Documenting Riemann's impact on the theory of complex functions

E NEUENSCHWANDER

Documenting Riemann's

Impact on the Theory of

Complex Functions

long WLth Euler, Gauss, and Htlbert, Bernhard Rwmann ts one of the most

tllus-tnous mathematLcwns of all tLme Hts promlnence

m

the fwld of complex

analy-sts may be apprecwted stmply by nottng that tn the

Mathematlsches Worterbuch

to Rtemann's functwn-theorettcal work (Rtemann map-pmg theorem, Rtemann dtfferentml equatwn, Rtemann sur-face, Rtemann-Roch theorem, Rtemann theta-functton, Rtemann sphere, Rtemann zeta-functwn, etc ) take up al-most as much space as those related to all the work of Euler or Gauss m total Rtemann's contnbutwn to com-plex analysts rests, on the one hand, on hts pubhcatwns, especmlly hts maugural dtssertatwn on the foundatwns of complex analysts (1851) and hts arttcles on the theory of hypergeometnc and Abehan functtons (1856-57), but hts several lecture courses m thts fteld are also very tmpor-tant

These were giVen m the years from 1855 to 1862, and regularly began wtth an mtroductory general part Rtemann then turned etther to the theory of elhpttc and Abe han func-tions or to hypergeometnc senes and related transcen-dental functwns Much of the more advanced parts of Rwmann's courses were pubhshed m hts Collected Papers [Rtemann 1990, 599-692] and m a book by H Stahl [Rtemann 1899], but not hts mtroductory lectures on gen-eral complex analysts Thus the latter gradually fell mto obhvton, desptte thetr mtnnstc mterest, and desptte thetr dectstve mfluence on later developments through the

closely related wntmgs of Durege, Hankel, Koemgsberger, Neumann, Prym, Roch, and Thomae It therefore seemed

appropnate to pubhsh them m a cnttcal edttion [Neuenschwander 1996), m order to make them accesstble to a broader ctrcle of readers For thts edttton, I also pre-pared an extenstve btbhography of the htstory of the rm-pact and mfluence of Rtemann's functwn theory

This newly assembled btbhography ts mamly mtended to close-at least m the field of complex analysts-the con-stderable gap eX1stmg between the btbhographles of Purkert and Neuenschwander, which were appended to the repnnt of Bernhard Ru:mann 's Gesammelte Mathemat~sche Werke [Rtemann 1990) The penod from 1892 to 1944, not system-atically covered there, was scrutmtZed, usmg the mdexes of names m Bzblwtheca Mathematzca (1887-1914), and the RelJUe semestnelle des Publu:atwns mathemattques

(1893--1934/35), and exammmg the sections on "Geschlchte und Phllosophle" and "Funkttonentheone" (or "Analysts") m the abstractmg JOurnal Jahrbuch uber dw Fortschntte der Mathemaflk (1868/71-1942/44) Addtttonally, I consulted the holdmgs of older books m the hbranes of the Institutes of Mathemattcs at the Uruverstties of Gottmgen and Zunch Generally, I mcluded only pubhcatwns whlch contamed ex-plzczt reference to Rtemann's work (e g, quotatwns wtth

exact mdicatwn of locatwn), but even this cntenon left more than 1,000 out of those 8,000 titles proVIsiOnally se-lected on the basis of the reVIews and the systematic li-brary mspectwn, the ongmals were then looked up, to make sure that they met the condltwns for mcluswn

Naturally, the new bibliography IS m no way comprehen-sive The literature which refers to Riemann's pwneenng work IS almost boundless Purkert, for example, exammmg about ten JOurnals from the first 25 years after the death of Bernhard Riemann, already found more than 500 publicatlons refernng to hiS work Accordmg to database surveys, the contlnuatlon and extenswn of Purkert's research up to the present would

have to take mto account about 30,000 publicatlons, all of

which obVIously cannot be exammed mdlVIdually Withm a rea-sonable tlme Nevertheless, I hope that the new bibliography

will become a useful tool for further mvestlgatlons

In the folloWing I Will try to Illustrate some of Its

pos-Sible applicatwns, by surveymg the Impact of Riemann's functiOn-theoretical work m the four most Important European countnes Germany, France, Italy, and Great Bntam Special attentwn Will be giVen to the developments m Great Bntam because they have not yet been analysed m detail For more specific mformatwn, the reader may turn to the bibliography Itself

Germany: Early and Sustained Reception of Riemann's Methods

For a prellmmary 1mpresswn of the situatlon m Germany, let us first look at the references to Riemann's work m August Leopold Crelle's mfluentlal Journal fur dw reme und ange-wandte Mathemat~k It IS noteworthy that the first of these references goes back to Helmholtz [CreUe 55 (1858), 25-55], who, as we know, later discussed Riemann's hypotheses concenung the foundatwns of geometry Helmholtz was followed m chronological order by Lipschitz, Clebsch, Chnstoffel, Schwarz, Bnll, Fuchs, Gordan, Luroth, and Weber, all of whom, even though they were not Riemann's Immediate pupils, did a great deal to dlssemmate hls Ideas, as dld his own students Roch, Thomae, and Prym [Grelle

61 (1863)-70 (1869)] Clebsch and Bnll-like Klem and Noether-published their later work pnmanly m the Mathemat~sche Annalen, which started to appear m 1869, thus, consultmg Crelle's Journal alone proVIdes only m-complete results for them Accordmg to our bibliography, other Important early promoters of Riemann's Ideas m the German-speakmg world were Cantor, Dedekmd, Du Bms-Reymond, Durege, Hankel, Koemgsberger, Neumann, Schlafli, and, m his later years, Schottky I

Italy: Enthusiastic Appreciation of Riemann's Work An Impressive picture of the extent to which Riemann's work was appreciated m Italy IS giVen by a smular study of the ref-erences to hiS wntlngs m the Annall d~ MatematUJa pura ed apphcata, a JOurnal which played an outstanding role m

the dlSsemmatwn of Riemann's thoughts As early as 1859, Ennco Betti, who was to become Riemann's fnend, trans-lated hiS dlssertatlon [Annalz 2 (1859), 288--304, 337-356], to which he soon returned m an extensive article on the the-ory of elhptlc functwns [Annah 3 (1860), 65-159, 298-310, 4 (1861), 26-45, 57-70, 297-336] In the same volumes, we also find a report by Betti on Riemann's treatise concern-mg the propagatiOn of planar mr waves [Annah 3 (1860), 232-241), and a report by Angelo Genocchi on Riemann's mvestigatwn of the number of pnmes less than a giVen bound [Annah 3 (1860), 52-59) In a later volume of the same JOurnal [Annal~ (2) 3 (1869-70), 309-326], we find a French translatwn of Riemann's hypotheses on the foun-dations of geometry by the French mathematician Jules Houel, who typically did not pubhsh his translatwn m a French Journal, but m the Annah We should also mentwn Eugemo Beltrami and Felice Casorati, the latter haVIng al-ready presented Riemann's theones m 1868 m a book [Casorati 1868] and m special lectures giVen m Milan [Armenante & Jung 1869, Casorati & Cremona 1869] In VIew

of this excellent mtroductwn and trml-blazmg, It IS not sur-pnsmg that Riemann's theones were Widely known m Italy, and that they were quoted m more than thlrty articles m the

Annalz alone up to 1890 2 _{As to Riemann's own stays m Italy,} and other followers of Riemann there, see the articles of Bottazzim, Dieudonne, Lona, Neuenschwander, Schenng, Tnconu, Volterra, and Well cited m the bibliography France: Hesitant Reception

In France, the Situation was radlcally dlfferent Sklffimmg

through the pages of the Journal de Matlu?rrwtLques pures et apphquees edlted by Joseph Liouville, one finds almost no ref-erences to Riemann's papers before 1878, and m other French

publications up to 1880 they also seem to be relatively sparse Furthermore, a certmn cntical reserve as to the usefulness of Riemann's methods qmte often comes through Bnot and Bouquet, for Instance, wnte m the foreword to the second editwn of their 1'1Wone des fonctwns elhpt~ques [Bnot &

Bouquet 1875, I f

I

In Cauchy's theory, the path of the zmagmary [complex]

vanable ~s charactenzed by the movement of a pmnt on

a plane To represent those functwns whzch assume sev-eral values for the same value of the varzable, R~emann regarded the plane as formed of several sheets, supenm-posed and welded together, m order to allow the vanable

to pass from one sheet to another, wh~le t1 aversmg a con-nectmg [branch] hne The concept of a many-sheeted sur-face presents some dzff~cultws, despzte the beaut~ful re-sults wh~ch R~emann achLeved by th~s method, ~t d~d not seem to us to offer any advantage for our own obJectwe Cauchy's ~dea ~s very suLtable for representmg mult~­ valued functwns, ~t LS suff~czent to JOin to the value of the van able the correspondmg value of the functwn, and

1 _{For b1bliograph1cal details on part1cular art1cles by these authors c111ng R1emann see [Neuenschwander 1996 131-232]}

2The authors of other art1cles 1n the Annalt conta1n1ng references to R1emann are Ascoli Beltrami Casorat1 Cesaro Chnstoffel D1n1 LIPschitz Pascal Schlafl1 Schwarz Tonelli Volterra etc For deta1ls see [Neuenschwander 1996]

when the vanable has descr"'bed a closed curve and the value of the functwn has thereby changed, to mdzcate thzs change by an mdex

Only Houel, mentiOned above, tned qmte early to promote Rtemann's tdeas, and, to-gether wtth Gaston Darboux, he several ttmes deplored thetr not bemg better known m France [Gtspert 1985, 386-390 et passzm] A revtew of Houel's p10neenng book m thts field [Houel1867-1874, parts 1 and 2 (1867/68)] con-cludes wtth the followtng, qmte mstructtve passage [Nouvelles Annates de Mathema-tzques (2) 8 (1869), 136-143]

May the ~eceptwn of thzs work encowage the

autlw~ to keep hzs promzse to gwe us, m a third part, an exposltwn of Rzemann's theorws, whzch up to now have been almost unknown zn ow country, and whzch ow nezghbours cul-twate 101th so much ardour and success'3

In sptte of Houel's effoiTh, the situation m

France seems to have changed fundamentally only after 1880, when a new generation of French mathematicians took over, the most promment among them were Henn Pomcare (1854-1912), Paul Appell (1855-1930), Emile Ptcard (1856-1941), and Edouard Goursat (1858--1936) Wtth these authors the reception of the works of Riemann's followers (mamly Clebsch and Fuchs) now began to lead to a more thorough readmg of Riemann's own wnt-mgs-as numerous quotations m therr pubhca-tiOns prove As a marufestation of the mcreas-mg apprectation of Rtemann's methods among French mathematicians at that time, we may take the fact that, m 1882, Georges Slillart wrote a dissertation of 123 pages, now almost

com-F1gure 1 D1scuss1on and representation of a branch pomt of an algebraic function on a R1emann surface, taken from Riemann's lectures 1n the summer semester of 1861 The reproduced page comes from a set of lecture notes made by Eduard Schultze w1th annotations by Hermann Amandus Schwarz, 1t IS kept 1n the Schwarz Nachlass at the Arch1v der Berhn-Brandenburg1schen Akadem1e der W1ssenschaften 1n Berhn There ex1sts a s1m1lar representation 1n the (probably) ongmal notes by Schultze wh1ch are now kept 1n the Rare Book and Manuscnpt Library at Columbia Umvers1ty 1n New York Both manuscnpts were discovered by the author, 1n 1979 and 1986, respectively For further 1nformat1on and a transcnpt1on of the text, see Neuenschwander 1988 and 1996, pp 75 f and 84, MS Sand 55

pletely forgotten, m which he tned to acquamt French math-ematiCians wtth Rtemann's funct10n-theoret1cal treattses In

Slillart's mtroductlon, we find the followmg statement, which confinns my own assessment [Slillart 1882, 1]

We know the magmfwent ~esults whzch Rzemann arrwed

at m hzs two Memous concernmg the genewl themy of analytzc functwns and the theoriJ of Abelwn functwns,

but the methods whnh he used-and explamed pe~haps

too succmctly--me not well known zn F~ance [ ] At

the same tzme, however, Rzemann's method contznues to have many adhe~entt. zn Germany, 1t serves as a bas1s for many Important publ1catwns by famous geometers,

such as Koemgsberger, Carl Newmann [szc], Klezn,

Dedekznd, Weber, Prym, Fuchs, etc The ~eadzng of these

Memmrs 1eqwres a knowledge of Rwmanman surfaces [and thus, fmaUy, of Rzemann's own wrztzngsj, the use of whzch has become standard at certazn German um-versztws

Great Britain: A Forgotten Interest

In contrast to what ts known about the recept10n of

Rtemann's theones m the three countnes already surveyed, thetr receptiOn m Great Bntam 1s httle studied Let me therefore provtde a more detailed account

In Great Bntatn Rtemann's theones seem to have be-come known and more wtdely dtssemmated only after his death One of the first Bnttsh mathemattctans who ctted Rtemann frequently was Arthur Cayley, who, already m 1865/66, ment10ned the mvestigat10ns of Clebsch and

3_{0ther scattered early c1tat1ons of R1emann s wntlngs are to be found 1n Bertrand Darboux Ell1ot Emmanuel Herm1te JonqUieres Jordan Mane and Tannery For}

Riemann on Abelian Integrals ill his papers on the trans-formation and higher sillgulanties of a plane curve Cayley also discussed Riemann's work ill later years ill great de-trul and With appreciatiOn This can be seen from his note on Riemann's posthumously published early student paper on generalized illtegratwn and differentiatiOn (1880), or from his presidential address to the Bntish AssociatiOn for the Advancement of Science (1883) Other early citatiOns of Riemann's papers ill Great Bntaill are by W Thomson (1867 ff) and J C Maxwell (1869 ff) They occur ill con-nectiOn With Helmholtz's papers on vortex motwn and With Listillg's studies on topology

A few years later, W K Chfford, H J S Smith, and J W L Glrusher also thoroughly analysed Riemann's papers and, on several occaswns, emphasized their great Importance Clifford, for example, as early as 1873, translated Riemann's famous paper on the hypotheses which he at the base of geometry illto English for the JOurnal Nature, and ill 1877 he published his illfluentlal memmr on the canomcal form and dissectiOn of a Riemann surface In November 1876 Smith, ill his presidential address On the Present State and Prospects of some B;anches of Pu;e Mathematzcs to the London Mathematical Society, reported ill detrul on Riemann's achievements, takillg illto account as well the newly published fragments ill Riemann's Collected Mathematical Works Smith's elaborate address seems to mark an Imtial highpoillt ill the appreciatiOn of Riemann's works ill Great Bntaill Because It has not yet been stud-Ied ill this context, I will treat It here ill more detrul

At the beginnillg of his address Smith outlined recent progress ill number theory and ill particular the illvestiga-tions on the number of pnmes less than a giVen bound He wrote [Snuth 1876-77, 16-18]

As to ow knowledge of the senes of the pnme numbms themselves, the advance smce the tzme of Euler has been g;eat, zjwe thmk of the dtffzculty of the problem, but ve;y smallzf we compme what has been done wzth what stzll remams to do We may mentwn, zn the jzrst place, the undemonst;ated, and zndeed conJectu;al, theorems of Gauss and Legend;e as to the asymptotzc value of the numbe; of przmes mjerwr to a gwen lzmzt x [ ] The memmr of Bernhard Rzemann, "Ueber dze Anzahl de; fumzahlen unter emer gegebenen Grosse," contazns (so jar as I am aware) the only znvesttgatwn of the

asymp-totzc frequency of the pnmes whzch can be ;ega;ded as ngorous [ ] No less zmpmtant than the znvestzgatwn of Rzemann, but approachzng the p;oblmn of the asymp-totzc law of the serzes of prunes j; om a d zjjerent szde, zs

the celebwted memozr, "Sur les Nomb;es Ptemze;s," by M Tchebychef [ ] The method of M Tchebychef, pro-found and zmmztable as zt zs, zs zn pomt of fact of a very

elementa;y character, and m thzs respect cont;asts st;ongly wzth that of Rzemann, whzch depends tluough-out on ve;y abstruse theorems of the Integral Calculus

Smith then went on to speak of some branches of analy-SIS which appeared to him to promiSe much for the

unmedi-ate future Here he focussed on the advancement of the "Integral Calculus " He mentioned, among other works, Riemann's memoir on the hypergeometnc senes and the un-fuushed memoir on hnear differential equations With algebrmc coefficients In doillg so he stressed the "great beauty and ong-illahty" of Riemann's reasonmg and the "fertility of the con-ceptions of Cauchy and of Riemann " In hiS closillg remarks

Smith descnbed the work of English mathematicians ill the field durmg the last ten years, nanung among others GlaiSher, Cayley, and Chfford He was convmced that nothmg so

hm-dered the progress of mathematical science m England as the want of advanced treatises on mathematical subJects, and

that there are at least th;ee t;eatzses whzch we g;eatly need-one on Dejznzte Integrals, one on the Themy of Functwns zn the sense m whzch that phrase zs

under-stood by the school of Cauchy and of Rzemann, and one (though he should be a bold man who would unde1 take the task) on the Hyperelhptzc and Abelzan Integrals

Smith called on his colleagues to close this gap, and to some extent they did 4

As early as about 1871 the Bntish Assocmtwn for the Advancement of Science set up a special Committee on Mathematical Tables to which belonged, besides Smith, A Cayley, G G Stokes, W Thomson, and J W L Glrusher The purposes for which the Committee was appoillted were (1) to form as complete a catalogue as possible of existillg mathematical tables, and (2) to repnnt or calculate tables necessary for the progress of the mathematical sciences The Committee decided to begill With the first task, and ill 1873, presented by the care of Glrusher a huge catalogue of 175 pages on mathematical tables which was pnnted ill

4_{1n a letter addressed to I Todhunter Smth g1ves a s1m1lar more d1rect and succ1nct est1mate of the present state of Mathemat1cs 1n Great Bnta1n France and Germany}

It proves once aga1n how h1ghly he regarded the work of R1emann and We1erstrass But I so heartily agree w1th much or rather w1th most of your book [Conflict of

Stud1es 1873] that I should not have troubled you With thiS letter 1f 1t were not that I cannot wholly subscnbe to your est1mate of the present state of MathematiCS

All that we have one may say comes to us from Cambndge for Dublin has not of late qu1te kept up the prom1se she once gave Further I do not th1nk that we have anyth1ng to blush for 1n a companson w1th France but France 1s at the lowest ebb IS consc1ous that she 1s so and IS mak1ng great efforts to recover her lost place 1n Sc1ence

Aga1n 1n Mixed Mathematics I do not know whom we need fear Adams Stokes Maxwell Ta1t Thomson w1ll do to put aga1nst any l1st even though 1t may con-tain Helmholtz and Claus1us

But 1n Pure MathematiCS I must say that I th1nk we are beaten out of s1ght by Germany and I have always felt that the Quarterly Journal IS a miserable spectacle as compared w1th Grelle [ s Journa~ or even Clebsch and Neumann [Mathemat1sche Annalen] Cayley and Sylvester have had the lion s share of the modern Algebra (but even 1n Algebra the whole of the modern theory of equat1ons substitUtions etc IS French and German) But what has England done 1n Pure Geometry 1n the Theory of Numbers 1n the Integral Calculus? What a tnfle the symbolic methods wh1ch have been developed 1n England are compared w1th such work as that of R1emann and We1erstrass1 [H J S Smth Collected Mathematical Papers Introduction Vol 1 lxxxv-lxxxv1]

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F1gure 2 D1scuss1on and representation of the branch pomts of the R1emann surface

defmed by the equat1on s3 - s + z = 0 from Casorat1's notes of conversations w1th

Gustav Roch 1n Dresden m October 1864 The notes compnse 12 numbered pages At the end they g1ve some 1nformat1on on h1s encounter w1th Roch (October 8-13) and Casorat1's w1shes to have a copy of Riemann's lectures The last page reproduced above from Casorat1's notes of conversations w1th Roch contams one of the f1rst schematiC representations of a R1emann surface 1n cross-sect1on ("Sez1one normale aile fronte") and v1ewed from above ("Veduta d1 fronte") outs1de Riemann's own man-uscnpts Such representations later became very Widespread, as can be seen from

s1m1lar drawings 1n Neumann 1865, Houel 1867-1874 Clifford 1877, Bobek 1884,

Amstem 1889, etc Th1s way of representmg a R1emann surface goes back 1n fact to

R1emann himself, as can now be Inferred from page 52 from Schultze's lecture notes, wh1ch IS published here for the f1rst t1me (Casorat1's conversation notes were diS-covered, stud1ed, and m part ed1ted by the author dunng research m Pav1a from Spnng 1976 onwards) For further mformat1on on Casoratl's notes and the Casorat1 Nachlass, see Neuenschwander 1978, especially pp 4, 19, 73 ff

ber 4, 1876 In these papers Glmsher thanked Smith for his help on the relevant literature, and m a note referred explicitly back to Smith's above-mentioned address to the London Mathematical Society m November

1876 for specific references

Further details on the very mtense later sci-entlfic mterchange between Sffilth and Glmsher and their common adffilratlon for Riemann can be gathered from Glmsher's IntroductiOn to his edltwn of Sffilth's Collected Mathematical Papms and from Smith's

pa-pers themselves 5 _{From 1877 onwards J W L}

Glmsher's father James was engaged m the proJect of the constructiOn of factor tables of numbers from 3,000,000 up to 6,000,000, and he once agmn documented thereby the great supenonty of Riemann's formula for the num-bers of pnmes as compared With those of Legendre and Tchebycheff James Glmsher's tables were c1ted m their tum by vanous math-ematlcmns m ScandinaVIan countnes (Opper-mann 1882 f, Gram 1884 ff, Lorenz 1891),

where Riemann's functwn-theoret1cal work had already excited mterest before, as IS shown by the pubhcatwns of Bonsdorff, Mittag-Leffler, and some others Moreover, m

1884 W W Johnson published a detailed ac-count of James Glmsher's factor tables and the distnbutwn of pnmes m the Amencan JOurnal Annals of Mathmnat1cs, wh1ch

mtro-duced Riemann's mvestigatwns and the work of the Bntish mathematicians m the New World

Fmally, attention should be pmd to Andrew Russell Forsyth, who went to hve m the town of Cambndge and studied nothmg but mathematics m the same year, 1876, that Sffilth and Glmsher started to draw attentwn to Riemann's work In his obituary notice, E T Whittaker descnbed Forsyth's Themy of Functwns (1893) as haVIng had a greater m-the annual Report of the Bn t1sh Assocw twn In 1875 a

con-tmuatwn of tills report was presented, compiled by Cayley

It mcluded, under the headmg "Art 1 DIVIsors and Pnme Numbers," new additiOnal references to the table of the fre-quency of pnmes m Gauss's Collected Wm ks and to the

re-lated approximate formulae by Gauss and Legendre Both reports contamed no reference to Riemann's famous paper

on pnme numbers It IS only mentwned m later Assocwtwn Repo1 ts on Mathematical Tables (1877 ff ), and m a senes of papers on the enumeratiOn of the pnmes and on factor tables whlch Glmsher presented to the Cambndge Philosophical Society, the first one bemg read on

Decem-fluence on Bntlsh mathematics than any work smce Newton's Prznc1pw Accordmg to Whittaker [1942, 218],

perhaps the most ongmal feature of this work was the meldmg of the three methods associated With the name& of Cauchy, Riemann, and Weierstrass, which m the conti-nental books were regarded as separate branches of math-ematics Furthermore, one can read m the Royal Society Bwgraphical Memmr of Henry Fredenck Baker by W V D Hodge that Forsyth "rendered Cambndge mathematics great serVIce by his efforts to bnng about closer co-opera-tiOn between mathematicians m this country and those on the contment of Europe, and he made It easier for his

5_{1n add1t1on to Sm1th s London address d1scussed above see also h1s paper On the mtegratlon of discontinuous functions ( 1875) w1th a deta1led analys1s of R1emann s}

memo1r on the representation of a funct1on by a tngonometnc senes h1s paper On some d1scont1nuous senes cons1dered by R1emann (1881) and h1s Memo1r on the

successors, mcludmg Baker, to get the full benefit of the work of the great German masters of the late nmeteenth century"

As can be seen from Wluttaker's obituary notice and Forsyth's own recollectiOns of his undergraduate days [Forsyth 1935], there IS a defirute lmk between Forsyth's later work m complex analysis and hls early studies m Cambndge, where he came mto repeated contact With Glaisher, Cayley, and the works of Smith, who taught at Oxford Cayley and Glmsher were also to be very mstru-mental m Forsyth's career m pure mathematics and helped

hlm to overcome that "Cambndge atmosphere" m which

"all were reared to graduation on applied mathematics "6 Accordmg to Whittaker, It was Glmsher who suggested to Forsyth that he wnte his first book, the Treat~se on dtf-ferentwl equatwns (1885) His first two pnncipal memmrs

on theta functiOns (1881/1883) and on Abel's theorem and Abelian functiOns (1882/1884) were presented by Cayley, on the other hand, to the Royal Society In the first mem-mr Forsyth gives a list of 22 pnncipal papers m the field, mcludmg among others Riemann's Theone der Abelschen Functwnen as well as twelve other papers by the German mathematicians Jacobi, Richelot, Rosenhmn, Gopel, Weierstrass, Koemgsberger, Kummer, Borchardt, and Weber The second memmr on Abel's theorem and Abelian functiOns contained a large sectiOn on Weierstrass's ap-proach, whlch clearly shows that around 1882 Forsyth was already fully aware of the achievements of the German mathematicians m this field

Conclusion

These bibliographlcal mvestigatwns make It eVIdent that Riemann's Ideas were more positively accepted m Great Bntmn, apparently even earlier, than m France It seems they were among the more liDportant stlffiuh whlch, through the efforts of Cayley, Clifford, Sffilth, Glmsher, etc , later led to the flounshmg of Enghsh pure mathematics and functiOn theory under Hobson, Forsyth, Mathews, Baker, Barnes, Hardy, Littlewood, Tltchmarsh, and many others In England

and Italy Riemann's theones entered open temtones, where they found mathematical commuruties eager to catch up m pure mathematics In England this mterest served to

broaden to the "Cambndge atmosphere," whlch was nearly totally onented towards natural philosophy and applied mathematics, m Italy It fitted m With a desrre to bmlt up a modem mathematical educatiOn for the newly founded na-tiOn ( cf Neuenschwander 1986) In France, on the other

hand, Riemann's Ideas first faced a cold reception from the well-established tradltwn of Cauchy and Bnot & Bouquet

Allowmg for a certam politically motivated enthusiasm, Weierstrass was therefore probably nght when, m a letter to Casorati dated 25 March 1867 ( wntten m the early pe-nod before the publicatiOn of Riemann's Collected Mathematical Works m 1876), he especially emphasized Italy's role m the dissemmatwn of Riemann's Ideas In that letter (see [Neuenschwander 1978, 72]f he wntes

The happy nse ofsc~ence m your fatherland w~ll nowhere else be followed wah more vwzd mterest than here m

North Germany, and you may be certam that the State

of Italy has nowhere else so many smcere and d~smter­ estedfnends Wtth pleasure we are therefore ready to con-tmue the allwnce between you and us-whzch

m

overcome, w1th whtch unhappy polzt~cs has for so long

sep-arated two generally congenwl natwns The papm whzch you sent to me proves to me once agam that our sczent~f~c

e'ltdeavours are better understood and apprecwted m Italy than zn F~ance and England, part~cularly m the latter country, where an overwhelmzng formal1,sm threateJtS to sttjle any feelmg for deeper mvest~gatwn How szgmfzcant tt zs that our Rzmnann, lDhose loss we cannot suffzczently

deplore, ts studwd and honoured outstde Germany, only

m Italy, m France he ts certamly acknowledged externally but l1ttle understood, and m England [at least before 1867],

he has rmnamed almost unknown

Note added m proof In a recent Issue of this JOurnal

(vol 19, no 4, Fall 1997) there appeared an article by Jeremy Gray whlch giVes the reader a very welcome sum-mary m English of Riemann's mtroductory lectures on gen-eral complex analysis Gray's paper IS largely based on my prepnnt Rtmnanns Vorlesungen zur Funktwnentheone Allgmnetner Tetl, Darmstadt 1987 and on Roch 1863/65 ( cf Neuenschwander 1996, p 15) but does not mentiOn the substantially expanded version which appeared m Spnng 1996 and which has been reVIewed m MR 97d 01041 and Zbl 844 01020 The published verswn also contams, be-Sides what was m the ongmal prepnnt, the above-men-tioned extensive bibliography of papers and treatises that were mfluenced by Riemann's work, as well as a list of all known lecture notes of hls courses on complex analysis From this list [Neuenschwander 1996, 81 ff ], one can m-fer that Cod Ms Riemann 37 compnses notes of three dif-ferent courses (not Just one, as suggested by Gray), and Its first part IS thereby qmte easily datable to the Wmter se-mester 1855/56 On page 111 of Cod Ms Riemann 37 one

6_{Accord1ng to Forsyth s own recollections as early as h1s student years he took an exceptional Interest 1n pure mathematiCS and went for one term 1n h1s th1rd year to}

Cayley s lectures where at the beg1nn1ng the very word plunged him 1nto complete bewilderment From other passages of Forsyth s recollections one may see that already at that t1me he made h1s f1rst t1m1d ventures outs1de the range of Cambndge textbooks and ploughed through among others a large part of Durege s Ellipttsche Functtonen Further deta1ls may be gathered from the following personal confess1on by Forsyth Someth1ng of d1fferent1al equat1ons beyond mere examples the ele

ments of Jacob1an ellipt1c funct1ons and the mathematics of Gauss s method of least squares I had learnt from Gla1sher s lectures and by work1ng at the matter of the one course by Cayley wh1ch had been attended I began to understand that pure mathematiCS was more than a collect1on of random tools ma1nly fash1oned for use 1n the Cambndge treatment of natural philosophy Otherw1se very nearly the whole of such knowledge of pure mathematics as 1s m1ne began to be acqu1red only after my T npos degree In that Cambndge atmosphere we all were reared to graduat1on on applied mathematics [Forsyth 1935 1 72]

7

AUTHOft

ERWIN NEUENSCHWANDER

lnstrtut fur Mathematik Abt Retne Mathematik UntverS!tat Zunch-lrchel

W1nterthurerstrasse 190

8057 Zunch Switzerland

Erw1n Neuenschwander ts Professor of the Htstory of MathematiCS at the Umverstty of Zunch He took hts degrees under B L van der Waerden tn Zunch He has pubhshed sev-eral papers on anctent, medteval, and ntneteenth-century mathemattcs, and espectally on Rtemann, to whom hts latest book ts devoted Dunng the past couple of years, he has also taken an tnterest tn the htstory of the other exact sctences and on the tnteracttons between sc1ence, phtlosophy, and soctety The ptcture shows htm before the tomb of Rtemann at Selasca {near Verbanta, Italy) on the occaston of the Verbanta Rtemann Memonal Conference tn July 1991

reads "Fortsetzung tm Sommersemester 1856 [Contmua-tlon m the Summer semester of 1856]," and on page 193 of the same manuscnpt "Theone der Functwnen complexer GroBen mtt besonderer Anwendung auf dte GauB'sche Rethe F(a, {3, y, x) und verwandte Transcendenten Wmter-semester 1856/57 " Furthermore, tt should be noted that Rtemann's lectures on general complex analysts developed gradually from 1855 to 1861 [Neuenschwander 1996, 11 f], and that m 1861 he also treated Weterstrass's factonzatwn theorem for enttre functwns wtth prescnbed zeros, mtro-ducmg to thts end loganthms as convergence-promtro-ducmg terms [Neuenschwander 1996, 62-65), whtch ts qmte re-markable, although not mentwned by Gray 1997

Acknowledgments

I am mdebted to Robert B Burckel (Manhattan, Kansas), Ivor Grattan-Gmnness (Enfield), and Samuel J Patterson (Gottmgen) for vanous suggesttons concemmg both con-tent and language The present arttcle was wntten under a grant from the Swtss Natwnal Foundatwn

BIBLIOGRAPHY

Armenante, A Jung, G [Relaztone sulle Leztont complementali date nell lstttuto tecntco supenore a Milano nello scorso anno (1868-69) daglt

egregt Professon Bnoscht Cremona e Casoratt] G1ornale d1

Mate-matlche 7 (1869), 224-234

Bottazztnt, U Rtemanns EtnfluB auf E Bettt und F Casoratt ArchiVe

for H1story of Exact Sc1ences 18 {1977/78), 27-37

Bottazztnt U II calcolo sub/me stona dell'anai!SI matemat1ca da Euler

a We1erstrass Bonnghten, Tonno 1981 Revtsed and enlarged Engltsh

verston Spnnger, New York 1986

Bottazztnt U Ennco Bettt e Ia formaztone della seuola matemattca

ptsana In 0 Montaldo, L Grugnettt (eds ) La stona delle matemat1che

m ltal!a Attt del Convegno, Caglian, 29-30 settembre e 1 ottobre 1982

Untverstta dt Caglian, Cagltan 1983 229-276

Bottazztnt, U Rtemann tn ltalia In Conferenza lnternazlonale ne/125°

Anmversano della morte d1 Georg Fnednch Bernhard R1emann 20 Lugl!o 1991 Verbama Att1 del convegno, 31-40

Bnot, C , Bouquet, J -C Theone des fonct1ons ell1pt1ques Deuxteme

edttton Gauthter-Vtllars Pans 1875

Bnot C Theone des fonctlons abel!ennes Gauthter-Vtllars Pans

1879

Casoratt F Teonca delle funz1om d1 vanab1i1 camp/esse Fust, Pavta,

1868

Casoratt, F , Cremona L lntomo al numero det moduli delle equaztont o

delle curve algebnche dt un dato genere Reale /stltuto Lombardo d1

Sc1enze e Lettere Rend1cont1 (2) 2 (1869), 62Q-625 = F Casoratt Opere

vol 1, 313-315 L Cremona, Opere rnatematlche, vol 3, 128-132

Dteudonne J The begtnntngs of Italian algebratc geometry In

Sympos1a mathemat1ca, vol 27, Academtc Press, London/New York

1986, 245-263

Forsyth, A R Old Tnpos days at Cambndge The Mathematical Gazette

19 (1935), 162-179

Gtllispte, C C (ed ) 01ctlonary of sCientific b1ography, 16 vols Scnbner,

New York 1970-1980

Gtspert, H Sur Ia productton mathemattque franc;;atse en 1870 (Etude

du tome premter du Bullettn des Sctences mathemattques) ArchiVes

/nternatlonales d'H!sto1re des SCiences 35 {1985), 380-399

Gtspert, H La France mathemattque La soctete mathemattque de

France (1870-1914) SuiVI de ctnq etudes par R Bkouche, C Gtlatn

C Houze!, J -P Kahane, M Zemer Cah1ers d'H!sto1re et de Ph!losoph1e

des Sc1ences (2), No 34 Blanchard, Pans, 1991

Houel, J Theone etementa1re des quantltes complexes

Gauthter-Vtllars Pans, 1867-187 4

Kletn F Rtemann und setne Bedeutung fur dte Entwtckelung der mo

dernen Mathernattk Jahresbencht der Deutschen

Mathemat1ker-Vere1mgung 4 (1894-95) [Berlin 1897], 71-87 = Gesammelte

mathe-mat!sche Abhandlungen, Bd 3, 482-497 English translation Bulletm of the Amencan Mathematical SoCiety 1 (1895), 165-180

Kletn, F Vor/esungen uber d1e Entw1cklung der Mathematik 1m 19

Jahrhundert Tetle 1 und 2, Grundlehren der mathemattschen

Wtssenschaften, Bde 24 und 25 Spnnger Berltn, 1926-1927 Repnnt

Berltn/Hetdelberg/New York 1979 Engltsh translatton Brookltne,

Mass 1979

Lona, G (ed) Lettere al Tardy dt Genoccht Bettt e Schlafli Att1 della

RealeAccadem1a de1 Lmce1 Sene qutnta Rendtcontt, Classe dt sctenze

ftstche, maternattche e naturali 241 (1915), 516-531

Lon a, G Stona delle matemat1che 3 vol STEN Ton no 1929-1933

Seconda edtztone nveduta e aggtornata Hoepli, Mtlano, 1950

Naas, J , Schmtd, H L (Hrsg ) Mathematlsches Worterbuch m1t

Embez1ehung der theoretlschen Physik, 2 Bde Akademte-Verlag/

Teubner, Berlin/Stuttgart, 1961

Neuenschwander, E Der NachlaB von Casoratt (1835-1890) tn Pavta

Neuenschwander E Uber d1e Wechselw1rkungen zw1schen der franzo-Sischen Schule, R1emann und We1erstraB E1ne Ubers1cht m1t zwe1

Ouellenstud1en Arch1ve for History of Exact Sc1ences 24 (1981),

221-255 English vers1on 1n Bulletin of the Amencan Mathematical

Soc1ety, New Ser 5 (1981) 87-105

Neuenschwander, E Der Aufschwung der 1tal1en1schen Mathematik zur Ze1t der pol1tlschen E1n1gung ltal1ens und se1ne Ausw1rkungen auf

Deutschland In Symposta Mathemaflca 27 (1986), 213-237

Neuenschwander, E A bnef report on a number of recently discov-ered sets of notes on R1emann's lectures and on the transmiSSion of the R1emann Nachlass H1stona Mathemattca 15 (1988), 101-113 Repnnted 1n R1emann 1990, 855-867

Neuenschwander E Secondary literature on B R1emann In R1emann 1990, 896-910

Neuenschwander, E R1emanns Emfuhrung m d1e Funkflonentheone

Eme quellenkntJsche Edttton semer Vorlesungen mtt emer B1bliograph1e zur W!rkungsgesch1chte der R1emannschen Funkttonentheone

Abhandlungen der Akadem1e der W1ssenschaften 1n Gott1ngen, Mathemat1sch-Phys1kalische Klasse, Dntte Folge Nr 44 Vandenhoeck

& Ruprecht, Gott1ngen 1996

Purkert, W Arbe1ten b1s 1891 In R1emann 1990, 869-895

R1emann, B Elltpflsche Funct1onen Vorlesungen von Bernhard

R1emann Mit Zusatzen herausgegeben von Hermann Stahl Teubner, LeipZig 1899

R1emann, B Gesammelte mathemattsche Werke w1ssenschaftlicher

Nachlass und Nachtrage Nach der Ausgabe von H Weber und A

Dedek1nd neu herausgegeben von A Naras1mhan Spnngerffeubner, Berlin, etc / Le1pz1g, 1990

Schenng, E Zum Gedachtn1s an B R1emann In Gesammelte

mathe-maflsche Werke von Ernst Schenng Bd 2 Berlin 1909 367-383

434-447 Partially repnnted 1n R1emann 1990, 827-844

Smart G Commental(e sur deux Memol(eS de R1emann relat1fs a Ia

theone generate des fonct1ons et au pnnctpe de Dmchlet Theses

presen-tees a Ia Faculte des Sc1ences de Pans Gauth1er-V1IIars, Pans 1882

Sm1th, H J S On the present state and prospects of some branches

of pure rnathernat1cs Proceedmgs of the London Mathematical Soctety

8 (1876-77). 6-29 =Collected Mathematical Papers, val 2, 166-190

Tncom1 F G Bernhard R1ernann e l'ltalia Umverstta e Po/itecn1co d1

Tonno, Rend1cont1 del Semmano Matemat1co 25 {1965-86) , 59-72 Volterra, V Bett1, Bnosch1, Casorat1, tro1s analystes 1tahens et tro1s man1eres d'env1sager les quest1ons d'analyse In Compte rendu du

deux1eme Congres International des Mathemat1ctens tenu a Pans du 6 au 12ao0t 1900 Gauth1er-V1IIars, Pans, 1902,43-57 =

Operematem-atJche, val 3, 1-11

We1l A R1emann, Bett1 and the b1rth of topology Archtve for History

of Exact Sctences 20 (1979), 91-96 21 (1979/80), 387

Whittaker, E T Andrew Russell Forsyth In Obituary Not1ces of Fellows

of the Royal Soc1ety 1942-1944 , vol 4 no 11 (1942) 209-227

W1rt1nger W R1emanns Vorlesungen uber die hypergeometnsche Re1he und 1hre Bedeutung In Verhandlungen des dntten lntematJonalen

Mathemat1ker-Kongresses m Hetdelberg vom 8 b1s 13 August 1904,

Teubner, Lepz1g, 1905 121-139 Repnnted 1n Riemann 1990 719-738

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