The rise and development of the theory of series up to the early 1820s / Giovanni Ferraro.

Format:

Book

Author/Creator:

Ferraro, Giovanni, 1957-

Series:

Sources and studies in the history of mathematics and physical sciences

Language:

English

Subjects (All):

Series--History--17th century.

Series.

Series--History--18th century.

Mathematics--History--17th century.

Mathematics.

Mathematics--History--18th century.

History.

Physical Description:

xv, 389 pages : illustrations ; 25 cm.

Place of Publication:

New York, NY : Springer, [2008]

Summary:

The theory of series in the 17th and 18th centuries poses several interesting problems to historians. Most of the results of 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.

Contents:

I From the beginnings of the 17th century to about 1720: Convergence and formal manipulation 1

1 Series before the rise of the calculus 3

2 Geometrical quantities and series in Leibniz 25

2.1 The capacity of series to express quantities and their manipulation 25

2.2 Power series 36

3 The Bernoulli series and Leibniz's analogy 45

4 Newton's method of series 53

4.1 The expansion of quantities into convergent series 54

4.2 On Newton's manipulations of power series 67

5 Jacob Bernoulli's treatise on series 79

6 The Taylor series 87

7 Quantities and their representations 93

7.1 Quantity and abstract quantity 93

7.2 Continuous quantities, numbers and fictitious quantities 100

8 The formal-quantitative theory of series 115

9 The first appearance of divergent series 121

II From the 1720s to the 1760s: The development of a more formal conception 131

10 De Moivre's recurrent series and Bernoulli's method 133

11 Acceleration of series and Stirling's series 141

12 Maclaurin's contribution 147

13 The young Euler between innovation and tradition 155

13.1 The search for the general term 155

13.2 Analytical and synthetical methods in series theory 160

13.3 The manipulation of the harmonic series and infinite equations 165

14 Euler's derivation of the Euler-Maclaurin summation formula 171

15 On the sum of an asymptotic series 181

16 Infinite products and continued fractions 185

17 Series and number theory 193

18 Analysis after the 1740s 201

18.1 Eighteenth-century analysis as nonfigural and symbolic investigation of the real 201

18.2 Functions, relations, and analytical expressions 205

18.3 On the continuity of curves and functions 211

19 The formal concept of series 215

19.1 Criticisms to the infinite extension of finite rules 215

19.2 The impossibility of the quantitative approach 219

19.3 Euler's definition of the sum 222

III The theory of series after 1760: Successes and problems of the triumphant formalism 231

20 Lagrange inversion theorem 233

21 Toward the calculus of operations 239

22 Laplace's calculus of generating functions 245

23 The problem of analytical representation of nonelementary quantities 251

24 Inexplicable functions 257

25 Integration and functions 263

26 Series and differential equations 267

27 Trigonometric series 275

28 Further developments of the formal theory of series 283

29 Attempts to introduce new transcendental functions 297

30 D'Alembert and Lagrange and the inequality technique 303

IV The decline of the formal theory of series 311

31 Fourier and Fourier series 315

32 Gauss and the hypergeometric series 323

33 Cauchy's rejection of the 18th-century theory of series 347.

Notes:

Includes bibliographical references (pages 363-382) and indexes.

ISBN:

9780387734675

0387734678

OCLC:

166357938

Online:

Publisher description