The Poincare-Volterra Theorem: From Hyperelliptic ...robbin/751dir/poincare-volterra-ullrich.pdf · The “Poincare-Volterra theorem” was formulated for the ﬁrst time in connection´

Arch. Hist. Exact Sci. 54 (2000) 375–402.c© Springer-Verlag 2000

The Poincare-Volterra Theorem: From HyperellipticIntegrals to Manifolds with Countable Topology∗

Peter Ullrich

Communicated byU. Bottazzini

The talk of Adolf Hurwitz (1859–1919) “Uber die Entwickelung der allgemeinenTheorie der analytischen Funktionen in neuerer Zeit” (“On the development of the gen-eral theory of analytic functions in recent times”) [39] before the First InternationalCongress of Mathematicians at Zurich in 1897 is generally considered as the “officialrecognition” of set theory by the mathematical community, in particular, by complexanalysts.1

In the present article we will study the history of one of the results which led tothis recognition and which Hurwitz also mentions in his lecture.2 The text deals witha development which took place mainly between the years 1888 and 1925 and duringwhich set theory or, strictly speaking, set theoretic topology appeared within the theoryof functions of one complex variable or, seen from a different point of view, crystallizedat problems of complex analysis.

The “Poincare-Volterra theorem” was formulated for the first time in connectionwith the definition of an analytic function that was promoted by Karl Weierstraß (1815–1897).3 One starts off with convergent power series of one complex variable. If twoof these power series with different centers of expansion attain the same values on theintersection (of the open interiors) of their discs of convergence, then, by the identitytheorem for power series, the two series resulting from re-expanding the given serieswith respect to the same point in the intersection coincide. Hence, one can join together

* Partial results of this article have been published under the title “Georg Cantor, Giulio Vivantiund der Satz von Poincare-Volterra” in theTagungsband des IV.Osterreichischen Symposions zurGeschichte der Mathematik, Christa Binder, Ed.Osterreichische Gesellschaft fur Wissenschafts-geschichte: Wien 1995, pp. 101–107.

My thanks go to the Handschriftenabteilung der Niedersachsischen Staats- und Universitatsbib-liothek Gottingen for the permission to publish from the letter of Georg Cantor to Giulio Vivantidated June 26, 1888 (Cod. Ms. Cantor 16, draft No. 84, pp. 180–181) and to the Akademiearchivof the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin, for the permission to quotefrom the letter of Karl Weierstraß to Hermann Amandus Schwarz dated March 14, 1885 (Schwarzestate, No. 1175).

1 cf. [27, pp. 213–214], [28, pp. 471–472], [64, p. 100], also [23, p. 247], [30, pp. 362–363].2 [39, pp. 99–100].3 e.g., [80, pp. 93–97].

376 P. Ullrich

such discs with non-empty intersection step by step, thus forming the Weierstraßian“Kreisketten” (“chains of discs”). Now, an analytic function in the Weierstraßian senseis defined by analytic continuation along these chains of discs. Each value at a pointc in the complex planeC which can be obtained in such a way is calledone valueof the analytic function atc. Here “multi-valued” functions are entirely admitted, i.e.,functions for which continuation along different chains of discs leads to different valuesat c.4

The question now arises: How large can the cardinality of the set of values bewhich are attained at the pointc? An obvious – nowadays, to be sure – translationof the problem to the language of Riemann surfaces reads as follows: Let an (alwaysconnected) concrete Riemann surface be given, i.e., a Riemann surface together with anon-constant analytic map to the Riemann sphereC. How many points of the surface canat most lie over a fixed pointc ∈ C? The answer to this question is that the cardinality canonly be countable. Proofs of this fact were published independently by Henri Poincare(1854–1912) [52] and Vito Volterra (1860–1940) [73] in 1888, who gave their names tothis theorem.

The problem to consider the “largeness” – measured in some vague sense – of theset of values of a multi-valued analytic function had, however, appeared much ear-lier in mathematics, long before the first investigations on denumerability by GeorgCantor (1845–1918) in 1873 [10] and even before the first definition of an analyticfunction by Weierstraß. This prehistory will be considered in Sect. 1 of the presentarticle.

The temporal coincidence of the proofs given by Poincare and Volterra of that the-orem was by no means by pure accident, but a reaction of them on the explicit posingof the problem by Giulio Vivanti (1859–1949) in a note [69] in the very year 1888,cf. Sect. 2. An inquiry of unpublished letters in Sects. 3 and 4 reveals, however, thatCantor and, after him, Weierstraß were in possession of the statement of the “Poincare-Volterra theorem” years ago, Weierstraß certainly in March 1885, possibly even in1878.

The publication of the “Poincare-Volterra theorem” was not too straightforward: Atfirst Vivanti gave a proof [70] which soon turned out to be fallacious (see Sect. 5). The(correct!) proofs of Poincare and Volterra are discussed in the Sects. 6 and 7, wherefor the Volterra part of the story we mainly refer to the study on the Vivanti-Volterracorrespondence by Giorgio Israel and Laura Nurzia [41].

In Sect. 8 we investigate the “reaction” of the mathematical community to the result,in particular its representation in standard texts both on function theory of one complexvariable and on topology until the 1930ies, its reformulation to the effect that a concreteRiemann surface has a countable basis of topology and its generalization to arbitraryabstract Riemann surfaces. The concluding Sect. 9 then shows why mathematicianssoon gave up generalizing the above statement on one-dimensional complex analyticmanifolds to other cases: Despite the simplicity of statement and proof of the Poincare-Volterra theorem there is no further generalization, neither in the real nor in the complex

4 [80, p. 97].

The Poincare-Volterra Theorem 377

case. Only in the complex one-dimensional caseall analytic (connected) manifolds havea countable basis of topology.

1. Abelian integrals of genus greater than or equal to2

The elementary functions known at the end of the 18th century pose no problemswith respect to the “largeness” of the set of values of a multi-valued function at a point.For algebraic functions, e.g. roots, the sets are always finite. (On the other hand, alreadythese functions prove that all finite cardinalities really can appear.) Furthermore, thelogarithm shows that the set can also by countably infinite; in this case the differentvalues differ by integer multiples of 2πi as already Leonhard Euler (1707–1783) knewin 1749 [26]. Also the inverse functions of trigonometric functions behave in this way.However, these examples are “harmless” in the sense that here the set of values attainedat a pointc always isdiscretein C (and by a result of Cantor from 18835 one knows thatdiscrete subsets of a real number spaceRn, hence in particular ofC, are automaticallycountable).

The story becomes more intricated when one starts considering the integrals

I (c) :=∫ c

c0

dx√p(x)

with p(x) a polynomial inx without multiple roots andc0 a fixed point inC. The valueof I (c) at a pointc depends on the choice of the path of integration fromc0 to c. And,as with the logarithm, one gets all possible values by adding to one single value all thelinear combinations of the periods of the integral with integer coefficients. In the ellipticcase, whenp(x) has degree 3 or 4, the integral possesses exactly two periods, which arelinearly independent over the real numbers, so that one again gets only a discrete set ofvalues.6

The picture changes drastically if one passes over to hyperelliptic Abelian integralswhere the polynomialp(x) has a degree greater than or equal to 5 (so that the corre-sponding Riemann surface has genus greater than or equal to 2). In fact, Jacobi foundout in 1834 that already if the degree equals 5 or 6 (and hence the genus is 2), theseintegrals have four periods, which in general are linearly independent over the ratio-nal numbersQ.7 Three linearly independent complex numbers are enough, however, togenerate an additive subgroup ofC which has an accumulation point at 0,8 by whichfact the set of valuesI (c) is not discretein C for any pointc. Even more, this set –though obviously countable – isdensein C: In the situation of areal polynomialp(x),

5 [12, Theorem I].6 This fact was clear to Carl Friedrich Gauß (1777–1855) already in 1798 [29, Vol. 3, pp. 433–

435, p. 492, p. 494; Vol. 10.1, pp. 194–206, pp. 274–278] and was published by Niels Henrik Abel(1802–1829) and Carl Gustav Jacob Jacobi (1804–1851) in the 1820ies.

7 [42, §§ 4–6].8 [42, §§ 2–3].

378 P. Ullrich

which Jacobi studies, this easily follows from the fact that of the four periods (whichare linearly independent overQ) two are real and two are purely imaginary.9

Of course, the notations “discrete” and “dense” cannot be found in Jacobi’s publi-cation [42]. But he remarkably clearly points out this difference in quality:

quemadmodum integralia elliptica pro eodem sinu amplitudinis numero valorum du-pliciter infinito gaudent . . . inter se aequidistantes: ita integralia . . . hyperelliptica . . .tantam multiplicatem [sic!] valorum secum ferunt, ut . . . e numero valorum, quos idemintegrale pro iisdem datis limitibus quibuslibet induere potest, semper sint, qui a datoquolibet valore reali aut imaginario minus differant quam ulla quantitate data quan-tumvis parva.10

By this discovery Jacobi was led to attacking the problem of the inverse functionfor hyperelliptic integrals in a different manner than for elliptic integrals: The inversefunctions of the latter are the elliptic functions, i.e., analytic – with the exception ofpoles –, doubly periodic, and univalent functions ofonecomplex variable. Contrary to

this, in the case of degree 5 or 6 Jacobi considered not only the integralc∫

c0

(1/√

p(x)) dx

but also the integralc∫

c0

(x/√

p(x)) dx and studied the inverse function as a function of

twocomplex variables.11 By this, “in hac quasi desperatione”,12 as he himself wrote, hefounded the theory of functions ofseveralcomplex variables. Speaking in modern terms,in this case, which later on was sometimes called the “ultraelliptic” one, Jacobi embeddedthe compact Riemann surface of

√p(x), where the holomorphic differentialdx/

√p(x)

is defined, into its “Jacobian”, the complex two-dimensional torus (C2 modulo its periodlattice), by means of the mapping

c 7→ c∫

c0

1√p(x)

dx,

c∫c0

x√p(x)

dx

.

On the other hand, at two places in his article Jacobi passes the verdict “absurd(um)”on attempting to solve the inversion problem for hyperelliptic integrals by functions ofonly onevariable, which would have to be fourfold periodic.13

Jacobi’s judgement gave rise to a longlasting debate in the 19th century on theinverse functions of hyperelliptic integrals. For example, on February 28, 1856 BernhardRiemann (1826–1866) wrote with respect to Jacobi’s verdict:

9 [42, § 1, §§ 7–8].10 “just as the elliptic integrals attain doubly infinitely many values for the same value of the

sine of the amplitude, . . . which are equidistant, the hyperelliptic integrals. . . carry with themsuch a strong multiplicity of values that. . . among the values that the same integral can attainfor the same arbitrary given boundaries there will always be some which differ from an arbitrarygiven real or imaginary value less than any given quantity, however small it may be.” [42, § 8].

11 [42, §§ 9–11], cf. [36, pp. 78–80].12 “in this so to speak desperate situation” [42, § 8].13 [42, § 4, p. 61 resp. p. 32 and § 7, p. 71 resp. p. 43], cf. also [63, esp. pp. 89–91].


Vielleicht etwas unvorsichtig [von Jacobi zu behaupten], es lasse sich keine mehr alszweifach periodische Funktion von einer Variabeln denken.14

(This remark was only published in 1902 in the “Nachtrage” of Riemann’s “Gesam-melte mathematische Werke”. Note also that Riemann explicitly speaks about “einwer-thige” (“univalent”) functions in his letter to Weierstraß of October 26, 1859 with the“Beweis des Satzes, dass eine einwerthige mehr als 2nfach periodische Function vonn Veranderlichen unmoglich ist”.15)

This debate has already been treated several times in the literature.16 Therefore atthis place we only discuss the aspect which will turn out to be of importance for thePoincare-Volterra theorem in the sequel. In the course of the edition of the second vol-ume of Jacobi’s collected works in the year 1882 Weierstraß commented in a footnotethat Jacobi’s verdict could no longer be upheld “[v]om Standpunkte der heutigen Func-tionenlehre aus”.17 Weierstraß must have found this criticism of Jacobi, whom he highlyrevered otherwise, easier insofar as “heutigen Functionenlehre” meant nothing else thanthe theory of, possibly multi-valued, functions of one complex variable which he him-self had co-founded to a decisive extent. Namely, his “analytische Gebilde” (“analyticformations”) – or, equivalently, the “Flachen” (“surfaces”) introduced by Riemann –made it possible to define the inverse function of a hyperelliptic integral as an analytic,be it multi-valued, function ofonecomplex variable.18

Encouraged by Weierstraß’ remark,19 Felice Casorati (1835–1890) took up anewhis investigations on the inversion of hyperelliptic integrals, which he had laid aside foralmost twenty years, and brought them to an end in the form of two articles [19], [20] in“Acta mathematica” in 1886 in which he solved the inversion problem in the languageof Riemann surfaces by means of explicit calculation.

2. The formulation of the problem by Vivanti

A discussion of Abelian integrals of genus greater than or equal to 2 and their inversefunctions is also the starting point of a note [69] by Vivanti, which he finished on June22, 1888 and which was communicated at the session of the “Circolo Matematico diPalermo” on July 8, 1888. Here one finds the question which cardinalities are possiblefor the set of values of a multi-valued analytic function at a given point posed in printfor the first time.

14 “Maybe a little bit imprudent [of Jacobi to claim] that it is not possible to imagine a functionof one variable which is more than doubly periodic.” [62, p. 709].

15 “Proof that a univalent more than 2nfold periodic function ofn variables is impossible”[60, p. 197 resp. p. 326].

16 [6], [50, pp. 8–21], also [63], [74], and [81, pp. 332–334, Notes 13–17].17 “from the point of view of today’s function theory” [43, Vol. 2, p. 516].18 For a longer exposition on this theme by Weierstraß himself see [79, pp. 123–144].19 [6, pp. 48–49], [50, pp. 13–14].

380 P. Ullrich

At first, Vivanti rightly criticizes20 the fact that the statement of the denseness ofvalues ofI (c) in C – which is precisely formulated in Jacobi’s original paper21 – wasreproduced in parts of the (for Vivanti) contemporary literature in an incorrect version,namely, that the integralI (c) would attaineachcomplex number as a value. As examples,he refers to the “Theorie der Abelschen Functionen”22 by Alfred Clebsch (1833–1872)and Paul Gordan (1837–1912) and the second edition of the “Neue Theorie der ultrael-liptischen Functionen”23 by Friedrich Prym (1841–1915).

Referring to “ricerche di G.C a n t o r” (“G. C a n t o r ’s investigations”), Vivantisubstantiates in his article24 that the set of values ofI (c) is only countably infinite,hence of “1a potenza” (“first cardinality”) whereas the setC of all complex numbers isof “2a potenza” (“second cardinality”).25,26

Now Vivanti defines in general that an analytic function (in the sense of the Weier-straßian definition) has “first cardinality” or is “of first class” (“una funzione ha la 1a

potenza ode della 1a classe”)27 if for each fixed pointc the set of values of the functionat c is of first cardinality, i.e., countable.

He immediately proves that this coining of a terminology is – a priori, at least – nofutile enterprise: On the one hand he shows by consideration of the associated Riemannsurface that each analytic function of first cardinality possesses an inverse function inthe sense of the Weierstraßian theory which again is of first cardinality.28 Here he doesnot waste a word on the existence of the inverse function, but remarks that for the case ofAbelian integrals of genus greater than or equal to 2 one can directly read off his resultfrom the explicit representation [19], [20] given by Casorati. On the other hand, Vivantishows that each analytic function which is defined by an algebraic differential equationhas first cardinality.29

This result brings him to the last third of his article, where he discusses Poincare’sresult of 1883 [51] on the uniformization of analytic functions.30 The task was toparametrize a given multi-valued analytic functiony = y(x) by means of a new variablez in such a way thaty = y(z) andx = x(z), respectively, are univalent analytic functionsof z. Contesting with Felix Klein (1849–1925), Poincare had made great progress to-wards the solution of this problem in his article of 1883, where he made use of countably

20 [69, no1].21 [42, § 8].22 “Theory of Abelian functions”, see [22, p. 134].23 “New theory of ultraelliptic functions”, see [55, pp. 1–2].24 [69, no2].25 [69, p. 136].26 Thus he presupposes the continuum hypothesis as Cantor had stated it at the end of his article

[11] in 1878 and also still at the end of the 1884 paper [14]. Cantor himself, on the other hand,had spent the summer and autumn of the year 1884 in a hot-and-cold bath of proofs and rejectionsof the continuum hypothesis (cf., e.g., [23, pp. 99–101, p. 118, pp. 135–137], [30, pp. 356–357],[49, pp. 139–141], [56, pp. 67–71, p. 81] for this aspect).

27 [69, p. 136].28 [69, no3, A].29 [69, no3, B].30 [69, no4–6].


infinitely multi-valued elliptic modular functions.31 This, however, had the consequencethat Poincare’s method could only by used in order to uniformise functions of first car-dinality as Vivanti was the first to point out, concluding from the results mentionedabove: “Dunque la dimonstrazione diP o i n c a re vale solo per le funzioni aventi la1a potenza.”32 Yet Vivanti abstains in his article [69] from any supposition whether theproperty of being of first cardinality holds foreachanalytic function.

3. Cantor’s letter to Vivanti

This did not mean that Vivanti was not occupied with this question: Already on May15, 1888 he had written to Cantor – with whom he was corresponding at least since De-cember 3, 188533 – a letter whose second part was treating this problem. Unfortunately,this letter has not been preserved in the Cantor estate in the Handschriftenabteilung derNiedersachsischen Staats- und Universitatsbibliothek in Gottingen.34

Cantor’s answer, however, can be found in his letter book for the years 1884–1888under the date of June 26, 1888.35 The relevant part for the question under considerationreads as follows:

Geehrter Herr Vivanti.Entschuldigen Sie freundlichst, daß ich erst heute Zeit finde, Ihr werthes Schreiben v.15 Mai zu beantworten. . . .Was den zweiten Theil Ihres Schreibens betrifft, so haben Sie Recht, daß “jede durcheine algebraische Differentialgleichung definirte Function die Eigenschaft hat, fur jedenWerth der unabh. Variablen nur eine abzahlbare Menge von Werthen zu erhalten.”Dieser Satz ist aber nur ein Spezialfall eines andern, den ich vor mehreren Jahren HerrnWeierstraß, dem er neu war, mittheilte, namlich des Satzes:“Jede analytischeFunction (im Weierstraßschen Sinne) hat, wenn sie unendlich vieldeutigist,. . . [deletions by Cantor] nothwendig eine Vieldeutigkeit nur von der erstenMachtigkeitω.”Weierstraß, der sich fur diesen Satz sehr interessirte, theilte mir einige Jahre spater mit,daß auch er einen Beweis dieses Satzes mit Hulfe seiner Theorie der Minimalflachengefunden hatte.Ich hatte gehofft, daß er seinen Beweis publicieren wurde. Allein dies ist unterblieben, ver-muthlich weil er in diesem Falle meine Theorie des Transfiniten hatte erwahnen mussen,was aber aus Rucksicht auf Kronecker und Helmholtz [subsequently inserted by Cantor:in Deutschland] bekanntlich nicht geschehen darf.

31 [51, p. 115 resp. p. 60].32 “Therefore Poincare’s proof is valid only for functions which have first cardinality.” [69,

no4, p. 138].33 [18, p. 251].34 Cod. Ms. Cantor 12 with the letters from Vivanti to Cantor only contains writings from the

years 1892 to 1894.35 Cod. Ms. Cantor 16, draft No. 84, pp. 180–181.

382 P. Ullrich

Wenn Sie einen Beweis fuhren, so lassen Sie sich hoffentlich nicht auch abhalten, ihn zuveroffentlichen.36, 37

Some comments on Cantor’s letter seem to be apt: Weierstraß’ esteem of Cantor’s de-numerability arguments is well-known. Already on December 22 and 23, 1873, Cantorhad reported to Weierstraß on his results concerning the denumerability of the alge-braic and the non-denumerability of the real numbers and Weierstraß had “veranlasst”(“caused”) Cantor to publish the article [10], as the latter informed Richard Dedekind(1831–1916) in a letter dated December 25, 1873.38 Shortly after that Weierstraß hadused the method to denumerate the rational and even the algebraic numbers, e.g. in lec-ture courses, in order to give examples of “pathological” functions.39 Furthermore, in aletter in English to Philip Jourdain (1879–1919) dated March 29, 1905 Cantor gives anextremely positive resumee of his relations to Weierstraß:

With Mr. Weierstrass I had good relations. . . . Of theconception of enumerability of whichhe heared from me at Berlin on Christmas holydays 1873 he became at first quite amazed,but one or two days passed over, it became his own and helped him to an unexpecteddevelopment of his wonderful theory of functions.40

(Since it does not seem to be known to which result of the Weierstraßian theory offunctions Cantor alludes here,41 the letter quoted above offers the tempting suggestionthat it is the theorem on the denumerability of the set of values of an analytic functionat a point.)

36 “Honoured Mr. Vivanti.Would you kindly excuse that I only find time today in order to answer your esteemed letter ofMay 15. . . .Regarding the second part of your letter, you are right that “each function which is defined by analgebraic differential equation has the property of attaining only a countable set of values for eachvalue of the independent variable.”This theorem, however, is only a special case of another, which I have communicated to Mr. Weier-straß – for whom it was new – several years ago, namely of the theorem:“Each analyticfunction (in the Weierstraßian sense) which is infinitely multi-valued necessarilyhas . . . [deletions by Cantor] a multiplicity only of the firstcardinalityω.”Weierstraß, who was very interested in this theorem, has informed me some years later that healso had found a proof of this theorem by means of his theory of minimal surfaces.I had hoped that he would publish his proof. But this did not take place, probably because he wouldhave had to mention my theory of the transfinite in this case, which, as is known, is not allowed[subsequently inserted by Cantor: in Germany] in deference to Kronecker and Helmholtz.If you give a proof then it is to be hoped that you will not be deterred from publishing it, too.”

37 Joseph Dauben has pointed out in [23, p. 342, Note 37] that this letter contains the firstinstance of Cantor using the superscripted bar in order to denote the cardinality of a set, in thiscaseω for the cardinality of the setω of natural numbers.

38 [17, pp. 16–17] For the positive reaction of Weierstraß on these results see also [64, p. 99,esp. footnote 1].

39 cf. [66, pp. 155–156].40 [30, p. 384] or [31, p. 124].41 [31, p. 124, footnote 17].


Furthermore, in a letter to Sofja Kowalewskaja (1850–1891) dated March 24, 1885,Weierstraß comments upon the problem of Abelian integrals of genus greater than orequal to 2 and their inverse functions:

. . . die zu einem und demselben Werthe vonc gehorigen Werthe vonI (c) bilden eineabzahlbareMenge, von der Cantor, wie ichuberzeugt bin, in unanfechtbarer Weise be-wiesen hat, daß es nicht nur unendlich viele Werthe giebt, die nicht nur nichtdarin en-thalten sind, sondern eine Menge von hoherer Machtigkeit bilden . . .42

(Note that this letter is three years older than the article [69], in which Vivanti usesthe same argument.)

Besides these aspects which strongly back the statements made in Cantor’s letter thereare, however, other arguments which may call for some reservation. First of all, in theCantor estate at Gottingen one can find no evidence for Cantor communicating his resultto Weierstraß: In the letter book Cod. Ms. Cantor 16, which is the only one containingdrafts of 1888 or earlier, there is only one letter to Weierstraß, dated May 16, 1887, andthis deals with the non-existence of infinitely small quantities. And Cod. Ms. Cantor 13with the letters from Weierstraß to Cantor merely contains three writings, all dating fromthe years 1881–82, none of which concerns the problem under consideration. One hasto admit, however, that the Cantor estate has been preserved in a rather incomplete stateand that Cantor may well have communicated the result to Weierstraß during a personalmeeting. Actually, in two of the three writings from Weierstraß mentioned above thedate of a visit of Cantor is discussed.

Furthermore, if Weierstraß was really “very interested in this theorem” as Cantorwrites, then one might expect that he would have mentioned the result in his letter toKowalewskaja of March 24, 1885 quoted above. This, however, is not the case. Evenmore, in summer 1886 Weierstraß gave a lecture course on “Ausgewahlte Kapitel ausder Funktionenlehre” (“Selected chapters from the theory of functions”). According tothe notes taken there [79], he elaborated upon how to repeal Jacobi’s “absurdum” verdictby means of his “analytic formations”.43 By means of logarithms he even constructedexamples of analytic functions whose value set at each point lies dense inC and whichare somewhat easier than the hyperelliptic integrals.44 There is, however, no indicationin the notes that Weierstraß had mentioned anything about the cardinality of the value setin general. The text has even led Reinhard Siegmund-Schultze to the hypothesis that atthat time – 1886 – Weierstraß did not yet know the Poincare-Volterra theorem,45 which,however, is in slight contradiction to Cantor’s statement of 1888 that he had informedWeierstraß “several years ago”.

Even more serious is the fact that the relation between Cantor and Weierstraß wasby no means always as happy and unproblematic as Cantor’s letter to Jourdain seems

42 “. . . the values ofI (c) which belong to one and the same value ofc form a countableset, ofwhich Cantor, as I am convinced, has shown in an indisputable way that there are not only infinitelymany values which are notcontained in it, but that [these] form a set of higher cardinality. . . ”[5, p. 71] or [81, p. 329], mathematical symbols adapted to the use above.

43 [79, pp. 123–144].44 [79, pp. 125–126].45 [79, p. 253, Note 194].

384 P. Ullrich

to indicate. From the beginning 1880ies until the death of Weierstraß Cantor expressedin several letters his feeling that Weierstraß did not pay due credit to him and his settheory, attributing this in part to Weierstraß’ own opinion and in part to his deference toother mathematicians.46 In particular, one is tempted to assume that Cantor’s hypothesison the reason for Weierstraß not publishing his proof says more about Cantor’s rela-tions to Leopold Kronecker (1823–1891) and Hermann (von) Helmholtz (1821–1894)than about Weierstraß. The notoriously bad relation between Cantor and Kronecker isdepicted in any biography of Cantor. The problems between Cantor and Helmholtz arenot that obvious since Dedekind had been successful in convincing Cantor to weakenhis comments on Helmholtz’ contributions on the foundations of geometry before Can-tor’s article [11] was published.47 However, Cantor’s opinion on Helmholtz had notimproved in the meantime, as letters from Cantor to Gosta Mittag-Leffler (1846–1927)dated December 30, 1883 and to Giuseppe Veronese (1854–1917) dated November 17,1890 reveal.48

4. Weierstraß’ use of the “Theorie der Minimalflachen”

The reservations given above against some of the claims in Cantor’s letter were, soto speak, from the point of history of mathematics. However, there is also a formulationin it which might make one purse one’s brow because of a purely mathematical reason:Cantor writes that Weierstraß had given his own “Beweis dieses Satzes mit Hulfe sei-ner Theorie der Minimalflachen” (“proof of this theorem by means of his theory ofminimal surfaces”).

Of course, Weierstraß had worked on minimal surfaces, even as early as in the1860ies [75], [76], [77]. But such a proof would seem overly complicated if one looksat the proofs of the theorem given by Poincare [52] (cf. Sect. 6) and by Volterra [73] (cf.Sect. 7) and compares them with the analytic machinery involved in minimal surfaces.

These doubts, however, are unfounded since Weierstraß made use of the term “theoryof minimal surfaces” in a sense different from the one presently current. This becomesclear from a letter of Weierstraß to Schwarz dated March 14, 1885,49 which also provesthat Weierstraß was in possession of the statement of the Poincare-Volterra theorem onthe denumerability of the value set as early as March 1885:

Daran knupft sich eine Frage. Kann man fur eine beliebige analytische Funktionf (u)

beweisen, daß fur jeden Werth vonu die zugehorigen Werthe vonf (u) eine abzahlbareMenge bilden, also zu einer Reihe geordnet werden konnen? Ohne Zweifel wird dieseFrage zu bejahen sein. In einer Minimalflache, die durch die bekannten Formeln, in de-nens, f (s) figuriren, definirt werden, entspricht jedem Werthpaare(s, f (s)) ein Punkt,in der Art, daß in demselben die complexen Großens, f (s) eine bestimmte geometrische

46 For details see [5, esp. p. 73–75], also [23, p. 138, p. 162, p. 164].47 Compare, for example, the letter from Cantor to Dedekind dated June 25, 1877 [17, pp. 33–

34] with the published text [11, p. 244 resp. pp. 120–121].48 [18, p. 162] resp. [18, p. 330].49 Akademiearchiv of the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin,

Schwarz estate, No. 1175.


Bedeutung haben. Durch eine solche Minimalflache mache ich die Gesammtheit der Wer-thepaare(s, f (s)), also, was ich ein monogenes Gebilde erster Stufe im Gebiete zweiercomplexer Variabeln nenne, meiner Vorstellung viel anschaulicher wie durch die Rie-mann’sche Flache – und mit Hulfe dieser Anschauungsweise habe ich mir einen Beweisfur die Bejahung der aufgeworfenen Frage zurecht gelegt. Gewiß wird es auch ohne einsolches geometrisches Hulfsmittel gehen. Nimmt man den Satz von Poincare als richtigan, daß jedes monogene Gebilde im Gebiete zweier Veranderlichenx, y sich durch dieGleichungen

x = f1(t), y = f2(t)

definiren lasse, wof1(t), f2(t) eindeutige Funktionen vont bedeuten, so laßt sich derfragliche Beweis sehr leicht fuhren.50

Here, “the known formulas, in whichs, f (s) appear” are the nowadays so-called“Weierstraß-Enneper representation formulas”:

x(s) = x0 + <∫ s

s0

(1 − s2)f (s) ds ,

y(s) = y0 + <∫ s

s0

i(1 + s2)f (s) ds ,

z(s) = z0 + <∫ s

s0

2sf (s) ds ,

wherex0, y0, z0 are fixed real numbers,s0 is a fixed complex number and< denotes thereal part. For each complex analytic functionf (s), these formulas parametrize a minimalsurface in the real three-dimensional space with coordinatesx, y, z as a function of thecomplex variables.51

50 “There is a question connected with this. Can one prove for an arbitrary analytic functionf (u) that for each value ofu the corresponding values off (u) form a countable set, so that theycan be arranged into a series? No doubt, the answer to this question is affirmative. In a minimalsurface which is given by the known formulas, in whichs, f (s) appear, to each pair of values(s, f (s)) there corresponds one [and only one] point in such a way that in it the complex quantitiess, f (s) have a certain geometrical meaning. By means of such a minimal surface I make the totalityof pairs of values(s, f (s)), thus what I call a monogenic formation of first grade in the domainof two complex variables, much more visual to my imagination than by the Riemann surface –and by the help of this way of visualization I have figured out for myself a proof for the positiveanswer to the proposed question. Certainly, it will also be possible without such a geometricalexpedient. If one takes Poincare’s theorem for granted – that each monogenic formation in thedomain of two variablesx, y can be defined by the equations

x = f1(t), y = f2(t)

with f1(t), f2(t) denoting univalent functions oft – then the proof in question can easily be given.”51 These formulas were published by Weierstraß in 1866 [75, § 1], but had been known to him

already in 1861 [75, p. 39]. At about the same time equivalent representation formulas were alsofound independently by Alfred Enneper (1830–1885) [25, pp. 107–108] and Riemann [61, § 9].

386 P. Ullrich

So, the use of the “theory of minimal surfaces” that Weierstraß makes is only arepresentation of the given analytic function (or the “analytic formation”) as an objectin the space of three real variables and, by this, a visualization of it.52 Furthermore, theabove letter of Weierstraß confirms that he was not content with his proof of the theoremof the countability of the value set. But the reason he gives here is that he would haveto use either “a geometrical expedient” or Poincare’s uniformization result [51]. By nomeans does he express any reservation against Cantor’s “theory of the transfinite” as thelatter had assumed in his letter to Vivanti.

Of course, Weierstraß does not mention Cantor’s name in the above quote, but asReinhard Bolling points out,53 Weierstraß hardly ever mentions Cantor in his correspon-dence; seemingly the problems in their relation were not one-sided.

One question that still remains open is, when did Cantor find the theorem and com-municate it to Weierstraß? The trivial lower bound is December 1873, of course, whenCantor started his investigations on denumerability. The above letter of Weierstraß givesas upper bound March, 1885, in accordance with Cantor writing in 1888 that it tookplace “several years ago”.

In any case, in the notes of the introductory lecture course of Weierstraß on thetheory of analytic functions from summer 1878 which were taken by Hurwitz one findsafter the definition of an analytic functiong by means of chains of discs with centersc1, c2, . . . , cn the following text:

Es findet nun der folgende wichtige Satz statt: “Entwederg(x|b) ist vollkommen un-abhangig von den vermittelnden Stellenc1, c2 . . . cn oder doch nur verschieden fur eineendliche Anzahl von Werthsystemenc1, c2 . . . cn.” 54

Admittedly, the statement is vague and somewhat cryptic.55 But it is definitely nota version of the monodromy theorem since this is proven some lectures later for star-shaped domains.56 So the above text must be another statement on the size of the valueset of the analytic functiong at the pointb and one might speculate that it is nothingelse than a distorted version of the theorem on the denumerability of this set.57

5. Vivanti’s attempted proof

Getting back to Cantor and his letter to Vivanti of June 26, 1888, one finds a re-markable illustration of Cantor’s distinction between the reception of his “theory of the

52 For a more detailled analysis of the use of geometric imagination in the mathematics ofWeierstraß see [67].

53 [5, p. 71].54 “Now the following important theorem takes place: “Eitherg(x|b) is completely independent

of the intermediate pointsc1, c2 . . . cn or just different for a finite number of systems of valuesc1, c2 . . . cn.” ” [80, p. 96].

55 Cf., e.g., the somewhat insecure comments in [80, pp. xviii–xix].56 [80, pp. 143–144].57 If this would be the case then Hurwitz, who obviously cited the statement verbatim, would

be the last to blame for the distortion: Since 1862 Weierstraß did not personally write at theblackboard but had this done by an advanced student to whom he was dictating the text [44, p. 61].


transfinite” in Germany and abroad.58 Already on July 30, 1888 – only about one monthafter finishing the note [69] and receiving Cantor’s writing – Vivanti completed anotherarticle [70], which was read at the session of the “Circolo Matematico di Palermo” onAugust 12, 1888. In this note he does not only state the theorem that each analytic func-tion is of first cardinality (i.e., has only a countable number of different values at eachpoint),59 but he also attempts to prove it. As to the origin of this theorem he states in afootnote:

Questo teorema, di cui io sospettava l’esistenza, mi fu comunicato recentemente dal ch.mo

prof. G i o r g i o C a n t o r , ilquale nello stesso tempo mi esortava, a tentarne dal miocanto la dimonstrazione.60

Similar reports on the genesis of the theorem are given by Volterra and Abraham(Adolf) Fraenkel (1891–1965).61 Neither, however, mentions that Vivanti had alreadyconjectured the statement. On the other hand, Fraenkel adds that Cantor had informedVivanti by means of a letter.

Here the specification of the exact creator of the proof given in Vivanti’s article [70]is of importance for an unpleasant reason (at least for Vivanti): To be sure, the statementof the theorem is correct, but he gives a fallacious proof of it!

He considers the concrete Riemann surfaceover C that corresponds to the givenmulti-valued analytic function which is defined on a domainin C.62 From the present(mathematical) point of view one might think that this construction was unproblematicand well-known at that time. For example, the Riemann surfaces which one has to usefor this purpose are nothing else than the “analytic formations” which Weierstraß usedto define in lengthy detail in his introductory lecture course on the theory of functions ofa complex variable, including the treatment of the branch points.63 On the other hand,one should keep in mind that the notion of a Riemann surface had by no means beenaxiomatized at the end of the 19th century.64 Much of the analysis of Vivanti’s argumentis therefore left to interpretation since Vivanti gives no definition of the notion of aRiemann surface which he uses and also is rather short in his exposition.

In any case, by taking resort to geometrical imagination and the picture of “elicoidale”(“winding stairs”), Vivanti argues that only countably many “fogli” (“sheets”) of theRiemann surface can meet at a branch point.65 Here a “sheet” of a Riemann surfaceshould be interpreted as something like a (maximal) open connected domain of thesurface where the function remains univalent. On the other hand, on such a sheet, at

58 cf. also [49, pp. 176–177].59 [70, p. 150, Teorema].60 “This theorem, whose existence I had conjectured, has recently been communicated to me

by ProfessorG e o r g C a n t o r , who at thesame time called upon me to try a proof on my own.”[70, p. 150].

61 [73, p. 359] resp. [27, p. 254], [28, p. 467].62 [70, p. 150].63 e.g., [80, pp. 159–165].64 Cf. Volterra’s criticism of Vivanti’s argument as decribed in [41, pp. 172–173, pp. 178–182],

and briefly here in Sect. 7.65 [70, a)].

388 P. Ullrich

least on its interior, the branch points lie isolated.66 Then Vivanti can quote Cantor’sresult of 1883 that each subset of a number spaceRn consisting only of isolated pointsis countable,67 by which he gets that on any sheet of the Riemann surface there are onlycountably many branch points.68

Up to this point Vivanti’s argument was all right – or can at least be interpreted in away that evades faults. The really delicate point comes now: Hence, he concludes, eachsheet of the Riemann surface is in direct connection with only countably many otherones and therefore, “in base ai principi della teoria degli aggregati”,69 also in indirectconnection with only countably many other sheets. The problem here is the argumentthat each sheet is in direct connection with only countably many other ones. Vivanti’stext is rather laconic: “Dalle due osservazioni precedenti risulta che ciascun foglio . . .e in comunicazione immediata con un insieme enumerabile di fogli.”,70 but it seemsthat the argument is meant as follows: There are only countably many branch pointson each sheet, only countably many sheets meet at each of the branch points, so eachsheet is in direct connection via branch points with only countably many other sheets.Though this line of thought is correct, Vivanti seems to have failed to notice that sheetsmay have connection with other ones also in a different way than via branch points or,putting things somewhat differently, that also a domain inC whose complement doesnot only consist of isolated points, e.g., an annulus, may have a multi-sheeted coveringand hence be the domain of definition of a non-univalent analytic function.

So the defects of Vivanti’s argument, which, by the way, were immediately pointedout by Hurwitz in his review [37], come from the theory of functions of a complex vari-able, whereas the set theoretic conclusions are correct. Therefore, in order to completethe proof, it is sufficient to give a class of (connected) open subsets of the Riemannsurface in such a way that the analytic function is univalent on each of these sets, thatthey cover the whole surface and that each of them has a non-empty intersection withonly countably many other ones. Then the canonical denumerability argument gives thatthere are only countably many of these sets at all and that, therefore, the function canattain only countably many values at each point.

Poincare and Volterra were the two mathematicians who, evidently independentlyof one another, had the necessary idea for this.

6. Poincare’s letter

In the meantime, on July 17, 1888, Vivanti’s first note [69] had been printed andpublished in the “Rendiconti del Circolo Matematico di Palermo”. Since it contained

66 supposing that a “Riemann surface” is defined in the way usual today, e.g., [40, 4. edition,III. 5. § 3, esp. p. 388].

67 [12, Theorem I].68 [70, b)].69 “on the basis of the principles of set theory” [70, p. 151].70 “From the above two observations it follows that each sheet . . . is in immediate connection

with a countable set of sheets.” [70, p. 151].


the proof that Poincare’s uniformization theorem [51] holds only for analytic functionsof first cardinality – leaving open whether this property holds for each function –, itcan be easily imagined that, in Poincare’s own wording, the reading of this note “m’avivement interesse et m’a inspire diverses reflexions”.71 Poincare’s enthusiasm is themore understandable because he was acquainted with set theory since his research onautomorphic functions, in which perfect, non-dense sets appear as sets of singularities,and he even knew Cantor personally since the beginning of 1884.72

Poincare laid down the abovementioned “reflections” [52] in a letter to Giovanni-Battista Guccia (1855–1914), the editor of the “Rendiconti”. The letter dates of October27, 1888, it was read in excerpts at the session of the “Circolo” at November 11, 1888,and published in the same volume of the “Rendiconti” as Vivanti’s two articles [69],[70]. Here Poincare proves that each analytic function is of first cardinality, but does notrefer to Vivanti’s attempted proof [70].73

Poincare’s exposition is complete, convincing and, yet, only three printed pages long.First, he explains the notion of denumerability and recalls the Weierstraßian definitionof analytic functions by means of power series and chained discs. Then he notes that theset ofall power series which define a given analytic function is uncountable.74 (At theend of the last Section a certain set of subdomains of the Riemann surface was required.Poincare’s remark has the consequence that one cannot satisfy this requirement by takingthe set ofall subdomains of a Riemann surface on which the function is described by apower series.)

Poincare’s decisive observation is that one does not need to consider all power seriesin order to determine the analytic function: If one starts off with a given power seriesand can attain a value of the function at a point by means of any chain of discs thenthe same can be performed by means of a chain of discs whose centers haverationalreal and imaginary parts. Hence it suffices to consider power series whose centers ofexpansion have rational coordinates.75

For a given power series and a given point inC the identity theorem implies that thereis at most one power series with the given point as center of expansion that coincides withthe given power series on the non-empty intersection of the open discs of convergence,i.e., a power series that is an immediate continuation of the given power series. Sincethere are only countably many points inC with rational coordinates, this implies thatthere are only countably many power series with rational points of expansion whichare immediate continuations of a given one. By use of the denumerability argumentwhich has already been mentioned in connection with Vivanti’s article [70] it followsthen that there are altogether only countably many power series with rational center of

71 “has vividly interested me and has inspired me to diverse reflections” [52, p. 197 resp. p. 11].72 [23, p. 280].73 As Vivanti’s second note [70] was printed almost a month after the first one [69], it is possible

that Poincare – who only in 1890 became member of the “Circolo” – was not aware of the secondnote when writing his letter.

74 [52, p. 198 resp. p. 12].75 [52, p. 198 resp. p. 12] Translated into the language of the Riemann surface associated to the

analytic function – which Poincare does not use, however –, this means that the images of powerseries with rational centers of expansion cover the whole Riemann surface.

390 P. Ullrich

expansion which are continuations of a given one, whether directly or indirectly. Hence,in particular, only a countable number of values can be attained at a given point.76

While developing his line of thought Poincare even gives an explicit – and veryelegant – proof of the set theoretic argument used, viz., that the set of all sequences ofnatural numbers of arbitrary finite length is countable: To each sequence(α1, α2, . . . , αn)

with n ∈ N, α1, α2, . . . , αn ∈ N arbitrary he assigns the finite continued fraction

α1 + 1

α2 + 1

. . . + 1

αn

.

This gives rise to a bijection between the set of finite sequences of natural numbersand the set of finite continued fractions. The latter one, however, is nothing else than theset of positive rational numbers, hence countable.77

7. Volterra’s article

In that very year 1888, when the articles [69], [70] by Vivanti and [52] by Poincareappeared, also Volterra published a proof of the theorem that each multi-valued analyticfunction attains only countably many values at each point. This proof, however, was notpublished in the “Rendiconti del Circolo Matematico di Palermo” but in the “Atti dellaReale Accademia dei Lincei” [73].

As to the contents of this article, Hurwitz writes in his review in the “Jahrbuchuber die Fortschritte der Mathematik”: “Die Betrachtungen des Verfassers stimmen imwesentlichen mit denen der Herren Vivanti und Poincare uberein” (“The considera-tions of the author essentially coincide with those of Messrs. Vivanti and Poincare”),from these expositions, however, “die Arbeit des Verfassers . . . sich durch Grundlichkeitauszeichnet” (“the author’s work distinguishes itself by its meticulousness”) [38].Whereas Poincare’s note bears the character of a clear but terse notice by letter, Volterrahas worked out the theory in all details, just as if he had wanted to publish a correctedversion of Vivanti’s second note [70]. He develops the Weierstraßian method of analyticcontinuation.78 In particular, he studies “dominii di monodromia” (“domains of mon-odromy”)79 on which the function remains univalent. He lists the properties of thesedomains in detail,80 noting especially that the power series expansions around centerswith rational coordinates give rise to a countable number of “domains of monodromy”which have the property that each value of the function is attained on one of them andthat each “punto regolare di diramazione”, i.e., interior branch point, of the functionlies on the boundary of such a domain.81 From this he does not only conclude the de-

76 [52, p. 200 resp. p. 13].77 [52, pp. 199–200 resp. p. 13].78 [73, pp. 356–357].79 [73, p. 357].80 [73, pp. 358–359, esp. Teorema I to V].81 [73, p. 358, Teorema I and p. 359, Lemma].


numerability of the values at a point but also that the set of (interior) branch points iscountable.82

Concerning the relation of Volterra’s article to those of the other two authors, Hurwitzin his review comes to the conclusion that “die Arbeit . . . offenbar unabhangig von denPublicationen der Herren Vivanti und Poincare entstanden [ist]” (“the work . . . wasapparently created independently of the publications of Messrs. Vivanti and Poincare”)[38]. Indeed, Volterra does not quote any of the three other articles which treat theproblem. Only in a footnote to his statement on the denumerability of the values hemakes the annotation

Questa proprieta e dovuta al prof. G. Cantor, che la comunico senza dimonstrazione aldott. G. Vivanti.83

This is, so to speak, the official and published version of the story. The Volterraestate in the “Accademia Nazionale dei Lincei”, however, reveals that Volterra’s part isdefinitely more complicated. (The following account is based on the study [41] by Israeland Nurzia.) After the printing of his second article [70] Vivanti had sent an offprintof it to Volterra with whom he was in correspondence since 1887.84 Following this, onAugust 21, 1888 Volterra wrote a letter – not to Vivanti but to Cantor in order to askfor the latter’s opinion concerning his, Volterra’s, objections against Vivanti’s method ofproof. The points of criticism have essentially already been mentioned in the discussionof Vivanti’s paper [70]: the possibility that sheets are in connection in other ways thanvia branch points, the question whether branch points are always isolated and, in general,the problem of constructing the Riemann surface corresponding to an arbitrary analyticfunction.85 Cantor’s answer to this on August 25, 1888 is rather evasive. But remarkablyhe again mentions that he had communicated the theorem in question several years ago toWeierstraß who had found a proof of it by means of the theory of minimal surfaces sometime later on.86 After this reply, in the time between end of August and mid-October,1888, Volterra worked on his article [73] and confronted Vivanti with his criticism onlyon October 13.87 From this a lively correspondence between the two resulted in thecourse of which Vivanti indeed could solve the problem of the sheets which are inconnection otherwise than via branch points. But at last he had to admit his failure togive an exact definition of the Riemann surface corresponding to an arbitrary analyticfunction.88 Then, on November 18, 1888 Vivanti informed Volterra that Poincare’s letter[52] had been read at a session of the “Circolo”,89 whereupon Volterra, who seems tohave at first had in mind to publish his article in the “Rendiconti”, submitted it to the“Atti” of the “Accademia Nazionale dei Lincei” on November 20, whose corresponding

82 [73, p. 359, Corollario], [73, p. 359, Teorema V].83 “This property is due to Professor G. Cantor who has communicated it without proof to

Doctor G. Vivanti.” [73, p. 359, footnote (3)].84 [41, p. 171], [41, p. 177].85 [41, pp. 171–172, p. 175].86 [41, p. 176].87 cf. [41, pp. 172–173]; [41, pp. 177–178].88 [41, pp. 172–173, pp. 178–182].89 [41, p. 183].

392 P. Ullrich

member he had become recently.90 The paper was read at the session on December 2,1888.

8. Further developments

After untangling the parsimonious remarks on Cantor and Vivanti in the two proofsof the Poincare-Volterra theorem, we turn to the reception of this theorem within themathematical community. Starting with Vivanti himself, this result is surprisingly notmentioned in his textbook on complex function theory [71] which appeared for the firsttime in 1901. Of course, one might be tempted to assume that this is a consequence ofhis negative experiences with that complex of problems. But, on the other hand, onehas to take into account a reason coming from within mathematics: Following a generaltendency at the turn of the century, Vivanti abandons the Weierstraßian definition ofmulti-valued analytic functions in his book. When in 1906 August Gutzmer (1860–1924) revised and translated it into German, the attribute “eindeutigen” (“univalent”)was explicitly added to its title, and Gutzmer writes in connection with the definition ofan analytic function:

Was man gewohnlich einen i c h t e i n d e u t i g eoder auch einem e h r d e u t i g eFunk-tion nennt, ist nach unserer Auffassung keine Funktion, sondern der Inbegriff von mehreren,bezw. unendlich vielen Funktionen.91

Contrary to this, the first volume of the textbook on complex function theory [3] byLudwig Bieberbach (1886–1982) still contains that “schonen vonPoincareundVolterraherruhrendenS a t z”.92 The first editon of this volume appeared in 1921. In the fourthedition, published in 1934, even a complete subsection was devoted to the theorem underthe title “Analytische Funktionen sind hochstens abzahlbar vieldeutig.”93As late as 1962,the book [35] of Einar Hille (1894–1980) discusses the Poincare-Volterra theorem inconnection with multi-valued analytic functions.94

On the whole, the Poincare-Volterra theorem did by no means fall into oblivioneither when formulated in terms of a multi-valued analytic function or when formulatedin terms of the associated Riemann surface, quite the contrary. In his first note [69]Vivanti had already pointed out the connection with Poincare’s uniformization result[51]. Now, when in 1907 the uniformization theorem was proven in full generality byPaul Koebe (1882–1945) and, independently, by Poincare, both authors referred in theirarticles to the Poincare-Volterra theorem, Koebe in a footnote without giving a source:

90 [41, p. 172], [41, p. 173], [41, p. 177].91 “What usually is called an o n - u n i v a l e n t oralso a m u l t i - v a l u e d function is no

function according to our conception, but the collection of several respectively infinitely manyfunctions.” [72, p. 109, footnote 1].

92 “beautiful t h e o r e mstemming fromPoincareandVolterra” [3, 1. edition, pp. 203–204].93 “Analytic functions are at most countably multi-valued.” [3, 4. edition, pp. 202–203].94 [35, p. 14, Theorem 10.3.2].


R i e m a n nsche Flachen mit nichtabzahlbarer Blatterzahl oder nicht abzahlbarer Anzahlder inneren Windungspunkte gibt es bekanntlich nicht.95,

Poincare referring to his own paper [52] of 1888:

Pourquoi cet ensemble doit-iletredenombrable, c’est ce que j’ai explique au tome 2 desRendiconti del Circolo Matematico di Palermo.96

Also Bieberbach in his proof of the uniformization theorem in the second volumeof his “Lehrbuch der Funktionentheorie”97 mentions this result. Here, by the way, heexplicitly talks about the Riemann surface of a (multi-valued) analytic function.

The theory of Riemann surfaces had been axiomatized in the meantime, mainly byHermann Weyl’s (1885–1955) monograph [82], which appeared for the first time in1913. Here he also proves the

von Poincare und Volterra ausgesprochene. . . Theorem. . . , daßes in einem analyti-schen Gebilde hochstens abzahlbar unendlichviele regulare Funktionselemente . . . mitvorgeschriebenem Mittelpunkt . . . gibt.98

In his proof Weyl in the main follows the lines of the articles by Poincare [52] andVolterra [73] and also cites them. (The names Cantor and Vivanti do not appear.)

Theorem and proof are also found – without any mentioning of the authors, how-ever – in the textbook on complex function theory [40] by Hurwitz and Richard Courant(1888–1972) from the second edition of 1925 onwards. Remarkably, they are not in thefirst part “Allgemeine Theorie der Funktionen einer komplexen Veranderlichen” (“Gen-eral theory of functions of a complex variable”) which was written by Hurwitz who hadbeen involved in the events of 1888, if only as a reviewer. But they are given in thethird part on “Geometrische Funktionentheorie” (“Geometric theory of functions”) byCourant, viz., in the fifth chapter, § 3. (At the beginning of this very section there is areference to Weyl’s book [82]. Since this reference can also only be found in the secondand later editions of the book of Hurwitz and Courant, it seems probable that the theoremhad found its way to their book via Weyl’s book [82].)

Within the function theoretic setting, the books of Weyl and of Hurwitz and Courantmark the definite shift from the arithmetically defined multi-valued analytic functions ofWeierstraß to their geometric counterparts, the associated (concrete) Riemann surfaces.Yet, the method of proof of the Poincare-Volterra theorem essentially remained the sameas in the original articles by Poincare [52] and Volterra [73]: One uses the fact that thecountable set of rational numbers lies dense in the set of real numbers and then arguesthat one can connect two arbitrary points by using only points with rational coordinatesas intermediate points.

95 “It is known that there are noR i e m a n nsurfaces with an uncountable number of sheetsor a non-countable number of interior branch points.” [47, p. 198, footnote 2].

96 “In volume 2 of the Rendiconti del Circolo Matematico di Palermo I have explained whythis set iscountable.” [53, p. 4 resp. p. 73].

97 [4, p. 161 resp. p. 162].98 “theorem stated by Poincare and Volterra. . . , that in an analytic formation there are only

countably infinitely many regular function elements . . . with prescribed center” [82, here 1. edition,pp. 14–15].

394 P. Ullrich

This proof hardly uses any analytic property of the function or its Riemann surfaceat all. In fact, nowadays the Poincare-Volterra theorem is usually considered as atopo-logical statement, namely that each concrete Riemann surface has a countable basis oftopology. A topological space is said to have a countable basis of topology (sometimeseven in short: a countable topology) if there exists a countable set of open subsets of thespace such that for each point of the space arbitrarily small neighborhoods belong to thisset. Since each open subset of a real number spaceRn has a countable basis of topology– just take the balls with rational radii centered at points with rational coordinates whichare contained in the open subset –, one only needs to check whether a given (real orcomplex) manifold, e.g., a Riemann surface, contains a dense countable subset in orderto show that it has a countable basis of topology. Therefore, also in the new clothingof the theorem, the idea of proof remains the same even though the wording differsconsiderably from the versions of 1888.

The abstraction of the notions involved in the statement and the proof of the Poincare-Volterra theorem, i.e., the translation into the language of set theoretic topology beganin the beginning of the 20th century. In his monograph [32] of 1914 Felix Hausdorff(1868–1942) already discusses the “zweite Abzahlbarkeitsaxiom” (“second axiom of de-numerability”) for topological spaces, i.e., that the topology of the space has a countablebasis.99

Surprisingly, however, in [32] Hausdorff does not mention the Riemann surface of ananalytic function as an example of a space for which the second axiom of denumerabilityholds. The same applies for the revised versions of this book which appeared in 1927and 1935100 and for the monograph by Paul Alexandroff (1896–1982) and Heinz Hopf(1894–1971) of 1935.101 Surely, spaces with a countable dense subset are discussedin these sources, in particular in connection with Paul Urysohn’s (1898–1924) resultsconcerning whether a given topological space carries a metric. But the result of Poincareand Volterra is not mentioned.

A result connected with their names was mentioned, though, when Bourbaki revisedhis “Topologie generale” [7] for the third edition of 1961. There he (or: they) added toChapitre 1, § 11 a section no 7 entitled “Application: le theoreme de Poincare-Volterra”.Of course, the topic is no longer analytic functions or Riemann surfaces: “Corollaire 3(theoreme de Poincare-Volterra)” states that a separated and connected topological spaceX has a countable basis of topology if there is a locally compact and locally connectedtopological spaceY with a countable topology and a continuous mapp: X → Y whichis locally homeomorphic onX. (The following seems to be an appropriate dictionaryfor translating this back to the original situation of a multi-valued analytic function:TakeY as the domain of definition of the function inC, X as the Riemann surface or“analytisches Gebilde” defined by it andp as the projection to the domain of definition,i.e., the local inverse of the analytic function.)

99 [32, Kap. VIII, §§ 1, 3] The “erste Abzahlbarkeitsaxiom” (“first axiom of denumerability”)states that each point has a countable basis of neighborhoods [32, Kap. VIII, §§ 1, 2]. This axiomapplies for Riemann surfaces and, more generally, for real topological manifolds.

100 [33], [34], in both cases § 25, § 30, 4., § 40, III.101 [2, Kap. I, § 7].


It should be recalled that the Poincare-Volterrra theorem only makes a statementonconcreteRiemann surfaces, i.e., Riemann surfaces on which a non-constant analyticmap to the Riemann sphereC is given. Already Klein, however, had studied Riemannsurfaces without such a concretization. And at least since Weyl’s topological defini-tion of a Riemann surface it made perfect sense to study also suchabstractRiemannsurfaces.102

For the case of concrete Riemann surfaces, i.e., multi-valued analytic functions,Vivanti had noted in his article [69] that the denumerability of the values of a functionat a point is a necessary prerequisite for Poincare’s uniformization result [51] to hold.Similarly, the problem whether an abstract Riemann surface has a countable basis oftopology arose in connection with the question of uniformization. Now, in order to useexhaustion procedures in the construction of uniformising functions, it was necessaryto secure thateach(connected) Riemann surface possesses a triangulation, i.e., a divi-sion into topological triangles. It is immediately clear that a triangulated surface has acountable topology. The converse also holds, but an exact proof is by no means trivial.A short proof that only each compact real 2-dimensional manifold can be triangulatedwas published as late as 1968 and in no less a journal than “Inventiones mathematicae”[24].

Weyl ridded himself of this problem in his “Idee der Riemannschen Flache” [82] byraising the existence of a triangulation in the first two editions and the denumerabilityof the topology in the third edition to an additional axiom for Riemann surfaces.103

Also Koebe a priori required the existence of a triangulation for the “RiemannschenMannigfaltigkeiten” (“Riemannian manifolds”) which he considered in his article of1917.104

This solution surely was effective but also not too elegant. Even more, it was un-necessary as Tibor Rado (1895–1965) first noted in 1922. He stated the theorem andalso sketched a proof thateach(abstract!) Riemann surface possesses a triangulation.105

Rado’s explanations were sketchy for the only reason that Heinz Prufer (1896–1934)had communicated his conjecture that each surface, even if only topological, can betriangulated to Rado who, therefore, was insecure whether his proof for the special caseof a Riemann surface was worth being published at all.106

9. Counterexamples

In January 1923 Prufer, however, revoked his conjecture that each topological sur-face can be triangulated and informed Rado of an example of a surface which cannotbe triangulated and hence has no countable topology.107 Following this, Rado published

102 It is still subject to debate whether Riemann himself thought of his surfaces as concrete orabstract, cf., e.g., the exposition [59, pp. 184–185].

103 [82, 1., 2. edition, p. 21], [82, 3. edition, pp. 21–22].104 [48, pp. 70–71].105 [57, pp. 35–36].106 [57, p. 35].107 [57, p. 35, footnote 9], [58, p. 102, footnote 1].

396 P. Ullrich

another article in 1925 where he exhibited Prufer’s counterexample, showed that eachsurface with countable topology can be triangulated, and finally proved that each Rie-mann surface has countable topology and hence can be triangulated.108,109

Prufer’s real 2-dimensional counterexample consists of a halfplane with half a unitdisc attached at each point of the boundary line, in the following way: First, in order toadd half a unit disc at one pointa of the boundary line, think ofa as fixed, shift eachpoint of the half plane one unit length away froma (in the same direction in that it lies,as seen froma), and insert half a unit disc in the space thus created. This gives, for eacha of the boundary line, a half plane which consists of the shifted points of the originalhalf plane and the inserted half disc. Now, make one manifold out of all these halfplanes by identifying again the points of the original half plane in the way as they weresituated before the respective shifts.110 The resulting manifold is a connected surfacewith uncountable topology. Even more – although this fact is not stated in [58] – bywriting down the construction in a convenient way, one can easily see that this surfaceis real analytic.111 Therefore, this a priori 2-dimensional example implies that in anyreal dimension greater than or equal to 2 there are real analytic manifolds which haveno countable basis of topology.

If Prufer had only been interested in topological spaces without additional conditions,however, then he could have simply asked Hausdorff concerning his conjecture on thecountability of the topology. Indeed, a recent examination of Hausdorff’s mathematicalestate in the Universitatsbibliothek at Bonn by Egbert Brieskorn and Walter Purkert hasbrought to light that as early as May 30, 1915 Hausdorff knew an example of a connectedtopological surface which does not fulfil the second axiom of denumerability, i.e., hasno countable topology.112

Contrary to Prufer’s construction, Hausdorff’s counterexample is the cartesian prod-uct of a common real interval with the so-called “long (half)line”. By use of the lex-icographical order one “inserts” at each point of the usual real line a copy of the realunit interval. This gives rise to a connected real one-dimensional topological manifoldwhich is “longer“ than the usual line (and, in fact, has no countable basis of topol-ogy).

The basic idea for this construction can already be found with Cantor in 1883:

Die erweiterte ganze Zahlenreihe kann, wenn es die Zwecke fordern, ohne Weiteres zueiner continuirlichen Zahlenmenge vervollstandiget werden, indem man zu jeder ganzenZahlα alle reellen Zahlenx, die grosser als Null und kleiner als Eins sind, hinzufugt.Es wird nun vielleicht hieran die Frage geknupft werden, ob man, da doch auf dieseWeise eine bestimmte Erweiterung des reellen Zahlengebietes in das Unendlichgrosse

108 [58, § 2], [58, § 3], [58, § 4].109 Nowadays, Rado’s result on the denumerability of the topology of Riemann surfaces is

usually proven by considering the surface with a disc removed, constructing a non-constant holo-morphic function on the universal covering of this “punched surface” by the help of methods fromfunction theory and then applying the Poincare-Volterra theorem, e.g, in the Bourbaki versioncited above.

110 For explicit formulas see [58, pp. 107–108].111 cf., e.g., [9, p. 337].112 Kapsel 31, Fasz. 121, sheet 2, cf. [8, p. 2] and [59, p. 188].


erreicht ist, nicht auch mit gleichem Erfolge bestimmte unendlich kleine Zahlen, oder wasauf dasselbe hinauslaufen mochte, endliche Zahlen definiren konnte, welche . . . sich anmuthmaasslichen Zwischenstellen inmitten der reellen Zahlen ebenso einfugen mochten,wie die irrationalen Zahlen in die Kette der rationalen oder wie die transcendenten Zahlenin das Gefuge der algebraischen Zahlen sich einschieben?113

Though the description is somewhat vague, one easily recognizes the idea of insertingfurther numbers, e.g., the unit interval, into the real line.

In are more detailed way, Leopold Vietoris (*1891) has treated the “long halfline” inhis doctoral dissertation which originates from the years 1913 to 1919 and was publishedin 1921.114 He, however, says nothing concerning the denumerability of its topology.One should note that Vietoris’ treatment of the “long line”, in particular his givingcredit to Cantor, is one of the paradoxes of the history of mathematics. He writes on hisconstruction:

Nach C a n t o r sAnleitung interpolieren wir zwischen je zwei aufeinanderfolgende untereiner beliebigen Schrankeσ liegende Ordnungszahlen einschließlichσ selbst ein demKontinuum [0, 1] ahnliches Linearkontinuum. . . .115

But “Cantor’s instruction”, i.e., his question quoted above, is purely rhetorical and theintended answer isnegative. Cantor strictly refused to accept the possibility of infinitelysmall numbers as they appear in the construction of the “long (half)line”. In his articleof 1883116 this is not as clearly expressed as in his paper of 1887117 and in his disputeby letter with Veronese in 1890.118

As mentioned above, Hausdorff had a clear perception of the topological propertiesof the “long halfline” already in May 1915. However, he did not publish his result,but communicated it to Heinrich Tietze (1880–1964).119 The latter included this as anexample120 in a manuscript which he had essentially completed in summer 1922. But

113 “If the ends demand it, the extended series of the integer numbers can be completed withoutfurther ado to a continuous set of numbers by adding to each integer numberα all real numbersx which are greater than zero and smaller than one.Maybe, the following question will be posed as a consequence to this: Since one has reached acertain extension of the domain of real numbers to the infinitely large in this way, why shouldone not with the same success insert certain infinitely small numbers or, what may amount tothe same, finite numbers which . . . fit into the conjectural intermediate places in the midst of thereal numbers, just like the irrational numbers insert themselves into the chain of rational or thetranscendental numbers into the fabric of algebraic numbers?” [13, § 4, p. 552 resp. p. 171].

114 [68, pp. 183–184].115 “Following C a n t o r ’ s instruction, we interpolate between any two successive ordinal

numbers lying below an arbitrary boundσ , includingσ itself, a linear continuum which is similarto the continuum [0, 1] . . . ” [68, p.183].

116 [13, p. 552 resp. pp. 171–172].117 [15, VI, pp. 407–409].118 cf. [18, pp. 326–332], also [5, p. 78], [23, pp. 128–132, pp. 233–238].119 [65, p. 217].120 [65, pp. 217–218].

398 P. Ullrich

he submitted it only in October 1923 to the “Mathematische Annalen” since he wantedfirst to get himself informed “uber auslandische Literatur”.121

Already on August 1, 1923, however, a manuscript by Alexandroff had reachedthis journal in which also the “long halfline” was discussed.122 (In fact, Alexandroffhad reported on the main results of [1] already in 1922 in Moscow and on June 26,1923 in Gottingen.123) Although Alexandroff’s paper [1] was published one volume of“Mathematische Annalen”later than Tietze’s, a tradition stems from these circumstancesto denote the “long (half)line” as the “Alexandroff (half)line”.124

Remarkably, this means that the existence of a real one-dimensional manifold with-out countable basis of topology was documented in print only after the correspondingfact for the 2-dimensional situation, i.e., Prufer’s example. Rado mentioned it already inhis article [57] that appeared in volume 90 of the “Mathematische Annalen”, whereasthe articles of Tietze and Alexandroff were published only one and two volumes later,respectively.

Furthermore, whereas it is more or less evident from its construction that Prufer’sexample is real analytic, it was not so clear whether the “long (half)line” bears not onlya topological but also a real analytic structure. Only in 1957 Hellmuth Kneser (1898–1973) proved this fact.125 Hence, there are real analytic manifolds of any dimensionwhich admit no countable basis of topology.

In the complex analytic situation Rado’s theorem says that for (complex) dimen-sion 1 each connected complex analytic manifold has a countable basis of topology.For higher dimensions, however, Eugenio Calabi and Maxwell Rosenlicht have givencounterexamples in 1952 [9]. Putting things the other way round: Rado’s result – or thePoincare-Volterra theorem – deals with the only real or complex analytic case in whichone always has a countable basis of topology.126

So, seen from the point of view of mathematics, the Poincare-Volterra theorem con-cerns the exceptional case and it looks like pure accident that mathematicians cameupon it. But the history depicted above shows this was by no means the case. The ad-vances of the complex analysis of one variable in the middle of the 19th century hadsubstantiated the hypothesis that the value set of a multi-valued analytic function couldnot be so large that it would prevent analytic inversion, e.g., of a hyperelliptic integral.Cantor’s concept of denumerability then supplied a language to express this belief. Andaltogether five mathematicians at least to tried to give proofs of the denumerability ofthe value set, even if the result nowadays is connected with the names of only two ofthem.

121 “on foreign literature” [65, p. 224].122 [1, p. 295, footnote 2].123 [1, p. 301].124 cf., e.g., [45, p. 4], [59, p. 189].125 [45, Satz 2] There exist even continuously many non-isomorphic real analytic structures on

the Alexandroff (half)line as he showed in a joint paper with his son Martin (*1928) in 1960 [46].126 For a further discussion of the interplay between countable topology and complex structure

on a manifold see [59, esp. pp. 189–191].


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Mathematisches InstitutWestfalische Wilhelms-Universitat

48149 MunsterGermany

(Received May 19, 1999)

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The Poincare-Volterra Theorem: From Hyperelliptic ...robbin/751dir/poincare-volterra-ullrich.pdf · The “Poincare-Volterra theorem” was formulated for the ﬁrst time in connection´