Mathematics emerging a sourcebook 1540 1900 1st Edition by Jacqueline Stedall 1435697855 9781435697850
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ISBN 10: 1435697855
ISBN 13: 9781435697850
Author: Jacqueline Stedall
Aimed at students and researchers in Mathematics, History of Mathematics and Science, this book examines the development of mathematics from the late 16th Century to the end of the 19th Century. Mathematics has an amazingly long and rich history, it has been practised in every society and culture, with written records reaching back in some cases as far as four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries.This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary mathematics.
Table of contents:
1 BEGINNINGS
1.1 Beginnings of arithmetic
1.1.1 Large number calculations, c. 1800 BC
1.1.2 Sacrobosco’s Algorismus, c. 1230 AD
1.2 Beginnings of geometry
1.2.1 Euclid’s definitions, c. 250 BC
1.2.2 Euclid’s construction of proportionals, c. 250 BC
1.2.3 Archimedes on circle measurement, c. 250 BC
1.2.4 Apollonius’ Conics, c. 185 BC
1.3 Beginnings of a theory of numbers
1.3.1 Euclid’s definitions of number, c. 250 BC
1.3.2 Euclid’s proof of the infinity of primes, c. 250 BC
1.3.3 The Arithmetica of Diophantus, (after 150 AD)
1.4 Beginnings of algebra
1.4.1 Completing the square, c. 1800 BC
1.4.2 Al-Khwārizmī’s Al-jabr, c. 825 AD
2 FRESH IDEAS
2.1 Improvements in calculation
2.1.1 Stevin’s decimal fractions, 1585
2.1.2 Napier’s logarithms, 1614
2.2 Improvements in notation
2.2.1 Harriot’s notation, c. 1600
2.2.2 Descartes’ notation, 1637
2.3 Analytic geometry
2.3.1 Viète’s introduction to the analytic art, 1591
2.3.2 Fermat and analytic geometry, 1636
2.3.3 Descartes and analytic geometry, 1637
2.4 Indivisibles
2.4.1 Cavalieri’s theory of indivisibles, 1635
2.4.2 Wallis and Hobbes on indivisibles, 1656
3 FORESHADOWINGS OF CALCULUS
3.1 Methods for tangents
3.1.1 Fermat’s tangent method, 1629
3.1.2 Descartes’ tangent method, 1637
3.2 Methods of quadrature
3.2.1 Fermat’s quadrature of higher hyperbolas, early 1640s
3.2.2 Brouncker and the rectangular hyperbola, c. 1655
3.2.3 Wallis’ use of indivisibles, 1656
3.2.4 Mercator and the rectangular hyperbola, 1668
3.3 A method of cubature
3.3.1 Torricelli’s infinite solid, 1644
3.4 A method of rectification
3.4.1 Neile and the semicubical parabola, 1657
4 THE CALCULUS OF NEWTON AND OF LEIBNIZ
4.1 The calculus of Newton
4.1.1 The chronology of Newton’s calculus
4.1.2 Newton’s treatise on fluxions and series, 1671
4.1.3 Newton’s first publication of his calculus, 1704
4.2 The calculus of Leibniz
4.2.1 Leibniz’s first publication of his calculus, 1684
5 THE MATHEMATICS OF NATURE: NEWTON’S PRINCIPIA
5.1 Newton’s Principia, Book I
5.1.1 The axioms
5.1.2 Ultimate ratios
5.1.3 Properties of small angles
5.1.4 Motion under a centripetal force
5.1.5 Quantitative measures of centripetal force
5.1.6 The inverse square law for a parabola
6 EARLY NUMBER THEORY
6.1 Perfect numbers
6.1.1 Euclid’s theorem on perfect numbers, c. 250 BC
6.1.2 Mersenne primes, 1644
6.1.3 Fermat’s little theorem, 1640
6.2 ‘Pell’s’ equation
6.2.1 Fermat’s challenge and Brouncker’s response, 1657
6.3 Fermat’s final challenge
7 EARLY PROBABILITY
7.1 The mathematics of gambling
7.1.1 Pascal’s correspondence with Fermat, 1654
7.1.2 Jacob Bernoulli’s Ars conjectandi, 1713
7.1.3 De Moivre’s calculation of confidence, 1738
7.2 Mathematical probability theory
7.2.1 Bayes’ theorem, 1763
7.2.2 Laplace and an application of probability, 1812
8 POWER SERIES
8.1 Discoveries of power series
8.1.1 Newton and the general binomial theorem, 1664–1665
8.1.2 Newton’s ‘De analysi’, 1669
8.1.3 Newton’s letters to Leibniz, 1676
8.1.4 Gregory’s binomial expansion, 1670
8.2 Taylor series
8.2.1 Taylor’s increment method, 1715
8.2.2 Maclaurin’s series, 1742
8.2.3 Functions as infinite series, 1748
8.3 Convergence of series
8.3.1 D’Alembert’s ratio test, 1761
8.3.2 Lagrange and the remainder term, 1797
8.4 Fourier series
8.4.1 Fourier’s derivation of his coefficients, 1822
9 FUNCTIONS
9.1 Early definitions of functions
9.1.1 Johann Bernoulli’s definition of function, 1718
9.1.2 Euler’s definition of a function (1), 1748
9.1.3 Euler’s definition of a function (2), 1755
9.2 Logarithmic and circular functions
9.2.1 A new definition of logarithms, 1748
9.2.2 Series for sine and cosine, 1748
9.2.3 Euler’s unification of elementary functions, 1748
9.3 Nineteenth-century definitions of function
9.3.1 A definition from Dedekind, 1888
10 MAKING CALCULUS WORK
10.1 Uses of calculus
10.1.1 Jacob Bernoulli’s curve of uniform descent, 1690
10.1.2 D’Alembert and the wave equation, 1747
10.2 Foundations of the calculus
10.2.1 Berkeley and The analyst, 1734
10.2.2 Maclaurin’s response to Berkeley, 1742
10.2.3 Euler and infinitely small quantities, 1755
10.2.4 Lagrange’s attempt to avoid the infinitely small, 1797
11 LIMITS AND CONTINUITY
11.1 Limits
11.1.1 Wallis’s ‘less than any assignable’, 1656
11.1.2 Newton’s first and last ratios, 1687
11.1.3 Maclaurin’s definition of a limit, 1742
11.1.4 D’Alembert’s definition of a limit, 1765
11.1.5 Cauchy’s definition of a limit, 1821
11.2 Continuity
11.2.1 Wallis and smooth curves, 1656
11.2.2 Euler’s definition of continuity, 1748
11.2.3 Lagrange’s arbitrarily small intervals, 1797
11.2.4 Bolzano’s definition of continuity, 1817
11.2.5 Cauchy’s definition of continuity, 1821
11.2.6 Cauchy and the intermediate value theorem, 1821
12 SOLVING EQUATIONS
12.1 Cubics and quartics
12.1.1 Cardano and the Ars magna, 1545
12.2 From Cardano to Lagrange
12.2.1 Harriot and the structure of polynomials, c. 1600
12.2.2 Hudde’s rule, 1657
12.2.3 Tschirnhaus transformations, 1683
12.2.4 Lagrange and reduced equations, 1771
12.3 Higher degree equations
12.3.1 Lagrange’s theorem, 1771
12.3.2 Aftermath: the unsolvability of quintics
13 GROUPS, FIELDS, IDEALS, AND RINGS
13.1 Groups
13.1.1 Cauchy’s early work on permutations, 1815
13.1.2 The Premier mémoire of Galois, 1831
13.1.3 Cauchy’s return to permutations, 1845
13.1.4 Cayley’s contribution to group theory, 1854
13.2 Fields, ideals, and rings
13.2.1 ‘Galois fields’, 1830
13.2.2 Kummer and ideal numbers, 1847
13.2.3 Dedekind on fields of finite degree, 1877
13.2.4 Dedekind’s definition of ideals, 1877
14 DERIVATIVES AND INTEGRALS
14.1 Derivatives
14.1.1 Landen’s algebraic principle, 1758
14.1.2 Lagrange’s derived functions, 1797
14.1.3 Ampère’s theory of derived functions, 1806
14.1.4 Cauchy on derived functions, 1823
14.1.5 The mean value theorem, and ε, δ notation, 1823
14.2 Integration of real-valued functions
14.2.1 Euler’s introduction to integration, 1768
14.2.2 Cauchy’s definite integral, 1823
14.2.3 Cauchy and the fundamental theorem of calculus, 1823
14.2.4 Riemann integration, 1854
14.2.5 Lebesgue integration, 1902
15 COMPLEX ANALYSIS
15.1 The Complex Plane
15.1.1 Wallis’s representations, 1685
15.1.2 Argand’s representation, 1806
15.2 Integration of complex functions
15.2.1 Johann Bernoulli’s transformations, 1702
15.2.2 Cauchy on definite complex integrals, 1814
15.2.3 The calculus of residues, 1826
15.2.4 Cauchy’s integral formulas, 1831
15.2.5 The Cauchy–Riemann equations, 1851
16 CONVERGENCE AND COMPLETENESS
16.1 Cauchy sequences
16.1.1 Bolzano and ‘Cauchy sequences’, 1817
16.1.2 Cauchy’s treatment of sequences and series, 1821
16.1.3 Abel’s proof of the binomial theorem, 1826
16.2 Uniform convergence
16.2.1 Cauchy’s erroneous theorem, 1821
16.2.2 Stokes and ‘infinitely slow’ convergence, 1847
16.3 Completeness of the real numbers
16.3.1 Bolzano and greatest lower bounds, 1817
16.3.2 Dedekind’s definition of real numbers, 1858
16.3.3 Cantor’s definition of real numbers, 1872
17 LINEAR ALGEBRA
17.1 Linear equations and determinants
17.1.1 An early European example, 1559
17.1.2 Rules for solving three or four equations, 1748
17.1.3 Vandermonde’s elimination theory, 1772
17.1.4 Cauchy’s definition of determinant, 1815
17.2 Eigenvalue problems
17.2.1 Euler’s quadratic surfaces, 1748
17.2.2 Laplace’s symmetric system, 1787
17.2.3 Cauchy’s theorems of 1829
17.3 Matrices
17.3.1 Gauss and linear transformations, 1801
17.3.2 Cayley’s theory of matrices, 1858
17.3.3 Frobenius and bilinear forms, 1878
17.4 Vectors and vector spaces
17.4.1 Grassmann and vector spaces, 1862
18 FOUNDATIONS
18.1 Foundations of geometry
18.1.1 Hilbert’s axiomatization of geometry, 1899
18.2 Foundations of arithmetic
18.2.1 Cantor’s countability proofs, 1874
18.2.2 Dedekind’s definition of natural numbers, 1888
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