Page:A History of Mathematics (1893).djvu/228

 Quadrature of Curves of 1704, the infinitely small quantity is completely abandoned. It has been shown that in the Method of Fluxions Newton rejected terms involving the quantity 0, because they are infinitely small compared with other terms. This reasoning is evidently erroneous; for as long as is a quantity, though ever so small, this rejection cannot be made without affecting the result. Newton seems to have felt this, for in the Quadrature of Curves he remarked that "in mathematics the minutest errors are not to be neglected" (errores quam minimi in rebus mathematicis non sunt contemnendi).

The early distinction between the system of Newton and Leibniz lies in this, that Newton, holding to the conception of velocity or fluxion, used the infinitely small increment as a means of determining it, while with Leibniz the relation of the infinitely small increments is itself the object of determination. The difference between the two rests mainly upon a difference in the mode of generating quantities.[35]

We give Newton's statement of the method of fluxions or rates, as given in the introduction to his Quadrature of Curves. "I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux: and so on in other quantities. These geneses really take place in the nature of things, and are daily seen in the motion of bodies.&hellip;

"Fluxions are, as near as we please (quam proxime), as the increments of fluents generated in times, equal and as small as possible, and to speak accurately, they are in the prime ratio of nascent increments; yet they can be expressed by any lines whatever, which are proportional to them."