The Progress of the Century

by Unknown

SOME ACHIEVEMENTS OF MATHEMATICAL ANALYSIS

Uranus, a planet unknown to the ancients, was discovered by its movement among the stars by William Herschel in 1781. It was not until 1846 that another major planet was added to the solar system, and this discovery was one of the sensations of the century.

The story of the independent discovery of Neptune by Adams and Le Verrier, who were both driven to the conclusion that certain apparent regularities in the motion of Uranus were due to the attraction of another body travelling on an orbit outside it, has been often told. The subsequent discovery of the external body not far from the place at which their mathematical analysis had led them to believe it would be seen, will forever be regarded as a fine triumph of the human intellect.

But the results of the inquiries which now concern us are generally of not so sensational a character, although they lie at the root of our knowledge of celestial motions. They more often take the shape of tables and discussions relating to the movements of the bodies which make up our solar system.

Gauss may be said to have led the way during the nineteenth century by his Theoria molus corporum coelestium solem ambientium. This was a worthy sequel to the Méchanique Céleste, in which work, towards the end of the eighteenth century, Laplace had enshrined all that was known on the planetary results of gravitation.

In later years Le Verrier and Newcomb have been among the chief workers on whom the mantle of such distinguished predecessors has fallen. From them the planet and satellite tables now in use have been derived.

But the motion of our own satellite, the moon, has had fascinations for other analysts besides those we have named.

The problem, indeed, of the moon’s motion is one of the most difficult, and has taxed the ingenuity of astronomers from an early date. Even at the present day it is impossible to predict the exact position of the moon at any one moment owing to inequalities and perturbations, the exact varying values of which are not known.

The two most important theories of the motion of the moon completed towards the middle of the century were due to Hansen and Delaunay. The former’s appeared in 1838, the lunar tables being published later (1857), while the latter’s was published in 1860.

Hansen’s theory had for its chief object the formation of tables; to avoid the inconvenience of using in his calculations series which slowly converge, he inserted numerical values throughout. In Hansen’s solution the problem is one actually presented by nature, allowance being made for every known cause of disturbance. There is one disadvantage, namely, that should observations demand a change in any of the constants used, there is no means of making any correction in the results.

Delaunay’s theory surmounted this difficulty, but at the expense of still greater inconvenience for making an ephemeris. The slow convergence of certain series involved an immense amount of labor to give sufficiently approximate results.

More recently, as the century was closing, Dr. Brown took up the subject and made a fresh attempt to calculate the motion of our satellite. It may be stated that he adopts all Delaunay’s modifications of the problem and works them out algebraically; but there are many technical differences which it would be out of place to mention here.

Enough has been stated to show that there is not likely to be any breach of continuity in the treatment of this most important problem.

Another attack on the moon, and, incidentally, its motion, has recently been made by another analyst, Professor George Darwin; grappling with all the consequences of tidal friction, he has been able to present to us the past and future history of our satellite. Beginning as a part of the material congeries from which subsequently some fifty million years ago both earth and moon, as separate bodies, were formed, it has ever since been extending its orbit, and so retreating farther away from its centre of motion, while the period of the earth’s rotation has been increasing at the same time, from a possible period of some three hours when the moon was born, to one of one thousand four hundred hours when the day and month will be equal, something like one hundred and fifty million years being required for the process.