, in Mechanics, vibration, or the reciprocal ascent and descent of a pendulum.

If a simple pendulum be suspended between two semicycloids BC, CD, that have the diameter CF of the generating circle equal to half the length of the string, so that the string, as the body E Oscillates, folds about them, then will the body Oscillate in another cycloid BEAD, similar and equal to the former. And the time of the Oscillation in any arc AE, measured from the lowest point A, is always the same constant quantity, whether that arc be larger or smaller. But the Oscillations in a circle are unequal, those in the smaller arcs being less than those in the larger; and so always less and less as the arcs are smaller, but still greater than the time of Oscillation in a cycloidal arc; till the circular arc becomes very small, and then the time of Oscillation in it is very nearly equal to the time in the cycloid, because the circle and cycloid have the same curvature at the vertex, the length of the string being the common radius of curvature to them there.

The time of one whole Oscillation in the cycloid, or of an ascent and descent in any arch of it, is to the time in which a heavy body would fall freely through CF or FA, the diameter of the generating circle, or through half the length of the pendulum string, as the circumference of a circle is to its diameter, that is as 3.1416 to 1. So that if l denote the length of the pendulum CA, and g = 16 1/12 feet = 193 inches, the space a heavy body falls in the 1st second of time, and p = 3.1416 the circumference of a circle whose diameter is 1: then by the laws of falling bodies, it is , the time of falling through CF or (1/2)l; therefore , which is the time of one vibration in any arch of the cycloid which has the diameter of its generating circle equal to (1/2)l. Or, by extracting the known numbers, the same time of an Oscillation becomes barely (4/25)√l or (16/100)√l very nearly, l being the length of the pendulum in inches. And therefore this is also very nearly the time of an Oscillation in a small circular arc, whose radius is l inches.

Hence the times of the Oscillation of pendulums of different lengths, are directly in the subduplicate ratio of their lengths, or as the square roots of their lengths.

The more exact time of Oscillating in a circular arc, when this is of some finite small length, is ; where h is the height of the vibration, or the versed sine of the single arc of ascent, or descent, to the radius l.

The celebrated Huygens first resolved the problem concerning the Oscillations of pendulums, in his book De Horologio Oscillatorio, reducing compound pendulums to simple ones. And his doctrine is founded on this hypothesis, that the common centre of gravity of several bodies, connected together, must ascend exactly to the same height from which it fell, whether those bodies be united, or separated from one another in ascending again, provided that each begin to ascend with the velocity acquired by its descent.

This supposition was opposed by several, and very much suspected by others. And those even who believed the truth of it, yet thought it too daring to be admitted without proof into a science which demonstrates every thing.

At length Mr. James Bernoulli demonstrated it, from the nature of the lever; and published his solution in the Mem. Acad. of Scienc. of Paris, for the year 1703. After his death, which happened in 1705, his brother John Bernoulli gave a more easy and simple solution of the same problem, in the same Memoirs for 1714; and about the same time, Dr. Brook Taylor published a similar solution in his Methodus Incrementorum: which gave occasion to a dispute between these two mathematicians, who accused each other of having stolen their solutions. The particulars of which dispute may be seen in the Leipsic Acts for 1716, and in Bernoulli's works, printed in 1743.

Axis of Oscillation, is a line parallel to the horizon, supposed to pass through the centre or fixed point about which the pendulum oscillates, and perpendicular to the plane in which the Oscillation is made.

Centre of Oscillation, in a suspended body, is a certain point in it, such that the Oscillations of the body will be made in the same time as if that point alone were suspended at that distance from the point of suspension. Or it is the point into which if the whole weight of the body be collected, the several Oscillations will be performed in the same time as before: the Oscillations being made only by the force of gravity of the oscillating body. See Centke of Oscillation.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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ORTELIUS (Abraham)
OUGHTRED (William)