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1902. ] HILBERT : MATHEMATICAL PROBLEMS. 437

node curve is fundamental in this theory. A number ofCremona's theorems on ruled surfaces with straight line

directrices are generalized to apply to all ruled surfaces.

Br. Wilczynski gives both analytic and synthetic proofs ofthese theorems. The third covariant furnishes another

congruence associated with a given surface, and in partic

ular a third ruled surface associated with the original oneand the one already mentioned. A few brief remarks aremade, showing how these covariant surfaces may serve tosimplify the integration of the original system of differen

tial equations. This paper will be combined with the previous paper on covariants for publication in the Transactions.

E. J. WILCZYNSKI.

MATHEMATICAL PKOBLEMS.*

LECTURE DELIVERED BEFORE THE INTERNATIONAL C O N GRESS OF MATHEMATICIANS AT PARIS IN 1900.

B Y P R O F E S S O R D A V I D H I L B E R T .

W H O of us would not be glad to lift the veil behind

which the future lies hidden; to cast a glance at the next

advances of our science and at the secrets of its development

during future centuries ? What part icular goals will there

be toward which the leading mathematical spirits of coming

generations will strive ? What new methods and new facts

in the wide and rich field of mathematical thought will thenew centuries disclose ?

History teaches the continuity of the development ofscience. We know that every age has its own problems,

which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea

of the probable development of mathematical knowledge inthe immediate future, we must let the unsettled questions

pass before our minds and look over the problems which

the science of to-day sets and whose solution we expect

from the future. To such a review of problems the present

day, lying at the meeting of the centuries, seems to mewell adapted. For the close of a great epoch not only

invites us to look back into the past but also directs ourthoughts to the unknown future.

* T ranslated for the B U L L E T I N , with the author's permission, by Dr.M A R Y W I N S T O N N E W S O N . The original appeared in the Göttinger Nach-richten, 1900, pp. 253-297, and in the Archiv der Mathernatik una P hysik,3dser. , vol. 1 (1901), pp. 44-63 and 213-237.

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4 3 8 HILBERT : MATHEMATICAL PROBLEMS. [J u ty ?

T h e d e e p s i g n i f i c a n c e o f c e r t a i n p r o b l e m s f o r t h e a d v a n c eof m a t h e m a t i c a l s ci e nc e i n g e n e r a l a n d t h e i m p o r t a n t r ô lew h i c h t h e y p l a y i n t h e w o r k o f t h e i n d i v i d u a l i n v e s t i g a t o ra r e no t to be de n ied . A s long as a b r an ch of sc ience o ffe rsa n a b u n d a n c e of p r o b l e m s , so l o n g is i t a l i v e ; a la c k of p r o b l e m s f o r e s h a d o w s e x t i n c t i o n o r t h e c e s s a t i o n o f i n d e p e n d e n td e v e l o p m e n t . J u s t a s e v e r y h u m a n u n d e r t a k i n g p u r s u e sc e r t a i n o b j e c t s , s o a l s o m a t h e m a t i c a l r e s e a r c h r e q u i r e s i t sp r o b l e m s . I t i s b y t h e s o lu t i o n of p r o b l e m s t h a t t h e i n v e s t i g a t o r t e s t s t h e t e m p e r o f h i s s t e e l ; h e f i n d s n e w m e t h o d s

a n d n e w o u t l o o k s , a n d g a i n s a w i d e r a n d f r e e r h o r i z o n .I t i s d i f f icu l t and o f ten imposs ib le to judge the va lue o f ap r o b l e m c o r r e c t l y i n a d v a n c e ; fo r t h e f in a l a w a r d d e p e n d su p o n t h e g a i n w h i c h s c i e n c e o b t a i n s f r o m t h e p r o b l e m .N e v e r t h e l e s s w e c a n a s k w h e t h e r t h e r e a r e g e n e r a l c r i t e r i aw h i c h m a r k a go o d m a t h e m a t i c a l p r o b le m . A n o ld F r e n c hm a t h e m a t i c ia n s a i d : " A m a t h e m a t i ca l t h e o r y i s n o t t ob e c o n s i d e r e d c o m p l e t e u n t i l y o u h a v e m a d e i t s o c l e a r t h a ty o u c a n e x p l a i n i t t o t h e firs t m a n w h o m y o u m e e t o n t h es t r e e t . " T h i s c l e a rn e s s a n d e a s e of c o m p r e h e n s i o n , h e r ei n s i s t e d o n f or a m a t h e m a t i c a l t h e o r y , I s h o u l d s t i l l m o r e

d e m a n d f o r a m a t h e m a t i c a l p r o b l e m i f i t i s t o b e p e r f e c t ;fo r w h a t i s c le a r a n d e a s i ly c o m p r e h e n d e d a t t r a c t s , t h e c o m p l i c a t e d r e p e l s u s .

M o r e o v e r a m a t h e m a t i c a l p r o b l e m s h o u l d b e d iffi cu lt i no r d e r t o e n t i c e u s , y e t n o t c o m p l e t e l y i n a c ce s s i b le , l e s t i tm ock a t ou r e f fo r ts . I t shou ld be to us a gu id e po s t ont h e m a z y p a t h s t o h i d d e n t r u t h s , a n d u l t i m a t e l y a r e m i n d e ro f o u r p l e a s u r e i n t h e s u c c e s s f u l s o l u t i o n .

T h e m a t h e m a t i c i a n s o f p a s t c e n t u r i e s w e r e a c c u s t o m e dto de vo te them se lv es to th e so lu t ion of d i fficu l t jpa r t i eu la rp r o b l e m s w i t h p a s s i o n a t e ze a l. T h e y k n e w t h e v a l u e of

d iffi cu lt p r o b l e m s . I r e m i n d y o u o n l y of t h e " p r o b l e m o ft h e li n e of q u i c k e s t d e s c e n t / ' p r o p o s e d b y J o h n B e r n o u l l i .E x p e r i e n c e t e a c h e s , e x p l a i n s B e r n o u l l i i n t h e p u b l i c a n n o u n c e m e n t of t h i s p r o b l e m , t h a t l o ft y m i n d s a r e le d t os t r i v e f o r t h e a d v a n c e o f s c i e n c e b y n o t h i n g m o r e t h a nb y l a y i n g b e fo r e t h e m d iffi cu lt a n d a t t h e s a m e t i m e u s e fu lp r o b l e m s , a n d h e th e r e f o r e h o p e s t o e a r n t h e t h a n k s of t h em a t h e m a t i c a l w o r ld b y fo l lo w i n g t h e e x a m p l e of m e n l i k eM e r s e n n e , P a s c a l , F e r m â t , V i v i a n i a n d o t h e r s a n d l a y i n g b e f o r e t h e d i s t i n g u i s h e d a n a l y s t s o f h i s t i m e a p r o b l e m b y w h i c h , a s a t o u c h s t o n e , t h e y m a y t e s t t h e v a l u e o f

t h e i r m e t h o d s a n d m e a s u r e t h e i r s t r e n g t h . T h e c a l cu l u so f v a r i a t i o n s o w e s i t s o r i g i n t o t h i s p r o b l e m o f B e r n o u l l ia n d t o s i m i l a r p r o b l e m s .

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1 9 0 2 . ] H I L B E R T : M A T H E M A T IC A L P R O B LE M S . 4 3 9

F e r m â t h a d a s s e r t e d , a s i s w e l l k n o w n , t h a t t h e d i o p h a n -t i n e e q u a t i o n

xn

+ yn=z

n

(x, y a n d z i n t e g e r s ) i s u n s o l v a b l e — e x c e p t i n c e r t a i n self-e v i d e n t ca s e s. T h e a t t e m p t t o p r o v e t h i s i m p o s s i b il it yof fe r s a s t r ik ing example o f the in sp i r ing e f fec t wh ich sucha v e r y s p e c i a l a n d a p p a r e n t l y u n i m p o r t a n t p r o b l e m m a yh a v e u p o n s ci en c e. F o r K u m m e r , i n ci te d b y F e r m â t ' sp r o b l e m , w a s l e d t o t h e i n t r o d u c t i o n o f i d e a l n u m b e r s a n d

t o t h e d i s c o v e r y o f t h e l a w o f t h e u n i q u e d e c o m p o s i t i o n o ft h e n u m b e r s of a c i r c u l a r fi el d i n t o i d e a l p r i m e f a c to r s — al a w w h i c h t o - d a y , i n i t s g e n e r a l i z a t i o n t o a n y a l g e b r a i cfie ld b y D e d e k i n d a n d K r o n e c k e r , s t a n d s a t t h e c e n t e r oft h e m o d e r n t h e o r y of n u m b e r s a n d w h o s e s i gn i fi ca n ce e x t e n d s f a r b e y o n d t h e b o u n d a r i e s o f n u m b e r t h e o r y i n t o t h er e a l m o f a l g e b r a a n d t h e t h e o r y o f f u n c t i o n s .

To speak o f a ve ry d i f fe ren t r eg ion o f r esea rch , I r emindy o u of t h e p r o b l e m of t h r e e b o d i e s . T h e f ru i t fu l m e t h o d sa n d t h e f a r - r e a c h i n g p r i n c i p l e s w h i c h P o i n c a r é h a s b r o u g h ti n t o c e le s t i a l m e c h a n i c s a n d w h i c h a r e to - d a y r e co g n i ze d a n d

a p p l i e d i n p r a c t i c a l a s t r o n o m y a r e d u e t o t h e c i r c u m s t a n c et h a t h e u n d e r t o o k t o t r e a t a n e w t h a t d i f f i c u l t p r o b l e m a n dt o a p p r o a c h n e a r e r a s o l u t i o n .

T h e t w o l a s t m e n t i o n e d p r o b l e m s — t h a t o f F e r m â t a n dt h e p r o b l e m of t h e t h r e e b o d ie s — s e e m t o u s a l m o s t l i k e o p p o s i t e p o l e s— t h e f o r m e r a fr ee i n v e n t i o n of p u r e r e a s o n , b e l o n g i n g t o t h e r e g i o n o f a b s t r a c t n u m b e r t h e o r y , t h e l a t t e rfo r ce d u p o n u s b y a s t r o n o m y a n d n e c e s s a ry t o a n u n d e r s t a n d i n g of t h e si m p l e s t f u n d a m e n t a l p h e n o m e n a of n a t u r e .

B u t i t o f te n h a p p e n s a l s o t h a t t h e s a m e s p e c ia l p r o b l e mfin d s a p p l i c a t i o n i n t h e m o s t u n l i k e b r a n c h e s of m a t h e

m a t i c a l k n o w l e d g e . S o , fo r e x a m p l e , t h e p r o b l e m o f t h es h o r t e s t l i n e p l a y s a ch ie f a n d h i s t o r i ca l l y i m p o r t a n t p a r ti n t h e f o u n d a t i o n s o f g e o m e t r y , i n t h e ' t h e o r y o f c u r v e dl i n e s a n d s u r f a ce s , i n m e c h a n i c s a n d i n t h e c a l c u l u s ofv a r i a t i o n s . A n d h o w c o n v i n c in g l y h a s F . K l e i n , i n h i sw o r k o n t h e i c o s a h e d r o n , p i c t u r e d t h e s i g n i f i c a n c e w h i c ha t t a c h e s t o t h e p r o b l e m o f t h e r e g u l a r p o l y h e d r a i n e l e m e n t a r y g e o m e t r y , i n g r o u p t h e o r y , i n t h e t h e o r y o f e q u a t i o n s a n d i n t h a t o f l i n e a r d i f f e r e n t i a l e q u a t i o n s .

I n o r d e r t o t h r o w l i g h t o n t h e i m p o r t a n c e o f c e r t a i np r o b l e m s , I m a y a ls o r e fe r to W e i e r s t r a s s , w h o s p o k e of i t

a s h i s h a p p y f o r t u n e t h a t h e fo u n d a t t h e o u t s e t of h i s sc ie n t if ic c a r e e r a p r o b l e m s o i m p o r t a n t a s J a co b i n s p r o b l e mo f i n v e r s i o n o n w h i c h t o w o r k .

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440 HILBERT : MATHEMATICAL PROBLEMS. [July,

H a v i n g n o w r e c a l le d t o m i n d t h e g e n e r a l i m p o r t a n c eof p r o b l e m s i n m a t h e m a t i c s , l e t u s t u r n t o t h e q u e s t i o nfr o m w h a t s o u r c e s t h i s s c ie n c e d e r i v e s i t s p r o b l e m s . S u r e l yt h e f ir st a n d o l d e s t p r o b l e m s i n e v e r y b r a n c h of m a t h e m a t i c s s p r i n g f r o m e x p e r i e n c e a n d a r e s u g g e s t e d b y t h ew o r l d of e x t e r n a l p h e n o m e n a . E v e n t h e r u l e s of ca l cu l a t i o n w i t h i n t e g e r s m u s t h a v e b e e n d i s c o v e r e d i n t h i s f a s h i o n i n a l o w e r s ta g e of h u m a n c i v i li z a t io n , j u s t a s t h e c h i l dof t o - d a y l e a r n s t h e a p p l i c a t io n of t h e s e l a w s b y e m p i r i c a lm e t h o d s . T h e s a m e i s t r u e of t h e fir st p r o b l e m s of g e o m

e t r y , t h e p r o b l e m s b e q u e a t h e d u s b y a n t i q u i t y , s u c h a s t h ed u p l i c a t i o n o f t h e c u b e , t h e s q u a r i n g o f t h e c i r c l e ; a l s o t h eo l d e s t p r o b l e m s i n t h e t h e o r y of t h e s o l u t i o n of n u m e r i c a le q u a t i o n s , i n t h e t h e o r y of c u r v e s a n d t h e d i ff e re n t i a la n d i n t e g r a l c a l c u l u s , i n t h e c a l c u l u s of v a r i a t i o n s , t h et h e o r y of F o u r i e r s e r ie s a n d t h e t h e o r y of p o t e n t i a l — t os a y n o t h i n g o f t h e f u r t h e r a b u n d a n c e o f p r o b l e m s p r o p e r l yb e l o n g i n g t o m e c h a n i c s , a s t r o n o m y a n d p h y s i c s .

B u t , i n t h e f u r th e r d e v e l o p m e n t of a b r a n c h of m a t h e m a t i c s , t h e h u m a n m i n d , e n co u r a g e d b y t h e s u cc e ss of i t ss o l u t i o n s , b e c o m e s c o n s c io u s of i t s i n d e p e n d e n c e . I t e v o l v e s

f rom i t s e l f a lo ne , o f ten w i th ou t app rec iab le in f luence f romw i t h o u t , b y m e a n s of l o g i ca l c o m b i n a t i o n , g e n e r a l i z a t i o n ,s p e c i a l i za t i o n , b y s e p a r a t i n g a n d c o l le c t in g i d e a s i n f o r t u n a t e w a y s , n e w a n d f r u i t f u l p r o b l e m s , a n d a p p e a r s t h e n i t s elf a s t h e r e a l q u e s t i o n e r . T h u s a r o s e t h e p r o b l e m of p r i m en u m b e r s a n d t h e o t h e r p r o b l e m s o f n u m b e r t h e o r y , G a l o i s ' st h e o r y o f e q u a t i o n s , t h e t h e o r y o f a l g e b r a i c i n v a r i a n t s , t h et h e o r y of a b e l i a n a n d a u t o m o r p h i c f u n c ti o n s ; i n d e e d a l m o s ta l l t h e n i c e r q u e s t i o n s of m o d e r n a r i t h m e t i c a n d f u n ct io nt h e o r y a r i s e i n t h i s w a y .

I n t h e m e a n t i m e , w h i l e t h e c r e a t i v e p o w e r o f p u r e r e a s o n

i s a t w o r k , t h e o u t e r w o r l d a g a i n c o m e s i n t o p l a y , f o r c e su p o n u s n e w q u e s t i o n s f r o m a c t u a l e x p e r i e n c e , o p e n s u pn e w b r a n c h e s of m a t h e m a t i c s , a n d w h i l e w e se e k t o c o n q u e rt h e s e n e w f i e l d s o f k n o w l e d g e f o r t h e r e a l m o f p u r e t h o u g h t ,w e oft en fin d t h e a n s w e r s t o o l d u n s o l v e d p r o b l e m s a n d t h u sa t t h e s a m e t i m e a d v a n c e m o s t s u c ce s s fu l ly t h e o l d t h e o r i e s .A n d i t s e e m s t o m e t h a t t h e n u m e r o u s a n d s u r p r i s i n g a n a l o g i e s a n d t h a t a p p a r e n t l y p r e a r r a n g e d h a r m o n y w h i c h t h em a t h e m a t i c i a n s o o f t e n p e r c e i v e s i n t h e q u e s t i o n s , m e t h o d sa n d i d e a s o f t h e v a r i o u s b r a n c h e s o f h i s s c i e n c e , h a v e t h e i ro r i g i n i n t h i s f e v e r - r e c u r r i n g i n t e r p l a y b e t w e e n t h o u g h t a n d

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

m a y b e j u s t l y l a id d o w n f or t h e s o l u t i o n of a m a t h e m a t i c a l

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1 9 0 2 . ] HILBERT : MATHEMATICAL PROBLEMS. 4 4 1

pro b le m . I sho u ld s ay fi rs t of a l l , th i s : th a t i t sh a l l bep o s s i b l e t o e s t a b l i s h t h e c o r r e c t n e s s o f t h e s o l u t i o n b y m e a n sof a f in i te num be r of s te ps base d up on a f in i te n um be r ofh y p o t h e s e s w h i c h a r e i m p l i e d i n t h e s t a t e m e n t o f t h e p r o b l e m a n d w h i c h m u s t a l w a y s b e e x a c t ly f o r m u l a t e d . T h i sr e q u i r e m e n t o f l o g i c a l d e d u c t i o n b y m e a n s o f a f i n i t e n u m b e r of p r o c e s se s i s s i m p l y t h e r e q u i r e m e n t of r i go r i n r e a s o n i n g . I n d e e d t h e r e q u i r e m e n t o f: r i g o r , w h i c h h a s b e c o m ep r o v e r b i a l i n m a t h e m a t i c s , c o r r e s p o n d s t o a u n i v e r s a l p h i l o s o p h i c a l n e c e s s i t y o f o u r u n d e r s t a n d i n g ; a n d , o n t h e o t h e r

h a n d , o n l y b y s a t is fy i n g t h i s r e q u i r e m e n t d o t h e t h o u g h tc o n t e n t a n d t h e s u g g e s t i v e n e s s o f t h e p r o b l e m a t t a i n t h e i rfu l l e ffect . A new p rob lem , espec ia l ly w he n i t com es f romt h e w o r l d of o u t e r e x p e r i e n c e , i s l i k e a y o u n g t w i g , w h i c ht h r i v e s a n d b e a r s f r u i t o n l y w h e n i t i s g r a ft e d c a r e fu l l y a n di n a c c o r d a n c e w i t h s t r i c t h o r t i c u l t u r a l r u l e s u p o n t h e o l ds t e m , t h e e s t a b l is h e d a c h i e v e m e n t s of o u r m a t h e m a t i c a lsc ience .

B e s i d e s i t i s a n e r r o r t o b e l i e v e t h a t r i g o r i n t h e p r o o f i st h e e n e m y of s i m p l i c it y . O n t h e c o n t r a r y w e f in d i t c o n firm e d b y n u m e r o u s e x a m p l e s t h a t t h e r i g o r o u s m e t h o d is

a t t h e s a m e t i m e t h e s i m p l e r a n d t h e m o r e e a s i ly co m pr eh en de d . T h e ve ry e ffo r t fo r r igo r fo rces us to find ou ts i m p l e r m e t h o d s o f proof. I t a l s o f r e q u e n t l y l e a d s t h e w a yt o m e t h o d s w h i c h a r e m o r e c a p a b l e o f d e v e l o p m e n t t h a n t h eo l d m e t h o d s of l e s s r i g o r . T h u s t h e t h e o r y of a l g e b r a i cc u r v e s e x p e r i e n c e d a c o n s i d e r a b l e s i m p l i f i c a t i o n a n d a t t a i n e dg r e a t e r u n i t y b y m e a n s o f t h e m o r e r i g o r o u s f u n c t i o n - t h e o r e t i c a l m e t h o d s a n d t h e c o n s i s t e n t in t r o d u c t i o n of t r a n s c e n d e n t a l d e v i c e s. F u r t h e r , t h e p ro o f t h a t t h e p o w e r s e r ie sp e r m i t s t h e a p p l i c a t i o n o f t h e f o u r e l e m e n t a r y a r i t h m e t i c a lo p e r a t i o n s a s w e l l a s t h e t e r m b y t e r m d i ff e re n t ia t io n a n d

i n t e g r a t i o n , a n d t h e r e c o g n i t i o n o f t h e u t i l i t y o f t h e p o w e rs e r i e s d e p e n d i n g u p o n t h i s p r o o f c o n t r i b u t e d m a t e r i a l l y t ot h e s i m p l i fi c a ti o n o f a l l a n a l y s i s , p a r t i c u l a r l y of t h e t h e o r yof e l i m i n a t i o n a n d t h e t h e o r y of d i f fe r e n t ia l e q u a t i o n s , a n da l s o o f t h e e x i s t e n c e p r o o f s d e m a n d e d i n t h o s e t h e o r i e s .B u t t h e m o s t s t r i k i n g e x a m p l e fo r m y s t a t e m e n t i s t h e ca l c u l u s of v a r i a t i o n s . T h e t r e a t m e n t of t h e fi rs t a n d s e c o n dv a r i a t i o n s of d e f in i te i n t e g r a l s r e q u i r e d i n p a r t e x tr e m e l yc o m p l i c a t e d c a l c u l a t i o n s , a n d t h e p r o c e s se s a p p l i e d b y t h e o l dm a t h e m a t i c i a n s h a d n o t t h e n e e d fu l r ig o r . W e i e r s t r a s ss h o w e d u s t h e w a y t o a n e w a n d s u r e fo u n d a t i o n of t h e

c a l c u l u s of v a r i a t i o n s . B y t h e e x a m p l e s of t h e si m p l e a n dd o u b l e i n t e g r a l I w i l l sh o w b ri e fl y , a t t h e c lo s e of m y l e c t u r e ,h o w t h i s w a y l e a d s a t o n c e t o a s u r p r i s i n g s i m p l i f i c a t i o n o f

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4 4 2 HILBERT : MATHEMATICAL PROBLEMS. [ J u l y ,

t h e c a l c u lu s of v a r i a t i o n s . F o r i n t h e d e m o n s t r a t i o n of t h eneces sa ry and suf f ic ien t c r i t e r ia fo r the occur rence o f a maxi m u m a n d m i n i m u m , t h e c a l c u l a t i o n o f t h e s e c o n d v a r i a t i o na n d i n p a r t , i n d e e d , t h e w e a r i s o m e r e a s o n i n g c o n n e c t e d w i t ht h e fi rs t v a r i a t i o n m a y b e c o m p l e t e ly d i s p e n s e d w i t h — t o s a yn o t h i n g o f t h e a d v a n c e w h i c h i s i n v o l v e d i n t h e r e m o v a l o fthe r es t r ic t ion to va r ia t ions fo r wh ich the d i f fe ren t ia l coef f i c i e n t s o f t h e f u n c t i o n v a r y b u t s l i g h t l y .

W h i l e in s i s t i n g o n r i g o r i n t h e p ro o f a s a r e q u i r e m e n t fo ra p e r f e c t s o l u t i o n o f a p r o b l e m , I s h o u l d l i k e , o n t h e o t h e r

h a n d , t o o p p o s e t h e o p i n i o n t h a t o n l y t h e c o n c e p t s o f a n a l y s is , o r e v e n t h o s e of a r i t h m e t i c a l o n e , a r e s u s c e p t i b l e of a f u ll yr i g o r o u s t r e a t m e n t . T h i s o p i n i o n , o c ca s io n a l ly a d v o c a t e db y e m i n e n t m e n , I c o n s id e r e n t i r e l y e r r o n e o u s . S u c h a o n e s i d e d i n t e r p r e t a t i o n o f t h e r e q u i r e m e n t o f r i g o r w o u l d s o o nl e a d t o t h e i g n o r i n g o f a l l c o n c e p t s a r i s i n g f r o m g e o m e t r y ,m e c h a n i c s a n d p h y s i c s , t o a s t o p p a g e of t h e flo w o f n e wm a t e r i a l fr o m t h e o u t s i d e w o r l d , a n d f in a ll y, i n d e e d , a s alas t conse quen ce , to th e r e jec t ion of the idea s o f th e cont i n u u m a n d of t h e i r r a t i o n a l n u m b e r . B u t w h a t a n im p o r t a n t n e r v e , v i t a l t o m a t h e m a t i c a l s ci e n ce , w o u l d b e c u t b y

t h e e x t i r p a t i o n o f g e o m e t r y a n d m a t h e m a t i c a l p h y s i c s I O nt h e c o n t r a r y I t h i n k t h a t w h e r e v e r , fr om t h e s i d e of t h et h e o r y o f k n o w l e d g e o r i n g e o m e t r y , o r f r o m t h e t h e o r i e s o fn a t u r a l o r p h y s i c a l s ci e n ce , m a t h e m a t i c a l i d e a s c om e u p ,t h e p r o b l e m a r i s e s f o r m a t h e m a t i c a l s c i e n c e t o i n v e s t i g a t et h e p r i n c i p l e s u n d e r l y i n g t h e s e i d e a s a n d s o t o e s t a b l i s ht h e m u p o n a s i m p l e a n d co m p l e t e s y s t e m of a x io m s , t h a t t h ee x a c t n e s s of t h e n e w i d e a s a n d t h e i r a p p l i c a b i l i t y t o d e d u c t i o n s h a l l b e i n n o r e s p e c t i n f e r i o r t o t h o s e o f t h e o l d a r i t h m e t i c a l c o n c e p t s .

T o n e w c o n c e p t s c o r r e s p o n d , n e c e s s a r i l y , n e w s i g n s .

T h e s e w e c h o o s e i n s u c h a w a y t h a t t h e y r e m i n d u s o f t h ep h e n o m e n a w h i c h w e r e t h e o c ca s io n fo r t h e f o r m a t i o n oft h e n e w c o n c e p t s . S o t h e g e o m e t r i c a l fig u re s a r e s i g n s o rm n e m o n i c s y m b o l s o f s p a c e i n t u i t i o n a n d a r e u s e d a s s u c hb y a l l m a t h e m a t i c ia n s . W h o d o es n o t a lw a y s u s e a l o n gw i t h t h e d o u b l e i n e q u a l i t y a > b > c t h e p i c t u r e of t h r e ep o i n t s fo l lo w i n g o n e a n o t h e r o n a s t r a i g h t l in e a s t h eg e o m e t r ic a l p i c t u r e of t h e i d e a " b e t w e e n ' ' ? W h o d o e sn o t m a k e u s e of d r a w i n g s of s e g m e n t s a n d r e c t a n g l e s e n c lo s e d i n o n e a n o t h e r , w h e n i t i s r e q u i r e d t o p r o v e w i t hper fec t r igo r a d i f f icu l t theo rem on the con t inu i ty o f func

t i o n s o r t h e e x i s t e n c e of p o i n t s o f c o n d e n s a t i o n ? W h ocou ld d i spense w i th the f igure o f the t r i ang le , the c i rc lew i t h i t s c e n t e r , o r w i t h t h e c ro s s of t h r e e p e r p e n d i c u l a r

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1902.] HILBERT : MATHEMATICAL PROBLEMS. 443

a x e s ? O r w h o w o u l d g i v e u p t h e r e p r e s e n t a t i o n of t h ev ec to r f i e ld , o r th e p ic tu r e o f a f ami ly o f cu rv es o r su r f acesw i t h i t s e n v e l o p e w h i c h p l a y s s o i m p o r t a n t a p a r t i n d i f f e r e n t i a l g e o m e t r y , i n t h e t h e o r y of d i ff e re n t i a l e q u a t i o n s , i nt h e f o u n d a t i o n of t h e c a l c u l u s of v a r i a t i o n s a n d i n o t h e rp u r e l y m a t h e m a t i c a l s c ie n ce s ?

T h e a r i t h m e t i c a l s y m b o l s a r e w r i t te n d i a g ra m s a n d t h eg e o m e t r i c a l f i g u r e s a r e g r a p h i c f o r m u l a s ; a n d n o m a t h e m a t i c i a n c o u l d s p a r e t h e s e g r a p h i c f o r m u l a s , a n y m o r e t h a n i nc a l c u l a t i o n t h e i n s e r t i o n a n d r e m o v a l o f p a r e n t h e s e s o r t h e

u s e o f o t h e r a n a l y t i c a l s i g n s .T h e u s e of g e o m e t r i c a l s i g n s a s a m e a n s of s t r i c t p r oo fp r e s u p p o s e s t h e e x a c t k n o w l e d g e a n d co m p l e te m a s t e r y oft h e a x i o m s w h i c h u n d e r l i e t h o s e f i g u r e s ; a n d i n o r d e r t h a tt h e s e g e o m e t r i c a l f i g u r e s m a y b e i n c o r p o r a t e d i n t h e g e n e r a lt r e a s u r e of m a t h e m a t i c a l s i g n s , t h e r e i s n e ce s s a r y a r i go r o u s a x io m a t i c i n v e s t i g a t i o n of t h e i r c o n c e p t u a l c o n t e n t .J u s t a s i n a d d i n g t w o n u m b e r s , o n e m u s t p l a c e t h e d i g i tsu n d e r e a c h o t h e r i n t h e r i g h t o r d e r , s o t h a t o n l y t h e r u l e so f ca lcu la t io n , i. e ., t h e a x i o m s o f a r i t h m e t i c , d e t e r m i n e t h eco r r ec t u se o f th e d ig i t s , so th e u se o f geo met r ica l s ign s i s

d e t e r m i n e d b y t h e a x i o m s of g e o m e t r ic a l co n c e p t s a n d t h e i rc o m b i n a t i o n s .T h e a g r e e m e n t b e t w e e n g e o m e t r i c a l a n d a r i t h m e t i c a l

t h o u g h t i s s h o w n a l s o i n t h a t w e d o n o t h a b i t u a l l y f ollo wt h e c h a i n o f r e a s o n i n g b a c k t o t h e a x i o m s i n a r i t h m e t i c a l ,a n y m o r e t h a n i n g e o m e t r ic a l d i s cu s s i o n s . O n t h e co n t r a r yw e a p p l y , e s p e c i a l ly i n fir st a t t a c k i n g a p r o b l e m , a r a p i d ,u n c o n s c i o u s , n o t a b s o l u t e l y s u r e c o m b i n a t i o n , t r u s t i n g t o ac e r t a i n a r i t h m e t i c a l f e e l i n g f o r t h e b e h a v i o r o f t h e a r i t h m e t i c a l s y m b o l s , w h i c h w e c o u l d d i s p e n s e w i t h a s l i t t l e i na r i t h m e t i c a s w i t h t h e g eo m e t ri ca l i m a g i n a t i o n i n g e o m e t r y .

A s a n e x a m p l e of a n a r i t h m e t i c a l t h e o r y o p e r a t i n g r i g o ro u s l yw i t h g e o m e t r ic a l id e a s a n d s i gn s , I m a y m e n t i o n M i n k o w s k i ' s w o r k , D i e G e o m e t r i e d e r Z a h l e n . *

S o m e r e m a r k s u p o n t h e d iffi cu lt ie s w h i c h m a t h e m a t i c a lp r o b l e m s m a y o ffe r, a n d t h e m e a n s of s u r m o u n t i n g t h e m ,m a y b e i n p l a c e h e r e .

I f w e d o n o t s u cc ee d i n so l v i n g a m a t h e m a t i c a l p r o b l e m ,t h e r e a s o n f r e q u e n t l y c o n s i s t s i n o u r f a i l u r e t o r e c o g n i z et h e m o r e g e n e r a l s t a n d p o i n t f r o m w h i c h t h e p r o b l e m b e f o r eu s a p p e a r s o n l y a s a s i n g l e l i n k i n a c h a i n o f r e l a t e d p r o b l e m s . A f te r f in d in g t h i s s t a n d p o i n t , n o t o n l y i s t h i s p r o b

l e m f r e q u e n t l y m o r e a c ce s s i b le t o o u r i n v e s t i g a t i o n , b u t a tt h e s a m e t i m e w e c o m e i n t o p o s s e s si o n of a m e t h o d w h i c h

* Leipzig, 1896.

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444 H I L B E E T : M A T H E M A T I C A L P R O B L E M S . [ J u l y ,

i s a p p l i c a b l e a l so t o r e l a t e d p r o b l e m s . T h e in t r o d u c t i o n ofc o m p l e x p a t h s of i n t e g r a t i o n b y C a u c h y a n d of t h e n o t i o nof the I D E A L S i n n u m b e r t h e o r y b y K u m m e r m a y s e r v e a se x a m p l e s . T h i s w a y fo r fin d i n g g e n e r a l m e t h o d s i s c e r t a i n l yt h e m o s t p r a c t i ca b l e a n d t h e m o s t c e r t a i n ; fo r h e w h o s e e k sfo r m e t h o d s w i t h o u t h a v i n g a d e f in i te p r o b l e m i n m i n d s e e k sf o r t h e m o s t p a r t i n v a i n .

I n d e a l i n g w i t h m a t h e m a t i c a l p r o b l e m s , s p e c i a l i z a t i o np l a y s , a s I b e l i ev e , a s ti ll m o r e i m p o r t a n t p a r t t h a n g e n e r a l i z a ti o n . P e r h a p s i n m o s t c a s e s w h e r e w e s e e k in v a i n t h e

a n s w e r t o a q u e s t i o n , t h e c a u s e o f t h e f a i l u r e l i e s i n t h ef ac t t h a t p r o b l e m s si m p l e r a n d e a s i er t h a n t h e o n e i n h a n dh a v e b e e n e i t h e r n o t a t a l l o r i n c o m p l e t e ly s o l v ed . A l ld e p e n d s , t h e n , o n fi nd in g o u t t h e s e e a s ie r p r o b l e m s , a n do n s o l v i n g t h e m b y m e a n s o f d e v i c e s a s p e r f e c t a s p o s s i b l ea n d of c o n c e p t s c a p a b l e of g e n e r a l i z a t i o n . T h i s r u l e i s o n eof t h e m o s t i m p o r t a n t l e v e r s fo r o v e r co m i n g m a t h e m a t i c a ld i f f i c u l t i e s a n d i t s e e m s t o m e t h a t i t i s u s e d a l m o s t a l w a y s ,t h o u g h p e r h a p s u n c o n s c i o u s l y .

O c c a s io n a l l y i t h a p p e n s t h a t w e s e e k t h e s o l u t io n u n d e ri n s u ff ic ie n t h y p o t h e s e s o r i n a n i n c o r r e c t s e n s e , a n d f or t h i s

r e a s o n d o n o t s u c c e e d . T h e p r o b l e m t h e n a r i s e s : t o s h o wt h e i m p o s s i b i li ty of t h e s o l u t io n u n d e r t h e g i v e n h y p o t h e s e s ,o r i n t h e s e n s e c o n t e m p l a t e d . S u c h p r o o fs of i m p o s s i b i l i t yw e r e e ffe cte d b y t h e a n c i e n t s , fo r i n s t a n c e w h e n t h e y s h o w e dt h a t t h e r a t i o o f t h e h y p o t e n u s e t o t h e s i d e o f a n i s o s c e l e sr i g h t tr i a n g l e i s i r r a t i o n a l . I n l a t e r m a t h e m a t i c s , t h e q u e s t i o n a s t o t h e i m p o s s i b i l i t y o f c e r t a i n s o l u t i o n s p l a y s ap r e e m i n e n t p a r t , a n d w e p e r c e i v e i n t h i s w a y t h a t o l d a n dd i f f icu l t p rob lems , such as the p roof o f the ax iom of pa ra l l e l s , t h e s q u a r i n g o f t h e c i r c l e , o r t h e s o l u t i o n o f e q u a t io ns o f th e fi fth d egree b y r ad ica l s h av e fina lly found

f u l l y s a t i s f a c t o r y a n d r i g o r o u s s o l u t i o n s , a l t h o u g h i n a n o t h e rs e n s e t h a n t h a t o r i g in a l l y i n t e n d e d . I t i s p r o b a b l y t h i si m p o r t a n t f ac t a l o n g w i t h o t h e r p h i l o so p h i c a l r e a s o n s t h a tg i v e s r i s e t o t h e c o n v i c ti o n ( w h i c h e v e r y m a t h e m a t i c i a ns h a r e s , b u t w h i c h n o o n e h a s a s y e t s u p p o r t e d b y a p ro o f) t h a te v e r y d e f in i te m a t h e m a t i c a l p r o b l e m m u s t n e c e s s a ri l y b e s u s c e p t i b l e o f a n e x a c t s e t t l e m e n t , e i t h e r i n t h e f o r m o f a na c t u a l a n s w e r t o t h e q u e s t i o n a s k e d , o r b y t h e p r o o f o f t h ei m p o s s i b i l i t y o f i t s s o l u t i o n a n d t h e r e w i t h t h e n e c e s s a r y f a i l u r e of a l l a t t e m p t s . T a k e a n y d e f in i te u n s o l v e d p r o b l e m ,s u c h a s t h e q u e s t i o n a s t o t h e i r r a t i o n a l i t y of t h e E u l e r -

M a s c h e r o n i c o n s t a n t C , o r t h e e x i s t e n c e of a n i n f i n i ten u m b e r o f p r i m e n u m b e r s o f t h e f o r m 2 W + 1 . H o w e v e ru n a p p r o a c h a b l e t h e s e p r o b le m s m a y s ee m t o u s a n d h o w -

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e v e r h e l p l e ss w e s t a n d b e fo r e t h e m , w e h a v e , n e v e r t h e l e s s ,t h e f i r m c o n v i c t i o n t h a t t h e i r s o l u t i o n m u s t f o l l o w b y af in i t e number o f pu re ly log ica l p roces ses .

I s t h i s a x i o m o f t h e s o l v a b i l i t y o f e v e r y p r o b l e m a p e c u l i a r i t y c h a r a c t e r i s t i c of m a t h e m a t i c a l t h o u g h t a l o n e , o r i s i tp o s s ib l y a g e n e r a l l a w i n h e r e n t i n t h e n a t u r e of t h e m i n d ,t h a t a l l q u e s t i o n s w h i c h i t a s k s m u s t b e a n s w e r a b l e ? F o r i no t h e r s c ie n c e s a l s o o n e m e e t s o l d p r o b l e m s w h i c h h a v e b e e ns e t t l e d i n a m a n n e r m o s t s a ti s fa c t o r y a n d m o s t u s e fu l t os c ie n c e b y t h e p r oo f of t h e i r i m p o s s i b i l i t y . I i n s t a n c e t h e

p r o b l e m of p e r p e t u a l m o t i o n . A f te r s e e k i n g i n v a i n fo r t h ec o n s t r u c t i o n of a p e r p e t u a l m o t i o n m a c h i n e , t h e r e l a t i o n sw e r e i n v e s t i g a t e d w h i c h m u s t s u b s i s t b e t w e e n t h e f o r c e s o fn a t u r e if s u c h a m a c h i n e i s t o b e i m p o s s i b le ; * a n d t h i s i n v e r t e d q u e s t i o n l e d t o t h e d i s c o v e r y o f t h e l a w o f t h e c o n s e r v a t i o n o f e n e r g y , w h i c h , a g a i n , e x p l a i n e d t h e im p o s s i b i l i t y of p e r p e t u a l m o t i o n i n t h e s en s e o r i g in a l l y i n t e n d e d .

T h i s c o n v i c t i o n of t h e s o l v a b i l i t y of e v e r y m a t h e m a t i c a lp r o b l e m i s a p o w e r f u l i n c e n t i v e t o t h e w o r k e r . W e h e a rw i t h i n u s t h e p e r p e t u a l c a ll : T h e r e i s t h e p r o b l e m . S e e ki t s s o l u t i o n . Y o u c a n fin d i t b y p u r e r e a s o n , fo r i n m a t h e

m a t i c s t h e r e i s n o ignorabimus.T h e s u p p l y o f p r o b l e m s i n m a t h e m a t i c s i s i n e x h a u s t i b l e ,a n d a s so o n a s o n e p r o b l e m i s so l v ed n u m e r o u s o t h e r s c o m ef o r th i n i t s p l a c e . P e r m i t m e i n t h e fo l lo w i n g , t e n t a t i v e l ya s i t w e r e , t o m e n t i o n p a r t i c u l a r d e fi n it e p r o b l e m s , d r a w nf r o m v a r i o u s b r a n c h e s o f m a t h e m a t i c s , f r o m t h e d i s c u s s i o no f w h i c h a n a d v a n c e m e n t o f s c i e n c e m a y b e e x p e c t e d .

L e t u s l o o k a t t h e p r i n c i p l e s o f a n a l y s i s a n d g e o m e t r y .T h e m o s t s u g g e s t i v e a n d n o t a b l e a c h i e v e m e n t s o f t h e l a s tc e n t u r y i n t h i s f i e l d a r e , a s i t s e e m s t o m e , t h e a r i t h m e t i c a l f o r m u l a t i o n o f t h e c o n c e p t o f t h e c o n t i n u u m i n t h e

w o r k s o f C a u c h y , B o l z a n o a n d C a n t o r , a n d t h e d i s c o v e r yof n o n - e u c l i d e a n g e o m e t r y b y G a u s s , B o l y a i , a n d L o b a -c h e v s k y . I t h e r e f o r e fi rs t d i r e c t y o u r a t t e n t i o n t o s o m ep r o b l e m s b e l o n g i n g t o t h e s e f i e l d s .

1 . C A N T O R ' S P R O B L E M O F T H E C A R D I N A L N U M B E R O FT H E C O N T I N U U M .

T w o s y s t e m s , i. e , t w o a s s e m b l a ge s of o r d i n a r y r e a l n u m b e r s o r p o i n t s , a r e s a i d t o b e ( a c c o r d i n g t o C a n t o r ) e q u i v a len t o r o f equa l cardinal number, i f t h e y c a n b e b r o u g h t i n t oa r e l a t i o n t o o n e a n o t h e r s u c h t h a t t o e v e r y n u m b e r o f

t h e o n e a s s e m b l a g e c o r r e s p o n d s o n e a n d o n l y o n e d e f i -*S ee Helm holtz , " Ueber die Weohselwirkung der Naturkrâefte i iüd

die darauf bezüglichen neuesten Erm itte lungen der Ph ys ik " ; Vortrag,gehalten in Königsberg, 1854.

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n i t e n u m b e r of t h e o t h e r . T h e i n v e s t i g a t io n s of C a n t o ro n s u c h a s s e m b l a g e s o f p o i n t s s u g g e s t a v e r y p l a u s i b l et h e o r e m , w h i c h n e v e r t h e l e s s , i n s p i t e of t h e m o s t s t r e n u o u s e ffo r ts , n o o n e h a s s u c c e e d e d i n p r o v i n g . T h i s i s t h et h e o r e m :

E v e r y s y s t e m o f i n f i n i t e l y m a n y r e a l n u m b e r s , i. e., e v e r ya s s e m b l a g e of n u m b e r s (o r p o i n t s ) , i s e i t h e r e q u i v a l e n t t ot h e a s s e m b l a g e o f n a t u r a l i n t e g e r s , 1 , 2 , 3,.•• o r t o t h e a s s e m b l a g e of a l l r e a l n u m b e r s a n d t h e r e f o r e t o t h e c o n t i n u u m ,t h a t i s , t o t h e p o i n t s of a l i n e ; as regards equivalence there

are, the refore, only two asse mblages of nu mbers, the coun table a s-semblage and the cont inuum.F r o m t h i s t h e o r e m i t w o u l d f o l l o w a t o n c e t h a t t h e c o n

t i n u u m h a s t h e n e x t c a r d i n a l n u m b e r b e y o n d t h a t of t h ec o u n t a b l e a s s e m b l a g e ; t h e p r o of of t h i s t h e o r e m w o u l d ,t h e r e f o r e , f o r m a n e w b r i d g e b e t w e e n t h e c o u n t a b l e a s s e m b l a g e a n d t h e c o n t i n u u m .

L e t m e m e n t i o n a n o t h e r v e r y r e m a r k a b l e s t a t e m e n t ofC a n t o r ' s w h i c h s t a n d s i n t h e c lo s e st c o n n e c t i o n w i t h t h et h e o r e m m e n t i o n e d a n d w h i c h , p e r h a p s , o ffe rs t h e k e y t oi t s proof. A n y s y s t e m of r e a l n u m b e r s i s s a i d t o b e o r d e r e d ,

if f o r e v e r y t w o n u m b e r s of t h e s y s t e m i t i s d e t e r m i n e d w h i c ho n e i s t h e e a r l i e r a n d w h i c h t h e l a t e r , a n d if a t t h e s a m e t i m et h i s d e t e r m i n a t i o n i s of s u c h a k i n d t h a t , if a i s befo re 6 andb i s b efore c, th en a a l w a y s c o m e s b e fo r e c. T h e n a t u r a la r r a n g e m e n t of n u m b e r s of a s y s t e m i s de fi n ed t o b e t h a ti n w h i c h t h e sm a l l e r p r e c e d e s t h e l a r g e r . B u t t h e r e a r e ,a s i s e a s i l y s e e n , i n f i n i te l y m a n y o t h e r w a y s i n w h i c h t h en u m b e r s o f a s y s t e m m a y b e a r r a n g e d .

I f w e t h i n k of a d e fi n it e a r r a n g e m e n t of n u m b e r s a n ds e le c t fr om t h e m a p a r t i c u l a r s y s t e m of th e s e n u m b e r s , as o - c a l l e d p a r t i a l s y s t e m o r a s s e m b l a g e , t h i s p a r t i a l s y s t e m

w i l l a l s o p r o v e t o b e o r d e r e d . N o w C a n t o r c o n s i d e r s ap a r t i c u l a r k i n d of o r d e r e d as s e m b l a g e w h i c h h e d e s i g n a t e sa s a w e l l o r d e r e d a s s e m b l a g e a n d w h i c h i s c h a r a c t e r i z e di n t h i s w a y , t h a t n o t o n l y in t h e a s s e m b l a g e i ts e lf b u ta l s o i n e v e r y p a r t i a l a s s e m b l a g e t h e r e e x i s t s a f i r s t n u m b e r .T h e s y s t e m of i n t e g e r s 1 , 2 , 3 , ••• i n t h e i r n a t u r a l o r d e r i se v i d e n t l y a w e l l o r d e r e d a s s e m b l a g e . O n t h e o t h e r h a n dt h e s y s t e m o f a l l r e a l n u m b e r s , L e. , t h e c o n t i n u u m i n i t sn a t u r a l o r d e r , i s e v i d e n t l y n o t w e l l o r d e r e d . F o r , if w et h i n k o f t h e p o i n t s o f a s e g m e n t o f a s t r a i g h t l i n e , w i t h i t si n i t i a l p o i n t e x c l u d e d , a s o u r p a r t i a l a s s e m b l a g e , i t w i l l h a v e

no f i r s t e lemen t .T h e q u e s t i o n n o w a r i s e s w h e t h e r t h e t o t a l i t y o f a l l n u m

b e r s m a y n o t b e a r r a n g e d i n a n o t h e r m a n n e r so t h a t e v e r y

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1902.] HILBERT : MATHEMATICAL PROBLEMS. 4 4 7

p a r t i a l a s s e m b l a g e m a y h a v e a f i r s t e l e m e n t , L e . , w h e t h e rt h e c o n t i n u u m c a n n o t b e c o n s id e r e d a s a w e l l o r d e r e da s s e m b l a g e — a q u e s t i o n w h i c h C a n t o r t h i n k s m u s t b e a n s w e r e d i n t h e a f fi rm a t iv e . I t a p p e a r s t o m e m o s t d e s i r a b l et o o b t a i n a d i r e c t p r o o f o f t h i s r e m a r k a b l e s t a t e m e n t o fC a n t o r ' s , p e r h a p s b y a c t u a l l y g i v in g a n a r r a n g e m e n t ofn u m b e r s s u c h t h a t i n e v e r y p a r t i a l s y s te m a firs t n u m b e rc a n b e p o i n t e d o u t .

2 . T H E C O M P A T I B IL IT Y O F T H E A R I T H M E T I C A L A X I O M S .

"W h en w e a r e e n g a g e d i n i n v e s t i g a t i n g t h e f o u n d a t i o n s ofa s c ie n c e , w e m u s t s e t u p a s y s t e m of a x i o m s w h i c h c o n t a i n s a n e x a c t a n d c o m p l e t e d e s c r i p t i o n o f t h e r e l a t i o n ss u b s i s t i n g b e t w e e n t h e e l e m e n t a r y i d e a s of t h a t s ci e n ce .T h e a x i o m s s o s e t u p a r e a t t h e s a m e t i m e t h e d e f in i ti o n sof t h o s e e l e m e n t a r y i d e a s ; a n d n o s t a t e m e n t w i t h i n t h er e a l m o f t h e s c i e n c e w h o s e f o u n d a t i o n w e a r e t e s t i n g i s h e l dt o b e c o r r e c t u n l e s s i t c a n b e d e r i v e d f r o m t h o s e a x i o m s b ym ea ns of a f in i te n um be r of log ica l s t e ps . Up on c lose r cons i d e r a t i o n t h e q u e s t i o n a r i s e s : Whether, in any way, certainstatements of s ingle axioms depend upon one another , and whether

the axioms may not therefore contain certain parts in comm on,which must be isolated i f one wishes to arrive at a system ofaxiom s that sha ll be altogether independent of one another .

B u t a b o v e a l l I w i s h t o d e s i g n a t e t h e f o l l o w i n g a s t h em o s t i m p o r t a n t a m o n g t h e n u m e r o u s q u e s t io n s w h i c h c a nb e a s k e d w i t h r e g a r d t o t h e a x i o m s : To prove that they arenot contradictory, that is , that a finite number of logical steps basedupon them can never lead to contradictory results.

I n g e o m e t r y , t h e p r o o f o f t h e c o m p a t i b i l i t y o f t h e a x i o m sc a n b e e f f e c t e d b y c o n s t r u c t i n g a s u i t a b l e f i e l d o f n u m b e r s ,s u c h t h a t a n a l o g o u s r e l a t i o n s b e t w e e n t h e n u m b e r s of t h i sfie ld c o r r e s p o n d t o t h e g e o m e t r i c a l a x i o m s . A n y c o n t r a d i c t i o n in t h e d e d u c t i o n s fr om t h e g e o m e t r i c a l a xi o m s m u s t t h e r e -u p o n b e r e c o g n i z a b l e i n t h e a r i t h m e t i c o f t h i s f i e l d o f n u m b e r s . I n t h i s w a y t h e d e s i r e d p r o o f f o r t h e c o m p a t i b i l i t y o ft h e g e o m e t r i c a l a x i o m s i s m a d e t o d e p e n d u p o n t h e t h e o r e mo f t h e c o m p a t i b i l i t y o f t h e a r i t h m e t i c a l a x i o m s .

O n t h e o t h e r h a n d a d i r e c t m e t h o d i s n e e d e d fo r t h ep r o o f of t h e c o m p a t i b i l i t y of t h e a r i t h m e t i c a l a x i o m s . T h ea x i o m s of a r i t h m e t i c a r e e s s e n t i a ll y n o t h i n g e ls e t h a n t h ek n o w n r u l e s o f c a l c u l a t i o n , w i t h t h e a d d i t i o n o f t h e a x i o m

of c o n t i n u i t y . I r e c e n t l y c o l le c t e d t h e m * a n d i n s o d o i n g* Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 8 (1900),

p. 180.

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r e p l a c e d t h e a x i o m of c o n t i n u i t y b y t w o s i m p l e r a x io m s ,n a m e l y , t h e w e l l- k n o w n a x io m of A r c h i m e d e s , a n d an e w a x i o m e s s e n t i a l ly a s fo ll ow s : t h a t n u m b e r s f or m as y s t e m of t h i n g s w h i c h i s c a p a b l e of n o f u r t h e r e x t e n s i o n ,a s l o n g a s a l l t h e o t h e r a x i o m s h o l d ( a x io m of c o m p l e t e n e s s ) .I a m c o n v i n c e d t h a t i t m u s t b e p o s s ib l e t o fi nd a d i r e c tp r o o f f o r t h e c o m p a t i b i l i t y o f t h e a r i t h m e t i c a l a x i o m s , b ym e a n s of a ca r e fu l s t u d y a n d s u i t a b l e m o d i f ic a ti o n of t h ek n o w n m e t h o d s o f r e a s o n i n g i n t h e t h e o r y o f i r r a t i o n a ln u m b e r s .

T o s h o w t h e s ig n if ic a n ce of t h e p r o b l e m fr om a n o t h e rpo in t of v iew , I ad d th e fo l low ing ob serv a t ion : I f cont r a d i c t o r y a t t r i b u t e s b e a s s i gn e d t o a c o n c e p t , I s a y , t h a tmathem atically the concept oes not exis t So , fo r example , ar e a l n u m b e r w h o s e s q u a r e i s — 1 d o e s n o t e x i st m a t h e m a t i c a l ly . B u t if i t c a n b e p r o v e d t h a t t h e a t t r i b u t e sa s s i g n e d t o t h e c o n c e p t c a n n e v e r l e a d t o a c o n t r a d i c t i o nby th e ap p l ica t io n of a f in it e n um be r o f log ica l p roces ses ,I s a y t h a t t h e m a t h e m a t i c a l e x i s t e n c e of t h e c o n c e p t ( fo rexample , o f a number o r a func t ion which s a t i s f ies ce r t a i n c o n d i t i o n s ) i s t h e r e b y p r o v e d . I n t h e c a se b e f o re u s ,

w h e r e w e a r e c o n c e r n e d w i t h t h e a x i o m s of r e a l n u m b e r si n a r i t h m e t i c , t h e p r o o f o f t h e c o m p a t i b i l i t y o f t h e a x i o m si s a t t h e s a m e t i m e t h e p r o o f o f t h e m a t h e m a t i c a l e x i s t e n c eof t h e c o m p l e t e s y s t e m o f r e a l n u m b e r s o r of t h e c o n t i n u u m . I n d e e d , w h e n t h e p ro o f fo r t h e c o m p a t i b i li t y oft h e a x i o m s s h a l l b e f u l l y a c c o m p l i s h e d , t h e d o u b t s w h i c hh a v e b e e n e x p r e s s e d o c c a s i o n a l l y a s t o t h e e x i s t e n c e o f t h ec o m p l e t e s y s t e m of r e a l n u m b e r s w i l l b e c o m e t o t a l l yg r o u n d l e s s . T h e t o t a l i t y of r e a l n u m b e r s , i. e . , the cont i n u u m a c c o r d i n g t o t h e p o i n t o f v i e w j u s t i n d i c a t e d , i s n o tth e to ta l i ty o f a l l pos s ib le s e r ies in dec im al f r ac t ions , o r o f a l l

p o s s i b le l a w s a c c o r d i n g t o w h i c h t h e e l e m e n t s of a f u n d a m e n t a l s e q u e n c e m a y p r o c e e d . I t i s r a t h e r a s y s t e m of t h i n g sw h o s e m u t u a l r e l a t i o n s a r e g o v e r n e d b y t h e a x i o m s s e t u pa n d fo r w h i c h a l l p r o p o s i t io n s , a n d o n l y th o s e , a r e t r u e w h i c hcan be der ived f rom the ax ioms by a f in i t e number o f log ica lp r o c e s s e s . I n m y o p i n i o n , t h e c o n c e p t of t h e c o n t i n u u m i ss t r i c t l y l o g ic a ll y t e n a b l e i n t h i s s e n s e o n l y . I t s e e m s t o m e ,i n d e e d , t h a t t h i s co r r e s p o n d s b e s t a ls o t o w h a t e x p e r i e n c e a n di n t u i t i o n t e l l u s . T h e co n c e p t of t h e c o n t i n u u m o r e v e ntha t o f the sys tem of a l l func t ions ex i s t s , then , in exac t lyt h e s a m e s e n s e a s t h e s y s t e m o f i n t e g r a l , r a t i o n a l n u m b e r s ,

fo r e x a m p l e , o r a s C a n t o r ' s h i g h e r c la s s e s of n u m b e r s a n dc a r d i n a l n u m b e r s . F o r I a m c o n v i n c e d t h a t t h e e x i s t e n c e oft h e l a t t e r , j u s t a s t h a t of t h e c o n t i n u u m , ca n b e p r o v e d i n

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1 9 0 2 .] HILBERT I MATHEMATICAL PROBLEMS. 4 4 9

t h e s e n s e I h a v e d e s c r i b e d ; u n l i k e t h e s y s t e m of all c a r d i n a l n u m b e r s o r of all C a n t o r ' s a l e p h s , fo r w h i c h , a s m a y b es h o w n , a s y s t e m o f a x i o m s , c o m p a t i b l e i n m y s e n s e , c a n n o tb e s e t u p . E i t h e r of t h e s e s y s t e m s i s , t h e r e f o r e , a c c o r d i n gt o m y t e r m i n o l o g y , m a t h e m a t i c a l l y n o n - e x i s t e n t .

F r o m t h e f i e l d o f t h e f o u n d a t i o n s o f g e o m e t r y I s h o u l dl i k e t o m e n t i o n t h e f o l l o w i n g p r o b l e m :

3 . T H E E Q U A L I T Y O F T H E V O L U M E S O F T w o T E T R A H E D R AO F E Q U A L B A S E S A N D E Q U A L A L T I T U D E S .

I n t w o l e t t e r s t o G e r l i n g , G a u s s * e x p r e s s e s h i s r e g r e t t h a tc e r t a i n t h e o r e m s o f s o l i d g e o m e t r y d e p e n d u p o n t h e m e t h o do f e x h a u s t i o n , i. e., i n m o d e r n p h r a s e o l o g y , u p o n t h e a x i o mof c o n t i n u i t y ( o r u p o n t h e a x i o m of A r c h i m e d e s ) . G a u s sm e n t i o n s i n p a r t i c u l a r t h e t h e o r e m o f E u c l i d , t h a t t r i a n g u l a rp y r a m i d s o f e q u a l a l t i t u d e s a r e t o e a c h o t h e r a s t h e i r b a s e s .N o w t h e a n a l o g o u s p r o b l e m i n t h e p l a n e h a s b e e n s o l v e d . fG e r l i n g a l s o s u c c e e d e d i n p r o v i n g t h e e q u a l i t y o f v o l u m e o fs y m m e t r ic a l p o l y h e d r a b y d i v i d in g t h e m i n t o c o n g r u e n tp a r t s . N e v e r t h e l e s s , i t s e e m s t o m e p r o b a b l e t h a t a g e n e r a l

p r o o f o f t h i s k i n d f o r t h e t h e o r e m o f E u c l i d j u s t m e n t i o n e di s i m p o s s i b l e , a n d i t s h o u l d b e o u r t a s k t o g i v e a r i g o r o u s p r o o f of i t s i m p o s s i b i l it y . T h i s w o u l d b e o b t a i n e d , a ss o o n a s w e s u c c e e d e d i n specifying two tetrahedra of equalbases and equal alti tudes which can in no way be split up into con*gruent te trahedra, and wh ich cannot be combined with congruenttetrahedra to form two polyhedra which themselves could be split u pinto congruent tetrahedra. %

4 . P R O B L E M O F T H E S T R A I GH T L I N E A S T H E S H O R T ES T D I S T AN C E B E T W E E N T W O P O I N T S .

A n o t h e r p r o b l e m r e l a t i n g t o t h e f o u n d a t i o n s of g e o m e t r y i s t h i s : I f fr o m a m o n g t h e a x i o m s n e c e s s a r y t oe s t a b l i s h o r d i n a r y e u c l id e a n g e o m e t r y , w e e x c l u d e t h ea x i o m o f p a r a l l e l s , o r a s s u m e i t a s n o t s a t i s f i e d , b u t r e t a i na l l o t h e r a x i o m s , w e o b t a i n , a s i s w e l l k n o w n , t h e g e o m e t r y of L o b a c h e v s k y ( h y p e r b o l i c g e o m e t r y ) . W e m a yt h e r e f o r e s a y t h a t t h i s i s a g e o m e t r y s t a n d i n g n e x t t o

* W erke, vol. 8, pp. 241 and 244.f Cf., beside earlier litera ture , H ub ert, Grundlagen der Geometrie, Leip

zig, 1899, eh. 4. [Translation by Townsend, Chicago, 1902.]

% Since this was written Herr Dehn has succeeded in proving this im possibility. See his note: "Ueber raumgleiche Polyeder," in Nach-richten d. K. Gesellsch. d. Wiss. zu Q'ôttingen, 1900, and a paper soon toappear in the Math. Annalen [vol. 55, p p . 465-478],

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4 5 0 H I L B E B T : M A T H E M A T I C A L P R O B L E M S . [ J u l y ,

e u c l id e a n g e o m e t r y . I f w e r e q u i r e f u r t h e r t h a t t h a t a x io mb e n o t s a t i s f i e d w h e r e b y , o f t h r e e p o i n t s o f a s t r a i g h tl i n e , o n e a n d o n l y o n e l i e s b e t w e e n t h e o t h e r t w o , w e o b t a i nB i e m a n n ' s ( e l l ip t i c) g e o m e t r y , so t h a t t h i s g e o m e t r y a p p e a r s t o b e t h e n e x t a f te r L o b a c h e v s k y ' s . I f w e w i s h t oc a r r y o u t a s i m i l a r i n v e s t i g a t i o n w i t h r e s p e c t t o t h e a x i o mo f A r c h i m e d e s , w e m u s t l o o k u p o n t h i s a s n o t s a t i s f i e d , a n dw e a r r i v e t h e r e b y a t t h e n o n - a r c h i m e d e a n g e o m e t r ie s w h i chh a v e b e e n i n v e s t i g a t e d b y V e r o n e s e a n d myself. T h e m o r eg e n e r a l q u e s t i o n n o w a r i s e s : W h e t h e r fr om o t h e r s u g ge s

t i v e s t a n d p o i n t s g e o m e t r i e s m a y n o t b e d e v i s e d w h i c h , w i t he q u a l r i g h t , s t a n d n e x t t o e u c l i d e a n ge o m e t r y . H e r e Is h o u l d l i k e t o d i r e c t y o u r a t t e n t i o n t o a t h e o r e m w h i c hh a s , i n d e e d , b e e n e m p l o y e d b y m a n y a u t h o r s a s a d e f i n i t i o n o f a s t r a i g h t l i n e , v i z . , t h a t t h e s t r a i g h t l i n e i s t h es h o r t e s t d i s t a n c e b e t w e e n t w o p o i n t s . T h e e s s e n t i a l c o n t e n t of t h i s s t a t e m e n t r e d u c e s t o t h e t h e o r e m o f E u c l i d t h a ti n a t r i a n g l e t h e s u m of t w o s id e s i s a l w a y s g r e a t e r t h a n t h et h i r d s i d e — a t h e o r e m w h i c h , a s i s e a s i l y s e e n , d e a l s s o l e l yw i t h e l e m e n t a r y c o n c e p t s , i. e., w i t h s u c h a s a r e d e r i v e d d i r e c t l y f r o m t h e a x i o m s , a n d i s t h e r e f o r e m o r e a c c e s

s i b le t o l o g ic a l i n v e s t i g a t i o n . E u c l i d p r o v e d t h i s t h e o r e m ,w i t h t h e h e l p of t h e t h e o r e m of t h e e x t e r i o r a n g l e , o n t h eb a s i s of t h e c o n g r u e n c e th e o r e m s . N o w i t i s r e a d i l y sh o w nt h a t t h i s t h e o r e m o f E u c l i d c a n n o t b e p r o v e d s o l e ly o nt h e b a s i s o f t h o s e c o n g r u e n c e t h e o r e m s w h i c h r e l a t e t o t h ea p p l i c a t i o n of s e g m e n t s a n d a n g l e s , b u t t h a t o n e of t h et h e o r e m s o n t h e c o n g r u e n c e of t r i a n g l e s i s n e c e s s a r y . W ea r e a s k i n g , t h e n , f o r a g e o m e t r y i n w h i c h a l l t h e a x i o m s o fo r d i n a r y e u c l i d e a n g e o m e t r y h o l d , a n d i n p a r t i c u l a r a l lt h e c o n g r u e n c e a x i o m s e x c e p t t h e o n e o f t h e c o n g r u e n c e o ft r i a n g l e s ( o r a l l e x c e p t t h e t h e o r e m o f t h e e q u a l i t y o f t h e

b a s e a n g l e s i n t h e i so s c e le s t r i a n g l e ) , a n d i n w h i c h , b e s i d e s ,t h e p r o p o s i ti o n t h a t i n e v e r y t r i a n g l e t h e s u m of t w o s i d e s i sg r e a t e r t h a n t h e t h i r d i s a s s u m e d a s a p a r t i c u l a r a x i o m .

O n e fi nd s t h a t s u c h a g e o m e t r y r e a l l y e x i s t s a n d i s n oo t h e r t h a n t h a t w h i c h M i n k o w s k i c o n s t r u c t e d i n h i s b o o k ,G e o m e t r i e d e r Z a h l e n , * a n d m a d e t h e b a s i s of h i s a r i t h m e t i c a l i n v e s t i g a t i o n s . M i n k o w s k i ' s i s t h e r e f o r e a l so ag e o m e t r y s t a n d i n g n e x t t o t h e o r d i n a r y eu c li d e a n g e o m e t r y ;i t i s e s s e n t i a l l y c h a r a c t e r i z e d b y t h e f o l l o w i n g s t i p u l a t i o n s :

1 . T h e p o i n t s w h i c h a r e a t e q u a l d i s t a n c e s f ro m a fi xe dp o i n t 0 l i e o n a c o n v e x c l o s e d s u r f a c e o f t h e o r d i n a r y e u c l i

d e a n s p a c e w i t h 0 a s a c e n t e r .

* Leipzig, 1896.

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1 9 0 2 . ] HILBERT I MATHEMATICAL PROBLEMS. 4 5 1

2 . T w o s e g m e n t s a r e s a i d t o b e e q u a l w h e n o n e c a n b ec a r r ie d i n t o t h e o t h e r b y a t r a n s l a t i o n o f t h e o r d i n a r ye u c l i d e a n s p a c e .

I n M i n k o w s k i ' s g e o m e t r y t h e a x i o m o f p a r a l l e l s a l s oh o l d s . B y s t u d y i n g t h e t h e o r e m o f t h e s t r a i g h t l i n e a s t h es h o r t e s t d i s t a n c e b e t w e e n t w o p o i n t s , I a r r i v e d * a t ag e o m e t r y i n w h i c h t h e p a r a l l e l a x i o m d o e s n o t h o l d , w h i l ea l l o t h e r a x i o m s of M i n k o w s k i 7 s g e o m e t r y a r e s a ti sf ie d . T h et h e o r e m of t h e s t r a i g h t l i n e a s t h e s h o r t e s t d i s t a n c e b e t w e e nt w o p o i n t s a n d t h e e s s e n t i a l l y e q u i v a l e n t t h e o r e m o f E u c l i da b o u t t h e s i d e s of a t ri a n g l e , p l a y a n i m p o r t a n t p a r t n o to n l y i n n u m b e r t h e o r y b u t a l s o i n t h e t h e o r y of s u r fa c e sa n d i n t h e c a l cu l u s of v a r i a t i o n s . F o r t h i s r e a s o n , a n db e c a u s e I b e l ie v e t h a t t h e t h o r o u g h i n v e s t i g a t i o n of t h ec o n d i t i o n s f o r t h e v a l i d i t y o f t h i s t h e o r e m w i l l t h r o w a n e wl i g h t u p o n t h e i d e a o f d i s t a n c e , a s w e l l a s u p o n o t h e re l e m e n t a r y i d e a s , e. g.9 u p o n t h e i d e a o f t h e p l a n e , a n dthe pos s ib i l i ty o f i t s def in i t ion by means o f the idea o ft h e s t r a i g h t l i n e , the construction an d systematic treatment ofthe geometries here possible seem to me desirable.

5 . L I E ' S C O N C E P T O F A C O N T IN U O U S G R O U P O F T R A N S FO R M A TIO N S W I T H O U T T H E A S S UM P T I O N O F T H E

D I F FE R E N T I A B I L I T Y O F T H E F U N C T I O N SD E F I N I N G T H E G R O U P .

I t i s w e l l k n o w n t h a t L i e , w i t h t h e a i d o f t h e c o n c e p t o fc o n t i n u o u s g r o u p s o f t r a n s f o r m a t i o n s , h a s s e t u p a s y s t e m o fg e o m e t r i c a l a x i o m s a n d , f r o m t h e s t a n d p o i n t o f h i s t h e o r yof g roups , has p roved tha t th i s sys tem of ax ioms suf f ices fo rg e o m e t r y . B u t s i n ce L i e a s s u m e s , i n t h e v e r y f o u n d a t i o nof h i s t h e o r y , t h a t t h e f u n c t i o n s d e f i n in g h i s g r o u p c a n b ed i ff e re n t i a te d , i t r e m a i n s u n d e c i d e d i n L i e ' s d e v e l o p m e n t ,w h e t h e r t h e a s s u m p t i o n of t h e d i f fe r e n t i a b i li t y i n c o n n e c t i o n w i t h t h e q u e s t i o n a s t o t h e a x i o m s o f g e o m e t r yi s a c t u a l l y u n a v o i d a b l e , o r w h e t h e r i t m a y n o t a p p e a rr a t h e r a s a c o n s e q u e n c e o f t h e g r o u p c o n c e p t a n d t h e o t h e rg e o m e t r i c a l a x i o m s . T h i s c o n s i d e r a t i o n , a s w e l l a s c e r t a i no t h e r p r o b l e m s i n c o n n e c t i o n w i t h t h e a r i t h m e t i c a l a x i o m s ,b r i n g s b e f o r e u s t h e m o r e g e n e r a l q u e s t i o n : H ow far Lie

1s

concept of cont inuous groups of transformations is approachable inour investigations without the assum ption of the differen tiability of

th e functions.

* Math. Annalen, vol. 46, p. 91.

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452 HILBERT : MATHEMATICAL PROBLEMS. [ J u t y ,

Lie defines a finite continuous group of transformationsas a system of transformations

x! =fi(

xv ->

xn ;

av ->

ar) (* = h - , *0

having the property that any two arbitrarily chosen trans

formations of the system, as

Xi :=fi{ XV '"•)Xn 5 a

V "*>ar)?

applied sucessively result in a transformation which also

belongs to the system, and which is therefore expressible in

the form

*/W«{/i(*><0> ->ƒ„(>>a) ;

6i> -> M = / i(% - ^ w 5

ci> "*>

cr)>

where cv •••, cr are certain functions of av •••, ar and 6X, •••, 6r.

The group property thus finds its full expression in a sys

tem of functional equations and of itself imposes no ad

ditional restrictions upon the functions fv —,/w; ov •••, cr.

Yet Lie's further treatment of these functional equations,

viz., the derivation of the well-known fundamental differ

ential equations, assumes necessarily the continuity anddifferentiability of the functions defining the group.

As regards continuity : this postulate will certainly be

retained for the present—if only with a view to the geo

metrical and arithmetical applications, in which the contin

uity of the functions in question appears as a consequence of

the axiom of cont inuity. On the other hand the differenti

ability of the functions defining the group contains a postu

late which, in the geometrical axioms, can be expressed

only in a rather forced and complicated manner. Hence

there arises the question whether, through the introduction

of suitable new variables and parameters, the group canalways be transformed into one whose defining func

tions are differentiable; or whether, at least with the help

of certain simple assumptions, a transformation is possible

into groups admitting Lie's methods. A reduction to anal

ytic groups is, according to a theorem announced by Lie * but

first proved by Schur,f always possible when the group is

transitive and the existence of the first and certain second

derivatives of the functions defining the group is assumed.

* Lie-Engel, Theorie der Transformationsgruppen, vol. 3, Leipzig,1893, §§82, 144.

t " Ueber den analytischen Charakter der eine endliche Kontinuier-l iehe Transformat ionsgruppen dars te llenden F un kt io ne n," M ath . Annalen, vol. 41.

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F o r i n f in i t e g r o u p s t h e i n v e s t i g a t i o n of t h e c o r r e s p o n d i n g q u e s t i o n i s , I b e l i e v e , a l s o of i n t e r e s t . M o r e o v e rw e a r e t h u s l e d t o t h e w i d e a n d i n t e r e s t i n g f i e l d o ff u n c ti o n a l e q u a t i o n s w h i c h h a v e b e e n h e r e t o fo r e i n v e s t i g a t e d u s u a l l y o n l y u n d e r t h e a s s u m p t i o n o f t h e d i f f e r e n t i a b i l i t y of t h e f u n c t i o n s i n v o l v e d . I n p a r t i c u l a r t h e f u n c t i o n a l e qu a t i o n s t r e a t e d b y A b e l * w i t h so m u c h i n g e n u i t y ,t h e d i f f e r e n c e e q u a t i o n s , a n d o t h e r e q u a t i o n s o c c u r r i n g i nt h e l i t e r a t u r e o f m a t h e m a t i c s , d o n o t d i r e c t l y i n v o l v ea n y t h i n g w h i c h n e c e s s i ta t e s t h e r e q u i r e m e n t of t h e diff e r e n t i a b i l i t y of t h e a c c o m p a n y i n g f u n c t i o n s . I n t h e s e a r c hf o r c e r t a i n e x i s t e n c e p r o o f s i n t h e c a l c u l u s o f v a r i a t i o n sI ca m e d i r e c t l y u p o n t h e p r o b l e m : T o p r o v e t h e d i ffe r en t i a b i l i t y o f t h e f u n c t i o n u n d e r c o n s i d e r a t i o n f r o m t h e e x i s t ence o f a d i ffe rence equ a t io n . In a l l the se cases , th en ,t h e p r o b l e m a r i s e s : In how far are the assertions which we canmake in the case of differentiable functions true under propermodifications without this assumption f

I t m a y b e f u r th e r r e m a r k e d t h a t H . M i n k o w s k i in h i sa b o v e - m e n t i o n e d G e o m e t r ie d e r Z a h l e n s t a r t s w i t h t h e f u n ct i o n a l e q u a t i o n

M + VV "'J Xn + Vn) = / O l > '"J Xn) + KVv ~> Vn)

a n d f ro m t h i s a c t u a l l y s u c ce e d s i n p r o v i n g t h e e x i s t e n c eof c e r t a i n d i ff e re n t i a l q u o t i e n t s fo r t h e f u n c t io n i n q u e s t i o n .

O n t h e o t h e r h a n d I w i s h to e m p h a s i ze t h e fa ct t h a t t h e r ec e r t a i n l y e x i s t a n a l y t i c a l f u n c t i o n a l e q u a t i o n s w h o s e s o les o l u t i o n s a r e n o n - d i f f e r e n t ia b l e f u n c t i o n s . F o r e x a m p l e au n i f o r m c o n t i n u o u s n o n - d i f f e re n t i a b l e f u n c t i o n <p(x) ca nb e c o n s t r u c t e d w h i c h r e p r e s e n t s t h e o n l y s o l u t i o n o f t h et w o f u n c t io n a l e q u a t i o n s

y(x + a) — <p(x) = ƒ ( » ? <p(x + £ ) — <p(x) = 0 ,

w h e r e a a n d fi a r e t w o r e a l n u m b e r s , a n d f(x) d e n o t e s , fo r a l lt h e r e a l v a l u e s o f x , a r e g u l a r a n a l y t i c u n i f o r m f u n c t i o n .S u c h f u n ct io n s a r e o b t a i n e d i n t h e s i m p l e s t m a n n e r b y m e a n so f t r i g o n o m e t r i c a l s e r i e s b y a p r o c e s s s i m i l a r t o t h a t u s e d b yB o r e l ( a c c o r d i n g t o a r e c e n t a n n o u n c e m e n t of P i c a r d ) f fo rt h e c o n s t r u c t i o n o f a d o u b l y p e r i o d i c , n o n - a n a l y t i c s o l u t i o no f a c e r t a i n a n a l y t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n .

*W erke , vol . 1 , pp . 1 , 61 , 389.f " Quelques théor ies fondamenta les dans l ' ana lyse mathémat ique ,"

Conférences faites à Clark University, Bévue générale des Sciences, 1900, p . 22.

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6 . M A T H E M A T I C A L T R E A T M E N T O F T H E A X I O M S of P H Y S I C S .

T h e i n v e s t i g a t i o n s o n t h e f o u n d a t i o n s of g e o m e t r y s u g g e s tt h e p r o b l e m : To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part ; in the first rank are the theory of probabilit ies and m echanics.

A s t o t h e a x i o m s o f t h e t h e o r y o f p r o b a b i l i t i e s , * i t s e e m st o m e d e s i r a b l e t h a t t h e i r lo g i ca l i n v e s t i g a t i o n s h o u l d b ea c c o m p a n i e d b y a r i g o r o u s a n d s a t i sf a ct o r y d e v e l o p m e n to f t h e m e t h o d o f m e a n v a l u e s i n m a t h e m a t i c a l p h y s i c s , a n d

i n p a r t i c u l a r i n t h e k i n e t i c t h e o r y o f g a s e s .I m p o r t a n t i n v e s t i g a t i o n s b y p h y s i c i s t s o n t h e f o u n d a t i o n sof m e c h a n i c s a r e a t h a n d ; I r e f e r t o t h e w r i t i n g s of M a c h , fH e r t z , J B o l t zm a n n § a n d V o l k m a n n . | | I t i s t h e r e f o r e v e r yd e s i r a b l e t h a t t h e d i s c u s s i o n o f t h e f o u n d a t i o n s o f m e c h a n i c sb e t a k e n u p b y m a t h e m a t i c ia n s a l so . T h u s B o l t z m a n n ' sw o r k o n t h e p r i n c i p l e s o f m e c h a n i c s s u g g e s t s t h e p r o b l e mo f d e v e l o p i n g m a t h e m a t i c a l l y t h e l i m i t i n g p r o c e s s e s , t h e r em e r e l y i n d i c a t e d , w h i c h l e a d fr om t h e a t o m i s t ic v i ew t o t h el a w s of m o t i o n of c o n t i n u a . C o n v e r s e l y o n e m i g h t t r y t od e r i v e t h e l a w s o f t h e m o t i o n o f r i g i d b o d i e s b y a l i m i t i n g

p r o c e s s f ro m a s y s t e m of a x i o m s d e p e n d i n g u p o n t h e i d e aof c o n t i n u o u s l y v a r y i n g c o n d i t i o n s of a m a t e r i a l fi lli nga l l s p a c e c o n t i n u o u s l y , t h e s e c o n d i t i o n s b e i n g d e f i n e d b yp a r a m e t e r s . F o r t h e q u e s t io n a s t o t h e e q u i v a l e n c e ofd i ff e re n t s y s t e m s of a x i o m s i s a l w a y s of g r e a t t h e o r e t i c a li n t e r e s t .

I f g e o m e t r y i s t o s e r v e a s a m o d e l f or t h e t r e a t m e n t o fp h y s i c a l a x i o m s , w e s h a l l t r y f ir st b y a s m a l l n u m b e r ofa x i o m s t o i n c l u d e a s l a r g e a c l a s s a s p o s s i b l e o f p h y s i c a lp h e n o m e n a , a n d t h e n b y a d j o in i n g n e w a x io m s t o a r r i v eg r a d u a l l y a t t h e m o r e s p e ci a l t h e o r i e s . A t t h e s a m e t i m eL i e ' s a p r i n c i p l e o f s u b d i v i s i o n c a n p e r h a p s b e d e r i v e d f r o mp r o f o u n d t h e o r y of i n f in i t e t r a n s f o r m a t i o n g r o u p s . T h em a t h e m a t i c i a n w i l l h a v e a l s o t o t a k e a c co u n t n o t o n l y oft h o s e t h e o r i e s c o m i n g n e a r t o r e a l i t y , b u t a l s o , a s i n g e o m e t r y ,of a l l l o g i ca l l y p o s s i b le t h e o r i e s . H e m u s t b e a l w a y s a l e r tt o o b t a i n a c o m p l e t e s u r v e y o f a l l c o n c l u s i o n s d e r i v a b l ef r o m t h e s y s t e m o f a x i o m s a s s u m e d .

*Cf. Bohlmann, "Ueber Versicherungsmathematik", from the collection : Kle in and Riecke, Ueber angewandte Mathematik und Physik,Leipz ig, 1900.

f Die Mechanik in ihrer Entwickelung, Leipzig, 4th edition, 1901.% Die Prinzipien der Mechanik, Leipzig, 1894.§ Vorlesungen über die Principe der Mechanik, Leipzig, 1897..|| Einführung in das Studium der theoretischen Physik, Leipzig, 1900.

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1 9 0 2 . ] HILBERT I MATHEMATICAL PROBLEMS. 4 5 5

F u r t h e r , t h e m a t h e m a t i c ia n h a s t h e d u t y to t e s t e x a ct ly ine a c h i n s t a n c e w h e t h e r t h e n e w a x i o m s a r e c o m p a t i b l e w i t ht h e p r e v i o u s o n e s . T h e p h y s i c i s t, a s h i s t h e o r i e s d e v e l o p ,o f ten f inds h imse l f fo rced by the r esu l t s o f h i s exper imen tst o m a k e n e w h y p o t h e s e s , w h i l e h e d e p e n d s , w i t h r e s p e c tt o t h e c o m p a t i b i l i t y o f t h e n e w h y p o t h e s e s w i t h t h e o l da x i o m s , s o l e l y u p o n t h e s e e x p e r i m e n t s o r u p o n a c e r t a i np h y s i c a l i n t u i t i o n , a p r a c t i c e w h i c h i n t h e r i g o r o u s l y lo g i c a lb u i l d i n g u p of a t h e o r y i s n o t a d m i s s i b l e . T h e d e s i r e d p r oo fof t h e c o m p a t i b i l i t y o f a l l a s s u m p t i o n s se e m s t o m e a l s o ofi m p o r t a n c e , b e c a u s e t h e e f f o r t t o o b t a i n s u c h p r o o f a l w a y sfo rces us mos t e f fec tua l ly to an exac t fo rmula t ion o f thea x i o m s .

S o f ar w e h a v e c o n s i d e re d o n l y qu e s t i o n s c o n c e r n i n g t h ef o u n d a t i o n s of t h e m a t h e m a t i c a l s ci e n ce s . I n d e e d , t h e s t u d yo f t h e f o u n d a t i o n s o f a s c i e n c e i s a l w a y s p a r t i c u l a r l y a t t r a c t i v e , a n d t h e t e s t i n g o f t h e s e f o u n d a t i o n s w i l l a l w a y s b ea m o n g t h e f o re m o s t p r o b l e m s of t h e i n v e s t i g a t o r . W e i e r -s t r a s s o n c e s a i d , " T h e f in a l o b j e ct a l w a y s t o b e k e p t i nm i n d i s t o a r r i v e a t a c o r r e c t u n d e r s t a n d i n g o f t h e f o u n d a

t i o n s of t h e s c ie n c e . * * * B u t t o m a k e a n y p r o g r e s s i nt h e s c i e n c e s t h e s t u d y o f p a r t i c u l a r p r o b l e m s i s , o f c o u r s e ,i n d i s p e n s a b l e . " I n f a ct , a t h o r o u g h u n d e r s t a n d i n g of i t ss p e c i a l t h e o r i e s i s n e c e s s a r y t o t h e s u cc e ss fu l t r e a t m e n t o ft h e f o u n d a t i o n s of t h e sc i e n ce . O n l y t h a t a r c h i t e c t i s i n t h ep o s i t io n t o la y a s u r e fo u n d a t i o n fo r a s t r u c t u r e w h o k n o w si t s p u r p o s e t h o r o u g h l y a n d i n d e t a i l . S o w e t u r n n o w t ot h e s p e c ia l p r o b l e m s of t h e s e p a r a t e b r a n c h e s of m a t h e m a t i c sa n d c o n s i d e r f i r s t a r i t h m e t i c a n d a l g e b r a .

7 . I R R A T I O N A L I T Y A N D T R A N S C E N D E N C E O F C E R T A I N

N U M B E R S .

H e r m i t e ' s a r i t h m e t i c a l th e o r e m s o n t h e e x p o n e n t i a l f un c t i o n a n d t h e i r e x t e n s i o n b y L i n d e m a n n a r e c e r t a i n o f t h ea d m i r a t i o n of a l l g e n e r a t i o n s of m a t h e m a t i c i a n s . T h u s t h et a s k a t o n c e p r e s e n t s it se lf t o p e n e t r a t e f u r t h e r a l o n g t h ep a t h h e r e e n t e r e d , a s A . H u r w i t z h a s a l r e a d y d o n e i nt w o i n t e r e s t i n g p a p e r s , * " U e b e r a r i t h m e t i s ch e E i g e n s ch a ft e n g ew i s se r t r a n s z e n d e n t e r F u n k t i o n e n . " I s h o u l dl i k e , t h e r e f o r e , t o s k e t c h a c l a ss of p r o b l e m s w h i c h , i n m yo p i n i o n , s h o u l d b e a t t a c k e d a s h e r e n e x t i n o r d e r . T h a t

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

*Math, Annalen, vols. 22, 32 (1883, 1888) .

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4 5 6 HILBERT Î MATHEMATICAL PROBLEMS. [ J u t y j

s is , t a k e a l g e b r a i c v a l u e s fo r c e r t a i n a l g e b r a ic a r g u m e n t s ,s e e m s t o u s p a r t i c u l a r l y r e m a r k a b l e a n d w o r t h y o f t h o r o u g hi n v e s t i g a t i o n . I n d e e d , w e e x p e ct t r a n s c e n d e n t a l f u n c t i o n st o a s s u m e , i n g e n e r a l , t r a n s c e n d e u t a l v a l u e s f o r e v e n a l g e b r a ic a r g u m e n t s ; a n d , a l t h o u g h i t i s w e l l k n o w n t h a t t h e r ee x i s t i n t e g r a l t r a n s c e n d e n t a l f u n c ti o n s w h i c h e v e n h a v e r a t i o n a l v a l u e s f o r a l l a l g e b r a i c a r g u m e n t s , w e s h a l l s t i l l c o n s i d e r i t h i g h l y p r o b a b l e t h a t t h e e x p o n e n t i a l fu n c t i o n ei7r*,f o r e x a m p l e , w h i c h e v i d e n t l y h a s a l g e b r a i c v a l u e s f o r a l l r a t i o n a l a r g u m e n t s s, w i l l o n t h e o t h e r h a n d a l w a y s t a k e

t r a n s c e n d e n t a l v a l u e s fo r i r r a t i o n a l a l g e b r a ic v a l u e s of t h ea r g u m e n t z. W e c a n a l so g i v e t h i s s t a t e m e n t a g e o m e t r ic a lform, as fo l lows :

If in an isosceles triangle, the ratio of the base angle to theangle at the vertex be algebraic but not rational, the ratio betweenbase and side is always transcendental .

I n s p i t e o f t h e s i m p l i c i t y o f t h i s s t a t e m e n t a n d o f i t ss i m i l a r i t y t o t h e p r o b le m s s o lv e d b y H e r m i t e a n d L i n d e -m an n , I cons id er t h e p roof o f th i s the o re m ve ry d i ff icu lt ;a s a l s o t h e p ro o f t h a t

The expression «^, for an algebraic base a and an irrational al

gebraic ex ponent /5, e. g., the number 2l/

* or en

= i~*\ always represents a transc ende ntal or at least an irrational n umber.I t i s c e r t a i n t h a t t h e s o l u t i o n o f t h e s e a n d s i m i l a r p r o b l e m s

m u s t l e a d u s t o e n t i r e l y n e w m e t h o d s a n d t o a n e w i n s i g h ti n t o t h e n a t u r e of s p e ci a l i r r a t i o n a l a n d t r a n s c e n d e n t a ln u m b e r s .

8. P R O B L E M S O F P R I M E N U M B E R S .

E s s e n t i a l p r o g r e s s i n t h e t h e o r y o f t h e d i s t r i b u t i o n o fp r i m e n u m b e r s h a s l a t e ly b e e n m a d e b y H a d a m a r d , d e l aV a l l é e - P o u s s i n , V o n M a n g o l d t a n d o t h e r s . F o r t h e c om

p l e t e s o l u t i o n , h o w e v e r , of t h e p r o b l e m s s e t u s b y B i e m a n n ' sp a p e r " U e b e r d i e A n z a h l d e r P r i m z a h l e n u n t e r e i n e r ge g e-b e n e n G r o s s e , " i t s t i l l r e m a i n s t o p r o v e t h e c o r r e c t n e s s ofa n e x c e e d i n gl y i m p o r t a n t s t a t e m e n t o f R i e m a n n , v i z ., thatthe zero points of the fun ction C(s) defined by the series

all have the real part \, e x c e p t t h e w e l l - k n o w n n e g a t i v e i n teg ra l r ea l ze ros . A s soon as th i s p roof ha s been su cces s

f u l l y e s t a b l i s h e d , t h e n e x t p r o b l e m w o u l d c o n s i s t i n t e s t i n gm o r e e x a c t l y R i e m a n n ' s i n f i n i t e s e r i e s f o r t h e n u m b e r o fp r i m e s b e lo w a g i v e n n u m b e r a n d , e s p e ci a ll y , to decide

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1 9 0 2 . ] H I L B E R T : M A T H E M A T IC A L P R O B L EM S . 4 5 7

whether the difference between the num ber of primes below a number x an d the integral logarithm of x does in fact become infiniteof an order not greater than \ in # .* F u r t h e r , w e s h o u l dd e t e r m i n e w h e t h e r t h e o c c a s i o n a l c o n d e n s a t i o n o f p r i m en u m b e r s w h i c h h a s b e e n n o t i ce d i n c o u n t i n g p r i m e si s r e a l l y d u e t o t h o s e t e r m s of R i e m a n n ' s f o r m u l a w h i c hdepend upon the f i r s t complex zeros o f the func t ion £ ( s ) .

A f t e r a n e x h a u s t i v e d i s c u s s i o n o f R i e m a n n ' s p r i m e n u m b e r f o r m u l a , p e r h a p s w e m a y so m e t im e b e i n a p o s i ti o n t oa t t e m p t t h e r i g o r o u s s o l u ti o n o f G o l d b a c h ' s p r o b l e m , fv i z . , w h e t h e r e v e r y i n t e g e r i s e x p r e s s i b l e a s t h e s u m o f t w op o s i t iv e p r i m e n u m b e r s ; a n d f u r t h e r t o a t t a c k t h e w e l l -k n o w n q u e s t io n , w h e t h e r t h e r e a r e a n i n f in i t e n u m b e r ofp a i r s of p r i m e n u m b e r s w i t h t h e d i ffe r en c e 2 , o r e v e n t h em o r e g e n e r a l p r o b l e m , w h e t h e r t h e l i n e a r d i o p h a n t i n e e q u a t i o n

ax + by + c = 0

( w i t h g i v e n i n t e g r a l c oe ffi ci en ts e a c h p r i m e t o t h e o t h e r s )i s a l w a y s s o l v a b l e i n p r i m e n u m b e r s x a n d y.

B u t t h e f o l l o w i n g p r o b l e m s e e m s t o m e o f n o l e s s i n t e r e s ta n d p e r h a p s of s ti l l w i d e r r a n g e : To apply the resu lts obtained for the distribution of rational prime num bers to the th eoryof the distribution of idea l primes in a given num ber-field 1c —a p r o b l e m w h i c h lo o k s t o w a r d t h e s t u d y of t h e fu n c t i o n C ft(s)b e l o n g i n g t o t h e f i e l d a n d d e f i n e d b y t h e s e r i e s

w h e r e t h e s u m e x t e n d s o v e r a l l i d e a l s j of the g iven r ea lm1c7 a n d n(j) d e n o t e s t h e n o r m o f t h e i d e a l j .

I m a y m e n t i o n t h r e e m o r e sp e c ia l p r o b l e m s i n n u m b e r

t h e o r y : o n e o n t h e l a w s o f r e c i p r o c i t y , o n e o n d i o p h a n t i n ee q u a t i o n s , a n d a t h i r d f r o m t h e r e a l m o f q u a d r a t i c f o r m s .

9 . P R O O F O F T H E M O S T G E N E R A L L A W O F R E C I P R O C I T Y

I N A N Y N U M B E R F I E L D .

For any field of nu mbers the law of reciprocity is to be provedfor the residues of th e Ith. power, w h e n I d e n o t e s a n o d d p r i m e ,a n d f u r t h e r w h e n I i s a power o f 2 o r a power o f an oddp r i m e .

* C f. a n a r t i c l e b y H . v o n K o c h , w h i c h is s o o n t o a p p e a r i n t h e Math ,Annalen [ V o l . 5 5 , p . 4 4 1 ] .

f C f . P . S t a c k e l : " Ü b e r G o l d b a c h ' s e m p i r is c h e s T h e o r e m , " Nach *richten d. K. Ges. d. Wiss. zu G ottingen, 1 8 9 6 , a n d L a n d a u , ibid., 1 9 0 0 .

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T h e l a w , a s w e l l a s t h e m e a n s e s s e n t i a l t o i t s proof, w i l l ,I b e l i e v e , r e s u l t b y s u i t a b l y g e n e r a l i z in g t h e t h e o r y of t h efi el d of t h e 7th r o o t s of u n i t y , * d e v e l o p e d b y m e , a n d m yt h e o r y o f r e l a t i v e q u a d r a t i c f i e l d s , f

1 0. D E T E R M I N A T I O N O F T H E S O L V A B IL IT Y O F A B I O P H A N -

T I N E E Q U A T I O N .

G i v e n a d i o p h a n t i n e e q u a t i o n w i t h a n y n u m b e r o f u n k n o w n q u a n t i t i e s a n d w i t h r a t i o n a l i n t e g r a l n u m e r i c a lcoefficients : To d evise a process according to which i t can be deter

mined by a finite nu mber of operations wheth er the equation issolvable in rational integers.

1 1 . Q U A DR A T IC F O R M S W I T H A N Y A L G E B R A I C N U M E R I C A L

C O E F F I C I E N T S .

O u r p r e s e n t k n o w l e d g e o f t h e t h e o r y o f q u a d r a t i c n u m b e rfields % p u t s u s i n a p o s i t i o n to attach suc cessfully the theory ofquadrat ic forms with any n umber of variables and with a ny algebraic numerical coefficients. T h i s l e a d s i n p a r t i c u l a r t o t h ei n t e r e s t i n g p r o b l e m : t o s o lv e a g i v e n q u a d r a t i c e q u a t i o n

w i t h a l g e b r a i c n u m e r i c a l c o e ffi ci en ts i n a n y n u m b e r ofv a r i a b l e s b y i n t e g r a l o r f r a c t i o n a l n u m b e r s b e l o n g i n g t o t h ea l g e b r a i c r e a l m o f r a t i o n a l i t y d e t e r m i n e d b y t h e c oe ffi ci en t s.

T h e f o ll o w i n g i m p o r t a n t p r o b l e m m a y f o rm a t r a n s i t i o nt o a l g e b r a a n d t h e t h e o r y o f f u n c t i o n s :

1 2. E X T E N S I O N O F K B O N E C K E R ' S T H E O R E M O N A B E L I A N

F I E L D S T O A N Y A L G E B R A I C E E A L M O F E A T I O N A L I T Y .

T h e t h e o r e m t h a t e v e r y a b e l i a n n u m b e r f i e l d a r i s e sfr o m t h e r e a l m of ra t i o n a l n u m b e r s b y t h e c o m p o s i ti o n o f

fi eld s of r o o t s of u n i t y i s d u e t o K r o n e c k e r . T h i s f u n d a m e n t a l t h e o r e m i n t h e t h e o r y o f i n t e g r a l e q u a t i o n s c o n t a i n s t w o s t a t e m e n t s , n a m e l y :

F i r s t . I t a n s w e r s t h e q u e s ti o n a s t o t h e n u m b e r a n d

*Jahresber. d. Deutsckm Math.-Vereinigung, u Ueber d ie Theorie deralgebraisohen Zahlkörper," vol . 4 (1897), Part V.

f Math. Annalen, vol. 51 and Nachrichten d. K. Ges. d. Wiss. zu Got-tingen, 1898.% Hubert, " Ueber den Dirichlet'schen biquadratischen Zahlenkörper, "

Math. Annalen, vol. 4 5 ; " Ueber die Theorie der relativquadratischenZahlkörper," Jahresber. d. Deutschen Mathematiker-Vereinigung, 1897, andMath. Annalen, vol. 5 1 ; " Ueber die Theorie der relativ-Abelschen

Körper ," Nachrichten d. É. Ges. d. Wiss. zu Göttingen, 1898 ; Grundlagender Geometrie, Leipzig, 1899, Chap. VIII, I 83 [Translation by Town-send, Chicago, 1902] , Cf. also the dissertation of G. Rückle, Göttingen,1901.

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e x i s t e n c e of t h o s e e q u a t i o n s w h i c h h a v e a g i v e n d e g r e e , ag i v e n a b e l i a n g r o u p a n d a g i v e n d i s c r i m i n a n t w i t h re s p e c tt o t h e r e a l m o f r a t i o n a l n u m b e r s .

S e c o n d . I t s t a t e s t h a t t h e r o o t s of s u c h e q u a t i o n s f o rma r e a l m o f a l g e b r a i c n u m b e r s w h i c h c o i n c i d e s w i t h t h er e a l m o b t a i n e d b y a s s i g n i n g t o t h e a r g u m e n t z i n t h e e x p o n e n t i a l f u n c t i o n e"™ a l l r a t i o n a l n u m e r i c a l v a l u e s i n s u c ces s ion .

T h e f i r s t s t a t e m e n t i s c o n c e r n e d w i t h t h e q u e s t i o n o f t h ed e t e r m i n a t i o n o f c e r t a i n a l g e b r a i c n u m b e r s b y t h e i r g r o u p sa n d t h e i r b r a n c h i n g . T h i s q u e s t io n c o r r e sp o n d s , th e r e f o r e ,t o t h e k n o w n p r o b l e m o f t h e d e t e r m i n a t i o n o f a l g e b r a i cf u n c t io n s c o r r e s p o n d i n g t o g iv e n E i e m a n n s u r fa c e s. T h es e co n d s t a t e m e n t f u r n is h e s t h e r e q u i r e d n u m b e r s b y t r a n s c e n d e n t a l m e a n s , n a m e l y , b y t h e e x p o n e n t i a l f u n c t i o n e

i1Tz.

S i n c e t h e r e a l m o f t h e i m a g i n a r y q u a d r a t i c n u m b e r f i e l d si s t h e s i m p l e s t a f t e r t h e r e a l m o f r a t i o n a l n u m b e r s , t h ep r o b l e m a r i s e s , t o e x t e n d K r o n e c k e r ' s t h e o r e m t o t h i s c a s e .K r o n e c k e r h i m s e l f h a s m a d e t h e a s s e r t i o n t h a t t h e a b e l i a ne q u a t i o n s i n t h e r e a l m o f a q u a d r a t i c f i e l d a r e g i v e n b y t h ee q u a t i o n s o f t r a n s f o r m a t i o n o f e l l i p t i c f u n c t i o n s w i t h s i n

g u l a r m o d u l i , s o t h a t t h e e l li p t ic f u n ct io n a s s u m e s h e r e t h es a m e r ô l e a s t h e e x p o n e n t i a l f u n c t i o n i n t h e f o r m e r c a s e .T h e p r o o f o f K r o n e c k e r ' s c o n j e c t u r e h a s n o t y e t b e e nf u r n i s h e d ; b u t I b e l ie v e t h a t i t m u s t b e o b t a i n a b l e w i t h o u tvery g rea t d i f f icu l ty on the bas i s o f the theory o f complexm u l t i p l i c a t i o n d e v e lo p e d b y H . W e b e r * w i t h t h e h e l p of t h ep u r e l y a r i t h m e t i c a l t h e o r e m s o n c l a ss fie ld s w h i c h I h a v ee s t a b l i s h e d .

F i n a l l y , t h e e x t e n s i o n o f K r o n e c k e r ' s t h e o r e m t o t h e c a s et h a t , in place of th e realm of rational numbers or of th e imaginaryquadratic field, any algebraic ield whatever is laid d own as realm

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

T h e p r o b l e m i s f o u n d t o b e a c c e ss i b le fr o m m a n y s t a n d p o i n t s . I r e g a r d a s t h e m o s t i m p o r t a n t k e y t o t h e a r i t h m e t ica l pa r t o f th i s p rob lem the genera l l aw of r ec ip roc i ty fo rr e s i d u e s o f Ith p o w e r s w i t h i n a n y g i v e n n u m b e r f i e l d .

A s t o t h e f u n c t i o n - t h e o r e t i c a l p a r t o f t h e p r o b l e m , t h e i n v e s t i g a t o r i n t h i s a t t r a c t i v e r e g i o n w i l l b e g u i d e d b y t h er e m a r k a b l e a n a l o g i e s w h i c h a r e n o t i c e a b l e b e t w e e n t h et h e o r y o f a l g e b r a i c f u n c t i o n s o f o n e v a r i a b l e a n d t h e t h e o r y

* Ell ipt ische Funotionen u nd algebraisobe Zahlen. Braunschweig,1891.

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4 6 0 H I L B E R T : M A T H E M A T IC A L P R O B L EM S . [ J u l y ,

of a l g e b r a i c n u m b e r s . H e n s e l * h a s p r o p o s e d a n d i n v e s t i g a t e d t h e a n a l o g u e i n t h e t h e o r y of a lg e b r a i c n u m b e r s t ot h e d e v e l o p m e n t i n p o w e r s e r i e s o f a n a l g e b r a i c f u n c t i o n ; a n d L a n d s b e r g f h a s t r e a t e d t h e a n a l o g u e of t h eE i e m a n n - K o c h t h e o r e m . T h e a n a l o g y b e t w e e n t h e d efic i e n c y o f a R i e m a n n s u r f a c e a n d t h a t o f t h e c l a s s n u m b e r o fa fie ld of n u m b e r s i s a l s o e v i d e n t . C o n s i d e r a R i e m a n nsurface of deficiency p = 1 ( t o t o u c h o n t h e s i m p l e s t c a s eo n l y ) a n d o n t h e o t h e r h a n d a n u m b e r f i e l d o f c l a s s h = 2 .T o t h e p r o of of t h e e x i s t e n c e of a n i n t e g r a l e v e r y w h e r e

f i n i t e o n t h e R i e m a n n s u r f a c e , c o r r e s p o n d s t h e p r o o f o ft h e e x i s t e n c e o f a n i n t e g e r a i n t h e n u m b e r fie ld s u c h t h a tt h e n u m b e r V a r e p r e s e n t s a q u a d r a t i c fie ld , r e l a t i v e l y u n -b r a n c h e d w i t h r e s p e c t t o t h e f u n d a m e n t a l fie ld . I n t h et h e o r y o f a l g e b r a i c f u n c ti o n s , t h e m e t h o d o f b o u n d a r y v a l u e s(Randwerthaufgabe) s e r v e s , a s i s w e l l k n o w n , fo r t h e p r o ofof R i e m a n n ' s e x i s te n c e t h e o r e m . I n t h e t h e o r y of n u m b e rf ie lds a l so , the p roof o f the ex i s tence o f jus t th i s number aoffe rs th e g rea tes t d i ff icu l ty . T h is p roof succeeds w i th in d i s p e n s a b l e a s s is t a n c e fr om t h e t h e o r e m t h a t i n t h e n u m b e rfie ld t h e r e a r e a l w a y s p r i m e i d e a l s c o r r e s p o n d i n g t o g i v e n

r e s i d u a l p r o p e r t i e s . T h i s l a t t e r f ac t i s t h e r e f o re t h e a n a l o g u e i n n u m b e r t h e o r y t o t h e p r o b l e m o f b o u n d a r y v a l u e s .T h e e q u a t i o n of A b e l ' s th e o r e m i n t h e t h e o r y of a l g e b r a i c

f u n c t i o n s e x p r e s s e s , a s i s w e l l k n o w n , t h e n e c e s s a r y a n ds u f f i c i e n t c o n d i t i o n t h a t t h e p o i n t s i n q u e s t i o n o n t h eR i e m a n n s u r f a ce a r e t h e z e r o p o i n t s of a n a l g e b r a i cf u n c t i o n b e l o n g i n g t o t h e s u r f a ce . T h e e x a c t a n a l o g u e ofA b e l ' s t h e o r e m , i n t h e t h e o r y o f t h e n u m b e r f i e l d o f c l a s sh = 2 , i s t h e e q u a t i o n o f t h e l a w o f q u a d r a t i c r e c i p r o c i t y J

w h i c h d e c l a r e s t h a t t h e , id e a l j i s t h e n a n d o n l y t h e n ap r i n c i p a l i d e a l o f t h e n u m b e r f i e l d w h e n t h e q u a d r a t i c r e s i d u e o f t h e n u m b e r a w i t h r e s p e c t t o t h e i d e a l j i s p o s i t i v e .

I t w i ll b e s e e n t h a t i n t h e p r o b l e m j u s t s k e t ch e d t h e t h r e ef u n d a m e n t a l b r a n c h e s of m a t h e m a t i cs , n u m b e r t h e o r y , a l g e b r a a n d f u n c t i o n t h e o r y , c o m e i n t o c l o s e s t t o u c h w i t h

*,Tahresber. d.Deutschen M ath.-Ver einigung, vo l . 6 , a n d a n a r t i c l e soont o a p p e a r i n t h e Math . Annalen [ V o l . 5 5 , p . 3 0 1 ] : " U e b e r d i e E n t w i o k e l -

u n g d e r a l g e b ra i s ch e n Z a h l e n i n P o t e n z r e i h e n . "f M ath. Anna len, v o l . 5 0 ( 1 8 9 8 ) .t C f. H u b e r t , " U e b e r d i e T h e o r i e d e r r e l a ti v - A b e l s o h e n Z a h l k ö r p e r , "

Gött . Nachrichten, 1898 .

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o n e a n o t h e r , a n d I a m c e r t a i n t h a t t h e t h e o r y o f a n a l y t i c a lf u n c t io n s of s e v e r a l v a r i a b l e s i n p a r t i c u l a r w o u l d b e n o t a b l ye n r i c h e d i f o n e s h o u l d s u c c e e d in finding and discussing thosefunctions wh ich play the part for any algebraic num ber field corresponding to that of the expon ential function in the field of rationalnu mbers and of the elliptic modular functions in the ima ginaryquadrat ic nu mber field.

P a s s i n g t o a l g e b r a , I s h a l l m e n t i o n a p r o b l e m fr o m t h et h e o r y of e q u a t i o n s a n d o n e t o w h i c h t h e t h e o r y of a l g e b r a i c i n v a r i a n t s h a s l e d m e .

1 3 . I M P O S S I B I L IT Y O F T H E S O L U T IO N O F T H E G E N E R A L

E Q U A T I O N O F T H E 7 T H D E G R E E B Y M E A N S O F

F UN C T I O N S O F O N LY T W O A R G U M E N T S .

J S T o m o g r a p h y * d e a l s w i t h t h e p r o b l e m : t o s o l v e e q u a t i o n s b y m e a n s o f d r a w i n g s o f f a m i l i e s o f c u r v e s d e p e n d i n go n a n a r b i t r a r y p a r a m e t e r . I t is s e e n a t o n ce t h a t e v e r yr o o t o f a n e q u a t i o n w h o s e c o e f f i c i e n t s d e p e n d u p o n o n l yt w o p a r a m e t e r s , t h a t i s , e v e r y f u n c t i o n o f t w o i n d e p e n d e n tv a r i a b l e s , c a n b e r e p r e s e n t e d i n m a n i f o l d w a y s a c c o r d i n g t o

t h e p r i n c i p l e l y i n g a t t h e f o u n d a t i o n of n o m o g r a p h y . F u r t h e r , a l a r g e c l a ss of f u n c t i o n s of t h r e e o r m o r e v a r i a b l e sc a n e v i d e n t l y b e r e p r e s e n t e d b y t h i s p r i n c i p l e a l o n e w i t h o u tt h e u s e o f v a r i a b l e e l e m e n t s , n a m e l y a l l t h o s e w h i c h c a n b eg e n e r a t e d b y f o r m i n g f i r s t a f u n c t i o n o f t w o a r g u m e n t s ,t h e n e q u a t i n g e a c h o f t h e s e a r g u m e n t s t o a f u n c t i o n o f t w oa r g u m e n t s , n e x t r e p l a c i n g e a c h o f t h o s e a r g u m e n t s i n t h e i rt u r n b y a f u n c ti o n o f tw o a r g u m e n t s , a n d s o o n , r e g a r d i n ga s a d m i s s i b l e a n y fi ni te n u m b e r of i n s e r t i o n s o f f u n c t i o n sof t w o a r g u m e n t s . S o , fo r e x a m p l e , e v e r y r a t i o n a l fu n c t i o nof a n y n u m b e r of a r g u m e n t s b e l o n g s t o t h i s c la s s of f u n c

t i o n s c o n s t r u c t e d b y n o m o g r a p h i c t a b l e s ; f o r i t c a n b e g e n e r a t e d b y t h e p r o c e s s e s o f a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n a n d d i v i s i o n a n d e a c h of t h e s e p r o c e ss e s p r o d u c e s af u n c t i o n o f o n l y t w o a r g u m e n t s . O n e s e e s e a s i l y t h a t t h er o o t s o f a l l e q u a t i o n s w h i c h a r e s o l v a b l e b y r a d i c a l s i n t h en a t u r a l r e a l m of r a t i o n a l i t y b e l o n g t o t h i s cl a s s of f u n c t i o n s ; fo r h e r e t h e e x t r a c t i o n of r o o t s i s a d jo i n e d t o t h ef o u r a r i t h m e t i c a l o p e r a t i o n s a n d t h i s , i n d e e d , p r e s e n t s af u n c ti o n of o n e a r g u m e n t o n l y . L i k e w i s e t h e g e n e r a l e q u a t i o n s o f t h e 5 t h a n d 6 t h d e g r e e s a r e s o l v a b l e b y s u i t a b l en o m o g r a p h i c t a b l e s ; fo r, b y m e a n s of T s c h i r n h a u s e n t r a n s

f o r m a t i o n s , w h i c h r e q u i r e o n l y e x t r a c t i o n o f r o o t s , t h e y c a n

* d'Ocagne, Trai té de Nomographic, Paris , 1899.

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462 HILBERT : MATHEMATICAL PROBLEMS. [J u t y ?

be reduced to a form where the coefficients depend upon two

parameters only.

Now it is probable that the root of the equation of the

seventh degree is a function of its coefficients which does

not belong to this class of functions capable of nomographic

construction, i. e., that it cannot be constructed by a finite

number of insertions of functions of two arguments. In

order to prove this, the proof would be necessary that the

equation of the seventh degree ƒ7

+ xf* + yf2

4- % f + 1 = 0 is

not solvable with the help of any continuous functions of only two

arguments. I may be allowed to add that I have satisfied

myself by a rigorous process that there exist analytical

functions of three arguments x, y, z which cannot be ob

tained by a finite chain of functions of only two arguments.

By employing auxiliary movable elements, nomography

succeeds in constructing functions of more than two argu

ments, as d'Ocagne has recently proved in the case of the

equation of the 7th degree.*

14. P R O O F OF THE F I N I T E N E S S OF C E R T A I N COMPLETE

S Y S T E M S OF F U N C T I O N S .

In the theory of algebraic invariants, questions as to the

finiteness of complete systems of forms deserve, as it seems

to me, particular interest. L. Maurer f has lately suc

ceeded in extending the theorems on finiteness in invariant

theory proved by P. Gordan and myself, to the case where,

instead of the general projective group, any subgroup is

chosen as the basis for the definition of invariants.

An important step in this direction had been taken al

ready by A. Hurwitz,J who, by an ingenious process, suc

ceeded in effecting the proof, in its entire generality, of the

finiteness of the system of orthogonal invariants of an arbitrary ground form.

The study of the question as to the finiteness of invari

ants has led me to a simple problem which includes that

question as a par ticular case and whose solution probably

requires a decidedly more minutely detailed study of the

theory of elimination and of Kronecker's algebraic modular

systems than has yet been made.

* " S u r la résolution nomographique de Péquation du septièmedegré ." Comptes rendus, Paris, 1900.

f Cf. Sitzungsber. d. K. Acad. d. Wiss. zu München, 1899, and an ar

ticle about to appear in the Math. Annalen.t " Ueber die Erzeugung der In varianten durch Integ ration ," Nach-

richten d. K. Gesellschaft d. Wiss. zu Göttingen, 1897.

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1 9 0 2 . ] HILBBRT : MATHEMATICAL PKOBLEMS. 4 6 3

L e t a n u m b e r m of i n t e g r a l r a t i o n a l fu n c t io n s X v X 2, —, XM

of the n v a r i a b l e s x v x 2, —,x n b e g i v e n ,

E v e r y r a t i o n a l i n t e g r a l c o m b i n a t i o n o f X v —, Xm m u s t e v i d e n t l y a l w a y s b e c o m e , a f t e r s u b s t i t u t i o n of t h e a b o v e e x

p r e s s i o n s , a r a t i o n a l i n t e g r a l f u n c t i o n o f x v —, x n . N e v e r t h e l e s s , t h e r e m a y w e l l b e r a t i o n a l f r a c t io n a l f u n c t io n s ofX v —, Xm w h i c h , b y t h e o p e r a t i o n o f t h e s u b s t i t u t i o n 8,b e c o m e i n t e g r a l f u n ct i o n s i n ^ , - , xn . E v e r y s u c h r a t i o n a lfunc t ion o f X v —, Xm9 w h i c h b e c o m e s i n t e g r a l i n x v —, x n

a f t e r t h e a p p l i c a t i o n o f t h e s u b s t i t u t i o n S , I p ropose to ca l la relatively integral func t ion o f X v —, Xm. E v e r y i n t e g r a lfunc t ion o f X v —, Xm i s e v i d e n t l y a l s o r e l a t i v e l y i n t e g r a l ;f u r t h e r t h e s u m , d i f f e r e n c e a n d p r o d u c t o f r e l a t i v e i n t e g r a lf u n c t i o n s a r e t h e m s e l v e s r e l a t i v e l y i n t e g r a l .

T h e r e s u l t i n g p r o b l e m i s n o w t o d e c i d e w h e t h e r i t i sa l w a y s p o s s i b le to find a finite system of relatively integral fun ction Xv—,Xm by which every other re latively integral fun ction ofX,,---,X w may be expressed rationally and integrally.

W e c a n fo r m u l a t e t h e p r o b l e m s t i l l m o r e si m p l y if w e i n t ro du ce th e ide a of a fin it e f ie ld of in te gra l i ty . B y a f in i tef ie ld o f in tegra l i ty I mean a sys tem of func t ions f rom whicha f i n i t e n u m b e r o f f u n c t i o n s c a n b e c h o s e n , i n t e r m s o fw h i c h a l l o t h e r f u n c t io n s of t h e s y s t e m a r e r a t i o n a l l y a n di n t e g r a l l y e x p r e s si b l e . O u r p r o b l e m a m o u n t s , t h e n , t o t h i s :t o s h o w t h a t a l l r e l a t i v e l y i n t e g r a l f u n c t i o n s of a n y g iv e n

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

r e s t r ic t i o n s d r a w n fro m n u m b e r t h e o r y , b y a s su m i n g t h ecoeff ic ients of the given funct ions fv—Jm t o b e i n t e g e r s a n di n c l u d i n g a m o n g t h e r e l a t i v e l y i n t e g r a l f u n c ti o n s of X 1 ? - - ,X mo n l y s u c h r a t i o n a l f u n c t i o n s o f t h e s e a r g u m e n t s a s b e c o m e ,b y t h e a p p l i c a t i o n o f t h e s u b s t i t u t i o n s 8, r a t i o n a l i n t e g r a l f u n c t i o n s o f x ly"',xn with r a t iona l in tegra l coef f i c i e n t s .

T h e f o ll o w i n g i s a s i m p l e p a r t i c u l a r c a s e of t h i s r e fi n e d

p r o b l e m : L e t m i n t e g r a l r a t i o n a l f u n c t i o n s Xv~,Xm of onev a r i a b l e x w i t h i n t e g r a l r a t i o n a l c o e f f i c i e n t s , a n d a p r i m en u m b e r p b e g i v e n . C o n s i d e r t h e s y s t e m of t h o s e i n t e g r a l

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4 6 4 H I L B B R T : M A T H E M A T I C A L P R O B L E M S . [ J u l y ,

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

G {S v- ,X m)

p "

w h e r e 6? i s a r a t i o n a l i n t e g r a l f u n ct io n of t h e a r g u m e n t sX v • • •, Xm a n d p

h i s a n y p o w e r of t h e p r i m e n u m b e r p. E a r l i e ri n v e s t i g a t i o n s of m i n e * sh o w i m m e d i a t e l y t h a t a l l s u c h e x p res s ions fo r a f ixed exponen t h fo rm a f in i t e domain o f in t e g r a l i t y . B u t t h e q u e s t i o n h e r e i s w h e t h e r t h e s a m e i s t r u e

f o r a l l e x p o n e n t s h, i. e., w h e t h e r a f i n i t e n u m b e r o f s u c he x p r e s s i o n s c a n b e c h o s e n b y m e a n s o f w h i c h f o r e v e r y e x p o n e n t h e v e r y o t h e r e x p r e s s i o n of t h a t f o rm i s i n t e g r a l l ya n d r a t i o n a l l y e x p r e s s i b l e .

F r o m t h e b o u n d a r y r e gi o n b e t w e e n a l ge b r a a n d g e o m e t ry ,I w i l l m e n t i o n t w o p r o b le m s . T h e o n e c o n ce r n s e n u m e r a -t i v e g e o m e t r y a n d t h e o t h e r t h e t o p o l o gy of a l g e b r a ic c u r v e sa n d s u r f a c e s .

1 5 . B I GO R O U S F O U N D A T I O N O F S C H U B E R T ' S E N U M E R A T I V E

C A L C U L U S .

T h e p r o b l e m c o n s i s t s i n t h i s : To e stablish rigorously and withan exact determination of the limits of their validity those geometrical numbers which Schubert f especially has determined on thebasis of the so-called principle of special position, or conserv ationof number, by means of the enumerative calculus developed byhim.

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

d e c i d e d l y m o r e is r e q u i s i t e , n a m e l y , t h e a c t u a l c a r r y i n g o u tof the p roces s o f e l imina t ion in the case o f equa t ions o fs p e c i a l f o r m i n s u c h a w a y t h a t t h e d e g r e e o f t h e f i n a l e q u a t i o n s a n d t h e m u l t i p l i c i t y o f t h e i r s o l u t i o n s m a y b e f o r e s e e n .

1 6 . P R O B L E M O F T H E TO P O L O G Y O F A L G E B R A I C C U R V E S

A N D S U R F A C E S .

T h e m a x i m u m n u m b e r of clo se d a n d s e p a r a t e b r a n c h e sw h i c h a p l a n e a l g e b r a i c c u r v e o f t h e nth. o r d e r c a n h a v e h a sb e e n d e t e r m i n e d b y H a r n a c k . J T h e r e a r is e s t h e f u r th e r

*Math. Annalen, vol. 36 (1 890), p . 485.f Kalkül der abzâhlenden Geometrie, Leipzig, 1879.X Math , Annalen, vol. 10.

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1902.] HILBERT : MATHEMATICAL PROBLEMS. 465

q u e s t i o n a s t o t h e r e l a t i v e p o s i t i o n of t h e b r a n c h e s i n t h ep l a n e . A s t o c u r v e s of t h e 6 t h o r d e r , I h a v e s a ti sf ie d m y s el f— b y a c o m p l i c a te d p r o c e s s , i t i s t r u e — t h a t of t h e e l e v e nb r a n c h e s w h i c h t h e y c a n h a v e a c c o r d i n g t o H a r n a c k , b y n om e a n s a l l c a n l ie e x t e r n a l t o o n e a n o t h e r , b u t t h a t o n e b r a n c hm u s t e x i s t i n w h o s e i n t e r i o r o n e b r a n c h a n d i n w h o s e e x t e r i o r n i n e b r a n c h e s l i e , o r i n v e r s e l y . A thorough investigation of the relative position of the sepa rate branches when theirnu mber is the m aximu m seems to me to be of very great interest,and not less Bo the corresponding invest igat ion as to the number,

form, and position of the shee ts of an algebraic surface in space.T i l l n o w , i n d e e d , i t i s n o t e v e n k n o w n w h a t i s t h e m a x i m u m n u m b e r o f s h e e t s w h i c h a s u r f a c e o f t h e 4 t h o r d e r i nt h r e e d i m e n s i o n a l s p a c e c a n r e a l l y h a v e . *

I n c o n n e c t i o n w i t h t h i s p u r e l y a l g e b r a i c p r o b l e m , I w i s ht o b r i n g f o r w a r d a q u e s t i o n w h i c h , i t s e e m s t o m e , m a y b ea t t a c k e d b y t h e s a m e m e t h o d o f c o n t i n u o u s v a r i a t i o n o fcoef f ic ien t s , and whose answer i s o f co r respond ing va lue fo rthe topo logy of f ami l ies o f cu rves def ined by d i f fe ren t ia le q u a t i o n s . T h i s i s t h e q u e s t io n a s t o t h e m a x i m u m n u m b e r a n d p o s i t i o n of P o i n c a r é ' s b o u n d a r y c y c le s ( c y c le s

l im i te s ) fo r a d i ffe ren t ia l e qua t ion of th e fi rs t o r de r an ddegree o f the fo rmdy_Y

dx~~ X'

w h e r e X a n d Y a r e r a t i o n a l i n t e g r a l f u n c t i o n s o f t h e n t hd e g r e e i n x a n d y. W r i t t e n h o m o g e n e o u s ly , th i s i s

x(4-ê)+r(>i-4)+*(4-4)-<>-w h e r e X , F , a n d Z a r e r a t i o n a l i n t e g r a l h o m o g e n e o u s f u n c t i o n s o f t h e n t h d e g r e e i n x, y, z, a n d t h e l a t t e r a r e t o b ed e t e r m i n e d a s f u n c t i o n s o f t h e p a r a m e t e r t.

1 7. E X P R E S S I O N O F D E F I N I T E F O R M S B Y SQ U A R ES .

A r a t i o n a l i n t e g r a l f u n c t i o n o r f o r m i n a n y n u m b e r o fv a r i a b l e s w i t h r e a l c oe ffi cie n ts s u c h t h a t i t b e c o m e s n e g a t i v ef o r n o r e a l v a l u e s o f t h e s e v a r i a b l e s , i s s a i d t o b e definite.T h e s y s t e m of a l l d e f in i t e fo r m s i s i n v a r i a n t w i t h r e s p e c t

t o t h e o p e r a t i o n s of a d d i t i o n a n d m u l t i p l i c a t io n , b u t t h e*Cf. Eohn, ''Flachen vierter Ordnung," Preisschriften der Fürstlich

Jablonowskischen Gesellschaft, Leipzig, 1886.

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4 6 6 H I L B E R T : M A T H E M A T I C A L P R O B L E M S . [July,

q u o t i e n t of t w o d e f i n it e f o rm s — i n c a s e i t s h o u l d b e a ni n t e g r a l f u n c t io n of t h e v a r i a b l e s — i s a l s o a d e fi n i te f o r m .T h e s q u a r e o f a n y f o r m i s e v i d e n t l y a l w a y s a d e f i n i t ef o rm . B u t s i n c e , a s I h a v e s h o w n , * n o t e v e r y d e f i n it ef o rm c a n b e c o m p o u n d e d b y a d d i t i o n fr o m s q u a r e s o ff o r m s , t h e q u e s t i o n a r i s e s — w h i c h I h a v e a n s w e r e d a f f i r m a t i v e l y fo r t e r n a r y f o rm s f — w h e t h e r e v e r y d e fi n it e fo r mm a y n o t b e e x p r e s s e d a s a q u o t i e n t of s u m s o f s q u a r e s off o r m s . A t t h e s a m e t i m e i t i s d e s i r a b l e , fo r c e r t a i n q u e s t io ns as to th e po s s ib i l i ty o f ce r ta in geom et r ica l cons truc t

i o n s , to know whether the coef f ic ien t s o f the fo rms to beu s e d i n t h e e x p r e s s i o n m a y a l w a y s b e t a k e n f r o m t h e r e a l mof r a t io n a l i ty g ive n by th e coef ficient s o f th e fo rm re p re s e n t e d , x

I m e n t i o n o n e m o r e g e o m e t r i c a l p r o b l e m :

1 8. B U I L D I N G U P O F S P A C E F RO M C O N G R U E N T P O L Y H E D R A .

I f w e e n q u i r e f o r th o s e g r o u p s of m o t i o n s i n t h e p l a n e f o rw h i c h a f u n d a m e n t a l r e g io n e x i s ts , w e o b t a i n v a r i o u s a n s w e r s , a c co r d i n g a s t h e p l a n e c o n s id e r e d is R i e m a n n ' s

( e l l i p t i c ), E u c l i d ' s , o r L o b a c h e v s k y ' s ( h y p e r b o l i c ) . I n t h ec a s e o f t h e e l l i p t i c p l a n e t h e r e i s a f i n i t e n u m b e r o f e s s e n t i a l l y d i ff e re n t k i n d s of fu n d a m e n t a l r e g i o n s , a n d a fin itenumber o f congruen t r eg ions su f f ices fo r a comple te cover i n g of t h e w h o l e p l a n e ; t h e g r o u p c o n s i s t s i n d e e d of afi nit e n u m b e r o f m o t i o n s o n l y . I n t h e c a s e of t h e h y p e r b o l i c p l a n e t h e r e i s a n i n f i n i t e n u m b e r o f e s s e n t i a l l y d i f f e r e n t k i n d s o f f u n d a m e n t a l r e g i o n s , n a m e l y , t h e w e l l - k n o w nP o i n c a r é p o l y g o n s . F o r t h e c o m p l e te c o v e r i n g of t h e p l a n ea n i n f in i t e n u m b e r of c o n g r u e n t r e g io n s i s n e c e s s a r y . T h ec a s e o f E u c l i d ' s p l a n e s t a n d s b e t w e e n t h e s e ; f o r i n t h i s c a s e

t h e r e i s o n l y a fi ni te n u m b e r of e s s e n t i a l l y d i ff e re n t k i n d so f g r o u p s of m o t i o n s w i t h f u n d a m e n t a l r e g i o n s , b u t f or ac o m p l e t e c o v e r i n g o f t h e w h o l e p l a n e a n i n f i n i t e n u m b e r o fc o n g r u e n t r e g i o n s i s n e c e s s a r y .

E x a c t l y t h e c o r r e s p o n d i n g f a c t s a r e f o u n d i n s p a c e o fth re e d im en s ion s . T h e fact of th e fin it enes s of th e g ro up s o fm o t i o n s i n e l l i p t i c s p a c e i s a n i m m e d i a t e c o n s e q u e n c e of af u n d a m e n t a l t h e o r e m of C . J o r d a n , § w h e r e b y t h e n u m b e r of

* Math . Annalen , vol . 3 2 .f Acta Mathemat ica , vol. 1 7 .%

Cf. Hilbert : Grundlagen der Geometrie, Leipzig, 1899, Chap. 7 andin particular \ 38.\ Crelle 's Journal, v o l . 8 4 ( 1 8 7 8 ) , and Atti d. Reale Acad, di Napoli

1880.

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1 9 0 2 . ] H I L B E R T : M A T H E M A T I C A L P R O B L E M S . 46 7

es se n t ia l ly d i ffe ren t k in d s of fin it e g rou ps of l in ea r su bs t i tu t i o n s i n n v a r i a b l e s d o e s n o t s u r p a s s a c e r t a i n f in it e l i m i td e p e n d e n t u p o n n. T h e g r o u p s of m o t i o n s w i t h f u n d a m e n t a l r e g i o n s i n h y p e r b o l i c s p a ce h a v e b e e n i n v e s t i g a t e db y F r i c k e a n d K l e i n i n t h e l e c t u r e s o n t h e t h e o r y o f a u t o -m o r p h i c f u n c t i o n s , * a n d f i n a l l y F e d o r o v , f S c h o e n f l i e s % a n dl a t e l y R o h n § h a v e g iv e n t h e p r o o f t h a t t h e r e a r e , i n e u c l i-dean space , on ly a f in i t e number o f es sen t ia l ly d i f fe ren tk i n d s o f g r o u p s o f m o t i o n s w i t h a f u n d a m e n t a l r e g i o n .N o w , w h i l e t h e r e s u l t s a n d m e t h o d s of p ro o f a p p l i ca b l e t o

e l l i p t i c a n d h y p e r b o l i c s p a c e h o l d d i r e c t l y fo r ^ - d i m e n s i o n a ls p a c e a l s o , t h e g e n e r a l i z a t i o n of t h e t h e o r e m fo r e u c l i d e a nspace s eem s to o ffe r dec ide d d i ff icu lt ie s . T h e inv es t iga t iono f t h e f o l l o w i n g q u e s t i o n i s t h e r e f o r e d e s i r a b l e : Is there inn-dim ensional euclidean spa ce also only a finite nu mber of essent ially different kinds of groups of motions with a fundamentalregion f

A f u n d a m e n t a l r e g i o n o f e a c h g r o u p o f m o t i o n s , t o g e t h e rw i t h t h e c o n g r u e n t r e gi o n s a r i s i n g fr om t h e g r o u p , e v i d e n t l yfi lls u p s p a c e c o m p l e t e l y . T h e q u e s t i o n a r i s e s : Wh ether poly-hedra also exist which do not appear as fundam ental regions oj

groups of motions, by means of which nevertheless by a suitablejuxtaposition of con gruen t copies a complete filling up of allspace is possible. I p o i n t o u t t h e f o l l o w i n g q u e s t i o n , r e l a t e dt o t h e p r e c e d in g o n e , a n d i m p o r t a n t to n u m b e r th e o r y a n dp e r h a p s s o m e t im e s u s e fu l t o p h y s i c s a n d c h e m i s t r y : H o wc a n o n e a r r a n g e m o s t d e n s e l y i n s p a c e a n i n f i n i t e n u m b e r o fequa l so l ids o f g iven fo rm, e. </., sph eres w i th g iven r a d i io r r e g u l a r t e t r a h e d r a w i t h g i v e n e d g e s ( o r i n p r e s c r i b e dp o s i t i o n ) , t h a t i s , h o w c a n o n e so fit t h e m t o g e t h e r t h a t t h era t io of th e f il led to th e unf i l led space m ay be as g r ea t a sp o s s i b l e ?

I f w e l o o k o v e r t h e d e v e l o p m e n t of t h e t h e o r y of f u n c t i o n s i n t h e l a s t c e n t u r y , w e n o t i c e a b o v e a l l t h e f u n d a m e n t a l i m p o r t a n c e o f t h a t c l a s s o f f u n c t i o n s w h i c h w e n o wd e s i g n a t e a s a n a l y t i c f u n c t i o n s — a c l a s s o f f u n c t i o n s w h i c hw i l l p r o b a b l y s t a n d p e r m a n e n t l y i n t h e c e n t e r o f m a t h e m a t i c a l i n t e r e s t .

T h e r e a r e m a n y d i f f e r e n t s t a n d p o i n t s f r o m w h i c h w em i g h t c h o o s e , o u t o f t h e t o t a l i t y o f a l l c o n c e i v a b l e f u n c t i o n s , e x t e n s i v e c l a s s e s w o r t h y o f a p a r t i c u l a r l y t h o r o u g h

* Le ipzig, 1897. Cf. especially A bsch nitt I, Chap ters 2 an d 3.t Symmetrie der regelmâssigen Système von Figuren, 1890.t Krysta l lsysteme un d K rysta l l s t ruktur , Le ipzig, 1891.§ Math . Annalen, vol. 53.

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4 6 8 HILBERT : MATHEMATICAL PROBLEMS. [ J ^ t y ,

i n v e s t i g a t i o n . C o n s i d e r , fo r e x a m p l e , the class of functionscha racterize d by ordin ary or partial algebraic differen tial equations. I t s h o u l d b e o b s e r v e d t h a t t h i s c l a s s d o e s n o t c o n t a i nt h e f u n c t i o n s t h a t a r i s e i n n u m b e r t h e o r y a n d w h o s e i n v e s t i g a t i o n i s of t h e g r e a t e s t i m p o r t a n c e . F o r e x a m p l e , t h ebefo re -men t ioned func t ion C(s ) s a t i s f ies no a lgebra ic diff e r e n t i a l e q u a t i o n , a s i s e a s i l y s e e n w i t h t h e h e l p o f t h ew e l l - k n o w n r e l a t i o n b e t w e e n C ( s ) a n d C ( l — s ) , i f one referst o t h e t h e o r e m p r o v e d b y H o l d e r , * t h a t t h e f u n c ti o n JP (#)sa t is fies no a lgeb ra ic d i f fe ren t ia l equ a t io n . A gain , th e

f u n c t i o n o f t h e t w o v a r i a b l e s s a n d x def ined by the in f in i t ese r iesx^ x^ x^

C(«, x) == x + - + ^ + - + - f

w h i c h s t a n d s i n c l o s e r e l a t i o n w i t h t h e f u n c t i o n C ( s ) , p r o b ab ly s a t is fies no a lgeb ra ic p a r t i a l d i f fe ren t ia l e qu a t io n . I nt h e i n v e s t i g a t i o n o f t h i s q u e s t i o n t h e f u n c t i o n a l e q u a t i o n

3C (s, x)x — ~ — - = C (s — 1, x)

ox

w i l l h a v e t o b e u s e d .I f , o n t h e o t h e r h a n d , w e a r e l e a d b y a r i t h m e t i c a l o r g e o m e t r i c a l r e a s o n s t o c o n s i d e r t h e c l a s s o f a l l t h o s e f u n c t i o n sw h i c h a r e c o n t i n u o u s a n d i n d e f i n i t e ly d i f fe r e n t ia b l e , w es h o u l d b e o b l i g e d i n i t s i n v e s t i g a t i o n t o d i s p e n s e w i t h t h a tp l i a n t i n s t r u m e n t , t h e p o w e r s e r ie s , a n d w i t h t h e c ir c u m s t a n c e t h a t t h e f u n c t i o n i s f u l l y d e t e r m i n e d b y t h e a s s i g n m e n t of v a l u e s i n a n y r e g io n , h o w e v e r s m a l l . W h i l e , t h e r e fore , the fo rmer l imi ta t ion o f the f i e ld o f func t ions was toon a r r o w , t h e l a t t e r s e e m s t o m e t o o w i d e .

T h e i d e a of t h e a n a l y t i c f u n ct io n o n t h e o t h e r h a n d i n

c l u d e s t h e w h o l e w e a l t h o f f u n c t i o n s m o s t i m p o r t a n t t o s c i e n c e , w h e t h e r t h e y h a v e t h e i r o r i g i n i n n u m b e r t h e o r y , i nthe theory o f d i f fe ren t ia l equa t ions o r o f a lgebra ic func t iona le q u a t i o n s , w h e t h e r th e y a r i s e i n g e o m e t r y o r i n m a t h e m a t i c a lp h y s i c s ; a n d , t h e r e f o r e , i n t h e e n t i r e r e a l m o f f u n c t i o n s , t h ea n a l y t i c f u n c t i o n j u s t l y h o l d s u n d i s p u t e d s u p r e m a c y .

1 9 . A R E T H E S O LU T IO N S O F B E G U L A R P R O B L E M S I N T H E C A L

C UL US O F V A R I A T I O N S A L W A Y S N E C E S S A R I L Y A N A L Y T I C ?

O n e of t h e m o s t r e m a r k a b l e f a ct s i n t h e e l e m e n t s of t h e

t h e o r y of a n a l y t i c fu n c t io n s a p p e a r s to m e t o b e t h i s : T h a tt h e r e e x i s t p a r t i a l d i f f e r e n t i a l e q u a t i o n s w h o s e i n t e g r a l s a r e

* Math. Annalen, vol. 28.

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a l l o f n e c e s s i t y a n a l y t i c f u n c t i o n s o f t h e i n d e p e n d e n t v a r i a b l e s , t h a t i s , i n s h o r t , e q u a t i o n s s u s c e p t i b le of n o n e b u ta n a l y t i c s o l u t i o n s . T h e b e s t k n o w n p a r t i a l d i f fe r e n t ia le q u a t i o n s of t h i s k i n d a r e t h e p o t e n t i a l e q u a t i o n

9a;2 ^ dtf

a n d c e r t a i n l i n e a r d i f f e r e n t i a l e q u a t i o n s i n v e s t i g a t e d b yP i c a r d ; * a l s o t h e e q u a t i o n

dx2 +

df ~ '

t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n o f m i n i m a l s u r f a c e s , a n do t h e r s . M o s t of t h e s e p a r t i a l d i ff e re n t i a l e q u a t i o n s h a v e t h ec o m m o n c h a r a c t e r i s t i c of b e i n g t h e l a g r a n g i a n d i ff e re n t i a le q u a t i o n s of c e r t a i n p r o b l e m s of v a r i a t i o n , v i z ., of s u c h p r o b l e m s o f v a r i a t i o n

J j F (p y q, z; x, y) dx dy = m i n i m u m

[ dz dz-\

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

d2F d

2F ( d

2F

dp2

' dq2

m> o

-i ts e lf b e i n g a n a n a l y t i c f u n c t i o n . W e s h a l l c a l l t h i s s o r to f p r o b l e m a regular v a r i a t i o n p r o b l e m . I t i s c h ie fl y t h er e g u l a r v a r i a t i o n p r o b l e m s t h a t p l a y a r ô l e i n g e o m e t r y , i nm e c h a n i c s , a n d i n m a t h e m a t i c a l p h y s i c s ; a n d t h e q u e s t i o nn a t u r a l l y a r i s e s , w h e t h e r a l l s o l u t i o n s o f r e g u l a r v a r i a t i o np r o b l e m s m u s t n e c e s s a ri ly b e a n a l y t i c f u n c ti o n s . I n o t h e rw o r d s , does every lagran gian partial differential equ ation of aregular va riation problem have the property of admitting analyticintegrals exclu sively t A n d i s t h i s t h e c a se e v e n w h e n t h efunc t ion i s con s t r a ine d to as s um e, as , e . <jr., in D i r ic h le t ' sp r o b l e m o n t h e p o t e n t i a l f u n c t i o n , b o u n d a r y v a l u e s w h i c ha r e c o n t i n u o u s , b u t n o t a n a l y t i c ?

I m a y a d d t h a t t h e r e e x i s t s u r fa c e s of c o n s t a n t negativeg a u s s i a n c u r v a t u r e w h i c h a r e r e p r e s e n t a b l e b y f u n c t i o n st h a t a r e c o n t i n u o u s a n d p o s se s s i n d e e d a l l t h e d e r i v a t i v e s ,

*Jour. de VEcole Polyteeh.y 1890.

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4 7 0 H I L B E R T : M A T H E M A T I C A L P R O B L E M S . [ J u l y ,

a n d y e t a r e n o t a n a l y t i c ; w h i l e o n t h e o t h e r h a n d i t i s p r o b a b l e t h a t e v e r y s u r f a c e w h o s e g a u s s i a n c u r v a t u r e i s c o n s t a n t a n d positive i s n e c e s s a r i l y a n a n a l y t i c s u r fa c e . A n d w ek n o w t h a t t h e s u r fa c e s of p o s it iv e c o n s t a n t c u r v a t u r e a r em o s t c lo s e l y r e l a t e d t o t h i s r e g u l a r v a r i a t i o n p r o b l e m : T op a s s t h r o u g h a c lo s e d c u r v e i n s p a c e a s u r fa c e of m i n i m a la rea which sha l l inc lose , in connec t ion w i th a f ixed su r facet h r o u g h t h e sa m e c lo s ed c u r v e , a v o l u m e of g i v e n m a g n i t u d e .

2 0 . T H E G E N E R A L P R O B L E M O F B O U N D A RY V A L U E S .

A n i m p o r t a n t p r o b l e m c l o se ly c o n n e c t e d w i t h t h e fo r eg o i n g i s t h e q u e s t i o n c o n c e r n i n g t h e e x i s t e n c e o f s o l u t i o n sof p a r t i a l d i ff e re n t ia l e q u a t i o n s w h e n t h e v a l u e s o n t h eb o u n d a r y of t h e r e g io n a r e p r e s c r ib e d . T h i s p r o b l e m i ss o l v e d i n t h e m a i n b y t h e k e e n m e t h o d s o f H . A . S c h w a r z ,C . N e u m a n n , a n d P o i n c a r é f o r t h e d i f f e r e n t i a l e q u a t i o n o ft h e p o t e n t i a l . T h e s e m e t h o d s , h o w e v e r , s e em t o b e g e n e r a l l y n o t c a p a b l e o f d i r e c t e x t e n s i o n t o t h e c a s e w h e r e a l o n gt h e b o u n d a r y t h e r e a r e p r e s c r i b e d e i t h e r t h e d i f f e r e n t i a lc oe ffi ci en ts o r a n y d e l a t i o n s b e t w e e n t h e s e a n d t h e v a l u e s o ft h e f u n c ti o n . N o r c a n t h e y b e e x t e n d e d i m m e d i a t e l y t o t h e

c a s e w h e r e t h e i n q u i r y i s n o t fo r p o t e n t i a l s u r f a c e s b u t , s a y

fo r su r faces of l eas t a r ea , o r su r faces of con s ta n t pos i t iveg a u s s i a n c u r v a t u r e , w h i c h a r e t o p a s s t h r o u g h a p r e s c r i b e dt w i s t e d c u r v e o r t o s t r e t c h o v e r a g i v e n r i n g s u r f a c e . I ti s m y c o n v i c t i o n t h a t i t w i l l b e p o s s i b l e t o p r o v e t h e s ee x i s te n c e t h e o r e m s b y m e a n s of a g e n e r a l p r i n c i p l e w h o s en a t u r e i s i n d i c a t e d b y D i r i c h l e t ' s p r i n c i p l e . T h i s g e n e r a lp r i n c i p l e w i l l t h e n p e r h a p s e n a b l e u s t o a p p r o a c h t h eq u e s t i o n : Has not every regular va riation problem a solution,provided certain assumptions regarding the given boundary conditions are satisfied ( s a y t h a t t h e f u n c ti o n s c o n c e rn e d i n

t h e s e b o u n d a r y c o n d i t i o n s a r e c o n t i n u o u s a n d h a v e i ns e c t i o n s o n e o r m o r e d e r i v a t i v e s ) , and provided also if needbe that the n otion of a solution shall be suitably extend ed f *

2 1 . P R O O F O F T H E E X I S T E N C E O F L I N E A R D I F F E R E N T I A L

E Q U A T IO N S H A V I N G A P R E S C R I B E D M O N O D R OM I C G R O U P .

I n t h e t h e o r y o f l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h o n ei n d e p e n d e n t v a r i a b l e z, I w i s h t o i n d i c a t e a n i m p o r t a n tp r o b l e m , o n e w h i c h v e r y l i k e l y B i e m a n n h i m s e l f m a y h a v eh a d in m i n d . T h i s p r o b l e m i s a s fo l lo w s : T o show that there

always exists a linear differential equation of the Fu chsian class,* Cf. my lecture on Dirichlet ' s principle in the Jahresber. d. Deutschen

Math.-Vereinigung, vol. 8 (1900), p. 184.

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with given singular points and monodromic group. T h e p r o b le mr e q u i r e s t h e p r o d u c t i o n o f n f u n c t i o n s o f t h e v a r i a b l e z,r e g u l a r t h r o u g h o u t t h e c o m p l e x z p l a n e e x c e p t a t t h eg i v e n s i n g u l a r p o i n t s ; a t t h e s e p o i n t s t h e f u n c t io n s m a ybecome in f in i t e o f on ly f in i t e o rder , and when z d e s c r i b e sc i r c u i t s a b o u t t h e s e p o i n t s t h e f u n c t i o n s s h a l l u n d e r g o t h ep r e s c r i b e d l i n e a r s u b s t i t u t i o n s . T h e e x i s t e n c e of s u c h diff e r e n t ia l e q u a t i o n s h a s b e e n s h o w n t o b e p r o b a b l e b y c o u n t i n g t h e c o n s t a n t s , b u t t h e r i g o r o u s p r o o f h a s b e e n o b t a i n e du p t o t h i s t i m e o n l y i n t h e p a r t i c u l a r c a se w h e r e t h e fu n

d a m e n t a l e q u a t i o n s o f t h e g i v e n s u b s t i t u t i o n s h a v e r o o t sa l l of a b s o l u t e m a g n i t u d e u n i t y . L . S c h l e si n g er h a s g i v e nt h i s proof,* b a s e d u p o n P o i n c a r é ' s t h e o r y o f t h e F u c h s i a nC-funct ions . T h e th eo ry of l ine ar d i f fe ren t ia l equ a t io nsw o u l d e v i d e n t l y h a v e a m o r e f i n i s h e d a p p e a r a n c e i f t h ep r o b l e m h e r e s k e t c h e d c o u l d b e d i sp o s e d of b y s o m e p e r f e c t l y g e n e r a l m e t h o d .

2 2 . T T N IFO R M I ZA T IO N O F A N A L Y T I C R E L A T I O N S B Y M E A N S

O F A U T O M O R P H I C F U N C T I O N S .

A s P o i n c a r é w a s t h e fir st t o p r o v e , i t i s a l w a y s p o s s i b le t or e d u c e a n y a l g e b r a i c r e l a t i o n b e t w e e n t w o v a r i a b l e s t o u n i f o r m i t y b y t h e u s e o f a u t o m o r p h i c f u n c t i o n s o f o n e v a r i a b l e .T h a t i s , i f a n y a l g e b r a i c e q u a t i o n i n t w o v a r i a b l e s b e g i v e n ,t h e r e c a n a l w a y s b e f o u n d fo r t h e s e v a r i a b l e s t w o s u c hs i n g l e v a l u e d a u t o m o r p h i c f u n c t i o n s o f a s i n g l e v a r i a b l et h a t t h e i r s u b s t i t u t i o n r e n d e r s t h e g i v e n a l g e b r a i c e q u a t i o na n i d e n t i t y . T h e g e n e r a l i z a ti o n of t h i s f u n d a m e n t a l t h e o r e mt o a n y a n a l y t i c n o n - a l g e b r a i c r e l a t i o n s w h a t e v e r b e t w e e nt w o v a r i a b l e s h a s l i k e w i s e b e e n a t t e m p t e d w i t h s u c c e s s b yP o i n c a r é , f t h o u g h b y a w a y e n t i r e l y d i ff e re n t fr o m t h a tw h i c h s e r v e d h i m i n t h e s p e c i a l p r o b l e m f i r s t m e n t i o n e d .F r o m P o i n c a r é ' s p r o o f of t h e p o s s i b i l i ty of r e d u c i n g t ou n i f o r m i t y a n a r b i t r a r y a n a l y t i c r e l a t i o n b e t w e e n t w ov a r i a b l e s , h o w e v e r , i t d o e s n o t b e c o m e a p p a r e n t w h e t h e rt h e r e s o l v i n g f u n c t io n s c a n b e d e t e r m i n e d t o m e e t c e r t a i na d d i t i o n a l c o n d i t i o n s . N a m e l y , i t i s n o t s h o w n w h e t h e rt h e t w o s i n g l e v a l u e d f u n c t i o n s o f t h e o n e n e w v a r i a b l e c a nb e s o c h o se n t h a t , w h i l e t h i s v a r i a b l e t r a v e r s e s t h e regulard o m a i n o f t h o s e f u n c t i o n s , t h e t o t a l i t y o f a l l r e g u l a r p o i n t sof t h e g i v e n a n a l y t i c fie ld a r e a c t u a l l y r e a c h e d a n d r e p r e s e n t e d . O n t h e c o n t r a r y it se e m s t o b e t h e c a se , fr o m P o i n -

* H and bu ch der Theorie der linearen Differentialgleichungen, vol . 2,par t 2 , No. 366.

-[Bul l , de la Soc. M ath, de France , vol. 11 (1883).

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c a r e ' s i n v e s t ig a t io n s , t h a t th e r e a r e b e s id e t h e b r a n c h p o i n t sc e r t a i n o t h e r s , i n g e n e r a l i n f i n i t e l y m a n y o t h e r d i s c r e t ee x c e p t i o n a l p o i n t s o f t h e a n a l y t i c f i e l d , t h a t c a n b e r e a c h e do n l y b y m a k i n g t h e n e w v a r i a b l e a p p r o a c h c e r t a i n l i m i t i n gp o i n t s of t h e fu n c t io n s . In view of the fundam ental importanceof P oincare 's formulation of the question it seem s to me that anelucidation and resolution of this difficulty is extrem ely desirable.

I n c o n j u n c ti o n w i t h t h i s p r o b l e m c o m e s u p t h e p r o b l e mo f r e d u c i n g t o u n i f o r m i t y a n a l g e b r a i c o r a n y o t h e r a n a l y t i cr e l a t i o n a m o n g t h r e e o r m o r e c o m p l e x v a r i a b l e s— a p r o b l e m

w h i c h i s k n o w n t o b e s o l v a b le in m a n y p a r t i c u l a r c a se s .T o w a r d t h e s o l u t i o n o f t h i s t h e r e c e n t i n v e s t i g a t i o n s o f P i c a r d o n a l g e b r a i c f u n c t i o n s o f t w o v a r i a b l e s a r e t o b er e g a rd e d a s w e lc om e a n d i m p o r t a n t p r e l im i n a r y s t u d i e s .

2 3 . F U R T H E R D E V E LO P M E N T O F T H E M E T H O D S O F T H E

C A L C U L U S O F V A R I A T I O N S .

S o f a r , I h a v e g e n e r a l l y m e n t i o n e d p r o b l e m s a s d e f i n i t ea n d s p e c i a l a s p o s s i b l e , i n t h e o p i n i o n t h a t i t i s j u s t s u c hd e f i n i t e a n d s p e c i a l p r o b l e m s t h a t a t t r a c t u s t h e m o s t a n df r o m w h i c h t h e m o s t l a s t i n g i n f l u e n c e i s o f t e n e x e r t e d u p o nsc ience . N e v e r t h e l e s s , I s h o u l d l i k e t o cl o se w i t h a g e n e r a lp r o b l e m , n a m e l y w i t h t h e i n d i c a t i o n o f a b r a n c h o f m a t h e m a t i c s r e p e a t e d l y m e n t i o n e d i n t h i s l e c t u r e — w h i c h , i n s p i t eof t h e c o n s i d e r ab l e a d v a n c e m e n t l a t e ly g iv e n i t b y W e i e r -s t r a s s , d o e s n o t re c e i v e t h e ge n e r a l a p p r e c i a t io n w h i c h , i n m yo p i n i o n , i s i t s d u e — I m e a n t h e c a l c u l u s o f v a r i a t i o n s . *

T h e l a c k o f i n t e r e s t i n t h i s i s p e r h a p s d u e i n p a r t t o t h en e e d of r e l i a b l e m o d e r n t e x t b o o k s . S o m u c h t h e m o r ep r a i s e w o r t h y i s i t t h a t A . K n e s e r i n a v e r y r e c e n t ly p u b l i s h e d w o r k h a s t r e a t e d t h e c a l c u l u s o f v a r i a t i o n s f r o m t h em o d e r n p o i n t s of v i e w a n d w i t h r e g a r d t o t h e m o d e r nd e m a n d fo r r i g o r , f

T h e c a l c u l u s o f v a r i a t i o n s i s , i n t h e w i d e s t s e n s e , t h et h e o r y o f t h e v a r i a t i o n o f f u n c t i o n s , a n d a s s u c h a p p e a r s a sa nece ssa ry ex tens io n of t h e d i f fe ren t i a l an d in t e g ra l ca lcu lus .

* T ext-books: Moigno-Lindelöf, Leçons duoaloul des variations, Paris,1861, and A. Kneser, Lehrbuch der Variations-reohnung, Braunschweig,1900.

t As an indication of th e contents of this work, i t may here be notedth a t for the simples t problem s Kneser derives sufficient condition s ofthe extreme even for th e case th at one l imit of integration is variable,and employs the envelope of a family of curves satisfying the differential

equations of the problem to prove the necessity of Jacobi 's condit ionsof the extreme . M oreover, i t should be noticed t ha t Kneser app liesW eierstrass 's theory also to the inqu iry for the extreme of such qua nti t ie sas are defined by differential equations.

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I n t h i s s e n s e , P o i n c a r é ? s i n v e s t i g a t i o n s o n t h e p r o b l e m o ft h r e e b o d i e s , f o r e x a m p l e , f o r m a c h a p t e r i n t h e c a l c u l u s o fv a r i a t i o n s , i n s o f a r a s P o i n c a r é d e r i v e s f r o m k n o w n o r b i t sb y t h e p r i n c i p l e o f v a r i a t i o n n e w o r b i t s o f s i m i l a r c h a r a c t e r .

I a d d h e r e a s h o r t j u s t i f i c a t i o n o f t h e g e n e r a l r e m a r k su p o n t h e c a l c u l u s of v a r i a t i o n s m a d e a t t h e b e g i n n i n g ofm y l e c t u r e .

T h e s i m p l e s t p r o b l e m i n t h e c a l c u l u s o f v a r i a t i o n s p r o p e ri s kno w n to con s i s t in f ind ing a func t ion y o f a v a r i a b l e xs u c h t h a t t h e d e f i n i t e i n t e g r a l

J=£F(y„y;x)dx, 3,, = g

a s s u m e s a m i n i m u m v a l u e a s c o m p a r e d w i t h t h e v a l u e s i tt a k e s w h e n y i s r ep laced by o ther func t ions o f x w i t h t h es a m e i n i t i a l a n d f i n a l v a l u e s .

T h e v a n i s h i n g o f t h e f i r s t V a r i a t i o n i n t h e u s u a l s e n s e

dj=0

gives fo r the des i r ed func t ion y t h e w e l l - k n o w n d i f f e r e n t i a le q u a t i o ndF

CD - ^ - * , - o ,

I n o r d e r t o i n v e s t i g a t e m o r e c l o s e l y t h e n e c e s s a r y a n dsuf lic ien t c r i t e r ia fo r th e occur re nce of th e r equ i red m ini m u m , w e c o n s i d e r t h e i n t e g r a l

J* = £{F+(y x-v)F p\dx,

[F^F(Pjy,x),F^ dJ^)l

Now we inquire how p is to be chosen as function of x ) y in orderthat the value of this integral J * shall be independ ent of th e pathof integration, i. e., of the choice of the func tion y of the variable x*T h e i n t e g r a l J * h a s t h e f o r m

J * = C\Ayu-B\dx,

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w h e r e A a n d B d o n o t c o n t a i n y x a n d t h e v a n i s h i n g of t h ef i r s t va r ia t ion

d j * = 0

i n t h e s e n s e w h i c h t h e n e w q u e s t i o n r e q u i r e s g i v e s t h ee q u a t i o n

dx+

dy '

i. e., w e o b t a i n f o r t h e f u n c t i o n p o f t h e t w o v a r i a b l e s x , y

t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n o f t h e f i r s t o r d e r

(i*) ffi + a W - J O . o .ox dy

T h e o r d i n a r y d i ff e re n t i a l e q u a t i o n of t h e s e c o n d o r d e r ( 1 )a n d t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 1 * ) s t a n d i n t h ec lo s e s t r e l a t i o n t o e a c h o t h e r . T h i s r e l a t i o n b e c o m e s i m m e d i a t e l y c l e a r t o u s b y t h e fo l lo w i n g s i m p l e t r a n s f o r m a t i o n

dJ

* =fjF

vd

y+F

*d

v + (ty- - * >F

* + ( & - * ) « ; }d x

= Ç\Fyôy + ôyxFp + ( y . - j p ) àF,\ dx

= 3J+ Ç\y,-p)àF,dx.

W e d e r i v e fr o m t h i s , n a m e l y , t h e f o ll o w i n g f a ct s : I f w ec o n s t r u c t a n y simple fam i ly of in teg ra l cu rv es of th e o rd i n a r y d i f f e r e n t i a l e q u a t i o n ( 1 ) o f t h e s e c o n d o r d e r a n d t h e n

f o r m a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n o f t h e f i r s t o r d e r( 2 ) y x = p(%,y)

w h i c h a l so a d m i t s t h e s e i n t e g r a l c u r v e s a s s o l u t i o n s , t h e nt h e f u n c t i o n p(x, y) i s a l w a y s a n i n t e g r a l o f t h e p a r t i a l diff e r e n t i a l e q u a t i o n ( 1 * ) of t h e f ir st o r d e r ; a n d c o n v e r s e l y ,if p(x, y) d e n o t e s a n y s o l u t i o n o f t h e p a r t i a l d i f fe r e n t i a le q u a t i o n ( 1 * ) o f t h e fir st o r d e r , a l l t h e n o n - s i n g u l a r i n tegra l s o f the o rd inary d i f fe ren t ia l equa t ion (2 ) o f the f i r s to r d e r a r e a t t h e s a m e t i m e i n t e g r a l s of t h e d i ff e r e n ti a l

e q u a t i o n ( 1 ) o f t h e s e c o n d o r d e r , o r i n s h o r t i f y x=s

p(x J y)i s a n in t eg ra l equ a t io n of th e f ir st o rd er of th e d i f fe ren t ia le q u a t i o n ( 1 ) of t h e s e c o n d o r d e r , p(x, y) r e p r e s e n t s a n

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1 9 0 2 . ] HILBERT : MATHEMATICAL PROBLEMS, 4 7 5

i n t e g r a l o f t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 1 * ) a n d c o n v e r s e l y ; t h e i n t e g r a l c u r v e s of t h e o r d i n a r y d i f fe r e n t ia le q u a t i o n o f t h e s e c o n d o r d e r a r e t h e r e f o r e , a t t h e s a m et i m e , t h e c h a r a c t e r i s t i c s o f t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n(1*) o f the f i r s t o rder .

I n t h e p r e s e n t c a s e w e m a y f i n d t h e s a m e r e s u l t b y m e a n sof a s im ple ca lcu la t io n ; fo r th i s g ives us th e d i f fe ren t ia le q u a t i o n s ( 1 ) a n d ( 1 * ) i n q u e s t i o n i n t h e f o r m

(1} vF 4- v F 4- F — 7 ^ = 0v y ifxx yxyx » iJx yxy • y^x y v ?

( 1 * ) ( A + PPy) F „ + pFpy + F px - F y = 0 ,

w h e r e t h e l o w e r i n d i c e s i n d i c a t e t h e p a r t i a l d e r i v a t i v e s w i t hr e s p e c t t o x, y, p, y x. T h e cor re c tnes s of th e a ffi rmed r e la t i o n i s c l e a r f r o m t h i s .

T h e c lo s e r e l a t i o n d e r i v e d b e fo r e a n d j u s t p r o v e d b e t w e e nt h e o r d i n a r y d i ff e re n t i a l e q u a t i o n ( 1 ) o f t h e s e co n d o r d e ra n d t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 1 * ) o f t h e f i r s t o r d e r ,is , as i t s eem s to m e , of fun da m en ta l s ign i f icance for th ec a l c u l u s of v a r i a t i o n s . F o r , fr o m t h e f a ct t h a t t h e i n t e g r a lJ * i s i n d e p e n d e n t o f t h e p a t h o f i n t e g r a t i o n i t f o l l o w s t h a t

(3) JT V (P ) + (y x-p)F p(p)\dx= £FQ x)dx,

i f w e t h i n k o f t h e l e f t h a n d i n t e g r a l a s t a k e n a l o n g a n y p a t hy a n d t h e r i g h t h a n d i n t e g r a l a l o n g a n i n t e g r a l c u r v e y oft h e d i ff e re n t i a l e q u a t i o n

y a = 2>0&, y ) .

W i t h t h e h e l p of e q u a t i o n ( 3 ) w e a r r i v e a t W e i e r s t r a s s ' s

f o r m u l a(* ) Ç

hF(yx)dx - ÇFÇy x)dx = f £(</*, p)dx,

% /a * / a %/a

w h e r e u n d e s ig n a te s W e i e r s t r a s s ' s e x p re s s io n , d e p e n d i n g u p o nv . * P > y > * >

F (y x, P ) = F {y x) - F{p) - (y . - p) F p(p).

S i n c e , t h e r e f o r e , t h e s o l u t i o n d e p e n d s o n l y o n f i n d i n g a n i n t e g r a l p (x, y) w h i c h i s s i n g l e v a l u e d a n d c o n t i n u o u s i n a ce r

t a i n n e i g h b o r h o o d o f t h e i n t e g r a l c u r v e y , w h i c h w e a r e c o n s i d e r i n g , t h e d e v e l o p m e n t s j u s t i n d i c a t e d l e a d i m m e d i a t e l y— w i t h o u t t h e i n t r o d u c t i o n of t h e s ec o nd v a r i a t i o n , b u t o n l y

8/8/2019 S0002-9904-1902-00923-3



b y t h e a p p l i c a t i o n of t h e p o l a r p r o c e s s tothe d i f fe ren t ia le q u a t i o n ( 1 ) — t o t h e e x p r e s s i o n of J a c o b i ' s c o n d i t i o n a n dt o t h e a n s w e r t o t h e q u e s t i o n : H o w f ar t h i s c o n d i t i o n ofJ a c o b i ' s i n c o n j u n c t io n w i t h W e i e r s t r a s s ' s c o n d i t i o n JE J> 0i s neces sa ry and suf f ic ien t fo r the occur rence o f a m i n i m u m .

T h e d e v e l o p m e n t s i n d i ca t e d m a y b e t ra n s f e r re d w i t h o u tn e c e s s i t a t i n g f u r t h e r c a l c u l a t i o n t o t h e c a s e o f t w o o r m o r er e q u i r e d f u n c t i o n s , a n d a l s o t o t h e c a se of ad o u b l e o r a m u l t i p l e i n t e g r a l . S o , fo r e x a m p l e , in t h e c a s e of a d o u b l ei n t e g r a l

J=JF(zx,zy,z;x,y)d«,, [«.-£, *,= | ]

t o b e e x t e n d e d o v e r a given r eg ion w, t h e v a n i s h i n g o f t h ef i r s t v a r i a t i o n ( t o b e u n d e r s t o o d i n t h e u s u a l s e n s e )

<5JT=0

g i v e s t h e w e l l - k n o w n d i f f e r e n t i a l e q u a t i o n of t h e s e c o n do r d e r

dF z dFg

r F -dF

F -d F

F-dFl

f o r t h e r e q u i r e d f u n c t i o n « of a; a n d y.O n t h e o t h e r h a n d w e c o n s i d e r t h e i n t e g r a l

J* =ƒ \F + (zx-p)Fp + ( s , - q)F t\ da,,

[F=F(P,q,z ;X,y),F P =

dF

^**>*>y\

„ _dF(p,q,z;x,y)1dq J '

and inquire, how p and q are to be taken a s functions of x, y an dz in order that the value of this integral ma y be indepe nden t ofthe choice of the surface passing through the given closed twistedcurve, i. e., of the choice of the function z of the variables x and y.

T h e i n t e g r a l J * h a s t h e f o r m

8/8/2019 S0002-9904-1902-00923-3



and the vanishing of the f i rs t var iat ion

<5J* = 0,

in the sense which the new formulat ion of the quest iondemands, gives the equat ion

dx dy dz '

i . e., we find for th e functions p an d q of the three var iablesx, y an d z the differential equation of the first order

dF p 3F q d(pF, + gF t-F ) = Q^

dx dy dx

If we ad d to this differential equation th e p artial different ial equat ion

( I * ) P* + 2P* — ?. + «fo

resul t ing from the equat ions

the partial differential equation (I) for the function z of thetwo var iables x an d y and the s imul taneous sys tem of thetwo partial differential equations of the first order (I*) forthe two funct ions p and q of the three var iables x, y, an d zstand toward one another in a relat ion exact ly analogous toth a t in w hich th e differential equat ions (1) and (1 *) s toodin the case of the simple integral.

I t follows from th e fact th a t th e integra l J * is indep en

dent of the choice of the surface of integration z t h a t

f\F(p, q) + (* u-p)F,(p, q) + (zy-q)F q(p, q)}du>

i f we think of the r ight hand integral as taken over an integral surface i of the partial differential equations

and with the help of this formula we arr ive at once at theformula

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( I V ) fF(zx, s s)d<o - fF(zx, zv)du> - ƒ E (

[£(>„, % „ p , q) = F (z x, zy) - F(p, q) - (z x -p)F p(p, q)

-(*y-q)F q(p,q)],

w h i c h p l a y s t h e s a m e r ôl e fo r t h e v a r i a t i o n of d o u b l e i n t e g r a l s a s t h e p r e v i o u s l y g i v e n f o r m u l a ( 4 ) f o r s i m p l e i n t e g r a l s . W i t h t h e h e l p of t h i s f o r m u l a w e c a n n o w a n s w e r t h eq u e s t i o n h o w fa r J a c o b r s c o n d i ti o n i n c o n j u n c t i o n w i t h

W e i e r s t r a s s ' s c o n d i t i o n U > 0 i s n e c e s s a r y a n d s u ffi ci en t fo rt h e o c c u r r e n c e o f a m i n i m u m .C o n n e c t e d w i t h t h e s e d e v e l o p m e n t s i s t h e m o d i fi e d f o r m

i n w h i c h A . K n e s e r , * b e g i n n i n g f r o m o t h e r p o i n t s o f v i e w ,h a s p r e s e n te d W e i e r s t r a s s 's t h e o r y . W h i l e W e i e r s t ra s s e m p l o y e d t o d e r i v e s u f f i c i e n t c o n d i t i o n s f o r t h e e x t r e m e t e h o si n t e g r a l c u r v e s of e q u a t i o n ( 1 ) w h i c h p a s s t h r o u g h a fi xe dp o i n t , K n e s e r o n t h e o t h e r h a n d m a k e s u s e o f a n y s i m p l ef a m i l y o f s u c h c u r v e s a n d c o n s t r u c t s f o r e v e r y s u c h f a m i l ya s o l u t i o n , c h a r a c t e r i s t i c f o r t h a t f a m i l y , o f t h a t p a r t i a ld i ff e re n t i a l e q u a t i o n w h i c h i s t o b e c o n s i d e r e d a s a g e n e r a l i

z a t i o n o f t h e J a c o b i - H a m i l t o n e q u a t i o n .T h e p r o b l e m s m e n t i o n e d a r e m e r e l y s a m p l e s o f p r o b l e m s ,

ye t th ey w i l l suffice to sh ow how r ich , how man i fo ld an dh o w e x t e n s i v e t h e m a t h e m a t i c a l s ci en c e o f t o - d a y i s , a n dt h e qu e s t i o n i s u r g e d u p o n u s w h e t h e r m a t h e m a t i c s i sd o o m e d t o t h e f a t e of t h o s e o t h e r s c ie n c e s t h a t h a v e s p l i tu p i n t o s e p a r a t e b r a n c h e s , w h o s e r e p r e s e n t a t i v e s s ca r ce l y u n d e r s t a n d o n e a n o t h e r a n d w h o s e c o n n e c t io n b e co m e s e v e rm o r e l o o s e . I d o n o t b e l i e v e t h i s n o r w i s h i t . M a t h e m a t i c a l s c i e n c e i s i n m y o p i n i o n a n i n d i v i s i b l e w h o l e , a n

o r g a n i s m w h o s e v i t a l i t y i s c o n d i t i o n e d u p o n t h e c o n n e c t i o nof i t s p a r t s . F o r w i t h a l l t h e v a r i e t y of m a t h e m a t i c a lk n o w l e d g e , w e a r e s t i l l cl e a r l y c o n s c io u s of t h e s i m i l a r i t yof the log ica l dev ices , the relationship of the ideas i n m a t h e m a t i c s a s a w h o l e a n d t h e n u m e r o u s a n a l o g i e s i n i t s d i f f e r e n t d e p a r t m e n t s . W e a ls o n o t ic e t h a t , t h e f a r th e r a m a t h e m a t i c a l t h e o r y i s d e v e l o p e d , t h e m o r e h a r m o n i o u s l y a n du n i f o r m l y d o e s i t s c o n s t r u c t i o n p r o c e e d , a n d u n s u s p e c t e dr e l a t i o n s a r e d i sc lo s e d b e t w e e n h i t h e r t o s e p a r a t e b r a n c h e sof t h e s c ie n c e . S o i t h a p p e n s t h a t , w i t h t h e e x t e n s i o n o fm a t h e m a t i c s , i t s o r g a n i c c h a r a c t e r i s n o t lo s t b u t o n l y

m a n i f e s t s i t s e l f t h e m o r e c l e a r l y .*Cf. his above-mentioned textbook, J J 14, 15, 19 and 20.

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1902.] REPLY TO MU. COOLIDGE. 479

B u t , w e a s k , w i t h t h e e x t e n s i o n of m a t h e m a t i c a l k n o w l edge wi l l i t no t f ina l ly become imposs ib le fo r the s ing le in v e s t i g a t o r t o e m b r a c e a l l d e p a r t m e n t s of t h i s k n o w l e d g e ?I n a n s w e r l e t m e p o i n t o u t h o w t h o r o u g h l y i t i s i n g r a i n e di n m a t h e m a t i c a l s ci e n ce t h a t e v e r y r e a l a d v a n c e g o es h a n di n h a n d w i t h t h e i n v e n t i o n of s h a r p e r t o o l s a n d s i m p l e rm e t h o d s w h i c h a t t h e s a m e t i m e a s s i s t i n u n d e r s t a n d i n ge a r l i e r t h e o r i e s a n d c a s t a s i d e o l d e r m o r e c o m p l i c a t e d d e v e l o p m e n t s . I t i s t h e r e f o r e p o s si b le fo r t h e i n d i v i d u a li n v e s t i g a t o r , w h e n h e m a k e s t h e s e s h a r p e r to o l s a n d s i m p l e rm e t h o d s h i s o w n , t o f i n d h i s w a y m o r e e a s i l y i n t h e v a r i o u sb r a n c h e s o f m a t h e m a t i c s t h a n i s p o s s i b l e i n a n y o t h e rsc ience .

T h e o r g a n i c u n i t y of m a t h e m a t i c s is i n h e r e n t i n t h e n a t u r eof t h i s s c ie n c e , fo r m a t h e m a t i c s i s t h e f o u n d a t i o n of a l l e x a c tk n o w l e d g e of n a t u r a l p h e n o m e n a . T h a t i t m a y co m p l e t e lyf u l f i l t h i s h i g h m i s s i o n , m a y t h e n e w c e n t u r y b r i n g i t g i f t e dm a s t e r s a n d m a n y z e a l o u s a n d e n t h u s i a s t i c d i s c i p l e s .

R E P L Y T O M R . J . L . C O O L I D G E 'S R E V I E W O FH I L L ' S E U C L I D .

I D E S I R E t o t h a n k t h e e d i t o r s o f t h e B U L L E T I N f o r t h e i rc o u r t e s y i n a c c e d i n g t o m y r e q u e s t t h a t t h e y s h o u l d i n s e r ta r e p l y t o t h e r e v i e w of m y e d i t i o n of t h e fi ft h a n d s i x t hb o o k s of E u c l i d ' s E l e m e n t s b y M r . C o o l id g e, p u b l i s h e d i nt h e F e b r u a r y n u m b e r o f t h e B U L L E T I N , a s i t c o n t a i n s s t a t e m e n t s w h i c h g i v e a n e r r o n e o u s i m p r e s s i o n of t h e c o n t e n t so f t h e b o o k .

T h e b o o k d i f f e r s f r o m p r e v i o u s e d i t i o n s i n t w o i m p o r t a n tp a r t i c u l a r s . T h e s e a r e :1 . T h e e x p l a n a t i o n s o f t h e f u n d a m e n t a l d e f i n i t i o n s o f t h e

f if th book of Eucl id .2 . T h e re m o v a l of t h e i n d i r e c t n e s s fr om E u c l i d ' s l i n e of

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

o n p a g e v i i i of t h e p r e f a ce , h a s b e e n p a s s e d o v e r w i t h o u tn o t i c e b y t h e r e v i e w e r . T h e d i s c o v e r y of t h i s i n d i r e c t n e s sa n d t h e p o s s i b i l i t y o f r e m o v i n g i t , w e r e p u b l i s h e d b y m e i nt h e Ca mbridge P hilosophical Transac tions, v o l u m e 1 6 , p a r t 4 ;

a n d t h e i m p o r t a n c e of t h e w o r k w a s r e co g n iz e d i n t h e r e v i e wof t h a t p a p e r i n t h e Jahrbuch uber die Fortschrit te der Mathe-matik , v o l u m e 2 8 ( 1 8 9 7 ) , p a g e 1 5 2 .

s0002-9904-1902-00923-3

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