306 THE MATHEMATICAL PRINCIPLES [BOOK. 11 ratio. But at the beginning of the motion, when the bodies begin to de scend and describe those arcs, the forces, which at that time are proportional to the arcs, will generate velocities proportional to the arcs. Therefore the velocities will be always as the whole arcs to be described, and there fore those arcs will be described in the same time. Q,.E.D. PROPOSITION XXVII. THEOREM XXII. If fnnependulous bodies are resisted in the duplicate ratio of their velocities, the differences between the times of the oscillations in a re sisting medium, and the times of the oscillations in a non-resisting medium of the same specific gravity, will be proportional to the arcs described in oscillating nearly. For let equal pendulums in a re sisting medium describe the unequal arcs A, B ; and the resistance of the body in the arc A will be to the resist ance of the body in the correspondent part of the arc B in the duplicate ra tio of the velocities, that is, as, A A to BB nearly. If the resistance in the arc B were to the resistance in the arc A as AB to AA, the times in the arcs A and B would be equal (by the last Prop.) Therefore the resistance AA in the arc A, or AB in the arc B, causes the excess of the time in the arc A above the time in a non-resisting medium ; and the resistance BB causes the excess of the time in the arc B above the time in a non-resisting medium. But those excesses are as the efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D. COR, 1. Hence from the times of the oscillations in unequal arcs in a resisting medium, may be knowrn the times of the oscillations in a non- re sisting medium of the same specific gravity. For the difference of the times will be to the excess of the time in the lesser arc above the time in a non-resisting medium as the difference of the arcs to the lesser arc. COR. 2. The shorter oscillations are more isochronal, and very short ones are performed nearly in the same times as in a non-resisting medium. But the times of those which are performed in greater arcs are a little greater, because the resistance in the descent of the body, by which the time is prolonged, is greater, in proportion to the length described in the descent than the resistance in the subsequent ascent, by which the time is contracted. But the time of the oscillations, both short arid long, seems to be prolonged in some measure by the motion of the medium. For retard ed bodies are resisted somewhat less in proportion to the velocity, and ac celerated bodies somewhat more than those that proceed uniformly forwards ;