with the interval AS, deſcribe the quadrant of a circle AE; and let CD be the right line of any arc AD; and the body A will in the time AD in falling deſcribe the ſpace AC, and in the place C will acquire the velocity CD.
This is demonſtrated the ſame way from prop. 10. as prop. 32. was demonſtrated from prop. 11.
cor. 1. Hence the times are equal in which one body falling from the place A arrives at the centre S, and another body revolving deſcribes the quadrantal arc ADE.
cor. 2. Wherefore all the times are equal in which bodies falling from whatſoever places arrive at the centre. For all the periodic times of revolving bodies are equal, by cor. 3. prop. 4.
Proposition XXXIX. Problem XXVII.
Supoſſing a centripetal force of any kind, and granting the quadratures of curvilinear figures; it it required to find the velocity of a body, aſcending or deſcending in a right line, in the ſeveral places through which it paſſes; as alſo the time in which it will arrive at any place; And vice verſa.
Suppoſe the body E (Pl. 17. Fig. 1.) to fall from any place A in the right line ADEC; and from its place E imagine a perpendicular EG
