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object, a fuller developement of the disturbances of the planets, both primary and secondary, than was compatible with the limits of the first part. After the ample detail into which we have entered concerning two of these subjects, the theory of the moon, and the perturbations of the primary planets, we need not enlarge on them further, though they are prosecuted in the second part of this work, and form the subject of the sixth and seventh books. In the second book, the inequalities had been explained, that depend on the simple power of the eccentricity: here we have those that depend on the second and higher powers of the same quantity; and such are the secular equations of Jupiter and Saturn, above mentioned. The numeral computations are then performed, and every thing prepared for the complete construction of astronomical tables, as the final result of all these investigations. The calculations, of course, are of vast extent and difficulty, and incredibly laborious. In carrying them on, Laplace had the assistance, as he informs us, of Delambre, Bouvard, and other members of the Institute. The labour is, indeed, quite beyond the power of any individual to execute.
The same may be said of the seventh book, which is devoted to a similar developement of the lunar theory. We can enter into no further detail on this subject. One fact we cannot help mentioning, which is to the credit of two British astronomers, Messrs Mason and Dixon, who gave
a new edition of Mayer's tables, more diligently compared with observation, and therefore more accurate, than the original one. Among other improvements, was an empirical equation, amounting to a little more than 20" when a maximum, which was not founded on theory, but was employed because it made the tables agree better with observation. As this equation, however, was not derived from principle (for the two astronomers just named, though accurate observers and calculators, were not skilled enough in the mathematics, to attempt deducing it from principle) it was generally rejected by other astronomers. Laplace, however, found that it was not to be rejected; but, in reality, proceeded from the compression of the earth at the poles, which prevents the gravitation to the earth from decreasing, precisely as the squares of the distances increase, and by that means produces this small irregularity. The quantity of the polar compression that agrees best with this, and some other of the lunar irregularities, is nearly that which was stated above, of the mean radius of the earth. The ellipticity of the sun does, in like manner, affect the primary planets; but, as its influence diminishes fast as the distance increases, it extends no further (in any sensible degree) than the orbit of Mercury, where its only effect is to produce a very small direct movement of the line of the apsides, and an equal retrograde motion of
the nodes, relatively to the sun's equator. We may judge from this, to what minuteness the researches of this author have extended: and, in general, when accuracy is the object to be obtained, the smaller the quantity to be determined, the more difficult the investigation.
The eighth book has for its object, to calculate the disturbances produced by the action of the secondary planets on one another; and particularly refers to the satellites of Jupiter, the only system of secondary planets on which accurate observations have been, or, probably, can be made. Though these satellites have been known only since the invention of the telescope, yet the quickness of their revolutions has, in the space of two centuries, exhibited all the changes which time developes so slowly in the system of the primary planets; so that there are abundant materials for a comparison between fact and theory. The general principles of the theory are the same that were explained in the second book; but there are some peculiarities, that arise from the constitution of Jupiter's system, that deserve to be considered. We have seen above, what is the effect of commensurability, or an approach to it, in the mean motion of contiguous planets; and here we have another example of the same. The mean motions of the three first satellites of Jupiter, are nearly as the numbers 4, 2, and 1; and hence a periodical system of inequali
ties, which our astronomer Bradley was sharpsighted enough to discover in the observation of the eclipses of these satellites, and to state as amounting to 437.6 days. This is now fully explained from the theory of the action of the satellites.
Another singularity in this secondary system, is, that the mean longitude of the first satellite minus three times that of the second, plus twice that of the third, never differs from two right angles, but by a quantity almost insensible.
One can hardly suppose that the original motions were so adjusted, as to answer exactly to this condition; it is more natural to suppose that they were only nearly so adjusted, and that the exact coincidence has been brought about by their mutual action. This conjecture is verified by the theory, where it is demonstrated that such a change might have been actually produced in the mean motion by the mutual action of those planetary bodies, after which the system would remain stable, and no further change in those motions would take place.
Not only are the mutual actions of the satellites taken into account in the estimate of their irregularities, but the effect of Jupiter's spheroidal figure is also introduced. Even the masses of the satellites are inferred from their effect in disturbing the motions of one another.
In the ninth book Laplace treats of Comets, of the methods of determining their orbits, and of the disturbances they suffer from the planets. We cannot follow him in this; and have only to add, that his profound and elaborate researches are such as we might expect from the author of the preceding investigations.
The tenth book is more miscellaneous than any of the preceding; it treats of different points relative to the system of the world. One of the most important of these is astronomical refraction.
rays of light from the celestial bodies, on entering the earth's atmosphere, meet with strata that are more dense the nearer they approach to the earth's surface; they are, therefore, bent continually toward the denser medium, and describe curves that have their concavity turned toward the earth. The angle formed by the original direction of the ray, and its direction at the point where it enters the eye, is called the astronomical refraction. Laplace seeks to determine this angle by tracing the path of the ray through the atmosphere; a research of no inconsiderable difficulty, and in which the author has occasion to display his skill both in mathematical and in inductive investigation. The method he pursues in the latter, is deserving of attention, as it is particularly well adapted to cases that occur often in the more intricate kinds of physical discussion.