Imatges de pàgina
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This was clear and juft, and shewed that the nature and idea of fluxions is agreeable to the nature and constitution of things. They can have no dependance upon any metaphyfical fpeculations, (fuch fpeculations as that anti-mathematician, my Lord of Cloyne, brought in, to cavil and difpute against principles he understood nothing of, and maliciously run the account of them into the dark;) but are the genuine offspring of nature and truth. An inftance or two may illuftrate the matter.

1. A heavy body defcends perpendicularly 16 feet in a second, and at the end of this time, has acquired a velocity of 32% feet in a fecond, which is accurately known. At any given distance then from the place the body fell, take the point A in the right line, and the velocity of the falling body in the point may be truly computed: but the velocity in any point above A, at ever fo fmall a distance, will be lefs than in A; and the velocity at any point below A, at the leaft poffible distance, will be greater than in A. Jt is therefore plain, that in the point A, the body has a certain determined velocity, which belongs to no other-point in the whole line. Now this velocity is the fluxion of that right line in the point A;

and

and with it the body would proceed, if gravity acted no longer on the body's arrival at A.

2. Take a glafs tube open at both ends, whofe concavity is of different diameters in different places, and immerse it in a stream, till the water fills the tube, and flows thro' it. Then, in different parts of the tube, the velocity of the water will be as the squares of the diameters, and of confequence different. Suppofe then, in any marked place, a plane to pass through the tube perpendicular to the axis, or to the motion of the water, and of confequence, the water will pass through this fection with a certain determinate velocity: But if another section be drawn ever fo near the former, the water, by reafon of the different diameters, will flow through this with a velocity different from what it did at the former, and therefore to one section of the tube, or fingle point only, the determinate velocity belongs. It is the fluxion of the space which the fluid defcribes at that section; and with that uniform velocity the fluid would continue to move, if the diameter was the fame to the end of the tube.

3. If a hollow cylinder be filled with water, to flow freely out through a hole at the bottom,

bottom, the velocity of the effluent will be as the height of the water, and fince the furface of the incumbent fluid defcends without stop, the velocity of the ftream will decreafe, till the effluent be all out. There can then be no two moments of time, fucceeding each other ever fo nearly, wherein the velocity of the water is the fame; and of confequence, the velocity, at any given point, belongs only to that particular indivisible moment of time. Now this is accurately the fluxion of the fluid then flowing; and if, at that inftant, more water was poured into the cylinder, to make the furface keep its place, the effluent would retain its velocity, and still be the fluxion of the fluid. Such are the operations of nature, and they vifibly confirm the nature of Fluxion. It is from hence quite clear, that the fluxion of a generated quantity, cannot retain any one determined value for the leaft fpace of time whatever, but the moment it arrives at that value, the fame moment it lofes it again. The fluxion of fuch quantity can only pafs gradually and fucceffively through the indefinite degrees contained between the two extreme values, which are the limits thereof, during the generation of the fluent, in cafe the fluxion be variable: But then, though a determinate degree of fluxion does not continue at all, yet, at

every determinate indivisible moment of time, every fluent has fome determinate degree of fluxion; that is, every generated quantity has every where a certain rate of increasing a fluxion, whose abstract value is determinate in itself, though the fluxion has no determined value for the least space of time whatever. To find its value then, that is, the ratio one fluxion has to another, is a problem ftrictly geometrical; notwithstanding the Right Rev. anti-mathematician has declared the contrary, in his hatred to mathematicians, and his ignorance of the true principles of mathematics.

If my Lord of Cloyne had been qualified to examine and confider the cafe of fluxions, and could have laid afide that unaccountable obftinacy, and invincible prejudice, which made him refolve to yield to no reason on the subject;-not to regard even the great Maclaurin's anfwer to his Analyft; he would have discovered, that it was very poffible to find the abstract value of a generated quantity, or the contemporary increment of any compound quantity. By the binomial theo rem, the ratio of the fluxion of a fimple quantity to the fluxion of that compound quantity, may be had in general, in the lowest term, and as near the truth as, we please, whilst we fuppofe fome very small incre

increment actually defcribed: And whereas the ratio of these fluxions is required for some one indivifible point of the fluid, in the very beginning of the increment, and before it is generated, we make, in the particular cafe, the values of the fimple increments nothing, which before was expreffed in general: then all the terms wherein they are found vanish, and what is left accurately fhews the relation of the fluxions for the point where the increment is fuppofed to commence. As the abstract value of the fluxion belongs only to one point of the fluent, the moments are made to vanish, after we have feen by their continual diminution, whither the ratio tends, and what it continually verges to; and this becomes as vifible as the very character it is written in.

But Dr. Berkley was unacquainted with mathematical principles, and out of his averfion to thefe fciences, and zeal for orthodoxy, cavilled and difputed with all his might, and endeavoured to bring the matter to a state unintelligible to himself, and every body else.Here Maria had done, and for near a quarter of an hour after, I fat filently looking at her, in the greatest aftonishment.

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