310 THE MATHEMATICAL PRINCIPLES [BOOK 11. COR. 1 . Therefore the resistance in the lowest place C is to the force OP of gravity as the area ^ ~ IEF to the area PINM. COR. 2. But it becomes greatest where the area PIHR is to the area IEF as OR to OQ. For in that case its moment (that is, PIGR — Y) becomes nothing. COR. 3. Hence also may be known the velocity in each place, as being in the subduplicate ratio of the resistance, and at the beginning of the mo tion equal to the velocity of the body oscillating in the same cycloid with out any resistance. However, by reason of the difficulty of the calculation by which the re sistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition following. PROPOSITION XXX. THEOREM XXIV. If a right line aB be equal to the arc of a cycloid which an oscillating body describes, and at each of its points D the perpendiculars DK be erected, which shall be to the length of the pendulum as the resistance of the body in the corresponding points of the arc to the force of grav ity ; I say, that the difference between the arc described in the whole descent and the arc described in the whole subsequent ascent drawn into half the sum of the same arcs will be equal to the area BKa which all those perpendiculars take up. Let the arc of the cycloid, de scribed in one entire oscillation, be expressed by the right line aB, equal to it, and the arc which would have been described in vaciw by the length AB. Bisect AB in C, and the point C will represent the lowest point of the cycloid, and CD Mill be as the force arising from gravity, with which the body in D i,s urged in the direction of the tangent of the cycloid, and will have the same ratio to the length of the pendulum as the force in D has to the force of gravity. Let that force, therefore, be expressed by that length CD, and the force of gravity by the length of the pendulum ; and if in DE you take DK in the same ratio to the length of the pendulum as the resistance has to the gravity, DK will be the exponent of the resistance. From the centre C with the interval CA or CB describe a semi-circle BEeA. Let the body describe, in the least time, the space Dd ; and, erecting the per pendiculars DE, de, meeting the circumference in E and e, they will be as the velocities which the body descending in vacuo from the point B would acquire in the places D and d. This appears by Prop, LII, Book L Let