BOOK II. OF THE MOTION OF BODIES. SECTION I. Of the motion of bodies that are resisted in the ratio of the velocity. PROPOSITION I. THEOREM I. Tf a body is resisted in the ratio of its velocity, the motion lost by re sistance is as the space gone over in its motion. For since the motion lost in each equal particle of time is as the velocity, that is, as the particle of space gone over, then, by composition, the motion lost in the whole time will he as the whole space gone over. Q.E.D. COR. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over, there will be given also the whole space which the body can de scribe in an infinite time. For that space will be to the space now de scribed as the whole motion at the beginning is to the part lost 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, (fee., and, by con version, 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 resisted in the ratio of its velocity, and moves, by its vis insita only, through a similar medium, and the times be taken equal, the velocities in the beginning of each of the times are in a geometri cal progression, and the spaces described 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 suppose the resistance to act with one single impulse which is as the velocity, the decrement of the velocity in each of