Cover -- Copyright page -- Title page -- Contents -- Introduction -- I The Calculus as Algebra: J.-L. Lagrange, 1736-1813 -- Preface to the Garland Edition -- Acknowledgements -- Introduction -- 1 The Development of Lagrange's Ideas on the Calculus: 1754-1797 -- General Sketch of the Development of Lagrange's Ideas -- Infinite Series and the Calculus: From 1754 up to 1772 -- Infinite Series and the Calculus: The 1772 Paper -- The Period of Indecision: The Berlin Prize -- The Period of Indecision: Arbogast's Sketch of a Solution -- Conclusion -- 2 The Algebraic Background of the Theory of Analytic Functions -- Introduction -- The Attractiveness of Algebra: Certainty -- The Attractiveness of Algebra: Methods -- The Algebraic Character of the Taylor Series -- Origins of the "Proof " that Every Function has a Taylor Series -- The Algebraic Background of the Lagrange Remainder: Approximations -- The Algebraic Background of the Lagrange Remainder: Error-EstimatesBefore Lagrange -- The Algebraic Background of the Lagrange Remainder: Lagrange and Bounds on Error -- 3 The Contents of the Fonctions Analytiques -- Introduction -- Lagrange's Critique of Earlier Methods -- The Results of the Calculus: By Means of Formal Power Series -- The Results of the Calculus: Those Needed to Derive the Remainder Term of the Taylor Series -- Results of the Calculus: Derivation of the Remainder Term -- Results of the Calculus: Application of the Remainder Term -- Refinements of Lagrange's Ideas: 1799-1813 -- Impact of the Fonctions Analytiques -- Conclusion -- 4 From Proof-Technique to Definition: The Pre-History of Delta-Epsilon Methods -- Introduction -- "Errours, tho' never so small, are not to be neglected in Mathematicks": Lagrange and the Lagrange Remainder -- Ampère's Forgotten Contribution: A New Definition -- Cauchy: Prolegomena to Any Future Mathematics.

Conclusion -- Appendix -- Bibliography -- Analytical Bibliography: 1966 -- I. Lagrange -- II. Articles about Lagrange. -- III. Works on the Calculus of a Lagrangian Tendency. -- IV. History of Mathematics -- V. General Background Works -- VI. Algebra -- VII. The Calculus -- Bibliography: 1966-Present -- II Selected Writings -- 1 The Mathematician, the Historian, and the History of Mathematics -- 2 Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus -- The Practice of Analysis: From Newton to Euler -- The Algebra of Inequalities -- Changing Attitudes Toward Rigor -- The Concepts of the Calculus -- Conclusion -- References -- 3 The Changing Concept of Change: The Derivative from Fermat to Weierstrass -- The Seventeenth-Century Background -- Finding Maxima, Minima, and Tangents -- Tangents, Areas, and Rates of Change -- Differential Equations, Taylor Series, and Functions -- Lagrange and the Derivative as a Function -- Definitions, Rigor, and Proofs -- Historical Development Versus Textbook Exposition -- References -- 4 The Centrality of Mathematics in the History of Western Thought -- 1. Introduction -- 2. Certainty -- 3. Applicability -- 4. More Than One Geometry? -- 5. Opposition -- 6. Conclusion -- References -- 5 Descartes and Problem-Solving -- Introduction -- A First Look at Descartes' Geometry -- The Background of Descartes' Geometry -- Descartes' Method in Action -- Beyond the Greeks -- The Power of Descartes' Methods: Tangents and Equations -- Conclusion -- References -- 6 The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and their Legacy -- Prelude: The Ways Mathematicians Think -- Introduction to the Geometric and Algebraic Approaches -- Maclaurin -- Lagrange -- Why the Difference? -- Influence -- Conclusion.

7 Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions -- 1. Introduction -- 2. The Standard Picture -- 3. The Nature of Maclaurin's Treatise of Fluxions -- 4. The Social Context: The Scottish Enlightenment -- 5. Maclaurin's Continental Reputation -- 6. Maclaurin's Mathematics and Its Importance -- a. Key Methods in the Calculus -- b. Ellipsoids -- c. The Euler-Maclaurin Formula -- d. Elliptic Integrals -- 7. Other Examples of Maclaurin's Mathematical Influence -- 8. Why a Treatise of Fluxions? -- 9. Why the Traditional View? -- 10. Some Final Reflections -- References -- 8 Newton, Maclaurin, and the Authority of Mathematics -- 1. Introduction: Maclaurin, the Scottish Enlightenment, and the "Newtonian Style" -- 2. What is the "Newtonian Style"? -- 3. Maclaurin's First Use of the Newtonian Style -- 4. Religion, Authority, and Mathematics for Newton and Maclaurin -- 5. Maclaurin's Mature Use of the Newtonian Style -- The Shape of the Earth -- "Gauging": Finding the Volumes of Barrels -- Social Agreement and Scientific Authority -- Actuarial Science -- "Method" and Authority -- 6. Religious Authority Revisited -- 7. Conclusion -- References -- 9 Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths -- A. Myth: The social history of mathematics is easy -- just determine what nation or group your mathematician comes from and generalize. -- B. Second Myth: All Modern Mathematics Comes from Men, Mostly White Christian Men in the Graeco-European Tradition -- C. Third Myth: There Was No Real Mathematics in the European Middle Ages. After the Decline of Greek Mathematics, Nothing Much Happened Mathematically in Europe Until the Renaissance -- D. Fourth Myth: Newton Invented the Calculus Just to Do His Mathematical Physics.

E. Fifth Myth: Colin Maclaurin, Because of His Old-Fashioned Geometrical Approach to the Calculus, Halted Mathematical Progress in 18th-Century Britain -- F. Sixth Myth: Lagrange was a Formalist. He Tried to Rigorize the Calculus, But Failed Because of His Unreflective Reliance on Formal Power Series -- G. Last Myth, Held by a Number of Past and Present Mathematicians: The Mathematical Approach Can be Applied to Solve Almost Any Major Question -- H. A conclusion in four parts -- 10 Why Did Lagrange "Prove" the Parallel Postulate? -- 1. Introduction -- 2. The Contents of Lagrange's 1806 Paper -- 3. Why Did He Attack the Problem This Way? -- 4. The Crucial Argument: Newtonian Physics -- 5. The Argument from Eighteenth-Century Mathematics and Science -- 6. Why Did It Matter so Much? -- 7. Conclusion -- References -- Index -- About the Author.