xiv CONTENTS 24.The Calculus of Variations in the Eighteenth Century,573 Least Action,579 5 Lagrange and Least Action,587 6The Second Variation,5 25.Algebra in the Eighteenth Century,592 I Status of the Number System.592 2.The Theory of Equations.597 3 Determinants and Elimination Theory,606 4 The 'Theory of Numbers,608 26.Mathematics as of 1800,614 1.The Rise of Analysis,614 2.The Motivation for the Eighteenth-Century Work, 616 3 The Problem of Proof,617 4 The Metaphysical Basis,619 5.Ihe Expansion of Mathematical Activity,621 6 A Glance Ahead,623 27.Functions of a Complex Variable,626 1.Introduction.626 2.The Beginnings of Complex Function 'Theory,626 3.The Gcometrical Representation of Complex Numbers,628 4 The Foundation of Complex FunctionTheory,632 5 Weierstrass's Approach to Function Theory,642 6 Elliptic Functions,644 7.Hyperelliptic Integrals and Abel's'Theorem,651 8 Riemann and Multiple-Valued Functions,655 9.Abelian Integrals and Functions,663 10 Conformal Mapping,666 11 The Representation of Functions and Excepional Values,667 28.Partial Differential Equations in the Nineteenth Century,671 1.Introduction,671 2.The Heat Equation and Fourier Series,671 3 Closed Solutions,the Fouricr Integral,679 4.The Potential Equation and Grcen's 'Theorem, 681 5 Curvilinear Coordinatcs,687 6 The Wave Equation and the Reduced Wave Equation,690 7 Systems of Partial Differential Equations,696 8 Existence Theorems,699 29.Ordinary Differential Equations in the Nineteenth Century,709 ccial Functions,709 3.Sturm- laritie unctions. Equations,730 onlinear Differential Equations The Qualitative Theory,732 30.The Calculus of Variations in the Nineteenth Century,739 of Va Variations Proper.745 31.Galois Theory,752 1.Introduction.752 2.Binomial E ork on the Solution o y Radic als,7544 quations.752.Abel's W onstruction Problems,763 6 The bstitution (;roups,764