FLUXION S.
TuE invention of the Method of Fluxions, as it is called in this country, or the Differential and Integral Calculus, as it is called by foreign mathematicians, goes no further back than the 17th century ; but the inquiries which have led to it, must have occurred to Geometers from the earliest period at which the science of geometry was cultivated. It appears from the writings of Euclid and Archimedes, that when the ancients considered curvilineal spaces, or the solids formed by the rotation of curves, they established the truth of their propositions by a peculiar mode of de monstration, which was indirect, and more subtile and pro lix than was used in ordinary cases. The second proposi tion of the xii. book of Euclid's Elements, is an instance of this kind of demonstration. It is there proposed to prove, that circles have to each other the ratio of the squares of their diameters. The preceding proposition proves, that similar polygons inscribed in the circles, have to each other that ratio ; and hence, by a mode of reasoning rather artificial, although quite accurate, the truth of the proposition is proved to extend to the circles themselves, by sliming, that the square of the one diameter cannot be to the square of the other diameter, as the one circle to a space either less, or greater than the other circle.
Although the ancients chose this mode of demonstrating the truth of such propositions, yet it may well be supposed, that they discovered them at first by a more simple mode of reasoning. In the instance we have quoted, as the ratio of similar polygons inscribed in the circles is altogether independent of the number of sides ; and as the greater the number of sides, the polygons became more nearly equal to the circles, from which at last they may differ by less than any assignable quantity, it is easy thence to infer the truth of the proposition. Here, however, there is a transition from a polygon of a finite number of sides to the circle, which is tacitly regarded as a polygon of an infinite number of sides : now this is the very circumstance, which in the end led to the invention of the method of fluxions.
When, after a long period of darkness, the light of science again shone forth in Europe, and the writings of Euclid and Archimedes were studied, with a view to detect the principles which had led to the discovery of the truths which they contain, it was soon observed, that these Geo meters had been more careful to convince, than enlighten their cotempo•aries ; and that however well the synthetic mode of demonstration was adapted, to place the truth of a proposition beyond doubt, yet it afforded little aid as an instrument of discovery. It was no doubt this view of the ancient geometry that induced Cavalerius to depart from its rigour, and invent his Method of Indivisibles, in which he considered lines as composed of an infinite number of points ; surfaces as composed of an infinite number of lines ; and solids as made up of an infinite number of sur faces. He appears to have possessed his theory in the year 1629, and he published it in 1635 with this title, Geometria inrlivisibiliunz continuoranz nova quddanz ratione promota. The accuracy of his method was attacked by Guildinus in 1640, and then he shelved, that at bottom it was the an cient theory of Exhaustions, but divested of its prolixity. In fact, these surfaces and lines, of which Cavalerius con sidered the ratios and the sums, are no other than the little solids, or the inscribed and circumscribed parallelograms of Archimedes, so numerous, as to differ from the figure, which is included between them, by less than any given quantity ; but while Archimedes, when lie demonstrates the ratio of a curvilineal figure to another known 013e, employs many words, and an indirect turn of demonstration ; the modern geometer, launching as it were into infinity, lays hold in imagination of the last term of these continued divisions and subdivisions, which should in the end annihi late the difference between the circumscribing and in scribed figures, and the curvilineal figure which they limit.