Contents -- Acknowledgements -- Introduction -- 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 -- People, institutions, and journals -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W.