8/12/2019 S0002-9904-1895-00271-2

1/16

1895] RIEMAKK AtfD MODERN MATHEMATICS. 165

R 1 E M A N N AND HIS S I G N I F I C A N C E POR THE DEV E L O P M E N T OF M O D E R N M A T H E M A T I C S .*ADDRESS DELIVERED AT THE GENERAL SESSION OF THE VER-SAMMLUNQ DEUTSGHER NATURFORSCHER UND AERZTE,IN VIENNA, SEPTEMBER 27, 1894,

BY PROFESSOR FELIX ISLEIN.I T is no do ub t unco m mo nly difficult to enter ta in a largeaudience with the discussion of any ma them atical questionor even of the general tendencies in the development ofm ath em atic al science. Th is difficulty arises from the factt h a t the very ideas with which themathem atic ian worksandwhose multifarious connectionsand interrelations he investigates are the p roduc t of long-co ntinued m ental laborandare therefore far removed from theth ingsof ordinary life.In spi te of thisI did nothesitatein accepting the honorconferred upon me by theExecut ive Com mit tee of yourAssociationin request ing me to address youto-day. In doingthisI was movedby thei l lustrious example of the greatinvestigator, so recently deceased,who hadorigina lly beenex

pected to speak heretoyou. It must always beregarded asa par t icular meri tofH e r m a n n vonHe lmh oltz t ha t, fromthevery beginning of his career, he took pains to present inlectures intell igible to a wider circle of scientific men theproblems and results of special work in all the manifoldbranchesof science that engaged hisa t ten t ion . Hethu s succeeded in being of assistance to each one of us in his ownspecial field.Whilefor pure m athemat icsit would,in the na tu re of thecase,beimpossibleto dothis completely,it is becoming moreand more recognized thatin thepresen t state of mathemat i cal scienceit isem inently desirableto try, at least, toaccomplishas m u c has can bea t ta ined in this respect. In sayingthisI do not express an individual opinion; I speak in thename of all the members of tiiat Mathematical Associationwhich was formed some years ago in connection with theAssociationofN atura l is t s andPhysicians and is practically,ifnot formally, identical with your Section I. Wecanno t helpfeeling thatin therapid developm ent of modern thoug htourscience is in danger of becoming more and more isolated.The int imate mutual re la t ion between mathematicsandtheoretical nat ur al science wh ich, to the lasting benefit of bothsides, existed ever sincetheriseof modern analysis, threatens

* Translated, with the permission of the author, by ALEXANDERZIWET.

8/12/2019 S0002-9904-1895-00271-2

2/16

166 RIEMAKN AND MODERN MATHEMATICS. [A p r i l ,to be disru pted . As mem bers of the M athem atical Association we desire to counteract this serious, ever-growing dangerwith all o ur po we r; it is for this reason th at we decided tomeet with the general Association of Natura lists. Th rou ghperso nal interc our se w ith you we wish to learn how scientificthought develops in your branches, and where accordinglyan opportunity may arise for applying the work of thema them atician. On the other ha nd, we desire and hope tofind in you a ready interest in and an intelligent appreciationof our aims and ideas.W ith these considerations in min d I shall now atte m pt togive you an idea of th e life-work of B E R K H A R D R I E M A N N , aman who more than any other has exerted a determining influence on the developmen t of mo dern mathe ma tics. I hopethat my remarks may prove of some interest to those, atleast, among you who are familiar with the general trend ofideas prevailing in mechanics and theoretical physics. Bu tall of you, I trust, will feel that the ideas of which I shallhave to speak form a connecting link between ma them aticsand natural science.

The outward life of Riemann may perhaps appeal to yoursym pa thy ; b ut it was too unev entful to arouse particu lar interest. Riem ann was one of those retirin g men of learningwho allow the ir profound tho ug hts to m atu re slowly in theseclusion of their study.H e was twenty-five years of age wh en, in 1 851 , he took hisdoctor 's degree in Gttingen, presenting a dissertation whichshowed rem arkab le power. It took thre e more years beforehe became a docen t at th e same university. At this t im ebega n th e app eara nce in rapid succession of all those impo rta n t researches of which I wish to speak. A t the d ea th ofDirichlet, in 1859, Riemann became his successor in the University of G tting en. Bu t already in 1863 he contrac tedth e fatal disease to which he succu mb ed in 1866, at th e earlyage of 40 years.His collected works, first published in 1876 by HeinrichW eber a nd D ede kind , and since issued in a second edition,are neither num erou s nor extensive. Th ey fill an octavovolume of 550 pages, and only about half of the mattercontained in this volume appeared during his lifetime.The remarkable influence exerted by Riemann's work in thepast and even to the present time is entirely due to theoriginality an d, of course, to th e penetrating power of hismathematical considerations.While the latter characteristic cannot be the object of myremarks to-day, I believe that you will understand the peculiar originality of Riem ann 's math em atical work if I point out

8/12/2019 S0002-9904-1895-00271-2

3/16

1 8 9 5 ] RIEM ANN AN D MODEBtf MATHEMATICS. 1 6 7to you rig ht here the unifying fund ame ntal idea to which asa common source all his developments can be traced.I mu st m ention, first of all , th at Kiem ann devoted mu chtime and tho ug ht to physical considerations. Grown upunder the great tradition which is represented by the combi-nation of the names of Gauss and Wilhelm Weber, influencedon the other hand by Herbart 's philosophy, he endeavoredagain and again to find a general mathematical formulationfor the laws und erlyin g all na tura l pheno men a. These investigations do n ot app ear to have been carried by him to asatisfactory completion; they are preserved in his posthumous papers in a very fragmen tary form. We find thereseveral incomplete attempts at a solution, all of which have,however, in common th e supposition which at the presentday, under the influence of Maxwell's electromagnetic theoryof lig ht, seems to be a dop ted by at least all you nger physicists, viz., th a t space is filled with a c on tinu ou s fluid w hichserves as the common medium for the propagation of optical,electrical, and gravitational phenomena.I shall not stop to explain the details, which at the presenttime could only have a historical interest. Th e poin t towhich I wish to call your attention is that these physicalviews are the mainspring of Biemann's purely mathema ticalinvestigations. T he same tenden cy which in physics discardsthe idea of action at a distance and explains all phenom enathrough the internal stresses of an all-pervading ether appears in ma them atics as the attem pt to und erstan d functionsfrom th e way they behave in th e infinitesimal, tha t is, fromth e differential equa tions satisfied by the m . An d jus t as inphysics any particula r phe nom enon depe nds on th e gene ralarran gem ent of the cond itions of the ex perim ent, thu s Kieman n particu larize s his functions by mean s of the specialboundary conditions which they are required to satisfy.Prom this point of view the formula required for the numerical calcu lation of th e func tion app ears as the final res ult ,and not as the starting-point, of the investigation.If I may be allowed to push the analogy so far, I shouldsay that the work of Riemann in m athematics offers a parallelto the work of Faraday in physics. Wh ile this comparisonhas reference in the first place to the qualitative co nte nt ofthe leading ideas due to the two men, I believe it to hold evenquantitatively ; i.e., th e results reached by Kiem ann are asimportant for mathem atics as the results of Fa rada y's workare for physical science.On the basis of this general conception let us now pass inreview the various lines of Biemann's mathematical researches.It will only be natural to begin with the theory of f unctions ofcomplex variables, which is most in timately connected with his

8/12/2019 S0002-9904-1895-00271-2

4/16

16 8 RIEMAKN AND MODERN MATHEMATICS, [ A p r il ,name, although he himself may have regarded it merely as anapplication of tendencies having a much wider range.Th e fundam ental idea of this theory is well know n. Toinvestigate the functions of a variable z we substitute for thisvariable a binomial quantity x + iy with which we operate soas to put always 1 for i\ Th e result is th at the prop ertiesof th e ord inary functions of sim ple variables become in telligible to a m uch higher degree tha n would otherwise be thecase. To rep eat th e words used by Rie ma nn himself in hisdissertation (1851) in which he laid down th e funda men talideas of his peculiar method of treating this theory: The introduction of complex values o f the variable brings out a certain harmony and regularity which otherwise would remainhidden.The founder of this theory is the great French mathematician Cauchy;* bu t only later, in G ermany, did this theoryassume its modern aspect which has made it the central pointof our pre sen t views of ma them atics. Th is was the resu lt ofthe simultaneous efforts of two mathematicians whom we shallhave to name together repeatedly,of Riemann and Weier-strass.While pointing to the same end, the particular methods ofthe se two men are as different as possible; they appear almostto be oppo sed to each o ther, tho ug h, from a high er po int ofview, this only means that they are complementary.Weierstrass defines the functions of a complex variableanalytically by means of a common formula, viz., by infinitepow er-series. In all his wo rk he avoids as far as possible th eassistance to be derived from the use of geometry; his specialachieve men t lies in the systematic rigor of his dem onstrations.Riemann, on the other hand, begins, in accordance withhis ge neral conce ption referred to above, with certain differen tial e qua tions satisfied by th e function s of x + iy* Thusthe problem at once assumes a physical form.L et us p u t (a; + iy) = u + iv . Ea ch of the two component parts of the function, u as well as v, then appears, owingto the differential equations, asa potential in th e space of th etwo variables x and y; and Rie ma nn 's m etho d can be brieflycharacterized by saying that he a pplies to these parts u and vthe principles of the theory of the potential. In other words,his starting point l ies in the domain of mathematical physics.

* In the text I refrained from men tioning Gauss, wh o, being in advanc eof his time in this as in other fields, anticipated many discoveries without publish ing wh at he had found. It is very rem arkable that in thepape rs of G auss we find occasional glimpses of method s in t he theo ryof functions wh ich are comp letely in line with the later methods ofRiemann, as if unconsciously a transfer of leading ideas had taken placefrom the older to the younger mathematician.

8/12/2019 S0002-9904-1895-00271-2

5/16

1 8 9 5 ] RIEMAHH AND MODERN MATHEMATICS. 16 9You will notice that even in mathematics free play is givento individuality of treatment.It should also be observed th at th e theo ry of th e pot en tial,wh ich in ou r day, owing to its impo rtanc e in th e theo ry ofelectricity and in o ther bra nch es of ph ysics, is qu ite universally known and used as an indispensable instrument ofresearch , was at tha t time in its infancy. It is tru e th a tGreen had written his fundamental memoir as early as in1828; bu t this paper rem ained for a long time almost un no ticed. In 1839 Gauss followed with his researches. As faras Germany is concerned, it is mainly due to the lectures ofDirichlet th at th e theo ry was farther developed an d becameknow n more ge nerally; and this is where Riem ann finds hisbase of operations.Riemann's specific achievement in this connection consists,of course, in the first place in the tendency to make thetheory of the potential of fundamental importance for thewhole of mathematics; and secondly, in a series of geometricconstructions, or, as I should prefe r to say, of geometric inventions, of wh ich you m us t allow me to say a few w ords.As a first step Riemann considers the equation u + iv= f(x-{-iy) throu gho ut as a representation (Abb ildung) ofth e xy-iplme on the w -p lan e. I t appears tha t this representation is conformai, i .e., th at i t preserves the m agn itud e ofthe ang les; ind eed, i t is directly characterized by this prop erty. We have thus a new mea ns of defining th e functionsof x - j - iy . Riem ann develops in this way th e elegant proposition t ha t th ere always exists a function th at transform sany simply-connected region of the ^ -p la n e in to any givensimply-connected region of the ^v-plane; this function isfully determined, apart from three constants which remainarbitrary.Next he introduces as a fundamental idea the conception ofwhat is now called a Riemann surface, i.e., a surface whichoverlies the plan e several time s, the different sheets ha ng ingtogeth er at th e so-called branch-po ints. Th is step in th edevelopment, while it must have been the most difficult totake , proved at the same time of the greatest consequence.Even at the present time we can daily notice how hard it isfor the beginner to understand the essential idea of the Riem an n surface and how he comes at once into th e possessionof th e whole th eory as soon as he has fully grasped th is fundam ental mode of conceiving the function. Th e Riemannsurface furnishes us a means of un ders tand ing many-valuedfunctions of x -f- iy in th e course of the ir variation . Fo r onthis surface there exist potentials just like those on the simpleplane , and their laws can be investigated by the same me tho ds;moreover, the representation (Abbildung) is again conformai.

8/12/2019 S0002-9904-1895-00271-2

6/16

170 RIEMANN AND MODERN MATHEMATICS. [A p r il ,As the primary principle of classification appears here theorder of connectivity of these surfaces, i.e., the number of thecross-sections or cuts that can be made without resolving thesurface into separate portion s. Th is is again an entirely newgeometrical method of attack, which in spite of its elementarycharacter had never been touched upon by anybody beforeEiemann.Pe rha ps I have gone too far into th e details with wh at Ihave said. I wish only to add t ha t all these new tools andmethods, created by Riemann for the purposes of pure mathematics out of the physical intuition, have again proved of thegreatest value for mathe matical physics. Th us , for instance,we now always mak e use of R ieman n's m ethods in trea tingth e stationary flow of a fluid within a two-dimensional region.A whole series of most interes ting prob lems, formerly regarde das insolvable, has th us been solved com pletely . One of th ebest know n problems of this kind is Helm holtz's determina tion of the shape of a free liquid jet.Perhaps less attention has been paid to another physicalapplication in which Riemann's ways of looking at things arelaid un de r contribu tion in a most attractive ma nner. I havein mind the theory of minimum surfaces. Riem ann's owninvestigations on this subject were not published till after hisde ath, in 1867, almost simultan eously with th e parallel investiga tion s of W eierstrass conc erning th e same question. Sinceth at t ime th e theory has been developed much farther bySchwarz and others. Th e problem is to determ ine the shapeof the least surface that can be spread out in a rigid frame,let us say, the form of equilibrium of a fluid lamina that fitsin a given contou r. I t is note wo rthy th at , with the aid ofRiem ann 's metho ds, the known functions of analysis are jus tsufficient to dispose of the more simple cases.These applications to which I here call particular attentionrep rese nt, of course, merely one side of th e m att er . T he recan be no doubt that the main value of these new methods inth e theo ry of function s is to be found in the ir use in puremathematics. I mus t try to show this more distin ctly, inaccordance with the importance of the subject, without presupposing much special knowledge.

Let me begin with the very general question of the presentstate of progress in th e dom ain of pu re mathem atics. To thelayman the advance of mathematical science may perhaps appear as something purely arbitrary because the concentrationon a definite g iven object is w antin g. Still the re exists aregulating influence, well recognized in other branches ofscience, tho ug h in a more limite d de gre e; it is the historicalcontinuity. Pure mathematics g rows as old problems are

8/12/2019 S0002-9904-1895-00271-2

7/16

18 9 5 ] RIEMAKK AND MODEBN MATHEMATICS. 171worked out by means of new methods. In proportion as a better understanding is thus gained for the older questions, newproblems must naturally arise.Gu ided by th is pr inc iple we mu st now briefly pass in review the working material that Kiemann found ready for usein the theory of functions when he entered upon his scientificcareer. I t ha d th en been recognized th at particu lar impo rtanc e attac hes to th ree classes among th e various k ind s ofana lytic func tions (i.e., func tions of x+ iy)- T h e first classcomprises the algebraic functions which are defined by afinite number of elementary operations (addition, multiplication, and division); in contrad istinction to thes e we have thetranscendental functions whose definition requires an infiniteseries of such operations. Am ong these trans cen den talfunctions the simplest are, on the one hand, the logarithms,on th e other th e trigon om etric functions, such as the sine,cosine, etc.Bu t in Kiem ann's time m athem atical science had alreadyadvanced beyond these elementary functions, first, to theelliptic functions derived from the inversion of the ellipticintegrals, and, second, to thef 'unctions connected with Gauss'shypergeometric series, viz., sphe rical h arm on ics, Bessel fun ct ions, gamma-functions, etc.Now what Eiemann accomplished may be stated briefly bysaying that for each of these three classes of functions hefound no t only new resu lts, b ut entirely new po ints of viewwhich have formed a continual source of inspiration up toth e present time. A few addition al rem arks may serve to explain this more in detail .The study of algebraic functions practically coincides w iththe study of algebraiccurves whose properties are investigatedby the geometer, wh ether h e calls himself an "a na ly st ," regarding the analytical formula as of primary importance, or a"synthet ic geometer ," as the term was understood by Steinerand von Staudt, who operates with the row of points and thepencil of ray s. T he essentially new po int of view, her e in tro duced by Eiemann, is that of the general single-valued transforma tion (or one-to-one corres pon denc e). Th is po int ofview allows to gro up the inn um era ble variety of algebraiccurves into large classes; and by ma king abstraction of thepeculiarities of shape of the individual curves, those generalprop erties can be studied tha t belong in comm on to all th ecurves of th e same class. T he geom eters have no t been slowto derive the results so obtained from their point of view andto develop them still farther ; Olebsch, in particular, worked inthis direction, and he even began to attack the correspondingproblems for algebraic configurations of more dimensions.But it must be insisted upon that the theory of curves should

8/12/2019 S0002-9904-1895-00271-2

8/16

172 RIEM AN N AN D MODERN MATHEMATICS. [April,try to assimilate themethods of Eiemann in their true essentialcha racte r. A first step will consist in co nstr uc ting on thecurve itself the analogue of the two-dimensional Eiemannsurface; and thi s can be done in various ways. A furtherstep would have to show us how to operate with the methodsof th e theo ry of function s in the configuration th u s defined.The theory of elliptic integrals finds its furth er develop me ntin the consideration of the general integrals of algebraic functions, a subject on which the Norwegian mathematician Abelpublished the first fundamental investigations in the twentiesof the prese nt cen tury. It m ust always be regarded as one ofthe greatest achievements of Jacobi that, by a sort of inspiration, he established for these integrals a problem of inversionwhich furnishes single-valued functions just as the simple inversion does in th e case of t he elliptic integ rals. T he actu alsolu tion of th is problem of inversion is th e cen tral task performed at the same time, but by different methods, by Weier-strass a nd Eie ma nn . T he great mem oir on th e Abelianfunctions in which Eiemann published his theory in 1857 hasalways been recognized as the most brilliant of all the achievem ents of his g enius. Ind eed , the result is here reached, no tby laborious c alcu lations , bu t in t he most direc t way, by aproper combination of the geometrical considerations just referred to. I have shown in ano ther place th at his resultscon cern ing th e in tegra ls, as well as th e conclusions th at followfor the algebraic functions, can be obtained in a very graphicmanner by considering the stationary flow of a fluid, say ofelectric ity, on closed surfaces situate d in any way in sp ace.This , however, has reference only to one half of Ei em an n'smem oir. Th e second half,which is concerned with the theta -series, is perh aps still more rem arkab le. Th e im po rtan t resultis here deduced that the theta-series required for the solutionof Jac obi 's problem of inversion are no t the general th eta -series; and this leads to the new problem of determining theposition of th e gen eral theta-se ries in our theory . Ac cord ingto an observat ion m ade by He rm ite, E ieman n m ust haveknow n th e proposition published at a later t ime by Weier-strass and recently discussed by Picard and Poincar, that thetheta-series are sufficient for defining the most general periodic functions of several variables.

B ut I m ust no t en ter in to these details. I t is difficult togive a connected accoun t of the further developmen t of Eie-mann's theory of Abelian functions because the far-reachinginvestigations of Weierstrass on the same subject are as yetkn ow n only from writte n lectures of his stu de nts . I willtherefore only me ntion th at the im po rtan t treatise publishedby Clebsch a nd Gordan in 1866 had in the main the objectof deriving Eiemann's results on the algebraic curve by means

8/12/2019 S0002-9904-1895-00271-2

9/16

1 8 9 5 ] B I E M A N N A N D M O D E RN M A T H E M A T IC S . 1 7 3of analytic geometry. A t th at t ime Riem ann 's meth ods werestill a sort of secret science confined to his immediate pupilsand were regard ed almost with distrus t by other ma them aticians. I can in this respect only repeat wha t I just rem arkedin speak ing of th e theo ry of curves, viz., th at t he gro wing de velopment of ma them atics leads with necessity to th e incorporation of Riem ann 's m ethods into th e general body ofma them atical science. It is intere sting to compare in this respect the latest French text-books.*The third class of functions referred to above comprisesthose laws of dependence that are connected with Gauss'shyper geometric series In a wider sense, we have here th efunc tions defined by linear differential equa tions with algebraic coefficients. Rie m ann pub lished on this subje ct du rin ghis lifetime only a preliminary study (in 1856) which is devoted entirely to the hypergeometric case and shows in a surprisin g way how all th e rem arka ble prop erties of th e hypergeometric function, that had been known before, can be derived w itho ut calcu lation from th e behavior of this func tionwhen th e variable passes arou nd th e singular points. Wenow know from his posthumous papers in what form he hadintended to carry out the corresponding general theory of thelinea r differential equ ation s of th e wth order. H ere , also, hewanted to take as a starting point and primary characteristicfor the classification th e grou p of those linear sub stitu tion swhich the integrals undergo when the variable passes aroundthe singular points.Th is idea, which in a certain sense corresponds to R iema nn'stre atm en t of th e Abelian integrals, has not y et been carriedout according to Riem ann's extensive plan. T he num erou sinvestigations on linear differential equations published during the last decades have really disposed only of certain partsof th e theory. In th is respect the researches of Fu ch s deserve special mention.As regards the differential equations of the second order,they are capab le of a simple geom etrical illustra tion . I t isonly necessary to consider the conformai representation whichthe quotient of two particular integrals of the differentialequation furnishes for the region of the independent variable.In the simplest case, i .e. that of the hypergeometric function,we obtain the repres entation of th e half-plane on a trian gleformed by circular arcs; and th is establishes a notewo rthy connection with spherical trigono me try. In th e general theorythe re are cases wh ich a dm it of on e-valued (eindeutig) inversion and thus produce those remarkable functions of a single

*See, for instance, PICARD, Trait d'analyse, and APPELL et Goun-S A T Thorie de s fonctions algbriques et de leurs intgrales.

8/12/2019 S0002-9904-1895-00271-2

10/16

17 4 EIEMANN AND MODERN MATHEMATICS. [A p r il ,variable which, l ike the periodic functions, are transformedinto them selves by an infinite n um be r of linear transfo rma tion s and wh ich I have therefore called automorphic functions.All these develop men ts which occupy th e ma them aticiansof our time appear more or less explicitly in Eiemann's posthumous papers , part icularly in the memoir on minimum surfaces referred to above. Fo r fur the r de tails I mu st refer youto Schwarz's memoir on the hypergeometric series and to theepoc h-m aking researches of Po incar in th e theory of au tomo rphic functions. W ith this grou p of investigations mu stalso be classed those on the elliptic modular functions andthe functions of the regular bodies.

I ca nno t leave the discussion of E iem ann 's work in th etheory of functions without mentioning an isolated paper inwhich the author makes interest ing contr ibutions to thethe ory of definite inte gra ls. Th is pape r has become celebratedmainly owing to the application which t he auth or makes ofhis resu lts to a problem in th e theo ry of nu m be rs, viz., the lawof distribution of the prime nu mbers in the natural series ofnum bers . Eiem ann arrives at an appro xim ate expression forthis law which agrees more closely with the results obtained byactual enum erat ions t ha n the empirical rules tha t had beendeduced up to that t ime from such enumerations.Two rem arks natu rally presen t themselves in th is connection . Fi rst , I should like to call your atte ntio n to the curious way in which the various branches of higher mathematicsare interwoven a problem appa rently belonging to the elementsof th e theory of nu mb ers is here in a most unexp ected ma nnerbrought nearer to its solution by means of considerations derived from the most intricate questions in the theory of functions. Second, I m ust observe th at th e proofs in Eie ma nn 'spaper, as he noticeshimself, are no t quite com plete; in spiteof n um erou s a ttem pts in recen t times, i t has not yet beenpossible to m ak e all these proofs perfectly satisfactory. I tappears tha t Eiem ann mu st have worked very largely by intui t ion.Th is, by th e way, is also tru e of his m ann er of establishingth e found ations of the theory of functions. Eiem ann herema kes use of a mo de of reason ing often employed in m ath ematical phy sics; he designated this m ethod as Dirichlefsprinciple, in hono r of his teach er Lejeune Dirichlet. W henit is required to determ ine a continu ous function th at m akesa certa in double integ ral a minim um , this princip le assertstha t the existence of such a func tion is eviden t from th e existence of the problem itself.* W eierstrass has shown tha t

* It appears from the context that, contrary to a common way of

8/12/2019 S0002-9904-1895-00271-2

11/16

1895] RIBMANN AND MODERN MATHEMATICS. 175this inference is faulty: it may happen that the requiredminimum represents a limit which cannot be reached withinthe region of the continuous functions. This considerationaffects the validity of a large portion of Riemann's developments.Nevertheless the far-reaching results based by Eiemann onthis principle are all correct, as has been shown later by CarlNeumann and Schwarz with the aid of rigorous methods.It must be supposed that Eiemann originally took his theorems from the physical intuition which here again proved tobe of the greatest value as a guide to discovery, and that heconnected them afterwards with the method of inferencementioned above, in order to obtain a connected chain ofmathematical reasoning. It appears from the long developments of his dissertation that in doing this he became conscious of certain difficulties ; but seeing that this mode of reasoning was used without hesitation in analogous cases bythose around him, even by Gauss, he does not seem to havepursued these difficulties as far as might have been desired.

So much about the functions of complex variables. Theyform the only branch of mathematics that Riemann hastreated as a connected whole; all his other works are separateinvestigations of particular questions* Still we should obtaina very inadequate picture of the mathematician Riemann ifwe were to disregard these latter researches. For, apart fromthe notable results reached by him, the consideration of theseresearches will place in better perspective the general conception that ruled his thoughts and the programme of workthat he had laid out for himself. Besides, every one of theseinvestigations has exerted a highly stimulating and determining influence on the further development of mathematicalscience.

To begin with, let me state more fully what I indicatedabove, viz., that Riemann's treatment of the theory of functions of complex variables, founded on the partial differentialequation of the potential, was intended by him to servemerely as an example of the analogous treatment of all otherphysical problems that lead to partial differential equations,or to differential equations in general. In every such case itshould be inquired what discontinuities are compatible withthe differential equations, and how far the solutions may bespeaking,I mean by the principle the mode of reasoning,and notthe results deduced thereby.I wish to use this opportunityto call attention to an articleby SirWm. Thomson, published in Liouville'sJournal vol. 12(1847),whichdoesnot seemtohave been sufficiently noticed by German mathematicians. In this articletheprincipleisexpressedin avery general form.

8/12/2019 S0002-9904-1895-00271-2

12/16

176 RIEMANN AND MODERN MATHEMATICS. [A p r il ,determ ined by th e discontinuities and accessory cond itionsocc urring in th e problem. T he execution of this programm ewhich has since been considerably advanced in various directions, and which has in recent years been taken up withpartic ular success by Fr en ch geometers, am oun ts to noth ingshort of a systematic reconstruction of the methods of integration required in mechanics and mathem atical physics.Riemann himself has treated on this basis only a singleproblem in gre ater detail, viz., th e pro pag ation of plane wavesof air of finite amplitude (1860).Two main types must be distinguished among the linearpartial differential equations of mathematical physics: theelliptic and the hyperbolic type, the simplest examples beingthe differential equation of the potential and that of thevibr ating s tring , respectively. As an interm ediate limitingcase we may distinguish the parabolic type to which belongsth e differential equa tion of th e flow of hea t. Th e rece ntinvestigations of Picard have shown that the methods of integration used in the theory of the potential can be extendedwith slight modifications to the elliptic differential equationsgenerally. How abou t the other types? Riem ann makes inhis paper an important contribution to the solution of thisque stion. H e shows wh at modifications m ust be mad e in thewell-known " boundary problem of the theory of the potential and in its solution by means of Green's function in orderto make this method applicable to the hyperbolic differentialequations.This paper of Riemann's is noteworthy for various otherreasons. T hu s, the redu ction of the problem named in thetitle to a linear differential equation is in itself no meanachievemen t. A no ther p oint to which I desire to call yourpart icular at tent ion is the graphical treatment of the problemwh ich evidently und erlies the whole mem oir. W hile thismode of treatment will have nothing surprising to the physicist, its value is in our day not infrequently underrated byth e ma them atician accustomed to more abstract metho ds. 1take , therefore, particu lar pleasure in pointin g to th e factthat an authority like Riemann makes use of this mode oftreatment and derives by its means most interesting results.

It rem ains to discuss th e two great memo irs presented byRie ma nn, at the age of 28 years, when he became a docentat (ro ttin ge n, in 1854, viz., th e essay On the hypotheses thatlie at the foundation of geometry, and the memoir On the pos-sibility of representing a function by means of a trigonometricseries. I t is rem arka ble how differently these two pap ershave been trea ted by th e wider scientific pub lic. T he importanc e of th e disqu isition on th e hypo theses of geom etry has

8/12/2019 S0002-9904-1895-00271-2

13/16

1 8 9 5 ] RIEMANSr AND MODERN MATHEMATICS. 17 7long since been adequately recognized, no doubt mainlyowing to th e p ar t th at He lmho ltz took in the discussion ofthe prob lem , as is probably know n to m ost of you. T he investigation of trigonometric series, on the other hand, has sofar not become known outside the circle of mathematicalreaders. Nev ertheless th e results contained in this latt erpap er, or ra th er th e conside rations to which it has given riseand with which its subject is intimately connected, must beregarde d as of th e hig hes t interes t from th e po int of view ofthe theory of knowledge.As regards the hypotheses of geometry, I shall not enterhere upon the discussion of the philosophical significance ofth e subject, as I should not have an yth ing new to add. Fo rthe mathematician the interest of the discussion centres notso much in the origin of the geometrical axioms as in theirm utu al logical interd epe nde nce . Th e most celebrated question is th a t as to th e na tu re of the axiom of para llels. It iswell known that the investigations of Gauss, Lobachevsky,and Bolyai ( to mention only the most prominent names) haveshown th at t h e axiom of pa rallels is certa inly no t a consequence of the other axioms, and that by disregarding theaxiom of parallels a more general, perfectly co nsistent geometry can be constructed which contains the ordinary geometry as a partic ular case. Eiem ann gave to these imp orta ntconsiderations a new tu rn by introd ucin g and applying themode of treatment of analytic ge om etry. H e regard s spaceas a par ticula r case of a triply-exten ded num erical manifold-ness in wh ich th e square of th e elem ent of arc is exp ressedas a qu adr atic form of the differentials of th e co-ordina tes.Without discussing the special geometrical results thus obtained and the subsequent development of this theory, I onlywish to po int out th at here again Kiem ann remains faithfulto his fundamental idea to interpret the proper ties of thingsfrom their behavior in the infinitesimal. H e has therebylaid th e fou nda tion for a new ch ap ter of th e differential calculus,viz., the theory of the qu adratic differential expressionsof any number of variables, and in particular of the invariants of such expressions under any transformations of thevariables. I mu st here dep art from th e prevailing characterof my remarks and call special attention to the abstract sideof the m atter. Th ere can be no doub t tha t in tryi ng to discover m ath em atica l rela tions it is by no me ans indifferentwhether we endow the symbols with which we operate witha definite me anin g or no t; for it is just thro ug h these concrete representations that we form those associations of ideaswhich lead us onw ard. T he best proof of thi s will be foundin almost everything that I have said to-day as to the intimate relation of Kiemann's mathematics to mathematical

8/12/2019 S0002-9904-1895-00271-2

14/16

17 8 RIEMAKN" AND MODERN MATHEMATICS. [ A p r i l ,physics. Bu t the final result of the ma them atical investigation is quite independent of these considerations and risesabove these special auxiliary methods; i t represents a generallogical framew ork whose co nte nt is indifferent and can beselected in various ways according to the nature of the case.Considered from thi s po int of view it will no lon ger appe arsur pris ing t h at at a later p eriod (1861), in the prize essaypresented to the Paris Academy, Riemann made an application of his investigation of differential expressions to thepro ble m of the flow of hea t, i.e., to a sub ject wh ich su rely h asno thin g to do with the hypotheses of geometry. In a similarway these researches of Riemann are connected with therecent investigations concerning the equivalence and classification of th e general problems of m echan ics. Fo r it is possible, according to Lagrange and Jacobi, to represent thedifferential equations of mechanics in such a way as to makethe m depe nd on a single qu ad ratic form of the differentialsof the co-ordinates.

I now pass to the consideration of the memoir On the trigonometric series, which I have inten tionally left to th e last,because it brings into prominence a final essential characteristic of Riem ann 's conception. In all the preceding remark sit has been possible for me to appeal to the current ideas ofphysics, or at least of geometry. Bu t Riem ann's pe netr atingmind was not satisfied with making use of the geometricaland physical in tu itio n; he wen t so far as to investigate thisintuition critically, and to inquire into the necessity of thema them atical relations flowing from it . Th e question atissue is noth ing less tha n thefundamental principles of theinfinitesimal calculus. In his other works Riemann hasnowhere expressed any definite opinions concerning theseque stions. I t is different in th e pap er on trigon om etricseries. U nfo rtun ately he considers only detached problems,viz., th e q uestions w heth er a function can be discon tinuou sat every po int, and wh ethe r in th e case of function s of sucha ge nera l na tu re it is still possible to speak of inte gra tion .But these problems he treats in so convincing a manner thatthe investigations of others on the foundations of analysishave received from him a most powerful impulse.Tradition has it that in later years Riemann pointed out tohis stude nts th e fact w hich m ust be regarded as th e m ostremarkable result of the modern critical spirit : the existenceof function s w hich are no t differentiable at any po int. Amo re detailed stud y of such " nonsensical " function s (as theyused to be called formerly) h as, how ever, been mad e only byWeierstrass, who has probably contributed most to give itspresent rigorous form to the theory of the real functions ofreal variables, as this whole field is now usually designated.

8/12/2019 S0002-9904-1895-00271-2

15/16

1 8 9 5 ] RIEMANSr AND MODERN MATHEMATICS* 17 9As I understand Kiemann's developments on trigonometricseries, he would, as far as the foundation is concerned, agreew ith the p rese ntatio n given by W eierstrass, which discards allspace-intuition and operates exclusively with arithmeticaldefinitions. B u t I could never believe th at Kiem ann shouldhave considered this space-intuition (as is now occasionallydone by extreme representatives of the modern school) assom ething antagonistic to m athem atics, which mus t necessarily involve faulty reasoning . I m ust insist on the positionthat a compromise is possible in this difficulty.We touch here upon a question which I am inclined to

conside r as of decisive imp orta nce for the fu rth er developme nt of ma them atical science in our time. Our stude nts areat pres ent introdu ced at the very begin ning to all those intricate relations whose possibility has been discovered bymodern analysis. Th is is no doub t very desirab le; bu t i tcarries with it the dangerous consequence that our youngma them aticians frequently hesitate to formulate any generalpropositions, th at the y are lacking in tha t freshness of tho ug htwithout which no success is possible in science as well as elsewhere.On the o ther ha nd , the majority of those engaged inapplied science believe that they may entirely leave asidethes e difficult inve stigatio ns. T hu s they detac h them selvesfrom rigorous science and develop for their private use aspecial kind of mathematics which grows up like a wild sprigfrom the root of th e grafted tr ee . We mu,st try with all ourmight to overcome this dangerous split between pure andapplied ma them atics. I may therefore be allowed to formulate my personal position concerning this m atte r in the following two propositions :First, I believe that those defects of space-intuition byreason of which it is objected to by mathematicians aremerely tem pora ry, and th at this in tuition can be so traine dth at with its aid the abstrac t developm ents of the analystscan be understood, at least in their general tendency.Second, I am of the opinion that, with this more highly-developed intu ition , the applications of ma them atics to thephe nom ena of th e outsid e world will, on th e whole, rema inunchanged, provided we agree to regard them throughout asa sort of interpolation which represents things with an appro xim atio n, lim ited , to be sure, bu t still sufficient for allpractical purposes.W ith these rem ark s I will close my add ress, which I hopehas not taxed y our indulg ence un du ly. You may have noticed that even in mathematics there is no standstill , thatthe same activity prevails there as in the natural sciences.And this, too, is a general law that while many workers con-

8/12/2019 S0002-9904-1895-00271-2

16/16

1 8 0 MULTIPLICATION OF SEMI-CONVERGENT SERIES [ A p r i l ,tri bu te to th e dev elop me nt of science, th e really new impu lsescan be traced back to b ut a small num be r of em inent men .B ut th e work of th ese men is by no m eans confined to th eshort span of their life; their influence continues to grow inproportion as their ideas become better understood in thecourse of time . Th is is certainly the case with R ieman n.For this reason you must consider my remarks not as the descriptio n of a past epoch, whose mem ory we c herish with afeeling of veneration, but as the picture of live issues whichare stil l at work in the mathematics of our time.

T H E M U L T I P L I C A T I O N O F S E M I - C O N V E R G E N TSER IES.B Y P R O F E S S O R F L O R I A N C A J O R I .

I N Math. Annalen, vol. 2 1, pp . 327-378 , A. Prings heim developed sufficient con ditions for th e convergence of the p rod uc tof two semi-convergent series, formed by Cauchy's multiplication rule, when one of the series becomes absolutely converge nt, if its term s are associated into gro ups with a finitenu m be r of term s in each gro up . T he necessary and sufficientcon dition s for c onvergenc e were obtained by A. Voss (Math.Annalen, vol. 24, pp . 42-47) in case th at the re are two term sin each grou p, and by the writer (Am. Jour. Math., vol. 15,pp . 339-343) in case that there are p terms in each group, pbeing some finite integ er. In this pap er it is proposed todeduce the necessary and sufficient conditions in the moregeneral case when the number of terms in the various groupsis not necessarily the same.

n nL et Un = 2an and Vn = 2bn be two semi-conv ergent series,and let the first become absolutely convergent when its termsare associated into groups with some finite number of termsin each grou p. Le t rn represent the number of terms in the(n -+- l ) t h gro up , and let gn represent the (n + l ) th groupembracing rn term s. Le t, moreover, a n represent the firstterm in the group gn, where EQ = 0 and Rn = r0 + rx + r2 +. . . + rw_ , then

nffn = (Rn + aRn+l + JB^+ . . + 28+r-l) a n d ^ = ^9^Since, by a theorem of M ertens, th e pro du ct of an absolutely convergent series and a semi-convergent series, formed