The References to the Plate are omitted in the printed Part of the first Sheet, but are fupplied by the Schemes themjelves, which refer to the Pages to which they belong.

Of the Motion of Bodies that are refifted in the ratio of the Velocity.

PROPOSITION I.

THEOREM I.

If a body is refifted in the ratio of its velocity, the motion loft by refiftance is as the Space gone over in its motion.

F

OR fince the motion loft in each equal particle of time is as the velocity, that is, as the particle of fpace gone over; then, by compofition, the motion loft in the whole time will be as the whole fpace gone over. Q.E.D.

VOL. II.

B

COR.

COR. Therefore if the body, deftitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and alfo the motion remaining after fome part of the way is gone over; there will be given alfo the whole fpace which the body can defcribe in an infinite time. For that space will be to the space now defcribed, as the whole motion at the beginning is to the part loft of that motion.

LEMMA I.

Quantities proportional to their differences are continually proportional.

Let A be to A-B as B to B-C and C to C-D, &c. and, by converfion, A will be to B as B to C and C to D, &c. Q.E.D.

PROPOSITION II.

THEOREM II.

If a body is refifted in the ratio of its velocity, and moves, by its vis infita only, through a Similar medium, and the times be taken equal; the velocities in the beginning of each of the times are in a geometrical progreffion, and the Spaces defcribed in each of the times are as the velocities.

CASE 1. Let the time be divided into equal particles ; and if at the very beginning of each particle we fuppofe the refiftance to act with one fingle impulfe which is as the velocity; the decrement of the velocity in each of the particles of time will be as the fame velocity. Therefore the velocities are proportional to their differences, and therefore (by Lem. 1. Book 2.)

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