Proposition 1

Proposition 1 Theorem 1

The areas, which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described.

The time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB.

In the second part of that time, the same would (by Law I.), if not hindered, so that by the radii proceel directly to c, alo ILJ; the line Be equal to

AS, BS, cS, draw. i to the centre, the equal areas ; ASB, BSc, would be de

But when the body scribed. arrived at B, suppose that a centripetal force acts at once with a great im is pulse, and, turning aside the body from the right line Be, compels afterwards to con it motion along the tinue its right line BC. Draw cC BC BS parallel to meeting and at the end of the in C ; second part of the time, the body (by Cor. I. of the Laws) will be found in C, in the same plane with the triangle A SB. Join SC, and, because s SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts successively in C, D, E. &c., and makes the body, in EF each single particle of time, to describe the right lines CD, DE, &c., they will all lie in the same plane and the triangle SCD will be equal to 7 : the triangle SBC, and SDE to SCD, and SEF to SDE. And immovable plane in equal times, equal areas are described in one therefore, : and, by composition, any sums SADS, SAFS, of those areas, are one to the other Now let the number of those as the times in which they are described. in wjinitum ; and be and breadth diminished their triangles augmented, will Cor. Lem. ADF be a curve line their ultimate 4, III.) perimeter (by and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually and any described areas SADS, SAFS, which are always proportional to the times of de Q.E.D. scription, will, in this case also, be proportional to those times. COR. 1. The velocity of a body attracted towards an immovable centre, : ; in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the veloci AB, BC, CD, DE, EF. ties in those places A, B, C, D, E. are as the bases of equal triangles and these bases are reciprocally as the perpendiculars let fall ; upon them. BC of two arcs, successively described in in spaces void of resistance, are completed equal times by the same body, of this parallelogram; and the diagonal into a parallelogram ABCV, COR. 2. If the chords AB, BV in the position which it ultimately acquires when those arcs are diminished in irifinitum, is produced both ways, it will pass through the centre of force. COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equalSEC. OF NATURAL PHILOSOPHY. II.] times, in spaces void of resistance, are completed DEFZ ABCV, : the forces in B and E 105 into the parallelograms are one to the other in the ulti ratio of the diagonals BV, EZ, when those arcs are diminished in and of the body (by Cor. 1 of the For the motions infinitum. but and Laws) are compounded of the motions Be, BV, and mate EF BC E/", EZ, which are equal to EZ BV : Cc and F/, in the demonstration of this Proposi tion, were generated by the impulses of the centripetal force in B and E and are therefore proportional to those impulses. COR. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times which versed sines tend to the centre of force, and bisect the chords when those ; ; For such versed sines are the halves of arcs are diminished to infinity. the diagonals mentioned in Cor. 3. COR. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those para bolic arcs which projectiles describe in the same time. COR. 6. And when the planes the same things do all hold good (by Cor. 5 of the Laws), in which the bodies are moved, together with the centres of force which are placed in those planes, are not at formly forward in right rest, but move uni lines.