The rise and development of the theory of series up to the early 1820s

There two reasons that have led me to write this book. The first reason is that there is a lack of a work which systematically describes the theory of series from around 1600 to the early 1820s although many pages have been written about various aspects of this topic. The second, and more important, reason is that the theory of series in the 17th and 18th century poses several interesting problems to historians. Mathematicians of the time derived numerous results, which range from the binomial theorem to the Taylor formula, from the power series expansions of elementary functions to trigonometric series, from Stirling's series to series solution of differential equations, from Euler-Maclaurin’s summation formula to Lagrange’s inversion theorem, from Laplace's theory of generating functions to the calculus of operations, etc. However, most of these results were derived using methods which would be found unacceptable today. Therefore, if we look back to the theory of series prior to Cauchy without reconstructing internal motivations and the conceptual background, it appears as a corpus of manipulative techniques lacking in rigour whose results seem to be the puzzling fruit of the mind of a magician or diviner rather than the penetrating and complex work of great mathematicians. In this monograph I describe the entire complex of 17th and 18th century procedures and results concerning series. My aim is to reconstruct the implicit and explicit principles upon which they based, draw attention to the underlying philosophy, highlight competing approaches, and investigate the mathematical context – applied mathematics included - where the series theory originated. In so doing I hope to improve our understanding of the framework of 17th and 18th century mathematics and avoid trivializing the complexity of historical development by bringing it into line with on modern concepts and views and by tacitly assuming that certain results belong, in some unproblematic sense, to a unified theory that has come down to us today. The initial and final points of my monograph require some clarification. The point of departure is the publication of a paper by Viète [1593] where geometrical series are discussed and Pi is expressed in the form of an infinite product. Even though previous tracks of infinite series can be found, Viète’s paper, when considered in the context of the new rising symbolic algebra, appears to be a step forwards in a path - very slow to begin with, but which developed much more rapidly after 1650- which has made series an essential instrument in mathematics. The point of arrival is the early 1820s when Cauchy published his Cours d’analyse and his Résumé, which can be considered to mark the definitive abandonment of the 18th century formal approach to the series theory. My main arguments can be summarised as follows. The mathematicians who first used series were interested in their capacity to represent geometrical quantities and only had an intuitive idea of convergence. They thought that series represented quantities if, and only if, they were convergent to this quantity (the quantitative aspect of the notion of series). However, a distinction between finite and infinite sums was lacking and this gave rise to formal manipulations; in other words, they used procedures which were the infinitary extension of finite procedures. In the works of mathematicians such as Newton and Leibniz, the quantitative and the formal aspect co-existed and formal manipulations were a tool for deriving convergent series. By the 1720s, this interlacing of the quantitative and the formal yielded several results that could not be reduced to the original concept. Mathematicians introduced recurrent series, which stressed the law of formation of coefficients, independently of the convergence of series The attempt to improve the acceleration of series subsequently led to the emergence of asymptotic series, which showed the possibility of using divergent series to obtain appropriate approximations. Furthermore, the investigation of continued fractions and infinite products and certain applications of series (for instance, in numerical analysis and in number theory) increasingly stressed the formal aspects. In this context, Euler offered a unitary interpretation of the complex of results concerning series, which even allowed the acceptance of those findings that did not form part of the early theory. A series was thought to be the result of a formal transformation of an analytical quantity expressed in a closed form. This transformation was sufficient to give a meaning to the series, even when the latter was not convergent. However mathematicians were not free to invent transformations by a free creative act. They limited themselves to using the same transformations that were used in the original theory or at least were compatible with it. This seemed to guarantee that the new more formal conception was a generalization of the earlier conception, which remained the essential basis from which all the parts of the series theory were subsequently generated. The more formal Eulerian approach was widely predominant during the second part of the 18th-century for two main reasons. Firstly, mathematicians who were critical of it were not able to eliminate the formal aspects of the early concept and found a really new theory: they always used the formal methodology that had led to asymptotic series and to the combinatorial use of series. Secondly, the formal concept of series contributed to the growth of mathematics. It led to many new discoveries and even to a new branch of analysis: the calculus of operations. The Eulerian approach became unsuited to most advanced mathematical research towards the end of the 18th century and the beginning of the 19th century. Applied mathematics encouraged investigations and introduction of new functions in analysis, but formal methodology was unable to treat quantities which were not elementary quantities and series which were not power series. The need to use trigonometric series to enable the analytical investigation of heat led Fourier to reject the formal concept of series and assert an entirely quantitative notion of series. The need to introduce hypergeometric and gamma functions into analysis and to have an adequate analytical theory of them forced Gauss to highlight the quantitative meaning of the sum of series and to reject formal manipulations. The new approach based only upon convergence was the basis of Cauchy's treatises. Given the purposes of this book, I cannot avoid dealing with some topics which are closely connected to series theory and are crucial to an understanding of its historical evolution: not only do these include other infinite processes (continued fractions and infinite product) but also certain basic mathematical notions (quantity, numbers, functions) and the eighteenth century concept of analysis. This book is divided into four parts. The first part starts with a chapter devoted to the use of series prior to the rise of the calculus (chapter 1), where I deal principally with Viète, Grégorie de Saint-Vincent, Mengoli, Wallis, and Gregory. I then move on to investigate the conception of the founders of the calculus (Leibniz in chapter 2; Newton in chapter 4). On the basis of this examination, and after discussing the contributions of Johan and Jacob Bernoulli (chapter 3 and 5) and the notion of a quantity and of a number (chapter 6), I offer an interpretative scheme of the early theory of series in chapter 7. The first part also includes the appearance of Taylor series in Newton and Taylor (chapter 8) and the rise of the problem of the sum of a divergent series in one of Grandi's writings and the ensuing debate in Leibniz, Varignon, Daniel Bernoulli and Goldbach (chapter 9).