Revue d'histoire des sciences
Cauchy elliptic and Abelian integrals
M Jeremy Gray
Abstract
SUMMARY. — This paper considers how the subjects of elliptic function theory and complex variable theory were increasingly
drawn together in the 1830s and 1840s. As the recognition of the importance of Abel and Jacobi's creation grew,
mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued
integrals. One alternative would have been to adopt and develop Cauchy 's theory of complex integrals; by and large this was
not done, for reasons which are discussed here. The paper concludes with the first tentative investigations of the integration of
algebraic functions.
Résumé
RÉSUMÉ. — Cet article étudie comment la théorie des fonctions elliptiques et celle de la variable complexe se rapprochèrent de
plus en plus dans les années 1830 et 1840. Alors que la conscience de l'importance de la création d'Abel et de Jacobi
grandissait, on en vint à penser qu'il n'était pas satisfaisant de fonder la théorie des fonctions elliptiques sur l'inversion
d'intégrales multiformes. Une alternative aurait été d'adopter et de développer la théorie des intégrales complexes de Cauchy ;
généralement il n'en fut pas ainsi, pour des raisons qui sont discutées ici. Pour finir, l'article analyse les premières recherches
concernant l'intégration des fonctions algébriques.
Citer ce document / Cite this document :
Gray Jeremy. Cauchy elliptic and Abelian integrals. In: Revue d'histoire des sciences, tome 45, n°1, 1992. pp. 69-82;
doi : https://doi.org/10.3406/rhs.1992.4232
https://www.persee.fr/doc/rhs_0151-4105_1992_num_45_1_4232
Fichier pdf généré le 08/04/2018
Cauchy
elliptic
and
Abelian
integrals
RÉSUMÉ. — Cet article étudie comment la théorie des fonctions elliptiques
et celle de la variable complexe se rapprochèrent de plus en plus dans les années
1830 et 1840. Alors que la conscience de l'importance de la création d'Abel et
de Jacobi grandissait, on en vint à penser qu'il n'était pas satisfaisant de fonder
la théorie des fonctions elliptiques sur l'inversion d'intégrales multiformes. Une
alternative aurait été d'adopter et de développer la théorie des intégrales
complexes de Cauchy; généralement il n'en fut pas ainsi, pour des raisons qui sont
discutées ici. Pour finir, l'article analyse les premières recherches concernant
l'intégration des fonctions algébriques.
SUMMARY. — This paper considers how the subjects of elliptic function
theory and complex variable theory were increasingly drawn together in the 1830s
and 1840s. As the recognition of the importance of Abel and Jacobi 's creation
grew, mathematicians came to feel that it was unsatisfactory to base the theory
of elliptic functions on the inversion of many-valued integrals. One alternative
would have been to adopt and develop Cauchy 's theory of complex integrals;
by and large this was not done, for reasons which are discussed here. The paper
concludes with the first tentative investigations of the integration of algebraic
functions.
The mathematical problem with which researchers were
confronted can be simply stated. Cauchy' s whole system of
definitions, based on his newly-defined concepts of limit, continuity,
differentiability, and integrability, was incompatible with talk of
many-valued functions. A many-valued function (to use the 19th
century term) is nowhere continuous in Cauchy's sense. But even
a naive treatment of elliptic integrals was felt by many to be fraught
with ambiguity because of the square root in the integrand. While
a doubly periodic function is a meromorphic function defined on
the whole of the complex plane, an elliptic integral only makes
sense on something like a Riemann surface (which is a torus in
this case). Thus the many-valued nature of an elliptic integral posed
a challenge to mathematicians throughout the 1830s and 1840s.
Rev. Hist. Sci., 1992, XLV/1
70
Jeremy Gray
In the winter of 1835/1836 Jacobi lectured on elliptic
functions. As is well-known, the central novelty of these lectures was
a new approach based on what are called ^-functions. Jacobi
commented:
"The definition of these transcendents by an infinite series has the
principle advantage that their values are represented in a definite way,
while there is no precise way in which they are represented by integrals
as soon as the limits of integration become imaginary, because we are
less accustomed to them in this form. Their theory has been described
by Cauchy, but it lacks the degree of clarity that an infinite series has.
Specifically, there is the obscurity that the square root under the integral
sign brings, particularly when the variable takes imaginary values..."
(Quoted in [Koenigsberger, 1904, 186-187].)
Jacobi's lectures of 1835 tell us a lot about his opinion of the
theory of functions of a complex variable. Koenigsberger, who
had access to Rosenhain's notes of Jacobi's lectures, wrote that
Jacobi ran into great difficulties
"with the idea of the infinitely many values of elliptic integrals which
he sought to explore using Cauchy's intuitive approach, and he came
in this way to a theorem he had proved earlier, namely that a function
of one variable cannot have more than two periods." [1904, 189.]
This was a result that Jacobi had published in his [1834], where
he had shown that
"if a function [of a single variable] has three periods then either
these periods are all obtainable from a pair of periods, or the function
has arbitrarily small periods. This being absurd, a function of three periods
does not exist." [1834, 32.]
The problem then is what to make of even the simplest hyperelliptic integrals:
o
[
dx
- k2x2)(\ - m2x2)
By analogy with the elliptic case, this must surely define by
inversion a function with four distinct periods, yet according to Jacobi's
theorem such a thing is absurd. Jacobi's way forward, the first
appearance of what has come to be called Jacobi inversion, was
the crucial move in the study of hyperelliptic and Abelian
integrals, opening up the way for the later insights of Riemann and
Elliptic and Abelian integrals
71
Weier strass. As such, it would be hard to underestimate its
importance.
Even so, Jacobi' s position tells us much about how complex
functions were regarded, and indeed were to be regarded for another
fifty years. For, according to any modern reading of Cauchy's
theory, (*) defines an analytic function of its upper endpoint, and
so there is an analytic inverse. Such may have been the opinion
of Gôpel, who asked in 1847, shortly before his untimely death,
why Jacobi believed that this function is absurd. Does the
absurdity reside in the very basis of the theory, or in the idea of integral
or in the idea of function? He did not answer his own question,
but it provoked Jacobi to this reply (unpublished in his lifetime).
The integral (*) as a function of its upper endpoint z can take
values arbitrarily close to any given one, merely by varying the
path of integration you choose [Werke, II, 516].
Given the difficulties Jacobi was experiencing understanding
the complex integrals underlying the theory of elliptic functions,
it is interesting to see what Cauchy himself had to say about the
foundations of elliptic function theory. This is a topic usually
omitted from even the best histories of the subject, although it
has been discussed, briefly, by Houzel [1978]. However, Cauchy
wrote quite extensively on just this aspect of their theory. His ideas
are set out in a series of papers written in the concluding months
of 1843 when, for some reason, his attention was drawn to the
subject of functions expressed by means of infinite products.
In the paper of 2 October 1843, Cauchy proposed to consider
only what he called geometrical factorials, expressions of the form
w(x, t) = (1 + x)(l + tx)(l + t2x)(l + t3x)...
He remarked:
"As each of the so-called elliptic functions reduces to the ratio of
two factorials, one should not be astonished to find that the formulae
deduced from the consideration of geometrical factorials provide, as
particular cases, the developments of elliptic functions in series." [1843a, 66.]
The details need not detain us, except to note that the calculus
of residues provided the bridge between the infinite products or
factorials and the infinite series. On 16 October, he obtained the
three elliptic functions "most commonly used" within the frame-
72
Jeremy Gray
work of his theory [1843£, 96]. In the next memoir, of 23 October,
he worked his way back to the elliptic integrals, and commented that
"M. Jacobi has reduced the evaluation of elliptic functions to the
determination of the relationships which exist between the functions we
call reciprocal factorials... One can with some advantage pursue the
opposite route and, having first established directly the remarkable
relationships which these reciprocal factorials enjoy and the formulae which
express these properties, one can deduce from these formulae those which
serve to establish the relationships of relevance to elliptic functions."
[1843c, 97.]
So news of Jacobi's lectures had not reached Paris. Cauchy was
particularly pleased with the fact that from his starting point it
was easy to study how the formulae about elliptic functions must
be modified when the variable x becomes imaginary. But he had
scarcely begun to consider the details when he was blown off that
course by the need to respond to Laurent's publication of what
we today call Laurent's theorem, and by the opportunity to
campaign for his re-election to the Bureau des longitudes. Although
he was unsuccessful in this he did not return to the topic of elliptic
functions until 1846.
These 70-odd pages written by Cauchy in barely two months
do not seem to have greatly changed the subject of elliptic
function theory. This may be because they do not, of themselves, bear
on any unsolved problems in the subject. But they do present a
reasonable foundation for it in the theory of particular kinds of
infinite products which, via the residue calculus, leads to all the
principal results. They show that Cauchy, at least, was aware that
Jacobi's method of simply replacing real values by complex ones
was not automatically valid. They also show that Cauchy was still
unable to include in his ideas about complex integration a theory
of the integrals of algebraic functions.
Further light is shed on contemporary attitudes to complex
integrals by some early papers of the English mathematician,
Arthur Cayley. In 1845 Cayley wrote a paper on elliptic functions
[1845a], designed to clear up the indeterminateness in the value
of the complete elliptic integrals
dx
cV)(l + e2x2)
Elliptic and Abelian integrals
73
v
2
Cayley saw that the indeterminateness depended on the path of
integration:
"Where the limits are real, it is tacitly supposed that the variable
passes through a succession of real values, and thus œ, v may be
considered as completely determined by these equations, but only in
consequence of this tacit supposition. If с and e are imaginary, there is
absolutely no system of values to be selected for co, v in preference to
any other system. The only remaining difficulty is to show from the
integral itself, independently of the theory of elliptic functions, that such
integrals contain an indeterminateness of two arbitrary integers ; and this
difficulty is equally great in the simplest cases. Why, a priori, do the
functions
contain a single indeterminate integer?" [CMP, I, 135.]
By raising this question but not answering it, Cayley showed very
clearly that he understood that a complex integral was a path
integral but that he did not have a theoretical understanding of the
implications of that view.
Cayley reformulated the theory of elliptic functions in two long
papers of 1845, written in one form for the Cambridge
Mathematical Journal [1845&] and in an only slightly more general form
for Liouville's Journal. In his view the best way to start was with
formulae Abel had put at the end of his great memoir, which
gave the expansions of the elliptic functions as quotients of two
infinite double products. These he called, following Cauchy but
without acknowledging the fact, "Abel's double factorial
expressions" whence the general factor contained two independent
integers. So this double factorial form displayed the reason for the
double periodicity (1).
(1) However, he noted [CMP I, 136] nothing like them appeared in Jacobi's Fundamenta Nova and this made it "difficult to trace the connection between Jacobi's formulae;
and in particular to account for the appearance of an exponential factor that runs through
them. "
74
Jeremy Gray
Cayley's new starting point had its analogue in the theory of
the trigonometric functions, specifically the infinite product
expansion for the sine function. Accordingly Cayley proposed to study
the function defined by the double product:
и =хПП|1
\
(mco + nvi)
where со and v are real positive quantities and the product is taken
over all integers m and n (the case m - n = 0 being tacitly
excluded). After making some remarks about the convergence of
such products which are of independent interest Cayley rapidly
and simply deduced a variety of interesting results. This is not
the place to go over the technicalities, which are formidable, but
his approach was successful and he eventually found that all the
functions Abel had introduced were at his fingertips. The hardest
problem involved passing from the doubly infinite products to
doubly infinite sums. Cayley relied on a theorem he took from
Cauchy's Exercices de mathématiques, II [ = OC(2), VII, 336],
which he interpreted for his readers as follows: "The integral residue
in question is the series of fractions that would be obtained by
the ordinary process of decomposition."
So we see that Cayley did not regard the inversion of an elliptic
integral as the best starting point for a theory of elliptic functions,
because the theory of such functions was not clear to him. In
particular, the reason for the periodicity of such functions was
not clear, although he knew that elliptic integrals were infinitely
many valued since they depended on the path of integration. Instead,
not unlike Cauchy, Cayley advocated starting with doubly infinite
products, and deriving Abel's elliptic functions from four such
products, cunningly chosen. This is intriguing testimony to the way
in which different parts of Cauchy's theory of complex functions
were gaining acceptance.
A similar approach to Cayley's to the study of elliptic
functions was taken at about the same time by the young German
mathematician Gotthold Eisenstein. In a short paper [1844] Eisenstein began by observing the analogy between the infinite product
expansion of the trigonometric functions and the infinite double
product for the elliptic transcendents. He proceeded to sketch
the details, contenting himself in this short paper with showing
Elliptic and Abelian integrals
75
how to obtain the infinite product expansions given by Jacobi (2).
His reasons for taking this new approach are worth quoting
in full.
"The usual definition that one gives of the elliptic functions, quite
contrary to the analogy with the exponential function, is that they are
the inverse functions of certain well-known integrals [...]. But since already
the clear meaning of such an integral, whose differential goes suddenly
from real to imaginary, is not easy to grasp, it may rather seem almost
impossible for the learner to obtain a clear idea a priori of the inversion
of such an integral function, while the geometrical intuition which one
can at least call on for the circular functions is here broken. There is
a particular difficulty with the periods. [...] If one wants to define sine
as that function y of x which is given by the integral
then to agree with the well-known properties of sine one must consider
that the integral takes infinitely many different values for each given
value of v, which contradicts the usual meaning one gives to such an
integral (e. g. for у < 1). In the same way for the elliptic integral [...]
sin am x one must say that for each value of у it takes so to speak
infinitely many different values infinitely many times." [1844, 31.]
What this quotation tells us is that Eisenstein's difficulties with
complex integrals were even greater than Cayley's, for he made
no mention of the way a complex integral depends on the choice
of path between its endpoints.
Eisenstein returned to the foundations of elliptic functions in
the long paper [1847]. Even its lengthy title is worth quoting: "A
precise study of the infinite double products by means of which
the elliptic functions can be represented as quotients, and the double
series connected with them (as a new foundation for the theory
of elliptic functions with particular respect to their analogy with
the circular functions)". Seldom can a title have said so much.
(2) Which, he said, can be written as the quotient of four similar products of the form
Щ1 -x/(A + A'A))
where each of A and A' runs through either the even or the odd integers (A = A' =0
excepted) and A is a given imaginary constant. Indeed, if one writes P(0, 0; x), P(l, 0;
x), P(0, 1 ; x), and P(l, 1 ; x) according as A, and A/ are respectively even or odd, then the
familiar elliptic functions are, he claimed, P(0, 0; x) , P(l, 0;x) , and P(l, l;x) .
P(0, 1 ; x) P(0, 1 ; x)
P(0, 1 ; x)
76
Jeremy Gray
Eisenstein then introduced elliptic functions by considering the
infinite double product:
ф(х) = л:П(1 - x/u)
where и = am + /to, and the product is taken over all integers
m and n; the associated doubly infinite sums yield the elliptic
functions (3). The basic periods of the functions are plainly a and /?,
so Eisenstein, like Cayley before him, had a general grasp of elliptic
functions with arbitrary periods, something which the then current
theory of integration made it hard to obtain. Eisenstein was also
pleased that his approach showed clearly why the elliptic functions
are doubly periodic, whereas Jacobi's approach of representing
them as quotients did not; the numerator and denominator are
each only singly periodic, which presents the periods unsymmetrically [1847, 418].
Eisenstein's ideas stimulated Cauchy to have second thoughts
about elliptic integrals, and he produced a series of papers that
ran through several memoirs in the Comptes rendus before coming
to a halt. He agreed in his [1846] that Eisenstein had quite rightly
pointed out some difficulties, and replied that
"what one calls the inverse integral of the definite integral
J(L0
is not the same as the value of x deduced from the complete integral
of the differential equation
dx = Vl - x2.dt
but the value of x that provides the complete integral of the differential
equation
dx = ydt
if one requires the new variable y
1) to satisfy the finite equation
x2 + y2 = 1
(3) I select just the essential example: £(x + w)~2 = (2, x). Eventually Eisenstein
showed, on setting y = (2, x), a = (2, a/2), a' = (2, 0/2), and a" = (2, (a + /8)/2), that
y' = 2{(y - a)(y - a')(y - a")}m,
so: "The function (2, x) is therefore in fact an elliptic function of the first kind in x. "
Elliptic and Abelian integrals
11
2) to vary with x by insensible degrees,
and if one requires, moreover, that x, y, t simultaneously take the initial
values
x = 0, y=l, t = 0."
[1846, 167.]
Similar considerations, he went on, pertained to elliptic integrals.
His conclusion was that in this way one could profitably hope
to avoid irrational functions and those involving radicals in favour
of differential equations.
It is not clear what to make, historically, of these few remarks.
On the one hand they are refreshingly clear, and seem to prefigure
the theory of Riemann surfaces. Yet that theory is properly named
for its German discoverer, and Cauchy did little more in this
direction. What he did do is confined to his theory of "lignes d'arrêt",
which displays just how "unRiemannian" Cauchy's ideas were,
if such a phrase be permitted. This theory was first presented in
January 1851 (see his [1851a]). It is based on the idea that
multivalued function can be thought of a single-valued upon a plane
from which some lines have been deleted. These lines emanate,
in Cauchy's theory, radially from some central point and each
one passes through one point where the function is infinite or takes
fewer than its expected number of values. The other values of
the function are obtained on crossing the "lignes d'arrêt", and
when evaluating a complex integral you must take account of the
lines, which are like so many frontiers between countries.
It is difficult to assess this idea, which may have done little
more than formulate the conventional wisdom of the day on what
to do with integrals of many-valued functions. It seems to be a
falling away from the remarks of 1846. Although Riemann's much
better formulation came out the same year, Riemann surfaces only
gradually made their way towards general acceptance even in
Germany, and in France Cauchy's formulation prevailed for quite
some time. Riemann's ideas were rejected by Briot and Bouquet
when they came to write the second edition of their book on elliptic
functions [1875], and in that context there is something to be said
for the French point of view (and something against: Riemann
himself lectured on elliptic function theory from his new point
of view). Much more tellingly, as late as 1881 no less a
mathematician than Hermite was advocating the use of "coupures". These,
78
Jeremy Gray
like "lignes d'arrêt", cut up the domain plane and replace a
continuous many-valued function with a family of continuous singlevalued ones and some discontinuous jumps across the cuts. Casorati wrote two scornful papers in his own journal rebutting this
approach [1887, 1888/1889], but Hermite, who once confessed to
Fuchs that he had never been happy with Riemann's ideas, was
the leading French mathematician of his generation.
A better way forward was found not by Cauchy but by Puiseux.
In his [1850] Puiseux presented two significant ideas: what has
come to be called the Newton-Puiseux polygon method for
evaluating the power-series expansions of an algebraic function in the
neighbourhood of a branch point; and his investigation of analytic
continuation around a branch point. Only the last will be discussed
here. Puiseux's alternative approach to many-valued functions was
expressly devised to avoid the discontinuities Cauchy was to
advocate in his theory of the "lignes d'arrêt". Puiseux stressed what
Cauchy had often remarked upon himself: a function only changes
its value when the z-variable is conducted around a branch point,
when the corresponding w-values are permuted in a number of
"systèmes circulaires"; and an integral along a closed curve is
only non-zero if the curve encloses either a simple pole or a branch
point. This formulation, as he observed, improved on Cauchy's
of 1846, which had only talked loosely of not containing points
where the function becomes discontinuous. Non-zero integrals taken
along closed curves Puiseux called periods. The value of an
integral between two points (specified by their z-values) can be varied
by adding arbitrary integer multiples of the periods.
Puiseux then sought to find all the distinct periods of an
integral, to see whether each one entered every value of the integral,
and to find the values of the integral without the periods being
taken into account. He was able to do this for the case when
the function can be written as:
polynomial in и = rational function in x.
In his second paper on this subject [1851], Puiseux took up the second
question in full generality. To answer it he established a theorem that
Cauchy, in his report on it [1851c, 381), called "very remarkable":
A single-valued algebraic function which is always continuous except
when it becomes infinite, is necessarily a rational function.
Elliptic and Abelian integrals
79
Using this theorem, Puiseux was able to show that for irreducible
functions, each period belongs to every value.
It is customary to observe that Cauchy left it to others to create
a theory of algebraic functions and their integrals. This is true,
but more can be said. As was remarked, Cauchy' s whole system
of definitions based on his newly-defined concepts of limit,
continuity, differentiability, and integrability, was incompatible with talk
of many-valued functions. He did not himself write that way, and
by and large he avoided those branches of mathematics which,
however attractive (as early elliptic function theory was) seemed
to be amenable to that way of thinking. As we have seen, other
leading younger mathematicians shared his concern when it came
to many- valued integrands. He and they were even prepared to
reformulate elliptic function theory in order to avoid the problems
the square root brought with it. But this proved premature, and
their alternative approaches have not driven out the direct approach.
Instead a way was found to extend Cauchy' s way of thinking about
functions (/. e. single-valued functions) to the integration of
algebraic functions. Cauchy did not do it, Riemann did. But it was
done within the framework for thinking about complex functions
Cauchy had devised, and it still is. That is surely the measure
of a considerable achievement, and one we are rightly remembering
today.
Faculty of mathematics,
Open University, Milton Keynes.
Jeremy Gray.
BIBLIOGRAPHIE
Briot (C.) et Bouquet (J.-C),
[1859] Théorie des fonctions doublement périodiques et, en particulier, des
fonctions elliptiques (Paris).
[1875] Théorie des fonctions elliptiques, 2nde éd. (Paris).
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[1887] Sopra le coupures del sig. Hermite, i Querschnitte e le superficie di
Riemann ed i concetti d'integrazione si reale che complessa, Annali
di matematica, (2) 15, 223-234.
[1888-1889] Idem, 2nd part, ibid., (2) 16, 1-20.
Cauchy (A.-L.),
[1843a] Mémoire sur une certaine classe de fonctions transcendantes liées entre
elles par un système de formules qui fournissent, comme cas parti-
80
Jeremy Gray
culiers, les développements des fonctions elliptiques en séries, Comptes
rendus, 17 (2oct.). ( = OC (1), VIII, 65-76.)
[18436] Mémoire sur les rapports entre les factorielles réciproques dont les
bases varient proportionnellement, et sur la transformation des
logarithmes de ces rapports en intégrales définies, ibid., 17 (16 oct.). (= OC
(1), VIII, 87-97).
[1843c] Sur la réduction des rapports de factorielles réciproques aux fonctions
elUptiques, ibid., 17 (23 oct.). (= OC (1), VIII, 97-110.)
[1846] Considérations nouvelles sur les intégrales définies qui s'étendent à tous
les points d'une courbe fermée, et sur celles qui sont prises entre des
limites imaginaires, ibid., 23 (12 oct.). (= OC (1), X, 153-168.)
[1851a] Mémoire sur les fonctions irrationnelles, ibid., 32 (20 janv.). ( = OC(1),
XI, 292-300.)
[1851b] Rapport sur un Mémoire présenté à l'Académie par M. Puiseux et
intitulé : Recherches sur les fonctions algébriques, ibid., 32 (25 fév.).
(= OC(1), XI, 325-335.)
[1851c] Rapport sur un Mémoire présenté à l'Académie par M. Puiseux et
intitulé : Nouvelles recherches sur les fonctions algébriques, ibid., 32
(7avr.). (= OC(1), XI, 380-382.)
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[CMP] The Collected Mathematical Papers, (New York : Johnson Reprint,
1963), 14 vol.
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and Journal of Science, 27, 424-427. (=[CMP\, I, 132-135.)
[18456] On the inverse elliptic functions, Cambridge Mathematical Journal,
4, 257-277. (=[CMP], I, 136-155.)
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[Werke] Mathematische Werke (New York : Chelsea, 1975), 2 vol.
[1844] Bemerkungen zu den elliptischen und Abelschen Transcendenten, Crelle's
Journal, 27, 185-191. ( = [Werke], I, 28-34.)
[1847] Genaue Untersuchung der unendlichen Doppelproducte, aus welchen
die elliptischen Functionen als Quotienten zusammengesetzt sind, ibid.,
35, 153-274. (= [Werke], II, 357-478.)
Hermite (C),
[1881] Sur quelques points de la théorie des fonctions, A cta Societatis Scientiarum Fennicae, \2, 1-83.
Houzel (C),
[1978] Fonctions elliptiques et intégrales abéliennes, in Dieudonné (J.), éd.,
Abrégé d'histoire des mathématiques, 1700-1900 (Paris : Hermann),
t. II, 1-113.
Jacobi (C. G. J.),
[Werke] Gesammelte Werke, 2nd éd. (New York : Chelsea, 1969), 8 vol.
[1834] De functionibus duarum variabilium quadrupliciter periodicis, Crelle's
Journal, 13, 55-78. (=[Werke], II, 23-50.)
Koenigsberger (L.),
[1904] Cari Gustav Jacob Jacobi (Leipzig : Teubner).
Elliptic and Abelian integrals
81
Puiseux (V.),
[1850] Recherches sur les fonctions algébriques, Journal de mathématiques,
(1) 15, 365-480.
[1851] Nouvelles recherches sur les fonctions algébriques, ibid., (1) 16, 228-240.
Riemann (В.),
[1851] Grundlagen fur eine allgemeine Théorie der Functionen einer veranderlichen complexen Grosse, Inauguraldissertation (Gôttingen).
(= Gesammelte Mathematische Werke, 3rd éd. (Berlin : Springer
Verlag/Leipzig : Teubner, 1990), 3-48.)
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evaluated by the editorial board and
the Institute and Museum of
two referees of international
the History of Science in
standing.
Florence, the National
The first, (and major),
Museum of Science and
section of each issue presents
Technology in Milan, the Italian
articles on original research,
Group for the History of
results and critical
Science and the publishers Leo S. authors. The languages of the new
commentary
of scholarly value.
Olschki, was the first Italian journal are English, Italian, The second
section deals with
periodical specifically
French, German and Spanish, documents, (with
dedicated to the History of
with each article accompanied commentary), letters, laboratory notes
Science. For thirty years the
by a long summary in English. and unpublished writings of
journal has followed and
The journal covers all major
of the past that are of
supported the growth in Italy of aspects of the field: the history scientists
historical value and
this discipline, which has still of astronomy, mathematics, particular
interest
In
the third section
to be fully recognised by the physics, chemistry, natural there are critical,
sometimes
University system and the philosophy, the earth sciences, provocative, discussions,
public.
medicine, natural history,
proposals and preliminary drafts
The situation today is biology and technology. The
of incomplete studies.The
very different, thanks to the journal is also open to
News and Information section
work of the Pisan Institute and contributions in the history of the
deals in the main with news and
the efforts of the many
human sciences, institutions and announcements concerning
scholars associated with it Many scientific instruments as well the History of Science in Italy:
Chairs in the History of Science as the historiography of science conferences and seminars, the
have been created in Italy over and the history of relationships activity of academic societies,
the last ten years, various
between science and
research groups, University
research centres have grown up philosophy, science and technology, departments and institutes. In
and the new Physis reflects this science and society and other the Book Reviews section a
enlarged institutional setting. crucial border-line areas.
choice of the more important
As in the previous
The journal does not publications in all the
series, the journal continues to specialise in any particular languages covered by the journal
welcome contributions from field or historical period, rather are reviewed by scholars from
both Italian and non-Italian it offers a wide range of
Italy and abroad.
1992 Subscription: Italy: Lire 75.000 • Foreign: Lire 96.000
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