s u r e t h at t h e y m e t t h e c o n d m o n s f o r i n c lu s i o n

N a t u r a ll y , t h e n e w b l b h o g r a p h y Ls m n o w a y c o m p r e h e n -

s w e T h e h t e r a tu r e w [ u c h r e f e r s t o R m m a n n ' s p i o n e e n n g w o r k

i s a l m o s t b o u n d l e s s P u r k e r t , fo r e x a m p l e , e x a m i n i n g a b o u t

t e n j o u r n a l s f r o m t h e f i r st 2 5 y e a r s a f t e r t h e d e a t h o f B e r n h a r d

R i e m a n n , a l r e a d y f o u n d m o r e t h a n 5 00 p u b h c a t m n s r e f e r ri n g

t o t u s w o r k A c c o r d i n g to d a t a b a s e s u r v ey s , t h e c o n t i n u a t i o n

a n d e x t e n s i o n o f P u r k e r t ' s re s e a r c h u p t o t h e p r e s e n t w o u l d

h a v e t o t a k e i n t o a c c o u n t a b o u t 3 0, 00 0 p u b h c a t m n s , a l l o f

w [ u c h o b w o u s l y c a n n o t b e e x a m m e d m c hv ~ du al ly w i t h i n a r e a -

s o n a b l e t u n e N e v e r t h e le s s , I h o p e t h a t t h e n e w b l b h o g r a p h y

w i l l b e c o m e a u s e f u l t o o l f o r f u r t h e r m v e s U g a t ~ o ns

I n t h e f o l l o w i n g I w i l l t r y t o i l l u s t r a t e s o m e o f i t s p o s -

s i b l e a p p li c a t i o n s , b y s u r v e y m g t h e i m p a c t o f R i e m a n n ' s

f u n c t i o n - t h e o r e t i c a l w o r k m t h e f o u r m o s t i m p o r t a n t

E u r o p e a n c o u n t r i e s G e r m a n y , F r a n c e , I t al y , a n d G r e a t

B n t m n S p e c i a l a t t e n t i o n w i ll b e g i v en t o th e d e v e l o p m e n t s

i n G r e a t B n t m n b e c a u s e t h e y h a v e n o t y et b e e n a n a l y s e d

i n d e t a i l F o r m o r e s p e c i f i c m f o r m a U o n , t h e r e a d e r m a y

t u r n t o t h e b i b l i o g r a p h y i ts e l f

G e r m a n y : E a r l y a n d S u s t a i n e d R e c e p t i o n o f

R i e m a n n s M e t h o d s

F o r a p r e h m m a r y n n p r e s s l o n o f t h e s i t u at i on m G e r m a n y , l e t

u s f i rs t l o o k a t t h e r e f e r e n c e s t o R m m a n n ' s w o r k m A u g u s t

L e o p o l d C r e l l e 's m f l u e n t l a l Journal fu r d~e re,he u nd ange-

wandte Mathemat~k I t ~s n o t e w o r t h y t h a t t h e f i r st o f t h e s e

r e f e r e n c e s g o e s b a c k t o H e l m h o l tz [Crelle 5 5 ( 1 8 5 8 ) , 2 5 - 5 5 ] ,

w h o , a s w e k n o w , l a t e r d i s c u s s e d R l e m a n n ' s h y p o t h e s e s

c o n c e r m n g t h e f o u n d a t io n s o f g e o m e t r y H e l m h o l t z w a s

f o l l o w e d i n c h r o n o l o g i c a l o r d e r b y l _ a p s c [ u t z , C l e b s c h ,

C h n s t o f f e l , S c h w a r z , B n l l , F u c h s , G o r d a n , L u r o t h , a n d

W e b e r , a l l o f w h o m , e v e n t h o u g h t h e y w e r e n o t R m m a I m ' s

i m m e d i a t e p u p i ls , d i d a g r e a t d e a l t o d ~ s s e m i n a t e tu s i d e a s ,

a s d i d [ u s o w n s t u d e n t s R o c h , T h o m a e , a n d P r y m

[Crelle

6 1 ( 1 8 6 3 ) - 7 0 ( 18 6 9) ] C l e b s c h a n d B r l l l - - h k e K l e in a n d

N o e t h e r - - p u b h s h e d t h e i r la t e r w o r k p r i m a r i l y in t h e

Mathemat~sche Annalen , w h i c h s t a r t e d t o a p p e a r m 1 86 9,

t h u s , c o n s u l t i n g C r e l l e ' s Journal a l o n e p r o w d e s o n l y m -

c o m p l e t e r e s u l t s f o r t h e m A c c o r d i n g t o o u r b i b l i o g r a p h y ,

o t h e r i m p o r t a n t e a r ly p r o m o t e r s o f R m m a n n 's i d e a s m t h e

G e r m a n - s p e a k i n g w o r l d w e r e C a n t o r , D e d e k m d , D u B o ls -

R e y m o n d , D u r ~ g e, H a n k e l, K o e m g s b e r g e r , N e u m a n n ,

S c h l a f h , a n d , m [ u s l a t e r y e a r s , S c h o t t k y i

I ta l y : E n t h u s i a s t i c A p p r e c i a t i o n o f R i e m a n n s W o r k

A n ~ m p r e ss i ve p m t u r e o f th e e x t e n t t o w [ u c h R m m a n n ' s w o r k

w a s a p p r e c i a t e d m I t a l y i s g w e n b y a s i m i l a r s t u d y o f t h e r e f-

e r e n c e s t o [ u s w r i t i n g s m t h e Annal~ dz Matematwa pura

ed apphcata,

a j o u r n a l w h i c h p l a y e d a n o u t s t a n d i n g r o l e m

t h e d m s e m m a t i o n o f R i e m a n n ' s t h o u g h t s A s e a r ly a s 18 59 ,

E n r i c o B e t t l, w h o w a s t o b e c o m e R i e m a n n ' s f r i en d , t r a n s -

l a t ed [ u s & s s e r t a t m n [Annalz 2 ( 1 8 5 9 ) , 2 8 8 - 3 0 4 , 3 3 7 - 3 5 6 ] , t o

w h i c h h e s o o n r e t u r n e d m a n e x t e n s i v e a r t i c le o n th e t h e -

o r y o f e l l i p t i c f u n c t i o n s [Annah 3 ( 1 8 6 0 ) , 6 5 - - 1 5 9 , 2 9 8 - 3 1 0 ,

4 ( 1 86 1 ), 2 6 - 4 5 , 5 7 - 7 0 , 29 7 - 3 3 6 ] I n t h e s a m e v o l u m e s , w e

a l s o f in d a r e p o r t b y B e t ti o n R i e m a n n ' s t r e a t i s e c o n c e r n -

m g t h e p r o p a g a t i o n o f p la n a r m r w a v e s

[Annah

3 ( 1 8 6 0 ) ,

2 3 2 -2 4 1 ], a n d a r e p o r t b y A n g e l o G e n o c c [ u o n R i e m a n n ' s

i n v e s t i g a ti o n o f th e n u m b e r o f p r i m e s l e s s t h a n a g i v e n

b o u n d

[Annals

3 ( 1 8 6 0 ), 5 2 - 5 9 ] I n a l a t e r v o l u m e o f t h e

s a m e j o u r n a l [Annalz ( 2 ) 3 ( 1 8 6 9 - 7 0 ) , 3 0 9 - 3 2 6 ] , w e f i n d a

F r e n c h t r a n s l a t i o n o f R m m a n n ' s h y p o t h e s e s o n t h e fo u n -

d a t io n s o f g e o m e t r y b y t h e F r e n c h m a t h e m a t i c i a n J u l e s

H o u e l, w h o t y p i c a l l y d i d n o t p u b h s h [ u s t r a n s l a t m n i n a

F r e n c h j o u r n a l , b u t i n t heAnnah W e s h o u l d a l s o m e n t i o n

E u g e m o B e l t r a m l a n d F e h c e C a s o r a t i , t h e l a t t e r h a v m g a l -

r e a d y p r e s e n t e d R m m a n n ' s t h e o r i e s m 1 86 8 m a b o o k

[ C a s o r a t i 1 86 8] a n d m s p e c i a l l e c t u r e s g i v e n m M i l a n

[ A r m e n a n t e & J u n g 1 86 9, C a s o r a U & C r e m o n a 1 86 9] I n w e w

o f t[ u s e x c e l l e n t m t r o d u c t i o n a n d t r m l - b l a zm g , i t i s n o t s u r -

p n s m g t h a t R m m a n n ' s t h e o r i e s w e r e w i d e l y k n o w n m I t a ly ,

a n d t h a t t h e y w e r e q u o t e d m m o r e t h a n t h u - ty a r t i c l e s m t h e

Annalz a l o n e u p t o 1 89 0 2 A s t o R m m a n n ' s o w n s t a y s m I t a ly ,

a n d o t h e r f o l l o w e r s o f R I e m a n n t h e r e , s e e t h e a r t i c l e s o f

B o t t a z z n u , D m u d o n n ( , L e n a , N e u e n s c h w a n d e r , S c h e r m g ,

T n c o n u , V o l t e r r a , a n d W e I1 c i t e d m t h e b i b l i o g r a p h y

F r a n c e : H e s i t a n t R e c e p t i o n

In France, the s i tualaon was ra hcaUy di f ferent Skmmung

t h r o u g h t h e p a g e s o f th e Journal de Mathdmat~ques pures et

apphqudes e & t e d b y J o s e p h L l o uv l ll e , o n e f i n d s a l m o s t n o r e f-

e r e n c e s t o R ] e m a n n ' s p a p e r s b e f o r e 1 87 8, a n d m o t h e r F r e n c h

p u b h c a t~ o n s u p t o 1 88 0 th e y a l s o s e e m t o b e r e l a a v e l y s p a r s e

F u r t h e r m o r e , a c e r t m n c n t ] c a l r e s e r v e a s t o t h e u s e f u l n e s s

o f R m m a n n ' s m e t h o d s q u i t e o f te n c o m e s t h r o u g h B n o t a n d

B o u q u e t, f o r i n s t a n c e , w r i t e i n t h e f o r e w o r d t o th e s e c o n d

e d i t i o n o f t h e i r Thdor~e des fo nc tw ns ell~pt~ques [ B n o t &

B o u q u e t 1 8 75 , I f ]

In Cauchy s theory, the path of the ~mag~nary [complex]

vartable ~s characterized by the movement of a point on

a plane To represent thosefunc tzons whzch assume sev-

eral values fo r the same value of the vamable, Rzem ann

regarded the plane as for med of several sheets, sup emm-

posed and welded together, ~n order to aUow the vartable

to pas s fro m one sheet to another, wh~le t~avers~ng a con-

nect~ng [branch] hn e The concept of a many-s heet ed sur-

fac e presen ts some d~ffienlt~es, despzte the beautiful re-

sults whwh R~emann achieved by th~s method, ~t dzd not

seem to us to offer any adv antage fo r our own ob2ectwe

Cauchy s zdea ~s very suitable for representing multi-

valued fu nc tw ns , ~t ~s suffw~ ent to 2ozn to the value of

the variable the corresponding value of the functzon, and

1For b~bl~ographrcal detatls on part icular ar t icles by these authors c~t~ng R~emann see [Neuenschwander 199 6 131 2 32]

2The authors o f other ar ttc les in the nnah containing references to R~emann are Asco h Beltramt C asoratl Cesaro Chnstoffe l Dtn l Ltpsch~tz Pascal Schlafh Schwarz

Tonelh Volterra etc For detai ls see [Neuenschwander 1996]

2

THE MATHEMATICAL NTELLIGENCER


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w h e n t h e v a ~ a b l e h a s d e s c r i b e d a c l o s e d

c u r v e a n d t h e v a l u e o f t h e f u n c t w n h a s

t h e r e b y c h a n g e d , t o ~n d z c a te t h z s c h a n g e b y

a n ~ n d e x

O n l y H o u e l, m e n t m n e d a b o v e , t r i e d q ui te

e a r l y t o p r o m o t e R l e m a n n s ~ d ea s, a n d , t o -

g e t h e r w i t h G a s t o n D a r b o u x , h e s e v e r a l U m e s

d e p l o r e d t h e i r n o t b e i n g b e t t e r k n o w n m

F r a n c e [ G l s p e r t 1 9 8 5 , 3 8 6 - 3 9 0 e t p a s s i m ] A

r e w e w o f H o u e l s p m n e e n n g b o o k m t h i s f ie ld

[ H o u e l 1 8 6 7 - 1 8 7 4 , p a r t s 1 a n d 2 (1 8 6 7 / 6 8 ) ] c o n -

c l u d e s w i t h t h e f o l l o w i n g , q u i t e m s t r u c t w e

p a s s a g e [ N o u ve l le s A n n a l e s d e M a t h e m a -

t ~ q u e s ( 2 ) 8 ( 1 8 6 9 ) , 1 3 6 - 1 4 3 ]

M a y t h e ~ e c e p t w n o f t h~ s w o r k e n c o u r a g e t h e

a u t h o ~ t o k e e p h ~s p r o m ~ s e t o g ~ v e u s, z n a t h u d

p ar t~ a n e x p o s z t w n o f R z e m a n n s t h eo r i es ,

w h i c h u p to n o w h a v e b e en a l m o s t u n k n o w n

~ n o u t c o u n t r y , a n d w h w h o u t n e i g h b o u r s c u l-

t w a t e u u t h s o m u c h a r d o u r a n d s u c c e s s ~

@

lO

I n s p i t e o f H o u e l s e f f o r t s , t h e s i t u a t i o n m ~ s ~

rance seems to have changed fundamentally

o n l y a f t e r 1 88 0, w h e n a n e w g e n e r a t i o n o f

F r e n c h m a t h e m a h c m n s t o o k o v e r, t h e m o s t / ~ /

p r o m i n e n t a m o n g t h e m w e r e H e n r i P o m c a r e ~

( 1 8 5 4 - 1 9 1 2 ) , P a u l A p p e l l (1 8 5 5 - 1 9 3 0 ) , E m i l e

P l c a r d ( 1 8 5 6 - 1 9 4 1 ), a n d l ~ d o u a r d G o u r s a t

( 1 8 5 8- 1 9 3 6) W i t h t h e s e a u t h o r s t h e r e c e p t m n

o f th e w o r k s o f R m m a n n s f o l lo w e r s ( m m n l y

C l e b s c h a n d F u c h s ) n o w b e g a n t o l e a d t o a

m o r e t h o r o u g h r e a d m g o f R m m a n n s o w n w n t -

m g s - - a s n u m e r o u s q u o t a t i o n s m th e i r p u b h c a -

t m n s p r o v e A s a m a m f e s t a t ] o n o f t h e m c r e a s -

m g a p p r e c l a O o n o f R m m a n n s m e t h o d s a m o n g

F r e n c h m a t h e m a t m m n s a t t h a t t u n e, w e m a y

t a k e t h e f a c t th a t , m 1 88 2, G e o r g e s S n n a r t w r o t e

a d i s s e r t a t a o n o f 1 23 p a g e s , n o w a l m o s t c o m -

p l e t e l y f o r g o t t e n , m w h i c h h e t r i e d t o a c q u a i n t F r e n c h m a t h -

e m a t m m n s w ~ t h R m m a n n s f u n c t m n - t h e o r e t~ c a l tr e a t i s e s I n

S L m a r t s m t r o d u e t m n , w e f i n d th e f o l l o w m g s t a t e m e n t , w t u c h

c o n f i r m s m y o w n a s s e s s m e n t [ S u n a r t 1 88 2, 1 ]

~

d c . / 4 Y 3 7

g z/ / ,Z. /7 3 J-

/ 1~ 3 . , / -,4- d 7

F , g u r e 1 D , s c u s s , o n a n d r e p r e s e n t a t i o n o f a b r a n c h p o i n t o f a n a l g e b r a i c f u n c t , o n

o n a R l e m a n n s u r f a ce , t a k e n f r o m R m m a n n s l e c t u r e s ,n th e s u m m e r s e m e s t e r o f

1 8 61 T h e r e p r o d u c e d p a g e c o m e s f r o m a s e t o f l e c t u re n o t e s m a d e b y E d u a rd

S c h u l t z e w , t h a n n o t a t i o n s b y H e r m a n n A m a n d u s S c h w a r z , , t , s k e p t , n t h e S c h w a r z

N a c h l a s s a t th e A r c h w d e r B e r | , n - B r a n d e n b u r g , sc h e n A k a d e m , e d e r W , s s e n s c h a f t e n

, n B e r l ,n T h e r e e x , s t s a s , m , l a r r e p r e s e n t a t m n m t h e ( p r o b a b l y ) o r , g , n a l n o t e s b y

S c h u l t z e w h , c h a r e n o w k e p t m t h e R a r e B o o k a n d M a n u s c r , p t L i b r a ry a t C o l u m b , a

U n , v e r s lt y , n N e w Y o r k B o t h m a n u s c r , p t s w e r e d , s c o v e r e d b y t h e a u t h o r , i n 1 9 79

a n d 1 9 8 6 , r e s p e c t , v e l y F o r f u r t h e r , n f o r m a t , o n a n d a t r a n s c r , p b o n o f t h e t e x t , s e e

N e u e n s c h w a n d e r 1 9 8 8 a n d 1 9 9 6 , p p 7 5 f a n d 84 , M S S a n d

s

M e m o i r s ~ eq u~ re s a k n o w le d g e o f R w m a n n ~ a n s u r f a ce s

[ a n d t h u s, f ~ n a U y , o f R ~ e m a n n s o w n w r t t ~ n g s ] , t h e u s e

o f w h i c h h a s b e c o m e s t a n d a r d a t c e r t a i n G e r m a n u r n -

w e t s 1 ~ e s

W e k n o w t he m a g m f i c e n t r e s ul ts w h z e h R z e m a n n a r r t v e d

a t ~ n h z s t w o M e m o i r s c o n c e r n z n g t h e g e n e r a l th eo ~? ] o f

a n a l y tz e f u n c t w n s a n d t he t he o r q o f A b e h a n f u n c t w n s ,

b u t th e m e t h o d s w h i c h h e u s e d - - a n d e x p l a i n e d p e r h a p s

t oo s u e c ~ n e t l y - -- a ~ e n o t w e l l k n o w n ~ n F ~ a n c e [ ] A t

t h e s a m e t z m e , h o w e v e r , R ~ e m a n n s m e t h o d c o n t i n u e s t o

h a v e m a n y a d h e r e n t s ~ n G e r m a n y , ~ t s e r v e s a s a ba s ~s

f o r m a n y z m p o r ta n t p u b h c a t w n s b y f a m o u s g e om e t er s ,

s u c h a s K o e m g s b e r g e r , C a r l N e w m a n n [ s~ c], K l e z n ,

D e d e k z n d , W e b e r , P r y m , F u e h s , e t c T h e r e a d i n g o f t h e s e

G r e a t B r it a in A F o r g o t t e n I n t e r e s t

I n c o n t r a s t to w h a t i s k n o w n a b o u t t h e r e c e p h o n o f

R m m a n n s t h e o r i e s m t h e t h r e e c o u n t r i e s a l r e a d y s u r v e y e d ,

t h e i r r e c e p t i o n m G r e a t B n t m n i s h t t l e s tu d m d L e t m e

t h e r e f o r e p r o v i d e a m o r e d e t m l e d a c c o u n t

I n G r e at B n t m n R m m a n n s t h e o r i es s e e m t o h a v e b e -

c o m e k n o w n a n d m o r e w i d e ly d i s s e m i n a t e d o n l y a f te r h is

d e a t h O n e o f t h e fi r st B n t m h m a t h e m a t i c i a n s w h o c i t e d

R m m a n n f r e q u e n t l y w a s A r t h u r C a y l e y , w h o , a l r e a d y m

1 8 65 /6 6 , m e n t i o n e d t h e i n v e s t i g a t i o n s o f C l e b s c h a n d

3Other scattered early ctta ons of R~emann s wnttngs are to be found tn Bertrand Darboux Elhot Emmanuel Herm~te Jonqu~eres Jordan Mane and Tannery For

further informa n see [Neuenschwander 1996]

VOLUME 2 NUMBER 3 1998 21


4/8

R m m a n n o n A b e h a n I n t e g r a l s m h is p a p e r s o n t h e t r a n s -

f o r m a t i o n a n d h i g h e r s m g u l a r ] t m s o f a p l a n e c u r v e C a y l e y

a l s o d i s c u s s e d R m m a n n ' s w o r k i n l a te r y e a r s m g r e a t d e -

t m l a n d w i t h a p p r e c m t m n T h i s c a n b e s e e n f r o m h is n o t e

o n R m m a n n ' s p o s t h u m o u s l y p u b h s h e d e a r ly s t u d e n t p a p e r

o n g e n e r a h z e d i n t e g r a t i o n a n d d i f f e r e n t m t m n ( 18 8 0) , o r

f r o m h i s p r e s ]d e n t m l a d d r e s s t o t h e B n t m h A s s o c l a t m n f o r

t h e A d v a n c e m e n t o f S c i e n c e ( 1 8 83 ) O t h e r e a r l y c ] t a t m n s

o f R m m a n n ' s p a p e r s m G r e a t B r i ta i n a r e b y W T h o m s o n

( 1 86 7 i f ) a n d J C M a x w e l l ( 18 6 9 i f ) T h e y o c c u r m c o n -

n e c t i o n w i t h H e l m h o l t z ' s p a p e r s o n v o r t e x m o r t o n a n d w ~ th

L m t m g ' s s t u d m s o n t o p o l o g y

A f e w y e a r s l a t e r , W K C h f f o r d , H J S S m i t h , a n d J W L

G l m s h e r a l s o t h o r o u g h l y a n a l y s e d R m m a n n ' s p a p e r s a n d ,

o n s e v e r a l o c c a s i o n s , e m p h a s i z e d t h e i r g r e a t i m p o r t a n c e

C l i f fo r d , f o r e x a m p l e , a s e a r l y a s 1 87 3, t r a n s l a t e d R l e m a n n ' s

f a m o u s p a p e r o n t h e h y p o t h e s e s w h i c h l ie at t h e b a s e o f

g e o m e t r y m t o E n g h s h f o r th e j o u r n a l N a t u r e , a n d m 1 8 7 7

h e p u b h s h e d h i s i n f l u e n ti a l m e m o i r o n t h e c a n o n m a l f o r m

a n d d m s e c t m n o f a R m m a n n s u r f a c e I n N o v e m b e r 1 8 76

S m i t h , m h i s p r e s i d e n t i a l a d d r e s s O n t h e P r e s e n t S t a t e a n d

P r o s p e c ts o f s o m e B ~ a n c h e s o f Pu ~ e M a t h e m a t z e s t o t h e

L o n d o n M a t h e m a t i c a l S o c m t y , r e p o r t e d m d e t m l o n

R m m a n n ' s a c h i e v e m e n t s , t a k m g m t o a c c o u n t a s w e l l th e

n e w l y p u b h s h e d f r a g m e n t s m R m m a n n ' s C o l l e c te d

M a t h e m a t i c a l W o r k s S m i t h ' s e l a b o r a t e a d d r e s s s e e m s t o

m a r k a n g u tm l h i g h p o m t i n t h e a p p r e c m t m n o f R m m a n n ' s

w o r k s m G r e a t B n t a m B e c a u s e L t h a s n o t y e t b e e n s t u d -

m d m t h i s c o n t e x t , I w i l l t r e a t I t h e r e m m o r e d e t m l

A t th e b e g m n m g o f h i s a d d r e s s S m i t h o u t l i n e d r e c e n t

p r o g r e s s m n u m b e r t h e o r y a n d m p a r t m u l a r t h e m v e st ~ ga -

t m n s o n t h e n u m b e r o f p r i m e s l e s s t h a n a g i v e n b o u n d H e

w r o t e [ S n u t h 1 8 7 6 - 7 7 , 1 6 - 1 8 ]

A s t o o u t s o f t h e s e r~ e s o f t h e p m m e n u m b e r s

t h e m s e l v e s , t h e a d v a n c e s ~ n ce t h e t z m e o f E u l e r h a s b e e n

g ~ e a t, z f w e t h ~ n k o f t h e d ~ f f z c u l t y o f t h e p r o b l e m , b u t v e~ :y

s m a l l z f w e c o m p a r e w h a t h a s b e e n d o n e w z t h w h a t s t~ U

r e m a i n s t o d o W e m a y m e n t i o n , ~ n t h e f i r s t p la c e , t h e

u n d e m o n s t ~ a t e d , a n d ~ n d e e d c o n g ec t u~ a l , t h e o r e m s o f

G a u s s a n d L e g e n d ~ e a s t o t h e a s y m p t o t z c v a l u e o f th e

n u m b e ~ o f p r i m e s ~ n f e r~ o r t o a g ~ v e n h m ~ t x [ ] T h e

m e m o i r o f B e r n h a r d R ~ e m a n n , U e b er d ~e A n z a h l d e~

P m m z a h l e n u n t e r e ~ n e r g e g e b e n en G r o s s e, c o n t a i n s ( s o

f a r a s I a m a w a r e ) t h e o n l y t n v es t zg a t ~ o n o f t h e a s y m p -

t o t w f r e q u e n c y o f th e p r i m e s w h z c h c a n b e ~ eg a~ d ed a s

~ g o r o u s [ ] N o l e s s ~ m p o ~ t a n t t h a n th e ~ n v e s t z g a h o n

o f R ~ e m a n n , b u t a p p r o a c h i n g t h e p ~ o b l e m o f t h e a s y m p -

t o tz c l a w o f t h e s e ~ e s o f p ~ m e s f l o r a a d ~ f fe r e n t s ~ de , ~ s

t h e c e l eb r a te d m e m o z r , S u r l e s N o t a b l e s t b e m ~ e ~ s , b y

M T c h g b y c h e f [ ] T h e m e t h o d o f M T c h d b y ch e f , p r o -

f o u n d a n d ~ n ~ m ~ ta b le a s s t ~ s, ~s ~ n p o i n t o f a c t o f a v e r y

e l e m e n t a r y c h a r a c te r , a n d z n th z s r es p e c t c o n h a s t s

s t r o n g l y w ~ t h th a t o f R ~ e m a n n , w h z c h d e p e n d s t h ~o u g h-

o u t o n v e r y a b s t r u s e t h e o r e m s o f th e I n te g r a l C a l c u l u s

S i m t h t h en w e n t o n t o s p e a k o f s o m e b r a n c h e s o f a n al y -

s is w h i c h a p p e a r e d t o h n n t o p r o i m s e m u c h f o r t h e u n m e d ]-

a t e fu t u r e H e r e h e f o c u s s e d o n t h e a d v a n c e m e n t o f t h e

I n t eg r a l C a l c u l u s H e m e n t m n e d , a m o n g o t h e r w o r k s ,

R m m a n n ' s m e m o i r o n t h e h y p e r g e o m e t n c s e r i e s a n d t h e u n-

f i m s h e d m e m o i r o n h n e a r d i f f e r e n ti a l e q u a t m u s w ~th a l g e b r m c

c o e f f i ci e n t s I n d o m g s o h e s t r e s s e d t h e g r e a t b e a u t y a n d o n g -

m a h t y o f t h e m a n n ' s r e a s o n m g a n d t h e f er t~ h ty o f t h e c o n -

c e p t l o n s o f C a u c h y a n d o f R m m a n n I n h i s cl o si n g r e m a r k s

S ~ u t h d e s c r i b e d t h e w o r k o f E n g h s h m a t h e m a t ] c m n s m t h e

f i el d d u n n g t h e l a s t t e n y e a r s , n a n u n g a m o n g o t h e r s G l m s h e r ,

C a y l ey , a n d C h f f o r d H e w a s c o n v m c e d t h a t n o t h m g s o h in -

d e r e d t h e p r o g r e s s o f m a t h e m a t m a l s c m n c e m E n g l a n d a s t h e

w a n t o f a d v a n c e d t r e a t i s e s o n m a t h e m a t m a l s u b j e ct s , a n d

t h a t t h e r e a r e a t l e a s t t h r e e t ~ ea t ~ se s w h i c h w e g ~ e a t l y

n e e d - - o n e o n D e f zn ~ t e I n t e g r a l s , o n e o n t h e T h e o r y o f

F u n c t i o n s ~n th e s e n s e ~n w h i c h t h a t p h r a s e z s u n d e r -

s t o od b y t h e sc h o o l o f C a u c h y a n d o f R ~ e m a n n , a n d o n e

( t h o u g h h e s h o u l d b e a bo ld m a n w h o w o u l d u n d e r t a k e

t h e t a s k ) o n t h e H y p e r e l l~ p t ~ c a n d A b e l ~ a n I n t e g r a l s

S m i t h c a l l e d o n h i s c o l l e a g u e s t o c l o s e t h i s g a p , a n d t o

s o m e e x t e n t t h e y d i d 4

A s e a r l y a s a b o u t 1 871 t h e B n t m h A s s o c m t m n f o r t h e

A d v a n c e m e n t o f S c m n c e s e t u p a s p e c i a l C o m m i t t e e o n

M a t h e m a t ] c a l T a b l e s t o w h i c h b e l o n g e d , b e s i d e s S m i t h, A

C a y l e y , G G S t o k e s , W T h o m s o n , a n d J W L G l m s h e r T h e

p u r p o s e s f o r w h i c h t h e C o m m i t t e e w a s a p p o i n t e d w e r e ( 1)

t o f o r m a s c o m p l e t e a c a t a l o g u e a s p o s s i b l e o f e x m t m g

m a t h e m a t m a l t a b l e s , a n d ( 2 ) t o r e p n n t o r c a lc u l a t e t a b l e s

n e c e s s a r y f o r t h e p r o g r e s s o f t h e m a t h e m a t i c a l s c m n c e s

T h e C o m m i t t e e d e c ~ d ed to b e g m w ~ th t h e f i rs t t a s k , a n d m

187 3, p r e s e n t e d b y t h e c a r e o f G l m s h e r a h u g e c a t a l o g u e

o f 175 p a g e s o n m a t h e m a t i c a l t a b l e s w h i c h w a s p n n t e d i n

4 In a l e t te r addressed to I T odhu n te r Smi th g~ves a s~m~la r more d t rec t and succ inc t es t ima te o f the p rese n t s ta te o f Ma them at i cs tn Grea t t3n ta~n France and G erma ny

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A l l t h a t w e h a v e o n e m a y s a y c o m e s t o u s f r o m C a m b n d g e f o r D u b hn h a s n o t o f l a te q ut te k e p t u p t h e p r o m t s e s h e o n c e g a v e F u r t h e r I d o n o t t h ~ nk t h a t w e h a v e

any th ing to b lush fo r ~n a com pans on wt th F rance bu t F rance ~s a t the l ow es t ebb ~s consc iou s tha t she ~s so and ~s mak ing g re a t e f fo r t s to recov er he r los t p lace

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s e e N e u e n s c h w a n d e r 1 9 7 8 , e s p e c i a ll y p p 4 , 1 9, 7 3 f f

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t m u a t m n o f t i u s r e p o r t w a s p r e s e n t e d , c o m p i l e d b y C a y l e y

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b e r 4 , 18 76 I n t h e s e p a p e r s G l a l s h e r t h a n k e d

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m a n n 1 88 2 f , G r a m 1 8 84 f f , L o r e n z 1 8 9 1) ,

w h e r e R m m a n n ' s f u n c t m n - t h e o r et m a l w o r k

h a d a l r e a d y e x c i t e d i n t e r e s t b e f o r e , a s i s

s h o w n b y t h e p u b l i c a t i o n s o f B o n s d o rf f ,

M l t t a g - L e ff l e r , a n d s o m e o t h e r s M o r e o v e r , m

1 88 4 W W J o h n s o n p u b h s h e d a d e t a i l e d a c -

c o u n t o f J a m e s G l a l s h e r ' s f a c t o r t a b l e s a n d

t h e d l s t n b u t m n o f p r i m e s m t h e A m e r i c a n

j o u r n a l Annals of Mathemahcs w h i c h m t r o -

d u c e d R m m a n n ' s m v e s t i g a t m n s a n d t h e w o r k

o f t h e B r i t is h m a t h e m a t l c t a n s m t h e N e w

W o r l d

F i n a l l y , a t t e n t i o n s h o u l d b e p r o d t o

A n d r e w R u s s e l l F o r s y t h , w h o w e n t t o l i v e m

t h e t o w n o f C a m b r i d g e a n d s t u d i e d n o t h i ng

b u t m a t h e m a t m s m t h e s a m e y e a r , 1 87 6, t h a t

S m i t h a n d G l m s h e r s t a r t e d t o d r a w a t t e n t m n

t o R l e m a n n ' s w o r k I n h i s o b i t u a r y n o t i c e,

E T W h i t t a k e r d e s c r i b e d F o r s y t h ' s Theory of

Functwns ( 1 8 93 ) a s h a w n g h a d a g r e a t e r i n -

f l u e n c e o n B r it i s h m a t h e m a t i c s t h a n a n y w o r k s i n c e

N e w t o n ' s Pnnc~p a A c c o r d i n g t o W h i t t a k e r [ 19 42 , 21 8 ],

p e r h a p s t h e m o s t o r i g i n a l fe a t u r e o f th ~s w o r k w a s t h e

m e l d i n g o f t h e t h r e e m e t h o d s a s s o c m t e d w i t h t h e n a m e s

o f C a u c h y , R m m a n n , a n d W e m r s t r a s s , w h i c h m t h e c o n t i-

n e n t al b o o k s w e r e r e g a r d e d a s s e p a r a t e b r a n c h e s o f m a t h -

e m a t l c s F u r t h e r m o r e , o n e c a n r e a d m t h e R o y a l S o c m t y

B m g r a p h l c a l M e m o i r o f H e n r y F r e d e r i c k B a k e r b y W V D

H o d g e t h at F o r s y t h r e n d e re d C a m b n d g e m a t h e m a t i c s

g r e a t s e r v i c e b y h i s e f f o rt s to b n n g a b o u t c l o s e r c o - o p e r a -

t i o n b e t w e e n m a t h e m a t ~ c i a n s m t h is c o u n t r y a n d t h o s e o n

t h e c o n t i n e n t o f E u r o p e , a n d h e m a d e i t e a s m r f o r h~ s

5In add i t i on t o Sm i t h s Lon don a ddres s d~ scussed a bove see a l so h~s pap er

On the ~ntegrat~on of d~scont~nuous functions

[1875) w=th a deta= led analys~s of R~emann s

m e m o i r o n t h e r e p r e s e n t a ti o n o f a f u n c t io n b y a t n g o n o m e t n c s e n e s h ~s p a p e r On some discontinuous senes considere d by R~ernann 1881) and h~s M e m o i r o n t h e

Theta and Omega unc tsons w h i c h w a s w n t t e n t o a c c o m p a n y t h e T a b l e s o f t h e T h e t a F u n c t io n s c a l c u l a t e d b y J W L G la ~ sh e r

JOLUME 20 NUMBER 3 199 8 2 3


6/8

successors, mclu(hng Baker, to get the full benefit of the

work of the great German masters of the late nmeteenth

century

As can be seen from W[uttaker's obituary notice and

Forsyth's own recollections of his undergraduate days

[Forsyth 1935], there is a defimte link betwee n For syt h's

later work m complex analysis and [us early studies in

Cambridge, where he came mto repeated contact w~th

Glmsher, Cayley, and the works of Smith, who taught at

Oxford Cayley and Glmsher were also to be very instru-

mental in Forsyth's caree r in pure mathematic s and helped

[urn to overco me that Cambridge atmos phere m which

all were reared to graduation on a pphed m athemat ics ,,6

Accorchng to Whittaker, it was Glmsher who suggested to

Forsyth that he write his fLrst book, the

T r e a t i s e o n d ~ f -

f e r en t ~ a l e q u a t w n s

(1885) His first two princ ipal memo irs

on the ta functions (1881/1883) and on Abel's the orem and

Abehan functions (1882/1884) were presented by Cayley,

on the other hand, to the Royal Society In the first mem-

oir Forsyth gives a list of 22 prmcipal papers in the field,

mcluchng among others Rmmann's T h e o r t e d e r A b e l s c h e n

F u n c t w n e n a s well as twelve other papers by the German

mathe mati cian s Jacobi , Rlchelot, Rosenhmn, Gopel,

Wemrstrass, Koemgsberger, Kummer, Borchardt, and

Weber The second memoir on Abel's theorem and Abeha n

functions contained a large section on Wemrstrass's ap-

proach, which clearly shows tha t around 1882 Forsyth was

already fully aware of the achievements of the German

mathematicians in this field

o n c l u s i o n

These bibhograplucal mvestlgaUons make it evident that

Rmmann's ideas were m ore posltavely acce pted m Great

Britain, apparently even earher, than m France It seems the y

were among the more important stimuh wluch, through the

efforts of Cayley, Clifford, Slmth, Glmsher, etc, later led to

the flounshmg of Enghsh pure mathematacs and function

theory under Hobson, Forsyth, Mathews, Baker, Barnes,

Hardy, IAttlewood, T~tchmarsh, and many othe rs In England

and Italy Rmmann's theories entered open temtorie s, where

they found mathematmal commumtms eager to catch up m

pure math ematics In England this interest serve d to

broad en to the Cambridge atmosphere, which was nearly

totally oriented towards natural philosophy and apphed

mathematics, m Italy it fitted m w~th a desire to built up a

modern mathematical education for the newly founded na-

tion (cf N euensc hwander 1986) In France, on the other

hand, Riemann' s ideas first faced a cold receptaon from the

well-esta bhshed tradition of Cauchy and Briot & Bouquet

Allowing for a certmn p olitical ly motiv ated enthus iasm,

Welerstrass was therefore proba bly right when, m a letter

to Casorati dated 25 March 1867 (written m the early pe-

riod before the publication of Rmmann's Collected

Mathemat ical Works m 1876), he espec ially empha sized

Italy's role m the dissemination of Rmmann's ideas In that

letter (see [Neuen schwan der 1978, 72]) 7 he writes

T h e h a p p y r ~ se o f s c w n c e ~n y o u r f a t h e r l a n d w a ll n o w h e r e

e l se b e f o l l o w e d w z t h m o r e v z v z d z n t e r e s t t h a n h e r e z n

N o r t h G e r m a n y a n d y o u m a y b e c e r t a i n t h a t t h e State

o f I t a l y h a s n o w h e r e e l se s o m a n y s ~ n c e r e a n d d ~ s~ n te r -

e s t e d f r i e n d s W z t h pl ec a~ u re w e a r e t h e r e f o r e r e a d y t o co n -

t ~ n u e t h e a l l i a n c e b e t w e e n y o u a n d u s - - w h w h ~ n t h e p o -

l ~ t w a l f i e l d h a s h a d s u c h g o o d r e s u l t s - - -a l s o ~ n s c w n c e s o

t h a t a ls o ~n t h a t a~ e a th e b a r r i e r s m a y m o r e a n d m o r e b e

o v e rc o m e w ~ t h w h w h u n h a p p y p o l i t i c s h a s f o r s o lo n g s e p-

a r a t ed t w o g e n e r a l ly c o n g en i a l n a t w n s T h e p a pe ~ w h z c h

y o u s e n t t o m e p r o v e s t o m e o n c e a g a i n t h a t o u r s c i e n t i f i c

e n d e a v o u r s a r e b e t te r u n d e r s t o o d a n d a p p r e c z a t e d ~ n I t a l y

t h a n ~ n F ~ a n c e a n d E n g l a n d p a r t z c u l a r l y ~ n t h e l a t t e r

c o u n t r y w h e r e a n o v e r w h e l m i n g f o r m a h s m t h r e a te n s t o

s t i f l e a n y f e e l z n g f o r d e e p e r ~ n ve s t~ g a t~ o n H o w s ~ g n ~ f i ca n t

~ t ~s t h a t o u r R w m a n n w h o s e lo s s w e c a n n o t s u f f i c i e n t l y

d e p lo r e ~ s s t u d ~ e d a n d h o n o u r e d o u t s i d e G e r m a n y o n l y

~ n I t a l y ~ n F r a n c e h e ~s c e r t a i n l y a c k n o w l e d g e d e x t e r n a l l y

b u t l~ t tl e u n d e r s t o o d a n d ~ n E n g l a n d [ a t l e a s t b e f o r e 1 8 6 7 ]

h e h a s r e m a i n e d a l m o s t u n k n o w n

N o t e a d d e d t n p r o o f

In a recent issue of this journal

(vol 19, no 4, Fall 1997) there app ear ed an article by

Jeremy Gray w[uch gives the r eader a very welcome sum-

mary m Enghsh of Riemaim's introduct ory lectures on gen-

eral complex analysis Gray's pape r is largely based on my

preprmt

R ~ e m a n n s V o r le s u ng e n z u r F u n k t ~ o n e n th e o m e

A l l g e m e ~ n e r T e ~ l

Darmstadt 1987 and on Roch 1863/65

(cf Neue nsch wand er 1996, p 15) but does not mentio n the

substantially expande d version which app eared m Sprmg

1996 and which has been reviewed m MR 97d 01041 and

Zbl 844 01020 The publis hed version also contams, be-

sides what was m the original prepnnt, the above-men-

tioned extensive bibliography of papers and treatis es that

were influenced by Rmmann's work, as well as a list of all

known lecture notes of [us courses on complex analysis

From this list [Ne uensc hwande r 1996, 81 ff ], one can in-

fer that Cod Ms Rmmann 37 comp rise s notes of three dif-

ferent courses (not just one, as suggeste d by Gray), and its

first part is thereby quite easily datable to the Winter se-

me st er 1855/56 On page 111 of Cod Ms Rmmann 37 one

6Accordtng to Forsyth s own recol lecttons as ear ly as h~s s tude nt years he to ok an exceptional ~nterest tn pure mathematics and w ent for one term in hts th ird year to

Cayle y s lectures whe re at the beginning the very wo rd plunge d him ~nto complete bewilderment From othe r passages of Forsyth s recol lections o ne may see that

already at that t~me he made h~s f i rs t t imid ventures outs ide the range of Cambridge textbo oks and plo ugh ed throug h amo ng others a large part of Durege s

lhpt~sche

Funct~onen

Further detai ls ma y be gathered from the fo l lowing personal confess ion by Forsyth Som ething of d~fferentta l equations be yon d mere examples the ele

mer i ts of Jacobean elhpttc functtons and the mathem attcs o f Gauss s metho d of least squares I had learnt from Glatsher s lectures and b y worktng at the matter of

the one course by Cayley whtch had be en attended I bega n to unde rstand that pure mathemattcs w as mo re than a col lectton of random to ols matnly fashioned for

use ~n the Cam bndg e treatment of natural phtlosophy Otherwtse ve ry near ly the whole of such k now ledge of pure m athematics as ~s m~ne bega n to be acquired only

a f te r my Tnpos degree In tha t Cambn dge a tmosphere we

ll

were reared to graduation on appl ied mathematics [Forsyth 1935 172]

7Nottce that h~s charactertzatton of th e s~tuat~on ~n England ts s tr tk tngly s tm~lar to th at o f Smtth see note 4) o r Forsyth see note 6)

2 4 THE MATHEMATICAL NTELLIGENCER


7/8

A U T H O I

ERWIN NEUENSCHW NDER

Instrtut fur Mathemattk

Abt Re~ne Mathemabk

Un~vers~tat Zunch-lrchel

Wtnterthurerstrasss 190

8057 Zunch Switzerland

Erw~n Neuenschwander ts Professor of the History of

Mathematics at the University of Zurich He took his degrees

under B L van der Waerden ~n Zurich He has published sev-

eral papers on ancient, medieval, and n~neteenth-century

mathematics, an d especially on Rtemann, to wh om h~s latest

boo k ~s devoted Dunng the pest cou ple of years, he has also

take n an ~nterest ~n he h~story of the othe r exact so|en ces and

on th e ~nteracttons between science, philosophy, and society

The p~cture shows h~m before the tom b of Rtemann at Selesca

near Verban~a, Italy) on the occes~on of t he Verban~a R~emann

Memo nal Co nferen ce ~n July 1991

reads Fortsetzung ;m Somme rsem este r 1856 [Continua-

tion m the Summe r semeste r of 1856], and on page 193 of

the same manuscript Theone der Functmnen complexer

Groigen mlt besonderer Anwendung auf die Gauf~'sche

Re;he

F a,/3,

% x) und verwandte Transcendenten Winter-

semester 1856/57 Furthermore, ;t should be noted that

Rmmann's lectures on general co mplex analys;s developed

gradually from 1855 to 1861 [Neuenschwander 1996, 11 f ],

and that m 1861 he also treated We|erstrass's fac tonzat mn

theorem for enUre functions w;th prescribed zeros, intro-

ducing to th;s end logarithms as convergence-producing

terms [Neue nschwa nder 1996, 62-65], which ;s quite re-

markable, although not mentioned by Gray 1997

c k n o w l e d g m e n t s

I am indebted to Ro bert B Burckel (Manhattan, Kansas),

Ivor Grattan-Gumness (Enfield), and Samuel J Patter son

(Gottmgen) for various suggestmns concernmg both con-

tent and language The present art;cle was written under a

grant from the Swiss National Foundation

BIBLIOGR PHY

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egreg~ Professort Br~osch~ Cre mo na e Casorat~] G/omale d~ Mate-

mat/che 7 1869), 224-234

Bottazzin~, U Rtemanns E~nflu6 auf E Bett~ und F Casoratt Archive

for History of Exact Sciences 18 1977/78), 27- 37

Bottazztnl U II calcolo subhme stona dell anahsl matematlca da Euler

a We~erstrass Bonnghterl, Tonno 1981 Revised and enlarged English

version Springer, New York 1986

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pisana In O Montaldo, L Grugnettl e ds) La stena delle matemat~che

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Unlverstta dl Cagharl, Cagharl 1983 229 -27 6

Bottazztnt, U Rtem ann tn Itaha In Conferenza Intemazlonale n el 125 ~

Ann~versano della morte d~ Georg Fnednch Bernha rd R~emann 20

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edit ion Gauthter-Wllars P ans 1875

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1879

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CasoratJ, F, Cre mon a L Intorno al num ero det moduh delle equaz~on~o

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vol 1,313-315 L Cremona, Opere matemat/che, vol 3, 128-132

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1986, 245-26 3

Forsyth, A R Old Tnpo s days at Cam brid ge The Math emabcal Gazette

19 1935), 162-179

GiIhsple, C C ed ) D~ctlonary of scientific b iogra phy, 16 vols Scnbner,

New York 1 970-1980

Gispert, H Sur la production mathe mattque frangatse en 1870 Etude

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Intemabonales d H~sto~re des Sciences 35 1985), 380-399

G~spert, H La France mathemat~que La soctete mathe mattq ue de

France 1870 -191 4) Sutw de ctnq etude s par R B kouohe, C Gtla~n

C Houzel, J -P Kahane, M Zerner Cah~ersd H~sto~reet d e Ph~losoph~e

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Houel, J Theone elementalre des quant~tes complexes Gauth~er-

Vdlars Pans, 1867-1874

Klein F R~ernann und seine Bed eutung fur d~e Entw~ckelung der mo

dernen Mathemattk Jahresbencht der Deutschen Mathemat/ker-

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mat/sche Abhandlungen, Bd 3, 482- 497 English translation Bulletin

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Klein, F Vorlesungen uber die Entw~cklung der Mathemaflk ~m 19

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Reale Acc ade mla de~ L~nce~ Sene qutnta Rendtcontt, Cla sse d~ sc~enze

fis~che, mate mat iche e natural~ 24 1 1915), 516 -531

Lorta, G Stona delle matemabche 3 vol STEN Torino 1 929-19 33

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Neuen schwand er, E Der Nachla8 ven Casoratt 1835 -189 0) ~n Pawa

Archive for History of Exact Sciences 19 1978), 1-89

VOLUME 2 NUMBER 3 19 98 25


8/8

Neue nschw ande r E Uber dte Wechselwtrku ngen zw~schen der franzo-

sische n S chule, Rte mann u nd Wererstral3 E~ne Ubers tcht m~t zwe~

Quellenstudten

Archive for History of Exact Sciences

24 (1981),

22 1- 25 5 Enghsh version tn

Bullet in of the Amencan Mathematical

Society

New Ser 5 (1981) 87- 105

Neuen schwan der, E Der Aufsch wung d er ~tahenlschen Math ema tik zur

Zest der pol~t~schen Etnigung Itallens und seine Ausw~rku ngen a uf

Deutschland In Symposia Mathemat~ca 27 (1986), 213-237

Neuen schwan der, E A bnef repor t on a numbe r of recently d~scov-

ered sets of notes on R~emann's lectures and on the transmtsston of

the R~emann Nachlass H~stona Mathemat~ca 15 (1988), 101-113

Reprinted ~n R~emann 1990, 855-867

Neue nschw ande r E Seco ndar y hteratu reon B R~emann In R~emann

1990, 896-910

Neuenschwander, E

R/emanns E/nfuhrung ~n die Funkt/onentheone

Etne quellenknt/sche Edit/on se/n er Vorlesunge n m /t e~ner B~bl~ograph~e

zur W~rkungsgeschlchte der RJem annsch en Funkt~onentheone

Abh and lun gen d er Akadem~e der W~ssenschaften ~n Gott~ngen,

Mathemabsch-Physikahsche Klasse, Dnt te Folge Nr 44 Vande nhoeck

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Elhpt~sche Funct~onen

Vorlesungen von Bernhard

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Leipzig 1899

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Gesammelte mathe -

mat/sche Werke von Ernst Schenng Bd 2 Berlin 1909 36 7-38 3

43 4-4 47 Partially reprinted ~n R~emann 1990, 827 -84 4

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Commenta/re sur deux Memo/res de R/emann relatlfs a la

theone generale des fonct~ons et au pnnc/pe de D/nchlet

Theses presen-

tees a la Faculte des Science s de Pans Gauthler-V~llars, Paris 1882

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of pure mathemabcs

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8 (1876-77), 6-29 =

Collected Ma thematical Papers

vol 2, 166-190

Tncomi F G Bernha rd Riemann e I ' l taha

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Tonno Rend~cont~ del Sem~nano Matematlco

25 (1965-66), 59-72

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deuxleme Co ngres Inte rnatio nal des MathematJclens tenu a Pans du 6

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at~che

vol 3, 1-11

We~l A R~emann, Beth and the b~rth of top olo gy Archive for H/story

of Exact Sciences 20 (1979), 91-9 6 21 (1979/80), 387

WhFttaker, E T An dr ew Russell Forsyth In Obitu ary Notices o f Fellows

of the Royal Society 19 42-1944

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Teubner, Letpzlg, 1905 121- 139 Repnnted in Riemann 1990 719 -73 8

5 THE MATHEMATICAL NTELLIGENCER

Documents

Documenting Riemann'SCOMPLEX