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to us, that we do not properly appreciate Kepler's merit in discovering them. If we view, however, the state of science, and Kepler's means and the inherent difficulty of the investigation, we must consider it to have been a great discovery. And even now availing ourselves of all the facilities of modern science, it is not easy briefly to shew, from a comparison of the observations of the sun, that the solar orbit is an ellipse." P. 444.
After the examination of the reasoning by which the elliptical form of the earth's orbit is shewn, and having stated the Jaws which Kepler deduced by which its motion in the ellipse is regulated, the next subject will naturally be the application of this knowledge to the determination of the place of a body in its elliptical orbit after a certain elapsed time from its being in the apside. This is what has been designated Kepler's problem, and its solution lays the first ground-work of the solar tables, or the knowledge of the sun's longitude throughout the different periods of a revolution. The solution of this important but difficult problem has exercised the ingenuity of the principal astronomers and mathematicians from the time of its great proposer to the present day; various solutions of great excellence have been at different times proposed. That however which our author adopts is the mode proposed by Cassini; and we cannot doubt that in this selection he has been guided by his usual judgment. We cannot however forbear strongly recommending to the notice of such of our mathematical readers as may not have met with it, the very admirable solution of this problem given by Professor Robertson, of Oxford, in the Philosophical Transactions for 1816. Part I.
The solution of Kepler's problem alone will not enable us to assign the place of the sun in his orbit at a given day. The place and motion of the aphelion of an orbit must in the first instance be determined. To this and some subjects depending on it, the reader's attention is now directed; and this being accomplished, the application of the problem to the determination of the sun's place is exhibited and explained at large, with numerous examples.
There are, however, certain inequalities in the earth's orbit and motion, which next become objects of enquiry, and which must obviously be essential in assigning the real law of the motion of the earth. It is at this point, then, that it becomes necessary to refer to the doctrine of "disturbing forces," as they are termed.
To investigate the mechanical principles of the celestial motions, and to trace the operation of the different forces acting upon the bodies of our system, is the province of what
is termed Physical Astronomy. Our author has devoted a
The discoveries of Kepler respecting the orbits of the planets were confined to the facts of their being elliptical in form, and of the proportion which subsists between the squares of their times of revolution and the cubes of their mean distances; an equable description of areas being constantly maintained.
It remained for Newton to bring in the aid of mechanical science, and to prove that such a state of things was the necessary result of the action of a certain force, according to a given law: to shew that a body being attracted to a central point, by a force acting inversely as the square of the distance, and projected at right angles to the line joining its position with the central point, would of necessity describe an ellipse having that point for one of its foci, and that the other conditions, laid down by Kepler, would by the same necessity take place.
This force, to which the name of attraction is given, was shewn, by the same philosopher, to depend for its intensity upon the relative masses of the central and the revolving body. If we now suppose another central body equal to the first, placed exactly at the same distance on the other side of the revolving body, and acting upon it precisely in the same manner, it is obvious that the revolving body will be equally urged to describe an ellipse round each centre, if we suppose it projected, as at first, at right angles to the line joining it with these centres. But as it cannot in this case describe an ellipse round either, the revolving motion will be entirely destroyed, and it will continue to be projected in a straight line. This may be considered the extreme case of what is called "a disturbing force."
If we now suppose either the mass of the new body to be diminished, or its distance from the revolving body increased, or both circumstances to take place together, then
the derangement or "perturbation" of the revolving body (as it is technically termed) will still continue, but in a less degree. An orbit or curvilinear path, concave towards the first central body, in the commencement of the motion, will be described; but it will neither be elliptical, nor of any other exact geometrical form.
The disturbing body, whatever be its mass or distance, will always derange the laws of the equable description of areas, and of elliptical motion. If its mass be considerable, and its distance not very great, when compared with the masses and distances of the other two bodies, the derangement will be so great as to render the knowledge of those laws useless in determining the real orbit and law of motion of the disturbed body. In such case Kepler's problem would become one of mere curiosity, and the place of the body must be determined by other means. If, however, the mass of the disturbing body be small, and its distance great, the perturbations may be so small, that the orbit shall be nearly, though not strictly elliptical; and the equable description of areas nearly, though not exactly true. Under such circumstances Kepler's problem will not be nugatory. It may be applied to determine the place of the revolving body, supposing it to revolve, which is not the case, but which is nearly so, in an ellipse. The erroneous supposition, and consequently erroneous results, being afterwards corrected, by supplying certain small "equations," that shall compensate the inequalities arising from the disturbing body.
In the predicaments just described are the bodies of the solar system. The mass of the sun, round which the earth is revolving, is 1300,000 times greater than that of the earth, and this 68 times greater than that of the moon, which, by the reciprocal action of gravity has a tendency to disturb the earth's motion. Similar considerations apply to the planets, which, though of greater masses, are at greater distances; consequently both the moon and the planets have but a small effect in disturbing the earth's eliptical motion. Kepler's problem then will afford a near approximation for finding the earth's, or in other words, the sun's place: subsequent corrections being applied for the disturbing effects.
The question then arises, how are the amounts of these corrections to be computed? The solution of this question is one of the greatest importance and difficulty which physical astronomy presents. It becomes a problem to find, for an assigned time, the place of a body attracted by one body and disturbed by another; the masses, distances, and posi
tions of the bodies being given. This, by way of distinction, has been termed "the problem of the three bodies."
In the solution of this problem the resources of physical astronomy have been called forth. An approximate solution is all that the case admits of, and this our author has exhibited in his second voluine; where this important and abstruse enquiry occupies a very prominent place, and necessarily takes up a considerable space in the detail of its various cases. It may, indeed, be considered as the most essential feature in a physical system, which shall account for the motions of the heavenly bodies. After the great original laws of gravitation, supposing no interfering causes to affect the exactness of the motions, the next in importance must be the enquiry as to the result, when, as we see in nature, many bodies simultaneously revolve round one central. If the original principle of attraction be true, these bodies, however relatively small or distant, must, in theory at least, exert some influence on each other. The investigation of the laws of such mutual action is, therefore, precisely what is wanted to complete the design of a perfect explanation of the phenomena of the universe; and as such constitutes the main bulk of the science, after the fundamental laws have once been established.
A knowledge of the corrections thus arising is, therefore, essential to the solar theory, and the construction of the solar tables; an important branch of astronomy, but which regards only the form of the earth's orbit, and the law of its motion. Such knowledge is equally essential in regard to the planetary theory.
"The perturbations," as our author observes, are as much a part of Newton's system, as the elliptical forms of the planetary orbits and the laws of the periods of their revolutions. They are as direct consequences of the principle of universal attraction, as the regularity of that system would be on the hypothesis of the abstraction of all disturbing forces. The quantities of the pertur bations are indeed small and not easily discerned but they are gradually detected as art continues to invent better instruments, and science better methods, and they so furnish not the most simple proof, perhaps, but the most irrefragable proof of the truth of Newton's theory." Part II. p. 496.
In the preface to his second volume, the author enters at considerable length upon a comparison of the different proofs which Newton's theory receives from the examination of the heavens. Speaking of the accordance of the phenomena of deviation with the principle of gravity, he ably remarks:
"These proofs are founded on the deviations from the elliptical system, the former on the system itself. Newton's theory might be true if a planet described an ellipse nearly: it could not be true, if it described an exact ellipse." Vol. II. Preface, p. xxxii.
We would particularly recommend to the attention of our readers the elaborate preface to the second volume of this work. It contains a very luminous and able view of the nu ture and objects of physical astronomy, and of the difficul, ties which it involves. Throughout this dissertation numerous highly original remarks are interspersed, and from the whole of it the student will derive the most useful instruction. In the study of the Principia, especially, there are many illustrations, both in the preface, and in other parts of the volume, which cannot fail to be of considerable use.
Our limits, however, will not permit any further remarks on this part of the subject, and we must hasten to resume and conclude our sketch where we broke off.
After the consideration of the solar motion naturally follows the subject of solar time, and hence the equation of time. We will not, however, follow our author by any remarks upon this part of the work, but proceed to notice the next grand division of it, which comprizes the planetary theory. The admirable clearness with which Mr. Woodhouse has the faculty of illustrating every subject of which he treats, is eminently displayed in this part of his work. He throughout adheres to the same simple and natural method with which he began, and explains first what we see of the planets, and thence deduces what we may know.
To fix with increasing accuracy the places of the fixed stars, we at first observed, was the primary object of observation: its next is the comparison of the motions of the planetary bodies, with reference to the points of the heavens so fixed from a knowledge of these motions to infer the real motions in space: to compare such deductions with those which the theory of gravitation would assign, and thus continually tend to the complete developement and confirmation of the true system of the universe.
When treating of the planetary theory, the method employed by M. Lalande, for deducing the period of the Herschel planet, is briefly described: a method founded upon trial and conjecture; such trials being repeated with new conjectural assumptions, till a sufficient approximation to the truth was obtained. On this our author makes a remark, which as it is of very general application in understanding the nature and principles of many astronomical processes, we bere subjoin.