A Short History of Astronomy - Arthur Berry (read along books .txt) 📗
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Ten years later he discussed in some detail the perturbations of the earth due to Venus and to the moon. This paper was remarkable as containing the first attempt to estimate masses of celestial bodies by observation of perturbations due to them. Clairaut applied this method to the moon and to Venus, by calculating perturbations in the earth’s motion due to their action (which necessarily depended on their masses), and then comparing the results with Lacaille’s observations of the sun. The mass of the moon was thus found to be about 1∕67 and that of Venus 2∕3 that of the earth; the first result was a considerable improvement on Newton’s estimate from tides (chapter IX., § 189), and the second, which was entirely new, previous estimates having been merely conjectural, is in tolerable agreement with modern measurements.139 It is worth noticing as a good illustration of the reciprocal influence of observation and mathematical theory that, while Clairaut used Lacaille’s observations for his theory, Lacaille in turn used Clairaut’s calculations of the perturbations of the earth to improve his tables of the sun published in 1758.
Clairaut’s method of solving the problem of three bodies was also applied by Joseph Jérôme Le François Lalande (1732-1807), who is chiefly known as an admirable populariser of astronomy but was also an indefatigable calculator and observer, to the perturbations of Mars by Jupiter, of Venus by the earth, and of the earth by Mars, but with only moderate success.
D’Alembert made some progress with the general treatment of planetary perturbations in the second volume of his Recherches, and applied his methods to Jupiter and Saturn.
236. Euler carried the general theory a good deal further in a series of papers beginning in 1747. He made several attempts to explain the irregularities of Jupiter and Saturn, but never succeeded in representing the observations satisfactorily. He shewed, however, that the perturbations due to the other planets would cause the earth’s apse line to advance about 13″ annually, and the obliquity of the ecliptic to diminish by about 48″ annually, both results being in fair accordance both with observations and with more elaborate calculations made subsequently. He indicated also the existence of various other planetary irregularities, which for the most part had not previously been observed.
In an essay to which the Academy awarded a prize in 1756, but which was first published in 1771, he developed with some completeness a method of dealing with perturbations which he had indicated in his lunar theory of 1753. As this method, known as that of the variation of the elements or parameters, played a very important part in subsequent researches, it may be worth while to attempt to give a sketch of it.
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.
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 limits140) 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 these141 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.
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During this period he produced an extraordinary series of papers on astronomy, on general dynamics, and on a variety of subjects in pure mathematics. Several of the most important of the astronomical papers were sent to Paris and obtained prizes offered by the Academy; most of the other papers—about 60 in all—were published by the Berlin Academy. During this period he wrote also his great Mécanique Analytique, one of the most beautiful of all mathematical books, in which he developed fully the general dynamical ideas contained in the earlier paper on libration. Curiously enough he had great difficulty in finding a publisher for his masterpiece, and it only appeared in 1788 in Paris. A year earlier he had left Berlin in consequence of the death of Frederick, and accepted an invitation from Louis XVI. to join the Paris Academy. About this time he suffered from one of the fits of melancholy with which he was periodically seized and which are generally supposed to have been due to overwork during his career at Turin. It is said that he never looked at the Mécanique Analytique for two years after its publication, and spent most of the time over chemistry and other branches of natural science as well as in non-scientific pursuits. In 1790 he was made president of the Commission appointed to draw up a new system of weights and measures, which resulted in the establishment of the metric system; and the scientific work connected with this undertaking gradually restored his interest in mathematics and astronomy. He always avoided politics, and passed through the Revolution uninjured, unlike his friend Lavoisier the great chemist and Bailly the historian of astronomy, both of whom were guillotined during the Terror. He was in fact held in great honour by the various governments which ruled France up to the time of his death; in 1793 he was specially exempted from a decree of banishment directed against all foreigners; subsequently he was made professor of mathematics, first at the École Normale (1795), the École Polytechnique (1797), the last appointment being retained till his death in 1813. During this period of his life he published, in addition to a large number of papers on astronomy and mathematics, three important books on pure mathematics,143 and at the time of his death had not quite finished a second edition of the Mécanique Analytique, the second volume appearing posthumously.
238. Pierre Simon Laplace, the son of a small farmer, was born at Beaumont in Normandy in 1749, being thus 13 years younger than his great rival Lagrange. Thanks to the help of well-to-do neighbours, he was first a pupil and afterwards a teacher at the Military School of his native town. When he was 18 he went to Paris with a letter of introduction to D’Alembert, and, when no notice was taken of it, wrote him a letter on the principles of mechanics which impressed D’Alembert so much that he at once took interest in the young mathematician and procured him an appointment at the Military School at Paris. From this time onwards Laplace lived continuously at Paris, holding various official positions. His first paper (on pure mathematics) was published in the Transactions of the Turin Academy for the years 1766-69, and from this time to the end of his life he produced an uninterrupted series of papers and books on astronomy and allied departments of mathematics.
Laplace’s work on astronomy was to a great extent incorporated
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