A Short History of Astronomy by Arthur Berry (best novels for students .TXT) 📕

title="295"> perihelion (the point of its orbit nearest the sun, P in fig. 80) about April 13th of the following year, though owing to various defects in his calculation there might be an error of a month either way. The comet was anxiously watched for by the astronomical world, and was actually discovered by an amateur, George Palitzsch (1723-1788) of Saxony, on Christmas Day, 1758; it passed its perihelion just a month and a day before the time assigned by Clairaut.

Halley’s brilliant conjecture was thus justified; a new member was added to the solar system, and hopes were raised—to be afterwards amply fulfilled—that in other cases also the motions of comets might be reduced to rule, and calculated according to the same principles as those of less erratic bodies. The superstitions attached to comets were of course at the same time still further shaken.

Clairaut appears to have had great personal charm and to have been a conspicuous figure in Paris society. Unfortunately his strength was not equal to the combined claims of social and scientific labours, and he died in 1765 at an age when much might still have been hoped from his extraordinary abilities.133

232. Jean-le-Rond D’Alembert was found in 1717 as an infant on the steps of the church of St. Jean-le-Rond in Paris, but was afterwards recognised, and to some extent provided for, by his father, though his home was with his foster parents. After receiving a fair school education, he studied law and medicine, but then turned his attention to mathematics. He first attracted notice in mathematical circles by a paper written in 1738, and was admitted to the Academy of Sciences two years afterwards. His earliest important work was the Traité de Dynamique (1743), which contained, among other contributions to the subject, the first statement of a dynamical principle which bears his name, and which, though in one sense only a corollary from Newton’s Third Law of Motion, has proved to be of immense service in nearly all general dynamical problems, astronomical or otherwise. During the next few years he made a number of contributions to mathematical physics, as well as to the problem of three bodies; and published in 1749 his work on precession and nutation, already referred to (chapter X., § 215). From this time onwards he began to give an increasing part of his energies to work outside mathematics. For some years he collaborated with Diderot in producing the famous French Encyclopaedia, which began to appear in 1751, and exercised so great an influence on contemporary political and philosophic thought. D’Alembert wrote the introduction, which was read to the Académie Française134 in 1754 on the occasion of his admission to that distinguished body, as well as a variety of scientific and other articles. In the later part of his life, which ended in 1783, he wrote little on mathematics, but published a number of books on philosophical, literary, and political subjects;135 as secretary of the Academy he also wrote obituary notices (éloges) of some 70 of its members. He was thus, in Carlyle’s words, “of great faculty, especially of great clearness and method; famous in Mathematics; no less so, to the wonder of some, in the intellectual provinces of Literature.”

D’Alembert and Clairaut were great rivals, and almost every work of the latter was severely criticised by the former, while Clairaut retaliated though with much less zeal and vehemence. The great popular reputation acquired by Clairaut through his work on Halley’s comet appears to have particularly excited D’Alembert’s jealousy. The rivalry, though not a pleasant spectacle, was, however, useful in leading to the detection and subsequent improvement of various weak points in the work of each. In other respects D’Alembert’s personal characteristics appear to have been extremely pleasant. He was always a poor man, but nevertheless declined magnificent offers made to him by both Catherine II. of Russia and Frederick the Great of Prussia, and preferred to keep his independence, though he retained the friendship of both sovereigns and accepted a small pension from the latter. He lived extremely simply, and notwithstanding his poverty was very generous to his foster-mother, to various young students, and to many others with whom he came into contact.

233. Euler, Clairaut, and D’Alembert all succeeded in obtaining independently and nearly simultaneously solutions of the problem of three bodies in a form suitable for lunar theory. Euler published in 1746 some rather imperfect Tables of the Moon, which shewed that he must have already obtained his solution. Both Clairaut and D’Alembert presented to the Academy in 1747 memoirs containing their respective solutions, with applications to the moon as well as to some planetary problems. In each of these memoirs occurred the same difficulty which Newton had met with: the calculated motion of the moon’s apogee was only about half the observed result. Clairaut at first met this difficulty by assuming an alteration in the law of gravitation, and got a result which seemed to him satisfactory by assuming gravitation to vary partly as the inverse square and partly as the inverse cube of the distance.136 Euler also had doubts as to the correctness of the inverse square. Two years later, however (1749), on going through his original calculation again, Clairaut discovered that certain terms, which had appeared unimportant at the beginning of the calculation and had therefore been omitted, became important later on. When these were taken into account, the motion of the apogee as deduced from theory agreed very nearly with that observed. This was the first of several cases in which a serious discrepancy between theory and observation has at first discredited the law of gravitation, but has subsequently been explained away, and has thereby given a new verification of its accuracy. When Clairaut had announced his discovery, Euler arrived by a fresh calculation at substantially the same result, while D’Alembert by carrying the approximation further obtained one that was slightly more accurate. A fresh calculation of the motion of the moon by Clairaut won the prize on the subject offered by the St. Petersburg Academy, and was published in 1752, with the title Théorie de la Lune. Two years later he published a set of lunar tables, and just before his death (1765) he brought out a revised edition of the Théorie de la Lune in which he embodied a new set of tables.

D’Alembert followed his paper of 1747 by a complete lunar theory (with a moderately good set of tables), which, though substantially finished in 1751, was only published in 1754 as the first volume of his Recherches sur différens points importans du système du Monde. In 1756 he published an improved set of tables, and a few months afterward a third volume of Recherches with some fresh developments of the theory. The second volume of his Opuscules Mathématiques (1762) contained another memoir on the subject with a third set of tables, which were a slight improvement on the earlier ones.

Euler’s first lunar theory (Theoria Motuum Lunae) was published in 1753, though it had been sent to the St. Petersburg Academy a year or two earlier. In an appendix137 he points out with characteristic frankness the defects from which his treatment seems to him to suffer, and suggests a new method of dealing with the subject. It was on this theory that Tobias Mayer based his tables, referred to in the preceding chapter (§ 226). Many years later Euler devised an entirely new way of attacking the subject, and after some preliminary papers dealing generally with the method and with special parts of the problem, he worked out the lunar theory in great detail, with the help of one of his sons and two other assistants, and published the whole, together with tables, in 1772. He attempted, but without success, to deal in this theory with the secular acceleration of the mean motion which Halley had detected (chapter X., § 201).

In any mathematical treatment of an astronomical problem some data have to be borrowed from observation, and of the three astronomers Clairaut seems to have been the most skilful in utilising observations, many of which he obtained from Lacaille. Hence his tables represented the actual motions of the moon far more accurately than those of D’Alembert, and were even superior in some points to those based on Euler’s very much more elaborate second theory; Clairaut’s last tables were seldom in error more than 1-1∕2′, and would hence serve to determine the longitude to within about 3∕4°. Clairaut’s tables were, however, never much used, since Tobias Mayer’s as improved by Bradley were found in practice to be a good deal more accurate; but Mayer borrowed so extensively from observation that his formulae cannot be regarded as true deductions from gravitation in the same sense in which Clairaut’s were. Mathematically Euler’s second theory is the most interesting and was of the greatest importance as a basis for later developments. The most modern lunar theory138 is in some sense a return to Euler’s methods.

234. Newton’s lunar theory may be said to have given a qualitative account of the lunar inequalities known by observation at the time when the Principia was published, and to have indicated others which had not yet been observed. But his attempts to explain these irregularities quantitatively were only partially successful.

Euler, Clairaut, and D’Alembert threw the lunar theory into an entirely new form by using analytical methods instead of geometrical; one advantage of this was that by the expenditure of the necessary labour calculations could in general be carried further when required and lead to a higher degree of accuracy. The result of their more elaborate development was that—with one exception—the inequalities known from observation were explained with a considerable degree of accuracy quantitatively as well as qualitatively; and thus tables, such as those of Clairaut, based on theory, represented the lunar motions very closely. The one exception was the secular acceleration: we have just seen that Euler failed to explain it; D’Alembert was equally unsuccessful, and Clairaut does not appear to have considered the question.

235. The chief inequalities in planetary motion which observation had revealed up to Newton’s time were the forward motion of the apses of the earth’s orbit and a very slow diminution in the obliquity of the ecliptic. To these may be added the alterations in the rates of motion of Jupiter and Saturn discovered by Halley (chapter X., § 204).

Newton had shewn generally that the perturbing effect of another planet would cause displacements in the apses of any planetary orbit, and an alteration in the relative positions of the planes in which the disturbing and disturbed planet moved; but he had made no detailed calculations. Some effects of this general nature, in addition to those already known, were, however, indicated with more or less distinctness as the result of observation in various planetary tables published between the date of the Principia and the middle of the 18th century.

The irregularities in the motion of the earth, shewing themselves as irregularities in the apparent motion of the sun, and those of Jupiter and Saturn, were the most interesting and important of the planetary inequalities, and prizes for essays on one or another subject were offered several times by the Paris Academy.

The perturbations of the moon necessarily involved—by the principle of action and reaction—corresponding though smaller perturbations of the earth; these were discussed on various occasions by Clairaut and Euler, and still more fully by D’Alembert.

In Clairaut’s paper of 1747 (§ 233) he made some attempt to apply his solution of the problem of three bodies to the case of the sun, earth, and Saturn, which on