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

1. Front Matter

   Pages I-XV

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2. From the beginnings of the 17th century to about 1720: Convergence and formal manipulation

   1. Front Matter

      Pages 1-2

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   2. Series before the rise of the calculus

      • Giovanni Ferraro

      Pages 3-24

   3. Geometrical quantities and series in Leibniz

      • Giovanni Ferraro

      Pages 25-44

   4. The Bernoulli series and Leibniz’s analogy

      • Giovanni Ferraro

      Pages 45-51

   5. Newton’s method of series

      • Giovanni Ferraro

      Pages 53-78

   6. Jacob Bernoulli’s treatise on series

      • Giovanni Ferraro

      Pages 79-85

   7. The Taylor series

      • Giovanni Ferraro

      Pages 87-92

   8. Quantities and their representations

      • Giovanni Ferraro

      Pages 93-113

   9. The formal-quantitative theory of series

      • Giovanni Ferraro

      Pages 115-120

   10. The first appearance of divergent series

       • Giovanni Ferraro

       Pages 121-130

3. From the 1720s to the 1760s: The development of a more formal conception

   1. Front Matter

      Pages 131-132

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   2. De Moivre’s recurrent series and Bernoulli’s method

      • Giovanni Ferraro

      Pages 133-140

   3. Acceleration of series and Stirling’s series

      • Giovanni Ferraro

      Pages 141-146

   4. Maclaurin’s contribution

      • Giovanni Ferraro

      Pages 147-153

   5. The young Euler between innovation and tradition

      • Giovanni Ferraro

      Pages 155-169

   6. Euler’s derivation of the Euler–Maclaurin summation formula

      • Giovanni Ferraro

      Pages 171-179

   7. On the sum of an asymptotic series

      • Giovanni Ferraro

      Pages 181-184

   8. Infinite products and continued fractions

      • Giovanni Ferraro

      Pages 185-192

   9. Series and number theory

      • Giovanni Ferraro

      Pages 193-199

The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Indeed, mathematicians of the time derived num- ous results that 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 di?erential equations, from theEuler–MaclaurinsummationformulatotheLagrangeinversiontheorem, from Laplace’s theory of generating functions to the calculus of operations, etc. Most of these results were, however, derived using methods that would be found unacceptable today, thus, if we look back to the theory of series priortoCauchywithoutreconstructinginternalmotivationsandtheconc- tual background, it appears as a corpus of manipulative techniques lacking in rigor whose results seem to be the puzzling fruit of the mind of a - gician or diviner rather than the penetrating and complex work of great mathematicians. For this reason, in this monograph, not only do I describe the entire complex of 17th- and 18th-century procedures and results concerning series, but also I reconstruct the implicit and explicit principles upon which they are based, draw attention to the underlying philosophy, highlight competing approaches, and investigate the mathematical context where the series t- ory originated. My aim is to improve the understanding of the framework of 17th- and 18th-century mathematics and avoid trivializing the complexity of historical development by bringing it into line with modern concepts and views and by tacitly assuming that certain results belong, in some unpr- lematic sense, to a uni?ed theory that has come down to us today.

• Fourier series
• Taylor series
• calculus
• differential equation
• number theory

From the reviews:

"Giovanni Ferraro’s book must be regarded as an important contribution to the history of mathematical analysis … . 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 organized. Therefore … 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, MathDL, April, 2008)

"The work is thoroughly referenced and should prove valuable to mathematical history researchers … . Summing Up: Recommended. Researchers/faculty and professionals." (C. Bauer, CHOICE, Vol. 46 (01), September, 2008)

"The author begins his detailed account with the arrival of the calculus with Newton and Leibniz, and their own treatments of series. … The range of series and functions treated is impressive … . The story stops with a brief notice of the start of the critical phase, especially concerning the convergence (or not) of infinite series, which was put forward by Cauchy and his followers … . The book is a valuable contribution, possibly the single most substantial source on the topic." (I. Grattan-Guinness, Mathematical Reviews, Issue 2008 k)

• Dipartimento STAT, Università del Molise, 86090 Pesche, Italy

  Giovanni Ferraro

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