A Treatise of Fluxions. In Two Books

MACLAURIN, Colin

Published by Printed by T.W. and T. Ruddimans, Edinburgh, 1742

About this Item

STOOD AS A MODEL OF RIGOR FOR ALMOST A CENTURY. First edition, a very fine large and thick paper copy, of "the earliest logical and systematic publication of the Newtonian methods. It stood as a model of rigor until the appearance of Cauchy s Cours d Analyse in 1821" (DSB). The Treatise was written partly (but only partly, see below) as a response to the attack on the foundations of the method of fluxions and infinitesimal calculus made by George Berkeley in The Analyst (1734). "[Berkeley] showed that many definitions in the infinitesimal calculus are paradoxical and cannot be justified by intuition. He explained the success of the new calculus by a repeated neglect of infinitely small quantities leading through a compensation of errors to a correct answer" (Jahnke, A History of Analysis, 127). "MacLaurin provided a rigorous foundation for the method of fluxions based on a limit concept drawn from Archimedean classical geometry. He went on to demonstrate that the method so founded would support the entire received structure of fluxions and the calculus, and to make advances that were taken up by continental analysts … The Treatise was generally cited by British fluxionists as the definitive answer to Berkeley s criticism, but Maclaurin had accomplished much more than this. Judith Grabiner [see below] has described Maclaurin s influence on the Continental analysts in detail. Maclaurin s work was cited with admiration by Lagrange, Euler, Clairaut, d Alembert, Laplace, Legendre, Lacroix, and Gauss. The influence of Maclaurin s use of the algebra of inequalities as a basis for his limit arguments can be seen in d Alembert, L Huilier, Lacroix and Cauchy. Maclaurin corresponded at length with Clairaut about the attraction of ellipsoids, and the latter in his La figure de la terre (1743) acknowledges his debt; Maclaurin s influence on this subject can be seen also in d Alembert, Laplace, Lagrange, Legendre, and Gauss … In addition, Maclaurin s use of infinite series in the analysis of functions, especially with the Euler Maclaurin formula, was known to Euler, Lagrange, and Jacobi; while his reduction of fluents to elliptic or hyperbolic curve length was used by d Alembert and extended by Euler, and Euler influenced Legendre s work on elliptic integrals" (Landmark Writings in Western Mathematics, pp. 143 & 157). The text block of this copy is about 50% thicker than that of an unpressed (and uncut) copy we handled recently. We know of only one other thick paper copy having appeared on the market, a presentation copy to the Earl of Morton (Sotheby s 1993). "The calculus was invented independently by Newton and Leibniz in the late seventeenth century. Newton and Leibniz developed general concepts differential and integral for Leibniz, fluxion and fluent for Newton and devised notation that made it easy to use these concepts. Also, they found and proved what we now call the Fundamental Theorem of Calculus, which related the two main concepts. Last, but not least, they successfully applied their ideas and techniques to a wide range of important problems. It was not until the nineteenth century, however, that the basic concepts were given a rigorous foundation. "In 1734 George Berkeley, later Bishop of Cloyne, attacked the logical validity of the calculus as part of his general assault on Newtonianism. Berkeley s criticisms of the rigor of the calculus were witty, unkind, and with respect to the mathematical practices he was criticizing essentially correct. Maclaurin s Treatise was supposedly intended to refute Berkeley by showing that Newton s calculus was rigorous because it could be reduced to the methods of Greek geometry. Maclaurin himself said in his preface that he began the book to answer Berkeley s attack, [p. i] and also to rebut Berkeley s accusation that mathematicians were hostile to religion. "The majority of Maclaurin s treatise is contained in its first Book, which is called "The Elements of the Method of Fluxions, d. Seller Inventory # 4222

Bibliographic Details

Title: A Treatise of Fluxions. In Two Books

Publisher: Printed by T.W. and T. Ruddimans, Edinburgh

Publication Date: 1742

Binding: Hardcover

Edition: First edition.