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

The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Most of the results derived from this time were derived using methods which would be found unacceptable today, and as a result, when one looks 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 rigor 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.

This monograph not only describes the entire complex of 17th and 18th century procedures and results concerning series, but it also reconstructs the implicit and explicit principles upon which they are based, draws attention to the underlying philosophy, highlights competing approaches, and investigates the mathematical context where the theory originated. The aim here 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 sense, to a unified theory that has come down to us today.

Giovanni Ferraro is Professor of Mathematics and History of Mathematics at University of Molise.

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The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Most of the results derived from this time were derived using methods which would be found unacceptable today, and as a result, when one looks 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 rigor 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.

 

This monograph not only describes the entire complex of 17th and 18th century procedures and results concerning series, but it also reconstructs the implicit and explicit principles upon which they are based, draws attention to the underlying philosophy, highlights competing approaches, and investigates the mathematical context where the theory originated. The aim here 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 sense, to a unified theory that has come down to us today.

 

Giovanni Ferraro is Professor of Mathematics and History of Mathematics at University of Molise.

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The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Most of the results derived from this time were derived using methods which would be found unacceptable today, and as a result, when one looks 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 rigor 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.

 

This monograph not only describes the entire complex of 17th and 18th century procedures and results concerning series, but it also reconstructs the implicit and explicit principles upon which they are based, draws attention to the underlying philosophy, highlights competing approaches, and investigates the mathematical context where the theory originated. The aim here 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 sense, to a unified theory that has come down to us today.

 

Giovanni Ferraro is Professor of Mathematics and History of Mathematics at University of Molise.

Content:
Front Matter....Pages I-XV
Front Matter....Pages 1-2
Series before the rise of the calculus....Pages 3-24
Geometrical quantities and series in Leibniz....Pages 25-44
The Bernoulli series and Leibniz’s analogy....Pages 45-51
Newton’s method of series....Pages 53-78
Jacob Bernoulli’s treatise on series....Pages 79-85
The Taylor series....Pages 87-92
Quantities and their representations....Pages 93-113
The formal-quantitative theory of series....Pages 115-120
The first appearance of divergent series....Pages 121-130
Front Matter....Pages 131-132
De Moivre’s recurrent series and Bernoulli’s method....Pages 133-140
Acceleration of series and Stirling’s series....Pages 141-146
Maclaurin’s contribution....Pages 147-153
The young Euler between innovation and tradition....Pages 155-169
Euler’s derivation of the Euler–Maclaurin summation formula....Pages 171-179
On the sum of an asymptotic series....Pages 181-184
Infinite products and continued fractions....Pages 185-192
Series and number theory....Pages 193-199
Analysis after the 1740s....Pages 201-214
The formal concept of series....Pages 215-229
Front Matter....Pages 231-231
Lagrange inversion theorem....Pages 233-237
Front Matter....Pages 231-231
Toward the calculus of operations....Pages 239-244
Laplace’s calculus of generating functions....Pages 245-250
The problem of analytical representation of nonelementary quantities....Pages 251-256
Inexplicable functions....Pages 257-262
Integration and functions....Pages 263-265
Series and differential equations....Pages 267-274
Trigonometric series....Pages 275-282
Further developments of the formal theory of series....Pages 283-295
Attempts to introduce new transcendental functions....Pages 297-301
D’Alembert and Lagrange and the inequality technique....Pages 303-309
Front Matter....Pages 311-313
Fourier and Fourier series....Pages 315-322
Gauss and the hypergeometric series....Pages 323-345
Cauchy’s rejection of the 18th-century theory of series....Pages 347-362
Back Matter....Pages 363-389

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The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Most of the results derived from this time were derived using methods which would be found unacceptable today, and as a result, when one looks 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 rigor 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.

 

This monograph not only describes the entire complex of 17th and 18th century procedures and results concerning series, but it also reconstructs the implicit and explicit principles upon which they are based, draws attention to the underlying philosophy, highlights competing approaches, and investigates the mathematical context where the theory originated. The aim here 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 sense, to a unified theory that has come down to us today.

 

Giovanni Ferraro is Professor of Mathematics and History of Mathematics at University of Molise.

Content:
Front Matter....Pages I-XV
Front Matter....Pages 1-2
Series before the rise of the calculus....Pages 3-24
Geometrical quantities and series in Leibniz....Pages 25-44
The Bernoulli series and Leibniz’s analogy....Pages 45-51
Newton’s method of series....Pages 53-78
Jacob Bernoulli’s treatise on series....Pages 79-85
The Taylor series....Pages 87-92
Quantities and their representations....Pages 93-113
The formal-quantitative theory of series....Pages 115-120
The first appearance of divergent series....Pages 121-130
Front Matter....Pages 131-132
De Moivre’s recurrent series and Bernoulli’s method....Pages 133-140
Acceleration of series and Stirling’s series....Pages 141-146
Maclaurin’s contribution....Pages 147-153
The young Euler between innovation and tradition....Pages 155-169
Euler’s derivation of the Euler–Maclaurin summation formula....Pages 171-179
On the sum of an asymptotic series....Pages 181-184
Infinite products and continued fractions....Pages 185-192
Series and number theory....Pages 193-199
Analysis after the 1740s....Pages 201-214
The formal concept of series....Pages 215-229
Front Matter....Pages 231-231
Lagrange inversion theorem....Pages 233-237
Front Matter....Pages 231-231
Toward the calculus of operations....Pages 239-244
Laplace’s calculus of generating functions....Pages 245-250
The problem of analytical representation of nonelementary quantities....Pages 251-256
Inexplicable functions....Pages 257-262
Integration and functions....Pages 263-265
Series and differential equations....Pages 267-274
Trigonometric series....Pages 275-282
Further developments of the formal theory of series....Pages 283-295
Attempts to introduce new transcendental functions....Pages 297-301
D’Alembert and Lagrange and the inequality technique....Pages 303-309
Front Matter....Pages 311-313
Fourier and Fourier series....Pages 315-322
Gauss and the hypergeometric series....Pages 323-345
Cauchy’s rejection of the 18th-century theory of series....Pages 347-362
Back Matter....Pages 363-389
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