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The Rise and Development of the Theory of Series up to the Early 1820s

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Giovanni Ferraro’s book must be regarded as an important contribution to the history of mathematical analysis, and I know of no other that concentrates so thoroughly on the historical theme of infinite series. In effect, it depicts the declining influence of geometrical thinking; it charts the emergence of powerful algebraic methods, and gives a good indication how improved understanding of number systems led to the sharpening of analytic ideas. In a sense, therefore, it constitutes a cross-sectional history of mathematics from about 1600 to around 1820, when Cauchy published hisCours d’analyse.

Most of the book concerns work on series that took place long before Cauchy’s rigorisation of analysis — and, in fact, long before the emergence of any coherent understanding of the number systems used today. Moreover, prior to Viète, there was no effective algebraic notation with which to establish analytical ideas with any generality; most mathematical reasoning still centred upon particular geometrical figures. In fact, it was only in Euler’s day that the concept of function began to crystallize, thereby enabling mathematics to further reduce its dependence upon geometrical thinking.

Part I explains the use of series by a host of mathematicians, from Archimedes to the time of Leibniz and Newton; the series under consideration are mainly geometric, harmonic and simple power series. Also, by way of summary, it concludes with a chapter providing epistemological discussion on the ‘The formal-quantitative theory of series’. So, given the relatively rudimentary state of mathematical analysis in that period, what sort of ‘theory’ could that be?

The fact is that there wasn’t any overriding theory at all, but merely a set of working practices that hinged upon the notion of ‘quantity’, which had its philosophical roots in the writings or Aristotle. However, Greek mathematicians conceived of specific geometrical quantities, such as straight or curved lines, angles, surfaces etc. On the other hand, quantity became more abstract when represented in Viète’s algebra, which was geometrically formulated in the sense that it depended upon the dimensional homogeneity of all terms in a particular equation

One particular practice, used when working with series, is referred to as the ‘principle of finite extension’. This meant that what was true for the a partial sum would be assumed true for the sum to infinity. Another mathematical principle of the period is described as the ‘generality of algebra’, which was the belief that, if an analytic expression was derived by using the rules of algebra, then it was thought to be universally valid. In modern parlance this meant that, if one proved that if f(x) had the property P in an interval I, then one could extend the property to values beyond this interval.

In short, the myriad of working practices applied to infinite series from the time of Archimedes to the era of Newton and Leibniz is described in this book as the ‘formal-quantitative theory of series’, which seems to be the author’s terminology for the panoply of techniques employed up to around 1700.

Part II, considers the development of a ‘more formal conception’ that took place in the first half of the 18th century. This involved the generalisation of procedures, rules that were valid for finite sums, or for certain intervals of values of the variables. There is close examination of ideas due to De Moivre, Maclaurin, Stirling and Euler etc, including discussion of asymptotic series, infinite products and continued fractions.

Parts III and IV are concerned with the era 1760 to 1820, when Cauchy rejected the approaches to analysis that had been employed up to the beginning of the 19th century. Within this period, there were the achievements of Lagrange, Laplace’s work on the calculus of generating functions, Fourier series and Gauss’s results with hypergeometric series, and so on.

In describing the emergence of a modern theoretical basis for infinite series, the book begins by showing how, prior to the 19th century, there were a range of speculative operational processes for combining infinite summations using the same algebraic operations that can be applied to finite expression. It also explains the incautious manner with which divergent series were handled, and how the notion of convergence gradually evolved from a position of intuitive understanding to the more precise version due to Cauchy. Curiously, however, there is no mention of the fact that Archimedes method for determining the limit of an infinite summation fits in with the Cauchy definition. Such an observation would have neatly tied together the beginning and end of the book.

The scope of the book as a whole is quite enormous, and it is mathematically very detailed. It analyses the historical origins of nearly every type of series that one would meet in contemporary books on analysis and calculus and it is thematically diverse (historically, mathematically, philosophically).

However, because the history of infinite series is so complex, and because the author’s coverage is so mathematically extensive, there has to be compression of the historical narrative. Consequently, there is no reference to Chinese mathematics nor to early work that took place in India. Also, readers will gain little insight into the notations employed by early mathematicians, and one has to speculate as to the reasons why certain ideas on series came under investigation.

Moreover, the book is not an easy read, even though it appears not to have been translated from Italian. This is partly due to the depth of mathematical and philosophical analysis and partly due to occasional opaqueness of the prose (on page viii of the Preface, for example).

In summary, this book is thoroughly researched; it is written with a high degree of accuracy, and the broad range of fascinating material is, in general, very well organised. Therefore, given the uniqueness of its coverage, every college and university library should have a copy. In fact, extracts from the book could be used to inject historical perspectives into courses on analysis, and it will certainly appeal to historians in general.

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.

Date Received: 
Saturday, December 29, 2007
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Giovanni Ferraro
Sources and Studies in the History of Mathematics and Physical Sciences
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P. N. Ruane

Preface.- Part I: From the beginnings of the seventeenth century to about 1720: convergence and formal manipulation.- Series before the rise of the calculus.- Geometrical quantities and series in Leibniz.- The Bernoulli series and Leibniz's analogy.- Newton's method of series.- Jacob Bernoulli's treatise on series.- The Taylor series.- Quantities and their representations.- The formal-quantitative theory of series.- The first appearance of divergent series.- Part II: From 1720s to 1760s: The development of a more formal conception.- De Moivre's recurrent seires and Bernoulli's method.- Acceleration of series and Stirling's series.- Maclaurin's contribution.- The young Euler between innovation and tradition.- Euler's derivation of the Euler-Maclaurin summation formula.- On the sum of an asymptotic series.- Infinite products and continued fractions.- Series and number theory.- Anaysis after the 1740s.- The formal concept of series.- Part III: The theory of series after 1760: Successes and problems of the triumphant formalism.- Lagrange inversion theorem.- Towards the calculus of operations.- Laplace's calculus of generating functions.- The problem of analytical representation of nonelementary quantities.- Inexplicable functions.- Integration and functions.- Series and differential equations.- Trigonometric series.- Further developments of the formal theory of series.- Attemps to introduce new transcendental functions.- D'Alembert and Lagrange and the inequality technique.- Part IV: The decline of formal theory of series.- Fourier and Fourier series.- Gauss and the hypergeometric series.- Cauchy's rejections of the eighteenth-century theory of series.

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Monday, April 7, 2008