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the plane. Now by the similarity of triangles EFC, and BFD, CF: FE :: DF : FB. That is, the absolute force of gravity is to the relative force on the plane, as the length of the plane DF, to its perpendicular altitude FB. If the plane therefore be elevated, so that its altitude is half of its length, one pound, hanging perpendicularly over the end of the plane, will be in equilibrio with two pounds resting on the plane.

Now since the absolute is to the relative force of gravity as the length of the plane is to its height, the spaces over which they would carry the body in the same time must be in the same proportion; for the effect must be proportional to the cause: that is the space passed over by the absolute force of gravity in a perpendicular descent is to the space rolled down the inclined plane by the relative force of gravity, as the length of the plane to the height of it; or as BF: FI. So that if BI be perpendicular to the plane, drawn from the horizon, FI will be the space the body will roll down the plane, in the time that it would fall perpendicularly through its altitude.

Hence, since every angle in a semicircle is a right angle, if a semicircle be described on the altitude of a plane as its diameter, it will intersect all the planes that have the same altitude in the points to which a heavy body would roll down, in the time it would fall through the perpendicular altitude or diameter of the circle. That is, a heavy body will fall through the diameter AB* of the circle, in the time that it would roll down AF on the plane AFC, or AG, on the plane AGD, or AI, on the plane AIE.

Hence a heavy body will roll down any of the chords of a circle, that are connected with the lowest or highest point of the diameter, in the same time that it falls

* See Plate I. Fig. 5, 6.

through the diameter. This is the fundamental proposition of the doctrine of pendulums, as we shall see, when we have said something farther of the inclined plane.

As the motion on an inclined plane is uniformly accelerated by gravity, it must observe the same laws with bodies falling freely by the force of gravity. That

is to say,

The spaces passed over, or the lengths of the planes, must be as the squares of the times, and as the squares of the last acquired velocities at the end of the descent: for the velocities are always as the times, when the planes are similarly situated.

As the velocity acquired at the lowest point, in perpendicular descents, is always the same from any given altitude, so the velocity acquired by rolling down any planes, that have the same perpendicular altitude, must also be the same. And hence it follows, that if a body roll down any number of planes through the same perpendicular altitude, the velocity acquired at last, will still be the same as if it had fallen perpendicularly; it may therefore roll down a curve, which may be considered as an infinite number of planes, having one given altitude, though not in the same time, it would fall through the perpendicular, yet acquiring the same velocity at last.

The times of rolling down planes of different lengths, but of the same altitude, must depend on the lengths of the planes: for the longer that the plane is, the less it is inclined to the horizon, and the more of the weight of the body will it sustain, leaving the relative velocity less, by which alone it is carried down the plane; and therefore as the force is less in proportion to the length of the plane, in the same proportion must the time necessary for rolling down such a plane be increased: that

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is, the times of descent along planes of the same altitude, but different lengths, must be proportional to their lengths. It is the same relative gravity, that acts on the body when rolling down planes similarly situated, and then the times of descent will be proportional to the square roots of the lengths of the planes. But when the body descends along planes of the same altitude and differently inclined to the horizon, the relative gravity lessens with the inclination of the plane, and the body must therefore move slower, in the same proportion. As the length of the plane or its inclination to the horizon does nothing but lessen the force of gravity, it follows, that the effect, viz. the space passed over, must be lessened in the same proportion. That is, if a certain force of gravity would carry a body down an inclined plane 10 feet in a second, half that force would carry it down but 5 feet in the same time; and therefore this half force must require two seconds to carry the body 10 feet down the plane; and this half force is occasioned by the increased length of the plane. Therefore the times of descent through planes of different lengths, are as the lengths of the planes, when the altitudes are the same.

The velocity acquired in falling down the different chords of a circle, connected together at the lowest point of the diameters, is as the lengths of these chords. Because these velocities are as the square roots of the perpendicular altitudes, or versed sines of these chords; and the square roots of these versed sines are as the lengths of the chords. For BF-ABxBH, and since AB is a given quantity, BF is proportional to BH, which is as the velocity.

We have now prepared the way for considering the important doctrine of the

MOTION OF PENDULUMS.

A PENDULUM is any body suspended by a thread and movable about any fixed point. The most material properties of the pendulum are such as these that follow, viz.

1. The vibrations in very small arches are performed nearly in equal times. Could the pendulum vibrate in the chord of a circle, then all its vibrations would be performed exactly in equal times; as the time of falling through all the chords of a circle is the same. And when the arches are very small, they nearly coincide with the chords. But when it vibrates in larger arches, it will lose time, as they are considerably longer than their correspondent chords, and require a longer time for their description by the pendulum.

2. The velocity of the pendulum at the lowest point is nearly as the length of the chord of the arch it has described in its descent: and is just sufficient to carry it up to the same altitude on the other side, did it meet with no resistance from the air.

3. The time of a complete vibration, from its greatest altitude on one side to its greatest altitude on the other, is equal to the time of descent through the diameter of a circle whose radius is four times the length of the pendulum, or through a space that is eight times the length of the pendulum.* For in half the time of a

* It is demonstrable, that the time of one entire vibration in a very small arch of a circle, or in any arch of a cycloid, is to the time of a heavy body's descending through half the length of the pendulum, as the circumference of a circle to its diameter. Or, which is the same thing, the constant number 4.93482 mL - pied by the length of the pendulum, will give the space which a body will descend by its own gravity in the time of one vibration.-Ed.

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vibration, a heavy body would descend through the diameter of a circle, whose radius is the length of the pendulum, and in the other half of the time, the heavy body continuing its motion, will have passed over four times the space counted from the beginning of the descent; that is eight times the length of the pendulum.

4. The times of vibrations of different pendulums are as the square roots of their lengths. Because they are proportional to the times of descent through spaces equal to eight times the lengths of the pendulums, and these times of descent are as the square roots of the spaces.

5. Since the lengths of pendulums are as the squares of the times of vibration, if a pendulum, whose length is 39.2 inches vibrate once in a second, as it is found to do, then a pendulum of one fourth of this length, that is 9.8 inches, will vibrate in half that time.

6. The rod of the pendulum is here supposed without weight; but if a uniform body, one third longer than the pendulum, were suspended by the end, it would vibrate in the same time with the pendulum. For the whole force of such a body is the same as if the whole matter was collected in a point two thirds distant from the point of suspension. This point is called the center of oscillation, or of percussion; because it will give the greatest stroke possible.

The length of a pendulum and the time of its vibration depending upon each other, it cannot be an exact measure of time, unless we could secure the continuance of the same length in all the varieties of heat and cold. If the rod of the pendulum be made of metal, it will expand and lengthen by heat, and contract by cold; so that in the first case the pendulum will lose time, and in the last gain time. By many experiments made with the pyrometer, it has been found, that with the same

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