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Also (if these two be granted) it follows that Kepler's third law is only another way of saying that the sun's force on different planets (besides depending as aboye on distance) is proportional to their masses. Having further proved the, for that day, wonderful proposition that, with the law of inverse squares, the attraction by the separate particles of a sphere of uniform density (or one composed of concentric spherical shells, each of uniform density), acts as if the whole mass were collected at the centre, he was able to express the meaning of Kepler's laws in propositions which have been summarised as follows: — The law of universal gravitation. — Every particle of matter in the universe attracts every other particle with a force varying inversely as the square of the distance between them, and directly as the product of the masses of the two particles." But Newton did not commit himself to the law until he had answered that question about * It must be noted that these words, in which the laws of gravitation are always summarised in histories and text-books, do not appear in the Principia; but, though they must have been composed by some early commentator, it does not appear that their origin has been traced. Nor does it appear that Newton ever extended the law beyond the Solar System, and probably his caution would have led him to avoid any statement of the kind until it should be proved. With this exception the above statement of the the apple; and the above proposition now enabled him to deal with the Moon and the apple. Gravity makes a stone fall 16. I feet in a second. The moon is 60 times farther from the earth's centre than the stone, so it ought to be drawn out of a straight course through 16. I feet in a minute. Newton found the distance through which she is actually drawn as a fraction of the earth's diameter. But when he first examined this matter he proceeded to use a wrong diameter for the earth, and he found a serious discrepancy. This, for a time, seemed to condemn his theory, and regretfully he laid that part of his work aside. Fortunately, before Newton wrote the Principia the French astronomer Picard made a new and correct measure of an arc of the meridian, from which he obtained an accurate value of the earth's diameter. Newton applied this value, and found, to his great joy, that when the distance of the moon is 6o times the radius of the earth she is attracted out of the straight course 16. I feet per approach to that statement occurs in the Seventh Proposition of Book III. : — Prop.: That gravitation occurs in all bodies, and that it is proportional to the quantity of matter in “. I.: The total attraction of gravitation on a planet arises, and is composed, out of the attraction on the separate parts. Cor. II.: The attraction on separate equal particles

law of universal gravitation contains nothing that is not to be found in the Principia; and the nearest

of a body is reciprocally as the square of the distance from the particles.

minute, and that the force acting on a stone or an apple follows the same law as the force acting upon the heavenly bodies." The universality claimed for the law — if not by Newton, at least by his commentators — was bold, and warranted only by the large number of cases in which Newton had found it to apply. Its universality has been under test ever since, and so far it has stood the test. There has often been a suspicion of a doubt, when some inequality of motion in the heavenly bodies has, for a time, foiled the astronomers in their attempts to explain it. But improved mathematical methods have always succeeded in the end, and so the seeming doubt has been converted into a surer conviction of the universality of the law. Having once established the law, Newton proceeded to trace some of its consequences. He saw that the figure of the earth depends partly on the mutual gravitation of its parts, and partly on the centrifugal tendency due to the earth's rotation, and that these should cause a flattening of the poles. He invented a mathematical method which he used for computing the ratio of the polar to the equatorial diameter. He then noticed that the consequent bulging * It is said that, when working out this final result, the probability of its confirming that part of his theory which he had reluctantly abandoned years before of matter at the equator would be attracted by the moon unequally, the nearest parts being most attracted; and so the moon would tend to tilt the earth when in some parts of her orbit; and the sun would do this to a less extent, because of its great distance. Then he proved that the effect ought to be a rotation of the earth's axis over a conical surface in space, exactly as the axis of a top describes a cone, if the top has a sharp point, and is set spinning and displaced from the vertical. He actually calculated the amount; and so he explained the cause of the precession of the equinoxes discovered by Hipparchus about 150 B.C. One of his grandest discoveries was a method of weighing the heavenly bodies by their action on each other. By means of this principle he was able to compare the mass of the sun with the masses of those planets that have moons, and also to compare the mass of our moon with the mass of the earth. Thus Newton, after having established his great principle, devoted his splendid intellect to the calculation of its consequences. He proved that if a body be projected with any velocity in free space, subject only to a central force, varying inversely as the square of the distance, the body must revolve in a curve which may be any one of the sections of a cone — a circle, ellipse, parabola, or hyperbola; and he found that those comets of which he had observations move in parabolae round the sun, and are thus subject to the universal law. Newton realised that, while planets and satellites are chiefly controlled by the central body about which they revolve, the new law must involve irregularities, due to their mutual action — such, in fact, as Horrocks had indicated. He determined to put this to a test in the case of the moon, and to calculate the sun's effect, from its mass compared with that of the earth, and from its distance. He proved that the average effect upon the plane of the orbit would be to cause the line in which it cuts the plane of the ecliptic (i.e., the line of nodes) to revolve in the ecliptic once in about nineteen years. This had been a known fact from the earliest ages. He also concluded that the line of apses would revolve in the plane of the lunar orbit also in about nineteen years; but the observed period is only ten years. For a long time this was the one weak point in the Newtonian theory. It was not till 1747 that Clairaut reconciled this with the theory, and showed why Newton's calculation was not exact. Newton proceeded to explain the other inequalities recognised by Tycho Brahe and older observers, and to calculate their maximum amounts as indicated by his theory. He further discovered from his calculations two new in

excited him so keenly that he was forced to hand over his calculations to a friend, to be completed by him.

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