"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."