Rev. Acad. Canar. Cierre., XX (Núms. 1-2), 119-134 (2008) (publicado en septiembre de 2009)
FOLLOWING THE STEPS OF SPANISH MATHEMATICAL
ANALYSIS: FROM CAUCHY TO WEIERSTRASS BETWEEN
1880 AND 1914
José M. Pacheco12, Francisco J. Pérez-Fernández**' & CarJos O. Suárez**4
* Departmento de Matemáticas. Universidad de Las Palmas de Gran Canaria
**Departamento de Matemáticas. Universidad de Cádiz
Abstract
Rigorous Mathematical Analysis in the Cauchy style was not accepted in a straightforward manner
by the European mathematical community of the central years of the 19'h Century. In average, only
around forty years after the 1821 Cours d'Analyse did Cauchy's treatment become a standard in the
more mathematically advanced countries, as a paradigm that remained in use until the
arithmetisation of Analysis by Weierstrass replaced it before the end of the century. In this paper
the authors show how rigorous Mathematical Analysis a la Cauchy was adopted in Spain quite late
-around 1880- and how in sorne more forty years, the Weierstrassian formulation became the usual
presentation in Spanish texts.
2000 MSC Numbers: 01A55, 01A72
1 Introduction
It is known that the definitive introduction of rigorous Mathematical Analysis in Spain was
achieved by Julio Rey Pastor (1888-1962) when he gave to print his two basic books
Elementos de Análisis Algebraico (Rey Pastor, 1917) and Teoría de las Funciones Reales
(Rey Pastor, 1918) after two stages in Germany: with Frobenius, Schottky and Schwartz in
Berlín, 1911-12, and with Caratheodory, Courant, HOider and Koebe in Gottingen, 1913-
1914. Inspired on the original theories by Cauchy, Weierstrass, Cantor, and Dedekind,
these two books incorporated to Spanish Mathematics the most rigorous standards of the
German mathematical schools.
Nevertheless, severa! attempts had been made in the presentation of Mathematical Analysis
in Spain to introduce rigour before Rey's books: The aim of this paper is to provide a
critical description ofthese mathematical activities.
According to Belhoste (Belhoste, 1991), Cauchy's viewpoints on Mathematical Analysis
were not accepted in a straightforward manner, neither in France nor elsewhere. When
Cauchy fled into exile the year 1830, his followers Navier, Sturm, Liouville and Duhamel
maintained bis ideas and used them in teaching and in mathematical writing, as a rule in a
less accurate way and even mixing them with other mathematical traditions (Grattan-
1 The first author wishes to thank the Max-Planck-lnstitut für Wissenschaftsgeschichte in Berlin for its hospitality during
the six-month stay (March-August 2008) when this paper was written. A first version is available in the Preprint Series of
the MPIWG (Nº 353, published in August 2008).
2 Corresponding author. Email: pacheco@dma.ulpgc.es
3 Email: javier.perez@uca.es
4 Email: carlososwaldo.suarez@uca.es
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Guinness, 2000, p.67). It was Duhamel who became one of the forerunners of the
introduction of the Cauchy style in Spain around 1880 through his Cours d'Analyse, a very
popular book in this country. The second step towards rigour, the Weierstrassian
revolution, was accepted in France when Camille Jordan (1838-1922) published his own
Cours d'Analyse de l'École Polytechnique (Jordan, 1893-96) and adopted the t:-ó style.
In spite of it, the older Cauchy presentation was still in use for severa] years: Jesper Lützen
points out that Stunn's text was used in Copenhagen as late as 1915 (Lützen, 2003).
Around 1840, the Spanish educational system was gradually recovering from the long reign
of Femando VII, where most universities and higher education centres were either closed or
dismantled. Soon the basic works of the mathematical scene were translated and/or adapted
into Spanish and the newer ideas of Gauss, Cauchy, and Abel on Mathematical Analysis, as
well as the birth of non-euclidean geometries, the development of projective geometry, and
the modem Funktionentheorie according to Riemann became available to the small Spanish
mathematical community, although in those days the books selected for translation or
adaptation did not yet include the ideas by Cauchy. In a previous paper (Pacheco et al.,
2007) the authors have shown how notions such as limits, functions, infinitesimals, etc.
were introduced in texts by authors like Vallejo and Feliu, but those terms and techniques
were not used in their proofs or would-be proofs.
As Grabiner points out, insistence in proof is a characteristic feature of the development
and foundations of Analysis according to the Cauchy style (Grabiner, 1981), and 1880 was
the year when the older pre-Cauchy Analysis really disappeared from Spanish higher
education and proofs actually entered the Spanish mathematical literature. The
mathematician Simón Archilla and the civil engineer Antonio Portuondo wrote the books
where Cauchy's vision of Analysis was presented in Spanish words for the first time. They
elaborated on the new ideas, but through Duhamel's books rather than by studying the
original work by Cauchy, even though these Iast ones were available at sorne Ieamed
libraries, like the one at the Real Observatorio de la Armada (Royal Navy Observatory) in
San Femando, close to Cádiz. This paper does not consider Portuondo's contribution
(Portuondo, 1880), essentially the small book Tratado sobre el infinito, which is the object
of another forthcoming paper by the same authors (Pacheco et al., 2009). More on other
mathematical activities of Portuondo can be found in (Pacheco 2008).
2 Archilla's Principios del Cálculo Diferencial
Simón Archilla (1836-1890) taught Mathematics at the Universities of Barcelona and
Madrid. His courses comprised Higher Algebra, Analytic Geometry, and Differential and
Integral Calculus. In order to cater for these last tapies, he published in 1880 the
fundamental book Principios del Cálculo Diferencial (The Principies of Differential
Calculus, Principios from now on). For availability reasons, in this paper the second edition
of 1894 compiled by his son Faustino Archilla will be used as the standard reference
(Archilla y Espejo, 1894). The authors also checked the first edition and realised that the
only difference between them are the page numbers: Nota single change was introduced in
the second edition which strictly speaking was only a second printing.
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Archilla was elected to the Real Academia de Ciencias Exactas, Físicas y Naturales,
(Royal Academy for the Exact, Physical, and Natural Sciences), and he read his inaugural
dissertation on June l01h 1888 with the title Sobre el concepto y principios fundamentales
del Cálculo Infinitesimal (On the concept and fundamental principies of Infinitesimal
Calculus) (Archilla y Espejo and Vicuña y Lazcano, 1888) where he made a most
interesting historical report on the idea of infinity, ranging from Archimedes to the date the
discourse was delivered. In page 61 of this report, Archilla acknowledges the role of
Cauchy:
"( ... ) there was a need to achieve what seemed so difficult from the very beginning: To force the
notion of infinity to serve the needs of Analysis. Opposing to this aim were not only the special
status of infinite quantities when considered as tools in mathematical applications to number and
distance, but also the philosophical criteria upon which the legitimate intervention of infinity in
Analysis would be based" (note 1).
Moreover, he points out how
"( ... ) under the light of the new doctrine, it has become common knowledge that the ratio of certain
infinitesimally small, although it is always finite quantity, <loes not tend to any lirnit whatsoever; it
is plain to see that the continuity of a function <loes not imply that the orders of its increment and
that of the variable be the same; and it is possible to conceive and to determine, as Weierstrass has
shown, continuous functions without a derivative at any point: Such things, if not unconceivable,
were difficult to understand and to explain with the old ideas" (note 2).
In the foreword to Principios the author presents the basics of the discourse in his book
through a quotation from the preface of Hoüel's Cours de Calcul Jnfinitésimal, a book
translated into Spanish in 1878:
"There is only one rigorous method to present Infinitesimal Calculus: It is the method of the
infinitely small, or of the lirnits, the method of Cauchy and Duhamel..."
This opinion is clearly detailed in the introduction to Principios:
"Our aim in this book is to summarise the most important principies of Infinitesimal Calculus,
trying to explain their natural interrelations, and to study the intimate relationships between the
fundamental notions upon which they are based and those that are legitimately obtained from them,
according to the doctrine first expounded by Cauchy and then developed by Duhamel..." (note 3).
The difference between this text and its predecessors is finally featured in the treatment
given to the ideas of continuity and differentiability in the light of infinitesimals
(Introduction, page VI):
"In the study of functions we have focused on the notion of continuity, by showing the difficulties
of directly studying it, and by relating it indirectly with the idea of infinitesimal quantities through
the lirnit notion ... "
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2.1 Archilla on infinitesimals
Archilla takes infinitesimals as the fundamental notion in his book, and he introduces the
concept of a variable in page 1: A variable quantity, or simply a variable, is a quantity that
can take a series of succesive values according to any prescribed law. He goes on by
explaining how to operate with variables according to the succesive values that determine
them, and he defines infinitely small and infinitely large quantities in page 4:
"A variable quantity that can take values smaller than any given quantity and can indefinitely
satisfy this condition is an infinitely small quantity, or simply an infinitely small. On the other hand,
if a variable quantity can take values larger than any given quantity and can indefinitely satisfy this
condition is an infinitely large quantity, or simply an infinitely large. Finally, quantities that are
constant orare neither infinitely small nor infinitely large are calledfinite quantities".
Limits are presented in a way that directly matches the definition of an infinitesimally
small, the limit of a variable quantity being a constant quantity such that the varying one
approaches it in the following sense: The difference between the constant and the
successive values of the variable becomes smaller than any given quantity, but never equals
zero. Therefore, from this definition it follows that:
(1) The difference between a variable and its limit is an infinitesimally small quantity.
(2) Infinitesimally small quantities have zero as their limit.
(3) Infinitely large quantities have no limit.
(4) The limit of a variable is a constant that can not be found among the successive values
of the variable.
This last observation is directly inherited from Cauchy and introduces a certain lack of
generality, for sequences tending to zero like 1,0, 3,0, {,O, ~, ... would not comply with such
a definition. Nevertheless, this observation can be considered a minor tlaw. More
interesting is the idea of an extended real line expounded in this paragraph, obtained from
page 6:
"Our aim has been to give more generality to this doctrine by encompassing in a common
classification finite quantities, infinitely small and infinitely large ones, thus establishing a general
theory of the orders of quantity."
Then Archilla proceeds to the definition of infinitely small, or large, or constant quantities
via the ratios between two quantities:
"When the ratio f is an infinitely small a , then x is said to be infinitely small with respect to y.
When the ratio f is an infinitely large A, then x is said to be infinitely largewith respect to y.
Finally, hen the ratio f is sorne constant a, then x is said to be of the same arder of y."
And he goes on by specifying a comparison unit:
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"Once a fundamental infinitesimally small quantity a has been arbitrarily chosen, we shall deem
every infinitesimal of the same order as first-order infinitesimals".
Thus, if /J is any other first order infinitesimal, then sorne finite quantity a will exist such
that ~ =a+ w, where w is an infinitesimally small, from where /J = a( a+ w) . In the same
way the definition of an n-th order infinitesimal with respect to the basic one is obtained as
/J,, = a"(a+ w)
Moreover, acknowledging that A=; is an infinitely large quantity, the orders for infinities
are likewise defined through the following equalities:
B,, = A"(a +w) =(~)"(a+ w) = a -"(a+ w) ,
and a whole scale of infinitesimals, ranging from the infinitely small to the infinitely large
through finite quantities, is obtained by allowing the exponent n to run over the real
numbers. In the above computations there is the implicit assumption that the product of an
infinitely small wand a finite quantity a is, again, an infinitely small. The proof of this fact
is found in page 15: In order to prove it, it suffices to show that am < t, where t is an
arbitrarily small quantity. But this inequality can be also written in the form aó < ~ and
since OJ is an infinitesimally small, its inverse is larger than any given number. Therefore
aOJ is an infinitely small quantity. With the above remarks, Archilla presents the usual
algebraic rules for infinitesimals and establishes the relationships between lirnits and
infinitesimals by noticing in page 20 that:
"If the difference between a variable x and a constant a is infinitely small, then this constant is the
limit of the variable. Therefore, the equation x = a+ w necessarily implies lim x = a"
2.2 Archilla on continuous functions
The notion of continuity is presented by Archilla in two stages. First, a general idea of
continuity for any variable quantity, is introduced (page 56):
"A variable quantity x varíes in a continuous manner if it necessarily takes every intermediate value
when going from a to b, and, moreover, the same property holds for any subinterval
[a1 ,b1 J ~ [a,b] ".
This definition has a geometric flavour, and the precision about the restnct10n to any
subinterval is a most interesting one. In a second stage the notion of continuous function is
introduced (page 57):
"A function f (x) of a variable x is said to be continuous between the values x =a and
x = b when x varies in a continuous way between these values, if the function values can not pass
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from sorne value m to any other one n :t m, both between f (a) and f (b), without taking ali
intermediate values between m and n".
This is Medvedev's "continuity in the sense ofDirichlet-Lobachevskii" (Medvedev, 1991,
p. 60-62), and it obviously needs one more condition, i.e. monotonicity of the function,
which is implicitly assumed as shown by the definition of discontinuity explained by the
figure in page 58. Nevertheless, when coming to practica! questions, Archilla does not
forget his programme on infinitesimals: In the same page 58 he recalls that
"( ... )a function is continuous on a given interval if, to any infinitely small increment of the
variable in that interval, there exists an associated infinitely small increment of the function".
Next !et us observe how the continuity of the logarithm is assessed: Let the increment of the
logarithm be log(x+ h)- log(x) = log .tif = log(l + ~). Now it is known -because it had been
proved earlier in the book- that for any fixed x, log(l + ~) and ~ differ in an infinitely small
OJ of order larger than ~, namely log(x+ h)- Iog(x)= ~+ OJ. Therefore it is plain that
Archilla considers that both the Cauchy and the Dirichlet-Lobachevskii viewpoints on
continuous functions one are equivalent ones, without taking care of proving it: To him,
that was simply true.
2.3 Archilla on derivable functions and on differentials
Differential calculus deals on how to establish infinitesimal relationships between
the increments of the independent variable(s) and that of the dependent variable or function.
Archilla denotes those increments by Llx and /}.qJ(x), and he writes (pages 65-66):
" Direct consideration of the limit of d~x) when Llx --7 O greatly simplifies the research whose aim
is to determine the infinitesimal relationships betweenLlx and !}.qJ(x) , and it introduces one of the
most important objects in the Mathematical Analysis. The limit is called the derivative, or derived
function, of qJ(x) ".
To show how Principios is still a mixture of old and new ideas, it is enough to no ti ce that
Archilla sticks to the old idea that the variable can not take the value of the limit when
approaching it and, moreover, he insists on a function being continuous for it to be
differentiable, even acknowledging that this condition is not sufficient. He explains it by
considering the infinitesimal orders of the two increments.
As a remark, the authors realise that there is no reference in Principios either to Weierstrass
or to the existence of continuous functions without a derivative at any point. Nevertheless,
Archilla might ha ve known about these functions after the first 1880 edition of Principios,
since he spoke about them in his 1888 discourse for theAcademia. It has been impossible
to know whether he thought of including such a topic in a possible second edition, as the
1894 one is simply a reprint, most possibly prepared by his son for monetary reasons.
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The treatment of the mean value theorems shows clearly the influence of Cauchy. In a
rather complicated paragraph (see note 4) in page 76, the author explains that if the
derivative <p'(x)is a continuous function for ali x betweenx0 and x0 +h, then it will take
finite values, their mean value μ(<p'(x)) being somewhere between the maximum and the
minimum ofthe derivative on that interval. Therefore, dueto the assumed continuity ofthe
derivative there, exists a value xd such that <p'(xd) = μ(<p'(xd)) and the equation he
obtained previously asan "interesting property" ofthe derivative:
can be rewritten as:
<p(x0 + h)-<p(x0 ) = h<p'(x11 ) = h<p'(x + Bh) ,
from which a completely modero version ofRolle's Theorem follows.
It must be noticed that the "interesting property" is simply an integral-free version of the
mean value for integrals as applied to the continuous derivative:
obtained by the author through a rather obscure argument in pages 74 and 75.
The largest part of Principios extends from page 115 until the end of the treatise, under the
title "Book III: Differential Calculus", and it starts with the concept ofthe differential ofa
function. Thus:
"When studying the form and general properties of the infinitely small increment of functions of
one or severa) variables, we realised that among the infinite number of quantities differing infinitely
little from the increment of the function, there existed one simpler than any other quantity, and it
was expressed through the derivative or partial derivatives of the function; moreover, we saw that
this unique form, in the case of a function of a single variable y= <p(x) , was <p'(x)Lix . This
infinitely small quantity, which is unique by his form among ali quantities that differ infinitely little
rrom ily , will be called the differential of y= <p(x) , and will be represented by the special
notation dy or d<p(x) ... "
This surprisingly modero definition lacks only the words "linear function" to be a fully
current one. Nevertheless, Archilla is well aware of the linearity of the differential as a
function of the increment of the independent variable:
"The values of the differential d<p(x) for sorne definite x are proportional to those of Lix = dx
and this is a property exclusive to the differential among ali quantities infinitely close to the
increment ily ."
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Principios ends without any consideration of the Integral Calculus. The authors do not
k:now whether the author intended to publish a seque) on this topic, but the fact is that no
evidence has been found to support this view.
3 Clariana and bis lecture notes on Mathematical Analysis: Complex Analysis
Lauro Clariana Ricart (1842-1916) spent his life in Catalonia, mostly in Barcelona and
Tarragona, the capital city of the neighbouring province of the same name. He studied in
Barcelona the courses needed to become Licenciado (1872) in the so called Ciencias
Exactas, the name given in Spain to Mathematics for more than a century. One year later he
obtained the Doctorate in the exact Sciences at the same university. Clariana was, for a long
period beween 1865 and 1916, a prolific writer on many topics ranging from Mathematical
Analysis and Rational Mechanics to Music and the construction of severa! machines. A
detailed biographical account and a complete Iist of Clariana's writings -sorne of them
unpublished- can be found in (Clariana-Clarós, 1993).
Clariana taught Integral and Differential Calculus, and Rational Mechanics, and was one of
the few Spaniards who participated in severa! Intemational Congresses and Meetings. He
travelled to Paris, Brussels, München, and Freiburg between 1888 and 1900, and in 1888 he
was awarded in Paris a prize for his Memoir "On the spirit of Mathematics in the modern
times". Clariana was the author of two books on Mathematical Analysis for the use of
students at the Escuela Superior de Ingeniería Industrial in Barcelona. They were
published in 1892 and 1893 under the titles Resumen de las lecciones de Cálculo
Diferencial e Integral (Clariana Ricart, 1892b) (Resumen from now on), and Complemento
a los elementos de los Cálculos (Clariana Ricart, 1892a) (Complemento in what follows).
They appeared as lithographed handwritten lecture notes, and a joint edition under the title
Conceptos fundamentales de Análisis Matemático appeared in a more normal printing style
the year 1903 (Clariana Ricart, 1903). A small favourable review of this last book appeared
in the Bulletin ofthe American Mathematical Society the year 1904 (McFarlane, 1904).
Resumen is a general introduction to Analysis, and at the very beginning, in page 5,
Clariana makes his position clear (note 5):
"Because the infinitely small and the infinitely large are the only elements that can become the basis
of quantity in Mathematics, synthesised in the finite ones, we shall admit three categories of
quantity under the following forms:
(!) That of the infinitely small.
(2) That of the finite.
(3) That of the infinitely large.
And these are the onJy true concepts of quantity that are directly connected to the Leibnizian idea of
a differential"
A very long introduction (Prolegómenos) of 45 pages is offered on the various classes of
numbers and functions, as well as on the foundations and the history of the infinitesimal
method, where the author summons Descartes, Johann Bemouilli and Cournot, and indeed
Newton, Leibniz, and D' Alembert. The rest of the book is a classical treatise on the usual
topics on Differential and Integral Calculus presented in a straightforward way. Theorems
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are not highlighted and proofs are not distinctly offered. The emphasis is on the succession
of useful and applicable formulas, with a few examples spread over the text. This mak:es the
book very readable and surprisingly modem even to today's standards. Complemento has a
different flavour: It is a compilation of loosely knit tapies in higher Analysis. Clariana
declares in a brief introduction:
"The aim of this book is to present what we should call 'modern theories of the infinitesimal and
integral calculus', not because sorne of them are recent ones, but because they have not yet been
presented in the Spanish education".
The first two chapters are devoted to "Infinitesimally Small Triangles" and "Orders of
Comparison for Curves", where the fundamental Leibnizian triangle is explained and
applied in depth, as well as the idea of the ordér of an infinitesimal is deeply applied to the
study of different elements of curves.
With this equipment, the book follows with a study of Classical Differential Geometry. On
the remaining chapters, a variety of tapies is included. There are the Euler-McLaurin
summation formula, special functions and elliptic integrals... To summarise; it is a
simplified version of the usual second volume in the classical French treatises that clearly
inspired the author, and the style is the same of Resumen. In basic questions, Clariana holds
the same opinions of Archilla. As an example, the definition of a continuous function on an
interval reads:
"The function y= F(x) is continuous between the values a and b attributed to x if its values pass
from one value to another through values that differ between them as little as desired."
But the main feature in Clariana's work is that he is the first author to introduce in print
Complex Analysis in Spain. He plainly states that "a complex quantity has the form x+ yi,
where x and y are real quantities" and goes on by explaining that Gauss was the first to
speak of yi as imaginary and that Cauchy denoted as imaginary the whole complex
quantity. Of course he points out that when x and y are variable quantities, then x+ yi is a
complex variable and that a complex quantity is infinitesimally small (large) if the real part
x and the imaginary part y are infinitesimally small (large) quantities. Continuity of a
complex quantity is, of course, assessed form the continuity of its component real variables.
Then, a standard theory follows.
It must be noted that befare Clariana no Spanish mathematician had studied complex
quantities as the object of Analysis. Only algebraic, geometric or arithmetic considerations
had been made in Spain on these numbers, and for nearly forty years the source book was
the rather obscure Teoría transcendental de las cantidades imaginarias, the posthumously
edited work (1865) of José María Rey Heredia (1818-1861) who inspired severa!
developments, especially in the presentation of Analytic Geometry. The work of Rey
Heredia and sorne of his followers has been extensively studied elsewhere by the authors
(Pacheco et al., 2006).
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4 Villafañe and the Tratado de Análisis Matemático
José María Villafañe Viñals (1830-1915) was of Cuban origin and taught Mathematics at
different Universities: Barcelona, Valencia, and Madrid. His only academic degree was
obtained at a very early age from a secondary school in his native Santiago de Cuba, and
once in metropolitan Spain he took advantage of various legal shortcuts and academic
tricks until he became a Catedrático (full professor) at the University of Valencia, where he
was a friend of the Spanish Nobel Prize winner for Medicine (1906) Santiago Ramón y
Cajal. A most complete biography of Villafañe can be found in (Llompart and Lorenzo,
2001).
In 1892 Villafañe published a treatise on Mathematical Analysis (Villafañe y Viñals, 1892)
in three parts, which he afterwards enlarged to four. The first one deals with the basic
techniques of Analysis: Functions, continued fractions, numerical congruencies, complex
numbers ... The second and most interesting part of the book contains the Infinitesimal
Analysis: limits, infinitesimals, continuous functions, derivatives, integrals, and infinite
series. Equation solving and algebraic forms conform the respective contents of the third
and fourth parts. Although Villafañe's book studies the same tapies presented in the above
analysed books, it must be noted that its presentation is much more formal and rigorous.
The authors shall dwell only in sorne remarkable differences. A first one is observed in the
definition of a limit (note 6):
"A fixed quantity will be called the limit of a variable quantity if this last one indefinitely
approaches the first one until the difference between them is smaller than any arbitrarily small
chosen quantity."
Indeed this formulation is much closer to the familiar e - ó definition, although the
absolute value is not employed. Nevertheless, a few pages onwards the absolute value is
used when trying to prove that t is larger than any finite quantity. In Part 11, Chapter III,
this definition of continuity is found (note 7):
"A function is continuous if (infinitely) small increments of the variable there yield (infinitely)
small increments -in absolute value- of the function".
Villafañe does not make the distinction between continuity and uniform continuity,
established by Heinrich Heine (1821-1881) (Heine, 1872), and already dealt with by
Dirich1et as early as 1854. What Villafañe really tried to do was to prove the equivalence
between the continuity definitions of Cauchy and of Dirichlet-Lobachevskii. Complex
Analysis is also considered in this treatise, as in the books by Clariana. But Villafañe goes
further by offering for the first time in a Spanish text a fairly good approximation to the
correct definition of continuity of a function of a complex variable, when he writes:
"( ... )the function u= f (z) of the imaginary variable z is continuous on the (plane) surface limited
by a closed contour if for each value of z within these bounds the modulus of the functional
increment liu = f (z + &)- f (z) tends to zero if the modulus of & tends to zero".
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Moreover, holomorphic functions are defined, in page 194 of Part II, before the derivative
of a complex function as continuous, monotropic and monogenic functions, or equivalently
adrnitting a well deterrnined derivative. Sorne bulk errors are found in Villafañe's treatise,
and here two of them are considered. When dealing with derivatives (Part II, page 211) this
paragraph is found:
"There exists a finite limit for the ratio of the infinitely small increments k and h of a continuous
variable and its independent variable, and only sorne exceptional values of the variable can make
the ratio grow indefinitely towards infinity or indefinitely decrease towards zero".
This is obviously wrong; since the statement affirms that every continuous function is
differentiable but for a set of exceptional points, and it contradicts the condensation
principie introduced by Hermann Hankel (1839-1873). With this method Hankel managed
to present in 1870 a construction of a continuous function without a derivative at any
rational point. Strangely enough, Villafañe knew, if not this construction, at least the 1872
example by Weierstrass of a continuous function without a derivative at any point.
Nevertheless, Part II, page 216 it reads:
"We do not deem it necessary to take into account the theorem of Weierstrass where he affirms the
existence of continuous functions without a derivative at any point, a matter still under discussion.
Even if we admitted the existence of such functions, it would not interfere with what we are going
to expound on the derivatives of continuous functions".
The calculus of complex functions is found from page 391 onwards, and the first result is
the assertion that the Cauchy conditions are necessary and sufficient conditions for a
function to have a correctly deterrnined derivative with respect to the complex variable z,
i.e. to be a monogenic function. The proof is essentially the same given by Cauchy
(Cauchy, 1882-1974, Sér. 1, Tome 1, p. 330) and continuity of the partial derivatives is used
without postulating it beforehand. Indeed, with this addition, the Cauchy conditions
become sufficient ones.
5 Pérez de Muñoz and his Elementos de Cálculo Infinitesimal
Ramón Pérez de Muñoz was a professor of Mathematics at the Escuela Superior de
Ingenieros de Minas (School of Mining Engineers) in Madrid. This Pérez de Muñoz should
not be rnistaken with his brother Francisco, a civil engineer who promoted the study of
quatemions in Spain and was a Professor at the University of Manila in the Philippines.
Pérez de Muñoz taught Calculus and Mechanics, and was the author of a book published in
1914 entitled Elementos de Cálculo Infinitesimal (Pérez de Muñoz, 1914). According to the
author' s foreword, the book is inspired in the works of Archilla, Villafañe, La V alléePoussin,
Duhamel, Cantor and others. In addition, he explains that his aim is to present a
rigorous and scientifically clear exposition, even at the cost of overflowing the usual
contents of books for engineers. Both Pérez de Muñoz and Rey Pastor were members of the
steering comrnittee for the year 1912 of the newly founded Sociedad de Matemáticas, as
shown in the February 1912 number of the Revista of the Society (pp. 223-233), when Rey
Pastor was about to come back to Spain after his first German period. Although the authors
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have not found documentary evidence of any written mathematical collaboration between
them, they hypothesise that the influence of Rey Pastor was determinant in Pérez de Muñoz
writing in the new mathematical style in Spain. A comparison with (Rey Pastor, 1918)
clearly points out in that direction.
In his text severa! novelties are found. The first chapters deal with number systems, being
this book the first Spanish one to offer a construction of the real field following Cantor and
Dedekind. It starts with the notions of a set and its cardinality, and a proof is offered of the
fact that between any two (rational) numbers there are infinitely many ones. After
considering these "cuts" in the sense of Dedekind, the continuity and non denumerability of
the real line are established, though in a rather rhetoric manner. The theorem on the
existence of a least upper bound for any bounded set is also presented. Limits are defined in
a modem way, although inaccessibility of the limit is still preserved. Nevertheless, the
proof that a monotonic and bounded sequence has a unique limit is a completely rigorous
one based on the previous ideas on real numbers. For the first time, the theorem asserting
that any function having a derivative at sorne point is also continuous at that point is
proved, although the hypothesis of continuity in sorne neighbourhood of the point is not
stated.
To summarise, most possibly due to the influential ideas brought from Germany by Rey
Pastor, there is a large qualitative leap forward in this book when compared with its
predecessors by Archilla, Clariana or Villafañe, thus paving the way to the mathematical
20th Century in Spain.
6 Conclusions and views
In this paper the authors show that the work of several mathematicians and engineers must
be acknowledged in the enterprise of introducing rigour in the teaching and spreading of
Mathematical Analysis in Spanish universities and engineering schools. The most
representative four personalities and books have been dealt with, highlighting their
achievements:
(1) Archilla is the first Spanish mathematician to introduce the Cauchy style in his book
Principios de Cálculo Diferencial.
(2) Clariana was the first to present complex analysis, as well as the definition of continuity
of a complex function.
(3) Villafañe made a most interesting effort when he presented
• The total derivative of a complex function.
• Several definitions related with complex functions, among them that of
a holomorphic function.
• The relationship between the existence of a derivative and the CauchyRiemann
conditions for continuous functions.
• Absolute values as a tool in proofs and definitions.
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(4) Pérez de Muñoz presented for the first time in Spanish a construction of the real field
and the proof of the continuity of a derivable function.
Indeed, the texts by these authors are not rnilestones in the road to Mathematical
knowledge, but they must be acknowledged since they represent the steps towards
introducing Spain in the mainstream of Mathematical Analysis during the first decades of
the 20th century, in particular between the years 1915 and 1935.
Notes
l. (. .. ) era necesario conseguir lo que desde un principio tan difícil aparecía: había necesidad de
obligar la noción del infinito a someterse dócil y servir de instrumento a las necesidades del
análisis; y a esto se oponía no sólo el carácter y sello especial con que la noción del infinito se
mostraba a la consideración matemática, cuando se aplicaba al número y a la distancia, sino
también el criterio filosófico que había de servir de fundamento a la legítima intervención del
infinito en el análisis.
2. (. .. )a la luz de la nueva doctrina, es ya noción vulgar la de la razón de ciertos infinitamente
pequeños, que, aunque siempre finita, no tiende a límite alguno; se ve claramente que la
continuidad de una función no implica que los incrementos de ésta y de su variable hayan de ser del
mismo orden; y se conciben y determinan, como lo ha hecho Weierstrass, funciones continuas que
no tienen derivada: cosas, si no inconcebibles, difíciles de entender y de explicar en el antiguo
orden de ideas.
3. Nos proponemos resumir en este libro los principios más importantes del Cálculo Diferencial,
procurando establecer su natural subordinación y dependencia, y estudiar las íntimas relaciones
quo existen entre las nociones fundamentales que les sirven de base y las que de éstas
legítimamente se derivan, conforme a la doctrina propuesta primero por Cauchy, y desarrollada
después por Duhamel.
4. Si la derivada qJ'(x) de la función de que se trata es también continua para todos los valores de
x comprendidos entre x0y x0 + h, los valores de la derivada serán finitos en este intervalo, y su
media μ(qJ'(x)) tendrá siempre un valor comprendido entre el mayor y menor de dichos valores
de la derivada; por consiguiente, hay un valor xd de la variable para el cual la derivada qJ'(xd) es
igual a dicha media, y la ecuación qJ(x0 + h)-qJ(x0 ) = hμ(qJ'(x)) puede escribirse con otra forma
y como xd es uno de los valores de la variable comprendidos entre x0 y x0 + h , sera xd = x0 +Oh,
en cuya expresión es () un número positivo, en general comprendido entre cero y Ía unidad, que
también en casos particulares podrá tomar uno de estos dos valores. la ecuación (B) puede, por lo
tanto, escribirse como sigue:
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5. Siendo los indefinidamente pequeños, así como los indefinidamente grandes, los únicos
elementos que pueden constituir la base de la cantidad en matemáticas, sintetizados en lo finito,
admitiremos en esta ciencia tres categorías de cantidad, bajo la forma siguiente:
( 1) Correspondiente a los indefinidamente pequeños.
(2) Correspondiente a lo finito.
(3) Correspondiente a los indefinidamente grandes.
Estos son los únicos y verdaderos conceptos de cantidad que se enlazan directamente con la
diferencial de Leibniz.
6. Se llama límite de una variable la cantidad fija, a que esta variable se aproxima indefinidamente
hasta poder ser la diferencia entre la variable y la cantidad fija menor que toda magnitud tan
pequeña como se quiera.
7. Una función es continua, cuando para incrementos pequeños de la variable, los incrementos
correspondientes de de la función son también infinitamente pequeños en valor absoluto.
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