I

or number of inftants being equal to the preceding time of acceleration, will be expreffed by the number of terms, or, in this feries, by the laft term. The fpace described will be therefore expreffed by the laft term multiplied by itself; that is to fay, it will be the double of the space defcribed by the accelerated motion in an equal time.

When it is faid that a number denotes any magnitude, it is to be understood that the number is part of a feries, whofe terms vary in a ratio always correfpondent, or equal to the ratio of the magnitudes denoted; that is to fay, the ratio between any two terms of the feries is always equal to the ratio between the two magnitudes that correspond to those terms. In this fenfe, any magnitudes, however unlike, may reprefent each other. Here follows a geometrical demonftration of the foregoing propofi tions, in which lines are made use of instead of numbers.

K In the triangle ABC, fig. 5, let the equal divi fions A, 1, 2, 3, &c. on the fide A B, reprefent equal parts of the time of an uniformly accelerated motion. Then the parallel lines, 1d, 2 e, 3 f, &c. may represent the velocities at the feveral instants, 1, 2, 3, &c. for they are in proportion to the times a I, A 2, A3, &c. And in like manner for any other part of the time as Am, the velocity generated will be represented by its correfpondent ordinate m n. And the fum of the ordinates correfponding with any part of the time will represent the fum of the velocities. But the fum of the ordinates, when taken indefinitely

indefinitely numerous, may be conceived to occupy the area contained between the ordinates of the firft and laft inftants of the time. And these areas, when taken from the beginning A, are as the fquares of the times A I, A 2, A 3, &c. or of the velocities 1 d, 2 e, 3 f, &c. by the property of fimilar figures. Therefore the fums of the velocities, and confe- L quently the spaces defcribed in any given terms of time taken from the beginning of an uniformly accelerated motion, are to each other as the fquares of the times, or of the laft acquired velocities. Hence M it likewise appears that the spaces described in equal fucceffive parts of time, are as the areas contained between A and I d, Id and 2 e, 2e and 3 f, &c. which areas are to each other as the odd numbers 1, 3, 5, 7, 9, &c. as appears by infpection from the number of equal and fimilar fmall triangles contained in each.

Again, fuppofe the motion at the end of the time N A B to become uniform with the last acquired velocity BC. Complete the parallelogram BDEC, making BD equal to A B, and the ordinates 1 i, 2 k, 31, &c. will denote velocities, and confequently spaces defcribed. Their fum will exprefs the whole space described in a time equal to A B, and will be denoted by the area BDE C. But this area is equal to twice the area A B C. Confequently the fpace defcribed by o an uniform motion with the laft acquired velocity, during a time equal to that of the acceleration from the beginning, will be double the space described by accelerated motion.

It

➤ It may readily be apprehended that an uniformly retarded motion is exactly the reverse of a motion uniformly accelerated. For fuppofe a constant force acting against a body in motion; as, for example, gravity acting against the motion of a body projected directly upwards, it will deftroy an equal part of the initial velocity in an equal particle of time. Now, if thefe equal deductions be called unities, and be fucceffively taken from any number whatsoever, till the last remainder be nothing, it is evident that the feries of remainders will be the natural numbers 1, 2, 3, &c. in a reverfed order, and every thing that was proved of the times, spaces, and velocities (28, F. 30, K), or of the parts of the triangle ABC, fig. 5, will be true, mutatis mutandis, that is to fay,

In the fame body in motion, and retarded by a conftant and equally acting force, the spaces defcribed in coming to reft, are as the fquares of the Rinitial velocities (29, G. 31, L), or as, the fquares s of the times during which they are defcribed: and

are equal to half the fpaces, which in an equal time would have been described by their respective initial velocities uniformly continued.

CHAP.

CHAP. V.

OF THE PRODUCTION, COMMUNICAtion, or

DESTRUCTION OF MOTION.

MOTION is produced, deftroyed, or changed in a T body, either by the impulfe, collision, or stroke, of another body in motion, or by the force of attraction. Repulfion being immediately the contrary to attrac- u tion, and not being perhaps fufficiently general and universal (6) to be admitted as a common property of bodies, need not be here confidered.

We do not know whether the diftinction be- v tween impulfe and attraction be real, and existing in the nature of things, or only relative to the imperfect state of our knowledge. The most obfervable w difference is, that impulse is a force which acts from place to place, or, in other words, cannot be without motion: but attraction can exert itself even though no motion is produced.

To exemplify this, fuppofe two bodies to meet x directly with equal quantities of motion; the effect of the stroke will be, that the whole motion will be destroyed, and the bodies will remain together. The forces will likewife be deftroyed, and the bodies may be moved apart; each with the fame facility as if the other did not exist. In this cafe, we have supposed no attraction to be exerted by them on each other. But let it now be supposed, that their motion, instead of being uniform, and the confequence of their inertia, is produced by a mutual attraction. They come together, and the motion is destroyed as beVOL. I. D fore.

Y

Z

fore. But the force of attraction, by which they were originally put in motion, remains, and is exerted in preffing them against each other.

The smallest finite impulfe can overcome the greatest finite preffure. For, let any preffure be fuppofed to produce acceleration, and the body, when in motion, will have more force than when it was merely preffed. In its acceleration from reft, it must pass through every poffible velocity lefs than the velocity laft acquired. Let the impelling body have a momentum expreffed by the product of its mass into its velocity (19, L). Whatever product this may be, it is poffible to affume a period of the acceleration of the body preffed, in which its velocity fhall be fo fmall, as that its product into its mass shall be still lefs. And it has already been. faid, that the force of mere preffure is yet less than this. Confequently, it is lefs than that of the impulfe.

From this it is inferred, if two bodies, perfectly hard, or unyielding, were to ftrike each other with any velocity, that they would be broken to pieces, provided the cohesion of their parts were less than A infinite. But if the cohesion were infinite, it is prefumed that the communication or deftruction of B motion would be inftantaneous. However, there

C

are no fuch bodies found in nature, and very confiderable difficulties arife in the abftract reafoning concerning them.

There appears to be the fame relation between preffure and momentum as between a line and a

furface.