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.
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.
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.
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.
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.
If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii
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.
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.
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.
From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle by means of a given centri describes in any time, is a mean proportional between the petal force, diameter of the circle, and the space which the same body falling by the force would descend through in the same given time. same given
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 to centripetal force decreasing in a duplicate ratio things which relate the centres.
from distances the of of the preceding Proposition and its Corollaries, we means Moreover, by discover the proportion of a centripetal force to any other known For if a body by means of its gravity re force, such as that of gravity. may volves in a circle concentric to the earth, this gravity is the centripetal But from the descent of heavy bodies, the time of one force of that body. entire revolution, as well as the arc described in any given time, is given such propositions, Mr. Huygens, in his Oscillatorio, has compared the force of
(by Cor. 9 of this Prop.).
De Horologio the with centrifugal forces of revolving bodies. gravity The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number excellent book And of sides. polygon, if a body, is reflected moved with a given from the velocity along the sides of the angular points, the force, circle at the several 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 ve that is (if the species of locity and the number of reflections conjunctly
the polygon be given), as the length described in that given time, and in creased or diminished in the ratio of the same length to the radius of the that is, as the square of that length applied to the radius and therefore the polygon, by having its sides diminished in inftnitum, coin cides with the circle, as the square of the arc described in a given time ap This is the centrifugal force, with which the body plied to the radius. circle
impels the circle and which the contrary to continually repels the body towards the centre,