The other force acting once for all would make it, were there no gravity acting, move in spaces proportioned to the times simply. The latter or projecting force would make it move through AB uniformly, or in spaces proportional to the times; the force of gravity would make it move

A

B

יך

M

through AP with a motion proportioned to the square of the times; therefore it will move in a curve passing through M, if P M is equal, and parallel to AB; and AP will be as the square of AB or PM, which is the property of the conic parabola m.AP=PM2, m being the parameter to the point A.

The Scholium concludes by stating some consequences of the equality of action and reaction, the third Law of Motion, with respect to oscillation and impact, and also with respect to mutual attractions; of which consequences the most important is that the attraction or weight of heavy bodies in respect of the earth, and of the earth in respect of them, is equal.

The great work itself, after these preliminary though essential matters, proceeds to its proper subject. But in order to show how the demonstrations are conducted, a short treatise is prefixed upon the method of Prime and Ultimate Ratios, in eleven Lemmas, with their corollaries. This method consists in considering all quantities as

generated by the uniform progression or motion of other quantities, and examining the relations which the smallest conceivable spaces thus generated by this motion bear to one another, and to the spaces generated at the moment of their inception, or when they are nascent, which is termed their prime ratio, and at the moment of their vanishing, or when they are evanescent, which is termed their ultimate ratio. Thus a point moving along in a straightforward direction generates a straight line; a line moving parallel to itself, or two lines moving at right angles to one another, generate a rectangle: one line moving, while a point in it moves along it so that its progress on the moving line always bears a given ratio to the progress the line has made (m.AP=PM), describes a triangle; the same motion, if the progress of the point bears a variable

relation to that of the line (x.AP=PM'; x.× AP being some function of AP), describes a curve line and curvilinear area; and so of solids, which are generated by the motion of planes.

It follows from this mode of generation that if the length of any curve line be divided into an infinite number of lines, the sum of these will not differ from the curve line by any assignable quantity, nor will each differ from a straight line; and if its area be divided into an infinite number of smaller areas by lines drawn parallel to the

line whose progressive motion generated the curvilinear area, the sum of these infinitely narrow areas will differ from the area of the curve by a difference less than any assignable quantity, nor will each differ from a rectangle ; in other words, the ratio of the nascent curve line and nascent curvilinear area will be that of equality with the small lines and small rectangles, and the ultimate ratio of the sums of the lines and rectangles to the whole curve line and curvilinear area, respectively, will be that of equality: Or to put it otherwise, if the axis of the curve be divided into parts PP, &c., and the area into spaces PMRP, &c., by ordinates PM, PR, &c., and the number of these spaces be increased, and their breadth PP be diminished indefinitely, which is the operation of the

generative motion of PM, the size of each of the small spaces MNRO (by which the curvilinear areas differ from the rectangles) diminishes indefinitely, and the ultimate ratio of all the curve areas PMRP, and all the rectangles PNRP, becomes that of equality, and therefore the sum of evanescent differences N MOR, NROR, &c., whereby the whole curvilinear area differs from the whole amount of the rectangles PNRP, becomes less than any assignable quantity, or the curvilinear area coincides with the sum of the rectangles. And so of the sum of all the diagonals MR, RR, &c., which becomes the curve line MRA. Hence we infer that the amount of these small spaces

or quantities N MOR, formed by multiplying together two evanescent quantities, is as nothing in comparison with the rectangles P MOP formed by only one evanescent quantity multiplied into a finite quantity, and may be neglected in any equation that expresses the relations of those rectangles with each other. But if some other

quantities be found which are, in comparison with these small ones, themselves infinitely small, the areas formed by multiplying this second set of small quantities may be rejected in any equation expressing the relations of those first small quantities.

Thus we have the origin and constitution of quantities which in the Newtonian scheme are called fluxions of different orders, because conceived to express the manner of the generation of quantities by the motion of others, and in Leibnitz's language are called infinitesimals or differences, because conceived to express the constant addition of one indefinitely small quantity to another. Obtaining the fluxions, or the differences, from the quantity generated by the motion or by the addition, is called the direct method; obtaining the quantity generated from the fluxions, or finding the sum of all the differences, is called the indirect method. The one theory calls the direct method that of finding fluxions, the indirect that of finding fluents; the other theory calls the former differentiation, or finding differentials, the latter integration, or finding integrals. The two systems, therefore, in no one respect whatever differ except in their origin and language; their rules, principles, applications, and results, are the

same.

A different symbol has been used in the two systems; Newton expressing a fluxion by a point or dot, and the fluxion of that fluxion, or a second fluxion, by two dots, and so on. Leibnitz prefixes the letter d, and its powers

d', d3, &c., instead, to express the differentials. In like manner f for sum is used by the latter to express the integral, and ƒ by the former for the fluent. Although the continental method of notation is now generally used, and is on the whole most convenient, yet it has its inconvenience, as the d is sometimes confounded with co-efficients of the variable quantities; it is in some respects, too, not very consistent with itself; as by making da2 mean the square of the fluxion, or differential of x; whereas it, strictly speaking, appears to denote the differential of 2. There can be no doubt, however, which notation is the most convenient in the extension of the system to the calculus of variations, where the symbol is ; for, although the variation of a fluxion or differential may perhaps even more conveniently be expressed by than by d d x, yet the fluxion of a variation can with no convenience be expressed

by or otherwise than by dô x. The expression of

second fluxions undeveloped is also far less convenient by the Newtonian notation. Thus the fluxion of

dy is

dx

sometimes required to be expressed without developement, as in the expression for the radius of curvature, where it is often expedient not to develope it in the general

dy

equation, but to find in terms of x or y before taking

dx

its fluxion; yet nothing can be more clumsy than to place

Several important considerations arise out of the nature and origin of these infinitesimal quantities as we have described them; and to these considerations we must now shortly advert, as they give the rules for finding the