If perturbations are ignored, a planet can be regarded as moving in an ellipse with the sun in one focus. The size and shape of the ellipse can be defined by the length of its axis and by the eccentricity; the plane in which the ellipse is situated is determined by the position of the line, called the line of nodes, in which it cuts a fixed plane, usually taken to be the ecliptic, and by the inclination of the two planes. When these four quantities are fixed, the ellipse may still turn about its focus in its own plane, but if the direction of the apse line is also fixed the ellipse is completely determined. If, further, the position of the planet in its ellipse at any one time is known, the motion is completely determined and its position at any other time can be calculated. There are thus six quantities known as elements which completely determine the motion of a planet not subject to perturbation.
When perturbations are taken into account, the path described by a planet in any one revolution is no longer an ellipse, though it differs very slightly from one; while in the case of the moon the deviations are a good deal greater. But if the motions of a planet at two widely different epochs are compared, though on each occasion the path described is very nearly an ellipse, the ellipses differ in some respects. For example, between the time of Ptolemy (A.D. 150) and that of Euler the direction of the apse line of the earth’s orbit altered by about 5°, and some of the other elements also varied slightly. Hence in dealing with the motion of a planet through a long period of time it is convenient to introduce the idea of an elliptic path which is gradually changing its position and possibly also its size and shape. One consequence is that the actual path described in the course of a considerable number of revolutions is a curve no longer bearing much resemblance to an ellipse. If, for example, the apse line turns round uniformly while the other elements remain unchanged, the path described is like that shewn in the figure.
Fig. 81.—A varying ellipse.
Euler extended this idea so as to represent any perturbation of a planet, whether experienced in the course of one revolution or in a longer time, by means of changes in an elliptic orbit. For wherever a planet may be and whatever (within certain limits[140]) be its speed or direction of motion some ellipse can be found, having the sun in one focus, such that the planet can be regarded as moving in it for a short time. Hence as the planet describes a perturbed orbit it can be regarded as moving at any instant in an ellipse, which, however, is continually altering its position or other characteristics. Thus the problem of discussing the planet’s motion becomes that of determining the elements of the ellipse which represents its motion at any time. Euler shewed further how, when the position of the perturbing planet was known, the corresponding rates of change of the elements of the varying ellipse could be calculated, and made some progress towards deducing from these data the actual elements; but he found the mathematical difficulties too great to be overcome except in some of the simpler cases, and it was reserved for the next generation of mathematicians, notably Lagrange, to shew the full power of the method.
237. Joseph Louis Lagrange was born at Turin in 1736, when Clairaut was just starting for Lapland and D’Alembert was still a child; he was descended from a French family three generations of which had lived in Italy. He shewed extraordinary mathematical talent, and when still a mere boy was appointed professor at the Artillery School of his native town, his pupils being older than himself. A few years afterwards he was the chief mover in the foundation of a scientific society, afterwards the Turin Academy of Sciences, which published in 1759 its first volume of Transactions, containing several mathematical articles by Lagrange, which had been written during the last few years. One of these[141] so impressed Euler, who had made a special study of the subject dealt with, that he at once obtained for Lagrange the honour of admission to the Berlin Academy.
In 1764 Lagrange won the prize offered by the Paris Academy for an essay on the libration of the moon. In this essay he not only gave the first satisfactory, though still incomplete, discussion of the librations (chapter vi., [§ 133]) of the moon due to the non-spherical forms of both the earth and moon, but also introduced an extremely general method of treating dynamical problems,[142] which is the basis of nearly all the higher branches of dynamics which have been developed up to the present day.
Two years later (1766) Frederick II., at the suggestion of D’Alembert, asked Lagrange to succeed Euler (who had just returned to St. Petersburg) as the head of the mathematical section of the Berlin Academy, giving as a reason that the greatest king in Europe wished to have the greatest mathematician in Europe at his court. Lagrange accepted this magnificently expressed invitation and spent the next 21 years at Berlin.