Proposition 4

Proposition 4 Theorem 4

The centripetal forces of bodies, which by equable motions describe different circles:

• tend to the centres of the same circles (Prop. 2 and Cor. 2, Prop. 50)
• are one to the other as the squares of the arcs described in equal times applied to the radii of the circles (by Cor. 4, Prop. 1)

Therefore, since those arcs are as arcs described in any equal times, and the diameters are as the radii. And so the forces will be as the squares of any area described in the same time applied to the radii of the circles. QED.

Corollary 1

Therefore, since those arcs are as the velocities of the bodies, the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the simple ratio of the radii inversely.

Corollary 2

Since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely, the centripetal forces, are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely.

Corollary 3

Whence if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii tke contrary.

Corollary 4

If the periodic times and the velocities are both in the subduplicate ratio of the radii, the centripetal forces will be equal among themselves and the contrary.

Corollary 5

If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii

Corollary 6

If the periodic times are in the sesquiplicate ratio of the radii, and therefore the velocities reciprocally in the subduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii inversely, and the contrary.

Corollary 7

Universally, if the periodic time is as any power Rn of the radius R, and therefore the velocity reciprocally as the power Rn-1 of the radius, the centripetal force will be reciprocally as the power R20-1 of the radius; and the contrary.

Corollary 8

The same things all hold concerning the times, the velocities, and forces by which bodies describe the similar parts of any similar figures that have their centres in a similar position with those figures; as appears by applying the demonstration of the preceding cases to those.

The application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii.

Corollary 9

From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time.

SCHOLIUM

The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.

Moreover, by means of the preceding Proposition and its Corollaries, we may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.

The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly: that is (if the species of the polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length applied to the radius; and therefore the polygon, by having its sides diminished in infinitum, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the body impels the circle; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal.