Towards the middle of the eighteenth century, European rather than British mathematicians began to investigate the remaining problems in the lunar and planetary theories. Although Newton had provided a solid foundation for the theory of the motions of the Moon and planets, he was not able to account for all the inequalities in their orbits. Newton’s calculated motion of the Moon’s apogee was only half the observed result. Similarly, perturbations in the planetary orbits, particularly for Jupiter and Saturn, could not be explained. It appeared that the Solar System might not be stable and therefore must from time to time require some divine intervention to balance out the inequalities.

The LUNAR THEORY was the least satisfactory. Newton himself was concerned by its disagreement with Flamsteed’s observations, and at the time of publication of the Principia’ had only given a qualitative account of lunar inequalities .The swiss mathematician Euler (1707-83) and the French mathematicians, Clairaut (1717-65) and D’Alembert (1717-83), all provided distinct solu¬tions to the problem of three bodies in a form suitable for the lunar theory. Clairaut first attempted an alteration in the inverse square law. i.e. in the form a/r2+b/r3 which gave a satisfactory result. Later, however, he discovered terms in the lunar equations that he had previously omitted which could account for the discrepancies in the Moon’s orbit, and Newton’s theory was saved. Euler and D’Alembert also reached satisfactory solutions without assuming any alteration in the inverse square law. The lunar tables of Tobias Mayer (1755), capable of giving longitude at sea to within half a degree, were based on Euler’s methods. Euler, Clairaut and D’Alembert had one distinct advantage over Newton in their treatment of lunar inequalities; they used analytical rather than geometrical methods. However, they still were not able to explain all aspects of the motion of the Moon satisfactorily. The secular acceleration was later accounted for by Laplace as being due to an indirect planetary effect, causing a slow increase in the Moon’s motion and thus a resultant decrease in the length of the month; his result agreed almost exactly with observations.

There are two types of inequality in an orbit – periodic and secular. By the middle of the eighteenth century, it was believed that the latter was cumulative and therefore that the Solar System was unstable. Such secular changes were known for the inclination of an orbit, the eccentricity and the length of the orbit axis. During the last quarter of the eighteenth century, a remarkable series of investigations on secular inequalities were undertaken by two French mathematicians, Lagrange in Berlin and Laplace in Paris. Laplace explained the slow change in the rates of motion of Jupiter and Saturn as being due to a periodic disturbance; since their times of revolution are nearly proportional to two whole numbers, a disturbing force is produced. He found this inequality to have a period of about 900 years. Lagrange obtained general expressions for the secular changes in the inclination of an orbit and the length of axis, and found these changes to be necessarily periodic. Laplace showed that allegedly secular variations of the eccentricity were also of a periodic nature. Their combined results indicated that any changes in axis, eccentricity and inclination of a planetary orbit are restricted within definite limits.

These results were summarized in Laplace’s masterpiece, the Mecanique Celeste”, published from 1799-1825. The Solar System was shown to be a stable system in which all the motions could be accounted for fairly accurately by means of the law of gravitation.

Thus it was that the eventual production of the theory of universal gravitation enabled scientists to give a harmonized view of motion within the heavens. One simple theory swept aside the spheres within spheres and wheels within wheels of the ancients, and this theory holds good even today, unless the velocities of the moving bodies are a significant fraction of the speed of light. Paralleling the long road to Newton’s theory was an even longer road to the distance scale. Newton needed to know some distances in order to test his theory, but as we shall see in the next section. scientists in Newton’s lifetime were a very long way from knowing the structure of the Universe on large scales.

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