follows, that thefe, or, in other words, the times of the vibrations of pendulums that defcribe fimilar arcs of circles, or which have equal angles of vibration, are as the fquare roots of the lengths of the pendulums; fo that if one pendulum be four times as long as another, the fhorter will vibrate in half the time, fo as to perform two vibrations in the fame time that the longer performs one. Again, if one pendulum be nine feet long, and another four, the fquare roots of which lengths are three and two, the fhort one will make three vibrations while the other is making two.

A difproportion in the length of two pendulums, occafions a great difference, you fee, in the times of their vibrations.

Of feveral pendulums vibrating in fimilar arcs, the vibrations of the longest are flower than thofe of the fhorter ones, and, confequently, if a pendulum is required that fhall vibrate feconds, it must have a determinate length, as the length of the pendulum fixes the time of it's ofcillation. This length has been afcertained to be thirty-nine inches.

From this principle, you may find how long a branch is which hangs down from the roof of a church; and, confequently, by measuring from the ball of the branch to the floor, and adding this to the length of the branch, you may find how high the church is. Let us fuppofe the branch to vibrate once in three feconds; then, fince the times of vibration are as the fquare root of a pendulum's length, it follows, that the lengths of pendulums are as the fquares of the times of vibration; and that the length of the branch is to the length of a pendulum which performs a vibration in one fecond, as the fquare of 3 to the fquare of 1, or as 9 to 1. Now the length of a pendulum that vibrates feconds

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feconds being 39 inches, multiplying this by 9, we obtain 352 inches 8 parts, for the length of the branch.

Pendulums which are of the fame length vibrate in the fame time, whatever be the proportion of their weights; or, in other words, the time of a pendulum's vibration is no way altered by varying the weight thereof. This follows from the property of gravity, which is always proportional to the quantity of matter; confequently all bodies in the fame circumstances are moved by the force of gravity with the fame velocity. To confirm this by experiment, here are two unequal weights, fo fufpended by two threads, as to conftitute two pendulums equal in length. Let them at the fame inftant of time fall from equal heights, they will keep pace together fo as to perform their vibrations in equal times.

From the motion of a pendulum, it is clear that, in any one place, the quantity of gravitating matter, in any body, is proportional to it's weight. For we find by experiment, that pendulums of equal length, whatever quantities of matter they contain, vibrate in the fame time. They have equal velocities in the fame time: the velocity and time being given, the quantity of matter is as the force of gravity.

What I have hitherto faid on this fubject, extends only to fimple pendulums; that is, those to which only one weight is fufpended, and where the thread by which it is fufpended is confidered as without gravity or weight: for when the rod by which the weight is fufpended is of any confiderable weight, the pendulum must be confidered as compounded, for the weight of the rod has the fame effect as a fecond weight faftened to the fame thread. And I defined a compound pendulum, as one to which feveral weights were fixed,

at

at invariable diftances one from the other, as well as from the point of fufpenfion.

Compound pendulums follow the fame laws as thofe that are fimple, only with fome modifi

cations.

OF THE CENTER OF OSCILLATION.

To determine the times of the vibrations of a compound pendulum, we must confider a thing of which I have not yet fpoken to you; that is, the center of ofcillation.

The center of ofcillation of a compound pendulum, is that point in which the efforts, or actions, of the weights which compofe it arè united, to cause the pendulum to vibrate in a certain time; or, in more technical terms, the center of ofcillation is that point of a pendulum, on each fide of which the quantities of motion are equal, or in which all the gravity of the pendulum might be collected, without altering the time of it's vibrations. Though the center of ofcillation be different from the center of gravity, yet is it evident, from thefe definitions, that they have a neceffary relation to each other.

The center of ofcillation of a fimple pendulum, whose thread is confidered without weight, is not in the center of gravity, but a little below it; but in the line of direction it is nearer or further from the center of gravity, according to a certain proportion between the radius of the ball forming the pendulum, and the length of the thread by which it is fufpended. Mr. Huygens has fhewn how to find the center of ofcillation from the proportion between the radius of the weight and the length of the pendulum.

The feal length of a fimple pendulum is not, therefore, the fame as the length of the thread

from

from the point of fufpenfion to the ball attached to it, nor even to the center of gravity thereof; but it's length is to be estimated from the point of fufpenfion to the center of ofcillation, which differs from the center of gravity, except when the length of the thread exceeds to a certain degree the radius of the ball, when the difference becomes insenfible.

When the line of the pendulum is poffeffed of weight which becomes fenfible with refpect to that which is attached to it, the center of ofcillation is no longer in the fufpended ball, but in a point fomewhat above it; and this point is further removed from the ball in proportion as the weight of the rod is heavier when compared to the weight of the fufpended ball.

In this cafe the 'true length of the pendulum, is the distance between the point of fufpenfion and the center of ofcillation, and the vibrations of this pendulum will be quicker than if the rod were without weight, because the pendulum is fhorter.

You have seen that the longer the pendulum, or the further the weight from the point of fufpenfion, the flower are it's vibrations. Thus if to an inflexible line A C, fig. 14, pl. 2, 4 feet long, from whofe extremity a weight P is fufpended, you add at Q a fecond weight B fomewhat higher than the other, as 3 feet from the point of fufpenfion; now the weight P, which is 4 feet from the point of fufpenfion, ought to make it's vibrations flower than the body B, which is only three feet therefrom: but as they are both attached to the fame inflexible line, they are forced to perform their vibrations in the fame time, which will be between the flowness with which it would have vibrated if there had been only the weight at A, and the greater velocity of it's vibrations if there had been

only

only the weight at Q. Thus the fecond weight quickens the vibrations of the first, and the first retards those of the fecond; and the center of ofcillation of this pendulum will be in that point in which if these two weights were united, the pendulum would perform it's vibrations in the fame time as that of the pendulum with the two weights.

To find therefore the center of ofcillation of à compound pendulum, is to find the length of a fimple pendulum that would perform it's vibrations in the fame time as the compound; and the real length of the compound pendulum is to that of a fimple ifochronous pendulum as the pendulum CR is to the pendulum CQA. Now as the lengths of pendulums are as the fquares of the times of their vibrations, it is easy to fee that the fimple pendulum CR, whofe vibrations are ifochronous to thofe of the compound pendulum CQA, fhould be more than three feet and lefs than four feet long; and confequently a fimple pendulum is always fhorter than a compound pendulum, whofe vibrations are performed in the fame time, and the center of ofcillation of the compound pendulum CQA will be between the two weights P and Q, that is, fomewhere about the point O.

It is eafy to perceive from what I have already explained to you, that in a compound pendulum CQ A, confifting of two weights, the nearer one of thefe weights is to the point of fufpenfion, or the further the two weights are from each other, the more the center of ofcillation approaches the point of fufpenfion; and, on the contrary, if the two weights are equally diftant from the point of fufpenfion, their centers of ofcillation would become the fame, and the compound pendulum might be confidered as a fimple one.

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