A Short History of Astronomy, Arthur Berry [large screen ebook reader .TXT] 📗

Again, the solar system happens to be so constituted that each body’s motion can be treated as determined primarily by one other body only. A planet, for example, moves nearly as if no other body but the sun existed, and the moon’s motion relative to the earth is roughly the same as if the other bodies of the solar system were non-existent.

The problem of the motion of two mutually gravitating spheres was completely solved by Newton, and was shewn to lead to Kepler’s first two laws. Hence each body of the solar system could be regarded as moving nearly in an ellipse round some one body, but as slightly disturbed by the action of others. Moreover, by a general mathematical principle applicable in problems of motion, the effect of a number of small disturbing causes acting conjointly is nearly the same as that which results from adding together their separate effects. Hence each body could, without great error, be regarded as disturbed by one body at a time; the several disturbing effects could then be added together, and a fresh calculation could be made to further diminish the error. The kernel of Newton’s problem is thus seen to be a special case of the so-called problem of three bodies, viz.:—

Given at any time the positions and motions of three mutually gravitating bodies, to determine their positions and motions at any other time.

Even this apparently simple problem in its general form entirely transcends the powers, not only of the mathematical methods of the early 18th century, but also of those that have been devised since. Certain special cases have been solved, so that it has been shewn to be possible to suppose three bodies initially moving in such a way that their future motion can be completely determined. But these cases do not occur in nature.

In the case of the solar system the problem is simplified, not only by the consideration already mentioned that one of the three bodies can always be regarded as exercising only a small influence on the relative motion of the other two, but also by the facts that the orbits of the planets and satellites do not differ much from circles, and that the planes of their orbits are in no case inclined at large angles to any one of them, such as the ecliptic; in other words, that the eccentricities and inclinations are small quantities.

Thus simplified, the problem has been found to admit of solutions of considerable accuracy by methods of approximation.131

In the case of the system formed by the sun, earth, and moon, the characteristic feature is the great distance of the sun, which is the disturbing body, from the other two bodies; in the case of the sun and two planets, the enormous mass of the sun as compared with the disturbing planet is the important factor. Hence the methods of treatment suitable for the two cases differ, and two substantially distinct branches of the subject, lunar theory and planetary theory, have developed. The problems presented by the motions of the satellites of Jupiter and Saturn, though allied to those of the lunar theory, differ in some important respects, and are usually treated separately.

229. As we have seen, Newton made a number of important steps towards the solution of his problem, but little was done by his successors in his own country. On the Continent also progress was at first very slow. The Principia was read and admired by most of the leading mathematicians of the time, but its principles were not accepted, and Cartesianism remained the prevailing philosophy. A forward step is marked by the publication by the Paris Academy of Sciences in 1720 of a memoir written by the Chevalier de Louville (1671-1732) on the basis of Newton’s principles; ten years later the Academy awarded a prize to an essay on the planetary motions written by John Bernouilli (1667-1748) on Cartesian principles, a Newtonian essay being put second. In 1732 Maupertuis (chapter X., § 221) published a treatise on the figure of the earth on Newtonian lines, and the appearance six years later of Voltaire’s extremely readable Éléments de la Philosophie de Newton had a great effect in popularising the new ideas. The last official recognition of Cartesianism in France seems to have been in 1740, when the prize offered by the Academy for an essay on the tides was shared between a Cartesian and three eminent Newtonians (§ 230).

The rapid development of gravitational astronomy that ensued between this time and the beginning of the 19th century was almost entirely the work of five great Continental mathematicians, Euler, Clairaut, D’Alembert, Lagrange, and Laplace, of whom the eldest was born in 1707 and the youngest died in 1827, within a month of the centenary of Newton’s death. Euler was a Swiss, Lagrange was of Italian birth but French by extraction and to a great extent by adoption, and the other three were entirely French. France therefore during nearly the whole of the 18th century reigned supreme in gravitational astronomy, and has not lost her supremacy even to-day, though during the present century America, England, Germany, Italy, and other countries have all made substantial contributions to the subject.

It is convenient to consider first the work of the three first-named astronomers, and to treat later Lagrange and Laplace, who carried gravitational astronomy to a decidedly higher stage of development than their predecessors.

230. Leonhard Euler was born at Basle in 1707, 14 years later than Bradley and six years earlier than Lacaille. He was the son of a Protestant minister who had studied mathematics under James Bernouilli (1654-1705), the first of a famous family of mathematicians. Leonhard Euler himself was a favourite pupil of John Bernouilli (the younger brother of James), and was an intimate friend of his two sons, one of whom, Daniel (1700-1782), was not only a distinguished mathematician like his father and uncle, but was also the first important Newtonian outside Great Britain. Like so many other astronomers, Euler began by studying theology, but was induced both by his natural tastes and by the influence of the Bernouillis to turn his attention to mathematics. Through the influence of Daniel Bernouilli, who had recently been appointed to a professorship at St. Petersburg, Euler received and accepted an invitation to join the newly created Academy of Sciences there (1727). This first appointment carried with it a stipend, and the duties were the general promotion of science; subsequently Euler undertook more definite professorial work, but most of his energy during the whole of his career was devoted to writing mathematical papers, the majority of which were published by the St. Petersburg Academy. Though he took no part in politics, Russian autocracy appears to have been oppressive to him, reared as he had been among Swiss and Protestant surroundings; and in 1741 he accepted an invitation from Frederick the Great, a despot of a less pronounced type, to come to Berlin, and assist in reorganising the Academy of Sciences there. On being reproached one day by the Queen for his taciturn and melancholy demeanour, he justified his silence on the ground that he had just come from a country where speech was liable to lead to hanging;132 but notwithstanding this frank criticism he remained on good terms with the Russian court, and continued to draw his stipend as a member of the St. Petersburg Academy and to contribute to its Transactions. Moreover, after 25 years spent at Berlin, he accepted a pressing invitation from the Empress Catherine II. and returned to Russia (1766).

He had lost the use of one eye in 1735, a disaster which called from him the remark that he would henceforward have less to distract him from his mathematics; the second eye went soon after his return to Russia, and with the exception of a short time during which an operation restored the partial use of one eye he remained blind till the end of his life. But this disability made little difference to his astounding scientific activity; and it was only after nearly 17 years of blindness that as a result of a fit of apoplexy “he ceased to live and to calculate” (1783).

Euler was probably the most versatile as well as the most prolific of mathematicians of all time. There is scarcely any branch of modern analysis to which he was not a large contributor, and his extraordinary powers of devising and applying methods of calculation were employed by him with great success in each of the existing branches of applied mathematics; problems of abstract dynamics, of optics, of the motion of fluids, and of astronomy were all in turn subjected to his analysis and solved. The extent of his writings is shewn by the fact that, in addition to several books, he wrote about 800 papers on mathematical and physical subjects; it is estimated that a complete edition of his works would occupy 25 quarto volumes of about 600 pages each.

Euler’s first contribution to astronomy was an essay on the tides which obtained a share of the Academy prize for 1740 already referred to, Daniel Bernouilli and Maclaurin (chapter X., § 196) being the other two Newtonians. The problem of the tides was, however, by no means solved by any of the three writers.

He gave two distinct solutions of the problem of three bodies in a form suitable for the lunar theory, and made a number of extremely important and suggestive though incomplete contributions to planetary theory. In both subjects his work was so closely connected with that of Clairaut and D’Alembert that it is more convenient to discuss it in connection with theirs.

231. Alexis Claude Clairaut, born at Paris in 1713, belongs to the class of precocious geniuses. He read the Infinitesimal Calculus and Conic Sections at the age of ten, presented a scientific memoir to the Academy of Sciences before he was 13, and published a book containing some important contributions to geometry when he was 18, thereby winning his admission to the Academy.

Shortly afterwards he took part in Maupertuis’ expedition to Lapland (chapter X., § 221), and after publishing several papers of minor importance produced in 1743 his classical work on the figure of the earth. In this he discussed in a far more complete form than either Newton or Maclaurin the form which a rotating body like the earth assumes under the influence of the mutual gravitation of its parts, certain hypotheses of a very general nature being made as to the variations of density in the interior; and deduced formulae for the changes in different latitudes of the acceleration due to gravity, which are in satisfactory agreement with the results of pendulum experiments.

Although the subject has since been more elaborately and more generally treated by later writers, and a good many additions have been made, few if any results of fundamental importance have been added to those contained in Clairaut’s book.

He next turned his attention to the problem of three bodies, obtained a solution suitable for the moon, and made some progress in planetary theory.