Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/239

 with the interval AS, decribe 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 decribe the pace AC, and in the place C will acquire the velocity CD.

This is demontrated the ame way from prop. 10. as prop. 32. was demontrated from prop. 11.

Hence the times are equal in which one body falling from the place A arrives at the centre S, and another body revolving decribes the quadrantal arc ADE.

Wherefore all the times are equal in which bodies falling from whatoever places arrive at the centre. For all the periodic times of revolving bodies are equal, by cor. 3. prop. 4.

Supoing a centripetal force of any kind, and granting the quadratures of curvilinear figures; it it required to find the velocity of a body, acending or decending in a right line, in the everal places through which it paes; as alo the time in which it will arrive at any place; And vice vera.

Suppoe 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