George William Hill

"For more than sixty years after the publication of the Principia astronomers were puzzled to account for the motion of the lunar perigee, simply because they could not conceive that terms of the second and higher orders, with respect to the disturbing force, produced more than half of it. For a similar reason, the great inequalities of Jupiter and Saturn remained a long time unexplained."

"It was not until 1748 that any computation of the perturbations of Jupiter and Saturn, in accordance with the theory of gravitation, was undertaken. This was by . He appears to have limited himself to the terms which have the mean elongation of the planets of the planets from each other as their argument. Later the terms factored by the simple power of the eccentricities were added by himself, , , , and . But these terms not bringing about a reconciliation between observation and theory, and were led to make their notable researches on the possibility of secular equations in the mean motions of the planet. At length the whole difficulty with Jupiter and Saturn was removed by discovery of the great inequalities in 1786. almost immediately constructed tables which far exceeded in accuracy any previously possessed. They are those that appear in the third edition of Astronomie."

"The application of mathematics to the solution of the problems presented by the motion of the heavenly bodies has had a larger degree of success than the same application in the case of the other departments of physics. This is probably due to two causes. The principal objects to be treated in the former case are visible every clear night, consequently the questions connected with them received earlier attention; while, in the latter case, the phenomena to be discussed must ofttimes be produced by artificial means in the laboratory; and the discovery of certain classes of them, as, for instance, the property of magnetism, may justly be attributed to accident. A second cause is undoubtedly to be found in the fact that the application of quantitative reasoning to what is usually denominated as physics generally leads to a more difficult department of mathematics than in the case of the motion of the heavenly bodies. In the latter we have but one independent variable, the time; while in the former generally several are present, which makes the difference of having to integrate ordinary differential equations or those which are partial."

"In the papers published by Hill in the American Journal of Mathematics there is introduced for the first time a very radical and important idea. Up to this time the orbits of the moon and planets were considered as being ellipses which continually change. The problem was to find the changes in the ellipses, or the deviations from the initial ellipses. That is, the ellipse was taken as a first approximation to the orbit of the body under consideration. Hill proposed to take a certain simple type of periodic orbit as a first approximation. He proved the existence of the periodic orbits by numerically integrating the differential equations in numerous special cases by a process known as mechanical quadratures. These were the first periodic orbits of the problem of three bodies having a practical use, and the first ones known to exist beyond the simple ones which were discovered by Lagrange. It should be added that Hill omitted a small part of the disturbing action of the sun, viz., that which is said to depend upon the solar parallax; but his method would have applied without sensible modification to the rigorous problem. In fact, in all his researches, on the problem of three bodies, Darwin used methods which differ from those of Hill only in the variables employed and in inconsequential details."