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BERKELEY S FOUNDATIONAL CRITIQUE OF THE CALCULUS . First edition of Berkeley s famous attack on the calculus of Newton and Leibniz, which the historian Florian Cajori described as "the most spectacular event of the century in the history of British mathematics" (History of the Calculus, p. 57), bound with Berkeley s important Theory of Vision (1733) and Defence of Free-Thinking in Mathematics (1735), and five further pamphlets. "The Analyst is a criticism of the calculus, in both its Newtonian and Leibnizian formulations, arguing that the foundations of the calculus are incoherent and the reasoning employed inconsistent. Berkeley s powerful objections provoked numerous responses, and the task of replying to them set the agenda for much of British mathematics in the 1730s and 1740s" (Jesseph, p. 121). Perhaps the most famous passage in the book (p. 59), and a vivid example of Berkeley s wit, is his response to the idea that fluxions could be defined using ultimate ratios of vanishing quantities: It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities? Modern historians have argued that the expression Ghosts of departed Quantities was intended to address both Leibnizian infinitesimals and Newtonian fluxions. "Berkeley s attack on the calculus pointed out real deficiencies … his attack was also incisive, witty, and infuriating. Many mathematicians were moved to try to answer it. In fact, several important eighteenth-century discussions of the foundations of the calculus can be traced back to Berkeley s attack. For instance, Maclaurin s monumental two-volume A Treatise ofFluxions began as a reply to Berkeley. Berkeley s attack had a more lasting effect than simply stimulating an immediate set of replies; it served to keep the question of foundations alive and under discussion, and it pointed to the questions which had to be answered if a successful foundation were to be given. D Alembert and Lazare Carnot both used some of Berkeley s arguments in their own discussions of foundations, and Lagrange took Berkeley s criticisms with the utmost seriousness" (Grabiner, p. 27). Despite these 18th century attempts, calculus was not placed on a secure foundation until around 1820, with the work of Bolzano and Cauchy on the theory of limits. The Infidel Mathematician in the title is thought to be Edmund Halley (1656-1742). "As a result [of the publication of The Analyst], there appeared within the next seven years some thirty pamphlets and articles which attempted to remedy the situation. The first appeared in 1734, a pamphlet by James Jurin, Geometry No Friend to Infidelity . Berkeley answered Jurin in 1735, in A Defence of Freethinking in Mathematics [bound in the offered volume], and justly asserted that the latter was attempting to defend what he did not understand. In this work Berkeley again appealed to the divergence in Newton s views as presented in De analysi, the Principia, and De quadratura to show a lack of clarity in the ides of moments, fluxions, and limits . In the meantime, however, numerous attempts, some noteworthy and others insignificant, were made to find new and more satisfactory forms and arguments in which to present Newton s method. By far the ablest and most famous of these was made by Colin Maclaurin. In his Treatise of Fluxions, in 1744, he aimed not to alter the conceptions invlolved in Newton s fluxions, but to d.
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