SEC. VI.] OF NATURAL PHILOSOPHY. 307 because the medium, by the motion it has received from the bodies, going forwards the same way with them, is more agitated in the former case, and less in the latter ; and so conspires more or less with the bodies moved. Therefore it resists the pendulums in their descent more, and in their as cent less, than in proportion to the velocity; and these two causes concur ring prolong the time. PROPOSITION XXVIII. THEOREM XXIII. If afunependvlous body, oscillating in a cycloid, be resisted in the rati > of the moments of the time, its resistance will be to the force of grav ity as the excess of the arc described in the whole descent above the arc described in the subsequent ascent to twice the length of the pen dulum. Let BC represent the arc described in the descent, Ca the arc described in the ascent, and Aa the difference of the arcs : and things remaining as they were constructed and demonstrated in Prop. XXV, the force with which the oscillating body is urged in any place D will be to the force of resistance as the arc CD to the arc CO, which is half of that difference Aa. Therefore the force with which the oscillating body is urged at the beginning or the highest point of the cycloid, that is, the force of gravity, will be to the resistance as the arc of the cycloid, be tween that highest point and lowest point C, is to the arc CO ; that is (doubling those arcs), as the whole cycloidal arc, or twice the length of the pendulum, to the arc Aa. Q.E.D. PROPOSITION XXIX. PROBLEM VI. Supposing that a body oscillating in a cycloid is resisted in a duplicate ratio of the velocity: to find the resistance in each place. Let Ba be an arc described in one entire oscillation, C the lowest point C O K O ,S P rR Q M of the cycloid, and CZ half the whole cycloidal arc, equal to the length of the pendulum ; and let it be required to find the resistance of the body is