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Joseph Louis Lagrange was skeptical of the infinitely small; for him, the accuracy of calculus results from a “compensation of errors.” And yet, he rejected the limit concept on account of its poor metaphysical grounding. The tangent was not a limit, for Lagrange, because after becoming the tangent, the secant could very well continue to the other side of the point and become a secant again. Nor did he accept the method of fluxions, because it made use of “the irrelevant notion of motion.” He rejected Euler’s presentation of dx and dy as 0, because Lagrange felt that we do not have a clear and precise notion of the ratio of two terms which become zero. As a result, Lagrange sought out a simple algebraic method that was free from these objections.
Lagrange found his means in the Taylor Series.
The series f (x + h) = f (x) + f ' (x)h + f '' (x)h^2/2! had been known at least from the time of Taylor, whose name it bears. In this series, the coefficients of the powers of h involve the ratios of differentials, or of fluxions. However, the series can be derived without reference to these notions. What would be more natural than to define differentials and fluxions in terms of the coefficients of such a series? This procedure would (only on the surface, as we know now) obviate the necessity of introducing either limits or infinitesimals into the work, and the calculus would thus be reduced to simple algebraic operations. (252c)
Taylor thought that such an approach would free calculus from "all metaphysics and of any theory of infinitely small or vanishing quantities" (Boyer quoting Lagrange 252d). Although his method was not completely satisfactory, it had the advantage that it did not make use of ideas from geometry, mechanics, or philosophy (253b).
And yet, Lagrange was later criticized for "giving up, in favor of mathematical formalism, the 'generative' concept which has frequently been felt to be the basis of the methods of fluxions and differentials" (253bc).
But Lagrange continued seeking a formalization of the notion of limit based on Euler's function concept.
Incidentally, in so doing he focused attention for almost the first time upon the quantity which is now the central conception in the calculus -- that of the derived function, or the derivative, or the differential coefficient. Lagrange, in this connection gave not only the name from which the word derivative was adopted, but also the notation f 'x, modifications of which are still conveniently used. (253d)
Newton did not interpret the ratio of infinitesimals as such a single number or quantity (derivative), for he considered it more as a ratio of increments or fluxions (254a).
Similarly, Leibniz did not consider the ratio of infinitesimals as a single number, but instead as a quotient of "inassignables" (254b).
Lagrange's method was first to properly make use of the notion of derivative as "merely a single coefficient of a term in an infinite series" that is also "completely divested of any idea of ratio, or limiting equality" (254c).
Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.
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