Bodies that are left to fall from any height, will move fafter and fafter the nearer they come to the furface of the earth; for if the force of gravity acted upon a body only at the commencement of

tween the four ftraight lines AH, AI, Ar, and AE. Secondly, it is also evident that if a plane furface be placed at E, parallel to the fquare OPBv, all that part of it which lies between the aforefaid ftraight lines; viz. IHEr, will be illuminated by the light which paffes through OPBv; but as the plane IHEr is larger than OPBʊ, the light upon it cannot be fo denfe as at OPBv; and for the fame reason, if a plane be fituated at D, parallel to OPBv, the light upon it will be lefs dense than at OPBv, but more denfe than at IHEr, &c. Thirdly, it is also evident that the planes IHEr, KGDs, LFCx, are fquare figures, fince the hole OPB has been fuppofed to be a fquare. Therefore, the only thing which remains to be proved, is, that if the distance AC be equal to twice the distance AB, the area of the fquare LFCx is four times as large as the area OPBv; that if AD be equal to three times AB, the area KGDs is nine times as large as OPBv; or, in fhort, that the areas OB, LC, KD, &c. are as the fquares of the distances from A, which is eafily done; for ABP, ACF, being equiangular triangles (Eucl. p. 29. B. I.) we have (Eucl. p. 4. B. VI.) AB: AC:: PB: FC; but PB and FC are the homologous fides of the fimilar plane figures OPBv, LFCx; and (Eucl. p. 20. B. VI.) thofe figures aré as the fquares, or in the duplicate proportion, of their homologous fides; therefore OPBv: LFCx:: PB2: FTP: : AB: A. And the like reafoning may be applied to the other fquares KGDs, &c.

its defcent, the body would, (according to the laws of fimple motion, Chap. IV.) continue to defcribe equal spaces in equal portions of time. But the very next moment the force of gravity impels the body again, in confequence of which the body's velocity must be doubled; fince the fecond impulfe is equal to the firft, and the firft remains unaltered. For the fame reafon on the third mo-. ment the body's velocity will be trebled, and so on. Or, fpeaking more properly, the velocity will increase as the time increases, viz. the velocity will be as the time; the meaning of which is, that the velocity at the end of two feconds is to the velocity at the end of three feconds, as two to three; or the velocity at the end of one minute is to the velocity at the end of one hour, as one is to fixty, &c. *.

The spaces defcribed by fuch defcending bodies cannot be proportionate fimply to the times of descent; for that would be the cafe if the velocity remained unaltered; but, the velocity increasing

* The velocities are as the times when the gravitating power remains unaltered, or with the fame gravitating power; but if two diftinct gravitating powers be compared together, then the velocities will be as the products of the times multiplied by the gravitating forces refpectively; it being evident that a double force will produce a double effect, a treble force will produce a treble effect, &c. Hence when the times are equal, or in the fame time, the velocities are as the gravitating, or the impelling, forces.

continually,

continually, it is evident that the spaces must be as the times multiplied by the velocities; for a double velocity will force the body to move through a double space in an equal portion of time, and through a quadruple fpace in twice that time; alfo a quadruple velocity will force the body to move through a quadruple space in an equal portion of time, and through eight times that space in twice that time; and fo on in any proportion. But it has been fhewn above that the velocities are as the times; therefore to say that the spaces are as the times multiplied by the velocities, is the fame thing as to fay that the spaces are as the times multiplied by the times, or as the fquares of the times; and for the fame reafon it is the fame thing as to fay that the spaces are as the velocities multiplied by the velocities, or as the fquares of the velocities *.

This property of defcending bodies, (viz. that they run through spaces which are as the fquares of the times) has been ufually demonftrated in a different way by the philofophical writers. Their demonstration may, perhaps, appear more fatisfactory than that of the preceding paragraphs to fome of my readers; I fhall therefore fubjoin it, efpecially as it proves at the fame time another law relative to the velocity of defcending bodies.

*Therefore in equal times the fpaces are as the im pelling, or gravitating, forces. See the laft note.

VOL. I

F

Let

Let AB, fig. 6. Plate I. represent the time, during which a body is defcending, and let BC represent the velocity acquired at the end of that time.

Complete the triangle ABC, and the parallelogram ABCD. Alfo fuppose the time to be divided into innumerable particles, ei, im, mp, pa, &c. and draw ef, ik, mn, &c. all parallel to the bafe BC. Then, fince the velocity of the descending body has been gradually increafing from the commencement of the motion, and BC represents the ultimate velocity; therefore the parallel lines ef, ik, mn, &c. will reprefent the velocities at the ends of the refpective times Ae, Ai, Am, &c. Moreover, fince the velocity during an indefinitely fmall particle of time, may be confidered as uniform; therefore the right line ef will be as the velocity of the body in the indefinitely small particle of time ei; ik will be as the velocity in the particle of time im, and fo forth. Now the space paffed over in any time with any velocity is as the velocity multiplied by the time; viz. as the rectangle under that time and velocity; hence the space paffed over in the time ei with the velocity ef. will be as the rectangle if; the space paffed over in the time im with the velocity ik, will be as the rectangle mk; the fpace paffed over in the time mp with the velocity mn, will be as the rectangle pn, and fo on. Therefore the space paffed over in the fum of all thofe times, will be as the fum of all thofe rectangles. But fince the particles of

time are infinitely. fmall, the fum of all the rectangles will be equal to the triangle ABC. Now fince the space paffed over by a moving body in the time AB with a uniform velocity BC, is as the rectangle ABCD, (viz. as the time multiplied by the velocity) and this rectangle is equal to twice the triangle ABC (Eucl. p. 31. B. I.) therefore the fpace paffed over in a given time by a body falling from reft, is equal to half the space paffed over in the fame time with an uniform velocity, equal to that which is acquired by the defcending body at the end of its fall.

Since the space run over by a falling body in the time represented by AB, fig. 7. Plate I. with the velocity BC is as the triangular ABC, and the space run over in any other time AD, and velocity DE, is represented by the triangle ADE; those fpaces must be as the fquares of the times AB, AD; for the fimilar triangles ABC, and ADE, are as the fquares of their homologous fides, viz. ABC is to ADE as the fquare of AB is to the fquare of AD, (Eucl. p. 29. B. VI.)

In fig. the 8th. Plate I. the spaces, which are defcribed by defcending bodies in fucceffive equal portions of time, are reprefented, for the purpose of impreffing with greater efficacy on the mind of the reader, the principal law of gravitation. The line AB reprefents the path of a body, which is let fall from A, and defcends towards the ground at B The divifions on the line AB denote the places of