=

From this value of the arc, the length of the line MA is found to be 161 feet when ME 240,000 miles. That is, a body at the Moon's distance falls as far in one minute as it would do on the Earth's surface in one second-that is, it falls a distance 60 times less. A body on the Earth's surface is 4,000 miles from the Earth's centre, whereas the Moon lies at a distance of 240,000 from that centre-that is, exactly (or exactly enough for our present purpose) 60 times more distant.

613. It is found, therefore, that the deflection produced in the Moon's orbit from the tangent to its path in one second is precisely of a foot. Here we see that, as the Moon is sixty times further from the Earth's centre than a stone at the Earth's surface, it is attracted to the Earth 60 X 60, or 3600 times less. In fact, the force is seen experimentally to vary inversely as the square of the distance of the falling body from the surface. It was this calculation that revealed to Newton the law of universal gravitation.

614. Long before Newton's discovery, Kepler, from observations of the planets merely, had detected certain laws of their motion, which bear his name. They are as follows:

I. Each planet describes round the Sun an orbit of elliptic form, and the centre of the Sun occupies one of the foci.

II. The areas described by the radius-vector of a planet are proportional to the time taken in describing them.

III. If the squares of the times of revolution of the planets round the Sun be divided by the cubes of their mean distances, the quotient will be the same for all the planets.

615. We have already in many places referred to the first law II. and III. require special explanation, which we will give in this place. We stated in Art. 293 that the planets moved faster as they approached the Sun; II. tells how much faster. The radius-vector of a planet is the line joining the planet and the Sun. If the planet were always at the same distance from the Sun, the radiusvector would not vary in length; but in elliptic orbits its length varies; and the shorter it becomes, the more rapidly does the planet progress. This law gives the exact measure of the increase or decrease of the rapidity.

PL

P

4

Fig. 67.-Explanation of Kepler's second law.

616. In Fig. 67 are given the orbit of a planet and the Sun situated in one of the foci, the ellipticity of the planet's orbit being exaggerated to make the explanation clearer. The areas of the three shaded portions are equal to each other. It is readily seen that where the radiusvector is longest, the path of the planet intercepted is shortest, and vice versa. This, of course, is necessary to

produce the equal areas. In the figure, the arcs P P1 P2 P3, and P4 P5, are those described at mean distances. perihelion and aphelion respectively, in equal times; therefore, as a greater distance has to be got over at perihelion and a less one at aphelion than when the planet is situated at its mean distance, the motion in the former case must be more rapid, and in the latter case slower, than in other parts of the orbit.

617. The third law shows that the periodic time of a planet and its distance from the sun are in some way bound together, so that if we represent the Earth's distance and periodic time by 1, we can at once determine the distance of, say, Jupiter from the Sun, by a simple proportion; thus

That is, whatever the distance of the Earth from the Sun may be, the distance of Jupiter is 2/140 times greater.

618. The following table shows the truth of the law we are considering :—

LESSON XLIX.

KEPLER'S SECOND LAW PROVED. CENTRIFUGAL TENDENCY. CENTRIPETAL FORCE. KEPLER'S THIRD LAW PROVED. THE CONIC SECTIONS. MOVEMENT IN AN ELLIPSE.

619. As these laws were given to the world by Kepler, they simply represented facts; for, owing to the backward state of the mechanical and mathematical sciences in his time, he was unable to see their hidden meaning. This was reserved for the genius of Sir Isaac Newton, after Kepler's time.

Fig. 68.-Proof of Kepler's second law.

620. Newton showed that all these laws established the truth of the law of gravitation, and flowed naturally from it. In Fig. 68, let S represent the centre of the Sun, and P a planet, at a given moment. During a very short time this planet will describe a part of its orbit PP', and its radius-vector will have swept over the area PSP'. If no new force intervene, in another similar interval the planet will have reached P", the area P'SP" being equal to PSP' according to Kepler's second law. But the planet will really describe the arc P'B, and the area P'SB will be equal to P'SP"; as the triangles are equal, and on the same base, the iine P'B will be parallel to P'S; and completing the parallelogram P'P" BC, we sce that the

planet at P' was acted upon by two forces, measured by P'P" and P'C—that is, by its initial velocity and a force directed to the Sun. Hence Kepler's second law shows that this force is directed towards the Sun.

621. A good idea of the tendency of bodies to keep in the direction of their original motion may be gained by attaching a small bucket, nearly filled with water, to a rope, and by swinging it round gently; the tendency of the water to fly off will prevent its falling out of the bucket; and it will be found that the more rapidly the bucket is whirled round, the greater will be this tendency, and therefore the tighter will be the rope.

622. The circular movement of the bucket is represented in Fig. 69. A represents the bucket, OA the rope ; let us suppose that the bucket receives an impulse which.

in the absence of the rope, would have sent it in the direction AB with an uniform motion. In a very short time, being held by the rope, it will arrive at c, and Ad measures the force applied by the rope. Call this force f,