By E. T. Whittaker
AC ourse of recent A nalysis: An advent to the final idea of countless sequence and of Analytic services; With an Account of the relevant Transcendental capabilities used to be written through E. T. Whittaker in 1902. it is a 393 web page e-book, containing 101264 phrases and 10 images. seek within is enabled for this title.
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Additional info for A Course of Modern Analysis. An Introduction to the General Theory of Infinite Series and of Analytic Functions; with an account of the Principal Transcendental Functions
He did not give explicit reasons until much later, but he may have become dissatisfied with the standard foundations- 39 The Status of Foundations in Eighteenth-Century Calculus infinitesimals, fluxions, and limits-through Berkeley's criticisms. 68 He clearly did not believe that the older foundations measured up to the standards of reasoning expected in mathematics. He believed that a suitable standard was met with only in algebra. On reading Lagrange's work, one is struck by his feeling for the general: his idea that any particular result of any interest, whether it is the unsolvability of the quintic equation, the accuracy of an approximation, or the equations of motion for a physical system, is-and must be shown to be-a special case of some more general principle.
Zeros The quantity h can be made less than any given quantity; hence, when added to a finite quantity like 2x, it is actually zero. But although h is zero when added to finite quantities, it is not zero when considered in ratios; thus (2xh + h2)/h is not the meaningless 0/0, but the welldefined 2x. This explanation was given by Leonhard Euler and is similar to that later used by Laplace. Lagrange said that this method is equivalent to that of limits,57 and therefore both his (and Berkeley's) criticisms apply to it.
Since for our example P(x) = 2x, 2x is the derivative. This method was used by John Landen, L. F. A. -P. Gruson, and most influentially, by Lagrange. 58 Its advantage is that the derivative is exactly 2x; we do not need to explain what happened to h. What must be explained, however, is how this definition corresponds to the usual 36 Chapter 2 one, especially in applications to finding tangents and rates of change, and how it is known that there is such a series expansion when the function y is not a rational function.