But Newton did not commit himself to the law until he had answered that question about the apple; and the above proposition now enabled him to deal with the Moon and the apple. Gravity makes a stone fall 16.1 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 16.1 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 60 times the radius of the earth she is attracted out of the straight course 16.1 feet per minute, and that the force acting on a stone or an apple follows the same law as the force acting upon the heavenly bodies.' gravitation contains nothing that is not to be found in the Principia; and the nearest 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 each. Cor. 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 of a body is reciprocally as the square of the distance from the particles. 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 excited him so keenly that he was forced to hand over his calculations to a friend, to be completed by him. 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 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 E 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 parabolæ 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 exact. calculation was not 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 inequalities, one of the apogee, the other of the nodes, and assigned the maximum value. Grant has shown the values of some of these as given by observation in the tables of Meyer and more modern tables, and has compared them with the values assigned by Newton from his theory; and the comparison is very remarkable. 19.43 ... 22.17 9.24 ... 9.0 Inequality of mean motion of apogee The only serious discrepancy is the first, which has been already mentioned. Considering that some of these perturbations had never been discovered, that the cause of none of them had ever been known, and that he exhibited his results, if he did not also make the discoveries, by the synthetic methods of geometry, it is simply marvellous that he reached to such a degree of accuracy. He invented the infinitesimal calculus which is more suited for such calculations, but had he expressed his results in that language he would have been unintelligible to many. Newton's method of calculating the precession of the equinoxes, already referred. to, is as beautiful as anything in the Principia. He had already proved the regression of the nodes of a satellite moving in an orbit inclined to the ecliptic. He now said that the nodes of a ring of satellites revolving round the earth's equator would consequently all regress. And if joined into a solid ring its node would regress; and it would do so, only more slowly, if encumbered by the spherical part of the earth's mass. Therefore the axis of the equatorial belt of the earth must revolve round the pole of the ecliptic. Then he set to work and found the amount due to the moon and that due to the sun, and so he solved the mystery of 2,000 years. When Newton applied his law of gravitation to an explanation of the tides he started a new field for the application of mathematics to physical problems; and there can be little doubt that, if he could have been furnished with complete tidal observations from different parts of the world, his extraordinary powers of analysis would have enabled him to reach a satisfactory theory. He certainly opened up many mines full of intellectual. gems; and his successors have never ceased in their explorations. This has led to improved mathematical methods, which, combined with the greater accuracy of observation, have rendered physical astronomy of to-day the most exact of the sciences. Laplace only expressed the universal opinion of posterity when he said that to the Principia is assured a pre-eminence above all the other productions of the human intellect." The name of Flamsteed, First Astronomer Royal, must here be mentioned as having supplied Newton with the accurate data required for completing the theory. The name of Edmund Halley, Second Astronomer Royal, must ever be held in repute, not only for his own discoveries, but for the part he played in urging |