Historia Mathematica Article Booklet - A4

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HISTORIA MATHEMATICA 20 (1993), 364-381
Hilda Geiringer-von Mises, Charlier Series, Ideology, and the
Human Side of the Emancipation of Applied Mathematics at
the University of Berlin during the 1920s
REINHARD SIEGMUND-SCHULTZE
Kastanienallee 12, 10119 Berlin, Germany
The controversy surrounding Hilda Geiringer's application for "Habilitation" (permission
to teach) at the University of Berlin (1925-1927) sheds some light on the struggle of "applied
mathematics" for cognitive and institutional independence. The controversy as well as
Geiringer's unpublished reminiscences reveal the decisive influence of Richard von Mises,
Geiringer's husband since 1943, on both her career and the course of applied mathematics
at the University of Berlin. Some more speculative remarks reflect on the possible ideological
background of this controversy in post-World War I Germany. The debate over Geiringer's
theses for Habilitation ("Habilitationsschriften") opens up a chapter of the history of mathematical statistics, namely, expansions of a discrete distribution with an infinite number of
values in a series in successive derivatives of the Poisson distribution with respect to the
parameter. These expansions were first proposed by the Swedish astronomer C. L. W.
Charlier (1862-1934) in 1905. © 1993AcademicPress, Inc.
Spor v 1925-1927 g. o prinyatii zhenshchiny-matematika, Khilda Geiringera, dotsentom
v Berlinskom universitete ( = "khabilitatsiya") otrazhaet borbu "prikladnoi matematiki" za
samostoyatelnost. Spor i neopublikovannye do sikh por vospominaniya Geiringera pokazyvayut znachitelnoe vliyanie Rikharda Fon Misesa, muzha G. s 1943 g., na kareru G. i
na razvitie prikladnoi matematiki. Neskolko legko spekulativnykh zamechanii kasayutsa
ideologicheskogo zadnogo plana spora vskore posle pervoi mirovoi voiny. Diskussiya "tesisa
za khabilitatsiyu" G. pozvolyaet ponimanie glavy istorii matematicheskoi statistiki. Rech
idet o razlozhenii diskretnykh raspredelenii s beskonechnom mnozhestvom znachenii veroyatnosti v ryady po proizvodnym (otnositelno parametra) raspredeleniya Puassona. Takoe
razlozhenie bylo predlozheno (no ne dokazano) pervyi raz shvedskim astronomom
K. L. V. Sharle (1862-1934) v 1905 g. © 1993AcademicPress, Inc.
Die Kontroverse um Hilda Geiringers Habilitationsverfahren an der Berliner Universit~it
(1925-1927) reflektiert Momente des Kampfes der"angewandten Mathematik" um kognitive
und institutionelle Unabhangigkeit. Die Kontroverse selbst wie auch unver6ffentlichte Erinnerungen von Geiringer zeigen den maBgeblichen Einflul3 von Richard von Mises, Geiringers
Ehemann seit 1943, aufihre Laufbahn und auf die Entwicklung der angewandten Mathematik
an der Berliner UniversitS.t. Einige spekulativere Bemerkungen sind dem mSglichen ideologischen Hintergrund jener Auseinandersetzungen in den Jahren nach dem Ersten Weltkrieg
gewidmet. Die Auseinandersetzung um Geiringers Habilitationsschriften erm6glicht Einblick in ein Kapitel der Geschichte der mathematischen Statistik. Es handelt sich dabei um
die Entwicklung von diskreten Verteilungen mit unendlich vielen Wahrscheinlichkeitswerten
in Reihen nach Ableitungen (hinsichtlich des Parameters) der Poisson-Verteilung, die
erstmals 1905 von dem schwedischen Astronomen C. L. W. Charlier (1862-1934) vorgeschlagen wurde. © 1993AcademicPress, Inc.
AMS 1991 subject classifications: 01A60, 01A72, 62-03.
KEY WORDS: Hilda Geiringer (1893-1973), Richard von Mises (1883-1953), applied mathematics
mathematical statistics, Charlier series, ideology in German academia after WWI.
364
0315-0860/93 $5.00
Copyright© 1993by AcademicPress, Inc.
All rightsof reproductionin any formreserved.
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HILDA GEIRINGER-VON MISES
365
INTRODUCTION
In a talk delivered in 1965 Alexander Ostrowski (1893-1986) e x p r e s s e d the view
that
Only with the appointment of Richard yon Mises [in 1920] to the University of Berlin did
the first mathematically serious German school of applied mathematics with a broad sphere
of influence come into existence. Von Mises was an incredibly dynamic person and at the
same time amazingly versatile like Runge. He was especially well versed in the realm of
technology. Because of his dynamic personality his occasionally major blunders were somehow tolerated. One has even forgiven him his theory of probability. At the same time the
mathematical atmosphere in Berlin was much more open and less tense than in G6ttingen. The
sovereign Olympian, Erhard Schmidt, Issai Schur's evident sense of what was mathematically
important, and Bieberbach's impulsive youthfulness created a mathematical climate that was
very favorable to yon Mises' activities. [Ostrowski 1966, 106]
O s t r o w s k i ' s statement, however, m a y well have been influenced by his conception
of what " a p p l i e d m a t h e m a t i c s " should be and does not seem to do justice to the
G6ttingen tradition [ I ]. M o r e o v e r , this article aims to show that Ostrowski painted
the mathematical a t m o s p h e r e in Berlin in s o m e w h a t too rosy colors. Applied
m a t h e m a t i c s in Berlin, notwithstanding the foundation of a n " Institute for Applied
M a t h e m a t i c s " (with only two p e r m a n e n t positions), was still in a situation of
struggle for institutional and cognitive independence. Financial conditions were
certainly tighter than in G6ttingen. Perhaps the image Ostrowski painted of applied
m a t h e m a t i c s in the 1920s is also s o m e w h a t distorted, due to the c o m p a r i s o n with
the far w o r s e situation in Nazi G e r m a n y , at least at the University of Berlin
[Siegmund-Schultze 1989].
The c o n t r o v e r s y surrounding Hilda Geiringer's application for '~Habilitation,"
i.e., the highest G e r m a n academic degree, connected with the " v e n i a legendi,"
the permission to teach at universities, reveals some of these problems and hints
at additional ones.
First, the shortage of adequately trained personnel for the new, more sophisticated d e m a n d s of applied mathematics is exemplified in the case of this young
w o m a n mathematician. Geiringer (1893-1973) c a m e from a rather narrow field of
pure mathematics and had to a c c o m m o d a t e herself in a very short period of time
to various fields of applied mathematics, which required versatility in methods.
At the same time she had to cope with a considerable teaching load (the wellk n o w n " P r a k t i k u m " at the Institute of Applied Mathematics), with her duties as
a single m o t h e r of a small child, and with a very exacting, sometimes even rude
teacher and friend, Richard von Mises. Geiringer's extremely self-critical attitude
until the end of her career [ 2 ] - - n o t an u n c o m m o n trait a m o n g mathematicians in
g e n e r a l - - w a s partly a reflection of these considerable burdens. At the same time
Geiringer's p r o b l e m s revealed the still unstable situation of applied m a t h e m a t i c s
within the G e r m a n mathematical culture.
Second, it was the " p u r e m a t h e m a t i c i a n s " at the University of Berlin who were
anxious to maintain the standards of their profession, drawing a borderline around
" a p p l i e d m a t h e m a t i c s " as a special, more restricted field. In the case of Ludwig
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Bieberbach, his "impulsive youthfulness" may have been coupled with additional
ideological motivations that seem to have sharpened his critical attitude toward
Geiringer's work. This was an additional burden for the first (in the end successful)
mathematical "Habilitation" of a woman in Berlin and certainly--after Emmy
Noether's in Grttingen--the second in Germany [Boedeker 1974].
Finally, it is the outstanding personality of Richard von Mises (1883-1953) and
his pioneering efforts for the promotion of applied mathematics as an independent
discipline that provide the background for the following story. While these pioneering efforts have been described in [Bernhardt 1979, 1980], von Mises' influence
on Geiringer's career in the 1920s has not been adequately discussed thus far.
Although von Mises and Geiringer worked for the most part on different topics
and coauthored only two papers [Binder 1992], there is no doubt that Geiringer
remained essentially in the role of von Mises' student until the end of their mathematical and personal relationship. As late as 1953 Geiringer considered the following remark of von Mises as typical of their (one-sided) partnership:
Du bist voreingenommen, mein Kind, meine treueste Bewunderin. [Binder 1992, 43]
It was her love for von Mises at least as much as her love for mathematics that
enabled Hilda Geiringer to surmount most of the problems described above and
to become one of the finest applied mathematicians of this century [3].
It is these three dimensions of Geiringer's career in the 1920s which are of
considerable general historical importance, and the main goal of this paper is to
provide an account of them. Although interesting in its own right, the mathematical
content of Geiringer's two "Habilitationsschriften," especially her contribution
to the theory of Charlier series in statistics, is discussed primarily with respect
to these more general questions.
GEIRINGER'S HABILITATION
On April 22, 1940, the renowned Polish-British-American statistician J. Neyman (1894-1981) responded to an inquiry of Warren Weaver, director of the
Rockefeller Foundation, concerning Hilda Geiringer, who had just sought refuge
in the U.S.:
Whether she is to be considered outstanding in ability or not, depends on the standard of
comparison. Among the present day mathematicians there are few, whose names undoubtedly
will remain in the history of mathematics . . . .
As for the newcomers in this country, I
have not the slightest doubt that von Mises is one of the men of such caliber . . . .
There
will perhaps be a dozen or perhaps a score of such persons all over the w o r l d . . , and Mrs.
Geiringer does not belong in this category.
But it may be reasonable to take another standard, that of an university professor of
probability and statistics, perhaps an author of now numerous books on statistical methods.
In comparison with many of those people Mrs. Geiringer is an outstanding person and I think
it would be in the interest of American science and instruction to keep her in some university.
[BUCB]
It should be noted that Neyman was referring to Geiringer's statistical work
exclusively, whereas her main mathematical achievements seem to belong to the
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theory of plasticity. It was Leopold Schmetterer (born 1919) who, on the occasion
of Geiringers doctorate-jubilee in Vienna 1967, mentioned the "fundamental Geiringer equations" in plasticity [Schmetterer 1967, 6].
In spite of Geiringer's indisputable importance and Neyman's support, she did
not get an adequate position in the United States [Binder 1992]. And just as 15
years earlier it was her devotion to Richard von Mises that influenced the course
of her career, this time, however--under the even more complicated conditions
of emigration--it resulted in serious personal sacrifices. In order to live close to
Cambridge, where von Mises, her husband since 1943, held a chair at Harvard
University, Geiringer accepted an appointment at Wheaton College in Norton
(Massachusetts) where, as she complained, "die Studentinnen definitely more
socially als scientifically minded sind" [Binder 1992, 31]. After von Mises' death
in 1953 she devoted her career almost exclusively to the edition of her husband's
works, especially his "Mathematical Theory of Probability and Statistics" [von
Mises 1964].
This tendency to sacrifice her own career in favor ofvon Mises' may be responsible for the fact that 20 years after Geiringer's death no comprehensive account
of her merits has yet been given [4].
In order to understand Geiringer's extraordinary feelings of indebtedness to
yon Mises it is necessary to go back to the beginnings of Geiringer's relations
with Richard von Mises in the early 1920s.
Hilda Geiringer was born in 1893 in Vienna, and took her doctorate from the
University of Vienna in 1917 with a thesis on Fourier series in two variables. The
reader ("Gutachter") was Wilhelm Wirtinger (1865-1945). Through her schoolmate Gerda Laski, who worked as a physicist in Heinrich Rubens' institute in
Berlin [5], Geiringer obtained an assistantship at von Mises' new institute in 1921.
In her "Mathematische Entwicklung" Geiringer writes:
Mises teilte mir gleich mit, ich k6nne nicht damit rechnen, mich zu habilitieren. Er wt~nschte
dringend, dab ich mich der 'angewandten' Mathematik zuwende, meinte, dann k6nne er mir
wissenschaftliche Anregungen geben. Er begrfindete dort ein gutes Programm der angewandten Mathematik, und ich h6rte alle Vorlesungen von Mises . . . .
Meine Aufgabe war vor
allem, ein 6-semestriges 'Praktikum' einzurichten und zu halten. [ME, 33/34]
Geiringer then refers to her short marriage to the statistician Felix Pollaczek
(1892-1981); her papers betwen 1923 and 1934 appeared under the hyphenated
name Pollaczek-Geiringer. (For the sake of simplicity this article refers to Geiringer
under her maiden name, which she resumed after 1934.) The following quotation
reveals Geiringer's active part in her personal relation with von Mises:
Im Jahre 1921 heiratete ich Felix Pollaczek, ein ausgezeichneter Mathematiker. Ich will von
unserer pers6nlichen Beziehung hier nicht sprechen, well es zu kompliziert ist. Noch 1922,
aber sp~itestens 1923, liel3 ich mich von Felix scheiden, da ich Mises lieber hatte als ihn. Von
1923 an war ich Mises sehr nahe und sowohl Gerda als auch die Wiener Freundin traten in
den Hintergrund. Doch liebte ich ihn sicher mehr als er mich. [ME, 35]
In 1922 Geiringer's daughter Magda was born in Berlin. Geiringer reports on her
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problems in dealing with all these personal and professional demands at the same
time: "Ich war nachher einige Monate in Wien, da das Leben in Berlin unendlich
schwer war. Mama war so giitig, Magda for eine Zeit (wohl nur einige Monate)
zu behalten." [ME, 35]
Geiringer's scientific relations with von Mises were not unproblematic either.
Referring to von Mises' comments on her paper [Pollaczek-Geiringer 1923] Geiringer wrote
Ich hielt diese Arbeit eigentlich immer ffir ganz interessant. Aber sie war es vielleicht nicht.
Irgendwann um diese Zeit sagte Mises, dab ihm scheine, als ob ich nicht imstande sei, mich
in irgendetwas wirklich einzuarbeiten . . . .
Es ist auch zu bemerken, dab Mi[ses] in Frank/
Mises [i.e. [Frank & Mises 1925/1927]] diese Arbeit nicht zitiert, obgleich sie dem Sinn nach
hineingeh6ren wtirde. Entweder hat e r - - d e r mich ja wissenschaftlich am besten kannte,-wirklich wenig von mir gehalten und im speziellen von dieser Arbeit, oder sein Urteil war
getriibt durch unsere pers6nliche Beziehung. [ME, 35]
Although von Mises, as seen above, did not initially encourage Geiringer's
aspirations for Habilitation, his attitude seems to have changed sometime around
1925.
On July 18, 1925, Geiringer applied for Habilitation at the Philosophische Fakult~it of the University of Berlin and submitted a paper on statics, "Ober starre
Gliederungen von Fachwerken." The faculty chose Richard von Mises and Ludwig
Bieberbach (1886-1982) as readers of this "Habilitationsschrift." In his nine-page
review ("Gutachten") of November 16, von Mises made some more general
remarks concerning the critical situation of applied mathematics with respect to
personnel in Germany:
Es ist jedem, der die heutige mathematische Situation in Deutschland tibersieht, bekannt,
wie auBerordentlich gering die Zahl der Gelehrten ist, die auf dem Gebiet der angewandten
Mathematik produktiv tfitig sind. Erledigte LehrstiJhle k6nnen nicht wieder besetzt werden.
Die an kleineren Hochschulen einige Jahrzehnte hindurch bestanden haben, gehen ein . . . .
So w0gte ich unter allen jiingeren Leuten, die in den letzten f0nf Jahren in Deutschland mit
eigenen Arbeiten auf dem Gebiete der angewandten Mathematik hervorgetreten sind, keine
zwei zu nennen, die ich hinsichtlich ihrer Eignung for eine Dozentur in diesem Fache Frau
Dr. Pollaczek zur Seite steUen k6nnte. [BA1, fol.258]
Von Mises stressed the special features of the field of "applied mathematics"
when he remarked that "es sich dem Wesen n a c h . . , um eine Habilitation for
angewandte Mathematik handelt, auch wenn die Fakultfit ihrer Obung gemaB die
venia 'ftir Mathematik' schlechthin bezeichnen sollte." [BA1, fol.257]
Partly a sign of professional self-confidence, this quotation should be understood
mainly as a preventive measure against possible objections on the part of the
" p u r e " mathematicians at Berlin. As a matter of fact, the "venia legendi" for
"mathematics" included "applied mathematics" but the latter title did not qualify
a person to teach courses in pure mathematics [6]. Also, in her "Mathematische
Entwicklung" Geiringer reinforced the commonly held view regarding pure and
applied mathematics in the following self-assessment, in which she referred to
her colleagues at Berlin university: "Die meisten dieser 'reinen' waren wesentlich
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begabter als ich; die 'angewandten' waren mehr von meinem Niveau, obwohl
Collatz und Schulz gr~ndlicher waren." [ME, 34]
Von Mises summarized the mathematical content of Geiringer's Habilitationsschrift in the following words: "Die vorliegende Arbeit gibt zum erstenmal,
u[nd] zw[ar] sowohl for das ebene wie ffir das r~umliche Fachwerk, die zugleich
notwendige und hinreichende Bedingung der Brauchbarkeit an." [BA1, fol. 250]
Geiringer showed in the first part of her paper, which was published later as
[Pollaczek-Geiringer 1927], that the absence of "accumulations" ("Anh~iufungen," that is, superfluous connecting rods) in any partial system of plane
frameworks is a necessary and sufficient condition for a framework with k nodes
( " K n o t e n " ) and (2k-3) rods ("St~ibe") not to be "useless" ("unbrauchbar") solely
due to its structure ("Gliederung"). "Uselessness" of the framework means that
the tension-problem does not have a finite solution; i.e., there exists a finite or
infinitesimal movability of the framework.
Geiringer tried to generalize the easily definable notion of "accumulation" in
plane frameworks (i.e., existence of more than (2k-3) rods in a partial system of
k nodes) to the case of spatial frameworks, introducing the notion of "Quasianh~tufung."
At this point, some problems of historical judgment have to be mentioned. As
a matter of fact the original version of Geiringer's Habilitationsschrift obviously
has not been preserved in [Pollaczek-Geiringer 1927] [7].
There is no doubt, though, that Geiringer's theorem was wrong in the case of
spatial frameworks, and the notion of"Quasianh~ufung" was therefore irrelevant.
Erhard Schmidt (1876-1959), although not appointed as a reader, was also interested in problems of applied mathematics [8] and read Geiringer's Habilitationsschrift. In a tactful manner he informed Geiringer of her mistake: "DAB das
Resultat auch algebraisch nicht trivial ist, sieht man aus dem nicht-Gelten im Raum,
auf das Erhard Schmidt mich an einem unvergef31ich schrecklichen Nachmittag
(obgleich er sich so vornehm und ritterlich wie m6glich benahm und mir . . .
Thee servierte) aufmerksam machte." [ME, 47]
In an additional review of Geiringer's paper Issai Schur (1875-1941) also confirmed the incorrectness of Geiringer's result [BA1, fol.247].
This came as a reaction to Bieberbach's report of March 4, 1926, which was
much more severe than Schmidt's and Schur's criticisms [BA1, fol.261-263].
Bieberbach wrote that he had gotten a "truly shattering impression" ("wahrhaft
niederschmetternden Eindruck") of Geiringer's "purely mathematical abilities
and achievements" ("rein mathematischen F~ihigkeiten und Leistungen"). The
second part of Geiringer's paper was--according to Bieberbach--a "collection of
mistakes" ("Fehlersammlung"). He, therefore, would not approve of Geiringer's
admission to any further stages of the Habilitation procedure as long as the problem
of the exact specification of Geiringer's venia legendi was not yet resolved.
After this Geiringer withdrew the second part of her paper. In a later publication
[Pollaczek-Geiringer 1932] she restricted the discussion to " a wide class of usable
spatial frameworks," namely frameworks with triangles as boundary surfaces.
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For her Habilitation Geiringer decided to submit a new paper on "The Poisson
distribution and the development of arbitrary distributions" half a year later, on
November 16, 1926. This is essentially [Pollaczek-Geiringer 1928a], although the
title as quoted in the Berlin University files suggests [Pollaczek-Geiringer 1928b].
This conclusion follows from the subsequent reviews as well as from Geiringer's
"Mathematische Entwicklung":
Da mir die L6sung der Brauchbarkeitsaufgabe fiir r~iumliche Fachwerke nicht gelungen war,
muBte ich, um mich zu habilitieren, noch eine Arbeit einreichen. Dies war die unter Druck
gemachte Arbeit . . . "Die Charliersche Entwicklung willki~rlicher Verteilungen," mit der
ich mich wieder der Wahrscheinlichkeitsrechnung zuwandte. [ME, 51]
In order to make clear the goals and content of this paper the following remarks
are necessary.
Richard von Mises considered Fechner's "KollektivmaBlehre" as an important
historical root of his own aspirations for an unification of the mathematical theories
of probability and statistics with particular emphasis on the notion of relative
frequency. In his 1931 book on probability and its applications in statistics and
physics von Mises wrote:
Stemming from the needs of practice, from problems of statistics and of the insurance business,
a new theory emerged, apparently alongside the theory of probability, as its empirical counterpart, which Theodor Fechner called "KollektivmaBlehre." Subsequently the astronomer
Heinrich Bruns (1848-1919) tried to unite both theories at least as teaching subjects. [Von
Mises 1931, 4]
Within the chapter "Beschreibende Statistik" ("Descriptive Statistics"), which
constitutes, according to von Mises' book, "in some sense a preparatory chapter
to 'theoretical statistics,' which is based on the theory of probability" [Von Mises
1931, 233], von Mises also discussed the so-called "Bruns' series". These are
expansions introduced by Bruns in 1906 " o f the 'Summenfunktion' [i.e., the
distribution function] in an infinite series analogous to the Fourier expansion of
an arbitrary function in certain fundamental functions" [Von Mises 1931,250], the
coefficients being certain linear functions of the moments of the given distribution.
Inspired by Bessel, Bruns chose as fundamental functions the integral (distribution function) of the Gaussian normal density function ~(x) = (1/V~-~)e -x2/z and
the integrals of its successive derivatives. Bruns' expansions applied to continuous
distributions and to discrete distributions with a finite number of values (i.e.,
"frequency distributions"). Geiringer considered the use of ~o(x) as a comparative
function for "arbitrary" distributions of this kind to be the "main idea of KollektivmaBlehre" [Pollaczek-Geiringer 1928b, 301]. In his book von Mises recommended
the so-called "Stetigkeitssatz des Momentenproblems" of G. Polya (1887-1985),
dating from 1920 [Polya 1920], as a "deep mathematical theorem" for proving
expansions of this kind since it allows for a "complete characterization of a
distribution by its moments" [Von Mises 1931, 249].
Geiringer's paper on Charlier series [Pollaczek-Geiringer 1928a] is to be judged
against this background. In a footnote she acknowledges her indebtedness to von
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Mises, who called her attention to the papers of the Swedish astronomer Carl
Ludwig Wilhelm Charlier (1862-1934) and to Polya's theorem as a method of
proof [Pollaczek-Geiringer 1928a, 98]. The expansions proposed (but not proven)
by Charlier in 1905, which were expansions of discrete distributions with an infinite
n u m b e r of values into series formed by successive derivatives with respect to
the p a r a m e t e r of the Poisson-distribution, were a counterpart to Bruns' series.
Charlier's expansions are sometimes called B-series and those of Bruns A-series.
(The notion of an A-series sometimes also includes expansions on the level of the
density functions, in the case of continuous distributions.) Von Mises saw the
justification for Charlier's approach in the fact that both the Poisson distribution
and the normal distribution are limits of the binomial distribution [Von Mises
1931, 265].
Geiringer's interest in Charlier series and in the question, still unsettled in 1926,
of expandability conditions for arbitrary discrete distributions (on the domain of
nonnegative integers 0, 1, 2 . . . . ) may have been stimulated by the topic of her
Vienna dissertation of 1917 on "Trigonometrische Doppelreihen." As a matter
of fact, the successive derivatives of the Poisson distribution,
So(x)
= (fir~x!)e -a
(X = O, 1, 2 . . . ) ,
constitute an orthogonal system of functions; they are obtained by multiplication
by the likewise orthogonal "Poisson-Charlier polynomials," which are still important in current probability theory [Pollaczek-Geiringer 1928b, 302]:
d"
Sn(x) = pn(x) . So(x) = da----;S°(x)
~c
S , , ( x ) S , ( x ) (1/S0(x)) = 0
(m # n)
x=0
= n!/a ~
( m = n).
The Poisson-Charlier polynomials are closely related to the Laguerre polynomials.
Gabor Szeg6 (1895-1985), who did fundamental work on orthogonal polynomials
during his time as a Privatdozent in Berlin (1922-1926), found a condition in
1926 for the expandability of distributions in Charlier series, based on Hilbert's
"Methode der unendlichvielen Variablen" [9]. Because Szeg6's condition was
less restrictive and more lucid than Geiringer's (cf. below [13]), her result became
obsolete practically the moment it was published, and Geiringer referred to Szeg6's
condition with his permission [Pollaczek-Geiringer 1928a, 110]. Von Mises, in his
book of 1931, did not even mention Geiringer's condition. Nevertheless, Geiringer's paper inspired Erhard Schmidt to undertake an investigation in 1928 of function-theoretic methods to determine necessary and sufficient conditions for the
convergence of Charlier series [Schmidt 1928, 1933]. Geiringer referred to this
fact in her "Mathematische Entwicklung" as follows:
Dann kam noch ein interessantes Nachspiel. Der grol3e Erhard Schmidt hatte seit Jahrzehnten
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(Tod seiner Frau) nicht ver6ffentlicht [10], obwohl er sich im Kolloquium etc. als unendlich
scharfsinnig zeigte. Da er nun 2. Referent meiner Arbeit war [he was, in fact, an additional
reader], begann das Problem ihn zu interessieren under fand die notwendigen und hinreichenden Bedingungen ffir die Konvergenz, was nat/3rlich groBes Aufsehen machte. [ME, 52]
With Schmidt's result the theoretical problem--at least with respect to ordinary
convergence [ l l ] - - o f the convergence of Charlier series was settled. As to the
applicability of Charlier's theorem within mathematical statistics, authors such
as [Boas 1949a,b], [Cram6r 1972], and [Kendall & Stuart 1963] stress the importance of the question of whether afinite number of terms in Charlier's expansion
gives a sufficient approximation to the distribution. This is all the more important,
since in general, "we cannot discuss the question of convergence or divergence
without supposing that all moments have known finite values" [Cram6r 1972,
206].
Geiringer's paper supplies no means for deciding this question of finite approximation. In general the B-series, sometimes also named after J. P. Gram (1850-1916)
and Charlier, have proved to be of rather limited use in statistics [Kendall/Stuart
1963, 163].
While Geiringer's publication [ 1928a]--in contrast to some of her other contributions to probability theory and statistics [12]--had only a limited impact on mathematics, her application for Habilitation in 1926 was adversely affected by another
circumstance: the incorrectness of her method of proof.
As reported above in her own words, Geiringer wrote the second part of her
Habilitationsschrift "under pressure." Using Polya's theorem, "one of the few
theorems I knew," she did not realize the restrictions of its applicability:
Die Arbeit hat eine peinliche Geschichte. Ich wollte den "Stetigkeitssatz des Momentenproblems" benutzen, weil das einer der wenigen S~itze war, die ich kannte. Beim Beweis machte
ich abet einen Fehler beziJglich Konvergenz einer Hilfsreihe--ich weiB nicht einmal mehr
was es w a r - - u n d es wurde von mir oder Mises oder E. Schmidt bemerkt, was schon wirklich
ein Skandal war nach dem (immerhin noch ehrenvollen) d6bacle mit den Raumfachwerken.
[ME, 52]
To give an idea of this mistake, some remarks are necessary concerning the
decisive method of proof of Polya's theorem on moments [Polya 1920]. This
theorem requires that the moments
MIt) = fo xtG(x)dx
(t = O, 1,2
)
of a given nonnegative function G(x), defined for all nonnegative real x, satisfy
the condition
< K,
if t is large enough.
(*)
Then the convergence of the moments iv,
n , which are defined for a sequence of
,~lt)
nonnegative functions Gn, to the moments M t'~ of G, i.e., the condition that
HM 20
373
lim .._,M
I'l = lim ~ xlt)Gn(x) dx = M ('~,
(**)
~0
allows Polya to demonstrate the uniform convergence of the indefinite integrals
of G, to the indefinite integral of G, provided (an additional problem, which
Geiringer did not discuss) the improper integrals M~'~ exist, that is,
lira fX: G, '(x) dx = f x2 G(x) dx.
n --~3¢
XI
X[
In order to apply Polya's theorem to discrete distributions, Geiringer had to
detour around t h e " s u m m e d - u p " density functions, i.e., the distribution functions.
Due to the special character of the "derivatives" of the Poisson distribution she
obtained very simple expressions for the moments of the differences. S,(x) 6(x) and V(x) - 6(x), where S~, V, and 6 denote the distribution functions of,
respectively, the partial sum s,, of the Charlier series, the given discrete distribution
V, and the Poisson distribution 60 [Pollaczek-Geiringer 1928a, 106]. Geiringer also
showed that the moments of these difference-functions fulfill Polya's conditions
(*) and (**).
Geiringer missed the point, however, that Polya's theorem requires all involved
functions to be nonnegative, which is generally not the case for these differencefunctions.
While von Mises did not notice this mistake, as is clear from his report of
November 12, 1926 [BA1, fol. 259/60], it did not escape the attention of the second
reader, Ludwig Bieberbach. Again casting serious doubts on the mathematical
abilities of the candidate, Bieberbach, in his undated review, revealed his rather
condescending views toward the field of "applied mathematics":
Soweit aber die Zulassung zu einem etwa ad hoc neu zu schaffenden Fach "Anwendungsgebiete der Mathematik" in Frage kommt, m6chte ich dem verantwortlichen Urteil des Herrn
ersten Referenten kein Gegenvotum entgegenstellen, zumalja ffir die Beurteilung des Wertes
einer Leistung im Rahmen eines Anwendungsgebietes noch andere als mathematische F~.higkeiten in Frage kommen, die sehr wohl M~ingel in mathematischer Hinsicht ausgleicben
k6nnen, F~higkeiten, fiber deren Vorhandensein ich ein eigenes Urteil nicht besitze. [BAI,
fol. 266]
The repeated mistakes of Geiringer resulted in the faculty's decision to solicit
two additional reports from Issai Schur and Erhard Schmidt. Both readers basically
supported Bieberbach's position, though without displaying a similar rudeness in
their choice of words, and favored the creation of a special field for Habilitation
called "Applied Mathematics." The final decision to grant Geiringer the Habilitation for this field became possible because of an addendum submitted by the
candidate [13]. Geiringer, in her "Mathematische Entwicklung," refers to her
indebtedness to Mises for finding this mathematical condition which saved the
situation. At the same time she shows her awareness of the fact that her original
mistake restricted the field of Habilitation:
Der Fehler wurde durch fieberhafte Arbeit von Mises in Ordnung gebracht und die Arbeit
374
HM 20
pr~isentiert und angenommen (in einer sehr angesehenen Zeitschrift, der Skandinavisk Aktuarietidskrift (1928). Damals wurden Arbeiten, die von einer guten Schule kamen, nicht vom
Herausgeber der Zs. beurteilt), und meine Habilitation ging durch, aber nur for 'Angewandte
Mathematik'. [ME, 52]
IDEOLOGICAL AND DISCIPLINARY ISSUES S U R R O U N D I N G
GEIRINGER'S HABILITATION
The question should be raised why Bieberbach showed such strongly antagonistic feelings throughout Geiringer's Habilitation procedure [14].
A particularly striking feature in Geiringer's memoir "Mathematische Entwicklung" [ME] is the fact that she never mentions Bieberbach, not even in
connection with her Habilitation, although he was certainly the one who was
primarily responsible for her troubles. Her assumption, quoted above, that the
mistake was found " b y myself, or Mises or Schmidt," is clearly erroneous, and
yet it is impossible that she was unaware of Bieberbach's role, even if she never
read his scathing reports. Moreover, the fact that Szeg6's condition [9] was first
mentioned in Bieberbach's report leads to the conclusion that Geiringer suppressed
in her "Mathematische Entwicklung" information about her relations with Bieberbach, perhaps because she simply found them too difficult to describe. A partial
explanation for this may be that she, as a Jewish emigr6e, wanted to exclude
Bieberbach from her memoir because of his role as the leading Nazi among German
mathematicians after 1933 [ 15]. Still, this explanation is not completely satisfactory
in view of Geiringer's close mathematical collaboration with another former Nazi,
Erhard Tornier (1894-1982), after World War II [16]. In any case, one cannot
ignore the fact that precisely during the time of Geiringer's Habilitation there must
have been contacts between Geiringer and Bieberbach on quite another level.
In 1926 the Teubner publishing house (Leipzig and Berlin) announced in a flyer
that Hilda Pollaczek-Geiringer's translation of "L'id6al scientifique des math6maticiens" of Pierre Boutroux (1880-1922) was "in press" ("unter der Presse").
When this translation appeared the following year in the series "Wissenschaft
und Hypothese," Geiringer remarked in the preface: "I undertook this translation
of the 1920 original at the suggestion of Professor Dr. Bieberbach, who also kindly
examined the manuscript." [Boutroux 1927]
Bieberbach's lively interest in Boutroux's book has been documented already
in [Mehrtens 1987, 206]. On February 15, 1926, Bieberbach addressed the Verein
zur F6rderung des mathematischen und naturwissenschaftlichen Unterrichts
(F6rderverein) with a lecture, entitled "Vom Wissenschaftsideal der Mathematiker." In this talk Bieberbach emphasized "intuition" as a decisive source of
mathematical thinking. While inspired by Boutroux in this respect, Bieberbach
went considerably further, alluding directly to Brouwer's philosophy of mathematics [17].
How should this scarcely documented [18] "collaboration" between Geiringer
and Bieberbach at the very time of her Habilitation be understood? I shall try to
give a tentative explanation. Certain facts seem to indicate that the origin of this
H M 20
HILDA GEIRINGER-VON
MISES
375
"collaboration" was an ideological discussion between Bieberbach and Geiringer
which was in some sense connected with Geiringer's application for Habilitation.
Against this background Geiringer's translation and her grateful acknowledgment
of Bieberbach's role in the preface to the book could be interpreted as an attempt
to reconcile the conflicts that arose during her Habilitation procedure.
As a matter of fact, in his review of Geiringer's first Habilitationsschrift, Richard
von Mises alluded to some "pedagogical and popular writings" of the candidate,
which, in von Mises' opinion, "included some immature judgments which can
only be understood as a result of the intellectual situation immediately after the
war." [BA1, fol.257]
That von Mises alluded to these papers of Geiringer's at all suggests that he
was preparing a defence strategy for his candidate, against the possibility that
some colleagues, especially Bieberbach, might react adversely to this kind of
political and philosophical writing [19]. That there were real reasons to fear such
troubles can be seen clearly from negotiations surrounding the appointment of
Hans Reichenbach (1891-1953) at the University of Berlin, which were taking place
at the same time as Geiringer's Habilitation. In the course of these negotiations
Bieberbach remarked on January 15, 1926--that is, two months before his first
report on Geiringer's Habilitationsschrift and one month before his talk at the
F6rderverein--with respect to Reichenbach's booklet"Student und Sozialismus"
of 1920 that "it lacks the objectivity and the appropriate tone to be expected of
a scholar." [Hecht/Hoffmann 1982, 654]
What was the " t o n e " of Geiringer's writing immediately after World War I and
the November revolution?
In 1922 Geiringer's booklet "The World of Mathematical Ideas" [Geiringer
1922b] appeared in a series which was largely inspired by the developing movement
of adult evening classes ("Volkshochschulwesen") after the war. In her book
Geiringer, who once attended Freud's lectures in Vienna [Geiringer 1967, III],
strongly emphasized the standpoints of psychoanalysis and of Mach's empiricism.
Von Mises, a noted Mach specialist, reviewed Geiringer's book very favorably
in his journal Zeitschrift fiir Angewandte Mathematik und Mechanik (ZAMM ),
recommending it as "worthy of being distributed, and above all, being read" [Von
Mises 1922]. As a matter of fact, this booklet of Geiringer's seems to have been
a kind of icebreaker in their personal relations: " A m Bemerkenswertesten finde
ich, dab Mises, den ich damals noch nicht gut kannte (Aber ich war 1922 schon
bei ihm in Berlin) und der ein sehr strenger Kritiker war, darOber eine sehr gute
Kritik schrieb." [ME, 27]
Geiringer's book includes passages such as the following, which may have
provoked Bieberbach's interest, in the event--indubitable to this author--that he
read the book:
Without any doubt the beginnings as well as the further development of mathematics were
always influenced by both driving forces, a biological--economical ("external") and a psychological ("internal"). It is still a question whether psychoanalytical investigations can shed
light on those internal forces and may thus explain the almost mystic enchantment which
lies in just those mathematical problems most remote from reality. [Geiringer 1922b, 160]
376
HM 20
Curiously enough, the very fact that Geiringer and Bieberbach shared a common
interest in disclosing the psychological-internal dimension of the history of mathematics may well have been a source of tension between these two Berlin mathematicians. In fact, Mach's philosophy had but limited influence on German scientists
[Holton 1992, 49], and Bieberbach, unlike von Mises, had no connections with
the Berlin circle of empiricist philosophy. Also Freud, Geiringer's other main
source, probably did not appeal to Bieberbach, who turned toward a different
kind of psychological theory, the typology of E. R. Jaensch, toward the end of
the 1920s [Mehrtens 1987]. Finally, Bieberbach was very touchy and at times selfrighteous [Biermann 1988, 220]. It may have been in the context of a discussion
of Geiringer's booklet that Bieberbach drew her attention to Boutroux's book,
which she did not cite in her publication of 1922. Instead, Geiringer quoted A.
Bogdanov's "Science and the Working Class," commenting on it in the following
words: "The author develops some interesting ideas concerning the origins and
goals of science, basing his discussion on Marx' theory and on the ideology of
class struggle." [Geiringer 1922b, 196]
In spite of a considerable revision of previous positions, Geiringer maintained
some strong views from a 1920 speech on "Reflections on the Teaching Method
at Adult Evening Classes." In this talk Geiringer called "our entire science alien
to the p e o p l e . . , the class-bound product of a tiny m i n o r i t y . . , full of pseudoknowledge" [Geiringer 1920, 100], and she called for a "real socialization of the
mind."
Passages like this no doubt had the potential to antagonize scholars such as
Bieberbach, who, for all his sometimes unconventional behaviour, remained at
that time in the spirit of the German "Bildungsbtirgertum." It is also clear that
at least Geiringer's earlier paper of 1920 expressed a mood of departure from the
present conditions and a longing for another world, a sentiment very similar to
that expressed in Reichenbach's "Student und Sozialismus" of the same year
[20].
In the above mentioned negotiations on Reichenbach's appointment, Bieberbach--partly supported by a review by Hermann Weyl--tried to dismiss Reichenbach's epistemological papers as superficial ("Arbeiten eines Popularphilosophen" [BA2, fol.327]) as well.
Therefore the conjecture may be allowed that in Geiringer's case it was Bieberbach who saw a connection between the political and scientific sides of her personality, and this may partly explain his extreme reactions throughout Geiringer's
Habilitation procedure. For lack of documentary evidence it remains an open
question to what extent additional ideological factors, anti-Semitism or sexism,
were possibly involved.
Before Geiringer obtained the Habilitation, some additional quarrels between
von Mises and "pure" mathematicians in Berlin seem to have taken place [Biermann 1988]. Von Mises, of course, must have felt humiliated by Bieberbach's
reports, which at the very least showed that von Mises had not read Geiringer's
papers carefully enough. Occasional "blunders" in his own writings, to which
H M 20
377
Ostrowski referred in his talk, may have been another reason for the fact that
von Mises was never proposed by any of his three prominent colleagues as a
member of the Prussian Academy of Sciences [Biermann 1988, 201]. Although
there seems to have been no open confrontation [19], the somewhat different
interests and values of the fields of pure and applied mathematics at least remained
a source of dispute.
Von Mises' statement on November 25, 1926, concerning the Habilitation of
Georg Feigl (1890-1945), who had been substituting for Erhard Schmidt by giving
introductory mathematical lectures for several years, cannot be understood except
against the background of the Geiringer controversy at the same time. Von Mises
wrote:
Neither the reviews of the Habilitationsschrift nor my personal acquaintance with the candidate allow the conclusion that he has achieved "outstanding" contributions according to our
stipulations. Opinions of other mathematicians whom I have asked privately for information
conform that the candidate's papers hardly reach the average of current achievements in this
field. Nevertheless I do not want to oppose the unanimous vote of the three representatives
of pure mathematics, because in my opinion it is the specialists who are the most responsible
for a Habilitation in their particular field. [Biermann 1988, 206]
It cannot be denied that von Mises' judgment with respect to Feigl was correct
[21]. Moreover, von Mises' statement points to a real conflict of interests between
two fields of teaching and research. Both Feigl and Geiringer were important
figures in the teaching of pure and applied mathematics, respectively. Due in part
to an overburden of teaching duties, their research had some defects. It is therefore
understandable that the representatives of pure mathematics in their reviews of
both Geiringer's and Feigl's Habilitationsschriften stressed the strength of the
methods they employed and did not go into the importance of the results. Thus,
for instance, Erhard Schmidt emphasized Feigl's systematic proofs of certain
"famous fundamental theorems of topology," which "generally are considered
true although their proofs have not been checked so far" [BA1, fol.66]. Von
Mises, on the other hand, rightly stressed the competence of the "specialists"
("engste Fachvertreter").
It was only on November 11, 1927, that the "venia legendi" was officially
bestowed on Hilda Geiringer. After the Berlin mathematicians had settled the
question of the proper delimitation of the fields in which she was to be granted
the permission to teach, the remaining parts of the Habilitation procedure were
merely formal:
Nun jedenfalls wurde ich zum mtindlichen Kolloquium zugelassen (ein Parterre von K6nigen,
auger den Mathematikern die weltberiihmten Physiker Planck, Laue . . . . ) was eine Formalitat war, und so wurde ich 1927 Privatdozentin for A[ngewandte] M[athematik]. Dies hat
mir noch nach Jahrzehnten entscheidend gen[itzt bei der Zuerkennung meiner ProfessorenPension. [ME, 52]
Richard von Mises, on the other hand, may well have viewed this recognition of
"applied mathematics" as a separate field of instruction with mixed feelings.
378
REINHARD
SIEGMUND-SCHULTZE
H M 20
ACKNOWLEDGMENTS
This paper is a revised and considerably extended version of a talk given at the Oberwolfach meeting
of historians of mathematics in 1988. I postponed publication because I hoped to find new archival
evidence for the main theses of that talk, in particular drafts of Geiringer's "Habilitationsschriften."
Although these hopes were not fulfilled, I did find a handwritten "Mathematische Entwicklung"
(German, 71 pp., around 1970) in the possession of Geiringer's daughter Magda Tisza in Chestnut
Hill, near Boston. This is a remarkable document, which although concerned primarily with Geiringer's
own mathematical career, sheds considerable light on the development of several fields of applied
mathematics in this century and particularly on the contribution of Richard von Mises. The information
in this document concerning Geiringer's personal and mathematical relations in 1920s Berlin with her
future husband, von Mises, gave the original paper an additional dimension. In order to preserve the
original form of the documents, I quote all unpublished sources without translation. All translations
of quotations, which had been published before, are mine.
I am indebted to Kurt-R. Biermann (Berlin), who gave me the inspiration to write this paper. I
gratefully acknowledge the courtesy of Mrs. Magda Tisza (Chestnut Hill), who provided access to
some up to now unknown papers of her mother, especially the "Mathematische Entwicklung." Cathryn
L. Carson (Harvard) and David E. Rowe (Mainz) kindly corrected my English. To the Archives of
the Berlin Humboldt-Universit~it and the Manuscript Division of Bancroft Library (Berkeley) go thanks
for permission to quote from the documents I cite in this paper. For several comments or advice I
am grateful to the late Hans Freudenthal (Utrecht), to Hilmar Grimm (Jena), and to William H. Kruskal
(Chicago). I also thank the old (pre-1991) Humboldt-Universit~it (East Berlin) and the Alexander-vonHumboldt-Stiftung (Bonn), which--at different times--supported this research. I am indebted to the
inspiring working atmosphere at the Harvard History of Science Department, where I finished this
paper as a fellow of the Humboldt Foundation in 1992.
NOTES
1. Ostrowski's judgment of yon Mises' theory of probability can only be understood as strong,
one-sided support for Kolmogorov's measure-theoretic approach. The allusion to the " t e n s e "
("verkrampft") atmosphere in G6ttingen seems to refer to the competitiveness there and the
fact that G/~ttingen was ahead of Berlin with respect to some aspects of the modernization of mathematics.
2. A typical remark by her in this respect alludes to [Geiringer 1922a], which was published under
the influence of L. Lichtenstein (1878-1933) during her time as his assistant at the "Jahrbuch fiber
die Fortschritte der Mathematik": Von der Arbeit gilt m.E. wie yon vielen meiner Arbeiten, dab sie
besser ist als ich, d.h. als mein Verst~indnis. Da ich aber andererseits immer viel weniger " b e k a m , "
als was mir gebfihrte, so ist auch wieder eine Art Gerechtigkeit hergestellt." [ME, 32] As late as 1967,
on the occasion of her doctoral jubilee in Vienna, Geiringer commented on her achievements: " E s
waren vielleicht zu viele Gebiete und in keinem tief genug." [Geiringer 1967, Ill]
3. The study of her private utterances, especially her "'Mathematische Entwicklung" [ME], led me
inevitably to this conclusion; cf. the acknowledgments above.
4. [Binder 1992] and [Richards 1989].
5. Geiringer alludes in ME to her contacts in Berlin with some other Austrians, among them Lise
Meitner: "Dort waren schon viele Wiener, vor allem am KWI in Dahlem. Die bedeutendste was Lise
Meitner, die Gerda und mir sehr nahe stand." [ME, 33]
6. In 1928 von Mises, perhaps still partly under the influence of the quarrels surrounding Geiringer's
Habilitation, insisted on restricting the venia of Robert Remak (1888-about 1944) to pure mathematics.
Von Mises' objection, however, was not upheld, and Remak went on, in fact, to teach courses on
actuarial mathematics [Biermann 1988, 210].
H M 20
HILDA GEIRINGER-VON
MISES
379
7. Neither in the Berlin University Archives, nor in Geiringer's partial estates at Harvard and in
the possession of her daughter Magda Tisza, could either of the two parts of her Habilitationsschrift
be found. As for the latter two locations, this is hardly surprising since academic theses at that
time were generally still submitted as handwritten manuscripts leaving the applicant without any
copies.
8. Schmidt's crucial role in the process of founding von Mises' institute, beginning in 1918, is
reported in [Biermann 1988, 186].
9. Szeg6's condition is
v2(x)/%(x) = e a ~ v2(x) x,/a x,
x=0
x-0
00(x) being the Poisson distribution, a its parameter, and v(x) the given distribution.
10. This is a slight exaggeration, although, in fact, Schmidt's most important papers on integral
equations dated back to 1905-1908. Between the death of his wife in 1918 and [Schmidt 1933], be
published only six short mathematical papers.
ll. Quadratic convergence was investigated by [Boas 1949a,b].
12. In [Pollaczek-Geiringer 1928b] she introduced the notion of a "zusammengesetzte PoissonVerteilung." [Schmetterer 1967, 4] emphasizes Geiringer's contribution to mathematical genetics.
13. This addendum was a rather cumbersome additional restriction for the moments of the given
distribution,
I~l
< a l n t',
e
with t' In t' = t, if t is large enough. This enables one to construct a nonnegative bound function for
the difference functions, making the application of Polya's theorem possible. Mises and Geiringer's
condition is stronger than Szegr's [9], as conceded by Geiringer in her publication.
14. It is the following somewhat speculative remarks of my 1988 talk concerning the political and
ideological background that I could not confirm by additional documentary evidence.
15. Mrs. Magda Tisza, Hilda Geiringer's daughter, informed me that Bieberbach was always considered a friend before 1933. His sudden turn to national socialism in 1933 came to Hilda Geiringer and
Richard von Mises, as well as to most of their colleagues, as a total surprise. Some irony but also
bureaucratic thoughtlessness can be seen in the fact that Geiringer had to rely on Bieberbach's
testimony after the war to obtain a German pension to which her Habilitation entitled her (1957). In
response to previous testimony of Bieberbach's, which led to a pension for Geiringer as von Mises'
widow, she wrote him a letter, dated July 13, 1955, in which she stated: "'Ich weiB, dab Sie meinem
Manne freundschaftlich zugetan waren, und dab Sie es darum gerne taten." I thank Mr. Ulrich
Bieberbach (Oberaudorf) for providing this information.
16. Geiringer acknowledged Tornier's crucial part in preparing the edition of [Von Mises 1964] on
page vii.
17. Also, it would do injustice to the fine book of Boutroux to hold it responsible for Bieberbach's
later racist aberrations.
18. In her "Mathematische Entwicklung" Geiringer, again, does not mention Bieberbach when she
refers to her translation of Boutroux's book.
19. Von Mises' political position in the 1920s is in itself far from evident, and probably changed
during the course of the decade. Von Mises sided with Bieberbach, Brouwer, and Erhard Schmidt in a
nationalist campaign against the participation of German mathematicians in the 1928 Bologna congress,
although participation was endorsed by Hilbert and the GSttingen mathematicians (cf. [Dalen 1990]
380
REINHARD
SIEGMUND-SCHULTZE
H M 20
and [Mehrtens 1987]). On the other hand, in his philosophical positions von Mises seems to have
differed considerably from the mainstream of German scientists.
20. Substantial personal contacts between Reichenbach and Geiringer at this time are unlikely,
however, according to Geiringer's daughter Magda Tisza. There is no correspondence between the
two, either in Geiringer's papers (Harvard and private) or in Reichenbach's (at the University of
Pittsburgh).
21. In a letter to the author, Feigl's friend and colleague in the 1920s, Hans Freudenthal, wrote
(September 30, 1984): "Ich betrachte es als ein Ehrenblatt fiir das Gremium der Berliner Mathematiker,
dab man ihn ausschlief~lich auf Grund seiner didaktischen Qualitaten zur Habilitation zulieB. In G/)ttingen w~ire das undenkbar gewesen."
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1928b. 1]ber die Poissonsche Verteilung und die Entwicklung willkiirlicher Verteilungen.
Zeitschrifi fiir Angewandte Mathematik und Mechanik 8, 292-309.
- 1932. Zur Gliederungstheorie r~iumlicher Fachwerke. Zeitschriftfiir Angewandte Mathematik
und Mechanik 12, 369-376.
Polya, G. 1920. 0 b e r den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung. Mathematische
Zeitschrift 8, 171-181.
Richards, J. L. 1989. Hilda Geiringer von Mises (1893-1973). In Women of Mathematics. A Biobibliographic Sourcebook, L. S. Grinstein & P. J. Campbell, Eds., pp. 41-46. New York/Westport/
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Schmetterer, L. 1967. Congratulatory address to Hilda Geiringer on the occasion of her doctorate-jubilee, 8 pp. Vienna: mimeographed. [In the possession of Mrs. Magda Tisza, Chestnut
Hill]
Schmidt, E. 1928. l]ber die Charliersche Entwicklung einer "arithmetischen Verteilung" nach den
sukzessiven Differenzen der Poissonschen asymptotischen Darstellungsfunktion fiir seltene Ereignisse. Sitzungsberichte der PreuJ3ischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, 148.
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How Probabilities Came to Be Objective and Subjective
LORRAINE DASTON*
Department of History, Unioersity of Chicago, 1126 E. 59th Street, Chicago, Illinois 60637
FOR DIRK STRUIK ON HIS 100TH BIRTHDAY
Between 1837 and 1842 at least six mathematicians and philosophers, writing in French,
English, and German, and working independently of one another, introduced distinctions
between two kinds of probability. Although the grounds, contents, and implications of these
distinctions differed significantly from author to author, all revolved around a philosophical
distinction between "objective" and "subjective" which had emerged ca. 1840. It was this
new philosophical distinction which permitted the revisionist probabilists to conceive of the
possibility of "objective probabilities," which would have been an oxymoron for classical
probabilists such as Jakob Bernoulli and Pierre Simon Laplace. Without relinquishing the
rigid determinism of the classical probabilists, the revisionists were nonetheless able to grant
chance an objective status in the world by opposing it to the subjective variability of the
mind. © 1994AcademicPress, Inc.
Zwischen 1837 und 1842 haben mindestens sechs Mathematiker and Philosophen, die auf
Franzrsisch, Englisch und Deutsch geschrieben haben und die unabh/ingig von einander
gearbeitet haben, Unterscheidungen zwischen zweierlei Arten von Wahrscheinlichkeiten
eingefiihrt. Obwohl die Begriindungen, Inhalte und Implikationen dieser Unterscheidungen
sehr verschiedend unter den Autoren waren, waren alle auf eine philosophische Unterscheidung zwischen "objektive" und "subjektiv," die rum 1840 erschien, zentriert. Diese philosophische Unterscheidung hat es ermfglicht, dab die revisionistische Probabilisten die M~3glichkeit von "objektiven Wahrscheinlichkeiten" zulassen konnten--eine Mrglichkeit die
for die klassische Probabilisten wie Jakob Bernoulli und Pierre Simon Laplace ein Widerspruch an sich gewesen w~ire. Ohne den strengen Determinismus der klassischen Probabilisten aufzugeben waren die revisionistischen Probabilisten bereit Zufall einen objektiven Status
in der Welt zu verleihen, als Gegengewicht der subjektiven Variabilit~it des Geistes. © 1994
AcademicPress. Inc.
Entre 1837 et 1842 au moins six mathrmaticiens et philosophes, 6crivant en fran~ais,
anglais, et allemand, et travaillant indrpendamment l'un de l'autre, ont introduit distinctions
entre deux esp~:ces de probabilitr. Quoique les raisons, qualitrs et implications de ces
distinctions se different beaucoup entre les auteurs, elles tournaient toutes autour d'une
distinction philosophique entre "objectif" et "subjectif" qui apparut ca. 1840. C'rtait cette
nouvelle distinction philosophique qui permit les probabilistes revisionnistes de concevoir
de la possibilit6 des "probabilitrs objectives," ce qui aurait 6t~ une alliance de mots pour
les probabilistes classiques comme Jacques Bernoulli et Pierre Simon Laplace. Sans abandonner le drterminisme strict des probabilistes classiques, les revisionnistes pourraient
* I thank Berna Eden for drawing my attention to the diversity of the frequentist tradition in
probability theory, and Joan Richards for pointing out the difficulties of a straightforward empiricist
reading of Robert Leslie Ellis's works on the foundations of probability theory.
330
0315-0860/94 $6.00
Copyright© 1994by AcademicPress, Inc.
All fightsof reproductionin any form reserved.
H M 21
OBJECTIVE AND SUBJECTIVE PROBABILITIES
331
donner au hasard une condition objective dans le monde, en l'opposant ~ la variabilit6
subjective de l'esprit. © 1994AcademicPress, Inc.
MSC 1991 subject classifications: 01A55, 60-03
KEY WORDS: Probability, Objectivity, Subjectivity, Poisson, Bolzano, Ellis, Fries, Mill, Cournot.
INTRODUCTION
In 1843 the French mathematician, philosopher, and economist A. A. Cournot
wrote o f " t h e double sense of probability, which at once refers to a certain measure
of our knowledge, and also to a measure of the possibility of things [possibilit~
des choses] independently of the knowledge we have of them." He christened
these two distinct senses "with the epithets of subjective and objective, which
were necessary in order for me to distinguish radically [between] the two meanings
of the term probability ''~ [9, 4-5]. Between 1837 and 1843 at least six authors-Simron-Denis Poisson, Bernard Bolzano, Robert Leslie Ellis, Jakob Friedrich
Fries, John Stuart Mill, and Cournot--approaching the topic as mathematicians
and philosophers, writing in French, German, and English, and apparently working
independently, made similar distinctions between the probabilities of things and
the probabilities of our beliefs about things. Some, though not all, attached the
terms "objective" and "subjective" to the two kinds of probability they were at
pains to pull apart. All warned against the dangers of ignoring such distinctions
in the application of mathematical probability, but they did not all agree on exactly
where the dangers lay--the probability of judgments? the probability of causes?
the Law of Large Numbers? Nor did they agree on exactly how to draw the
distinction between the two kinds of probability, or even to what kind of events
or entities probabilities could properly be said to apply.
In this essay I examine not the whys but the hows of this explosion of concern
among probabilists ca. 1840. 2 That is, I will be concerned not with why ca. 1840
so many authors became simultaneously exercised about what mathematical probability could and could not mean, 3 but rather with how they went about making
such distinctions. More specifically, I will be concerned with how such diverse
arguments and examples could eventually, by the late 19th century, converge
upon the same terminology of "objective" and "subjective" probabilities. What
i Unless otherwise noted, all translations are my own.
2 Although Jakob Friedrich Fries's Versuch einer Kritik der Principien der Wahrscheinlichkeitsrechnung (1842) certainly belongs to this cluster of works, I do not discuss it in this paper. Fries's account
of probability theory is embedded in an elaborate Kantian framework and must be set in the context
of his other, extensive writings on the philosophy of the natural sciences and applied mathematics in
order to appreciate its relation to his understanding of the meaning of objective and subjective. This
is an undertaking which would require at least a paper in its own right. For a summary of those aspects
relating to statistical regularities, see [37, 85-86].
3 There already exists an historical literature on the " w h y " problem: [37, 77-86] on the influence
of social statistics and criticisms of the probability of causes, [12, 210-224, 370-386] on the influence
of the decline of associationist psychology and criticism of the probability of judgments, and [25,
95-98] on the influence of Joseph Fourier's Recherches statistiques sur la ville de Paris (1829) on
Poisson in particular.
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LORRAINE DASTON
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had these terms come to mean in the early decades of the 19th century, and
why were they so eminently available to probabilists trained in such diverse
mathematical and philosophical traditions, and with such contrasting visions of
probability theory and its proper domain of applications? I want to argue three
claims: first, that it was ca. 1830 that the words "objectivity" and "subjectivity"
emerged in French, German, and English with new meanings that resonated with
the new distinctions of the probabilists; second, that the probabilists were far
from unanimous about the grounds for and the implications of their distinctions;
and third, that the novelty of their distinctions lay not in any unified view about
either the meanings or applications of mathematical probability, but rather in a
new ontology of chance events that made it possible to understand "objective
probability" as something other than a contradiction in terms.
THE NEW MEANINGS OF THE OBJECTIVE AND SUBJECTIVE
Cournot wrote that it was "the example of Jakob Bernoulli" that had stiffened
his resolve to import the "metaphysical" language of objectivity and subjectivity
into mathematics, and Bernoulli was indeed the first mathematical probabilist to
use the terms "objective" and "subjective" in relation to probabilities. However,
the sense in which he used them diverged significantly from Cournot's usage. In
the opening passages of Part IV of the Ars conjectandi (1713) it is certainty
(certitudo), not probability, which Bernoulli modifies by objectivO: "All things
under the sun, past, present, and future, in themselves and objectively [in se &
objective] always have the greatest certainty." In contrast, all probabilities, defined as "degrees of certainty that differ from [certainty] as part to whole" [3,
239], are incorrigibly subjective just because they fall short of total certainty.
God's knowledge is certain and therefore objective; human knowledge is objective
only insofar as it is certain. Galileo had daringly claimed that human knowledge
about mathematics "equals the Divine in objective certainty [certezza obiecttiva]"
because it is knowledge of necessity [21,129]; Bernoulli similarly argued that causal
knowledge of eclipses partook of the same necessity and therefore objectivity [3,
240]. Given this understanding of objectivity as certainty based on an understanding of necessary causes, "objective probabilities" would have been an oxymoron.
We have recourse to probabilities when forced by our ignorance to traffic in
contingencies rather than the objective necessity of things in themselves. For
classical probabilists from Bernoulli through Pierre-Simon de Laplace, probabilities were officially subjective in Bernoulli's sense, figments of human ignorance
for which an omniscient deity, or even a well-informed super-calculator, would
have no need.
This does not imply that classical probabilists did not regularly make use of
what after 1840 came to be known as objective probabilities. As Ian Hacking
has shown, from the earliest stirrings of mathematical probability the subjective
understanding of probabilities coexisted side by side with a conception of probabilities derived from observed frequencies (e.g., mortality statistics) or physical constitution (e.g., the symmetry of coins and dice) [24, 11-17]. Bernoulli himself, in
the theorem which crowned the Ars conjectandi, aimed to connect the probabilities
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333
of reasonable expectation with the actual frequencies of all manner of events,
from wine harvests to storms. Eighteenth-century probabilists slid easily between
sense of probabilities rooted in states of mind and in states of the world. Insofar
as they remarked at all upon the gap between these senses, philosophers and
mathematicians relied upon the associationist psychology of Locke, Hartley,
Hume, and Condillac to correlate experience with expectation [12, 191-210].
Classical probabilists did not lack for applications of either subjective or objective
probabilities in our sense, but they did lack any felt need to draw a sharp distinction
between the two, much less a need to describe that distinction in the language of
"objective" and "subjective," for them so redolent of late scholasticism.
For Galileo, Bernoulli, and other 17th-century writers the terminology of the
objective and subjective still bore some of the marks of its 14th-century coinage
in the works of Duns Scotus, William of Occam, and other nominalist Schoolmen
on the status of universals and particulars. The objective in this context referred
to the objects of thought, and the subjective to objects in themselves [35, A.2.a].
This (to modern ears) inverted sense survived well into the 18th century; witness,
for example, the entry for "Objective/objectivus" in the 1728 edition of Chamber's
Dictionary: "Hence a thing is said to exist OBJECTIVELY, objectivO, when it
exists no otherwise than in being known; or in being an Object of the Mind" [6,
649]. The meanings of the terms had, however, already branched and crisscrossed
in the 17th century in both Latin and in various vernaculars, although "objective"
still generally modified thoughts rather than external objects. A famous example
can be found in the Meditationes (1641) of Ren6 Descartes, in which he contrasted
the "objective reality" of an idea--whether it represents its cause by perfection
and/or content--with its "formal reality"--whether it corresponds to anything
external to the mind [15, 40-42; 8, 136-137; 33] By the mid-18th century certain
metaphysical texts drew an objective/subjective distinction along the lines of
things in themselves versus thoughts, but even in such cases the mentalist associations were still strong: "One divides the truth into the objective or metaphysical
[objektivische oder metaphysische], which is nothing other than the reality or
possibility of the object itself ... [a]nd into the subjective or logical [subjektive
oder logikalische], which is truth in a really existing mind . . . All objective truth
is thus in the divine mind a subjective truth" [11, 95].
But by the time these lines were published, the terminology of objectivity and
subjectivity had an archaic and pedantic ring to it, and was confined to obscure,
mostly German treatises on metaphysics and logic. In addition to its technical
scholastic and theological ("God is our objective beatitude" [16]) senses, the most
common eighteenth-century definition is the "objective" lens ("object glass") of
a microscope or telescope. A quick survey of major dictionaries in French, English,
and German reveals that the words surface again with something like their familiar
modern meaning only in the 1830s--somewhat earlier in German. Already in 1820
a German dictionary defines Objektivitiit as "relation to an external object" and
Subjektiv as "personal, inner, inhering in us, in opposition to objective" [27], and
numerous editions of the Grimm brothers' etymological dictionary traced the
newer philosophical senses of both objektiv and subjektiv directly to Kant [22;
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LORRAINE DASTON
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23]. The early French entries were equally pointed in linking the new meaning of
objectifas that "which is outside the thinking subject; all that is real and not at
all ideal" with " n e w systems of philosophy" [4]. Littrr's etymological dictionary
of 1863 was still more explicit in crediting the " n e w sense" of objectifas " e v e r y
idea which comes from objects exterior to the mind" as " d u e to the philosophy
of K a n t " [30].
English-language dictionaries were somewhat sluggish in picking up newfangled
philosophical meanings, tending to assimilate them to 18th-century logical definitions .4 But there is independent literary evidence that philosophical winds blowing
from Kant's Krnigsberg also carried the new usage of "objective" and "subjective" to British readers. The poet Samuel Taylor Coleridge seems to have reintroduced the term back into general English usage in 1817, in the context of an
exposition of Kantian and neo-Kantian philosophical systems he had learned about
during a stay in Germany: " N o w the sum of all that is merely OBJECTIVE we
will henceforth call N A T U R E , confining the term to its passive and material sense,
as comprising all the phenomena by which its existence is made known to us. On
the other hand the sum of all that is SUBJECTIVE, we may comprehend in the
name S E L F or I N T E L L I G E N C E . Both conceptions are in necessary antithesis"
[7, 1:174]. By 1856 Thomas De Quincey could remark of the word "objective"
that "[t]his word, so nearly unintelligible in 1821, so intensely scholastic, and . . .
yet, on the other hand, so indispensable to accurate thinking, and to wide thinking,
has since 1821 become too common to need any apology" [14, 265].
To summarize: although a philosophical distinction between "objective" and
"subjective" dated back to the 14th century, the terms languished in the late 17th
and 18th centuries. Revived by Kant in his Kritik der reinen Vernunft (1781, 1787),
which breathed wholly new meanings into them, 5 they became entrenched in
general usage in German, French, and English in the period 1820-1840. But by
1840 this usage bore almost as little resemblance to the Kantian philosophy which
had resurrected the terms as to the scholastic philosophy which had created them.
For Kant, " o b j e c t " (Gegenstand) and "objective validity" (objektive Giiltigkeit)
were quite distinct concepts, and it is the latter, understood as the synthetic a
priori categories such as time, space, and causation that are preconditions for
experience, which undergirds Kant's own distinction between the objective and
subjective [28, 251-252, A201-202/B246-247; 1, 134-155]. Just how remote this
usage is from our own is made clear by Kant's regular pairing of the "subjective"
4 SamuelJohnson's Dictionary of the English Language defines one sense of "objective" from 1755
onwards as "[b]elongingto the object; contained in the object," a definitionrepeated verbatim (along
with the illustrative quotation from Isaac Watts's Logick (1724)) by several other English-language
dictionaries until late in the 18th century, e.g., [34]. Some mid-19th-centuryentries incorporate the
new philosophical opposition of "objective" to "subjective," but preserve the 1724 quotation, e.g.,
[41].
5 In the Kritik der reinen Vernunft, Kant discusses "probability [Wahrscheinlichkeit]" as well as
the nature of the "objective" and "subjective," but the two discussions remain wholly unconnected.
Moreover, Kant uses "probability" in a nonquantitative,almost archaic sense, reminiscentof classical
treatises on rhetoric. Compare, for example, [28,705-706, A775/B803] with [2, 2:1730, 1094b19-28].
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with the "merely empirical." Yet the usage enshrined by 1840, despite its bows
in the direction of the "new philosophy," drew the distinction between an "objective" external reality independent of all minds and "subjective" internal states
dependent upon individual minds. This was the meaning of the distinction which
crystallized in French, German, and English in the 1830s and upon which almost
all of the distinctions between two kinds of probability made ca. 1840 relied, even
if they did not invoke the newly fashionable terminology of the objective and
subjective.
A PLURALITY OF DISTINCTIONS
An arresting temporal and conceptual coincidence links the emergence both of
the new philosophical distinction between objective and subjective realms, and
of various distinctions between two kinds of probability. But it would be rash to
conclude to the existence of any straightforward connection between the philosophical and probabilistic distinctions: some of the probabilists--such as Cournot
and Bolzano--explicitly couched their distinctions in the new philosophical terminology, but others did not. Moreover, even those who did speak the new language
of objective and subjective probabilities did not agree with one another as to what
these were. Upon closer inspection the relationship between the philosophical
and probabilistic distinctions seems to be one of a shared ontology, one which
seems to have become not only thinkable but self-evident only in the 1830s. This
ontology not only carved up the world into what was inside and outside human
minds--Descartes had already done as much--but also (pace Enlightenment rationalists) located reality and truth in the "outside" realm of objects, and (pace
Enlightenment empiricists) further insisted on the mismatch not only between
world and mind, but also among the minds of different individuals. That is, the
new subjectivity was a threat to knowledge not only because it was insufficiently
faithful to the reality of objects, but also because its contents and dictates varied
from person to person. In order to understand how this ontology shaped the
probabilistic distinctions, we must briefly examine the latter.
What is most striking upon a first composite reading of the works of Poisson,
Cournot, Bolzano, Ellis, and Mill is the sharp divergence of motivations, formulations, and consequences of the distinctions they each draw between kinds of
probability. Poisson, eager to rescue the mathematical theory of the probability
of judgments from attacks recently launched against it by mathematicians, philosophers, and politicians [12, 342-369], and impressed by the criminal statistics
recently gathered by the French Ministry of Justice [38, 186-194; 25, 87-104],
emphasized the empirical foundations of probability theory. Bernoulli's theorem
was but a special case of the Law of Large Numbers (Poisson's original coinage),
which was a "general and incontestable fact, resulting from experience" of both
physical and moral phenomena, and the "base of all applications of the calculus
of probabilities" [36, 12]. Yet Poisson reserved the traditional term "probability"
for its traditional definition as "the reason we have to believe that [an event] will
or will not occur" [36, 30], letting "chance" designate "events in themselves and
independent of the knowledge we have of them" [36, 31]. So, for example, the
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LORRAINE DASTON
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"chances" of heads or tails for a given coin are unlikely to be equal because of
the physical asymmetries of all such objects, but the "probability" is nonetheless
equal because we know nothing about the coin's constitution. For Poisson, both
the "chance" of underlying causes and the "probability" of our ignorance were
equally legitimate interpretations of what it was that mathematical probabilities
measured. He proposed to evict from mathematical probability theory neither the
probabilitds of belief, nor any of the much-maligned applications, such as the
probabilities of causes and judgments, which depended on them. Clarification
alone would suffice. All confusion and paradox would disappear, Poisson was
confident, if probabilists would only learn to differentiate between probabilit~ and
chance.
Cournot was considerably more severe about restricting both legitimate applications and interpretations of mathematical probabilities. Although he professed in
his preface to be in total agreement with Poisson's distinction (at which Cournot
had arrived independently) [9, 5-6], and although he defended the probability of
judgments [9, 231], Cournot sharply criticized Condorcet and Laplace's Bayesian
approach to the probability of causes, preferring a n " experimental determination"
based on Bernoulli's Theorem [9, 106]. The "subjective probabilities" based on
equal ignorance of outcomes were fit only for the "frivolous use of regulating the
conditions of a bet" [9, 11 I, 288], and were moreover the "cause of a crowd of
equivocations [which] have falsified the idea one ought to have of the theory of
chances and of mathematical probabilities" [9, 59]. What Cournot held against
subjective probabilities was not so much that they were not empirical givens as
that they varied "from one intelligence to another, according to their capacities
and the data with which they are provided" [9, 106]. Far from equating objective
probabilities with frequencies tout court as John Venn was later to do [40, 90],
Cournot exhorted statisticians to purify the "immediate data of observation" from
all features that depend "solely on the point of view in which the observer is
situated," calling on theory and principles to help transcend mere "compilations
of facts and figures" [9, 125].
None of the probabilists in this group, in fact, went so far as to identify probabilities baldly with frequencies, although John Stuart Mill came the closest in his
attacks on the classical theory of probability as "ignorance... coined into science"
by the mere manipulation of numbers 6 [31, 8:1142]. He roundly rejected both the
probabilities of causes and judgments, condemning them for having made the
calculus of probabilities "the opprobrium of mathematics." He further claimed
that if the theory was to be profitably turned to any "scientific purpose," its
foundations must rest on "the utmost attainable amount of positive knowledge"
[31, 7:539]. Ideally, that knowledge would be of causes, for frequencies alone
"can give rise to no other induction than that per enumerationem simplicem; and
the precarious inferences derived from this" [31, 7:542].
6 Mill's Logic went through seven editions during his lifetime: 1843, 1846, 1851, 1856, 1862, 1865,
1868. I here quote from the seventh edition, insofar as it agrees with the first edition. When the two
diverge, I have indicated this by adding "1843" to the reference.
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337
Still less reliable were estimates of equiprobability based on equal ignorance
about possible outcomes. In the first 1843 edition of his Logic, Mill had dismissed
all such applications of the Principle of Indifference as so much algebraic alchemy,
transmuting base ignorance into the true metal of knowledge [31, 8:1141]. But in
later editions, under the influence of John Herschel [39], Mill grudgingly conceded
that "as a question of prudence" we might rationally assume that "one supposition
is more probable to us than another supposition," and even bet on that assumption
"if we have any interest at stake" and if we were in the desperate (and rare)
situation of having no relevant experience whatsoever [31, 7:535-536].
Note that Mill's "scientific" probabilities did not precisely coincide with either
Poisson's chance or Cournot's probabilit(s objectives. The latter referred to real
states of the world, independent of all human knowledge, whereas Mill's probabilities, "scientific" or not, were always based upon experience, more or less complete, and therefore upon our knowledge: "We must remember that the probability
of an event is not a quality of the event itself, but a mere name for the degree of
ground which we, or some one else, have for expecting it" [31, 7: 535]. That
knowledge coincided with things as they are only when it was full, causal knowledge. Despite his loud and repeated calls for probabilities founded on experience,
Mill curiously remained the most traditional of the revisionists in his interpretation
of all probabilities as epistemic.
Neither Mill nor Robert Leslie Ellis, the other British probabilist in my ca. 1840
sample, used the word "objective" in relation to the probabilities of experience.
The omission perhaps reflects nothing more than the slight British lag-time in the
reception of the new, pseudo-Kantian use of the word: Mill did use the word by
1863 in the context of the alleged "objective reality" of moral obligations [32,
43], and Ellis invoked the "merely subjective" character of secondary qualities
in his 1857 introduction to the philosophical works of Francis Bacon [20, 75,
80-81]. Yet in Mill's case there may have been more principled reasons to deny
merely epistemic probabilities, even those grounded in experience, an accolade
reserved for things and events in the world. Mill believed these latter to be
absolutely certain: "Every event is in itself certain, not probable" [31, 7:535].
The case of Ellis is considerably more convoluted. His essay occasionally
glimmers with bits of neo-Kantian vocabulary, as when he insists on the importance
of "d priori truths" and "the ideal elements of knowledge," as well as on the
incompatibility of the classical theory of probabilities with "the views of the nature
of knowledge, generally adopted at present ''7 [18, 1, 6]. Although in later articles
on the Method of Least Squares Ellis seconded Mill's claim "that mere ignorance
[here, of the specific law of error which applies for a given set of observations]
is no ground for any inference whatever. Ex nihilo nihir' [19,325], his 1842 essay
7 The language of "ideal elements" is redolent of William Whewell's faintly Kantian Philosophy of
the Inductive Sciences (1840), and it is quite possible that Whewell's views influenced Ellis directly
through their association at Trinity College, Cambridge, where Whewell was appointed Master in
1841, and Ellis a fellow in 1840.
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LORRAINE DASTON
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on the foundations of probability theory was a scathing critique of at least one
kind of empiricism, the sensationalist philosophy of Condillac, because it "rejects
all reference to d priori truths as such" [18, I]. This would hardly have sat well
with Mill's steadfast opposition to all a priori truths, even in mathematics and
logic [31, 7:224-251]. Ellis, in contrast, argued that Bernoulli's theorem was not
even the result of a mathematical deduction, much less a fact of experience in
Poisson's sense. Rather, it was an a priori truth, established by "an appeal to
consciousness" that revealed the impossibility of judging otherwise [18, 1-2].
Ellis insisted that judgments of probability were nonetheless "founded, not on
the fortuitous and varying circumstances of each trial, but on those which are
permanent--on what is called the nature of the case" [18, 3]. But what was
permanent was the "fundamental axiom" that in the long run "the action of
fortuitous causes disappears," an axiom supplied by neither mathematics nor
experience but rather by "the mind itself, which is ever endeavouring to introduce
order and regularity among the objects of its perceptions" [18, 3].
Ellis's talk of mind was not about mere psychology, and he resisted the suggestion that probabilities were "the measure of any mental state." Rather, they
properly referred to "the number of ways in which a given event can occur, or
the proportional number of times it will occur on the long run" [18, 3]. Note
Ellis's cautious deployment of " c a n " and "will": probabilities cannot be simply
read off of observed frequencies. The objections Ellis leveled against Laplace's
"law of succession" and the probability of causes [ 12,277-279J--namely, that the
definition of what constituted the next event in a series of observations depended
crucially on a judgment of similarity according to " a point of view" [18, 4-5]--are
equally devastating for every attempt to gather statistics. Ellis did not so much
condemn the dependence of probability judgments "on the mind which contemplates [an event]" as require the specification of viewpoint. What Cournot would
have dismissed as the incorrigible subjectivity of the observer's particular perspective, Ellis saw as a necessary component of all valid probability judgments, which
even "the most perfect acquaintance with the nature of the case" could not do
without.
Bernard Bolzano's distinction between "real or objective [wirkliche oder objektire]" and "apparent or merely subjective [vermeintliche oder blofl subjektive]"
probabilities strayed farthest of all from statistical frequencies. For Bolzano,
probabilities measure the relationship between propositions, not events [5, 3:274,
sect. 317]: the relation between the number of cases in which propositions A, B,
C, D . . . . are true, and the number of cases in which A, B, C, D . . . . together
with the additional proposition M are true can be represented by a fraction which
equals one when M can be deduced from A, B, C, D . . . . [5, 2:171-173, sect.
161]. If the propositions A, B, C, D . . . . are all true, then the probability is
"objective"; if some are false, then it is "subjective" [5, 3:266-267, sect. 317].
Both objective and subjective probabilities are judgments made by a "certain
thinking being"; the objective/subjective distinction does not apply to probabilities
as they relate to propositions in se, regardless of whether they are true or false,
HM 21
339
or whether a thinking being can judge them to be so [5, 3:264, sect. 317]. That is,
both objective and subjective probabilities are relative to a thinking mind, which
can be objectively certain because the propositions it holds to be true are indeed
so, or subjectively certain, because it mistakes false propositions for true. For
God all probabilities are objective, because all that the divine mind holds to be
certain is also true [5, 3:264, sect. 317]. For Bolzano, probability remained a
measure of certainty, as it had been for the classical probabilists. But whereas
they had considered probability immiscible with the objective judgments of God,
Bolzano reconciles probability to even divine objectivity.
Bolzano was quite aware that his usage of "objective" with respect to a thinking
subject could be attacked as Pickwickian, for by 1837 the objective was usually
"understood as only that which can be thought [to be] without any relation to a
subject" [5, 3:272, sect. 317]. But he defended his terminology by noting that
rival candidates for the title "objective probabilities," such as the probabilities
in games of chance, in fact rest upon states of knowledge that might in time be
improved, so that eventually probabilities might become certainties, or at least
change their value [5, 3:272, sect. 317]. Nor was Bolzano content with a distinction
between quantitative and "philosophical" probabilities, of the sort Cournot elaborated in his later work [10, 1:71-101], for the boundary between the two seemed
to him "often very subjective and variable" [5, 3:274, sect. 317].
THE OBJECTIVITY OF CHANCE IN A DETERMINISTIC WORLD
Even this briefest of surveys suffices to show how divergent the views of the
ca. 1840 cluster of revisionist probabilists were on almost every significant point.
All attacked some aspect of classical probability theory (usually making a target
of some passage in Laplace's work), but not the same aspect. Poisson hoped to
rescue the probability of judgments from certain pernicious assumptions; Cournot
rejected the probability of causes; Mill lambasted both; Ellis challenged the mathematical demonstration of Bernoulli' s theorem and Laplacean error theory; Bolzano
objected to the application of probabilities to events rather than propositions.
Their attitudes toward the relationship between probability and experience were
similarly various. Mill insisted that "scientific" probabilities be anchored in the
experience of causes or observed frequencies, but retained the contrast between
the probabilities of the mind and the certainties of events; Cournot cautioned that
probabilities derived from frequencies must be refined by the regulative principles
of rationality; Ellis emphasized the role of those regulative principles in defining
a reference class in securing the foundations of the mathematical theory such
as Bernoulli's theorem; Poisson counted the Law of Large Numbers (of which
Bernoulli's theorem was a special case) as an observational fact and the basis of
all applications of mathematical probability, but nonetheless granted the probabilit~s of belief equal if separate status with the chances of events in themselves;
Bolzano protested that probabilities were not about events at all, although they
could be altered by advances in knowledge. Given this Babel of voices, it is
tempting to conclude that these writers, far from all simultaneously arriving at
340
LORRAINE DASTON
HM 21
the same distinction between kinds of probabilities, were not in agreement about
anything.
Yet there is a common theme which unites this cluster of ca. 1840 revisionists,
and it is one that bears directly on the new meanings of "objective" and "subjective" that emerged at approximately the same time. Although all insisted on
the principle of universal causation and the complete determinism of events in
themselves, they nonetheless also carved out a place for chance in the world
rather than in the mind. That is, in their writings, chance became objective without
for one moment admitting the existence of genuine randomness. This was a position
that would have been inconceivable for the classical probabilists, for whom objectivO, in the words of Jakob Bernoulli, was synonymous with certainty and necessity, and for whom chance, in the words of Abraham De Moivre, "can neither
be defined nor understood: nor can any Proposition concerning it be either affirmed
or denied, excepting this one, 'That it is a mere word' " [13,253].
The symbiosis between determinism and classical probability theory was a
deep and enduring one. What the revisionist probabilists did was not so much
to abolish that alliance as to reinterpret it. With one crucial exception, all the
elements of their reinterpretation were available in the writings of the classical
probabilists, particularly in those of Laplace. These elements were a reaffirmation of a seamless causal order which determined all events necessarily, and
a perturbational model of chance as the superimposition of weak, sporadic
causes upon the action of strong, constant ones. At the same time--and this
was the novelty--the revisionists opposed a stable objective order to a variable
subjective one.
The revisionist probabilists wholeheartedly embraced the ironclad determinism
famously set forth by Laplace in the Essai philosophique sur les probabilit~s
(1814): "All events, even those which because of their smallness do not seem to
hold to the great laws of nature, are a consequence of them as necessary as the
revolutions of the sun" [29, vi]. But whereas Laplace and other classical probabilists since Bernoulli had concluded that mathematical probabilities must therefore
measure degrees of belief, the revisionist probabilists did not draw this conclusion.
Not that they were any less staunch in their allegiance to universal causation than
Laplace had been: Mill elevated the "law of universal causation" to "the rigorous
certainty and universality" of geometric truths [31, 7:325]; Cournot proclaimed
that "no phenomenon or event is produced without [a] cause; this is the sovereign
and regulative principle of human reason, in the investigation of real facts" [9,
53]. Rather, the revisionists grounded the reality of chance in another Laplacean
dictum about ontology, this time about the nature ofhasard: what we call "chance"
is not randomness, but rather the interaction of uniform, constant causes with
variable causes. In the long run regularity emerges as "the variable causes of this
irregularity produce effects alternatively favorable and contrary to the regular
course of e v e n t s . . , mutually destroying one another in a large number of trials"
[29, xlvii].
Reflections of this image of constant causes ultimately triumphing over the
H M 21
341
fluctuations of variable ones can be found in the writings of almost all 8 of the
revisionists, although they differed wildly as to whether it was a regulative principle
of human reasoning, a mathematical theorem, or a fact derived from experience.
Poisson's notion of chance approximated a constant underlying cause, while
hasard encompassed "the collection of all causes which concur in the production
of an event, without influencing the magnitude of its change" [36, 79-80]; Cournot
described "fortuitous events" as the superimposition of independent causal chains
[9, 55]; Mill reduced chance to the "composition of Causes . . . [where] we have
now one constant cause, producing an effect which is successively modified by
a series of variable causes" [31, 7:530]. But whereas Laplace took his model to
be a warrant (and even a moral imperative, in the case of the principles of sound
government [29, xlviii-xlix]) for ignoring the effects of chance (in his sense of
perturbations), the revisionists took it as a warrant for the real existence of
chance.
What made it possible for the revisionists to reconcile the real existence of
chance with watertight determinism? They rejected randomness with as much
finality as Laplace--or for that matter, Jakob Bernoulli and De Moivre--had;
moreover, they took over Laplace's ontological model of the fortuitous lock,
stock, and barrel. But they imposed upon both determinist convictions and ontological model a new philosophical grid, Kantian in its inspirations if not in its
substance, that divided the stable, external, "objective" world from the variable,
internal, "subjective" world. Within the framework of classical probabilities
causes and reasons had been systematically conflated, just as within 18th-century
associationist psychology, experience and belief had been equally systematically
linked [12, 191-210]. The distinction between objective and subjective which also
emerged ca. 1840 destroyed the plausibility of any smooth meshing between the
world of things and the world of the mind. This was not merely scepticism redux,
for the champions of objectivity were buoyantly confident of human abilities to
know the world of things. Nor was it simply a renewed warning against the
Baconian idols of cave, tribe, marketplace, and theater: the main threat posed by
subjectivity was not distortion but variability, and no one looked to the methods
of "Tables of Essence and Presence" or "Prerogative Instances" as a suitable
defense. When Cournot worried about "modifications" introduced by the differing
viewpoints of observers [9, 125], or when Ellis thundered that probabilities cannot
"be taken as the measure of any mental state" [18, 3], they were not simply
affirming empiricism. On the contrary, both, as we have seen, argued vigorously
for the centrality of regulative mental principles and a priori axioms of the mind
to the precepts and practice of mathematical probability theory. Their target was
not the mind p e r se--mental universals and regulative principles of thought were
8 Bolzano is the one exception here, for his probabilities applied to the certainty of propositions
and only indirectly and elliptically to the reality of events [5, 3:274, sect. 317]. But even Bolzano
seems to have implicitly subscribed to such a model, claiming that if all circumstances for subsequent
casts of a die were truly identical, then "not only 'no man' but God himself could not give grounds"
why different faces turned up [5, 3:273, sect. 317].
342
LORRAINE DASTON
H M 21
still revered, as we have seen in the cases of Ellis and Cournot--but rather all
that was idiosyncratic or variable in the mind.
This was the new core meaning of subjective: a capricious, arbitrary quality of
the mind, responsible for not only inter- but also intra-individual differences. In
contrast, the variability of irregular causes did not disturb the revisionist probabilists, for none of them believed that such fluctuations could, in the long run,
occlude the constant causes that probabilities really measured. Much less did they
believe that objective chance, defined in terms of causal variability, could topple
the iron regime of determinism. Against this background it is perhaps not surprising
that randomness first became thinkable when Gustav Theodor Fechner's psychophysical parallelism challenged not only this double standard of variability, but
also the philosophical distinction between inner and outer experience [26]. It was
this distinction that had made the objective probabilities of ca. 1840 suddenly
conceivable, without abandonment of the determinism which had made subjective
probabilities seemingly inevitable in the classical theory.
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1983.
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3. Jakob Bernoulli, A rs conjectandi [ 1713], in Die Werke von Jakob Bernoulli, ed. Naturforschenden
Gesellschaft in Basel, Vol. 3, Basel: Birkhaiiser, 1975.
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languefran~aise, Paris: Simon/Garnier Fr~res, 1847, 2:679.
5. Bernard Bolzano, Wissenschaftslehre [1837], ed. Wolfgang Schultz, 4 vols., Leipzig: Felix
Meiner, 1929.
6. E. Chambers, Objective/objectivus, in Cyclopaedia: or, An Universal Dictionary o f Arts and
Sciences, London, 1728, 2:649.
7. Samuel Taylor Coleridge, Biographia Literaria [1817], ed. J. Shawcross, 2 vols., Oxford: Oxford
University Press, 1973.
8. John Cottingham, A Descartes Dictionary, Oxford: Blackwell, 1993.
9. A. A. Cournot, Exposition de la th~orie des chances et des probabilit~s [1843], ed. Bernard
Bru, Paris: Librairie J. Vrin, 1984.
10. A. A. Cournot, Essai sur lesfondements de nos connaissances, Paris: Librairie de L. Hachette,
1851.
11. C. A. Crusius, Weg zur Zuverl~issigkeit und Gewil3heit der menschlichen Erkenntnis [1747], in
Die philosophoschen Hauptwerke yon Crusius, ed. G. Tonelli, Vol. 3, Hildesheim: Georg Olms, 1965.
12. Lorraine Daston, Classical Probability in the Enlightenment, Princeton: Princeton University
Press, 1988.
13. Abraham De Moivre, The Doctrine o f Chances [1718], 3rd Ed., London, 1756.
14. Thomas De Quincey, The Confessions o f an Opium Eater [1821, 1856], in Works, 2nd ed., Vol.
1, Edinburgh: Adam and Charles Black, 1863.
15. Ren6 Descartes, Meditationes de prima philosophia [1641] in Oeuvres de Descartes, eds. Charles
Adam and Paul Tannery, Vol. 7, Paris: Lropold Cerf, 1904.
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343
16. Dictionnaire de l'Acad~mie Franqaise, Objectif, Paris: Firmin Didot Fr6res, 1835, 2:284.
17. Dictionnaire de I'Acad~mie Franfaise, Objectif, Nismes: Pierre Beaume, 1778, 2:163.
18. Robert Leslie Ellis, On the Foundations of the Theory of Probabilities [1842], Transactions of
the Cambridge Philosophical Society 8 (1849), 1-6.
19. Robert Leslie Ellis, Remarks on an Alleged Proof of the 'Method of Least Squares', Contained
in a Late Number of the Edinburgh Review, Philosophical Magazine, Ser. 3 37(1850), 321-328.
20. Robert Leslie Ellis, General Preface to Bacon's Philosophical Works [1857], in The Works of
Francis Bacon, ed. James Spedding, Robert Leslie Ellis, and Douglas D. Heath, Vol. 1, Boston:
Brown and Taggard, 1861.
21. Galileo Galilei, Dialogo sopra i due massimi sistemi del mondo [1632], in Le Opere di Galileo
Galilei. Edizione Nationale, ed. Antonio Favaro, Vol. 7, Florence: Tipographia di G. Barb/~ra,
1897.
22. Jacob and Wilhelm Grimm, Objectiv/Objektivisch, in Deutsches W6rterbuch, Leipzig: S. Hirzel,
1889, 7:1109.
23. Jacob and Wilhelm Grimm, Subjectiv, Subjectivit~it, in Deutsches W6rterbuch, Leipzig: S.
Hirzel, t942, 10:813-814.
24. Ian Hacking, The Emergence of Probability, Cambridge: Cambridge University Press, 1975.
25. Ian Hacking, The Taming of Chance, Cambridge: Cambridge University Press, 1990.
26. Michael Heidelberger, Fechner's Indeterminism: From Freedom to Laws of Chance, in The
Probabilistic Revolution, Voh I: Ideas in History, ed. Lorenz Krfiger, Lorraine Daston, and Michael
Heidelberger, Cambridge, Mass.: MIT Press, 1987, pp. 117-156.
27. Theodor Heinsius, Object; Subject, Volkthiimliches Wfrterbuch der deutschen Sprache, Hannover: Hahnschen Buchhandlung, 1820, 3:714, 922.
28. Immanuel Kant, Kritik der reinen Vernunft [1781, 1787], ed. Raymund Schmidt, Hamburg: Felix
Meiner, 1956.
29. Pierre Simon de Laplace, Essai philosophique sur les probabilit~s [1814], in Oeuvres compldtes
de Laplace, ed.Acad6mie des Sciences, Vol. 7, Paris: Gauthier-Villars, 1886.
30. E. Littr6, Objectif/Objectivit6, in Dictionnaire de la languefranqaise, Paris: Librairie Hachette,
1863, 2:775-776.
31. John Stuart Mill, A System of Logic, Ratiocinative and Inductive [1843], ed. J. M. Robson,
Vols. 7 and 8 of The Collected Works of John Stuart Mill, Toronto/London: University of Toronto
Press, 1974.
32. John Stuart Mill, Utilitarianism, London, 1863.
33. Calvin Normore, Meaning and Objective Meaning: Descartes and His Sources, in Essays on
Descartes' Meditations, ed. Amelie Oksenberg Rorty, Berkeley: University of California Press, 1986,
pp. 223-242.
34. John Ogilvie, Objective, in The Imperial Dictionary, London: Blackie and Son, 1850, 2:257.
35. Oxford English Dictionary. Compact Edition, Objective, Oxford: Oxford University Press, 1971,
1:1963.
36. Sim6on-Denis Poisson, Recherches sur la probabilit~ des jugements en mati~re criminelle et en
matiOre civile, Paris: Bachelier, 1837.
37. Theodore M. Porter, The Rise of Statistical Thinking 1820-1900, Princeton: Princeton University
Press, 1986.
38. Stephen M. Stigler, The History of Statistics. The Measurement of Uncertainty Before 1900,
Cambridge, MA: Harvard University Press, 1986.
344
LORRAINE DASTON
HM 21
39. John V. Strong, John Stuart Mill, John Herschel, and the 'Probability of Causes,' Proceedings
of the Philosophy of Science Association 1 (1978), 31-41.
40. John Venn, The Logic of Chance. An Essay on the Foundations and Province of the Theory of
Probability [1866], 3rd ed., London: Macmillan, 1888.
41. Noah Webster, Objective, in An American Dictionary of the English Language, Springfield,
Mass.: George and Charles Merriam, 1850, pp. 762-763.
Doubling the Cube: A New Interpretation of Its Significance
for Early Greek Geometry
KEN SAITO
Faculty of Letters, Chiba University, Yayoi-cho, lnage-ku, Chiba, 263 Japan
It is widely known that Hippocrates of Chios reduced the problem of doubling the cube
to the problem of finding two mean proportionals between two given lines. Nothing, however,
is known about how this reduction was justified. To answer this question, propositions and
patterns of arguments in Books VI, XI, and XII of the Elements are examined. A reconstruction
modelled after Archimedes' On Sphere and Cylinder, Proposition II-1, is proposed, and its
plausibility is discussed. © 1995AcademicPress, Inc.
I1 est admis qu'Hippocrate de Chio a r6duit la duplication du cube au probl~me de la
recherche de deux moyennes proportionnelles entre deux segments donn6s, mais rien n'est
certain quant ~ la justification de cette r6duction. Pour r6pondre ~ cette question, sont examin6es les propositions comme les argumentations caract6ristiques des livres VI, XI, et XII des
Elements. Une reconstruction sera propos6e d'apr~s De la sphere et du cylindre, proposition
II-1 d'Archimbde, et son caract6re plausible sera discut6. © 1995AcademicPress, Inc.
Es ist wohlbekannt, daft Hippocrates von Chios das Problem der Verdoppelung des Wtirfels
auf das Problem der Auffindung von zwei mittleren Proportionalen zwischen zwei gegebenen
Geraden zurtickfiihrte. Jedoch ist nichts bekannt tiber die Rechffertigung dieser Reduktion.
Um diese Frage zu beantworten, werden S~itze und Strukturen von Argumenten aus den
Btichern VI, XI, und XII der Elemente untersucht. Eine Rekonstruktion wird dann vorgeschlagen, die sich auf Satz II-1 in Archimedes Uber Kugel und Zylinder stiitzt, und ihre Plausibilitat
wird diskutiert.
© 1995 Academic Press, Inc.
MSC 1991 subject classifications: 01A20, 51-03.
KEY WORDS: Theory of proportion, Euclid, Archimedes.
1. I N T R O D U C T I O N
T h e p r o b l e m o f d o u b l i n g t h e c u b e is o n e o f t h e c e n t r a l p r o b l e m s in G r e e k
m a t h e m a t i c s . S i n c e it a t t r a c t e d m a n y m a t h e m a t i c i a n s o f t h e p e r i o d , w e p o s s e s s
e x c e p t i o n a l l y r i c h a n c i e n t m a t e r i a l s o n this p r o b l e m . A s is w e l l k n o w n , H i p p o c r a t e s
o f C h i o s (ft. ca. 440 B . C . ) is said t o h a v e o r i g i n a t e d t h e t r a d i t i o n o f i n v e s t i g a t i n g
c u b e d u p l i c a t i o n . S e v e r a l s o u r c e s c r e d i t h i m w i t h r e d u c i n g this p r o b l e m t o a n o t h e r ,
t h a t o f f i n d i n g t w o m e a n p r o p o r t i o n a l s b e t w e e n t w o g i v e n lines. F o r e x a m p l e , in
his c o m m e n t a r y
to A r c h i m e d e s '
O n Sphere and Cylinder ( h e r e a f t e r S C ) , E u t o c i u s
cites Eratosthenes:
And it was sought among the geometers in what way one could double the given solid, keeping
it in the same shape, and they called this sort of problem the duplication of the cube. And
when they all puzzled for a long time, Hippocrates of Chios first conceived that if, for two
119
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Copyright © 1995 by Academic Press, Inc.
All rights of reproduction in any form reserved.
120
KEN SAITO
HM 22
given lines, two mean proportionals were found in continued proportion, the cube will be
doubled. Whence he turned his puzzle into another no less puzzling)
Hereafter we call this transformation of cube duplication into the problem of
finding two mean proportionals "Hippocrates' reduction." It was in this reduced
form that the solution of cube duplication was sought during antiquity. The traditions
of these ancient studies have recently been so thoroughly studied by Knorr [9,
11-153] that it seems as if nothing further could be teased out of the extant materials.
In this article, however, we concentrate on a problem about which almost all the
ancient sources are strangely taciturn: how cube duplication was reduced to finding
two mean proportionals between two given lines. Strangely enough, in contrast to
the abundance of solutions for inserting two mean proportionals, we have little
testimony on how Hippocrates' reduction was proved in antiquity. No extant text
explains how Hippocrates arrived at this reduction. It would therefore seem worthwhile to try to understand the kind of arguments he was likely to have made. 2
Throughout this paper, we will be less concerned with the heuristic context that
motivated Hippocrates' work than with the justification of the discovery, the proof
of the equivalence of cube duplication to finding two mean proportionals. Our
concern will lead us to propose a new interpretation of the status of the theory of
proportion in the early stage of Greek mathematics, where we will highlight a group
of propositions which may have served as a tool for problem-solving (including
that of cube duplication) for the mathematicians of the period.
2. T H E P R O B L E M A N D ITS R E D U C T I O N
First, let us briefly analyze the problem of doubling the cube, taking into account
ancient testimonies and modern studies on its reduction. Cube duplication is a
problem that appears to have been originally stated as follows:
[CD] To find a cube which is twice as large (in size, not side) as a given one.
Fairly early on, however, this problem was studied in the following generalized form:
[CDx] To find a cube whose ratio to a given cube equals the ratio of two given
lines (not necessarily in the ratio 2:1) (see also [8, 23]).
This generalized problem was then reduced to the following problem:
1 [5, 3:88]. We cite Knorr's translation in [9, 147]. Knorr persuasively asserts the genuineness of
Eratosthenes' accounts. See [8, 17-24] and [9, 131-153].
2 It might be doubted if Hippocrates really proved the truth of his reduction, since Eratosthenes
simply says that Hippocrates "first conceived" the truth of the reduction. Therefore, the reader with
maximum reserve should read the phrase "Hippocrates' proof (or justification) of his reduction" in this
paper as "the first rigorous proof of Hippocrates' reduction." This proof was surely found very soon,
if not by Hippocrates himself, since it was concerned with such an important problem as cube duplication.
Archytas already contrived a method to insert two mean proportionals [5, 3:84-88; 9, 100-110], so that
we may assume that the proof was already available to Archytas. It seems to us that the attribution
of both the discovery and the proof of the "reduction" to Hippocrates himself fits better with other
testimonies which rank him as one of the great mathematicians of the period.
HM 22
D O U B L I N G T H E CUBE
121
[CDxr] T o find two m e a n proportionals b e t w e e n two given lines.
Thus given two lines a, b, one must find x, y, such that a : x = x : y = y : b. T h e last
problem, [CDxr], was usually investigated by the ancients u n d e r the n a m e o f cube
duplication, a l t h o u g h strictly speaking, this only covers the case b = 2a.
W e have a few sources that d e m o n s t r a t e the truth of H i p p o c r a t e s ' reduction,
and they are b a s e d on the use of the c o n c e p t of triplicate ratio (as used in a
proposition like Elements XI-33), or the c o m p o u n d i n g of ratios. W e can confirm
this in Diocles [15, 102], P a p p u s [6, 1:66-68] (see also [9, 90-91]) and Philoponus
[18, 104-105] (for translation, see [9, 20]). W e cannot, however, regard these arguments as reflecting the first p r o o f of H i p p o c r a t e s ' reduction for reasons that will
be explained below.
S o m e of the m o d e r n reconstructions follow the same lines as these authors of
late antiquity. F o r example, K n o r r explains the equivalence of cube duplication
[CDx] and its reduction [CDxr] as follows:
If for any two given lines, A and B, we can insert the two mean proportionals, X and Y, then
A : X = X : Y = Y: B. Thus, by compounding the ratios, one has (A : X) 3 = (A : X)(X : Y)(Y : B),
that is, A 3 : X 3 = A : B. Thus, X will be the side of a cube in the given ratio (B : A) to the given
cube ( m 3 ) . [8, 23]
In this passage, K n o r r is less interested in restoring H i p p o c r a t e s ' line of t h o u g h t
than in convincing m o d e r n readers of the truth of H i p p o c r a t e s ' reduction. Indeed,
he does n o t p r e t e n d to restore any ancient argument. Using m o d e r n notations, he
assumes the equivalence of triplicate ratio with the o p e r a t i o n of c o m p o u n d i n g the
same ratio three times, 3 and he further assumes that the triplicate ratio of two lines
is equal to the ratio of the cubes on them. T h e truth of the former, t h o u g h evident
to us, is n o t explicitly p r o v e d in the extant Elements, while the latter is a special
case of Elements XI-33. 4
In earlier studies [12; 13], we have a r g u e d that the concepts of duplicate and
triplicate ratio, as well as that of c o m p o u n d ratio, a p p e a r to have b e e n relatively
late arrivals in the corpus of G r e e k mathematics, and that the same holds true for
Proposition XI-33. If this thesis is correct, then these notions were certainly not
available to H i p p o c r a t e s w h e n he f o r m u l a t e d his reduction of cube duplication. It
would, therefore, be worthwhile to search for a p r o o f without these notions. ~
In this context, we should note a n o t h e r w a y to formulate cube duplication which
must have b e e n clear to G r e e k g e o m e t e r s f r o m the time o f Hippocrates. T h e
p r o b l e m [CDx] can easily be r e p r e s e n t e d in the following m o r e g e o m e t r i c form:
[CDxg] G i v e n a rectangular prism on a square base, to find a cube equal to it.
3 Knorr is far from the first to make this assumption. For example Heath, in his account of Hippocrates'
reduction, states that "Hippocrates could work with compound ratios" [3, 1: 200].
4 Another modern reconstruction in [1, 57ff.] is based on the corollary of XI-33.
5 In this respect, two modern reconstructions [16, 119; 14, 128] are to be noted. These authors do not
resort to multiplicate and compound ratios, but use instead more fundamental theorems. However, as they
did not intend to offer historical reconstructions, they offered no textual evidence for these arguments.
122
KEN SAITO
G
HM 22
E
C
B
FIGURE 1
Obviously, the solution of [CDxg] implies that of [CDx]. Let there be given a cube
AB and two lines a, b (see Fig. 1). To find a cube whose ratio to AB is the given
ratio a:b, one needs only to construct a rectangular prism FG with square base
H G congruent to BC and whose height FH is such that FH : AC = a : b. It follows
immediately that if cube DE is equal to prism FG, then DE : AB = a : b.
Thus, the problem [CDx] is reduced to [CDxg]. In this respect, cube duplication
is a natural analogue of the problem of squaring a rectangle in plane geometry.
We therefore begin by examining the problem of squaring, and its relationship to
cube duplication.
3. THE P R O B L E M OF S Q U A R I N G
This section examines the problem of squaring a rectangle, the counterpart of
cube duplication in plane geometry. The first problem is solved by finding one
mean proportional, whereas the second requires finding two mean proportionals
(see [3, 1:201; 8, 22]).
In the Elements, the problem of squaring a rectangle is treated in I1-14. In
Book VI, the same construction is presented as a method for finding one mean
proportional between two given lines. In VI-17, the problem of squaring a given
rectangle is reduced to another problem, that of finding a mean proportional between two given lines. This step corresponds to Hippocrates' reduction of cube
duplication. Then, the problem of finding one mean proportional is solved by
means of the construction in VI-13, which is identical with that given in 11-14. The
counterpart of this step in cube duplication is finding two mean proportionals. It is
important to note that proposition VI-17, the counterpart to Hippocrates' reduction,
does not depend on propositions concerning duplicate or compound ratios. In other
words, a reduction of cube duplication using triplicate ratio or compound ratio,
such as that given by Diocles or Pappus, is a different sort of solution from Euclid's
solution of squaring.
HM 22
DOUBLING THE CUBE
123
So what means did Hippocrates use in his reduction? To answer this question,
it is necessary to know which propositions he had at his disposal. Let us therefore
examine the propositions concerning ratios between figures both in Book VI (plane
figures) and in Books XI and XII (solid figures).
Book VI of the E l e m e n t s contains several propositions concerning ratios between
plane figures. Here we choose those relevant to the problem of squaring, and classify
these propositions according to the scheme introduced by Mueller [10]:
D
base-area proportionality
VI-1
F
equal-area propositions
VI-14,15,16,17
G
duplicate-ratio between similar figures
VI-19,20
J
compound-ratio proposition
VI-23
The letters in the first column indicate the symbols used by Mueller 6 to denote
each type of proposition. The second column gives a short explanation of each type
of proposition. Thus, type D refers to the proportionality relationship between base
and area in two parallelograms or in two triangles of equal height. Only Proposition
VI-1 belongs to this type and it constitutes the basis for all other propositions.
Although Proposition VI-1 is concerned with the relationship between base and
area, not with that involving height and area, the corresponding height-area proportionality can be established through VI-1, with the aid of VI-4. In the solid geometry
of Books XI and XII, however, D (base-volume proportionality) and E (heightvolume proportionality) are treated as distinct propositions.
Type F, which we call "equal-area propositions," deserves a closer look. The
first of these is:
In equal and equiangular parallelograms the sides about the equal angles are reciprocally
proportional; and equiangular parallelograms in which the sides about the equal angles are
reciprocally proportional are equal. [Elements Vl-14]
For the proof, Euclid begins with two equiangular parallelograms, AB and BC
(see Fig. 2). By using VI-1, he proves that if AB = BC then DB : BE = GB : BF,
and vice versa. Proposition VI-15 proves the corresponding theorem for triangles
that have one angle equal. Proposition VI-16 treats the special case of VI-14, where
the parallelograms are rectangles. VI-17 then handles the particular case of VI-16
in which one of the rectangles is a square. The proof of VI-17 depends on propositions VI-14 through VI-16.
Type G contains two propositions (VI-19, VI-20) proving that similar plane figures
are in the duplicate ratio of their corresponding sides.. Finally, type J contains
the only theorem in Book VI concerning compound ratios (VI-23: equiangular
parallelograms are in the ratio compounded of the ratios of their sides). This
result has the greatest affinity to the modern formula for the numerical area of a
parallelogram.
6 See [10, 217; Appendix 1]. Some types of propositions in Mueller's classification are omitted since
they are of little importance here.
124
KEN SAITO
HM 22
E
C
FmVR~2
De Morgan noted [4, 2:217, 233-234] that the two propositions of types D and
J are sufficient to establish all the other propositions concerning ratios between
parallelograms. In fact, VI-23 can be proved directly from VI-1, and it thus depends
on none of the intermediate results of types F and G. VI-19 is merely a special
case of VI-23, whereas the propositions in F could even seem to be trivial corollaries
that follow directly from VI-23 (see [10, 161-162]).
It is important to recognize, however, that the logical relations of these propositions, as presented in Book VI, are quite different. First, J (VI-23) is completely
isolated in the Elements: not only is it never used later but it also depends on the
notion of compound ratios, a concept whose definition in Book VI is spurious.
We believe that J was interpolated into an earlier version of Book VI which
existed before Eudoxus and Euclid [12, 33-35]. 7 Furthermore, the propositions of
type G (VI-19, VI-20) also seem to be out of place in Book VI. While VI-19 is
used in the proof of VI-20, neither VI-19 nor VI-20 are used anywhere else in Book
VI. Instead, the corollary (porism) to VI-19, which avoids the concept of "duplicate
ratio," is used to prove VI-22, VI-25, and VI-31. This odd avoidance of type G
suggests that the term "duplicate ratio" was also a later insertion into the original
version of Book VI.
The very proof of VI-19 supports the plausibility of this interpolation. If one
were to demonstrate VI-19 using the full power of the idea of duplicate ratio and
following the kind of argument presented in VIII-18 for numbers, it would be
natural to proceed as follows:
7This interpolation may have been inserted by Euclid himself or by somebody after him. We can
only speak of relative chronologyhere.
HM 22
D O U B L I N G T H E CUBE
125
A
B
Q
C
D
FIGURE3
L e t A B C and D E F be similar triangles (Fig. 3). Construct triangle P B Q c o n g r u e n t
with the triangle D E F , and join A Q . T h e n
ABC :ABQ = BC :BQ = BA :BP = ABQ :PBQ = ABQ :DEF,
so
ABC :ABQ = ABQ :DEF.
T h e r e f o r e , it is evident that the ratio of similar triangles A B C : D E F is the duplicate
ratio of their c o r r e s p o n d i n g sides B C : E F ( = B C : B Q ) . 8
T h e p r o o f of VI-19 in the E l e m e n t s , however, p r o c e e d s quite differently.
Using VI-11, one finds a third p r o p o r t i o n a l B G to B C and E F (Fig. 4):
BC:EF = EF:BG.
Since
AB :DE = BC:EF,
one has
8 This argument tacitly invokes ex aequali (V-22). For a more detailed discussion, see [13, 120].
126
KEN SAITO
HM 22
A
B
G
C
D
E
F
FIGVRE4
AB :DE = EF:BG.
B y F (VI-14), this leads to the e q u a l i t y of two triangles:
ABG = DEF.
Since, b y VI-1, A B C : A B G = B C : B G ,
A B C : D E F = BC : B G = 2 * (BC : EF). 9
T h u s , E u c l i d e v i d e n t l y p r e f e r r e d a n a r g u m e n t d e p e n d i n g o n F (VI-14) to a direct
p r o o f which he c o u l d have f o r m u l a t e d after the m o d e l of VIII-18. His p r o o f seems
to b e m o r e a p p r o p r i a t e for the corollary to VI-19 r a t h e r t h a n for VI-19 itself,
b e c a u s e he actually constructs the third p r o p o r t i o n a l in the given figure. 1°
T h e idiosyncracies we have n o t e d a b o v e in B o o k V I are difficult to a c c o u n t for
o n m a t h e m a t i c a l g r o u n d s alone. H o w e v e r , they b e c o m e u n d e r s t a n d a b l e if we as9 We use this notation for the duplicate ratio, and the triplicate ratio of BC:EF will be written
3 * (BC : EF).
10VI-19 and its corollary are as follows:
VI-19. Similar triangles are to one another in the duplicate ratio of the corresponding sides.
VI-19 Corollary. From this it is manifest that, if three straight lines be proportional, then, as
the first is to the third, so is the figure described on the first to that which is similar and similarly
described on the second.
The corollary refers explicitly to the third proportional, while in the proposition, it is hidden under the
term "duplicate ratio."
HM 22
DOUBLING THE CUBE
127
sume that the application of the theory of proportion to plane geometry began
with the problem of squaring simple figures, and that propositions of the types G
and J represent the fruits of later developments (for more discussion see [13]).
Although this interpretation may appear speculative, the contrary assumption
that the techniques of multiplicate ratios and compound ratios were available at
the time of Hippocrates poses a major difficulty: if these notions were familiar over
100 years before Euclid, why are compound ratios so isolated and why are duplicate
ratios not exploited more often in the extant text of Book VI of the Elements? If
we assume, on the other hand, that these concepts were not available to Hippocrates,
the question arises: what was Hippocrates' reduction like? To this question, the
solid geometry in the Elements provides valuable suggestions.
4. PROPOSITIONS C O N C E R N I N G T H E RATIOS OF SOLID F I G U R E S
IN T H E E L E M E N T S
We now consider Books XI and XII of the Elements and analyze the propositions
presented therein concerning the ratios between solid figures. The figures treated
in these propositions are parallelpipeds, prisms, pyramids, cylinders, cones, and
spheres. Here, as in the preceding section, we again make use of Mueller's classification:
D
E
F
G
base-volume
proportionality
height-volume
proportionality
equal-volume
proposition
triplicate-ratio
proposition
p
tpy
py
c
XI-32
XII-5
XII-6
XII-11
XI-32
s
XII-14
XI-34
XII-9
XII-15
XI-33
XII-8
XII-12
XII-18
In this table, the symbols p, tpy, py, c and s in the first line stand for parallelpiped,
triangular pyramid, pyramid (in general), cone (and cylinder) and sphere, respectively. Propositions like VI-23 (type J, using compound ratios) are absent in the
solid geometry of the Elements, while those of type E are often independent of
type D.
The most striking feature of solid geometry in the Elements is that the selections
and demonstrations of propositions are not as systematic as one might expect.
There is no use of compound ratios although this could greatly simplify the proof
of other propositions, as Ian Mueller has noted. In pointing out that Euclid could
have proven the theorem that pyramids are to one another in the same ratio as
the compound ratio of their bases and heights, Mueller states: "Euclid's failure to
prove this extension of XII,9 is perhaps some further confirmation of the view
that the connections among compounding, multiplying, and volumes were not so
128
KEN SAITO
HM 22
immediately clear to him as they are to us [10, 229]." We should add that the
propositions of type J are not the only ones that fail to appear in Book XII. As is
seen in the table, most of the propositions concerning pyramids in general are
lacking, though these are easily provable.
We now examine the propositions and the logical dependency among them. First,
let us analyze the propositions concerning parallelpipeds: Dp and Ep (XI-32), Fp
(XI-34), and Gp (XI-33). The basic results Dp and Ep are covered by XI-32, which
is the counterpart of VI-1 in solid geometry.11 Next, Gp (XI-33) and Fp (XI-34)
are proved from XI-32. What is striking in these propositions is that Gp appears
before Fp, unlike the propositions of types G and F for parallelograms. This reversed
order of propositions entails a difference in their proof, of course. Both Gp (XI33) and Fp (XI-34) are proved directly from Dp and Ep (XI-32). Gp therefore does
not depend on Fp. Neither Gtpy nor Gc depends on its corresponding theorem of
type F, since they are both proved from Gp.
In examining the propositions on parallelograms, we emphasized the logical
dependence of propositions of type G (VI-19, 20) on those of type F (VI-14 to
VI-17). We thence claimed that the reduction of squaring to finding one mean
proportional (VI-17, a special case of VI-14) was not proved through duplicate
ratio.
In solid geometry, this dependence no longer exists. Should we then reject F as
irrelevant to cube duplication and assume that the reduction of cube duplication
was carried out by means of G (XI-33), taking advantage of triplicate ratio? We
do not think so. The independence of Gp from Fp may reflect the period of the
draft of these books on solids, which are no doubt later than Book VI. Indeed, the
fact that propositions of type F appear in Books XI and XII is significant, because
they are no longer necessary. Propositions XI-34, XII-9, and XII-15, which are
never used in any substantial way in the Elements, 12 suggest again that the theorems
of type F reflect the mathematics of an earlier period, as their role in the extant
text of the Elements is otherwise difficult to explain. We believe this earlier context
was intimately connected with the problem of doubling the cube.
To support this thesis, let us now examine Book XII more closely. The propositions of Book XII deal with curved solids, and therefore require the method of
exhaustion, which is employed in the proofs of the propositions concerned with
cylinders and cones. The method of exhaustion is also used in the last proposition
XII-18 (spheres are to one another in the triplicate ratio of their respective diameters), the most sophisticated result on spheres before Archimedes. The propositions
on pyramids, which did not require the method of exhaustion, are only useful
lemmata for the proof of the propositions on cylinders, cones, and spheres. Clearly,
the aim of Book XII was not to provide a compendium of propositions concerning
the volumes of pyramids.
11AlthoughXI-32 does not explicitlystate E, this property is easilyderivedfrom XI-32 and VI-1, as
Euclid argues in XI-34.
lz The onlyuse of F, namelyof XI-34 in XII-9, does not explainthe significanceof F, sinceboth XI34 and XII-9 belongto F.
H M 22
DOUBLING
THE CUBE
129
The method of exhaustion is used in Propositions 2, 5, 11, 12, and 18, and most
of the other propositions in Book XII are used as lemmata to establish these five
theorems. Only four propositions, namely 9, 13, 14, and 15, have no connection
with the method of exhaustion, and they appear to be dead-end propositions with
no apparent purpose. 13We should not, however, simply disregard these propositions
as useless. We must first make every effort to discover their significance. What then
was the significance of these four propositions? Since they all establish F, the equalvolume propositions (XII-9 for triangular pyramids, and XII-15 for cylinders and
cones), 14 it may be assumed that the author had a special interest in establishing
propositions of type F.
Our examination of solid geometry in the Elements thus again reveals an emphasis
on F, the equal-volume propositions. In the next section, we will examine a document
that suggests that the equal-volume propositions underpinned Hippocrates' reduction of cube duplication.
5. ARCHIMEDES' O N S P H E R E A N D C YLIND E R , PROPOSITION II-1
In this section we examine Archimedes' SC II-1. Its solution is reduced to the
same problem as cube duplication: finding two mean proportionals between two
given lines. This coincidence inspired Eutocius to add a long commentary on cube
duplication. In fact, without Eutocius's commentary, we would have far poorer
documentation on the history of cube duplication. We concentrate on this problem
in SC because we believe that the coincidence of the solutions of this problem in
SC II-1 and of cube duplication reveals a more fundamental affinity.
The first proposition of the second book of Archimedes' On Sphere and Cylinder
solves the following problem:
SC II-1. Given a cone or cylinder, to find a sphere equal to it.
This proposition consists of two parts: the analysis and the synthesis (though Archimedes does not use these terms here). We examine the analysis, where Archimedes
first reduces the problem as follows:
SC 11-1'. Given a cylinder, to find an equal cylinder whose height is equal to
its diameter.
Archimedes' analysis of II-1' proceeds as follows.
Let E be the given cylinder (Fig. 5), with height EF, and base diameter CD, and
let K be the cylinder to be constructed, with height KL equal to its base diameter
GH. Then
(circle E) : (circle K), that is, sq(CD) : sq(HG) = KL: EF
(1)
and K L = HG. Therefore,
13 See Neuenschwander's diagram of the logical relationships between the propositions in Book XII
in [11, 116].
14The other propositions, namely XII-13 and XII-14, serve as lemmata for XII-15.
130
KEN SAITO
HM 22
M
F
C
(
H
FIGURE 5
sq(CD) : sq(HG) = HG :EF.
(2)
L e t [MN be a line such that]
sq(nG) = r(CD, MN)
(3)
C D : M N = s q ( C D ) : s q ( H G ) , that is, = H G : EF.
(4)
C D : H G = H G : M N = M N : EF.
(5)
Then
Then, alternately,
W e n o w e x a m i n e each step in A r c h i m e d e s ' argument, filling up small gaps in his
exposition. T h e first step (1) is based o n F (the e q u a l - v o l u m e proposition) for
cylinders (SC L e m m a 4 after 1-16; 15 Elements, XII-15), and the p r o p o r t i o n a l i t y o f
the squares and circles (Elements, XII-2). T h e next relation (2) is derived simply
f r o m the r e p l a c e m e n t of K L in (1) by an equal length, H G .
T h e introduction of the line M N in (3) deserves attention. It is c o n s t r u c t e d so
that the rectangle contained by M N and C D is equal to the square on H G . O n the
basis o f this relation (3), A r c h i m e d e s then j u m p s to (4). His small skip w o u l d
a d e q u a t e l y be filled as follows (our reconstruction): 16
15s c Lemma 4 after 1-16. In equal cones the bases are reciprocally proportional to the heights; and
those cones in which the bases are reciprocally proportional to the heights are equal.
16We have to note that Heiberg's reference to Definition 9 of the Elements in [5, 1:173, Note 2] is
misleading. First of all, Heiberg confuses two notions that are clearly different in Greek mathematics:
duplicate ratio (the ratio of the first term to the third when three magnitudes are proportional) and the
ratio of squares (see also the text to Note 4, in Section 2). His confusion seems to show that our modern
arithmetical approach to geometry using real numbers makes it difficult for us to distinguish them. For
further discussion, see [13].
Among modern translations, [2, 182] follows Heiberg in introducing duplicate ratio. We follow Vet
Eecke's interpretation in [17, 1:91, Note 2].
HM 22
DOUBLING THE CUBE
131
Since
sq(HG) = r(CD, M N )
(3)
sq(CD) :sq(HG) = sq(CD) :r(CD, M N ) = C D :MN.
Therefore,
CD :MN = sq(CD) : sq(HG).
This is the first half of (4). The rest of (4), sq(CD) : sq(HG) = H G : EF, is the very
content of (2).
The deduction of (5) also contains a leap which we supplement in the following
manner:
Applying Archimedes' indication "alternately" to the previous relation:
CD : MN = H G :EF,
(4)
CD : H G = MN : EF.
(5)
we obtain
The rest of the relation (5), i.e., CD : H G = H G : MN, would be obtained by applying
Elements VI-16 (another proposition of type F) to the equality (3): sq(HG) =
r(CD, MN).
We have now examined every step of Archimedes' analysis in SC II-1. It is worth
noting that no recourse is made to either multiplicate ratio (i.e., duplicate and
triplicate ratio) or compound ratio.
Archimedes' argument is thus completely clear, in the sense that every proposition
he used in each step has been identified. However, we cannot yet be satisfied
because we have not yet found an explanation which would make his argument
understandable as a whole. Since his argument seems at first sight tortuous, we
need to consider the context that may have motivated it.
The affinity of Archimedes' argument with Elements VI-19 offers a satisfactory
explanation of Archimedes' intention in his analysis of SC II-1. Let us examine this
point more closely. In his analysis, Archimedes does not use the propositions
concerning multiplicate ratio (type G), their role is replaced by SC L e m m a 4 after
1-16 which is identical to the Elements, XII-15 (type F). In this respect, his argument
has much in c o m m o n with the treatment of squaring in B o o k VI.
This affinity is far stronger than one might at first imagine. Let us further confirm
this point. Both Archimedes and Euclid make use of propositions of type F. Both
introduce an auxiliary line which performs an analogous role. This parallelism
deserves further explanation. In Archimedes' analysis, the introduction of the line
MN has a twofold significance: on the one hand, a proportion
CD : H G = H G : MN
is derived from the equality of areas through VI-16 (type F) of the Elements. On
132
KEN SAITO
HM 22
the other hand, since it supposes an equality of areas (3): s q ( H G ) = r(CD, MN),
it enables one to reduce the ratio between squares s q ( C D ) : s q ( H G ) to that between lines:
sq(CD) : s q ( H G ) = sq(CD) : r(CD, MN) = C D :MN.
Looking at VI-19 of the Elements (see Fig. 4), one notes that the auxiliary line
B G thereby has exactly the same twofold role in the proof: to introduce (i) a
proportionality ( B C : E F = E F : B G ) and (ii) an equality of areas which leads to
the reduction of a ratio between areas to that between lines ( A B C : D E F = B C : BG).
T h e r e is a perfect parallelism between these arguments. The only difference is that,
in VI-19, the line is introduced not through an equality of areas, as in step (3) of
A r c h i m e d e s ' analysis, but as the third proportional to two given lines, just as in
step (6) of the same analysis. This difference, however, does not signify a substantial
difference between these two arguments. Since VI-19 of the Elements is a theorem,
while A r c h i m e d e s ' passage is an analysis, the arguments are necessarily in reverse
order. Thus, SC II-1 and Elements VI-19 employ the same technique of introducing
an auxiliary line segment.
In a previous article, we have pointed out the use of this same technique in Euclid's
Data, Proposition 68, and in Apollonius's Conics, 1-43, calling this a "reduction to
linear ratio" [12, 35-48] .17 The function of this technique can be precisely described:
it applies a proposition of type F, thereby avoiding an argument by duplicate ratios
or c o m p o u n d ratios.
N o w we can better understand A r c h i m e d e s ' argument as a whole. It is a due
result of the systematic application of an established and widely diffused method,
"reduction to linear ratio."
A r c h i m e d e s ' choice of this technique is significant, because another solution by
triplicate ratio was certainly available to him. In a l e m m a in the first b o o k of SC
Archimedes states that similar cones are in triplicate ratio of the base diameters. TM
This is also part of XII-12 of the Elements, where the same property is also established for cylinders. Though Archimedes does not seem to have directly consulted
Euclid's Elements, we m a y certainly assume that Archimedes was also aware that
his l e m m a applied to cylinders as well as to cones.
With this lemma, p r o b l e m II-1 can be solved as follows:
A Possible Reconstruction of the Solution of 11-1'. T a k e a point O on E F such that
C D = E O (Fig. 6). Then f r o m the l e m m a above we have
cylinder C D O : cylinder G H L = 3 * (CD : G H )
(1)
Since cylinders which are on equal bases are as their axes (SC L e m m a 2 after 116; cf. Elements, XII-14),
17Of course it should be distinguished from "Hippocrates' reduction" which we investigate in the
present paper.
18SC I, Lemma 5 after Proposition 16: Cones whose diameters of the bases have the same ratio as
their axes are in the triplicate ratio of the diameters of their bases.
HM 22
DOUBLING THE CUBE
133
F
L
O
C
(
H
FIGURE 6
cylinder C D O : cylinder C D F = E O : E F = C D : EF.
(2)
Also it is supposed that
cylinder G H L = cylinder CDF.
(3)
F r o m (1), (2), and (3), we have
CD : E F = 3 * (CD : H G ) .
(4)
Then, from the definition of triplicate ratio, H G (the diameter of the base of the
cylinder to be found) is the first of the two m e a n proportionals between CD and EF.
This argument, though our reconstruction, depends only upon propositions available
to Archimedes and conforms completely to the style of G r e e k geometry at that
time. In short, we see no reason why Archimedes could not have carried out
this argument.
Archimedes, however, did not use the l e m m a concerning triplicate ratio, although
it was clearly within his reach, preferring the technique of "reduction to linear
ratio." His choice confirms the prevalence of this technique, which we have found
in both Euclid and Apollonius. A r c h i m e d e s ' case is particularly significant for two
reasons: first, he chose not to use c o m p o u n d and multiplicate ratios in this p r o b l e m
in spite of his perfect mastery of these concepts;19second, Archimedes could not
have failed to notice that his p r o b l e m (SC II-1) was essentially equivalent to cube
duplication. In fact, it is easy to see that, if one replaces the base circles of the
cylinders in SC I I - l ' with squares circumscribed on these circles, the p r o b l e m turns
19For compound ratio, see SC 11-4. As for multiplicate ratio, Archimedes uses it in SC 1-32, 33, and
34, he also refers to the results of his predecessors (cf. Elements XII-2 and 18) in terms of multiplicate
ratio in his preface to Quadrature of Parabolas. Moreover, he even goes on to introduce a "sesquialteral"
ratio in SC 11-8 (if a: b = b:c = c:d, then a:d is the sesquilateral ratio of a:c).
134
KEN SAITO
HM 22
M
G
B
F
D
H
FIGURE 7
into the geometrically represented form of cube duplication [CDxg]; Archimedes
indeed performs this very substitution in his argument.
Under these circumstances, Archimedes' argument in SC II-1 seems to suggest
something more than his preference for the technique of "reduction to linear ratio."
To make our point clear, let us pose a question. The equivalence of the problem of
doubling the cube and finding two mean proportionals was no doubt a commonplace
knowledge for mathematicians at Archimedes' time. What then was its proof at
that time? The assumption that it was based on triplicate ratio (this seems to
be the opinion of the majority of modern scholars) makes it difficult to explain
Archimedes' avoidance of triplicate ratio in SC II-1. It would seem plausible, on
the other hand, that the proof of Hippocrates' reduction of cube duplication was
similar to Archimedes' analysis in SC II-1 and that the latter was in fact an adaptation
of the former.
This latter assumption leads to a natural reconstruction of the reduction of cube
duplication modelled on Archimedes' analysis in SC II-1. This reconstruction goes
as follows:
Let there be given a rectangular prism AB, with square base BC (Fig. 7). It is
required to find a cube EF which is equal to prism AB. Let it be assumed that the
cube EF is constructed. Then, by Fp (XI-34),
sq(CD) :sq(GH) = E G : A C .
(1)
Since EF is a cube, GH = EG. Therefore,
sq(CD) : sq(GH) = G H : A C .
(2)
Let MN be a line such that
sq(GH) = r(CD, MN).
Then
(3)
HM 22
DOUBLING THE CUBE
135
sq(CD) : sq(GH) = sq(CD) : r(CD, MN) = CD :MN.
Therefore,
CD : MN = sq(CD) : sq(GH), that is, G H : AC.
(4)
Then, alternately,
CD : G H = MN : AC.
From (3), by F(VI-16), CD : G H = G H : MN. Therefore,
CD : G H = G H : MN = MN : AC.
(5)
Therefore, GH, the side of the cube to be constructed, is the first of the two mean
proportionals between given two lines CD and AB.
In the reconstruction above, we have given the same numbers to the corresponding relations so as to make the parallelism between the two arguments apparent.
To conclude, let us examine what we have established, and discuss the plausibility
of our analysis and the above reconstruction.
6. C O N C L U D I N G O B S E R V A T I O N S
First, we have established that Archimedes' argument in SC II-1, which seems
at first sight roundabout and tortuous, is in fact a systematic application of a
technique which we call "reduction to linear ratio." In fact, this proposition provides
further evidence that this technique enjoyed wide diffusion and long persistence.
At the same time, we have discovered the significance of equal-volume propositions
(those of type F) in Books XI and XII of the Elements. These results are indispensable for the technique of "reduction to linear ratio" (just as are their counterparts
in plane geometry), and this method enabled Greek mathematicians to avoid using
multiplicate and compound ratios. How old is this technique of "reduction" then?
Although the lack of extant documents prevents us from giving a definitive answer
to this question, the technique would seem fairly ancient. For Archimedes, both
the technique of "reduction" and the method of multiplicate and compound ratios
were clearly available, and certain types of problems could be solved by either of
them. In the Data and the Elements, however, we see a preference for the "reduction" method. There seems to be no reason to believe that this technique was first
introduced by Euclid, since it requires only basic theorems of proportion theory.
At any rate, one should be wary of invoking the use of the multiplicate and compound ratios in reconstructing early Greek geometry, since these methods appear
to have been developed later and are not directly supported by pre-Euclidean
documentary evidence.
Second, the proof of Hippocrates' reduction which we have reconstructed from
Archimedes' argument, or something quite close to this, very probably served as
a standard justification of Hippocrates' reduction. For although our reconstruction
has no direct historical evidence, the assumption that this argument was unfamiliar
would at once entail the following assertions which seem very difficult to sustain:
136
KEN SAITO
HM 22
• The technique of "reduction to linear ratio," which was widely used in a variety
of contexts, was not applied to cube duplication;
• In spite of this, Archimedes adopted the technique in SC II-1, a p r o b l e m similar
to cube duplication, when he could have just as easily utilized the m e t h o d of
multiplicate and c o m p o u n d ratios.
•
BooksXIandXIIofEuclid'sElementscontainsomeequal-volumepropositions
(type F) with no apparent purpose.
Taking these points into consideration, it would seem safe to assume that the
justification of H i p p o c r a t e s ' reduction of cube duplication existed in the form we
have reconstructed it, by no later than A r c h i m e d e s ' time.
Finally, let us briefly discuss the possibility of attributing our reconstruction to
Hippocrates himself. The technique of "reduction to linear ratio" m a y safely be
dated before Eudoxus (fl. ca. 368 B.C.), since Eudoxus's theory of proportion could
justify the use of multiplicate ratios and these were indeed used in XII-1 (see [13,
130-135]). If so, our reconstructed p r o o f would have originated by no later than
the first half of the fourth century B.C., which would be much less than a century
after Hippocrates.
If we m a y be permitted to m a k e some conjectures here, we would like to propose
that H i p p o c r a t e s himself gave a rigorous p r o o f similar to our reconstruction, within
the standards of his time, for his reduction of cube duplication, and that he developed
the technique of "reduction to linear ratio" based on the propositions of type F
contained in his Elements. In this connection, we would emphasize the importance
of the propositions of type F as a tool for the technique of "reduction to linear
ratio" in early G r e e k geometry. The greatest advantage of this approach is that it
is in accordance with the idiosyncrasies in the Elements and other G r e e k mathematical works.
Thus our study of how Hippocrates or his contemporaries m a y have justified
H i p p o c r a t e s ' reduction has resulted in a proposal for a new interpretation of the
d e v e l o p m e n t of the theory of proportion in early G r e e k geometry. We believe that,
despite the disturbing lack of d o c u m e n t a r y evidence, we can still hope to improve
our understanding of pre-Euclidean g e o m e t r y through careful analysis of the theorems and techniques used in the extant texts. W e hope that our approach will
inspire further researches in this direction.
ACKNOWLEDGMENTS
The author expresses his gratitude to John Stape, his ex-colleagueat Chiba University, who generously
offered his assistance in improving the linguistic expression of several of the author's recent articles,
including the present one. Further thanks go to David Fowler, who read earlier versions of this paper
and gave helpful suggestions for improving them. The meticulous and very thorough analysis of one of
the referees enabled me to produce a more concise and less speculative argument. The final version
owes a great deal to David Rowe, whose advice led me to emphasize the possibilityof a new interpretation
of early Greek geometry rather than to focus on one specific problem.
REFERENCES
1. A. Frajese, Attraverso la storia della matematica, Florence: Le Monnier, 1969.
2. A. Frajese, Ed., Opere di Archimede, Turin: UTET, 1974.
H M 22
DOUBLING THE CUBE
137
3. T. L. Heath, A History of Greek Mathematics, 2 vols, Oxford: Clarendon Press, 1921; reprint ed.,
New York: Dover, 1981.
4. T. L. Heath, Ed. The Thirteen Books of Euclid's Elements, 3 vols., 2nd ed., Cambridge: Cambridge
Univ. Press, 1926; reprint ed., New York: Dover, 1956.
5. J. L. Heiberg, Ed., Archimedis opera omnia cure commentariis Eutocii. 3 vols., Leipzig: Teubner,
1910-1915; reprint ed., Stuttgart: Teubner, 1972.
6. F. Hultsch, Ed., Pappus. Collectionis quae supersunt, 3 vols., Berlin, 1876-1878.
7. W. R. Knorr, Archimedes and the pre-Euclidean proportion theory, Archives internationales d'histoire des sciences 28(1978), 183-244.
8. W. R. Knott, The Ancient Tradition of Geometric Problems, Boston: Birkhauser, 1986.
9. W. R. Knorr, Textual Studies in Ancient and Medieval Geometry, Boston: Birkh~iuser, 1989.
10. I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge,
MA: The MIT Press, 1981.
11. E. Neuenschwander, Die stereometrische B0cher der Elemente Euklids. Archive for History of
Exact Sciences 9(1973), 91-125.
12. K. Saito, Compounded Ratio in Euclid and Apollonius, Historia Scientiarum 31(1986), 25-59.
13. K. Saito, Duplicate Ratio in Book VI of the Elements, Historia Scientiarum, 2nd ser., 3(1993), 115-135.
14. A. Szab6, Anfiinge der griechischen Mathematik, Munich: Oldenbourg, 1969.
15. G. J. Toomer, Ed., Diocles. On Burning Mirrors, Berlin: Springer-Verlag, 1976.
16. B. L. van der Waerden, Science Awakening. Groningen, 1954, 3d ed., New York: Oxford Univ.
Press, 1971.
17. P. ver Eecke, Ed., Les oeuvres completes d'Archimdde, 2 vols., Bruges, 1921; reprint ed., Liege:
Vaillant-Carmanne, 1960.
18. M. Wallies, Ed., loannes Philoponus. In Aristotelis Analytica Posteriora Commentaria, Commentaria
in Aristotelem Graeca, Vol. 13, pt. 2, Berlin, 1909.
Branch Points of Algebraic Functions and the Beginnings
of Modern Knot Theory 1
MORITZ EPPLE
AG Geschichte der Mathernatik, Fachbereich 17--Mathematik, Universitiit Mainz,
D-55099 Mainz, Germany
Many of the key ideas which formed modern topology grew out of "normal research" in
one of the mainstream fields of 19th-century mathematical thinking, the theory of complex
algebraic functions. These ideas were eventually divorced from their original context. The
present study discusses an example illustrating this process. During the years 1895-1905, the
Austrian mathematician, Wilhelm Wirtinger, tried to generalize Felix Klein's view of algebraic
functions to the case of several variables. An investigation of the monodromy behavior of
such functions in the neighborhood of singular points led to the first computation of a knot
group. Modern knot theory was then formed after a shift in mathematical perspective took
place regarding the types of problems investigated by Wirtinger, resulting in an elimination
of the context of algebraic functions. This shift, clearly visible in Max Dehn's pioneering work
on knot theory, was related to a deeper change in the normative horizon of mathematical
practice which brought about mathematical modernity. © 1996AcademicPress,Inc.
Viele der Ideen, die die moderne Topologie geprigt haben, stammen aus 'normaler
Forschung' in einer der Hauptstr6mungen des mathematischen Denkens des 19. Jahrhunderts,
der Theorie komplexer algebraischer Funktionen, und wurden erst allmihlich yon ihrem
urspringlichen Kontext abgelOst. Ein Beispiel dieser Entwicklung wird diskutiert. In den
Jahren 1895-1905 versuchte der Osterreichische Mathematiker Wilhelm Wirtinger, Felix Kleins
Sicht algebraischer Funktionen auf den Fall mehrerer Variabler zu verallgemeinern. Eine
Untersuchung des Monodromieverhaltens solcher Funktionen in der Nihe singulirer Punkte
fihrte zur ersten Berechnung einer Knotengruppe. Die moderne Knotentheorie entstand dann
nach einer Verschiebung der mathematischen Perspektive auf die von Wirtinger untersuchten
Fragen, die zu einer Elimination des Kontexts algebraischer Funktionen fihrte. Diese
Verschiebung, die in Max Dehns knotentheoretischen Pionierarbeiten deutlich sichtbar ist,
war mit jener tieferen Verinderung im normativen Horizont der mathematischen Praxis
verknipft, die die mathematische Moderne hervorbrachte. © 1996AcademicPress.Inc.
Beaucoup des iddes centrales qui ont form6es la topologie moderne ont leurs racines dans
la 'recherche normale' dans un des courants les plus importants de la pens6e math6matique
du 196me siBcle, Ces iddes ont alors ~tds s6par6es graduellement de leur contexte originale. Un
exemple de ce d~veloppement sera discut6. Pendant les ann6es 1895-1905, le mathdmaticien
autrichien, Wilhelm Wirtinger, essayait de g6n6raliser le point de vue de Felix Klein sur les
fonctions alg6briques complexes dans le cas de plusieurs variables. Une investigation de la
monodromie de telles fonctions lui amenait ~ la premibre calculation d'un groupe d'un noeud.
La th6orie des nceuds moderne a alors 6t6 form6e aprbs un changement de la perspective
Earlier versions of this paper have been presented at the Mathematical Colloquium at Heidelberg
and the History of Mathematics Meeting in Oberwolfach, April 1994. I have much profited from
interesting discussions on both occasions. The paper is part of a larger research project on the history
of knot theory.
371
0315-0860/96 $12.00
Copyright@1996by AcademicPress,Inc.
All rightsof reproductionin any formreserved.
372
MORITZ EPPLE
HM 22
math6matique sur les probl6mes 6tudi6s par Wirtinger. Ce changement, visible distinctement
dans le travail de Max Dehn sur la th6orie des neeuds, a abouti en une 61imination du contexte
des fonctions alg6briques. I16tait reli6 a un changement plus fondamental de l'horizon normatif
de la pratique math6matique qui amenait h ce qui a 6t6 appel6 la modernit6 math6matique.
© 1995AcademicPress,Inc.
MSC 1991 subject classifications: 01A55, 01A60, 01A80, 32-03, 57-03.
KEY WORDS: discipline formation, rationality, modernity, knots, algebraic functions.
INTRODUCTION
Due to the rapid development and application of new knot invariants in mathematics and physics following Vaughan Jones' discovery of a new knot polynomial,
knot theory has received growing attention within and even outside the mathematical community.2 In this context, it has often been asked why the knot problem--of
all topological problems--was among the first to be studied by early topologists of
our century such as Heinrich Tietze, Max Dehn, James W. Alexander, and Kurt
Reidemeister. This question appears all the more puzzling since 19-century work
on knots had certainly not been at the cutting edge of mainstream mathematical
research--unlike, for instance, the topological problems that arose in connection
with the theory of algebraic functions or algebraic geometry. In the following, an
answer to this question will be given. Using hitherto unpublished correspondence
between the Austrian mathematician, Wilhelm Wirtinger, and Felix Klein, it will
be shown that modern knot theory did in fact originate from these latter fields.
Furthermore, while they were familiar to the pioneers of modern topology, a series
of events gradually left these roots forgotten by their followers.
In considering the origins of modern knot theory, I will do more than merely
retrace the technical developments. Rather, this formation of a new field of mathematical research illustrates a certain pattern which, in a nutshell, may be characterized as follows:
Thesis. What appears, at first sight, to be the invention of a new mathematical
discipline, turns out, on closer inspection, to be the outcome of a rather complex
process of differentiation, and, as I would like to call it, a subsequent elimination
of contexts.
Here the term "differentiation" is taken from the Weberian tradition in sociology.
As is well known, Max Weber has described the formation of modern culture and
society as a process of progressive differentiation of cultural "value spheres" and
domains of social action, the most important of which are science (which Weber
links with technology and industrial production), ethics and religion (linked with
the institutions of law), and art. 3 This picture is interesting for the history of science
2 This breakthrough was first announced in [14]. Since then, a wealth of popular and scientific presentations of knot theory, old and new, have been streaming into the market. Contributors come from all
ranks of the scientific hierarchy, including authorities such as Michael Atiyah [37]. Some articles have
included historical comments, for instance the nice survey by de la Harpe [47]. Przytycki [55] has given
a presentation of some of the combinatorial ideas which led to polynomial knot invariants. The reader
should be aware that most of these treatments are not intended to be serious historical studies.
3 The locus classicus for this view is the "Zwischenbetrachtung" in Vol. 1 of [61].
HM 22
BEGINNINGS OF MODERN KNOT THEORY
373
because of Weber's idea of viewing the formation of modernity from the perspective
of a history of rationality. According to this view, a specific standard of rationality
is associated to each of the different "value spheres" which organizes social practice
in these respective domains. For Weber, the history of modernity is to a large extent
the history of the evolution of these rationality standards. 4 Robert Merton has
applied this perspective to an intermediate stage in the process of differentiation
of science and religion in his ground-breaking study [53]. In contrast to Weber's
and Merton's macrosociological approach, which treats science mostly from an
external perspective, the idea of "differentiation" will be used on a microscopic,
internal level in the context of the following study. This term will denote the
gradual separation of a certain bundle of p r o b l e m s - - p r o b l e m s appearing in a wellestablished field of what Kuhn called normal r e s e a r c h - - f r o m the mainstream of
that field. As we shall see, even on this microscopic level a gradual separation of
different standards of rationality is characteristic for such a process of differentiation.
An "elimination of contexts," on the other hand, marks a critical step in
mathematical (or, more generally, scientific) research. It puts, so to speak, a
previously differentiated complex of problems onto its own feet. As a conscious
or unconscious effect of active decisions taken by scientists, it leads to or
completes a modification of the network of scientific disciplines. Typically, the
decision to accept a new standard of rationality is central in an elimination of
contexts. A change of such standards implies a reevaluation and reorganization
of the manifold elements of scientific practice, including the perceived architecture
of the body of scientific knowledge. On the macroscopic level, an example of
this elimination of contexts is the gradual suppression of religious elements in
science. We shall see that similar phenomena may be observed on the internal
level of mathematical research.
Using these ideas from a history of rationality, 5 we shall be able to trace the
influence of the norms guiding the mathematical community not only in the way
in which mathematical research is embedded into general scientific and social culture, but also in the regulation of choices determining the constitution of the body
of mathematical knowledge itself. In particular, it turns out that we can perceive
in the early history of modern knot theory reflections of the broad changes in
mathematical culture around the turn of the century. This leads to another aspect
which will be central in the following.
The historical narrative to be presented is drawn from the history of topology,
that is, from the history of one of those mathematical disciplines which must be
called genuinely modern--if such a thing exists at all. The events in question
4 A modern presentation of Weber's theory of modernity along these lines is contained in Jttrgen
Habermas' influential [46] see in particular Chap. II.
5 A n important difference between a history of rationality on Weberian lines and attempts to give a
"rational reconstruction" of the history of science in the spirit of Lakatos [49] should be pointed out:
Whereas the latter import the relevant standards of rationality from a particular methodology of science,
the former considers these standards as historical data, to be traced and interpreted by the historian.
374
MORITZ EPPLE
HM 22
occurred during the two decades before and after the turn of the century. They
thus overlap in time with Poincard's writings on Analysis situs, which mark the
disciplinary threshold of topology. Moreover, they are contemporary with the onset
of what Herbert Mehrtens and other writers have called "mathematical modernity,"
marked by events such as the publication of Hilbert's Grundlagen der Geometrie
in 1899 and his famous talk on open mathematical problems delivered at Paris in
1900. (We shall see that both events had an influence on the story to be told.) While
these connections are striking, the present case study should not be understood as
a general theory about the pattern of differentiation and elimination of contexts
in the history of mathematics. Still, this pattern might be typical for the creation
of some modern mathematical theories.
Prelude: Poincar~'s Fundamental Group
Let me first give a brief illustration of this pattern. It concerns one of the basic
notions of topology, the "fundamental group" of a manifold. As is well known,
Poincar6 introduced the fundamental group in his paper on Analysis situs of 1895
[24] in the context of a discussion of the monodromy behavior of multivalued
functions on a manifold.6 In fact, he gave a motivation for his notion in terms of
monodromy and then a definition in terms of homotopy classes of paths. Instead
of simply defining his new notion, he explained the action of closed paths on the
set of values of a certain class of multivalued functions at a given point in the
manifold. (In more modern terms, he considered the action of the fundamental
group on the fiber of the covering associated with a given set of multivalued
functions. In fact, the only condition which Poincar6 required for the function set
implies that this covering is unbranched, so that there are no exceptional fibers.)
He then remarked that the resulting group of permutations of these values (which we
may call the global monodromy group associated with the given class of functions) is
always a homomorphic image of the group of path classes, which therefore is
rightly considered "fundamental," at least from the point of view of monodromy
considerations.
Poincar6's text documents the last step in a process of differentiation. Investigations of the monodromy behaviour of multivalued functions in the neighborhood
of singular points had been normal research problems in analytic (or algebraic)
function theory on surfaces since the time of Puiseux and Riemann. The term
"monodromy group" was coined by Camille Jordan in his Trait~ des substitutions
et des ~quations algObriques of 1870 [15]. The idea of the group had already been
implicit in Victor Puiseux's Recherches sur les fonctions algObriques of 1850 [27].
Therein, Puiseux had examined the permutations of the roots of a polynomial
equation with rational functions as coefficients induced by analytic continuation
along small loops around branch points. One year later, Charles Hermite identified
6A historical discussion of the developments leading to Poincar6's notion was given by vanden
Eynde [45].
HM 22
BEGINNINGS OF M O D E R N KNOT T H E O R Y
375
this set explicitly with the Galois g r o u p of the defining e q u a t i o n [11]. 7 T h e c o n c e p t
of m o n o d r o m y soon f o u n d i m p o r t a n t applications, for instance in the context of
differential equations. In 1895, Poincar6 was on the verge of separating a particular
c o m p l e x of questions f r o m this context by introducing the new n o t i o n of the
f u n d a m e n t a l group. H e was well aware that it was precisely the context which
m a t t e r e d to him and his c o n t e m p o r a r y mathematicians; this functioned as a source
of legitimation for the study of the topological questions involved. T h e r e f o r e , in
the a n n o u n c e m e n t of his w o r k on Analysis situs [23], which was i n t e n d e d to
convince readers of the i m p o r t a n c e of the new field for topics like analytic
functions or algebraic g e o m e t r y , Poincar6 restricted himself to explaining the
f u n d a m e n t a l g r o u p in terms of the m o n o d r o m y b e h a v i o r of multivalued functions
on a surface. There, it is simply the " m a x i m a l " global m o n o d r o m y g r o u p (belonging
to " t h e most g e n e r a l " set of multivalued functions, associated with what was not
yet called the universal covering of a surface). The g r o u p of path classes w a s
not even mentioned. 8
A t the same time, Poincar6's n o t i o n o p e n e d up the possibility of eliminating the
context of m o n o d r o m y considerations. A l r e a d y the text of [24] allows one to isolate
conceptually the n o t i o n of the f u n d a m e n t a l g r o u p f r o m its action on multivalued
functions. A n d this is exactly w h a t h a p p e n e d later on. 9 T h e central step which led
to this elimination was w h e n Poincar6 decided to redefine a certain class of manifolds
in a purely combinatorial way, a step which he took, m o t i v a t e d by Poul H e e g a a r d ' s
criticisms, in the first Compl~ment d l'Analysis situs of 1899 [26]. Here, we find the
first hints at a new s t a n d a r d of rationality for topological argumentation. It was
established after the turn of the century by those m a t h e m a t i c i a n s w h o a d v o c a t e d
an axiomatic, purely combinatorial a p p r o a c h to topology. While Poincar6's first
Compl~ment d l'Analysis situs did not m e n t i o n the f u n d a m e n t a l group, it was clear
that a combinatorial n o t i o n of manifolds offered new possibilities for viewing the
f u n d a m e n t a l group, too. It was Heinrich Tietze w h o t o o k this step in his Habilitationsschrift of 1908, by reducing all t h e n - k n o w n topological invariants of threedimensional manifolds to the f u n d a m e n t a l group.l° This g r o u p was n o w introduced
and investigated by m e a n s of a g r o u p presentation associated with a given combinatorial complex. A combinatorial n o t i o n of h o m e o m o r p h i s m was i n t r o d u c e d which
enabled Tietze to show the invariance of the f u n d a m e n t a l g r o u p by means of
combinatorial g r o u p theory. N o m e n t i o n was m a d e of algebraic or analytic functions
7 See [63, 118 f.]. It is a pity that Hermite's short paper escaped vanden Eynde's notice in [45] since
it shows, in fact, that group-theoretic thinking was in the air, at least immediately following Puiseux's
work. However, it was not the fundamental group, but the monodromy group which was studied then.
It seems to be a general shortcoming of vanden Eynde's article that she underestimates the role of
monodromy considerations in the developments leading to the notion of the fundamental group.
s The notion of equivalence classes of paths was familiar to Poincar6 by 1883 from his work on the
uniformization problem. This problem represents another important source of topological notions in
the context of complex function theory. See [45].
9 A thorough discussion of the relations between the fundamental group and coverings of a given
manifold was taken up by Reidemeister [28] and Seifert and Threlfall [30].
10[32]; see [38] for a survey of the combinatorial parts of Tietze's paper.
376
MORITZ
EPPLE
H M 22
and their monodromy behavior. (See [38, 160-162].) Only after this elimination of
the specific motivating context did the notion of the fundamental group of a manifold
acquire its broad significance in topological research. In particular, mathematicians
interested in the possibilities of the new discipline, such as Tietze and Dehn, could
now use the notion without necessarily knowing a good deal of function theory.
New problems could be posed and treated which did not presuppose a connection
with algebraic functions or algebraic geometry, a famous example being Poincar6's
conjecture regarding the 3-sphere. n
This early history of the notion of the fundamental group discloses more than
it may seem to at first glance. We shall see that the first steps of modern knot
theory were the outcome of a line of thought which is astonishingly close to
that just mentioned. Again, investigations of the monodromy behavior of algebraic
functions--in this case, of two complex variables--led to a topological notion,
which, after an elimination of the original context and a change in thought
style, became known as the group of a knot. Seen from a purely mathematical
perspective, this may come as no great surprise, since knot groups are special
cases of fundamental groups. Seen historically, however, the parallel is rather
instructive, since in the beginning at least, the two lines of thought evolved
independently. This underlines the relevance of the specific pattern of transformation, as well as the importance of late 19th-century research on algebraic functions
for the birth of topology.
DIFFERENTIATION
The First Result of Modern Knot Theory: The Impossibility of Disentangling the
Trefoil Knot
Let us begin with an explanation of what "modern knot theory" shall be taken
to mean in the following. The problem of classifying knots (or rather, plane knot
diagrams) apparently already puzzled Gauss back in the 1820's (see, e.g., [55]). We
also possess a remarkable letter from Betti, who reported on conversations with
Riemann that document the importance which Gauss attached to the knot problem
in his later days. (Betti to Tardy, 6.10.1863. See [62].) In fact, Gauss regarded this
as one of the paradigmatic problems of Analysis situs. The next main episode in
the history of knot theory was the beginning of knot tabulations by Peter Guthrie
Tait and his followers, working in the Scottish context of Lord Kelvin's speculations
about a theory of vortex atoms. 12 (In passing, note that again it was the context
u For information on the early history of this conjecture, see [60].
12 From a mathematical point of view, 19th-century knot tabulations have been discussed in detail
by Thistlethwaite [59]. On Thomson's speculations, a standard reference is [58]. So far, no detailed
treatment of the connections between this proposal of an atomic model and Tait's tabulations has been
given. A rather interesting line of topological thought links the Scottish physicists to Riemann's ideas
about connectivity. In particular, Thomson's papers contain, though in a rather vague way, the claim
that the first Betti number of knot complements equals one in all cases. I plan to investigate these ideas
on another occasion.
HM 22
377
that served to legitimize non-normal research.) However, the first serious published
proofs of results on knots date from the beginning of our century. They were
contained in the above-mentioned article by Tietze and in a series of pioneering
papers by Max D e h n that a p p e a r e d in the years 1910-1914. Soon thereafter, knot
theory attracted several talented young mathematicians. After the interruption
caused by World W a r I, Otto Schreier, Kurt Reidemeister, Emil Artin, and J a m e s
Wadell Alexander were the first to take up the knot problem. It became the subject
of journal a r t i c l e s - - m o s t of which were published in a new journal with a m o d e r n
outlook, the A b h a n d l u n g e n aus dem mathematischen S e m i n a r der H a m b u r g i s c h e n
Universitiit, founded in 1922. Ten years later, the first m o n o g r a p h appeared: Reidemeister's Knotentheorie [29]. It was during this period that " m o d e r n knot t h e o r y " that is, knot theory as part of mathematical m o d e r n i t y - - w a s formed. (In this study,
the problem of whether or when the " m o d e r n " period of knot theory ended will
be left open.)
In 1908, Heinrich Tietze published the first result of m o d e r n knot theory. In a
section of his Habilitationsschrift which contains a discussion of 3-manifolds embedded in ordinary 3-space (called " d e v e l o p p a b l e dreidimensionale Mannigfaltigkeiten" by Tietze), he presented an argument for the impossibility of disentangling
a trefoil knot. H e began by mentioning that the fundamental group of a solid torus
( e m b e d d e d in 3-space) is the infinite cyclic group. A h o m e o m o r p h i c manifold is
given, he continued, by boring a cylindrical channel out of a solid ball. " I f one
instead would bore a knotted channel out of the ball as in fig. 3, then the fundamental
group of the resulting manifold would be generated by two operations satisfying
the relation s t s = tst so that this manifold cannot be h o m e o m o r p h i c with that first
mentioned. ''13 (See Fig. 1.)
Tietze gave his result without proof, in fact without even giving a hint at the
method used to compute the group. H e merely mentioned an earlier note stating
this result in a local Viennese journal [31]. A closer reading of his Habilitationsschrift,
however, indicates where the result came from, namely, from another mathematician who was working in a completely different context. Tietze buried this information in a section far r e m o v e d from the passage cited above, and without calling
attention to the connection between the two passages. Later readers, such as Dehn,
who were interested mainly in combinatorial topology, appear to have hardly read
23"Greifen wir etwa die von einer einzigen Fl~iche vom Geschlechte 1 berandeten developpablen
Mannigfaltigkeiten heraus. Das einfachste Beispiel einer solchen Mannigfaltigkeit stellt der von einer
Torusflache berandete Teil des ~3 vor. Die Fundamentalgruppe dieser Mannigfaltigkeit ist die aus einer
Operation, fiir die keine definierende Relation besteht, erzeugte zyklische Gruppe unendlich hoher
Ordnung. Eine dieser Mannigfaltigkeit hom6omorphe erh~ilt man, wenn man aus einer Kugel einen
zylindrischen Kanal ausbohrt. Wt~rde man start dessen einen verknoteten Kanal wie in Fig. 3 aus
der Kugel ausbohren, so w~ire die Fundamentalgruppe der so entstandenen Mannigfaltigkeit aus zwei
erzeugenden Operationen mit der Relation sts = tst aufgebaut, so dab diese Mannigfaltigkeit mit der
erstgenannten nicht hom6omorph sein kann." [32, 81]
378
MORITZ EPPLE
H M 22
f
/
:,"
, I, ",
i
|
|
'f J
J%
t$
Rn
"--,I
Fio 1. Tietze's Fig. 3.
this section at all. 14 It treats what Tietze (following H e e g a a r d [10]) called " R i e m a n n
spaces," that is, t h r e e - d i m e n s i o n a l a n a l o g u e s of R i e m a n n surfaces [32, Sect. 18].
T h e r e we find the n a m e of W i l h e l m W i r t i n g e r , T i e t z e ' s o l d e r colleague, w h o actually
had i n d u c e d h i m to t u r n to t o p o l o g y for his Habilitation. I n a n a u t o b i o g r a p h y ,
Tietze w r o t e in 1960 a b o u t his early career:
[After finishing my dissertation] a strong impression was made on me by Wirtinger who had
come from Innsbruck to Vienna and who, when speaking of algebraic functions and their
integrals in lectures and seminars, pointed out that topological elements lie at the basis of
this theory. 15
M o r e o v e r , in T i e t z e ' s Sect. 18, we find n o t only the missing t e c h n i q u e b u t also the
missing context. Just o n e single s e n t e n c e refers to it: the " i n v e s t i g a t i o n of the
f u n c t i o n of two c o m p l e x variables r e p r e s e n t e d by C a r d a n ' s f o r m u l a " [32, 105].
T i e t z e ' s text r e p r e s e n t s the critical step in a striking e x a m p l e of " c o n t e x t - e l i m i n a t i o n . " His decision to state the k n o t - t h e o r e t i c a l result w i t h o u t specifying the c o n t e x t
in which it was r o o t e d led others to credit h i m with h a v i n g i n i t i a t e d m o d e r n k n o t
t h e o r y (at least b y h a v i n g asked the right questions). F o r a b o u t 20 years at least,
this virtually e l i m i n a t e d W i r t i n g e r ' s w o r k f r o m the collective c o n s c i o u s n e s s of the
14Tietze's article consisted of two parts. In Sects. 1-14, he developed a combinatorial theory of
topological invariants (Betti and torsion numbers, the fundamental group). In Sects. 15-22, he discussed
problems which did not yet seem tractable in a purely combinatorial fashion (e.g., embeddings of
manifolds), admitting that the standard of rigour reached in the first part could not be maintained in
the second; see [32, 80 ff]. Most readers appear to have concentrated on the first part. References to
the second part are extremely rare in later papers on knot theory, although parts of its content came
to be known later on via an oral tradition, see below.
15"[Nach meiner Dissertation] erhielt ich einen starken Eindruck von Wirtinger, der von Innsbruck
nach Wien gekommen war und in Vorlesungen und "Ubungen, wenn er auf algebraische Funktionen
und ihre Integrale zu sprechen kam, darauf hindeutete, dab es topologische Momente sind, die dem
Aufbau zugrunde liegen." Quoted from [42, 78]. In a footnote at the beginning of his study (no. 10),
Tietze explained that its starting point had been suggested to him by Wirtinger. The latter was also one
of the three editors of the Viennese Monatshefie, in which Tietze's Habilitationsschrifiwas published.
HM 22
379
mathematical community. Today, most knot theorists know the central parts of
Wirtinger's r e s u l t s - - b u t few know that they are due to Wirtinger and even fewer
that this piece of mathematics, rather than postdating the invention of knot theory,
actually furnished the basic tool for treating the subject, the group of a knot.
Wirtinger's Approach to Branch Points of Algebraic Functions
of Two Complex Variables
Before turning to the piece of mathematics in question, it seems appropriate to
include a short characterization of its author. Wilhelm Wirtinger was born in 1865
in the little town of Ybbs in Lower Austria, the son of a physician. Even in school
he seems to have read some mathematical classics, including some of R i e m a n n ' s
works. Whatever he may have understood from these, he was to b e c o m e a renowned
specialist in geometric function theory. H e took his doctorate in 1887 under the
Viennese mathematician, Emil Weyr, and continued his studies during a stay in
Berlin and GOttingen. In GOttingen, he participated in Felix Klein's seminar and
thus established one of the most important connections of his professional life. In
1890, he habilitated in Vienna, and after a period at the Technische Hochschule in
Innsbruck, he received a call to Vienna in 1903, where he remained for the rest of his
professional career. During his years in Innsbruck, he published widely appreciated
papers on Abelian and theta functions. This recognition from his colleagues culminated in 1907 when he was awarded the Sylvester Medal by the Royal Society of
London. Wirtinger's further academic career went smoothly (further details are
given below), and he retired in 1935. H e died in 1945 in his hometown, Ybbs.
Let us now discuss the piece of mathematical work that led to the first calculation
of a knot group. This was the discovery of the connection between knots and the
topology of singular points of algebraic curves, a finding usually attributed to Karl
Brauner, who published a three-part article on the subject in 1928 in the Hamburger
Abhandlungen [3]. 16 Brauner was one of Wirtinger's students, and this article was
his Habilitationsschrifi. ~7Moreover, the central idea of this article was clearly due
to Wirtinger, even though the latter never chose to publish it. In his report on
Brauner's tlabilitation, Wirtinger wrote: " M o r e than twenty years ago, the referee
showed the way in which these difficult, but basic problems may be dealt with. ''is
The only printed documentation of Wirtinger's earlier work is the title of a talk
he gave in 1905 at the annual meeting of the Deutsche Mathematiker-Vereinigung:
" O b e r die Verzweigungen bei Funktionen von zwei Ver~inderlichen" [35]. However,
we are in a good position to reconstruct his work. On the one hand, an oral tradition
dating back to Wirtinger's lectures in Vienna is documented in several early papers
on knot theory by Schreier, Artin, and Reidemeister, in addition to Tietze's and
16See for instance [36, Sect. 1.4; 56, 159; 54, 3 f.; 39, 415; or 47, 243].
17Brauner was definitely not a first-rate mathematician. After his Habilitationsschrifi,he published
nothing of importance. In the 1930s and 1940s, he became a convinced supporter of the Nazis, and in
1945 he was removed from his chair in Graz. See [42, 249].
~8"Der Berichterstatter hat vor mehr als zwanzig Jahren den Weg angegeben, auf welchem diesen
schwer zug~inglichen, abet grundlegenden Problemen beizukommen ist." Quoted from [42, 247].
380
M O R I T Z EPPLE
HM 22
B r a u n e r ' s texts. A c t u a l l y , all o f t h e s e m a t h e m a t i c i a n s h a d a t t e n d e d W i r t i n g e r ' s
l e c t u r e s at o n e t i m e o r a n o t h e r [42, 18]. C o m p a r i n g t h e a s c r i p t i o n s t h e s e texts m a d e
to W i r t i n g e r l e a d s to a r a t h e r c l e a r picture. F o r t u n a t e l y , this p i c t u r e is fully c o n f i r m e d
a n d e v e n e x t e n d e d b y a series o f l e t t e r s i n c l u d e d in W i r t i n g e r ' s c o r r e s p o n d e n c e
with t h e p o w e r f u l m a t h e m a t i c i a n w h o h a d f r o m e a r l y on g u i d e d a n d s u p p o r t e d his
c a r e e r , F e l i x Klein. 19 W h a t follows is a fairly d e t a i l e d d e s c r i p t i o n o f W i r t i n g e r ' s
ideas, b a s e d o n t h e s e letters. F r o m t h e m , o n e can follow the g r a d u a l p r o c e s s o f
d i f f e r e n t i a t i o n which e n d e d in t h e first t r e a t m e n t s o f k n o t groups.
T h e first l e t t e r to K l e i n w h i c h is r e l e v a n t h e r e d a t e s f r o m D e c e m b e r 22, 1894. It
c o n t a i n s a sort o f a n n u a l r e p o r t o n W i r t i n g e r ' s work. A m o n g o t h e r subjects, he
writes about a new research project:
For functions of several variables, I have another project, namely to investigate whether the
bilinear differential form in question can be determined in such a way that the real and
imaginary parts of such a complex function on an arbitrary manifold remain potentials, too. 2°
This p r o j e c t a m o u n t e d to n o t h i n g less t h a n an e x t e n s i o n o f K l e i n ' s view o f a l g e b r a i c
functions, b a s e d on the t h e o r y of p o t e n t i a l f u n c t i o n s on surfaces, to t h e case o f
s e v e r a l v a r i a b l e s . E v i d e n t l y , W i r t i n g e r ' s p r o p o s a l was to view, as K l e i n h a d successfully d o n e for o n e v a r i a b l e in his t r e a t i s e Uber R i e m a n n s Theorie der algebraischen
Funktionen u n d ihrer Integrale o f 1882, t h e v a r i e t y a s s o c i a t e d with an a l g e b r a i c
f u n c t i o n o f n c o m p l e x v a r i a b l e s as a 2 n - d i m e n s i o n a l real m a n i f o l d w h o s e c o m p l e x
s t r u c t u r e is d e t e r m i n e d b y a R i e m a n n i a n metric. T h e class o f r e a l p a r t s o f a l g e b r a i c
f u n c t i o n s d e f i n e d on such a v a r i e t y s h o u l d t h e n - - s o W i r t i n g e r h o p e d - - b e i n c l u d e d
in t h e class o f h a r m o n i c f u n c t i o n s o n this R i e m a n n i a n m a n i f o l d . 21
A t t h e time, t h e s t u d y o f a l g e b r a i c f u n c t i o n s o f two o r m o r e v a r i a b l e s was a still
y o u n g a n d flourishing field o f r e s e a r c h . W h e n W i r t i n g e r c o n c e i v e d his p r o j e c t , o n l y
a few s t u d i e s h a d a d d r e s s e d this n a t u r a l e x t e n s i o n o f R i e m a n n ' s w o r k on a l g e b r a i c
f u n c t i o n s a n d t h e i r integrals. F r o m a r o u n d 1870 o n w a r d s , M a x N o e t h e r a n d l a t e r
l~mile P i c a r d h a d s t u d i e d a l g e b r a i c f u n c t i o n s z o f two c o m p l e x v a r i a b l e s x a n d y
given b y a p o l y n o m i a l e q u a t i o n
f ( x , y, z) = O,
x, y, z E C.
I n p a r t i c u l a r , t h e y h a d discussed r e s o l u t i o n s o f singularities b y m e a n s o f r a t i o n a l
t r a n s f o r m a t i o n s . C l e b s c h a n d s e v e r a l I t a l i a n g e o m e t e r s h a d also c o n s i d e r e d special
19Wirtinger's letters to Klein are contained in the Klein Nachlass in NSUB G6ttingen, Cod. Ms.
Klein XII, 364-412. This includes ca. 50 letters dating from 1890 to 1924. At present, I do not know
whether the other half of this correspondence is still extant.
20 "FOr die Functionen mehrerer Variablen habe ich noch ein Project, n~imlich zu untersuchen, ob
sich die bewusste bilineare Differentialform nicht so bestimmen l~isst,dass der reelle u. imagin~ire Theil
einer solchen complexen Function auf beliebiger Mannigfaltigkeit auch Potentiale bleiben."
21 For the plane C n, it had been shown by Poincar6 in 1883 that a straightforward extension of the
approach to complex functions in one variable via harmonic functions was impossible. There exist
harmonic functions of several complex variables which are not real parts of analytic functions. Still, the
reverse inclusion holds, so that potential theory can be applied to complex functions of several variables.
See [22; 25].
HM 22
B E G I N N I N G S O F M O D E R N KNOT T H E O R Y
381
classes o f a l g e b r a i c f u n c t i o n s of t w o v a r i a b l e s , i n t e r p r e t i n g t h e s e as a l g e b r a i c surfaces. It s o o n b e c a m e c l e a r t h a t o n e of t h e m a j o r d i f f e r e n c e s b e t w e e n a l g e b r a i c
f u n c t i o n s o f o n e a n d o f two v a r i a b l e s was the m u c h m o r e i n v o l v e d t o p o l o g i c a l
s i t u a t i o n s t h a t arise in t h e l a t t e r case. A l g e b r a i c surfaces w e r e c o m p l i c a t e d o b j e c t s
of f o u r r e a l d i m e n s i o n s i m m e r s e d in a s p a c e o f six r e a l d i m e n s i o n s . F o r instance,
the set o f singularities consists n o t o f i s o l a t e d points, as is t h e case for o n e v a r i a b l e ,
b u t is an a l g e b r a i c curve, given b y the d i s c r i m i n a n t D of t h e defining p o l y n o m i a l f :
Dr(x, y) = O,
x, y ~ C.
In an influential m e m o i r [20], P i c a r d h a d e m p h a s i z e d this a s p e c t with r e g a r d to t h e
singularities o f a l g e b r a i c functions o f t w o v a r i a b l e s . W h e n in 1897 he a n d G e o r g e s
S i m a r t p u b l i s h e d t h e first m o n o g r a p h o n such functions, a s u b s t a n t i a l c h a p t e r o f
t h e i r b o o k was d e v o t e d to Analysis situs.
A l l this m a k e s c l e a r t h a t W i r t i n g e r ' s p r o j e c t - - h o w e v e r n a t u r a l in an e s t a b l i s h e d
line o f r e s e a r c h - - w a s a f o r m i d a b l e one. W e shall see t h a t W i r t i n g e r ' s a m b i t i o n s
s o o n b o i l e d d o w n to a m u c h m o r e l i m i t e d d o m a i n o f questions. H e , too, was a w a r e
o f t h e t o p o l o g i c a l p r o b l e m s w h i c h his p r o j e c t w o u l d pose. I n his l e t t e r to K l e i n ,
he c o n t i n u e d :
o f course, the faculty of imagination must here be educated and extended essentially. Let me
just mention as an example that in 4-dimensional space, a surface of integration and a surface
of singularities may be linked together like two rings in the three-dimensional domain. The
surface of integration may then be deformed arbitrarily but cannot be reduced to a point.
To grasp all this in a typical and general way will not be easy, but it must be done in the
end if the consideration of complex functions of several variables will not be restricted to the
most elementary facts.22
•
.
.
W e shall s o o n see t h a t W i r t i n g e r ' s s u g g e s t i o n t h a t the t o p o l o g i c a l difficulties of t h e
s u b j e c t c a l l e d for a t r a i n i n g of m a t h e m a t i c a l i n t u i t i o n was n o t just a p a s s i n g r e m a r k .
E x a c t l y o n e y e a r later, o n D e c e m b e r 22, 1895, in his n e x t " a n n u a l r e p o r t , "
W i r t i n g e r c o u l d write to K l e i n a b o u t his first successes. T h e c o m p l e t e text of t h e
letter, which t h r o w s an i n t e r e s t i n g light o n his r e l a t i o n s h i p to K l e i n , too, is given
in t h e a p p e n d i x . In this letter, w e find t h e first signs of t h e d i f f e r e n t i a t i o n o f a
c e r t a i n p r o b l e m f r o m t h e c o n t e x t of f u n c t i o n t h e o r y which l a t e r t u r n e d o u t to b e
decisive. W h a t W i r t i n g e r in t h e l e t t e r calls " t h e c o r e o f t h e w h o l e s u b j e c t ' - - t h e
i n v e s t i g a t i o n o f t h e b r a n c h i n g s i n g u l a r i t i e s o f a l g e b r a i c f u n c t i o n s - - w i l l b e transf o r m e d into the i n v e s t i g a t i o n o f k n o t g r o u p s 15 y e a r s later. H e r e I shall try to
i s o l a t e a n d i n t e r p r e t t h o s e a s p e c t s o f W i r t i n g e r ' s s t a t e m e n t s which p e r t a i n to this development.
22 "Freilich muss hier das Vorstellungsverm6gen wesentlich geschult u. erweitert werden. Ich erw~ihne
nur beispielsweise, dass eine Integrationsfl~iche u. eine Singularit~itenfl~icheim Raum yon 4 Dimensionen
so ineinander h~ingen kOnnen, wie zwei Ringe im dreidimensionalen Gebiet. Die Integrationsfl~iche
kann dann beliebig verschoben u. vefiindert werden, aber nicht auf einen Punct reducirt werden . . . .
Alles dieses typisch u. allgemein zu erfassen wird nicht leicht sein, aber doch schliesslich gemacht
werden mtissen, wenn man die Betrachtung complexer Functionen mehrerer Variablen nicht auf das
allerelementarste beschranken will. . . . "
382
MORITZ EPPLE
HM 22
As was usual at this time, Wirtinger viewed algebraic functions of two variables
as branched coverings of the complex plane C 2, in modern notation,
p:{(x,y,z) EC3:f(x,y,z)
= 0}---~ C 2,
(x,y,z)~--~(x,y).
(A similar situation obtains for more than two variables). He then distinguished
two kinds of branch points, according to whether the sheets of the covering are
permuted cyclically along a closed path around the branch point or not. H e realized
that at regular points of the branch curve, where an analogue of the Puiseux
expansion of the algebraic function is available, the sheets are permuted cyclically,
while the situation is unclear at its singular points. At such points, say (x0, Yo), he
interpreted the defining polynomial f as a polynomial in z with coefficients in the
"Rationalit~itsbereich" of power series a(x, y), convergent in a neighborhood of
(x0, Y0) (probably, he meant the quotient field of this ring):
f ( x , y, z) = ao(x, y) + al(x, y ) z + " ' " + an(x, y ) z n = O.
It was known that the Galois group of the corresponding equation for the case of
one variable, over the field of rational functions, coincides with the m o n o d r o m y
group of the unbranched covering of the Riemann number sphere with the set of
branch points removed. (This had first been recognized by Hermite; see above).
Evidently, Wirtinger extended this insight to his higher-dimensional, local situation,
claiming that the Galois group of his equation equals the "local" m o n o d r o m y group
of the associated covering, that is, the m o n o d r o m y group of the covering of a
neighborhood of (x0, Y0) with the branch curve taken out. Thus he could say that
"this group characterizes the branch p o i n t " - - w e should add, topologically, e3
Wirtinger then gave an example. It involves the general equation of order 3. He
considered the algebraic function z of x and y given by
f(x,y,z)
= z 3 + 3xz + 2y = O.
The equation of the branch curve then is
D f ( x , y) = x 3 + y2 = O.
This cubic has a cusp in (0, 0). Since the Galois group of a general equation is the
full symmetric group, there exist closed paths in the neighborhood of (0, 0), which
induce arbitrary permutations among the three branches of the algebraic function
defined by Wirtinger's equation. Thus the singular branch point is not of the cyclical kind.
This example remained paradigmatic for all later work. We shall see that while
23It is not clear to me what Wirtinger meant in the letter by saying that a neighborhood of a singular
branch point is not homeomorphic to an n-cell and has a "certain connectivity." Is he speaking of a
neighborhood of the branch point in the base--with the branching manifold taken out--or in the total
space of the covering? Certainly, there is an argument in the air concerning the first possibility. Since
the fundamental group of such a neighborhood has a non-abelian homomorphic image--the Galois
group--it cannot be abelian either. However, the sources do not ascribe such an argument to him.
H M 22
383
p u r s u i n g t h e q u e s t i o n s h e n o w h a d p u t to h i m s e l f - - t o s p e c i f y g e n e r a l c o n d i t i o n s
w h i c h a g r o u p o f p e r m u t a t i o n s h a d to satisfy in o r d e r to o c c u r as t h e local m o n o d r o m y g r o u p a s s o c i a t e d to a s i n g u l a r b r a n c h p o i n t - - W i r t i n g e r was l e d to t h e first
c a l c u l a t i o n o f a k n o t g r o u p , a r e s u l t m a d e p u b l i c b y T i e t z e in 1908.
I n t h e f o l l o w i n g y e a r s , h o w e v e r , W i r t i n g e r r e m a i n e d s i l e n t a b o u t his p r o j e c t . H e
h a d e n o u g h else to do, for i n s t a n c e i n w o r k i n g w i t h M a x N o e t h e r to p r e p a r e a n
e x t e n s i v e s u p p l e m e n t to R i e m a n n ' s c o l l e c t e d w o r k s , a n d w r i t i n g t h e article o n
a l g e b r a i c f u n c t i o n s a n d t h e i r i n t e g r a l s for t h e Enzyklopiidie der mathematischen
Wissenschaften [34]. F e l i x K l e i n , w h o t h o u g h t h i g h l y o f his A u s t r i a n a d m i r e r , p r o b a b l y h a d his h a n d s in t h e a r r a n g e m e n t o f b o t h tasks. S i g n i f i c a n t l y e n o u g h , t h e Enzyklopiidie article c o n t a i n e d n e a r l y n o t h i n g a b o u t f u n c t i o n s o f s e v e r a l v a r i a b l e s ,
a n d we find W i r t i n g e r w r i t i n g :
• . . in general, the theory of functions of several variables has not yet been developed very
far. In particular, one has not yet succeeded in determining a given algebraic variety by a finite
number of data in a similar way as is possible with the different forms of a Riemann surface.
These investigations presuppose a thorough treatment of analysissitus for several dimensions.24
W i r t i n g e r r e t u r n e d to t h e s u b j e c t in a l e t t e r d a t e d A u g u s t 26, 1903. F i n a l l y , h e
a n n o u n c e d a r e s u l t w h i c h h e i n t e n d e d to p r e s e n t p u b l i c l y :
For the branchings of algebraic functions of several variables I have made a completely elementary study whose only aim is to make clear how it happens, and how it must be imagined
topologically, that along the connected discriminant manifold only two branches are connected
in general, while in its singular points perhaps arbitrarily many [branches are connected]. I
wanted to report on these things in Kassel, but it has again become unclear whether I shall
be able to go there. 25
A c t u a l l y , W i r t i n g e r did n o t t r a v e l to t h e a n n u a l m e e t i n g of t h e Deutsche Mathematiker-Vereinigung in K a s s e l i n S e p t e m b e r 1903, d u e to p r o b l e m s w i t h his ear. 26
A n o t h e r t w o y e a r s p a s s e d b e f o r e h e g a v e a t a l k o n this s u b j e c t in 1905.
24 ,,... dass die Theorie der Funktionen mehrerer Variablen tiberhaupt noch wenig ausgebildet ist.
Es ist im besondern noch nicht gelungen, das einzelne algebraische Gebilde durch eine endliche Anzahl
von Bestimmungsstticken in ahnlicher Weise festzulegen, wie dies bei den verschiedenen Formen der
Riemannschen Fl~iche mOglich ist. Die Untersuchungen selbst setzen eine eingehende Bearbeitung der
Analysis situs ftir mehrere Dimensionen voraus" [34,174]. For this last task, Wirtinger refers to Poincar6's
texts [24; 26]. The middle sentence contains an allusion to his own work. One way to determine a
Riemann surface had been studied by Hurwitz [13]: by specifying the number of sheets, the position
of its branch points, and the local monodromy behavior at these points, i.e., a certain sheet permutation
associated with each branch point• Wirtinger's thoughts focused on a generalization of the last of these
three aspects.
25,,... l)ber die Verzweigungen algebraischer Functionen mehrerer Variablen babe ich eine ganz
elementare Untersuchung angestellt, welche nur den Zweck hat, klar zu stellen, wie es kommt u. wie man
sich topologisch vorzustellen hat, dass l~ings der zusammenh~ingenden Discriminantenmannigfaltigkeit
iiberall im allgemeinen nur zwei Zweige zusammenh~ingen, dagegen in deren singul~iren Punkten unter
Umstgnden beliebig viele, l]ber diese Dinge wollte ich in Cassel berichten, ob es mir abet m6glich sein
wird hinzukommen ist wieder zweifelhaft geworden . . . . "
26Wirtinger to Klein, Cod. Ms. Klein XII, 409; probably autumn 1903.
384
MORITZ EPPLE
HM 22
Taking together what we find in [1; 3; 32], it is not too difficult to reconstruct the
contents of that lecture. The main source is Sect. 18 of Tietze's Habilitationsschrift;
Brauner's paper also makes it possible to add some computational details. In the
talk, a crucial new idea came into play which, according to Tietze, Wirtinger had
taken from Heegaard's dissertation of 1898, entitled Forstudier til en topologisk
teori for de algebraiske fladers samrnenhceng [10]. As the title indicates, Heegaard
had developed similar interests to those of Wirtinger in a topological treatment of
algebraic functions of two complex variables, but unlike Wirtinger he preferred the
viewpoint of algebraic geometry. Heegaard's idea was to study singular points of
algebraic surfaces by looking at the restriction of the branched covering of C 2
defined by the equation of the surface to a 3-sphere bounding a small neighborhood
of the singular point in question. In this way, one gets a branched covering of the
3-sphere, which is precisely what Heegaard called a " R i e m a n n space" [10, Sect.
13]. In the situation of the paradigmatic example, one obtains (for small positive c)
{(x,y,z) e C3:z 3 -I- 3ZX + 2y = O& IX]2 + lyl 2 = c}
p ; (x, y, z) ~ (x,y)
{(x,y) E C2:lxl 2 + lyl2 = c}
It was now a matter of straightforward calculation for Wirtinger to see that the
restriction of the 'branching manifold to the 3-sphere is a trefoil knot/One simply
had to solve the system of equations
x 3 + y2 = 0
and
Ixl2 + 1212 = c.
The calculation makes clear that the result is a curve which lies on a torus and
winds twice round the first meridian and three times round the second.
Moreover, the restriction transforms the local m o n o d r o m y group characterizing
the original singularity into the global m o n o d r o m y group of the three-sheeted
covering of the exterior of the trefoil knot. Thus in order to study the topological
situation at the branch point, it was enough to look at the three-sheeted covering
of the exterior of the trefoil. As was usual in the case of Riemann surfaces, Wirtinger
now applied more or less intuitive cutting and pasting arguments in order to obtain
generators and relations for this m o n o d r o m y group. There is a picture in Artin's
article [1] which illustrates these techniques (Fig. 2). The same picture had been
described in words by Tietze, and it reappeared in Brauner's article. Finally, it was
reprinted in Reidemeister's Knotentheorie. In all of these cases, the use of this
picture in order to derive a presentation of the knot group is ascribed to Wirtinger. 27
(The idea of the picture itself was again taken from Heegaard's dissertation, where
it had been used to construct what Heegaard had called the "diagram" of a Riemann
space.) This alone suffices to establish the existence of an oral tradition--which in
this case transmitted a picture, an intuition! 28
27In Artin's case, Schreier was the go-between;see [1, 58].
28See [44] for another example and a more philosophical discussion of the importance of cognitive
acts related to such intuitions in the creation of mathematical knowledge.
HM 22
385
FIG 2. Wirtinger's semicylinder.
The first step toward determining a presentation of the m o n o d r o m y group was
to cut the covering into three simply connected sheets. This was achieved by joining
the points on the branch curve by straight lines to a point (conveniently chosen at
infinity in a direction along which the knot projects to a regular knot diagram) and
cutting along the resulting semicylinder with self-intersections. 29 Evidently, the
m o n o d r o m y group is then generated by the sheet permutations associated to penetrations of this semicylinder. In his example, Wirtinger used six generators, associated with the six parts of the semicylinder which lie between the three lines of selfintersection. By looking at small circles which do not circle around the trefoil, the
group relations could be determined. Three cases were to be distinguished: (i) only
one part of the semicylinder is traversed, giving no relation; (ii) two parts of the
semicylinder are penetrated (this may happen a b o v e a line of self-intersection and
reduces the number of generators to three, say r, s, t, associated to the three arcs
in the knot diagram); (iii) four parts of the semicylinder are passed through. This
leads to those relations which today are still called Wirtinger's relations. For the
trefoil, they are given by
1 = r s r - l t -~ = s t - ~ s
lr = t r - l t - ~ s .
29In one dimension lower, the same idea had been used in the 19th century, e.g., by Hurwitz [13].
386
M O R I T Z EPPLE
HM 22
Eliminating r = sts 1 one arrives at the following relation for the sheet p e r m u t a t i o n s
of the covering: sts = t s t - - w h i c h is exactly the relation which a p p e a r e d in Tietze's
a r g u m e n t for the knottedness of the trefoil. In fact, it is clear f r o m Wirtinger's
a r g u m e n t that (s, tlst = tst) is a presentation of the f u n d a m e n t a l g r o u p of the
exterior of the trefoil knot. As an i m m e d i a t e corollary, we obtain an a r g u m e n t for
the fact that this g r o u p is not infinite cyclic. By construction, it has the s y m m e t r i c
g r o u p of o r d e r three (the m o n o d r o m y group) as a h o m o m o r p h i c image. In this
way, Wirtinger had not only characterized the complicated singularity of his e x a m p l e
by m e a n s of the g r o u p of the trefoil, 3° but had also shown h o w to f o r m a very
intuitive picture of the topological situation a r o u n d the b r a n c h point. T h e r e m a r k s
to Klein in his letter of 1903 were thus fully justified.
F r o m the sources, it is not quite clear w h e t h e r Wirtinger was fully aware of the
fact that he had actually d e v e l o p e d m u c h m o r e than what he originally had b e e n
looking for. W h a t he had given was a m e t h o d for deriving not just necessary
conditions on the m o n o d r o m y g r o u p in question, but a presentation of the fundam e n t a l g r o u p of the exterior of an arbitrary knot! While it is not very p r o b a b l e
that a n y o n e w h o had read Poincar6 w o u l d o v e r l o o k this difference, it is astonishing
to realize that even in 1928 B r a u n e r seems to confuse exactly these two aspects of
Wirtinger's procedure. 31 T h e m e t h o d b e c a m e generally k n o w n u n d e r Wirtinger's
n a m e w h e n A r t i n described it in his widely read article on the braid g r o u p [1]. It
is interesting that Artin m a k e s no reference to Tietze's Habilitationsschrift, where
the m e t h o d already had b e e n described in detail. This once m o r e suggests that
Tietze's reference to his advisor escaped the notice of the m a t h e m a t i c a l c o m m u n i t y .
In any case, by 1908 the separation f r o m its original context of the central p r o b l e m
which was to constitute m o d e r n knot t h e o r y was complete. 32
Wirtinger's D e b t to Klein
B e f o r e turning to knot t h e o r y p r o p e r and the final elimination of contexts, it
should be e m p h a s i z e d that Wirtinger's c o m m i t m e n t to the context of a certain view
of algebraic functions was not just a question of mathematics. W e have seen that
w h e n he b e g a n w o r k on his project he viewed it as a natural generalization of an
a p p r o a c h to algebraic functions a d v o c a t e d by one of the mighty figures on the
30TO be precise, he had shown the way to arrive at a topological classification of singular points of
plane algebraic curves. Wirtinger's final result no longer concerned the covering he originally had
intended to consider, but the topology of the base of this covering. See the Prelude above.
31For Brauner, Wirtinger's relations are still monodromy relations and not relations in the fundamental
group of the knot complement. See [3, 4 if].
32Wirtinger's investigation not only made the connection between singularities and knots; it also
contained the first example of a knotted surface in a manifold of four real dimensions. This is exactly
the position of the complex branch curve in the complex plane associated with the given algebraic
function when looked at from the point of view of real manifolds. It is evident that Artin had this
example in mind when he inaugurated the study of knotted surfaces in his short paper [2]. Once more,
we face an elimination of contexts. A similar situation obtains in the case of Artin's alleged "invention"
of braids in [1]; they had originally been treated in the context of Riemann surfaces by Hurwitz in his
remarkable paper [13]. For details, see [40; 43].
HM 22
BEGINNINGS OF M O D E R N KNOT T H E O R Y
387
G e r m a n m a t h e m a t i c a l scene. Like m a n y others, Wirtinger was very clear as to the
influence which Klein had o n his professional career. Thus, it is hardly surprising
that he i n f o r m e d Klein a b o u t progress and promising perspectives in his research.
It m a y even be the case that Klein expected something of this kind f r o m m a t h e m a t i cians u n d e r his protection. W e have also seen that Wirtinger never chose to leave
the context of algebraic functions which was so dear to Klein.
In his general views on mathematics, too, he was strongly influenced by his
mentor. His letters to Klein express a strong a d h e r e n c e to Klein's values coflcerning
m a t h e m a t i c a l practice and the place of m a t h e m a t i c s in culture. Wirtinger shared
Kleinian values with respect to the status of geometric i n t u i t i o n - - a s evidenced by
the way in which Wirtinger a p p r o a c h e d his project on algebraic f u n c t i o n s - - a s well
as with regard to the social i m p o r t a n c e of higher m a t h e m a t i c a l education. In one
of a series of letters dealing with the issues put f o r w a r d by Klein in his talk on
" a r i t h m e t i z a t i o n " [17], Wirtinger expressed his general a g r e e m e n t by using a nice
m e t a p h o r to illustrate the m a t h e m a t i c i a n ' s task. H e imagined the m a t h e m a t i c i a n
of the 20th century, he said, like a painter w h o looks at the world with a painter's
eyes, thinking a b o u t the w a y in which he would like to paint it. Correspondingly,
the m a t h e m a t i c i a n should try to "see the m a t h e m a t i c a l p r o b l e m " in w h a t e v e r f o r m
she or he e n c o u n t e r s it. This perceptive faculty should be the result of general
m a t h e m a t i c a l education. Wirtinger then joins in with Klein's critique of the growing
trend t o w a r d abstraction in mathematics; an abstract definition, he says, is n o t h i n g
but a r~sum~ of a series of concrete instances, in which the real m a t h e m a t i c a l
interest must lie. 33
It is certainly apt to call Wirtinger a convinced s u p p o r t e r of a "Kleinian style"
of m a t h e m a t i c a l practice. Wirtinger m a d e this very explicit not only in his letters
but also in an article written on the occasion of Klein's 70th birthday, entitled Klein
und die Mathematik der letzten 50 Jahre [35].
O f particular i m p o r t a n c e in the present context is Wirtinger's reluctance to accept
and to p r o m o t e the general tendencies t o w a r d a growing differentiation of m a t h e matical fields. This hesitation, central to the style of m a t h e m a t i c a l practice a d v o c a t e d
by Klein and Wirtinger, is clearly expressed in the following passage f r o m Klein's
Entwicklung der Mathematik im 19. Jahrhundert which refers to the differentiation
of p r o b l e m fields and m e t h o d o l o g i c a l schools in c o n n e c t i o n with algebraic functions:
This tendency to dissect science not only into an ever greater number of individual disciplines,
but also to create schools based on differences with regard to methodology would, if it should
become prevalent, lead to the death of science. We have always aimed at the opposite ourselves.
33 "Ich stelle mir den Mathematiker des 20t Jahrhunderts so vor, dass er, wie der Maler, so oft er
will, die Welt malerisch sieht u. denkt wie er sie malen wtirde (u. nicht blos an classische Galeriebilder),
auch so oft er will das mathematische Problem sieht, wo u. in welcher Gestalt immer es entgegentritt.
Als Resultat der allgemeinen mathematischen Bildung, denke ich mir nun die F~ihigkeit dieses Sehens,
wenigstens im Princip. Es scheint mir, dass Sie mit der Bemerkung auf pag. 8 tiber die zu grosse
Abstraction, die nur hindert ein concretes Problem zu erfassen, die Wurzel des Obel's bezeichnet haben.
Mir pers6nlich war die Verbaldefinition nichts anderes als das Resumde tiber eine Reihe concreter F~ille
u. ohne Kenntnis derselben ganz ohne Interesse." [Wirtinger to Klein, May 22, 1896]
388
MORITZ EPPLE
HM 22
In our generation, we have kept 1. the theory of invariants, 2. the theory of equations, 3.
function theory, 4. geometry and 5. number theory more or less in contact and this was our
special pride.34
Elsewhere I have tried to show not only that this attitude represents a strong
normative commitment with respect to the organization of mathematical research,
but also that it was connected to a particular style of mathematical argumentation
which drew heavily o n - - a n d sought to draw o n - - c o n n e c t i o n s between different
areas of mathematics [43]. As such, Klein's statement may be interpreted as expressing one aspect of a specific standard of rationality in mathematical practice. This
standard is "integrative" in a strong sense. It is regarded as rational to promote
the coherence of mathematics as a whole, in professional politics as well as in doing
research which links different mathematical fields. It may well be that Wirtinger's
adherence to a similar standard was one of the factors which prevented him from
pursuing and publishing the purely topological parts of the results which he obtained
in his investigation of branch points of algebraic functions.
ELIMINATION OF CONTEXTS
Max Dehn's Research on Knots
At about the time when his Habilitationsschrifi was published, Heinrich Tietze
had an encounter in R o m e with another aspiring young mathematician interested
in topology and knot theory, Max Dehn. A t the time, Dehn was convinced that he
knew of a topological characterization of ordinary 3-space which would have been
more or less equivalent to a proof of the Poincar6 conjecture. 35 Tietze pointed out
the error in D e h n ' s argument, and Dehn had to withdraw a paper which he already
had sent to Hilbert with the urgent request for speedy publication in the GOttinger
Nachrichten. 36 D e h n ' s misconception prevented him from publishing the other half
of the paper immediately, and only in 1910 did the first of a series of papers appear
w h i c h - - a m o n g other things--definitively established knot theory as a promising
subfield of what was then called combinatorial topology ([5-8]). In fact, this paper
also contained a serious gap which would not be filled during D e h n ' s lifetime, the
notorious " D e h n lemma" (see, e.g., [51]).
The central notion in this work was again that of the fundamental group of the
34"Diese Tendenz,die Wissenschaftnicht nur in immer zahlreichere Einzelkapitel zu zerlegen, sondern
Schulunterschiede nach der Art der Behandlung zu schaffen, wOrde, wenn sie einseitig zur Geltung
k~ime, den Tod der Wissenschaft herbeiftihren. Wir selbst haben imrner das Umgekehrte angestrebt.
In unserer Generation haben wir 1. Invariantentheorie, 2. Gleichungstheorie, 3. Funktionentheorie, 4.
Geometrie und 5. Zahlentheorie mehr oder weniger in Kontakt gehalten, und das war unser besonderer
Stolz." [18, 327]
35The key to this characterization would have been "dab der gew6hnliche Raum die einzige 3dim.
Mannigfaltigkeit ist, in der jeder 'geschlossene Fl~ichenkomplex' [as Dehn had defined it] zersttickelt"
(Dehn to Hilbert, February 12, 1908). Compare also the final paragraph in [5].
36Dehn to Hilbert, February 12, and April 16, 1908. Dehn had feared that somebody,perhaps Poincar6
himself, might anticipate his result.
H M 22
BEGINNINGS
OF MODERN
KNOT THEORY
389
exterior of a knot, which Dehn simply called "the group of the knot." One of the
famous results of the first paper is the statement (which hinges upon " D e h n ' s
lemma") that a knot may be disentangled, i.e., is equivalent to the trivial knot, if
and only if its group is abelian? 7 Even more impressive was the proof, contained
in the fourth paper, showing that a left-handed trefoil knot cannot be deformed into
its right-handed mirror image. The proof involved classifying the automorphisms of
the group of the trefoil, which, as we have seen, had already been determined in
Wirtinger's and Tietze's work. Nevertheless, Dehn did not use their techniques. In
Dehn's papers, no idea related to algebraic functions or at least coverings of the
exterior of a knot was mentioned, and he ignored Wirtinger's method of deriving
a presentation of the knot group. Instead, Dehn gave another presentation starting
completely from s c r a t c h y The methods he used to prove his deeper results were
devoid of all vestiges of analysis, and consisted in a highly original fusion of combinatorial group theory and hyperbolic geometry (which contributed to topology and
group theory in almost equal parts).
Why did Dehn not take up the thread offered to him by Tietze? It is clear that
he had read at least those parts of Tietze's paper which seemed important to h i m . 39
The answer to this question accounts for why the context of algebraic functions is
no longer present in the first period of modern knot theory. Moreover, it also
reveals how this particular example of context-elimination reflects the rise of a
new style of mathematical practice dominated by another mighty figure on the
mathematical scene, David Hilbert, and connected to a new standard of mathematical rationality. In order to explain this fully, we must once more work our way
backwards in time. We begin with a detailed description of Dehn's way of defining
knots and his way of proving the knottedness of the trefoil. Since Dehn's work in
topology is better known than Wirtinger's, we need not go into the same detail as
in the previous section. 4°
Dehn's paper of 1910 began with some considerations which proved to be of
great influence for the development of combinatorial group theory. Dehn's subject
was finite group presentations, and he stated clearly the algorithmic problems
associated with them: to find methods that would enable one to decide in a finite
number of steps whether or not two words (a) represent the same group element,
or (b) represent conjugate elements. (A third algorithmic problem, namely, to
decide whether two given presentations determine isomorphic groups, had already
been stated and treated by Tietze in the context of showing the combinatorial
invariance of the fundamental group [32]; see above.) Dehn went on to translate
the first of these p r o b l e m s - - t h e "Identit~itsproblem" as he called it in [6]--into
37 A n o t h e r aim of this paper was to construct examples of 3-manifolds--in particular, of homology
spheres--using what was later called surgery on a knot.
38 For some time, the group of a knot was even called " D e h n ' s group," for instance, by Veblen [33]
and in a letter of Reidemeister to Hellmuth Kneser of 1925. The picture was corrected by Artin [1].
39 This is evident from the group-theoretical parts of his papers, for instance [6]. See also [40, 17 ft.].
4o For a historical treatment of Dehn's topological papers, see, e.g., [51] or [40]. Bollinger [38] and
vanden Eynde [45] have also devoted some attention to Dehn's work.
390
MORITZ
EPPLE
HM
22
another combinatorial problem, the construction of the "Gruppenbild" or Cayley
graph associated to a group presentation G := (al . . . . , an I rl . . . . . rn). The vertices
of this graph are the group elements. Two vertices g, h ~ G are connected by an
edge of the graph if and only if h = aig holds for some generator ai. Consequently,
each closed path in the graph represents a relation in the group. To solve the word
problem and to construct the graph are therefore equivalent.41 The "Gruppenbild"
is an example of what Dehn called a "Streckenkomplex," that is, an assembly of
basic elements (here, group elements) together with a set of pairings (here, equations
h = a i g ) . We shall see shortly how this combinatorial notion could be used to treat
knot groups.
After some further preparations (including his "lemma"), Dehn defined knots
as what he called "closed nonsingular curves," embedded in 3-space [5,153]. A closer
look shows that a "curve" is, for him, another example of a "Streckenkomplex. ''42 It
is a polygonal curve, given by a set of points in 3-space, paired according to the
line segments which make up the polygon. (For Wirtinger, by contrast, the trefoil
had been given by the intersection of an algebraic curve with a 3-sphere in the
complex plane!) A curve is nonsingular if no two line segments meet (except
neighboring segments at the vertices of the polygon). The group of a knot was then
introduced in the following way. Dehn's starting point was the "Streckenkomplex"
given by a regular projection of a knot (a knot diagram with self-crossings). To
such a graph, he had associated earlier in the paper [5, Part I, Sect. 2] a group
presentation which was nothing but the fundamental group of the surface bounding
a tubular neighborhood of the projection. Then, for each crossing of the projection,
new relations were introduced which account for the fact that, in contrast to its
projection, the knot has no self-crossings in 3-space. In this way, Dehn arrived at
a presentation of the group of equivalence classes of paths on the torus which
bounds a tubular neighborhood of the knot, where two such paths are considered
equivalent if and only if they are homotopic in the knot complement [5, 157]. This
completed Dehn's definition of the group of a knot. For the trefoil, Dehn found a
presentation that is easily shown to be equivalent to Wirtinger's presentation:
(C1,C2,C3,C41C1C41C2,C2C41C3,C3C41Cl).
Note that the knot group was introduced on a purely combinatorial basis, starting
from a "Streckenkomplex" and using no further information. It was not given
beforehand and then shown to possess a certain presentation, as in Wirtinger's
case. That it has an obvious interpretation in terms of equivalence classes of paths
is not essential for the definition itself. Interestingly enough, Dehn makes no attempt
to show that his group is in fact an invariant under a combinatorial version of
isotopy. This leads one to wonder how Dehn could have shown that a given knot
41 For further information on the '~Gruppenbild" and D e h n ' s use of it, see [40].
42 In [5], the reader is simply referred to the Enzyklopiidie article [4]; see below.
H M 22
391
4
.....i
2
13
i....
4
•..
3
3
...-
4
FIG 3. Dehn's "Gruppenbild" of the trefoil.
is non-trivial. D e h n did so by using the correct, but insufficiently proven result that
a knot is trivial if and only if its group is abelian. (Since the "only if" part is enough
for showing knottedness, the gap in Dehn's proof is not relevant here.) Let us
consider how this result appears in the case of the trefoil.
In this instance, Dehn actually managed to construct the "Gruppenbild." It can
be built from infinitely many "strips" of the form shown in Fig. 3 (left), where
vertical edges always represent c4, while oblique edges represent cl, c2, and c3 as
indicated in the figure. Different copies of such strips must then be pasted together
according to the scheme shown in Fig. 3 (right). Here, every line segment represents
one copy of the strip, as "seen from above," and in pasting one has to ensure that
at each vertex all four types of edges meet (this is indicated by the numbers).
It is easy to see that the resulting graph is in fact the " G r u p p e n b i l d " of the trefoil
group. Dehn's argument for the knottedness of the trefoil is now a matter of
inspection. In his own words: " N o w we recognize immediately that the group is
not isomorphic to the group {S ~} [i.e., 7/], that it is not abelian. For example, the
polygonal tract ClC4Calcg I is not closed. ''43
43 "Wir erkennen nun sofort, dab die Gruppe nicht isomorph mit der Gruppe {S"}, dab sie nicht
abelsch ist. Zum Beispiel ist der Streckenzug clc4c~1c5~ 1 nicht geschlossen" [5, 160]. Looking through
the right glasses, the figure also shows that the group of the trefoil acts by isometries on the hyperbolic
plane, a fact heavily exploited in Dehn's article of 1914 [8].
392
MORITZ EPPLE
HM 22
A Hilbertian Approach to Topology
Evidently, Dehn's way of dealing with knots and their groups depended on
a completely different conceptual outlook than Wirtinger's. This conceptual
framework was taken from an article that Dehn had written together with Poul
Heegaard for the Enzyklopi~die der mathematischen Wissenschaften in 1907.
There, they sketched a rigorously combinatorial approach to topology while
attempting to reformulate the classical problems of 19th-century topology in
this setting.
A m o n g these classical topics, the problem of classifying knots was taken up and
given a systematic place in the hierarchy of topological problems. Dehn and Heegaard distinguished between problems of "Nexus" and problems of "Connexus."
The former are problems of classifying manifolds up to homeomorphism, whereas
the latter deal with embeddings of manifolds of various dimensions into each other
[4, 170]. A m o n g " C o n n e x u s " problems, one finds problems of homotopy (Dehn
and Heegaard only mention closed curves in n-dimensional manifolds, which give
the problem of calculating the fundamental group), and problems of isotopy. According to Dehn and Heegaard, the first interesting case of the latter type of
problems is the knot problem [4, 207 ff.].44 They even sketched a (rather trivial)
"arithmetization" of the knot problem. This "arithmetization" consisted in considering knots as chains of nearest neighbors in a three-dimensional cubic lattice,
equivalence of knots being given by the obvious elementary deformations. 45
More interesting than the remarks on knots, however, is the general perspective
on topology which was advocated in the article. Unlike nearly all the other articles
in the Enzyklopiidie, the systematic parts of Dehn and Heegaard's text (for which
Dehn was primarily responsible [4, 153]) were written with an evident dependence
on Hilbert's epoch-making Grundlagen der Geometrie. Like Hilbert's geometry,
Analysis situs was presented as a theory dealing with aggregates of uninterpreted
elements, for which only combinatorial rules are specified. 46 Some of these rules
were taken directly from the "topological" parts of Hilbert's book, as for example
the notion of a "Streckenkomplex," which is a direct generalization of what Hilbert
had called a "Streckenzug. ''47 Dehn and Heegaard even formulated axioms (in
Sect. 8), without, however, treating problems of consistency or uniqueness. These
axioms were thought of rather as conditions which the combinatorial definitions
had to satisfy in order to allow for an intuitive interpretation of the theory, to give
it an "Anschauungssubstrat." After introducing this formal apparatus, the authors
went on to characterize Analysis situs as a "part of combinatorics, characterized
44Even though Dehn had worked on the Jordan curve theorem earlier, the isotopy problem of closed
curves on a surface was not mentioned.
45Eighteen years later, Artin also called his discussion of the braid group an "arithmetization" of
(topological) braids.
46For further details, see [38, 144-147].
47According to Hilbert [12, Sects. 3-6], a line segment is defined by a pair (AB) of points A, B; a
"Streckenzug" is a system of line segments ((AB)(BC)(CD) ... (KL)). Admitting arbitrary pairings
instead of chains yields a "Streckenkomplex" as defined by Dehn and Heegaard [4, 156].
HM 22
393
by its intuitive meaning." Moreover, in another striking parallel to Hilbert's
Grundlagen der Geometrie, D e h n and H e e g a a r d viewed Analysis situs as occupying
a rather fundamental place in the architecture of mathematical disciplines, namely,
as "the most primitive section of geometry, where the notion of a limit is still of
no importance. ''48 Consequently, not even the analytical notion of continuity has
a systematic place in D e h n and H e e g a a r d ' s treatment. In sum, the mathematical
discipline of topology which D e h n had in mind was from the beginning conceived
as an axiomatic, self-contained theory, very fundamental and far r e m o v e d from
analytic contexts.
C o m p a r e d to Wirtinger's and even Poincar6's ideas about the role of topological
problems in mathematics, D e h n and H e e g a a r d ' s approach represents a new start.
The decision to present topology as a part of combinatorics a m o u n t e d to establishing
a new standard of rationality in dealing with topological questions. On the one
hand, this standard aimed at perfect rigor. Topological arguments could be guided
by intuition, but essentially they should be reducible to arguments dealing with
combinatorial data on a formal, axiomatic basis. On the other hand, the new standard
differentiates topology from other mathematical disciplines. Topology has concepts,
techniques, and a hierarchy of problems defined internally, without reference to
either the origin of its basic concepts in or its application to other fields, like complex
function theory or algebraic geometry. O f course, this differentiated status of topology does not preclude, and very probably was not intended to preclude, applications
of topological ideas to other mathematical fields. What had changed, however, was
the type of relations between these fields, and the way of conceiving topology as a
subject on its own. 49
When, one year after D e h n and H e e g a a r d ' s article, Tietze published his
Habilitationsschrift, we find him wavering between the new standard and a m o r e
traditional outlook on topology. As far as possible, he tried to follow a rigorous
combinatorial approach. However, he was not yet p r e p a r e d to give up completely
connections with ideas originating in fields like algebraic functions or the kind
of intuitive arguments that had been usual in earlier treatments. His personal
solution to this conflict of rationality standards was diplomatic. H e proposed to
consider those topological issues not yet tractable from the combinatorial point
of v i e w - - a m o n g them " R i e m a n n spaces" and Wirtinger's approach to the knot
g r o u p - - a s suggesting the type of problems to be dealt with in the future [32,
48 Analysissitus is a "... durch seine anschauliche Bedeutung ausgezeichneter Teil der Kombinatorik,
... der primitivste Abschnitt der Geometrie, wo der Grenzbegriff noch nirgendwo von Bedeutung ist"
[4, 170 ff.].
49It would be interesting to compare this outlook on topology with that advocated by Listing 60 years
earlier [19]. In 1847, not even the separation of mathematics and physics was an established fact.
Accordingly, Listing's efforts to promote a new mathematical discipline included the attempt to convince
scientists of all sorts that topology had significant insights to offer in their contexts: in crystallography,
biology, physics, etc. Two levels of differentiation separate Listing's project from Dehn and Heegaard's:
first, the differentiation of pure mathematics from other scientificcontexts; and second, the differentiation
of topology from its contexts in pure mathematics.
394
MORITZ
EPPLE
H M 22
80 ft.].50 Whatever the merits of this early attempt to give topology an axiomatic,
combinatorial foundation may have been, one thing was clear: once Dehn,
Heegaard, and Tietze had published their papers, no reader of them would have
denied that topology had emerged as a discipline in its own right, endowed
with its own problems and standards of rigor.
It is easy enough to see where Dehn got the orientation which determined
his views on topology. A r o u n d the turn of the century, he had been one of
Hilbert's model pupils. In a letter to Hurwitz, Hilbert spoke in enthusiastic
terms about the results of Dehn's thesis on the foundations of geometry? 1 (It
was in this connection that Dehn first began working on hyperbolic geometry,
an interest which proved useful to him later on.) After having solved Hilbert's
third problem on polyhedra, Dehn could be sure of the future support of his
teacher. 52 From the correspondence between the two it is clear that Dehn was
deeply involved in the revisions that led to several new editions of Grundlagen
der Geometrie. 53 He even considered writing a book on the foundations of
geometry himself. 54 Although this project was never realized, he later did write
an historical appendix to the second edition of Pasch's classic Vorlesungen iiber
neuere Geometrie in which he sought to emphasize the lines of thought leading
to Hilbert's foundations of geometry [9]. Hilbert, in turn, contributed to the
career of his pupil by writing letters of r e c o m m e n d a t i o n Y Thus, when in the
years following 1908 Dehn turned to topology and knot theory, it was clear
that a genuine follower of Hilbert's axiomatic program in geometry had begun
research on knots. Little wonder, then, "that knot t h e o r y - - l i k e combinatorial
topology as a w h o l e - - w a s seen as a fundamental, self-contained theory which
a priori had nothing to do with higher analysis. 56
5~ In a way, Tietze's ambivalence was not new, and it was never resolved completely. Dehn and
Heegaard's radical hope to establish topology as a subfield of combinatorics never quite got off
the ground. During the 1920s and 1930s, a competition dominated the scene between topologists
favoring the combinatorial approach and topologists who elaborated analytical notions and methods.
Even when the possibilities of the different approaches could be assessed more clearly in the
language of categories, this still did not bring to an end the conflict between combinatorial and
analytic orientations shared by different members of the topological community. This ongoing debate
points to a conflict between what Gerald Holton has called a pair of antithetical themata in scientific
thought [48, Chapter 1].
51 Hilbert to Hurwitz, 5. and 12.11.1899, contained in the Hurwitz Nachlass in NSUB GOttingen.
52 See [51] for details.
53 The correspondence is contained in the Hilbert Nachlass in NSUB G6ningen and the Dehn Nachlass
in Austin, Texas.
54 Dehn to Hilbert, January 19, 1903.
55 Dehn to Hilbert, April 3, 1911, concerning Hilbert's recommendation of Dehn's call to Kiel;
March 9 and July 11, 1913, concerning an unsuccessful attempt to promote Dehn in Kiel and his
call to Breslau.
56 Still another influence of Hilbert on Dehn resulted in the awareness of algorithmic problems in
combinatorial disciplines. When, in 1910, Dehn decided to begin his topological paper with a statement
of the word and conjugacy problems of combinatorial group theory, this may well have been a response
to Hilbert's quest for an algorithmic solution of problems in number theory, expressed in Hilbert's 10th
problem. See [40, 54 ft.].
HM 22
395
Wirtinger, Dehn and the Memory of the Mathematical Community
This background allows us to understand why D e h n did not take up the line of
thought on knots which had been pursued by Wirtinger and his younger colleague,
Tietze. H e simply was educated and accustomed to think along a completely different horizon of mathematical values. D e h n adhered to a different standard of mathematical rationality. H e also was moving in a different social setting in the mathematical community, and he was influenced by a different constellation of power
relations. 57 Probably, D e h n simply did not read passages in Tietze's Habilitationsschrifi which had to do with things like " R i e m a n n spaces," the more so since
Tietze himself admitted that they were not treated in a rigorous combinatorial
fashion but rather depended heavily on intuitive arguments, situated in a peculiar
way between different mathematical theories. A n d even if D e h n had read these
passages, it must have been clear to him that there was no easy way to adapt their
contents into the picture of combinatorial topology to which he adhered.
The absence of the context of algebraic functions in D e h n ' s pioneering work on
knots m a y be explained along these same lines. This elimination of contexts need
not necessarily have been the the result of a conscious decision. Rather, it was
probably the unintended effect of aprior decision, the decision to accept a Hilbertian
style in mathematical practice together with its inherent standard of rationality.
Relative to that standard, the elimination was itself "rational." The combinatorial
approach to knots was simpler and m o r e streamlined since it carried no ballast
from the theory of algebraic functions. It promised a different level of rigor in proofs
as well as a clearcut separation between the knot problem and other mathematical
problems related to it. The tendency of the new style to foster a differentiation
within mathematical disciplines m a d e it also especially attractive for a young mathematician like Dehn. It allowed such a person to reach the frontiers of research
without the long years of education necessary to survey a whole network of mathematical fields as Klein would have liked. D e h n could do knot theory without knowing
much about Galois groups, analytic continuation, or singular points of algebraic
curves. It was no longer necessary to delve deeply into the classics of 19th-century
mathematical literature, reading works by Puiseux, Riemann, Jordan, etc.
It is equally understandable that when, after the interruption caused by World
W a r I, the next generation of m a t h e m a t i c i a n s - - S c h r e i e r , Reidemeister, and A r t i n - entered the scene, they started from D e h n ' s combinatorial approach, and not from
Wirtinger's (still unpublished) ideas. Thus, in his Cambridge Colloquium on Analysis
situs, Oswald Veblen summarized the history of the knot p r o b l e m in the following words:
A large number of types of knots have been described by Tait and others and a list of references
may be found in the Enzyklop~idiearticle on Analysis situs. But a more important step towards
developing a theory of knots was taken by M. Dehn, who introduced the notion of the group
of the knot, which is essentially the group of the generalized three-dimensional complex [sic!]
57The notion of "power relations" should be understood here in a neutral and descriptive way,
referring equally to the asymmetry of social relations in a community and to scientific leadership.
396
MORITZ EPPLE
HM 22
obtained by leaving out the knot from the three-dimensional space. Dehn gave a method for
obtaining the group of a knot explicitly . . . . [33, 150]
This passage suggests that by 1922 not only was the mathematical context of algebraic functions eliminated from knot theory but also that 10 years of an individual's
efforts had b e e n - - f o r the time b e i n g - - d e l e t e d from the collective memory of the
mathematical community.
However, this was not quite the truth. As mentioned earlier, Reidemeister,
Schreier, and Artin had attended Wirtinger's lectures in Vienna. In fact, there was
a most intimate connection between the newly founded "Mathematisches Seminar"
in Hamburg and the mathematicians in Vienna. Kurt Reidemeister, for example,
went to Vienna following the recommendation of Hamburg's Wilhelm Blaschke,
who himself had received most of his education in Vienna (in fact, Blaschke had
written his dissertation under Wirtinger). It was there that Reidemeister decided
to turn to knot theory, and a closer look at his mathematical contributions reveals
that he absorbed Wirtinger's approach to knots. In fact, it was precisely this knowledge that enabled him to go a step further than Dehn by giving the first effectively
computable knot invariants, the torsion numbers of cyclic coverings of knot exteriors.
Interestingly enough, even this remnant of the original context which had led to
the notion of a knot group was relegated to the last sections of Reidemeister's
Knotentheorie of 1932. There, too, Reidemeister's new invariants were introduced
in a purely combinatorial way, and their connection to covering spaces only appeared
as a secondary, though interesting, interpretation. 5s
CONCLUSION
Why was Wirtinger's work presented here in such detail? Certainly not to engage
in priority debates. To ask whether or not Wirtinger was the "real father" of modern
knot theory misses the point of the present study. Surely, it would be misguided
to accuse Tietze or Dehn of having temporarily suppressed an interesting piece of
mathematics. Rather, my intention has been to re-contextualize early work in
modern knot theory. Knot theory was neither an invention out of thin air nor an
application of general topological notions to a particular problem. 59 It emerged as
part of a gradual process of differentiation out of one of the mainstream disciplines
of 19th-century mathematical research, the theory of algebraic functions. With
regard to that process, Wirtinger was the key figure. Moreover, the status of knot
theory as a separate subfield of the new discipline of topology was attained only
after an elimination of the context which originally served to legitimize Wirtinger's
project. (Actually, algebraic function theory was not the only context of 19thcentury work on the knot problem which moved into the background during the
constitution of modern knot theory. Another was the physical context, from which
Tait had derived his justification for tabulating knots.)
An attempt to redescribe the invention of a mathematical theory or discipline
581 hope soon to present a study of this period in the formation of modern knot theory.
59As such, it appears in Dieudonn6's presentation [41,307-310].
HM 22
B E G I N N I N G S O F M O D E R N KNOT T H E O R Y
397
as a p r o c e s s o f d i f f e r e n t i a t i o n a n d / o r c o n t e x t - e l i m i n a t i o n s h o u l d n o t b e v i e w e d as
a r e a s s e s s m e n t o f t h e a c h i e v e m e n t s o f t h e p i o n e e r i n g figures. R a t h e r , it p l a c e s t h e s e
a c h i e v e m e n t s in a d i f f e r e n t light b y e x h i b i t i n g s o m e o f t h e c a u s a l links b e t w e e n
m a t h e m a t i c a l life b e f o r e a n d a f t e r t h e d i s c i p l i n a r y t h r e s h o l d has b e e n r e a c h e d .
M o r e o v e r , this a p p r o a c h m a y d i r e c t h i s t o r i a n s ' a t t e n t i o n to t h a t a s p e c t o f m a t h e m a t ical c u l t u r e w h i c h is r e s p o n s i b l e for m o s t o f its d e e p e r changes, t h e d o m a i n of
t h e n o r m s a n d v a l u e s g u i d i n g m a t h e m a t i c a l r e s e a r c h , i n c l u d i n g t h e field o f p o w e r
c o n s t e l l a t i o n s c o n n e c t e d with the n o r m a t i v e s t r u c t u r e o f e v e r y c o m m u n i t y . I h o p e
to h a v e s h o w n that, in t h e case o f e a r l y m o d e r n k n o t t h e o r y , this d o m a i n h a d an
influence n o t o n l y o n t h e r o l e a n d status o f m a t h e m a t i c a l k n o w l e d g e in scientific
c u l t u r e as a w h o l e , as has o f t e n b e e n discussed, b u t also o n t h e i n t e r n a l c o n s t i t u t i o n
o f t h e b o d y o f k n o w l e d g e itself. W h e n the c o m p l e x o f p r o b l e m s within t h e t h e o r y
o f a l g e b r a i c f u n c t i o n s w h i c h h a d l e d W i r t i n g e r to his t o p o l o g i c a l i n v e s t i g a t i o n o f
the complement of certain knots eventually differentiated into a new topological
( s u b - ) d i s c i p l i n e , c e n t e r e d a r o u n d a c o m b i n a t o r i a l t r e a t m e n t o f t h e k n o t g r o u p , this
r e f l e c t e d a shift in t h e r e l a t i v e v a l u a t i o n s o f m a t h e m a t i c a l p r o b l e m s a n d t h e o r y c o n s t r u c t i n g t e c h n i q u e s . T h e c o n t e x t - e l i m i n a t i o n w h i c h is so e v i d e n t in D e h n ' s a n d
R e i d e m e i s t e r ' s l a t e r w o r k was t h e c o n s e q u e n c e o f a series o f n o r m a t i v e decisions,
d e c i s i o n s w h i c h d e p a r t e d f r o m a K l e i n i a n view o f m a t h e m a t i c s a n d r e i n f o r c e d a
" m o d e r n i s t " view o f m a t h e m a t i c s as a c o m p l e x o f m o r e o r less s e l f - c o n t a i n e d ,
a x i o m a t i z e d t h e o r i e s . M o r e o v e r , t h e s e d e c i s i o n s w e r e n o t t a k e n in a social v a c u u m ,
b u t in a social s p a c e s t r u c t u r e d b y c o m p l e x n o r m a t i v e h o r i z o n s a n d p o w e r r e l a tions. 6°
APPENDIX
This a p p e n d i x p r e s e n t s t h e c o m p l e t e text o f W i r t i n g e r ' s l e t t e r to Klein, d a t e d
22.12.1895. T h e l e t t e r is c o n t a i n e d in t h e K l e i n N a c h l a s s in G 6 t t i n g e n , filed in Cod.
Ms. K l e i n X I I , 391. O r t h o g r a p h y a n d p u n c t u a t i o n a r e left u n c h a n g e d .
Innsbruck, 22./XII 1895.
Hochgeehrter Herr Collega!
Diesen Brief muss ich mit einer Entschuldigung einleiten. Ich habe n~imlichbis jetzt gehofft,
das Manuscript der Arbeit iiber Differentialgleichungen bis Neujahr fertigstellen zu k6nnen,
nun aber sehe ich, dass es doch nicht geht. Der Entwurf, ziemlich ausgearbeitet, ist fertig.
Die Schlussredaction werde ich hoffentlich im J/~nner fertigmachen k6nnen. Vorlesungen, ein
1/~ngeres Unwohlsein, Prtifungen u. was derlei sch6ne Dinge mehr sind haben mich nicht dazu
kommen lassen. Vergebliche Versuche diese und jene ber0hrte Frage weiter zu f0hren als es
601 leave it to the reader to draw the obvious connections from my narrative to Herbert Mehrtens'
discussion of the opposition beween what he calls "'modern" and "counter-modern" trends in mathematical practice around the turn of the century. One could even go further and speculate whether the shift
from an integrative standard of mathematical rationality as expressed by Klein toward a differentiative
standard as implicit in (at least the reception of) Hilbert's Grundlagen der Geometrie reflects the general
trend of modern culture toward differentiation noticed by Max Weber and others. For further information
on Klein, Hilbert and general matters, I refer to work by Herbert Mehrtens [52] and David Rowe [57].
It will be clear that my perspective on the developments presented in this article owes much to both
of them, a debt which I gratefully acknowledge.
398
MORITZ EPPLE
H M 22
geschehen ist haben auch ihren Anteil daran. Nun aber ist es wenigstens soweit, dass ich nur
mehr die letzte Redaction vor mir habe.
Die Functionen mehrerer Variablen sind auch wenig vorgertickt. Abet einiges habe ich mir
doch klar gemacht. Um Directiven for die allgemeinen Fragen zu gewinnen, bin ich daran
gegangen, mir das Verhalten gew6hnlicher algebraischer Functionen l~ings Verzweigungsmannigfaltigkeiten genauer auszudenken. Mein Ziel ist dabei zu erweisen, dass man ein beliebiges
System algebraischer Gebilde yon n - 1 Dimensionen mit zugeh6rigem Verzweigungsschema
immer als System yon Verzweigungsmannigfaltigkeiteneiner Function von n Variablen auffassen kann. Das gibt verschiedene algebraische und transcendente Formulirungen, die erst fassbar
werden, wenn ich tiber die singul~ren Stellen eine deutliche Vorstellung gewonnen habe. Der
Kern der ganzen Sache ist mir nun ziemlich deutlich. Bei einer Variablen liegt die Frage
deshalb so einfach, weil in der N~ihe eines Verzweigungspunktes die cyclischen Functionen
der Werte eindeutig werden, also die n-re Wurzel der Variablen selbst eindeutig ist. In dieser
Fassung liegt der Keim der Verallgemeinerung.
Das cyclische Verhalten bleibt aufrecht fiir beliebige Verzweigungsmannigfaltigkeiten die
durch Nullsetzen einer Potenzreihe mit Gliedern erster Ordnung gewonnen werden. Mit Hilfe
yon p1/, kann man dann alle Functionen darstellen. Sind mehrere solche P an einer Stelle
Null, so schadet das nichts. Anstelle des Schemas der Aufl6sung der einfachen Abel'schen
Gleichungen tritt dann nur das Schema der ~mehrf~iltigen' Abel'schen Gleichungen.
Ganz anders ist es abet wenn P keine Glieder erster Ordnung hat u. auch nicht zerlegbar
ist in Reihen mit Gliedern erster Ordnung. Dann ist n~imlich jede Galoissche Gruppe m6glich.
Und eben diese Gruppe charakterisiert den Verzweigungspunkt. Beispielsweise z 3 + 3 z x +
2y = 0 - z als Function von x u. y - - h a t in der N~ihe der Stelle 0 nur die symmetrische Gruppe.
Eine Darstellung aller Functionen, die auf der gegebenen Mannigfaltigkeit in der Umgebung
einer Stelle eindeutig sind ist dann m6glich durch eine Wurzel der Galoisschen Resolvente,
u. diese Wurzel ist es, welche an die Stelle von x 1/" in der Ebene tritt. Der Rationalit~itsbereich
ist hier der aller convergenten Potenzreihen von n Variablen.
Es ist im allgemeinen nicht m6glich, die Umgebung einer solchen Stelle stetig auf ein einfach
zusammenh~ingendes Raumsttick von n Dimensionen abzubilden, sondern die Umgebung einer
solchen Stelle hat selbst einen gewissen Zusammenhang! 61
Aus diesem Grunde halte ich eine allgemeine, iiberall bestimmte, ParameterdarsteUung des
Gebildes in der N~ihe der betrachteten Stelle, welche ausserdem jede Stelle nur einmal liefert,
ftir unm6glich. Dagegen k6nnen sehr wohl Dastellungen m6glich sein, welche das Gebilde
mehrfach tiberdecken, oder ausserwesentliche Singularit~iten aufweisen. In wie fern man vielleicht die letzteren durch Benutzung homogener Variablen unsch~ldlich machen kann ist mir
noch nicht ganz klar, ich setze aber nicht viel Hoffnung darauf.
Der Kern der ganzen Sache liegt jetzt fiir mich in der Eruirung der Gruppe eines Verzweigungspunktes, also eigentlich in einer Irreduzibilit~itsfrage im Gebiete der Potenzreihen einerseits,
andrerseits in der Frage: Kann man diese Gruppe willktirlich vorgeben, oder ist sie an Bedingungen gebunden, damit zugeh6rige Functionen existieren?
Das sind die Fragen und Gesichtspunkte, die ich nun genauer durcharbeiten will.
Mit meiner Lehrth~itigkeit in Innsbruck hat es eine eigene Bewandtniss. Der Menschenschlag
ist hier sehr z~ih,fleissig aber ungemein widerstandsf~ihig gegen jeden Versuch den Gesichtskreis
zu erweitern. Die Vorbildung l~isst viel zu wtinschen tibrig. Ich musste z.B. letzthin 3 Seminarstunden darauf verwenden, urn die Elemente der Kettenbriiche vorzuftihren, u. das Leuten
aus dem VII u. VIII Semester! Von Geometric will ich gar nicht reden, wurde ich doch letzthin
angegangen--von den n~imlichen Semestern--zu erkl~ren was eine I n v o l u t i o n ist!! Da k6nnen
Sie denken, wie viel Geduld ein auch nur geringer Lehrerfolg braucht.
61 Here Wirtinger amended "Zusammenhangszahl" into "Zusammenhang," showing that he was
aware of the nontrivial topological situation he was considering.
H M 22
399
Nun aber wtinsche ich Ihnen und dem mathematischen Lexikon besten Erfolg im neuen
Jahr. FOr mich war das vergangene Jahr nach aussen hin ein Glticksjahr u. das verdanke ich
zum gr6sstentheil Ihnen. Lassen Sie mich also nochmals recht herzlich far Ihre Freundschaft
u. F/Srderung danken u. bewahren Sie mir dieselbe auch im kommenden Jahr, wo ich Sie in
Frankfurt a.M. wieder zu sehen hoffe.
Die Herren Hilbert, Burkhardt, Sommerfeld bitte ich vielleicht bei Gelegenheit von mir
zu grtigen.
Mit den herzlichsten Wtmschen ftir Ihre Gesundheit verbleibe ich
Ihr dankbar ergebener
Wirtinger
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15. Camille Jordan, Trait~ des substitutions et des ~quations alg~briques, Paris: Gauthier-Villars, 1870.
16. Felix Klein, Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrale, Leipzig:
Teubner Verlag, 1882.
17. Felix Klein, Ober Arithmetisierung der Mathematik, reprinted in Gesammelte mathematische Abhandhmgen, 3 vols., Berlin: Springer-Verlag, 1921-1923, 2: 232-240.
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Springer-Verlag, 1926.
19. Johann Benedikt Listing, Vorstudien zur Topologie, G/Stringer Studien, 1847.
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400
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H M 22
21. Emile Picard and Georges Simart, Thdorie des fonctions alg~briques de deux variables ind~pendantes,
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22 (1898), 89-178.
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27. Victor Puiseux, Recherches sur les fonctions alg6briques, Journal des math~matiques pures et appliquOes (I) 15 (1850), 365-480.
28. Kurt Reidemeister, Fundamentalgruppe und Oberlagerungsrhume, Nachrichten der Akademie der
Wissenschaften GOttingen, Mathematisch-Physikalische Klasse II (1928), 69-76.
29. Kurt Reidemeister, Knotentheorie, Berlin: Springer-Verlag, 1932.
30. Herbert Seifert and William Threlfall, Lehrbuch der Topologie, Leipzig: Teubner Verlag, 1934.
31. Heinrich Tietze, Zur Analysis situs mehrdimensionaler Mannigfaltigkeiten, Sitzungsberichte der
Wiener Akademie 115 (1906), 841-846.
32. Heinrich Tietze, "Uber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefie far Mathematik und Physik 19 (1908), 1-118.
33. Oswald Veblen, The Cambridge Colloquium 1916, 11. Analysis Situs, American Mathematical Society
Colloquium Lectures, vol. 5, New York: American Mathematical Society, 1922.
34. Wilhelm Wirtinger, Algebraische Funktionen und ihre Integrale, in Enzyklopiidie der mathematischen Wissenschafien H, 2 B 2, Leipzig: Teubner Verlag, 1901, pp. 115-175.
35. Wilhelm Wirtinger, Ober die Verzweigungen bei Funktionen von zwei Ver~inderlichen. Jahresberichte der Deutschen M athematiker- Vereinigung 14 (1905), 517.
35. Wilhelm Wirtinger, Klein und die Mathematik der letzten 50 Jahre, Die Naturwissenschafien 7
(1919), 287-288.
36. Oscar Zariski, Algebraic Surfaces, Berlin: Springer-Verlag, 1935.
Se co n d a r y Literature
37. Michael Atiyah, The Geometry and Physics of Knots, Cambridge: Univ. Press, 1990.
38. Maja Bollinger, Geschichtliche Entwicklung des Homologiebegriffs, Archive for the History of Exact
Sciences 9 (1972), 94-170.
39. Egbert Brieskorn and Horst KnOrrer, Plane Algebraic Curves, Basel: Birkhauser, 1986.
40. Bruce Chandler and Wilhelm Magnus, The History of Combinatorial Group Theory, New York:
Springer-Verlag, 1982.
41. Jean Dieudonn6, A History of Differential and Algebraic Topology, 1900-1960, Boston: Birkhhuser, 1989.
42. Rudolf Einhorn, Vertreter der Mathematik und Geometrie an den Wiener Hochschulen, 1900-1940,
Dissertation, Technische Universit~t Wien, 1983.
43. Moritz Epple, Styles of Argumentation in Late 19th-Century Geometry and the Structure of Mathematical Modernity, in "Analysis and Synthesis: History and Philosophy," (Michael Otte and Marco
Panza, Eds.), Dordrecht: Kluwer, to appear.
44. Moritz Epple, Mathematical Inventions: Poincar6 on a "Wittgensteinian" Topic, preprint, Mainz,
1994, Proceedings of the International Congress Henri Poincar~, Nancy, 1994, to appear.
45. Ria vanden Eynde, Historical Evolution of the Concept of Homotopic Paths, Archive for History
of Exact Sciences 45 (1992), 127-188.
H M 22
BEGINNINGS
OF MODERN
KNOT THEORY
401
46. Jt~rgen Habermas, Theorie des kommunikativen Handelns, 2 vols., Frankfurt-am-Main: Suhrkamp, 1981.
47. Pierre de la Harpe, Introduction to Knot and Link Polynomials, in Fractals, Quasierystals, Chaos,
Knots and Algebraic Quantum Mechanics, Proceedings Acquafredda di Maratea 1987, ed. A. Amann
et al., Dordrecht: Kluwer, 1988, pp. 233-263.
48. Gerald Holton, The Scientific Imagination: Case Studies, Cambridge: Univ. Press, 1978.
49. Imre Lakatos, History of Science and Its Rational Reconstructions, in Method and Appraisal in the
Physical Sciences, ed. Colin Howson, Cambridge: Univ. Press, 1976, pp. 1-40.
50. Wilhelm Magnus, Braid Groups: A Survey, in Proceedings of the 2nd International Conference on
the Theory of Groups, Lecture Notes in Mathematics, vol. 372, Berlin: Springer-Verlag, 1974,
pp. 463-487.
51. Wilhelm Magnus, Max Dehn, The Mathematical lntelligencer 1 (1978), 132-142.
52. Herbert Mehrtens, Moderne--Sprache--Mathematik, Frankfurt-am-Main: Suhrkamp, 1990.
53. Robert K. Merton, Science, Technology and Society in Seventeenth-Century England, Osiris 4
(1938); reprint ed, New York: Howard Fertig, 1970.
54. John W. Milnor, Singular Points of Complex ttypersurfaces, Annals of Mathematics Studies, vol.
61, Princeton: Princeton Univ. Press, 1968.
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Historia Mathematica 30 (2003) 457–493
www.elsevier.com/locate/hm
L’intégration graphique des équations différentielles ordinaires
Dominique Tournès a,b
a IUFM de la Réunion, allée des Aigues Marines, Bellepierre, F-97487 Saint-Denis cedex, France
b CNRS, équipe REHSEIS, UMR 7596, Université Paris 7, 2 place Jussieu, F-75251 Paris cedex 05, France
Résumé
Dans la période qui précède l’apparition des ordinateurs, les besoins en calcul des scientiques et des ingénieurs
ont conduit à un développement important des méthodes graphiques d’intégration. Pour contribuer à l’étude de
ce phénomène peu connu, l’article présente les techniques et les instruments utilisés pour l’intégration graphique
des équations différentielles ordinaires, et recherche leurs origines historiques en remontant aux débuts du calcul
innitésimal : procédés de calcul par le trait reposant sur la méthode polygonale ou la méthode des rayons
de courbure, emploi du mouvement tractionnel pour la conception d’intégraphes, réduction à des quadratures
graphiques en nombre ni ou inni.
 2003 Elsevier Inc. All rights reserved.
Abstract
In the period which precedes the appearance of computers, needs in calculation of the scientists and engineers
led to an important development of graphic methods of integration. To contribute to the study of this little
known phenomenon, the article presents techniques and instruments used for the graphic integration of ordinary
differential equations, and looks for their historic origins by going back to the beginning of calculus: processes of
geometric calculation by the polygonal method or the method of radius of curvature, use of tractional motion for
the conception of integraphs, reduction to graphic quadratures in nite or innite number.
 2003 Elsevier Inc. All rights reserved.
MSC : 01A45 ; 01A50 ; 01A55 ; 01A60 ; 34-03 ; 65-03 ; 65S05
Keywords: Differential equations; Graphical integration; Integraphs
Introduction
Dans l’Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, les mathématiciens appliqués Carl Runge et Friedrich A. Willers publient en 1915 un article intitulé « NumeAdresse e-mail : tournes@univ-reunion.fr.
0315-0860/$ – see front matter  2003 Elsevier Inc. All rights reserved.
doi:10.1016/S0315-0860(03)00033-8
458
D. Tournès / Historia Mathematica 30 (2003) 457–493
rische und graphische Quadratur und Integration gewöhnlicher und partieller Differentialgleichungen »
[Runge et Willers, 1915]. La deuxième partie de ce texte commence par une phrase que l’on peut juger
aujourd’hui surprenante : « Pour l’intégration des équations différentielles ordinaires, ce sont les méthodes graphiques qui sont les plus rapides et les plus claires ; c’est pourquoi nous commencerons notre
étude par elles ».1 Pour saisir pleinement la portée d’une telle afrmation, il faut avoir à l’esprit qu’au
début du vingtième siècle, le calcul graphique était une composante importante de l’analyse numérique.
En effet, avant l’apparition des calculatrices électroniques, le calcul numérique à la main et aux tables de
logarithmes était si coûteux en temps et en énergie que les scientiques et les ingénieurs, du moins tant
qu’ils n’avaient pas besoin d’une grande précision, avaient tendance à lui préférer le calcul graphique
[Tournès, 2000].
Je me propose ici de dégager les grandes idées qui sous-tendent l’intégration graphique des équations
différentielles ordinaires et de rechercher leurs origines. Il y a là, pour l’historien, un vaste champ à
explorer car, depuis la synthèse de Runge et Willers, c’est un thème qui est resté quasiment absent de la
littérature secondaire.
En laissant de côté ce qui relève de l’anecdote, il me semble que se dégagent quatre lignes principales
de recherches à parcourir successivement. Je distingue, tout d’abord, deux approches qui relèvent du
calcul par le trait : lorsqu’on pratique des constructions à la règle et au compas, il est assez naturel
de discrétiser le phénomène étudié et de remplacer la courbe intégrale inconnue par une suite de petits
segments de tangentes (cf. Section 1) ou par une suite de petits arcs de cercles osculateurs (cf. Section 2).
Ensuite, il y a une idée qui vient de Leibniz : puisque la règle, le compas et, plus généralement, les
systèmes articulés ne permettent d’atteindre que des courbes algébriques, il est indispensable de faire
intervenir un élément physique dans la construction si l’on veut accéder aux courbes transcendantes
dénies par des équations différentielles. Cet élément physique, c’est le mouvement tractionnel, qui a
donné lieu à une longue lignée de travaux conduisant aux intégraphes modernes (cf. Section 3). Enn, je
regroupe dans une dernière catégorie les méthodes qui, sans chercher à mettre au point des techniques
particulières pour les équations différentielles, visent à ramener l’intégration de ces dernières à des
quadratures, en nombre ni ou inni, et à exploiter des procédés déjà connus de quadrature graphique
(cf. Section 4).
1. Les lignes polygonales
1.1. La méthode d’Euler
Considérons un problème différentiel de conditions initiales mis sous la forme la plus générale
y = f (x, y),
y(x0 ) = y0
(dans cette écriture, la lettre y peut représenter une fonction vectorielle en dimension p, ce qui permet de
ramener à la même forme un système différentiel de p équations d’ordre 1 ou une équation différentielle
scalaire d’ordre p).
1 « Bei der Integration gewöhnlicher Differentialgleichungen führen die graphische Methoden am schnellsten zum Ziel und
sind am übersichtlichsten; sie mögen daher hier zuerst besprochen werden » [Runge et Willers, 1915, 141].
459
La technique connue aujourd’hui sous le nom de « méthode d’Euler » consiste à prendre comme
courbe intégrale approchée une ligne polygonale dont les sommets sont déterminés de proche en proche,
à partir du point initial, en remplaçant l’équation différentielle par une équation aux différences nies.
Chaque point (x, y) est joint au point suivant (x + x, y + y) par un segment de droite, avec une
différence y déterminée explicitement par la formule y = f (x, y)x ou implicitement par l’équation
y = f (x + x, y + y)x. Autrement dit, la méthode d’Euler revient à remplacer, sur chaque petit
intervalle, la courbe intégrale par sa tangente au point initial (méthode explicite) ou par sa tangente au
point nal (méthode implicite).
Dans ses Institutiones calculi integralis de 1768 et 1769, Leonhard Euler a exposé pour la première
fois la méthode polygonale sous une forme purement numérique, mais, dans sa version géométrique
et graphique, cette méthode est beaucoup plus ancienne. Elle vient tout droit des débuts du calcul
innitésimal, intimement associée à la conception d’une courbe en tant que ligne polygonale à une innité
de côtés formés de segments de tangentes inniment petits.
1.2. Les premières méthodes polygonales
En 1638, Florimond Debeaune soumet quelques exemples de problèmes inverses des tangentes à
la sagacité des mathématiciens français, mais Fermat, Roberval, Beaugrand et Debeaune lui-même
échouent dans leurs tentatives pour les résoudre. Seul René Descartes donne une solution à l’un des
problèmes (le deuxième) dans sa lettre à Debeaune du 16 février 1639 [Descartes, 1639, 514–517]. En
termes modernes, il s’agit de construire la courbe AX dénie par
dy x − y
=
dx
b
et y(0) = 0,
où AY = y, YX = x et où AB = b est une longueur donnée (cf. Fig. 1).
Faute de trouver une construction exacte, Descartes propose une méthode de construction par points
qui, d’après lui, est très générale :
Il y a bien vne autre façon qui est plus generale, & à priori, à sçauoir par l’intersection de deux tangentes, laquelle se doit tousiours
faire entre les deux points où elles touchent la courbe, tant proches qu’on les puisse imaginer. Car en considerant quelle doit estre
cette courbe, an que cette intersection se fasse tousiours entre ces deux points, & non au deça ny au delà, on en peut trouuer la
construction. [Descartes, 1639, 514]
Explicitons la démarche : il s’agit de construire des points de la courbe, de proche en proche, à partir du
point initial A. Supposant le point V déjà construit, on cherche à construire le point suivant X. Pour cela,
Descartes considère les tangentes à la courbe en V et en X, qui se coupent en D, et son approximation
repose sur le fait que la projection F de D sur l’asymptote se trouve entre celles de V et de X, à savoir
P et R. Or, quand F est en R, cela revient à remplacer la courbe par sa tangente au point V, et, quand F
est en P, cela revient à remplacer la courbe par sa tangente au point X. On reconnaît, nalement, les
méthodes d’Euler explicite et implicite.
Si l’on reconstitue scrupuleusement les constructions géométriques suggérées dans le texte, on obtient
deux lignes polygonales qui encadrent la vraie courbe intégrale. Sur la Fig. 2, je suis parti d’un partage
de AB en huit. Dans sa lettre, Descartes évoque également un partage en seize et la possibilité d’une
approximation aussi précise que l’on veut :
460
Fig. 1. Le deuxième problème de Debeaune [Descartes,
1639, 515].
Fig. 2. Reconstitution des constructions de Descartes.
De façon que, diuisant AB en plus de parties, on peut approcher de plus en plus, à l’inny, de la iuste longueur des lignes A α, Aβ, &
semblables, & par ce moyen construire Mechaniquement la ligne proposée. [Descartes, 1639, 516]
Un autre texte remarquable est la lettre d’Isaac Newton à Robert Hooke du 13 décembre 1679 [Newton,
1679, 307–308]. Newton y parle de la courbe décrite par un mobile soumis à une force centrale d’intensité
constante, et fournit une gure sans explication (cf. Fig. 3). Il écrit simplement : « I might add something
about its description by points quam proximè » [Newton, 1679, 308]. En termes modernes, il s’agit
d’intégrer l’équation différentielle
−−−→
CM
d2 M
,
= −k
2
dt
CM
où k est une constante, et où le mobile M, attiré par le centre C, part du point A avec une vitesse initiale
donnée dans la direction de la tangente Am.
Comment Newton a-t-il pu construire cette courbe par points ? Ainsi que l’a montré Herman Erlichson
de manière convaincante [Erlichson, 1991, 1992], la réponse se trouve vraisemblablement dans la
méthode des impulsions instantanées qui est exposée dans les Principia de 1687 et, auparavant, dans
le petit traité De Motu de 1684. De façon générale, Newton analyse le mouvement en le discrétisant.
Pendant chaque petit intervalle de temps, il considère que le mouvement est composé de deux parties :
un mouvement inertiel sur la tangente et une impulsion dirigée vers le centre d’attraction (cf. Fig. 4). Si
l’on regarde bien, il s’agit d’une méthode d’Euler implicite puisque, sur chaque intervalle de temps, la
courbe est remplacée par sa tangente au point nal. Sur la Fig. 5, j’ai appliqué cette technique au cas de
la gravité constante : on retrouve bien une courbe analogue à celle de la lettre de 1679.
Une conrmation que c’est effectivement cette méthode d’intégration graphique qui était utilisée par
Newton et Hooke se trouve dans un manuscrit de Hooke de 1685, sur une gure où subsistent tous les
traits de construction (cf. Fig. 6). Cette fois, il s’agit d’étudier la trajectoire d’un corps soumis à une
Fig. 3. La solution de Newton au problème de la gravité
constante [Erlichson, 1992, 57].
Fig. 5. Reconstitution de la construction de Newton.
461
Fig. 4. La méthode des impulsions instantanées dans les
Principia [Newton, 1687 (Trad. fr. 1759), planche 1].
Fig. 6. Une construction de Hooke en 1685 [Erlichson,
1997, 173].
gravité proportionnelle à la distance. Hooke démontre graphiquement que c’est une ellipse ayant pour
centre le point attracteur [Erlichson, 1997].
On rencontre encore une utilisation de la méthode polygonale dans un article de Gottfried W. Leibniz
de 1694 sur le problème de l’isochrone paracentrique : trouver la trajectoire d’un point pesant qui
s’éloigne uniformément d’un point donné [Leibniz, 1694 (Trad. fr. 1989), 299–304]. Gardons les
notations de Leibniz (cf. Fig. 7) : on donne un cercle de centre A et de rayon AH = a ; le point C,
qui s’éloigne uniformément du point xe A à partir d’un point initial 1 C, est repéré par les coordonnées
462
Fig. 7. L’isochrone paracentrique [Leibniz, 1694 (Trad. fr. 1989), 460].
AC = t et AL = z ; celles-ci vérient l’équation différentielle
a dz
dt
√ =√
.
at
a 3 z − az3
Après une première solution, qui ramène l’intégration à la rectication d’une courbe algébrique,
Leibniz donne une seconde solution sous forme d’une ligne polygonale 1 C 2 C 3 C. . . construite à la règle
et au compas, dans laquelle on peut reconnaître la méthode d’Euler implicite. La conclusion de Leibniz
est des plus intéressantes pour comprendre le statut de la méthode polygonale, à la fois construction
graphique concrète de la courbe intégrale et justication de l’existence de celle-ci par une idéalisation du
processus :
463
Nous obtiendrons de la sorte un polygone 1 C 2 C 3 C etc. remplaçant la courbe inconnue, c’est-à-dire une courbe Mécanique tenant
lieu de courbe Géométrique, du même coup nous voyons bien qu’il est possible de faire passer la courbe géométrique par un point
donné 1 C puisqu’une telle courbe est la limite où en dénitive s’effacent progressivement les polygones convergents. Nous disposons
en même temps d’une série de grandeurs ordinaires convergeant vers la grandeur transcendante cherchée. [Leibniz, 1694 (Trad. fr.
1989), 304]
Il y a là, très clairement, l’idée qui sera approfondie par Cauchy, en 1824, et par Lipschitz, en 1868,
pour établir, avec les nouveaux critères de rigueur du dix-neuvième siècle, le théorème fondamental
d’existence dit « de Cauchy–Lipschitz ».
1.3. Les directrices de Bernoulli
La difculté de la méthode polygonale initiale, telle que nous en avons rencontré des avatars chez
Descartes, Newton ou Leibniz, est qu’il faut, pour chaque nouvelle équation, imaginer un procédé
spécique permettant, à chaque étape, de construire la pente de la tangente au point en lequel on est
parvenu. Le procédé serait plus performant si l’on pouvait tracer directement toute courbe intégrale en
évitant ces constructions auxiliaires. Pour cela, il serait bon de disposer à l’avance de toutes les tangentes,
c’est-à-dire de connaître graphiquement le champ de vecteurs déterminé par l’équation différentielle.
C’est Jean Bernoulli, en 1694, qui développe pour la première fois ce point de vue de portée universelle
[Jean Bernoulli, 1694a]. À cet effet, il introduit la notion de « lignes directrices », lieux des points où les
courbes intégrales ont une pente donnée. Pour l’équation différentielle y = f (x, y), il s’agit des courbes
d’équation f (x, y) = k, où k est une constante (aujourd’hui, à la suite des travaux de Junius Massau (cf.
Section 1.4), ces courbes sont appelées « isoclines »).
Par le moyen des directrices, Bernoulli dégage un procédé général de construction des courbes
intégrales : on construit préalablement un faisceau de directrices et la pente associée à chacune d’elles ;
pour obtenir ensuite la courbe intégrale issue d’un point A, il suft de connecter les directrices successives
par des segments ayant pour pentes les pentes associées à ces directrices.
Le procédé a suscité l’intérêt des contemporains, ainsi qu’il apparaît à l’examen des correspondances
des années 1694–1695. Leibniz, nullement surpris, déclare y avoir songé aussi [Leibniz, 1695, 178].
Pierre Varignon tente de comprendre la méthode (cf. Fig. 8), mais, insatisfait, demande des explications
complémentaires ainsi qu’un exemple [Varigon, 1695, 81].
Je ne sais pas si Varignon a obtenu ce qu’il souhaitait. Par contre, on trouve un tel exemple dans une
lettre de Bernoulli au marquis de l’Hospital [Jean Bernoulli, 1694b, 247–249]. Il s’agit de l’équation
xx dx + yy dx = aa dy
dont les directrices sont des cercles (cf. Fig. 9). Bernoulli en prote pour préciser l’intérêt et la portée de
son idée :
Voilà donc ma methode que j’ay trouvée pour la construction generale des équations differentielles ; elle pourra etre d’un grand usage
dans la pratique, lorsqu’on se contente d’une construction mechanique, car plus on fait de courbes directrices approchantes l’une de
l’autre, et plus on approchera de la veritable courbe cherchée ; Outre cela on construit par là avec une egale facilité toutes les equations
differentielles, sans employer aucune rectication ni quadrature, au lieu que la methode ordinaire aprés avoir surmonté la plus grande
difculté, qui est de separer les indeterminées (ce qui est pourtant le plus souvent impossible) demande encore une rectication ou
quadrature, ce qui rend la pratique pour ainsi dire impratiquable. [Jean Bernoulli, 1694b, 249]
464
Fig. 8. Les directrices de Bernoulli illustrées par Varigon [1695, 81].
Fig. 9. Un exemple de Bernoulli [1694b, 248].
1.4. Renouveau de la méthode polygonale à la n du dix-neuvième siècle
La méthode polygonale, dans sa version graphique, n’évolue plus jusque vers les années 1870. À ce
moment, les besoins de calcul des ingénieurs font que l’on assiste à un essor fulgurant des méthodes
graphiques et à la constitution du calcul graphique en tant que discipline autonome, avec la mise en place
d’enseignements spécialisés et la publication de nombreux traités [Tournès, 2000]. Ce phénomène global
prote, en particulier, aux méthodes d’intégration graphique, dont l’étude est reprise et approfondie par
465
Fig. 10. Étude graphique d’une équation différentielle à partir de ses isoclines [Massau, 1878–1887, planche 13].
l’ingénieur tchèque Josef Solín2 [1872], puis, surtout, par le Belge Junius Massau,3 professeur à l’école
d’ingénieurs de Gand. Dans une série de travaux élaborés entre 1873 et 1904, Massau développe tous les
aspects de l’intégration graphique, depuis les quadratures graphiques jusqu’à l’intégration graphique des
équations différentielles ordinaires et même des équations aux dérivées partielles [Massau, 1878–1887,
1887, 1889, 1900–1904].
Massau introduit le terme de « courbe isocline » à partir de la remarque que c’est le lieu des points où
l’intégrale a une même inclinaison [Massau, 1878–1887, livre VI, 501] (il semble que ce soit la première
occurrence du mot « isocline » dans ce contexte). Le principal apport de Solín et Massau est de prendre
les sommets de la ligne polygonale, non pas sur les isoclines, mais entre les isoclines, à peu près à égale
distance de deux isoclines consécutives (sur la Fig. 10, on peut voir un exemple de courbe intégrale
ABCD construite de cette manière). Suivant la façon dont on interprète l’expression « à peu près à égale
distance », on obtient des équivalents graphiques de la méthode du point milieu et de la méthode des
trapèzes, qui s’écrivent respectivement
1
f (x, y) + f (x + x, y + y)
1
y =
x.
y = f x + x, y x + x x,
2
2
2
On abandonne ainsi la méthode d’Euler, d’ordre 1, pour des méthodes plus efcaces d’ordre 2. Massau
proposera plus tard des traductions graphiques d’autres formules usuelles de quadratures approchées
[Massau, 1878–1887, livre VI, 502 ; 1889, 427]. Rappelons à ce propos que c’est également en essayant
d’adapter aux équations différentielles les méthodes du point milieu, des trapèzes et de Simpson que
2 Josef Marcell Solín (1841–1912), ingénieur et mathématicien tchèque, fut, à partir de 1876, professeur de mécanique de
construction à l’École polytechnique tchèque de Prague. Il appartient à la fameuse école géométrique tchèque qui brilla durant
la seconde moitié du dix-neuvième siècle et le début du vingtième.
3 Junius Massau (1852–1909) est né dans le Hainaut, en Belgique wallonne. Il reçut son diplôme d’ingénieur des ponts
et chaussées en 1874, à l’École du génie civil de l’université de Gand. Dès 1878, il devint professeur dans cette école, où
il enseigna la mécanique analytique, la théorie des machines et la graphostatique. Ses travaux sur la mécanique appliquée,
l’intégration graphique et la nomographie lui valurent une solide réputation en Belgique et à l’étranger. En 1906, l’Académie
des sciences de Paris lui décerna le prix Wilde.
466
les mathématiciens appliqués allemands Carl Runge, Karl Heun, et Wilhelm Kutta vont mettre au
point, entre 1895 et 1901, les algorithmes numériques connus aujourd’hui sous le nom de « méthodes
de Runge–Kutta ». Sous forme numérique, cette adaptation soulève de grandes complications car les
équations aux différences déterminant l’accroissement y sont implicites. Par contre, sous forme
graphique, la combinaison des pentes fournies par plusieurs isoclines consécutives est immédiate, et
a été abondamment exposée et commentée à la suite de Solín et Massau [Nehls, 1877, chap. 10 ; Ocagne,
1908, 155–157], parfois en lien avec des réexions sur les preuves du théorème fondamental d’existence
[Picciati, 1893; Cotton, 1905, 496 ; 1908, 122]. Il n’est donc pas absurde de penser que c’est peut-être
le savoir-faire accumulé préalablement par les calculateurs graphiques à partir de 1870 qui a inspiré
Runge et ses successeurs lors du passage à une approche numérique. En retour, après la mise au point
rigoureuse de la formule classique de Runge–Kutta d’ordre 4, on assiste à sa traduction sous forme
graphique [Runge, 1912, 132–135].
Dès la n du dix-neuvième siècle, on rencontre des applications concrètes d’envergure des versions
améliorées de la méthode des isoclines. Massau, tout d’abord, s’en sert pour déterminer la forme des
axes hydrauliques des cours d’eau non prismatiques [Massau, 1878–1887, livre VI]. Henry Léauté,4
après avoir utilisé l’ancienne méthode d’Euler pour l’étude des oscillations à longue période dans les
machines actionnées par des moteurs hydrauliques [Léauté, 1885, 122–124], passe aux méthodes plus
précises du point milieu et des trapèzes pour l’étude du mouvement troublé des moteurs consécutif à une
perturbation brusque [Léauté, 1891, 22–23].
Par ailleurs, pour Massau, les courbes isoclines ne servent pas seulement à la construction pratique
d’une courbe intégrale particulière. À partir de l’étude des isoclines, il s’intéresse aussi à diverses
propriétés géométriques du champ de vecteurs de l’équation différentielle : courbe des inexions, courbe
des rebroussements, points singuliers, etc. Il entreprend ainsi, à peu près à la même époque que Poincaré
et probablement de façon indépendante, une véritable étude qualitative des équations différentielles. Cette
étude est utile au calculateur graphique pour préciser le tracé des intégrales au voisinage des singularités,
là où les méthodes générales peuvent tomber en défaut.
1.5. Variantes et améliorations de la méthode polygonale
Václav Láska5 étudie les équations de la forme f (x + y dy/dx, y) = 0, qui se prêtent à une
construction simple et rapide des petits segments de tangente à partir de la courbe f (ξ, η) = 0 [Láska,
1890]. Dans la même veine, Emanuel Czuber6 [1899] simplie la méthode d’Euler pour les équations
linéaires du premier ordre
y + P (x)y = Q(x)
4 Henry Léauté (1847–1916), ingénieur français, fut professeur de mécanique à l’École Polytechnique et membre de
l’Académie des sciences. Il est connu pour ses travaux à caractère mathématique sur la régulation et la télécommande des
machines.
5 Václav Láska (1862–1943) était un géophysicien, astronome et mathématicien tchèque.
6 Emanuel Czuber (1851–1925), mathématicien tchèque, t ses études à l’École polytechnique allemande de Prague. Il
soutint son doctorat en 1876 dans le domaine de la géométrie pratique. En 1891, il devint professeur à l’École polytechnique
de Vienne. Il travailla sur la théorie des probabilités, le calcul des observations et les mathématiques des assurances. En 1894,
il créa un cours technique sur les assurances, dans lequel il introduisait l’usage des probabilités. C’est également lui qui écrivit
l’article sur le calcul des probabilités dans l’Encyklopädie der mathematischen Wissenschaften.
467
en remarquant que les tangentes passant par les points de même abscisse x concourent au point de
coordonnées
ξ =x+
1
,
P (x)
η=
Q(x)
.
P (x)
Il suft donc de construire préalablement par points la courbe (ξ, η) et de la graduer avec les abscisses x
pour pouvoir ensuite tracer la tangente passant par un point donné quelconque. On peut ainsi construire
les courbes intégrales à l’aide d’une seule courbe auxiliaire en lieu et place de tout un faisceau d’isoclines.
Tetsuzô Kojima approfondit la remarque de Czuber, d’une part en traduisant sous une forme graphique
facile à mettre en œuvre les méthodes numériques de Runge et de Kutta, d’autre part en étendant le
procédé à certaines équations non linéaires [Kojima, 1914]. Pour cela, il remplace les parallèles à l’axe
des ordonnées par un système de courbes telles que les tangentes passant par les points de chaque courbe
concourent en un même point. De manière indépendante, Richard Neuendorff 7 développe la même idée
pour construire les équations différentielles données en coordonnées polaires [Neuendorff, 1922].
En généralisant autrement la remarque de Czuber, Rudolf Mehmke8 propose, pour une équation
différentielle générale y = f (x, y), de construire l’enveloppe des tangentes passant par les points d’une
parallèle à l’axe des ordonnées [Mehmke, 1917, 119 ; 1930]. Ainsi, avec une règle passant par un point
donné et placée de manière à être tangente à l’enveloppe correspondante, on peut tracer directement
un petit élément de tangente. Willers9 étudie des cas, en particulier celui de l’équation de Riccati, où
l’enveloppe est une conique, ce qui permet d’obtenir des constructions simples et exactes de ces petits
éléments de tangente [Willers, 1918a, 1918b].
Plutôt que les isoclines k = f (x, y), où k est une constante, Theodore R. Running10 construit
préalablement les courbes y = f (x, k), en portant les valeurs de y en ordonnée [Running, 1913].
À partir du point représentant les conditions initiales, on trace des segments connectant chaque courbe à
la suivante, de sorte que l’aire sous chaque segment soit égale à la différence des valeurs correspondantes
de y. La ligne polygonale ainsi construite approche la courbe dérivée de la courbe intégrale cherchée,
et les coordonnées des points de cette dernière peuvent se lire directement sur la gure, comme sur
7 Richard Neuendorff (1877–1935) a soutenu sa thèse en 1908 à l’université Christian-Albrechts de Kiel et fut ensuite
professeur de mathématiques à l’université de Francfort. Il a écrit plusieurs livres de mathématiques pratiques, de calcul
graphique et de calcul mécanique à destination des élèves des écoles techniques et des ingénieurs. Ces ouvrages ont
apparemment eu du succès puisqu’ils ont fait l’objet de plusieurs rééditions entre 1911 et 1923.
8 Rudolf Mehmke (1857–1944) s’initia aux mathématiques et à l’architecture à Stuttgart. Il se rendit ensuite à Berlin où il
étudia sous la direction de Weierstrass, Kronecker et Kummer. Il devint professeur de mathématiques à l’École polytechnique
de Darmstadt en 1884, puis professeur de géométrie descriptive et projective à l’École polytechnique de Stuttgart en 1894. Avec
Carl Runge, Mehmke est considéré comme l’une des grandes gures des mathématiques appliquées du début du vingtième
siècle. Il s’est surtout occupé de géométrie descriptive, de méthodes graphiques et d’instruments mathématiques, mais il
s’intéressait aussi aux mathématiques pures (calcul vectoriel de Grassmann, fonctions elliptiques, séries trigonométriques,
théorie du potentiel, etc.).
9 Friedrich Adolf Willers (1883–1959) fut un élève de Carl Runge. En 1928, il obtint un poste à l’Institut de mathématiques
appliquées de l’université technique de Freiberg. En 1949, il devint directeur de l’Institut de mathématiques appliquées de
l’université technique de Dresde. Il se consacra à l’application des mathématiques aux sciences de l’ingénieur, notamment à
l’intégration graphique, à l’analyse pratique et aux instruments de mathématiques.
10 Theodore Rudolph Running (1866–1952) fut professeur de mathématiques à l’université du Michigan, à Ann Arbor.
468
un nomogramme. De son côté, Gustav Doetsch11 exploite les courbes y = f (k, y), qu’il trace sur un
transparent tournant et dont il déduit une approximation polygonale de la courbe intégrale dans un
système de coordonnées tangentielles, en représentant chaque couple (x, y) par la droite d’équation
ξ cos x + η sin x − y = 0 [Doetsch, 1921].
Paul Schreiber12 examine l’avantage que présente dans certains cas, pour le tracé des isoclines et
des courbes intégrales, l’utilisation des papiers logarithmiques et autres papiers spécialement gradués
qui commencent à être commercialisés à cette époque [Schreiber, 1922]. Enn, pour de larges classes
d’équations d’ordre n couramment rencontrées dans les applications, Victor A. Bailey13 et Jack M.
Somerville14 simplient considérablement la construction des lignes polygonales par l’emploi d’un
transparent que l’on fait glisser sur la feuille de dessin et sur lequel on a tracé préalablement des fonctions
auxiliaires liées aux coefcients de l’équation [Bailey and Somerville, 1938].
2. Les rayons de courbure
2.1. La méthode des rayons de courbure
Considérons une équation différentielle du second ordre, de la forme
dy
d2 y
,
= f x, y,
dx 2
dx
et posons dy/dx = tan α. On sait que le rayon de courbure d’une courbe solution est donné par
d2 y
1
= cos3 α 2 = cos3 α × f (x, y, tan α).
ρ
dx
Sous cette nouvelle forme, l’équation différentielle fournit le rayon de courbure en fonction de x, y et α,
c’est-à-dire en fonction de la position du mobile et de la direction du mouvement. On conçoit donc la
possibilité, à partir d’un point initial en lequel on connaît la pente de la tangente, d’obtenir une courbe
approchée en construisant une suite d’arcs de cercles osculateurs.
11 Gustav Heinrich Adolf Doetsch (1892–1977) étudia les mathématiques, la physique, la science des assurances et la
philosophie à Göttingen, Berlin et Munich. Après avoir participé à la Première Guerre mondiale en tant qu’ofcier de l’air,
il passa son Habilitation à Hanovre, puis enseigna les mathématiques appliquées à Halle, Stuttgart, et enn Fribourg à partir
de 1931. Ses travaux portent sur la transformation de Laplace : il a donné une structure solide à la théorie et en a fait un outil
central des techniques de l’ingénieur. Paciste dans les années 1920, il dut faire des concessions au national-socialisme pour ne
pas perdre son poste. En 1939, il fut mobilisé au service de la recherche de guerre. Après la guerre, il fut d’abord suspendu lors
de la dénazication avant d’être réinstallé dans son poste de professeur à Fribourg en 1951.
12 Carl Adolf Paul Schreiber (1848–1924) fut directeur de la station météorologique de Chemnitz, en Allemagne. Ses
nombreux travaux portent sur la météorologie, l’hydrographie, la géodésie et l’astronomie. C’est en cherchant à modéliser
des observations météorologiques à partir de théories hydrodynamiques et thermodynamiques qu’il a été amené à s’intéresser
à des moyens pratiques d’intégration des équations différentielles. Il a publié plusieurs autres articles sur l’usage du papier
logarithmique, en l’appliquant notamment à la trigonométrie sphérique et à l’étude du mouvement d’un ballon libre.
13 Victor Albert Bailey (1895–1964) est né à Alexandrie, en Égypte. Après des études au Queen’s College, à Oxford, il
enseigna la physique à l’université de Sydney à partir de 1926.
14 Jack Murielle Somerville (1912–1964) est né à Sydney où il commença ses études avant de rejoindre l’université de
Cambridge. Il enseigna ensuite les mathématiques et la physique à l’université de Sydney et à celle de Nouvelle-Angleterre.
469
Fig. 11. Méthode des rayons de courbure [Poncelet, 1827–1830, 2e partie, 46].
Sylvestre-François Lacroix décrit clairement cette construction dans la seconde édition de son Traité
du calcul différentiel et du calcul intégral :
Ayant pris arbitrairement le premier point M et la première tangente MT, ce qui donne les premières valeurs des quantités x, y, dy/dx,
on tire de l’équation proposée la valeur correspondante de d2 y/dx 2 , avec laquelle on calcule le rayon du premier cercle osculateur ;
on porte ce rayon sur la normale MF, et décrivant un cercle MN , on prend sur ce cercle un second point N , duquel résultent de
nouvelles valeurs de x, y et dy/dx, qui conduisent à une seconde valeur de d2 y/dx 2 , puis à un second cercle osculateur, et la courbe
cherchée se construit par une suite d’arcs de cercle. On pourrait appliquer ce procédé aux équations du premier ordre, mais la première
tangente ne serait plus arbitraire, et il faudrait différentier l’équation proposée pour en tirer l’expression de d2 y/dx 2 . [Lacroix, 1814,
451]
Dès les premiers temps du calcul innitésimal, on s’était intéressé à la détermination du centre de
courbure et du cercle osculateur en un point d’une courbe donnée. Je n’ai pourtant pas trouvé mention
du problème inverse, c’est-à-dire de la construction d’une courbe dénie par une équation différentielle à
partir de ses centres de courbure, avant ce texte de Lacroix. Le fait que cette construction apparaisse dans
la seconde édition de 1814 alors qu’elle est absente de la première édition de 1798 peut laisser penser
que la méthode des rayons de courbure était très récente à l’époque de Lacroix ou, tout au moins, encore
peu connue.
2.2. Poncelet et la balistique
La première application concrète des rayons de courbure aux équations différentielles se rencontre
chez Jean-Victor Poncelet, dans le cours de mécanique industrielle qu’il professait à l’École d’application
de Metz [Poncelet, 1827–1830, 2e partie, 46–49]. Poncelet s’intéresse à la trajectoire d’un projectile de
masse M, lancé d’un point A avec une vitesse initiale V dirigée suivant la tangente AT, et soumis à
une force motrice AP (cf. Fig. 11). La composante Ap perpendiculaire à la tangente équilibre la force
centrifuge F = M × V2 /AC, ce qui permet de construire le rayon de courbure AC. À partir du centre
de courbure C, on trace un petit arc de cercle AA . La nouvelle vitesse V est alors calculée grâce au
théorème des forces vives 12 M × (V2 − V 2 ) = Aq × AA , et tout est prêt pour recommencer.
Les idées de Poncelet ont été approfondies par un de ses élèves, le général Isidore Didion,15 qui a
développé plusieurs variantes graphiques de la méthode des rayons de courbure (cf. Fig. 12) en vue de la
15 Isidore Didion (1798–1878) entra à l’École Polytechnique de Paris en 1817 et choisit ensuite une carrière d’ofcier
d’artillerie. Il devint professeur d’artillerie à l’École d’application de Metz en 1837, puis examinateur d’admission à l’École
Polytechnique. Il fut promu général en 1858. De 1848 à 1875, il a publié divers travaux relatifs à la balistique et à l’artillerie.
470
Fig. 12. Construction de la trajectoire d’une bombe [Didion, 1848, planche 2].
résolution de nombreux problèmes de balistique extérieure [Didion, 1848, Section VI]. La méthode des
rayons de courbure se retrouve ensuite régulièrement dans les traités de balistique jusqu’à une époque
récente, sous le nom de « méthode de Poncelet–Didion » [Sonnet, 1867, 1355–1358 ; Charbonnier, 1921,
598–599].
2.3. Une méthode intimement associée aux travaux de Lord Kelvin
La technique des rayons de courbure a été également employée pour traiter le problème de la
capillarité : quelle est la courbe méridienne de la surface de révolution formée par un liquide dans un tube
ou par une goutte de liquide reposant sur une surface plane ? Lors d’une conférence donnée le 29 janvier
1886, William Thomson (Lord Kelvin) présente une solution de ce problème en intégrant graphiquement
l’équation différentielle de Laplace grâce aux rayons de courbure [W. Thomson, 1889]. Selon ses dires,
Thomson a conçu le principe de cette construction vers 1855, avant d’en coner l’exécution en 1874 à
l’un de ses assistants, John Perry,16 qui a réalisé de magniques dessins (cf. Fig. 13).
16 John Perry (1850–1920) enseigna les mathématiques et la mécanique aux ingénieurs. Il travailla constamment avec son
collègue William Edward Ayrton (1847–1908). Ensemble, ils publièrent environ 70 articles scientiques et techniques entre
1876 et 1891. On leur doit notamment l’invention d’un système de freinage pour les trains électriques, le premier tricycle
électrique, et de nombreux instruments de mesure électriques.
471
Fig. 13. Une courbe de capillarité dessinée par John Perry [W. Thomson, 1889, 34].
En 1892, Thomson emploie aussi la méthode des rayons de courbure pour la recherche d’orbites
périodiques du problème des trois corps, à la suite des travaux de l’astronome américain George W. Hill
sur la théorie de la Lune [W. Thomson, 1892]. À partir de là, pour les auteurs ultérieurs de traités de
calcul graphique, la méthode des rayons de courbure va être dénitivement associée à Lord Kelvin, qui
en est considéré comme le créateur [Willers, 1928 (Trad. angl. 1948), 394]. Tout se passe comme si les
recherches antérieures faites en balistique depuis Poncelet n’avaient jamais été diffusées en dehors du
milieu restreint des ingénieurs militaires.
Par ailleurs, il est intéressant de constater que la méthode graphique de Lord Kelvin a probablement
inspiré les astronomes britanniques qui ont mis au point les méthodes multipas pour l’intégration
numérique des équations différentielles, celles qui sont appelées aujourd’hui « méthodes d’Adams »
[Tournès, 1998, 43]. En effet, John C. Adams s’intéresse au problème de la capillarité en 1855, à l’époque
des premiers travaux de Thomson, George H. Darwin recherche des orbites périodiques du problème des
472
trois corps juste après l’article de Thomson de 1892, et tant Adams que Darwin mettent les équations
du second ordre qu’ils ont à intégrer sous une forme faisant apparaître le rayon de courbure. De plus,
à propos de son algorithme numérique, Darwin écrit explicitement : « This method is the numerical
counterpart of the graphical process described by Lord Kelvin in his Popular Lectures, but it is very
much more accurate » [Darwin, 1897, 125]. Comme nous l’avons vu plus haut pour les méthodes de
Runge–Kutta (cf. Section 1.4), il semble que les méthodes d’Adams proviennent en partie de la volonté
d’approfondir sous forme numérique des idées exploitées en premier lieu par les calculateurs graphiques.
2.4. Variantes et amélioration de la méthode des rayons de courbure
En 1893, Charles V. Boys17 apporte une simplication pratique à la méthode de Thomson : il
conçoit un dispositif assez simple, formé d’une règle spécialement graduée (an d’éviter certains calculs
auxiliaires) et d’un tripode dont chaque pied se termine par une aiguille [Boys, 1893]. La règle, en
celluloïd transparent, comporte un petit trou dans lequel on fait passer le crayon servant à tracer la courbe.
Un pied du tripode est planté sur la règle au niveau du premier centre de courbure et les deux autres sur
la feuille de papier, de sorte qu’on puisse tracer un petit arc par un mouvement de rotation de la règle.
En maintenant la règle xe, on déplace ensuite le tripode jusqu’au deuxième centre de courbure et ainsi
de suite. L’avantage de ce dispositif est que l’on trace les arcs de cercle successifs sans que la pointe du
crayon quitte la feuille de papier, d’où une courbe parfaitement lisse. En 1916, Rudolf Rothe18 améliore
l’instrument de Boys en ajoutant, sur une version élargie de la règle, un second axe perpendiculaire au
premier, ce qui permet, lorsqu’on travaille sur du papier quadrillé, de lire directement y en plus de x et y
[Rothe, 1916].
En 1913, Ernst Meissner19 utilise le système de coordonnées tangentielles où l’on représente le couple
(x, y) par la droite d’équation ξ cos x + η sin x − y = 0 (cf. Section 1.5) : pour l’équation du second
ordre y = f (x, y, y ), le rayon de courbure vaut tout simplement ρ = y + y , d’où une construction
particulièrement aisée dans ce mode inhabituel de représentation [Meissner, 1913]. Meissner s’en sert
avec brio (cf. Fig. 14) pour intégrer les équations du pendule simple et du pendule sphérique, et
pour étudier les oscillations amorties ou forcées d’un système mécanique. Plus tard, Rudolf Inzinger20
généralisera la méthode de Meissner aux équations linéaires d’ordre quelconque à coefcients constants
[Inzinger, 1947]. Entre-temps, Neuendorff fournit l’adaptation de la méthode des rayons de courbure à la
construction des équations données en coordonnées polaires, en 1922 pour le second ordre [Neuendorff,
1922, 135] et l’année suivante pour un ordre quelconque [Neuendorff, 1923].
En ce qui concerne l’amélioration de la précision, on a proposé très tôt de prendre, pour le tracé
d’un petit arc de courbe, non pas le rayon de courbure au point initial de l’arc, mais une moyenne, en
17 Sir Charles Vernon Boys (1855–1944) étudia la physique, les mines et la métallurgie à la Royal School of Mines de
Londres. Il travailla au Royal College of Science à partir de 1881 et fut élu Fellow of the Royal Society en 1888. On peut
signaler qu’il inventa un intégraphe indépendamment d’Abdank-Abakanowicz. Toutefois, sa contribution la plus importante
concerne la gravitation : en se servant de bres de quartz pour mesurer les petites forces, il reprit entre 1890 et 1895 les
expériences de Cavendish (1798) an d’améliorer la valeur de la constante gravitationnelle. Après 1897, Boys abandonna sa
situation de professeur pour occuper le poste de Metropolitan Gas Referee. En parallèle, il développa une carrière lucrative
d’expert en instruments de mesure. Il présida la Physical Society of London (1917–1918) et fut anobli en 1935.
18 Rudolf Ernst Rothe (1873–1942) fut professeur de mathématiques à l’École polytechnique de Berlin–Charlottenburg.
19 Ernst Franz Samuel Meissner (1883–1939) fut professeur de mécanique à l’École polytechnique de Zurich.
20 Rudolf Inzinger, né en 1907, fut professeur à l’université technique de Vienne.
Fig. 14. Méthode des rayons de courbures en coordonnées tangentielles [Meissner, 1913, 222].
473
474
Fig. 15. La tractrice, courbe de tangente constante MT = a et d’équation x = a ln
√
a+ a 2 −y 2
−
y
a2 − y 2 .
divers sens, des rayons de courbure aux extrémités [Didion, 1848, 200 ; W. Thomson, 1892, 444]. Si ces
variantes diminuent effectivement l’erreur, elles n’entraînent pas un changement d’ordre de convergence.
Un ingénieur argentin, José Babini,21 a étudié cette question d’un point de vue théorique et a tenté de
mettre au point, de façon analogue aux méthodes de Runge–Kutta, des méthodes graphiques d’ordre plus
élevé par des combinaisons savantes de rayons de courbure successifs [Babini, 1926]. Dans le même
esprit, Ernst Völlm22 a réussi à accélérer la convergence de la construction en coordonnées tangentielles
de Meissner [Völlm, 1939].
3. Le mouvement tractionnel
3.1. De la tractrice aux premiers intégraphes
En 1672, Leibniz rencontre à Paris le médecin Claude Perrault, qui lui soumet le problème suivant :
quelle est la courbe décrite par une montre lorsque l’on tire l’extrémité de la chaîne le long du bord
rectiligne de la table ? Cette courbe, la tractrice, a beaucoup intéressé les géomètres dans la mesure où elle
est liée algébriquement à la courbe logarithmique (cf. Fig. 15). On voyait là un moyen indirect de tracer
la courbe logarithmique d’un mouvement continu et, par suite, de légitimer l’emploi des logarithmes en
géométrie. Dans ce but, de nombreux savants, Christiaan Huygens en tête, tentèrent de mettre au point
un instrument pour tracer concrètement la tractrice. Tout cela a été étudié nement par Henk J.M. Bos,
auquel je renvoie pour davantage de détails [Bos, 1988].
Généralisant cette idée de mouvement tractionnel, Leibniz expose, en 1693, le principe d’une sorte
d’intégraphe universel [Leibniz, 1693 (Trad. fr. 1989), 263–267]. L’appareil (cf. Fig. 16) consiste
essentiellement en un l dont on tire une extrémité T le long d’une droite AT ou, plus généralement,
d’une courbe donnée (T) dans un plan horizontal et dont la longueur variable TC est déterminée par une
autre courbe donnée (E) dans un plan vertical. La seconde extrémité C du l trace alors, en l’enveloppant,
21 José Babini (1897–1984), ls d’immigrants italiens, est né à Buenos Aires, en Argentine. Après une carrière d’ingénieur
civil puis de professeur de mathématiques, il se tourna vers l’histoire des sciences dans les années 1940. Parallèlement, il occupa
des fonctions administratives dans plusieurs universités argentines avant d’accéder, en 1958, à la tête de la Direction nationale
de la culture. Entre 1920 et 1940, il fut le premier dans son pays à utiliser des machines à calculer pour son enseignement de
mathématiques appliquées.
22 Ernst Völlm soutint en 1933 une thèse sur la théorie de la nomographie à l’École polytechnique de Zurich.
475
Fig. 16. L’intégraphe de Leibniz [1693 (Trad. fr. 1989), 458].
une courbe (C). En choisissant convenablement les courbes (T) et (E), on devrait pouvoir, selon Leibniz,
tracer une courbe (C) dont les pentes obéissent à une loi donnée, c’est-à-dire résoudre n’importe quel
problème inverse des tangentes. Bien que ce programme puisse sembler assez vague, nous allons voir
qu’il a guidé la conception de nombreux instruments d’intégration jusqu’au début du vingtième siècle.
Contrairement aux méthodes des lignes polygonales et des rayons de courbure vues plus haut, il ne
s’agit plus ici de constructions approchées, mais bien de constructions exactes, du moins en théorie, sous
réserve d’ignorer l’imperfection matérielle des appareils.
Dès la n du dix-septième siècle, plusieurs géomètres imaginent et, parfois, construisent effectivement
des instruments tractionnels formés de ls guidés par des règles ou des poulies pour résoudre certains
problèmes inverses des tangentes. Un premier exemple est un problème posé par Jean Bernoulli en 1693
[Jean Bernoulli, 1693] : trouver une courbe dont la tangente PQ est proportionnelle à l’abscisse du point
Q (cf. Fig. 17). La condition PQ/OQ = p = EO/OF se traduit par l’équation
2
dx
dx
.
=p x −y
y 1+
dy
dy
La même année, son frère Jacques obtient une solution à l’aide d’une équerre que l’on déplace vers la
gauche et d’un l GQP dont l’extrémité G est maintenue sur la droite EF [Jacques Bernoulli, 1693].
On reconnaît, dans le dispositif de Bernoulli, l’équivalent de la courbe (E) de Leibniz (ici la droite
EF). Huygens, de son côté, aborde le même problème en imaginant divers systèmes avec des poulies
pour guider le l, en fonction de la valeur du paramètre p [Huygens, 1693, 550–552]. Dans des
écrits ultérieurs, Jacques Bernoulli suggère que, pour guider le l, on pourrait remplacer la droite EF
de la Fig. 17 par une courbe algébrique quelconque, ce qui permettrait d’intégrer d’autres équations
différentielles [Jacques Bernoulli, 1694 ; 1696].
476
Fig. 17. Un appareil tractionnel de Jacques Bernoulli [Bos, 1988, 42].
En 1706, l’Anglais John Perks,23 cherchant une quadratrice de l’hyperbole, est amené à étudier une
courbe telle que la somme de la tangente PR et de la sous-tangente RM soit constante [Pedersen, 1963,
5–11]. La condition PR + RM = a, qui se traduit par l’équation
2
dx
dx
= a,
−y
y 1+
dy
dy
est réalisée mécaniquement par l’instrument tractionnel schématisé sur la Fig. 18, où la règle SO est xe
et où l’on tire le point R vers la droite.
Les Italiens Giovanni Poleni24 [1728], professeur à l’université de Padoue, et son élève Giambatista
Suardi25 [1752, 26–36] construisent à leur tour des appareils pour tracer la tractrice et la courbe
23 John Perks est seulement connu pour trois articles publiés dans les Philosophical Transactions en 1699, 1706, et 1715.
Dans ces écrits, il s’intéresse à la quadrature des lunules, à la quadrature de l’hyperbole et à des problèmes de cartographie liés
à la projection de Mercator. De tels problèmes l’amènent à étudier des courbes transcendantes et à construire des instruments
pour les tracer mécaniquement.
24 Giovanni Poleni (1683–1761) suivit des études de philosophie et de théologie à Venise avant d’embrasser une carrière
juridique. Il se tourna ensuite vers les mathématiques et les sciences physiques : en 1719, il reçut la chaire de mathématiques
de l’université de Padoue laissée vacante par le départ de Nicolas Bernoulli ; en 1738, il créa un laboratoire de physique
expérimentale. Ses travaux, fort nombreux, se rapportent à des domaines variés comme la météorologie, la navigation,
l’hydraulique, l’archéologie et l’architecture. En particulier, il s’est beaucoup intéressé à la fabrication d’instruments
scientiques (baromètres, thermomètres, machines arithmétiques, etc.).
25 Giambatista Suardi (1711–1767), ou Giovanni Battista Soardi, étudia les mathématiques sous la direction de Poleni à
l’université de Padoue. Il s’illustra en tant qu’inventeur d’instruments de mathématiques.
477
Fig. 18. Un appareil tractionnel de John Perks [Pedersen, 1963, 6].
logarithmique, en traduisant mécaniquement la propriété de la tangente constante et celle de la soustangente constante. Pour la première fois, il s’agit d’instruments véritablement professionnels faisant
appel à de la mécanique de précision et où les ls, peu ables car difciles à garder tendus, sont
remplacés par des tiges rigides dont la longueur peut varier grâce à des glissières. Ce qui distingue
plus particulièrement les recherches de Poleni, c’est que, non content d’obtenir mécaniquement ces
courbes transcendantes usuelles, il engage une réexion sur la construction d’équations différentielles
plus générales. C’est ainsi qu’il modie son traceur de logarithmes pour intégrer l’équation
a dx + b dy = aa dy : y
ou qu’il construit une équation dont la solution est un cercle, ce qui lui permet d’afrmer que les
instruments tractionnels sont tout aussi légitimes en géométrie que le compas puisque les deux types
d’instruments permettent de tracer la même courbe ! D’autre part, il relance l’idée de Leibniz qu’en
guidant convenablement le l, on devrait pouvoir intégrer toute équation différentielle.
Parallèlement à ces tentatives quelque peu disparates, une théorie générale de la construction des
équations différentielles par l’emploi du mouvement tractionnel est commencée par Euler, en 1736, dans
le but d’intégrer par ce procédé la fameuse équation du comte Jacopo Riccati, équation qui résistait
jusque-là à toutes les tentatives [Euler, 1741]. La théorie d’Euler, utilisée ponctuellement par AlexisClaude Clairaut pour résoudre un problème de dynamique [Clairaut, 1745, 9] a été développée ensuite
par Vincenzo Riccati, le ls de Jacopo. Dans un mémoire de 1752, Vincenzo Riccati fait à peu près le
tour de la question et montre que, sous des hypothèses très générales, toute équation différentielle peut
être intégrée de manière exacte grâce à un mouvement tractionnel déterminé, comme chez Leibniz, par
deux courbes convenablement choisies [Riccati, 1752 ; Riccati et Saladini, 1767, chaps. 14–16].
3.2. Les intégraphes composés de la n du dix-neuvième siècle
La construction mécanique des équations différentielles passe ensuite de mode pendant plus d’un
siècle. En dehors de quelques tentatives peu convaincantes de Gustave-Gaspard Coriolis en 1836
[Coriolis, 1836], il faut attendre les années 1870 pour voir réapparaître les idées fécondes des géomètres
du dix-huitième siècle. Tout se passe comme s’il s’agissait d’une redécouverte. Dans un des premiers
478
Fig. 19. L’intégraphe d’Abdank-Abakanowicz
[Willers, 1928 (Trad. angl. 1948), 153].
Fig. 20. L’intégraphe de Pascal [Willers, 1911, 37].
intégraphes, celui de Bruno Abdank-Abakanowicz,26 on retrouve un mouvement tractionnel [AbdankAbakanowicz, 1886]. L’appareil est monté sur un cadre rigide qui se déplace sur des roulettes le long de
l’axe des abscisses (cf. Fig. 19). L’extrémité d’une tige est tirée le long d’une courbe donnée d’équation
y = f (x), de sorte que la pente de la tige soit précisément f (x). La roulette coupante située à l’autre
extrémité de la tige enveloppe ainsi une courbe dont la pente est constamment égale à f (x), ce qui permet
d’obtenir la primitive cherchée.
D’autres instruments tractionnels fabriqués peu après, beaucoup plus précis que ceux du dixhuitième siècle en raison des progrès de la mécanique industrielle, permettent d’intégrer certaines
classes d’équations différentielles. Dans le vocabulaire de l’époque, on les appelle des « intégraphes
composés », par opposition aux « intégraphes simples », comme celui d’Abdank-Abakanowicz, qui sont
seulement destinés aux quadratures, c’est-à-dire aux équations différentielles sans variables mêlées du
type y = f (x).
26 Le Polonais Bruno Abdank-Abakanowicz (1852–1900) s’est installé à partir de 1881 à Paris, au sein de la communauté
scientique polonaise en exil. C’était un élève de Wawrzyniec Zmurko, professeur aux Écoles polytechniques de Vienne et de
Lvov, qui s’était lui aussi intéressé à l’intégration mécanique.
479
Dans l’intégraphe d’Ernesto Pascal27 (cf. Fig. 20), inspiré de celui d’Abdank-Abakanowicz, un
pointeur C permet de suivre une courbe donnée, d’équation y = Q(x), tandis qu’un marqueur C trace
la courbe enveloppée par la roulette coupante D. La longueur GD est contrainte à garder une projection
constante sur l’axe des abscisses, égale à a (a = 1 sur la Fig. 20). La roulette D va donc envelopper une
courbe d’équation
Q(x) − y
,
a
ou encore ay + y = Q(x). L’intégraphe de Pascal permet ainsi d’intégrer toute équation du premier
ordre à coefcients constants [Pascal, 1910]. Pascal donne de nombreux exemples d’application de
son instrument à divers problèmes se ramenant à une ou plusieurs équations de ce type [Pascal,
1911]. Aussitôt, en s’inspirant à la fois de l’intégraphe de Pascal et de la construction de Czuber
(cf. Section 1.5), Willers conçoit un instrument voisin pour l’intégration des équations linéaires du
premier ordre à coefcients variables [Willers, 1911]. Pascal se lance alors dans l’étude systématique
de diverses variantes de son intégraphe et en tire de multiples applications à certaines classes d’équations
différentielles, aux équations intégrales du type de Volterra, aux intégrales elliptiques, à la balistique, etc.
En quelques années, il publie sur ces sujets une dizaine d’articles dont le contenu est repris, en 1914, dans
un mémoire de synthèse [Pascal, 1914]. On peut signaler également que Mario Merola28 a eu recours à
l’appareil de Pascal pour d’autres applications d’envergure en balistique [Merola, 1920].
Par ailleurs, dès 1899, l’ingénieur serbe Michel Petrovitch29 a imaginé d’utiliser un tractoriographe
pour l’intégration de certaines équations différentielles [Petrovitch, 1899]. Un tractoriographe est un
appareil qui permet de tracer la tractoire d’une courbe donnée, c’est-à-dire la courbe décrite par
l’extrémité libre d’un l posé sur un plan horizontal lorsque l’on tire l’autre extrémité le long de la
courbe donnée (l’ancienne tractrice de Perrault et Leibniz n’était autre que la tractoire d’une droite).
Si la démarche de Petrovitch est restée spéculative, l’ingénieur de la marine Louis-Frédéric Jacob30 a
développé concrètement cette idée au début du vingtième siècle [Jacob, 1907, 1908, 1909]. Pour les
besoins des problèmes de balistique qu’il rencontrait dans l’artillerie de marine, Jacob a construit deux
y =
27 Ernesto Pascal (1865–1940) t ses études à Naples, Pise et Göttingen, où il gagna l’estime de Felix Klein. Grâce à l’appui
de ce dernier, Pascal obtint en 1890, à 25 ans seulement, la chaire de calcul innitésimal de l’université de Pavie. En 1907, il
changea pour l’université de Naples, où il resta jusqu’à sa mort. Il fut membre de l’Accademia Nazionale dei Lincei. Parmi son
imposante production mathématique (près de 250 publications), on peut noter de nombreux travaux concernant les intégrales
d’équations différentielles particulières.
28 Mario Merola, qui a fait l’essentiel de sa carrière comme professeur de mathématiques et physique dans les écoles
secondaires, a été pendant une courte période, au milieu des années 1920, assistant à l’observatoire astronomique de
Capodimonte, à Naples. En dehors de quelques travaux d’astronomie, il s’est surtout intéressé à l’enseignement : il a publié
un manuel de trigonométrie plane pour les lycées et divers articles de réexion sur les programmes de sciences de l’école
élémentaire.
29 Michel Petrovitch (1868–1943), professeur à l’université de Belgrade (Serbie), a publié de nombreux travaux sur les
équations différentielles, la théorie des fonctions, le calcul et l’algèbre (ses œuvres complètes représentent quinze volumes).
Outre les instruments mécaniques dont il est question ici, Petrovitch a aussi construit des appareils hydrauliques et chimiques
pour intégrer graphiquement certains types d’équations différentielles [Petrovitch, 1897, 1898, 1900 ; Price, 1900]. Ces
appareils, à la fois ingénieux et marginaux, sont inclassables dans le cadre de mon étude.
30 Louis-Frédéric Jacob, né en 1857, fut ingénieur de l’artillerie navale, colonel d’artillerie coloniale et directeur du
Laboratoire central de la Marine à Paris. Il a publié plusieurs ouvrages sur la balistique, l’artillerie, les mécanismes et le
calcul mécanique.
480
Fig. 21. L’intégraphe de Jacob pour l’équation de Riccati [Jacob, 1909, 22].
Fig. 22. L’intégraphe de Jacob pour l’équation d’Abel [Jacob, 1909, 97].
appareils, l’un pour intégrer l’équation de Riccati y = Ay 2 + By + C (cf. Fig. 21), l’autre pour l’équation
d’Abel yy = Ay 2 + By + C (cf. Fig. 22).
Dans les deux cas, un transporteur formé de deux parallélogrammes articulés permet de mouvoir
l’appareil parallèlement à lui-même sur la table à dessin. Cet appareil est en fait un tractoriographe, avec
une pointe que l’on déplace le long d’une courbe donnée. Dans le cas de l’équation de Riccati, la longueur
de la tige est constante ; dans le cas de l’équation d’Abel, cette longueur est variable et déterminée par
un guide ayant la forme d’une seconde courbe donnée. Il est frappant de retrouver ici, une nouvelle
fois, le principe inépuisable de Leibniz, à savoir deux courbes choisies convenablement pour contrôler
à tout instant la pente de la courbe que l’on veut tracer et résoudre ainsi le difcile problème inverse
des tangentes. L’ingénieur espagnol Leonardo Torres Quevedo31 a également écrit sur ce principe pour
31 Leonardo Torres Quevedo (1852–1936) est considéré comme un précurseur de la cybernétique, du calcul analogique et de
l’informatique. Il débuta en 1876 une carrière d’ingénieur des ponts et chaussées, mais ce n’est qu’en 1893, à l’âge de 41 ans,
qu’il publia son premier travail scientique. Il déploya ensuite, pendant une trentaine d’années, une activité frénétique qui le
conduisit à s’occuper de dirigeables, de téléphériques, de télégraphie sans l ou de machines analogiques de calcul. En 1900,
il présenta à l’Académie des sciences de Paris un « Mémoire sur les machines à calculer ». Il construisit lui-même une série
de machines de type mécanique pour résoudre, par exemple, une équation du huitième degré à la précision du millième, ou
une équation du second degré à coefcients complexes. Il conçut également un automate joueur d’échecs et se lança dans la
théorie de l’automatique. Tout ceci lui valut un énorme prestige au niveau international, surtout en Espagne et en France. En
481
concevoir des machines permettant de résoudre toute équation différentielle, mais ces machines, bien
plus ambitieuses que celles de Jacob, sont restées purement théoriques [Torres, 1901].
4. Les quadratures
4.1. L’intégration par quadratures en tant que méthode graphique
À l’origine, la méthode qui consistait à ramener une équation différentielle aux quadratures était aussi
une technique graphique permettant de construire concrètement les courbes intégrales. C’est ainsi qu’elle
est présentée par Jean Bernoulli (cf. Section 1.3) ou dans le Traité de Lacroix :
Dans les premiers tems on chercha à déterminer par les aires ou même par les arcs de quelques courbes connues, l’ordonnée de la
courbe demandée ; depuis on a laissé ces constructions de côté, parce que, quelqu’élégantes qu’elles fussent dans la théorie, elles
étoient toujours moins commodes et sur-tout moins exactes dans la pratique, que les formules approximatives qui ont pris leur place.
[Lacroix, 1798, 196]
Dans la pratique ancienne, on effectuait la quadrature d’une courbe en décomposant approximativement l’aire sous la courbe en une réunion de petits triangles, rectangles ou trapèzes, que l’on transformait
par des opérations graphiques en un seul rectangle ou un seul carré (d’où le mot « quadrature », en un
sens classique depuis l’Antiquité). De même, la rectication d’une courbe s’opérait en approchant la
courbe par une ligne polygonale formée de petits segments que l’on mettait bout à bout pour former un
segment unique à mesurer. Ce sont ces opérations qui, dans le cas des courbes dénies par des équations différentielles, sont jugées impraticables par Bernoulli, insufsamment commodes et exactes par
Lacroix.
De fait, la résolution par quadratures des équations différentielles n’est pas si facile à réaliser
graphiquement. On s’en rend compte en lisant la description du procédé, pour le cas d’une équation
à variables séparables, dans les ouvrages de référence que sont l’Encyclopédie de Diderot et d’Alembert
[1754, 389], le Traité de Lacroix [1798, 297] ou l’Histoire des mathématiques de Jean-Étienne Montucla
[1802, 174–175]. Une fois qu’on a réussi à mettre l’équation sous la forme
Y (y) dy = X(x) dx,
il reste à construire successivement cinq courbes par points (cf. Fig. 23) :
x
f :
y = X(x),
h :
y=
X(t) dt,
0
y
0
m :
x = Y (y),
n :
x=
Y (t) dt,
0
x
Y (t) dt =
s :
y
X(t) dt.
0
1901, il devint directeur du nouveau Laboratoire de mécanique appliquée de Madrid et, en 1927, il fut élu membre associé de
l’Académie des sciences de Paris.
482
Fig. 23. Intégration graphique par quadratures d’une équation
différentielle [Montucla, 1802, planche 5].
Fig. 24. Schéma d’un planimètre d’Hermann [Maurer,
1998, 220].
Si l’on souhaite construire la courbe intégrale avec précision, la lourdeur d’une telle procédure est
évidente. Voilà pourquoi, vers la n du dix-huitième siècle, ramener les équations différentielles à des
quadratures semblait une impasse du point de vue pratique, faute de disposer de moyens rapides et
performants de quadrature graphique.
4.2. Progrès du dix-neuvième siècle pour les quadratures graphiques
La technique précédente, fastidieuse à réaliser par les seuls procédés du calcul par le trait, devient
plus praticable si l’on dispose d’instruments spéciaux pour effectuer les quadratures. Il peut s’agir des
intégraphes du type d’Abdank-Abakanowicz (cf. Section 3.2), fondés sur le mouvement tractionnel, ou
d’autres appareils mécaniques exploitant un principe différent : celui de la roulette intégrante.
Le premier à avoir construit un tel appareil semble être l’ingénieur bavarois Johann M. Hermann32 en
1814. Le mécanisme (cf. Fig. 24) est déplacé parallèlement à l’axe des abscisses, avec un pointeur H qui
suit une courbe y = f (x). Une roulette liée à H tourne par friction sur un cône C, de sorte que son angle
de rotation soit proportionnel à l’ordonnée du point H. Ainsi, pour un déplacement élémentaire dx le long
de l’axe des abscisses, la roulette tourne (à un coefcient près dépendant des dimensions de l’appareil)
d’un angle f (x) dx et, pour un déplacement le long du segment [x0 , x], elle tourne d’un angle total
x
F (x) =
f (t) dt.
x0
On peut par là mesurer l’aire sous une courbe et, plus généralement, en enregistrant les valeurs de l’angle
pour diverses valeurs de x, tracer par points une primitive.
32 Johann Martin Hermann (1785–1841) était un arpenteur bavarois qui travaillait pour le service du cadastre. Son invention
n’attira guère l’attention de ses supérieurs et ne fut pas publiée.
483
Grâce à la roulette intégrante, on matérialise la vision leibnizienne de l’intégrale en tant que somme
innie de rectangles innitésimaux. On s’appuie donc sur le second aspect de l’intégrale : avec le
mouvement tractionnel, on résolvait le problème inverse des tangentes ; avec la roulette intégrante, on
résout celui de la quadrature des courbes.
Après Hermann, d’autres ingénieurs imaginèrent de faire tourner la roulette intégrante sur un disque,
puis directement sur le plan de la feuille de papier pour aboutir notamment, en 1854, au fameux
planimètre polaire de Jakob Amsler33 où la roulette se déplace par une combinaison de mouvements
de rotation et de glissement. Ce planimètre, simple et relativement peu coûteux connut un grand succès
commercial : on estime que, sous ses différentes variantes, il a été fabriqué à plus de 500 000 exemplaires.
Mais ce n’est pas ici le lieu de refaire une histoire détaillée des instruments mécaniques d’intégration.
Je renvoie pour cela aux études récentes et très complètes de Joachim Fischer [1995, 2002], ainsi qu’à
quelques livres classiques sur les instruments de mathématiques [Dyck, 1892 ; Ocagne, 1893 ; Jacob,
1911 ; Galle, 1912 ; Morin, 1913 ; Horsburgh, 1914 ; Willers, 1926, 1951 ; Meyer zur Capellen, 1941].
Parallèlement au développement des instruments mécaniques d’intégration, le calcul des quadratures
par le trait fait également de grands progrès. Par la traduction sous forme graphique des formules de
quadratures numériques, on dépasse les méthodes anciennes qui consistaient à décomposer grossièrement
l’aire sous une courbe en petits rectangles ou petits trapèzes. Dès 1835, on rencontre chez Poncelet
une version graphique de la méthode de Simpson [Poncelet, 1835, 162], mais c’est surtout Massau (cf.
Section 1.4) qui donne une impulsion décisive à ce processus. Massau montre comment remplacer, sur
chaque subdivision de l’intervalle d’intégration, la courbe à intégrer par une courbe polynomiale de
degré quelconque, et comment construire exactement l’aire sous cette courbe polynomiale, de manière à
atteindre une grande précision. Par ailleurs, en s’attachant à la construction d’une primitive en tant que
courbe, plutôt qu’à la simple évaluation graphique d’une intégrale dénie en tant qu’aire, il ouvre la voie
à la réalisation d’intégrations graphiques itérées. Enn, il fait le lien avec certains procédés d’intégration
développés par les ingénieurs pour les besoins de la statique graphique, notamment l’usage du polygone
funiculaire [Massau, 1878–1887 ; Saviotti, 1883 ; Favaro, 1885 ; Ocagne, 1908 ; Runge, 1912 ; Runge et
Willers, 1915 ; Willers, 1920 ; Maurer, 1998].
4.3. Approximations successives et quadratures répétées
Grâce aux instruments mécaniques d’intégration, grâce aux nouvelles techniques d’intégration par
le trait popularisées par Massau, l’intégration par quadratures des équations différentielles devenait,
sous forme graphique, beaucoup plus facile à réaliser qu’au dix-huitième siècle. Tout cela reste
néanmoins d’un intérêt pratique limité, car on sait bien que la plupart des équations différentielles ne
33 Né en Suisse, Jakob Amsler (1823–1912) étudia d’abord la théologie en Allemagne avant de s’orienter vers les
mathématiques et la physique. De retour en Suisse, il obtint son Habilitation à Zurich et devint professeur au Gymnasium
de Schaffhausen. Il s’intéressa pendant quelques années à la physique mathématique (magnétisme, conduction de la chaleur,
attraction des ellipsoïdes), puis se consacra, à partir de 1854, à la construction d’instruments mathématiques de précision.
Il inventa le planimètre polaire, premier instrument mécanique d’intégration utilisant les coordonnées polaires. Cet appareil,
bien adapté à la détermination des moments statiques, des moments d’inertie et des coefcients de Fourier, fut largement
employé par les ingénieurs de la construction navale et des chemins de fer. À Schauffhausen, Amsler monta une fabrique
spécialement destinée à la production de son planimètre, d’où sortirent environ 50 000 appareils. Pour l’originalité et la qualité
de ses instruments, Amsler obtint des prix aux expositions universelles de Vienne (1873) et de Paris (1881, 1889), et fut élu en
1892 à l’Académie des sciences de Paris.
484
se laissent pas ramener à un nombre ni de quadratures. En fait, c’est la méthode des approximations
successives, étudiée d’un point de vue théorique par Cauchy et Liouville, dans les années 1830, pour les
équations différentielles linéaires, puis par Picard, dans les années 1890, pour les équations différentielles
quelconques, qui va donner un nouvel élan à cette direction de recherche.
Le problème de Cauchy
y = f (x, y),
y(x0 ) = y0
peut se mettre sous la forme intégrale équivalente
x
y = y0 +
f t, y(t) dt.
x0
Dans la méthode des approximations successives, on part d’une première solution approchée y1 (x),
obtenue par n’importe quel procédé, et on calcule les suivantes par les relations
x
yn+1 (x) = y0 +
f t, yn (t) dt.
x0
Du point de vue graphique, la construction de l’équation reviendrait, en théorie, à effectuer une
innité de quadratures. En pratique, après un nombre réduit d’étapes, on obtiendra, en général, une
bonne approximation de la courbe intégrale cherchée (vu l’épaisseur des traits de crayon, deux courbes
successives seront rapidement indiscernables). À partir du moment où l’on sait réaliser facilement et avec
précision les quadratures graphiques, il semble donc possible d’intégrer graphiquement toute équation
différentielle.
4.4. Mise en œuvre de l’idée des quadratures répétées
C’est William Thomson qui a eu, le premier, l’idée d’exploiter la méthode des approximations
successives autrement que sous forme numérique. L’occasion lui en a été fournie par l’invention, par
son frère James,34 d’un nouveau type d’intégraphe formé d’un disque, d’une sphère et d’un cylindre
[J. Thomson, 1876]. William Thomson se sert de deux de ces appareils pour intégrer pratiquement les
équations linéaires du second ordre à coefcients variables, omniprésentes en physique [W. Thomson,
1876a]. Pour cela, dans la lignée des travaux de Sturm et Liouville, il part du fait qu’une telle équation
peut être réduite à la forme
d 1 du
= u,
dx P dx
34 James Thomson (1822–1892) est beaucoup moins connu que son frère William (Lord Kelvin). À partir de 1840, James
poursuivit une carrière d’ingénieur à Dublin, Glasgow et Belfast. En 1857, il devint professeur de génie civil au Queen’s College
de Belfast, puis, en 1873, à l’université de Glasgow. En 1877, il fut élu Fellow of the Royal Society. Il étudia notamment l’effet
de la pression sur l’abaissement du point de congélation de l’eau et t diverses recherches en hydraulique.
485
où P est une fonction donnée de x, et qu’une solution qui s’annule en x = 0 est la limite de la suite
x
un+1 =
x
P C+
0
un dx dx,
0
avec une constante arbitraire C et une fonction u1 quelconque, par exemple u1 = x. Les deux intégraphes,
utilisés conjointement, permettent de réaliser d’un seul coup la quadrature double nécessitée par
l’itération : on suit avec le pointeur du premier intégraphe la courbe représentative de la fonction un et le
second intégraphe fournit, sur son cylindre d’enregistrement, le tracé de la fonction un+1 ; on recommence
jusqu’à ce qu’il n’y ait plus de différence sensible entre deux courbes successives.
Thomson eut même une illumination : si l’on couplait mécaniquement les deux intégraphes de sorte
que la fonction sortant du second coïncide automatiquement avec celle qui entre dans le premier, on
obtiendrait directement la solution exacte de l’équation ! Dans un autre article encore plus visionnaire
[W. Thomson, 1876b], Thomson donne le principe théorique d’appareils pouvant intégrer une équation
linéaire à coefcients variables de n’importe quel ordre, et même une équation différentielle quelconque,
avec l’idée qu’on pourrait s’en servir pour le problème des trois corps. Ces appareils sont tous formés
d’un grand nombre d’intégraphes simples connectés entre eux par des dispositifs mécaniques complexes.
Malheureusement, les moyens technologiques de son époque étaient insufsants pour permettre à
Thomson de réaliser valablement de tels instruments.
Cependant, l’idée initiale de Thomson, à savoir que l’on peut réaliser les quadratures successives
l’une après l’autre sur une feuille de papier, fait son chemin. Peu après la publication des travaux
de Picard, qui prouvent la convergence de la méthode itérative pour une équation différentielle très
générale y = f (x, y), Runge propose une traduction graphique du procédé en synthétisant au mieux
les techniques disponibles [Runge, 1907]. La première intégrale approchée y1 est construite à partir des
isoclines, avec la méthode du point milieu (cf. Fig. 25). La deuxième intégrale vient de l’intégration
graphique de la fonction f (x, y1 (x)), réalisée soit avec un intégraphe, soit par le trait, comme sur la
Fig. 25, avec les méthodes de Massau. On recommence ensuite autant de fois que nécessaire. Tout cela
se généralise aux systèmes d’équations du premier ordre ou aux équations d’ordre supérieur [Runge,
1912, 138–141], en construisant simultanément plusieurs courbes (cf. Fig. 26).
Après Runge, de nombreux auteurs, que nous avons déjà rencontrés plus haut et qui ont imaginé
une méthode originale pour le tracé d’une première courbe intégrale approchée, ne manquent pas
de souligner que la méthode des approximations successives peut permettre d’améliorer ensuite cette
construction [Rothe, 1916, 94 ; Doetsch, 1921, 465 ; Neuendorff, 1922, 135 ; 1933, 450]. D’autre part,
Eugène A. Kholodovsky,35 en adaptant les idées de Massau et de Runge, explique en détail comment
réaliser les quadratures graphiques en coordonnées polaires [Kholodovsky, 1929] et comment intégrer
graphiquement, par des quadratures répétées, les équations du premier ordre, les systèmes de deux
équations du premier ordre et les équations du second ordre lorsque les variables sont considérées comme
des coordonnées polaires [Kholodovsky, 1930].
35 Eugène A. Kholodovsky, né en 1876, a été professeur de mathématiques à l’Institut polytechnique de Petrograd. Émigré aux
États-Unis après la Révolution russe, on le retrouve au début des années 1920 comme assistant à l’observatoire astronomique
Lick de l’université de Californie, à Santa Cruz.
486
Fig. 25. Intégration par approximations successives d’une équation du premier ordre [Runge, 1912, 121].
Fig. 26. Cas d’un système [Runge, 1912, 139].
Le procédé itératif a été utilisé de façon systématique pour des travaux d’électricité industrielle dans
la première moitié du vingtième siècle. Par exemple, Thornton C. Fry36 s’en est servi pour intégrer
36 Thornton Carl Fry (1892–1991) reçut son Ph.D. en mathématiques, physique et astronomie en 1920 à l’université du
Wisconsin. Il fut l’un des premiers mathématiciens à travailler pour l’industrie. Il occupa le poste de chef du département de
mathématiques à la Western Electric Company (1916–1924), puis aux Bell Telephone Laboratories, où il eut l’occasion de
collaborer avec Claude Shannon. Pendant la Seconde Guerre mondiale, l’équipe de Fry fut réquisitionnée au service de l’effort
de guerre et se consacra à la défense anti-aérienne. Fry a joué un rôle pionnier pour introduire le calcul des probabilités dans les
sciences de l’ingénieur. Une fois à la retraite, il est resté actif en devenant consultant pour diverses entreprises comme Boeing
ou le National Center for Atmospheric Research.
487
Fig. 27. Intégration par approximations successives d’une équation du second ordre [Fry, 1928, 408].
des équations du second ordre apparaissant dans des problèmes de circuits électriques [Fry, 1928]. Fry
ramène classiquement une équation du second ordre à un système de deux équations du premier ordre
puis, partant de solutions approchées constantes égales aux valeurs initiales, il effectue les quadratures
successives à l’aide d’un intégraphe jusqu’à ce que deux courbes consécutives soient indiscernables sur
le papier (cf. Fig. 27).
C’est pour ces mêmes problèmes d’électricité industrielle que des analyseurs différentiels ont été
construits dans les années 1930–1950, réalisant enn le programme visionnaire de Lord Kelvin. En 1931,
au Massachussets Institute of Technology, Vannevar Bush37 réussit à construire un appareil (cf. Fig. 28)
formé de six intégraphes pouvant être connectés entre eux par des liaisons mécaniques et permettant
de résoudre toute équation différentielle jusqu’à l’ordre six (car les problèmes usuels font intervenir
jusqu’à trois équations simultanées du second ordre) [Bush et al., 1927 ; Bush and Hazen, 1927 ; Bush,
1931]. Les entrées et les sorties se font de manière graphique : des pointeurs sont guidés, mécaniquement
ou manuellement suivant les cas, le long des courbes construites pour représenter les coefcients de
l’équation, tandis qu’à la sortie, les solutions sont tracées automatiquement sur une table traçante.
Un autre analyseur différentiel fut construit en 1935, à l’université de Manchester, par Douglas
R. Hartree38 [1935, 1938, 1940, 1949, chap. 2]. Pendant la Seconde Guerre mondiale, une dizaine
37 Vannevar Bush (1890–1974) fut, entre les deux guerres mondiales, professeur d’ingénierie électrique au Massachussets
Institute of Technology. C’est là qu’à partir de 1931, il développa avec ses étudiants un premier analyseur différentiel et
d’autres machines analogiques destinées à surmonter les calculs difciles qui freinaient l’essor de domaines comme l’électricité
industrielle, la géophysique, l’étude des rayons cosmiques ou la mécanique quantique. En quelques années, des machines
comparables à celles de Bush se répandirent aux États-Unis et à l’étranger. Pendant la Seconde Guerre mondiale, Bush fut
appelé à Washington pour diriger un réseau de laboratoires voués à la recherche de guerre. Après 1945, il continua à conseiller
le gouvernement en matière de science et de défense, et rejoignit les directions de Merck et de AT&T.
38 Douglas Rayner Hartree (1897–1958) a fait ses études à Cambridge. Il obtint son doctorat en 1926 et devint professeur de
mathématiques appliquées puis de physique théorique à Manchester. En 1946, il revint à Cambridge comme Plummer Professor
of Mathematical Physics. Tout au long de sa carrière, il développa des méthodes puissantes d’analyse numérique pour intégrer
488
Fig. 28. Schéma de l’analyseur différentiel de Vannevar Bush [1931, 452].
d’appareils de ce type étaient en service et certains furent mobilisés pour des calculs balistiques. À partir
de 1942, de nouvelles versions apparurent, avec des intégraphes couplés entre eux par des liaisons
électriques et non plus mécaniques [Bush and Caldwell, 1945]. Cependant, les analyseurs différentiels
disparurent rapidement face à l’essor des calculateurs électroniques, dans lesquels les entrées et les
sorties ne se font plus sous forme graphique, mais sous forme numérique par l’intermédiaire de cartes
perforées [Goldstine, 1972, 1re partie, chap. 10]. D’une certaine manière, on peut voir dans les analyseurs
différentiels des sortes de « dinosaures » marquant à la fois l’apogée et la n de l’intégration graphique
des équations différentielles.
5. Conclusion
L’intégration graphique des équations différentielles est à situer dans une longue tradition géométrique
qui remonte à l’antiquité grecque. Dans cette tradition, de nombreux problèmes, quelle que soit leur
origine, étaient abordés avec une pensée et un langage géométriques : résoudre une équation, c’était
construire sa solution par le tracé et l’intersection de courbes. Sur le plan théorique, l’approche
géométrique s’efface vers le milieu du dix-huitième siècle au prot d’une approche algébrique. Après
Euler, on ne construit plus des courbes, on calcule des fonctions. On représente désormais les solutions
des équations par des algorithmes algébriques, éventuellement innis. Pourtant, la pensée géométrique
les équations différentielles, notamment en lien avec la défense anti-aérienne, la physique atomique, l’hydrodynamique et le
contrôle automatisé des usines chimiques. C’est après avoir étudié l’analyseur différentiel de Vannevar Bush lors d’une visite
aux États-Unis qu’Hartree projeta d’en construire un à son tour. Par ailleurs, pendant la Seconde Guerre mondiale, Hartree fut
impliqué dans le projet de l’ENIAC, l’un des premiers calculateurs numériques électroniques.
489
ne disparaît pas complètement après 1750. Elle continue notamment à vivre et à se développer entre
les mains des utilisateurs des mathématiques : physiciens, ingénieurs civils et militaires. C’est ainsi
qu’au sein de la science du calcul, les méthodes graphiques ont conservé une place importante pendant
deux siècles, jusqu’à l’apparition des calculateurs électroniques. Les recherches nombreuses et variées
évoquées dans cet article sont là pour en témoigner.
L’étude de l’intégration graphique est, tout d’abord, un moyen de mieux comprendre ce processus
historique général qui fait passer, dans une grande partie des mathématiques, d’une pensée géométrique
ancienne à la pensée algébrique moderne, jusqu’à la situation actuelle où l’on parle de « tout numérique ».
Plus spéciquement, c’est un moyen d’aborder autrement l’histoire de l’analyse numérique des équations
différentielles, en redonnant aux méthodes graphiques la juste place qu’elles occupèrent aux côtés des
algorithmes purement numériques.
Remerciements
Je remercie chaleureusement Christian Gilain et Ivor Grattan-Guinness, qui m’ont encouragé à écrire
cet article et qui ont accepté d’en relire attentivement la première version.
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