Mathematical thought from ancient to modern times. v. 3 / Morris Kline.

Includes bibliographical references.

1.1\x Mathematics in Mesopotamia : Where did mathematics begin? ; Political history in Mesopotamia ; The number symbols ; Arithmetic operations ; Babylonian algebra ; Babylonian geometry ; The uses of mathematics in Babylonia ; Evaluation of Babylonian mathematics -- Egyptian mathematics : Background ; The arithmetic ; Algebra and geometry ; Egyptian uses of mathematics ; Summary -- The creation of classical Greek mathematics : Background ; The general sources ; The major schools of the classical period ; The Ionian school ; The Pythagoreans ; The Eleatic school ; The Sophist school ; The Platonic school ; The school of Eudoxus ; Aristotle and his school -- Euclid and Apollonius : Introduction ; The background of Euclid's Elements ; The definitions and axioms of the Elements ; Books I to IV of the Elements ; Book V: the theory of proportion ; Book VI: similar figures ; Books VII, VIII, and IX: the theory of numbers ; Book X: the classification of incommensurables ; Books XI, XII, and XIII: solid geometry and the method of exhaustion ; The merits and defects of the Elements ; Other mathematical works by Euclid ; The mathematical work of Apollonius -- The Alexandrian Greek period: geometry and trigonometry : The founding of Alexandria ; The character of Alexandrian Greek mathematics ; Areas and volumes in the work of Archimedes ; Areas and volumes in the work of Heron ; Some exceptional curves ; The creation of trigonometry ; Late Alexandrian activity in geometry -- The Alexandrian period: the reemergence of arithmetic and algebra : The symbols and operations of Greek arithmetic ; Arithmetic and algebra as an independent development -- The Greek rationalization of nature : The inspiration for Greek mathematics ; The beginning of a rational view of nature ; The development of the belief in mathematical design ; Greek mathematical astronomy ; Geography ; Mechanics ; Optics ; Astrology -- The demise of the Greek world : A review of the Greek achievements ; The limitations of Greek mathematics ; The problems bequeathed by the Greeks ; The demise of the Greek civilization -- The mathematics of the Hindus and Arabs : Early Hindu mathematics ; Hindu arithmetic and algebra of the period A.D. 200-1200 ; Hindu geometry and trigonometry of the period A.D. 200-1200 ; The Arabs ; Arabic arithmetic and algebra ; Arabic geometry and trigonometry ; Mathematics circa 1300 -- The medieval period in Europe : The beginnings of a European civilization ; The materials available for learning ; The role of mathematics in early medieval Europe ; The stagnation in mathematics ; The first revival of the Greek works ; The revival of rationalism and interest in nature ; Progress in mathematics proper ; Progress in physical science ; Summary -- The Renaissance : Revolutionary influences in Europe ; The new intellectual outlook ; The spread of learning ; Humanistic activity in mathematics ; The clamor for the reform of science ; The rise of Empiricism -- Mathematical contributions in the Renaissance : Perspective ; Geometry proper ; Algebra ; Trigonometry ; The major scientific progress in the Renaissance ; Remarks on the Renaissance -- Arithmetic and algebra in the sixteenth and seventeenth centuries : Introduction ; The status of the number system and arithmetic ; Symbolism ; The solution of third and fourth degree equations ; The theory of equations ; The binominal theorem and allied topics ; The theory of numbers ; The relationship of algebra to geometry -- The beginning of projective geometry : The rebirth of geometry ; The problems raised by the work on perspective ; The work of Desargues ; The work of Pascal and La Hire ; The emergence of new principles -- Coordinate geometry : The motivation for coordinate geometry ; The coordinate geometry of Fermat ; Rene Descartes ; Descartes's work in coordinate geometry ; Seventeenth-century extensions of coordinate geometry ; The importance of coordinate geometry -- The mathematization of science : Introduction ; Descartes's concept of science ; Galileo's approach to science ; The function concept -- The creation of the calculus : The motivation for the calculus ; Early seventeenth-century work on the calculus ; The work of Newton ; The work of Leibniz ; A comparison of the work of Newton and Leibniz ; The controversy over priority ; Some immediate additions to the calculus ; The soundness of the calculus -- Mathematics as of 1700 : The transformation of mathematics ; Mathematics and science ; Communication among mathematicians ; The prospects for the eighteenth century -- Calculus in the eighteenth century : Introduction ; The function concept ; The technique of integration and complex quantities ; Elliptic integrals ; Further special functions ; The calculus of functions of several variables ; The attempts to supply rigor in the calculus -- Infinite series : Introduction ; Initial work on infinite series ; The expansion of functions ; The manipulation of series ; Trigonometric series ; Continued fractions ; The problem of convergence and divergence -- Ordinary differential equations in the eighteenth century : Motivations ; First order ordinary differential equations ; Singular solutions ; Second order equations and the Riccati equations ; Higher order equations ; The method of series ; Systems of differential equations ; Summary -- Partial differential equations in the eighteenth century : Introduction ; The wave equation ; Extensions of the wave equation ; Potential theory ; First order partial differential equations ; Monge and the theory of characteristics ; Monge and nonlinear second order equations ; Systems of first order partial differential equations ; The rise of the mathematical subject -- Analytic and differential geometry in the eighteenth century : Introduction ; Basic analytical geometry ; Higher plane curves ; The beginnings of differential geometry ; Plane curves ; Space curves ; The theory of surfaces ; The mapping problem -- The calculus of variations in the eighteenth century : The initial problems ; The early work of Euler ; The principle of least action ; The methodology of Lagrange ; Lagrange and least action ; The second variation -- Algebra in the eighteenth century : Status of the number system ; The theory of equations ; Determinants and elimination theory ; The theory of numbers -- Mathematics as of 1800 : The rise of analysis ; The motivation for the eighteenth-century work ; The problem of proof ; The metaphysical basis ; The expansion of mathematical activity ; A glance ahead -- Functions of a complex variable : Introduction ; the beginnings of complex function theory ; The geometrical representation of complex numbers ; The foundation of complex function theory ; Weierstrass's approach to function theory ; Elliptic functions ; Hyperelliptic integrals and Abel's theorem ; Riemann and multiple-valued functions ; Abelian integrals and functions ; Conformal mapping ; The representation of functions and exceptional values -- Partial differential equations in the nineteenth century : Introduction ; The head equation and Fourier series ; Closed solutions; the Fourier integral ; The potential equation and Green's theorem ; Curvilinear coordinates ; The wave equation and the reduced wave equation ; Systems of partial differential equations ; Existence theorems -- Ordinary differential equations in the nineteenth century : Introduction ; Series solutions and special functions ; Sturm-Liouville theory ; Existence theorems ; The theory of singularities ; Automorphic functions ; Hill's work on periodic solutions of linear equations ; Nonlinear differential equations: the qualitative theory -- The calculus of variations in the nineteenth century : Introduction ; Mathematical physics and the calculus of variations ; Mathematical extensions of the calculus of variations proper ; Related problems in the calculus of variations -- Galois theory : Introduction ; Binominal equations ; Abel's work on the solution of equations by radicals ; Galois's theory of solvability ; The geometric construction problems ; The theory of substitution groups -- Quaternions, vectors, and linear associative algebras : The foundation of algebra on permanence of form ; The search for a three-dimensional "complex number" ; The nature of quaternions ; Grassman's calculus of extension ; From quaternions to vectors ; Linear associative algebras -- Determinants and matrices : Introduction ; Some new uses of determinants ;

Determinants and quadratic forms ; Matrices -- The theory of numbers in the nineteenth century : Introduction ; The theory of congruences ; Algebraic numbers ; The ideals of Dedekind ; The theory of forms ; Analytic number theory -- The revival of projective geometry : The renewal of interest in geometry ; Synthetic Euclidean geometry ; The revival of synthetic projective geometry ; Algebraic projective geometry ; Higher plane curves and surfaces -- Non-Euclidean geometry : Introduction ; The status of Euclidean geometry about 1800 ; The research on the parallel axiom ; Foreshadowings of non-Euclidean geometry ; The creation of non-Euclidean geometry ; The technical content of non-Euclidean geometry ; The claims of Lobatchevsky and Bolyai to priority ; The implications of non-Euclidean geometry -- The differential geometry of Gauss and Riemann : Introduction ; Gauss's differential geometry ; Riemann's approach to geometry ; The successors of Riemann ; Invariants of differential forms -- Projective and metric geometry : Introduction ; Surfaces as models of non-Euclidean geometry ; Projective and metric geometry ; Models and the consistency problem ; Geometry from the transformation viewpoint ; The reality of non-Euclidean geometry -- Algebraic geometry : Background ; The theory of algebraic invariants ; The concept of birational transformations ; The function-theoretic approach to algebraic geometry ; The uniformization problem ; The algebraic-geometric approach ; The arithmetic appr

1.2\x oach ; The algebraic geometry of surfaces -- The instillation of rigor in analysis : Introduction ; Functions and their properties ; The derivative ; The integral ; Infinite series ; Fourier series ; The status of analysis -- The foundations of the real and transfinite numbers : Introduction ; Algebraic and transcendental numbers ; The theory of irrational numbers ; The theory of rational numbers ; Other approaches to the real number system ; The concept of an infinite set ; The foundation of the theory of sets ; Transfinite cardinals and ordinals ; The status of set theory by 1900 -- The foundations of geometry : The defects in Euclid ; Contributions to the foundations of projective geometry ; The foundations of Euclidean geometry ; Some related foundational work ; Some open questions -- Mathematics as of 1900 : The chief features of the nineteenth-century developments ; The axiomatic movement ; Mathematics as man's creation ; The loss of truth ; Mathematics as the study of arbitrary structures ; The problem of consistency ; A glance ahead -- The theory of functions of real variables : The origins ; The Stieltjes integral ; Early work on content and measure ; The Lebesgue integral ; Generalizations -- Integral equations : Introduction ; The beginning of a general theory ; The work of Hilbert ; The immediate successors of Hilbert ; Extensions of the theory -- Functional analysis : The nature of functional analysis ; The theory of functionals ; Linear functional analysis ; The axiomatization of Hilbert space -- Divergent series : Introduction ; The informal uses of divergent series ; The formal theory of asymptotic series ; Summability -- Tensor analysis and differential geometry : The origins of tensor analysis ; The notion of a tensor ; Covariant differentiations ; Parallel displacement ; Generalizations of Riemannian geometry -- The emergence of abstract algebra : The nineteenth-century background ; Abstract group theory ; The abstract theory of fields ; Rings ; Non-associative algebras ; The range of abstract algebra -- The beginnings of topology : The nature of topology ; Point set topology ; The beginnings of combinatorial topology ; The combinatorial work of Poincare ; Combinatorial invariants ; Fixed point theorems ; Generalizations and extensions -- The foundations of mathematics : Introduction ; The paradoxes of set theory ; The axiomatization of set theory ; The rise of mathematical logic ; The logistic school ; The intuitionist school ; The formalist school some recent developments.

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