ARTICLE XVIII.

An Essay on the first Elements of Fluxions, being the first Section of "the Principles of Fluxions, written by the Rev. S. Vince, A. M. F. R.S. Plumian Professor of Astronomy and Experimental Philosophy in the Univer sity of Cambridge in England." Edit. 1800. published at the request of Walter Folger, jun. Nantucket.

DEFINITIONS.

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1. Every quantity is here considered as generated by motion; a line by the motion of a point; a surface by the motion of a line; a solid by the motion of a surface.*

2. The quantity thus generated is called the fluent or flowing quantity.

13. The velocities with which flowing quantities. increase or decrease at any point of time, are called the fluxions of those quantities at that instant.

Cor. 1. As the velocities are in proportion to the increments or decrements uniformly generated in a given time, such increments or decrements will represent the fluxions.†

*Sir ISAAC NEWTON, in the Introduction to his Quadrature of Curves, observes, that "these geneses really take place in the nature of things, and are daily seen in the motion of bodies. And after this manner, the ancients, by drawing moveable right lines along immoveable right lines, taught the genesis of rectangles."

†This is agreeable to Sir I. NEWTON's ideas on the subject. He says, "I sought a method of determining quantities from the velocities of the motions or increments with which they are generated; and calling these velocities of the motions or increments, fluxions, and the generated quantities fluents, I fell by degrees upon the method of fluxions."

Introd, to Quad. Curves..

Cor. 2. Hence, as any given time may be assumed, the fluxion is not an absolute but a relative quantity. When we have several cotemporary fluxions, we may assume one fluxion what we please, and thence determine the values of the others. Thus, if x and y increase uniformly, and if x increase by p in the time that y increases by q, then the cotemporary increments of x and y will be p and q, 2p and 2 q, 3p and 34, &c. hence, if p be assumed the fluxion of x, the fluxion of will be q; if the former fluxion be 2p, the latter will be 2 q, &c. &c.

Cor. 3. A constant quantity has no fluxion.

¶ 4. The first letters, a, b, c, &c. of the alphabet are usually put for constant quantities, and the last, v, w, x, y, z for variable ones; and they are to be thus understood, unless the contrary be expressed.

5. The fluxion of a simple quantity, as x, is expressed by placing a point over it, thus x.

TO FIND THE FLUXIONS OF QUANTITIES.

PROP. I..

If two quantities increase or decrease uniformly, the increments or decrements generated in a given time will be as their fluxions.

6. This appears from ¶ 3. Cor. 1. 2.

PROP. II. .

If one quantity increase uniformly, and another of the same kind increase with an accelerated or retarded velocity, and trvo increments be assumed which are generated in the same time; if those increments be diminished till they vanish, that ratio to which they approach as their limit is the ratio of the fluxions of those quantities.

¶ 7. Let the line FK be described with uniformi velocity, and AZ with an accelerated velocity, and

let the increments Gs, Pm be generated in the same time; let also Po be the increments that would have

Been generated in the same time, if the velocity at P had been continued uniform; then by Prop. 1. the fluxions of FK, AZ at the points G and P will be represented by Gs and Pv. Let v be the velocity with which Gs is described, and the velocity with which Po is described, and let the velocity at m be V+r; then vm is the increment which is described in con-sequence of the increase r of velocity since the de-scribing point left P. Now let Pm be described with the uniform velocity V+w in the same time that Po and Pm are described; then it is manifest, that this uniform velocity must be between the velocities at P and at m, that is, Vw is greater than V and less than Vr, or w is greater than 0 and less than r.. Also, since the spaces described in the same time are as the velocities, V: V+w:: Po: Pm. Now diminish the times in which these increments are de-. scribed; then as the points and m approach to P, Po will continue to be described with the uniform ve-. locity ; but r will be diminished, and by diminishing the time till it becomes indefinitely small, r will become indefinitely small; but vm is described in consequence of this increase r of velocity; hence, when becomes indefinitely small in respect to the space om must become indefinitely small in respect to Po; therefore the ratio of Pv: Pm is, in that state, indefinitely near to a ratio of equality; but it is manifest that it never can become accurately a ratio of equality, because vm will not vanish until

Po and Pm vanish; consequently the ratio of the ac-tual increments Gs: Pm can never accurately express the ratio of the fluxions, that ratio being expressed by the ratio Gs: P. Let us then consider what ratio Po: Pm approaches to its limit, when we make the time in which the increments are described, and consequently the increments themselves vanish. In every state of these increments, V: V+w:: Po: Pm;: and by continually diminishing the time, and consequently the increments, we diminish r, and consequently w, but remains constant; it is manifest therefore that the ratio of VV+w, and consequently that of Pv: Pm, continually approaches to a ratio of equality, agreeable to what we have already shown;; and when the time, and consequently the increments, become actually=0, then r=0; consequently w=0; therefore the limit of the ratio of Po: Pm becomes that of V: V, a ratio of equality.* *Hence, the limit of the ratio of Gs: Pm is the same as the limit of the ratio of Gs: Pu, or as Gs: Pv, that ratio being con-stant; that is, the limiting ratio of the increments is the ratio of the fluxions.

The same is manifestly true for the limiting ratio> of the decrements of two quantities; for, conceiving the describing points to move backwards, and to be retarded by the same law, the decrements sG, mP in: this case become the same as the increments in the other; consequently their limiting ratio will express the ratio of the fluxions at G and P, or the rate at which FG, AP are, at that instant, decreasing.

Cor. 1. Hence, the limiting ratio of the increments or decrements of two quantities which are both gene-

By keeping the ratio of the vanishing quantities thus expressed by finite quantities, removes the obscurity which may arise when we consider the quantities themselves: this is agree able to the reasoning of Sir I. NEWTON in his Principia, lib. 1. sect. 1. lem. 7, 8, 9.

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rated by variable velocities, will be the ratio of their fluxions. And as the velocities with which these two lines increase or decrease may be made to agree with the rate of increase or decrease of any two like quantities, the proposition must be true for quantities of any kind.

Cor. 2. As the limiting ratio of the increments is the ratio of the fluxions, it is manifest that when the increments are in an increasing or decreasing state, the fluxions will be increasing or decreasing

¶ 8. It has been said, that when the increments are actually vanished, it is absurd to talk of any ratio between them. It is true; but we speak not here of any ratio then existing between the quantities, but of that ratio to which they have approached as their limit; and that ratio still remains. Thus, let the increments of two quantities be denoted by ax2+mx and bx2+nx; then the limit of their ratio, when x=0, is mn; for ax2+mx: bx2+nx :: ax+m: bx+n:: (when x=0) m: n. As the quantities therefore approach to nothing, the ratio approaches to that of mn as its limit. Hence, if m=n, the limit of this ratio is a ratio of equality. We must therefore be careful to distinguish between the ratio of two evanescent quantities, and the limit of their ratio; the former ratio never arriving at the latter, as the quantities vanish at the instant that such a circumstance is about to take place.

PROP. III.

If the fluxion of x be denoted by x, the fluxion of ax will be ax.

9. For if x increase uniformly, ax will also increase uniformly, and a times as fast: hence, by Prop. 1, the fluxion of the latter will be a times greater than that of the former, or it will be ax.