An Expository Sketch of a New Theory of the Calculus (Classic Reprint)

Excerpt from An Expository Sketch of a New Theory of the Calculus

The writer attempts, in the following pages, to re* construct run calculus upon a basis altogether in dependent of considerations, not only of limits, but also of infinitely small elements. Giving to the term differential a meaning somewhat different from the one commonly received, and regarding the value of dz, or the differential of the independent variable, as constant, and always equal to unity, he endeavours to determine the formulas for all the correlative finite values which dy, or the differential of the dependent variable, may assume in the different cases where the equation y f(w) denotes an ordinary algebraic or transcendental functional relation, and also to show the utility of those formulas in the solution of problems. If it is acknowledged that he has done something (no matter how little) towards vindicating mathematical science from the reproach of demonstrating many ofits most important principles by arguments so fanciful and so arbitrary that a they would not be allowed'm Theology n, the measure of his ambition is filled. He permits himself to add that his method (which is prs sumed to be new) turns neither upon the relations which subsist between time, space and velocity, no: upon the possibility of the analytical developmento values into series but solely upon the facts of direc tion curvature and tangency.

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