numerable equal particles, the fquares of the velocities at the beginning of each of the times will be proportional to the differences of the fame velocities. Let thofe particles of time be AK, KL, LM, &c. taken in the right line CD; and erect the perpendiculars AB, Kk, Ll, Mm, &c. meeting the hyperbola BklmG, defcribed with the centre C, and the rectangular afymptotes CD, CH, in B, k, l, m, &c.; then AB will be to Kk as CK to CA, and, by divifion, AB - Kk to Kk as AK to CA, and, alternately, AB - Kk to AK as Kk to CA; and therefore as AB × Kk to AB × CA. Therefore fince AK and AB × CA are given, AB Kk will be as AB × Kk; and, laftly, when AB and Kk coincide, as AB2. And, by the like reafoning, Kk - Ll, Ll - Mm, &c. will be as Kk2, Ll2, &c. Therefore the fquares of the lines AB, Kk, Ll, Mm, &c.. are as their differences; and, therefore, fince the fquares of the velocities were fhewn above to be as their differences, the progreffion of both will be alike. This being demonftrated it follows alfo that the areas described by these lines are in a like progreffion with the fpaces defcribed by thefe velocities. Therefore if the velocity at the beginning of the first time AK be expounded by the line AB, and the velocity at the beginning of the fecond time KL by the line Kk, and the length defcribed in the first time by the area AKKB, all the following velocitics will be expounded by the following lines Ll, Mm, &c. and the lengths defcribed, by the areas Kl, Lm, &c. And, by compofition, if the whole time be expounded by AM, the fum of its parts, the whole length defcribed will be expounded by AMmB the fum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. fo that CA, CK, CL, CM, &c. may be in a geometrical progreffion; and thofe parts will be in the fame progreffion, and the velocities AB, Kk, Ll, Mm, &c. will be in the fame progreffion inversely, and the spaces defcribed Ak, Kl, Lm, &c. will be equal. Q.E.D.

COR. 1. Hence it appears, that if the time be expounded by any part AD of the afymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time will be expounded by the ordinate DG; and the whole space defcribed by the adjacent hyperbolic

area ABGD; and the space which any body can defcribe in the fame time AD, with the first velocity AB, in a non-refifting medium, by the rectangle AB × AD.

COR. 2. Hence the space described in a refifting medium is given, by taking it to the space defcribed with the uniform velocity AB in a non-refifting medium, as the hyperbolic area ABGD to the rectangle AB x AD.

COR. 3. The refiftance of the medium is alfo given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-refifting medium, the velocity AB in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the afymptote in T, the right line AT will be equal to AC, and will exprefs the time in which the first refiftance, uniformly continued, may take away the whole velocity AB.

COR. 4. And thence is alfo given the proportion of this refiftance to the force of gravity, or any other given centripetal force.

COR. 5. And, vice verfa, if there is given the proportion of the refiftance to any given centripetal force, the time AC is alfo given, in which a centripetal force equal to the refiftance may generate any velocity as AB; and thence is given the point B, through which the hyperbola, having CH, CD for its afymptotes, is to be defcribed; as alfo the space ABGD, which a body, by beginning its motion with that velocity AB, can defcribe in any time AD, in a fimilar refifting medium.

PROPOSITION VI. THEOREM IV. Homogeneous and equal fpherical bodies, oppofed by refiftances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, defcribe equal spaces, and lofe parts of their velocities proportional to the wholes. (Pl. 1, Fig. 8.)

To the rectangular afymptotes CD, CH defcribe any hyperbola BbEe, cutting the perpendiculars AB, ab, DE, de in B, b, E, e; let the initial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Dd.

Therefore as Aa is to Dd, fo (by the hypothefis) is DE to AB, and fo (from the nature of the hyperbola) is CA to CD; and, by compofition, fo is Ca to Cd. Therefore the areas ABba, DEed, that is, the spaces defcribed, are equal among themfelves, and the first velocities AB, DE are proportional to the laft ab, de; and therefore, by divifion, proportional to the parts of the velocities loft, AB - ab, DE - de. Q.E.D.

PROPOSITION VII. THEOREM V. If Spherical bodies are refifted in the duplicate ratio of their velocities, in times which are as the first motions directly, and the first refiftances inversely, they will lofe parts of their motions proportional to the wholes, and will defcribe fpaces proportional to thofe times and the firft velocities conjunctly.

For the parts of the motions loft are as the refiftances and times conjunctly. Therefore, that thofe parts may be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the refistance inverfely. Wherefore the particles of the times being taken in that ratio, the bodies will always lofe parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their firft velocities. And because of the given ratio of the velocities, they will always defcribe spaces which are as the firft velocities and the times conjunctly. Q.E.D.

COR. 1. Therefore if bodies equally fwift are refifted in a duplicate ratio of their diameters, homogeneous globes moving with any velocities whatsoever, by defcribing fpaces proportional to their diameters, will lofe parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mafs conjunctly, that is, as the velocity and the cube of its diameter; the refiftance (by fuppofition) will be as the fquare of the diameter and the fquare of the velocity conjunctly; and the time (by this propofition) is in the former ratio directly, and in the latter inversely, that is, as the diameter directly and the velocity inversely; and therefore the space, which is proportional to the time and velocity, is as the diameter.

1