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was the first great astronomer that was able to establish a theory that enabled him to predict the places of the sun and moon with a degree of accuracy which answered the purposes of astronomy in the comparatively rude state of observation in his day. This he did by introducing the epicycle and demonstrating its value and correctness in representing the inequalities in the motions of the sun and moon. He calculated solar and lunar tables upon this hypothesis, and their accuracy was sufficient to enable him to predict eclipses. Delambre, one of the most critical historians of astronomy, says: " In Hipparchus we find one of the most extraordinary men of antiquity; the very greatest, in the sciences which require a combination of observation with geometry.”+ The theory of Hipparchus consists in the resolution of the real motions of a heavenly body into an assemblage of circular motions; as, for example, while the sun was supposed to revolve uniformly in the circunference of a sinall circle, the centre of this circle (epicycle) revolved uniforinly in the circumference of a larger circle of which the earth occupied the centre. We may remark that this method of resolving the true motion of a body into eqnable circular motions, so far from being a barren hypothesis, fitted to be employed only in the early and rude age the science, has never been superseded, but is still employed in physical astronomy, since geometers resolve all unequal motions into a series of terms which run throngh their periods in circles, these motions being expressed by sines and cosines of circular arcs, which increase uniformly with the time. Thus one of the inost improved methods of modern physical astronomy dates far back into antiquity, having been propounded in the school of Plato more than two thousand years ago.

Kepler was the next great theoretical astronomer. He did away with the epicycles of Hipparchus by proving that the planets revolve around the sun in ellipses, the sun occupying one of the foci. To find the position of the planet in its orbit, he supposed it to revolve uniformly in a circle supposed

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* Whewell's Hist. of the Inductive Sciences, Vol. II., p. 182. Ancienne, tom. i., p. 186. Hist. Ind. Sci., Vol. I., p. 196.

to be described on the major axis, and to have a period of revolution the same as that required for the ellipse. To the position thus found a correction was applied, called the equation of the centre. This is known as Kepler's problem. Kepler also proved that the radius-vecter of a planet, or the imaginary line drawn from the centre of the sun to the centre of the planet, passes over equal areas in equal times, and this law enabled him to find the position of the planet in its orbit. These are known . in physical astronomy as Kepler's first and second laws. The third law, that the squares of the times of revolution are proportional to the cubes of the major axis, cost him seventeen years of study. If he had divided the period of one planet by the period of another, and put the quotient equal to some unknown power of the quotient arising from dividing the distance of the former planet from the sun by the distance of the latter, he could have solved this exponential equation in an hour or two at least, even without the aid of logarithms. But the science of algebra was not then much advanced, nor was it studied then by mathematicians so extensively as at the present day, but the ancient geometry was employed in the investigations of physical problems. The application of algebra to geometry, by Descartes, has completely revolutionized the methods employed in physical research.

We can, at this day, look back to the problems which so taxed the intellectual powers of Kepler, and point out easy and direct methods for solving them ; but we must not forget that these methods have been perfected only by great labor, performed by some cf the ablest minds that the world has known. Nor have these things been developed all at once by some master genius. Step by step, and extending through many years, have they been advanced. Many minds are now able, by close application to the subject, to read and understand the great truths developed in the last two centuries, but only a few were able to advance into the unknown and develop the hidden truths and principles of nature. There is a vast difference between that mind which is able to develop new truths and thus advance knowledge, and the one that is only able to

follow and understand what has already been explained. Especially is this true in the mathematical and physical sciences.

Although philosophers in Kepler’s day obtained some faint glimpses of the great law of gravitation, afterwards discovered by Newton, yet too many other things remained to be developed to enable him to arrive at correct and definite views on the subject of attraction. The laws of motion were scarcely understood at that period. Galileo did not discover them till the early part of the seventeenth century. In Italy, Holland and England, mathematicians viewed the problem of the celestial motions by the light which the discovery of the real laws of motion threw upon it.* Borelli, one of Galileo's pupils, entertained some correct views of the nature of central action, in a work which he published in 1666.7 He referred to the whirling of a stone in a sling, to show that a circular motion gives a tendency to recede from the centre, by giving rise to centrifugal force. Huygens, afterwards, in 1673, at the end of his work, Horologium Oscillatorium, published some important theorems on centrifugal force, but without any demonstration. Newton afterwards gave the demonstrations in his Principia. While all these guesses and real disco veries served as so many aids to Newton, yet the step between them and the great discoveries of Newton was very great, and beyond any ordinary mind to accomplish.

All the discoveries to which we have now alluded pertain rather to mechanics than to astronomy; but after the nature of the planetary orbits, and the relations of the motion of one planet to another, had been made known, physical astronomy could make but little progress without the development of the science of mechanics. Newton added vastly to this department of knowledge. Even at this time, nearly two centuries after the publication of the Principia, the first two books of that immortal work contain almost as complete a treatise on dynamics as we have. The discovery of the law of universal gravitation by Newton, namely: that every particle of matter attracts every other particle with a force that is directly proportional to the mass, or quantity, of matter, and inversely proportional to the square of the distance, completely reduced the motions of the heavenly bodies to the science of mechanics, and now the mathematician is able to predict, ages in advance, the configuration of the planets of our system.

* Whewell's Hist. Ind. Sciences, Vol. II., p. 150. + Theories of the Medicean Planets, Florence, 1666.

The Principia was published in the year 1687, and in that work Newton applied the law of gravitation to the determination of the motions of the moon. With a skill in the nse of the ancient geometry which has never been equalled, he was able to account for the most of the irregularities then known in the motion of the moon.* Ile saw that the planets must be subject to similar irregularities in their motions, but the difficulties of the subject seem to have deterred him from undertaking the investigation, and this great problem he bequeathed to posterity. But the difficulty. of applying the ancient geometry entirely prevented mathematicians from making any progress beyond what Newton had done, in the great problem of planetary perturbations, for half a century after the publication of the Principia. Geometry has the advantage of always keeping every step, in the solution of a problem, distinctly before the mind; and when the conclusion is once reached, the successive steps carry the most complete conviction to the mind. But a more powerful instrument was needed to make further progress in the solution of the various problems which physical astronomy presented. The invention of the infinitesimal analysis by Newton and Leibnitz,t though in its infancy when Newton's work was published, afterwards supplied the necessary means for carrying on these investigations. This powerful engine of the human mind, the infinitesimal calculus, had to be perfected before it could be used

* Principia, Book I, Prop. 66.

+ Strictly speaking, Newton invented the fluxional calculus, and Leibnitz the differential calculus ; the former conceived all quantities to be generated by motion, and the latter supposed them to be made up of indefinitely small differences: the fluential and the integral calculus are the same. The motion of the planets probably suggested the idea of the fluxional calculus.

in physical research. While the continental mathematicians steadily opposed the Newtonian philosophy, they were supplying the best possible means for its verification, by improving and extending the recently invented calculus. Leibnitz and the two Bernonillis, John and his son Daniel, applied themselves with great energy to this subject. The nature of this branch of mathematics seemed to peculiarly adapt it to the tracing ont of complicated variable motion. ". The ponderous instrument of synthesis, so effective in Newton's hands, has never since been grasped by one who could use it for such purposes, and we gaze at it with admiring curiosity, as on some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden.”* In analysis “symbols think for us, without our dwelling constantly upon their meaning, and obtain for us the consequences which result from the relations of space and the laws of force, however complicated be the conditions under which they are combined.”

Mathematical analysis having been brought to a sufficient state of perfection to enable mathematicians to undertake the solution of the great problem presented by the law of universal gravitation in its application to the motions of the celestial bodies, a few of the great minds of the earlier part of the last century applied themselves to the work which had thus become a necessity for the further progress of physical astronomy. The motions of the moon seemed to offer to geometers the best facilities for testing, beyond what Newton had done, the law of attraction which he had found to hold in several cases, both because of the many irregularities which its motions presented, and its use in determining terrestrial longitude, if its place in the heavens could be calculated with sufficient accuracy.

Enler seems to have been the first geometer that attempted to develop the theory of gravitation beyond what was already done in the Principia. In the year 1745 he investigated the

* Whewell,

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