THE MATHEMATICAL PRINCIPLES [BOOK II COR. 1. Therefore, if, having the points A and G given, the time bo expounded by the hyperbolic area ABED, the velocity may be expounded by -r the reciprocal of GD. COR. 2. And by taking GA to GD as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, the point G will be found. And that point being found the ve locity may be found from any other time given. PROPOSITION XII. THEOREM IX. The same things being supposed, I say, that if the spaces described are taken in arithmetical progression, the velocities augmented by a cer tain given quantity will be in geometrical progression. In the asymptote CD let there be given the point R, and, erecting the perpendicular RS meeting the hyperbola in S, let the space de scribed be expounded by the hyperbolic area I RSED ; and the velocity will be as the length J GD, which, together with the given line CG, ** composes a length CD decreasing in a geo metrical progression, while the space RSED increases in an arithmetical [(regression. For, because the incre nent EDde of the space is given, the lineola DC?, which is the decrement of GD, will be reciprocally as ED, and therefore directly as CD ; that is, as the sum of the same GD and the given length CG. But the decrement of the velocity, in a time reciprocally propor tional thereto, in which the given particle of space D^/eE is described, is as the resistance and the time conjunctly, that is. directly as the sum of two quantities, whereof one is as the velocity, the other as the square of the velocity, and inversely as the veh city ; and therefore directly as the sum of two quantities, one of which is given, the other is- as the velocity. Therefore the decrement both of the velocity and the line GD is as a given quantity and a decreasing quantity conjunctly; and, because the decre ments are analogous, the decreasing quantities will always be analogous ; viz., the velocity, and the line GD. U.E.D. COR. 1. If the velocity be expounded by the length GD, the space de scribed will be as the hyperbolic area DESR. COR. 2. And if the point „ be assumed any how, the point G will be found, by taking GR to GD as the velocity at the beginning to the velo city after any space RSED is described. The point G being given, the space is given from the given velocity : and the contrary. Cotw 3. Whence since (by Prop. XI) the velocity is given from the given