OSCILLATION, in mechanics, the vi bration, or reciprocal ascent and descent of a pendulum. See PIINDULr14. It is demonstrated, that the time of a complete oscillation in a cycloid, is to the time in which a body would fall through the axis of that cycloid, as the circumference of a circle to its diameter; whence it follows : 1. That the oscillations in the cycloid are all performed in equal times, as being all in the same ratio to the time in which a body falls through the diameter of the generating circle. 2. As the middle part of the cycloid may be conceived to co incide with the generating circle, the time in a small arch of that circle will be nearly equal to the time in the cycloid : and hence the reason is evident, why the times in very little arches are equal. 3. The time of a complete oscillation in any little arch of a circle, is to the time in which a body would fall through half the radius, as the circumference of a circle to its diameter : that is, as 3.1416 to 1. If / denote the length of a pendulum, g= 16 = 193 inches, the space a heavy falls through in the first second of time, and p= 3.1416 = periphery of a circle whose diameter is 1, then, by the laws of falling bodies, it will be I In 4 T ___ =-- 137f6W Vrnearly the time of falling through 41: therefore 1 :p = 7 2g Which is the time of one vibration in any arch of the cycloid which has diameter of its generating circle equal to being the length of the pendulum in inches ; and since the latter time is half the time in which a body would fall through the whole diameter, or any chord, it follows, that the tune of an os cillation in any little arch, is to the time in which a body would fall through its chord, as the semicircle to the diameter.
4. The times of the oscillations in cycloids, or in small arches of circles, are in a sub duplicate ratio of the lengths of the pendu lums. 5. But if the bodies that oscillate be acted on by unequal accelerating forces, then the oscillation will be per formed in times that are to one another in the ratio compounded of the direct sub-duplicate ratio of the lengths of the pendulums, and inverse sub-duplicate ra tio of the accelerating forces. Hence it appears, that if oscillations of unequal pendulums are performed in the same time, the accelerating gravities of these pendulums must be as their lengths ; and thus we conclude, that the force of gravity decreases as you go towards the equator, since we find, that the lengths of pendulums that vibrate seconds, are al ways less at a less distance from die equa tor. 6. The space described by a falling body in any given time, may be exactly known : for, finding by experiments what pendulum oscillates in that time, the half of the pendulum will be to the space re quired, in the duplicate ratio of the dia meter of a circle to the circumference.