Finding the orbits from the focus given

PROPOSITION 54 PROBLEM 30

Granting the quadratures of curvilinear figures, it is required to find the times in which bodies by means of any centripetal force will descend or ascend in any curve lines described in, a plane passing through the centre of force. Let the body descend from any place S, and move in any curve ST/R given in a plane passing through the centre of force C. Join CS, and lei

Q

be divided into innumerable equal parts, and let be one of those parts. From the centre C, with the intervals CD, Cd, let the circles DT, dt be de it Dd scribed, ST*R meeting the curve line in T and t. And because the law of centripetal force is given. and also the altitude CS from which the at body there will be given the velocity of the body in any other altitude (by Prop. XXXIX). But the time in which the body describes the lineola Tt first fell, CT Lei, the ordinate DN, is as the length of that lineola, that is, as the secant of the angle /TC directly, and the velocity inversely. proportional to this time, be made perpendicular to because Dd is given, the rectangle the area DNwc?, will be proportional to the same time. Therefore if PN/?, be a curve line in which the point is perpetually found, and its asymptote be the right line SQ, standing upon the line CS at right the right line Dd X DN, CS that at the point D, and is, N D will be proportional to the time in which the angles, the area SQPJN and therefore that area in its descent hath described the line ; ST found, the time is also given. body being

PROPOSITION 55 THEOREM 19

If a body move in any curve superficies, whose axis passes through the centre of force, and from the body a perpendicular be let fall iipon the axis
and a line parallel and equal thereto be drawn from any given point of the axis ; I say, that this parallel line will describe proportional to the time. Let BKL revolving in an area T be a curve superficies, a body a trajectory which the it, STR body describes in the same, S the beginning OMK the axis of the curve of the trajectory, a line let fall perpendic right superficies, the to the from axis a line body ularly TN ; OP parallel and equal thereto drawn from the given point O in the axis AP ; the orthogra phic projection of the trajectory described by in which the the point P in the plane revolving line OP is AOP A the beginning found : of that projection, answering to the point S a part thereof a right line drawn from the body to the centre the body tends towards the which with force the to centripetal proportional a right line perpendicular to the curve superficies TI a centre C force of pressure with which the body urges part thereof proportional to the ; TO ; ; TM TG

193 the superficies, and therefore with which it is again repelled by the super a right line parallel to the axis and passing through towards ficies M PTF the body, and OF, IH ; G right lines let fall perpendicularly from the points upon that parallel PHTF. I say, now. that the area AGP, de scribed by the radius OP from the beginning of the motion, is proportional to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re solved into the forces TF, FG and the force TI into the forces TH, HI and I ; ; TF, TH, acting in the direction of the line PF perpendicular the plane AOP, introduce no change in the motion of the body but in a di but the forces to rection perpendicular to that plane. Therefore its motion, so far as it has same direction with the position of the plane, that is, the motion of the the point P, by which the projection plane, is the same as if the forces AP TF, HI wei e acted on by the forces FG, were to describe in the plane AOP of the trajectory is described in that TH were taken away, and same as is, the curve AP by means of a alone ; that the ,f the body the body centripetal and force tending to the centre O, and equal to the sum of the forces will be de HI. But with such a force as that (by Prop. 1) the area FG AOP

Corollary

By the same reasoning, if a body, acted on by forces tending to two or more centres in any the same right line CO, should describe in a free space any curve line ST, the area AOP would be always proportional scribed proportional to the time. to the time