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that the phenomena of the precession and nutation must be the same in the actual state of our terraqueous spheroid, as if the whole was a solid mass; and that this is true, whatever be the irregularity of the depth of the sea. He shows also, that currents in the sea, rivers, trade-winds, even earthquakes, can have no effect in altering the earth's rotation on its axis. The conclusions with regard to the constitution of the earth that are found to agree with the actual quantity of the precession of the equinoxes are, that the density of the earth increases from the circumference toward the centre; that it has the form of an ellipsoid of revolution, or, as we use to call it, of an elliptic spheroid, and that the compression of this spheroid at the poles is between the limits of and part of the radius of the equator.
The Second part of La Place's work, has, for its object, a fuller development 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, abovementioned. 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, La Place had the assistance, as he informs us, of De Lambre, 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 development 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. La Place, 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 aboved, 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 develops 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 ; and hence a periodical system of inequalities, which our astronomer Bradley was sharp-sighted 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 comcidence 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 La Place 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 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. The 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. La Place 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.
The path of the ray would be determined from the laws of refraction, did we know the law by which the density of the air decreases from the earth upwards. This last, however, is not known, except for a small extent near the surface of the earth, so that we appear here to be left without sufficient data for continuing the investigation. We must, therefore, ei
ther abandon the problem altogether, or resolve it hypothetically, that is, by assuming some hypothesis as to the decrease of the density of the atmosphere. Little would be gained by this last, except as an exercise in mathematical investigation, if it were not that the total quantity of the refraction for a given altitude can be accurately determined by observation. La Place, availing himself of this consideration, begins with making a supposition concerning the law of the density, that is not very remote from the truth, (as we are assured of from the relation between the density of air and the force with which it is compressed); and he compares the horizontal refraction calculated on this assumption with that which is known to be its true quantity. The first hypothesis which he assumes, is that of the density being the same throughout; this gives the total refraction too small, and falls on that account to be rejected, even if it were liable to no other objection. The second hypothesis supposes a uniform temperature through the whole extent of the atmosphere, or it supposes that the density decreases in geometrical proportion, while the distance from the earth increases in arithmetical. The refraction which results is too great; so that this supposition must also be rejected.
If we now suppose the density of the air to decrease in arithmetical progression, while the height does the same, and integrate the differential equation to the curve described by the ray, on this hypothesis, the horizontal refraction is too small, but nearer the truth than on the first hypothesis. A supposition intermediate between that which gave the refraction too great, and this which gives it too small, is therefore to be assumed as that which approaches the nearest to the truth. It is this way of limiting his conjectures by repeated trials, and of extracting from each, by means of the calculus, all the consequences involved in it, that we would recommend to experimenters, as affording one of the most valuable and legitimate uses of hypothetical reasoning. He then employs an intermediate hypothesis for the diminution of the density of the air; which it is not easy to express in words, but from which he obtains a result that agrees with the horizontal refraction, and from which, of course, he proceeds to deduce the refraction for all other altitudes. The table, so constructed, we have no doubt, will be found to contribute materially to the accuracy of astronomical observation.
The researches which immediately follow this, relate to the terrestrial refraction, and the measurement of heights by the barometer. The formula given for the latter, is more complicated than that which is usually employed with us in Britain, where
this subject has been studied with great care. In one respect, it is more general than any of our formulas; it contains an allowance for the difference of latitude. We are not sure whether this correction is of much importance, nor have we had leisure to compare the results with those of General Roy and Sir George Schuckborough. We hardly believe, that in point of accuracy, the two last can easily be exceeded.
The book concludes with a determination of the masses of the planets, more accurate than had been before given; and even of the satellites of Jupiter. Of all the attempts of the Newtonian philosophy,' says the late Adam Smith in his History of Astronomy, that which would appear to be the most above the reach of human reason and experience, is the attempt to compute the weights and densities of the sun, and of the several planets. What would this philosopher have said, if he had lived to see the same balance in which the vast body of the sun had been weighed, applied to examine such minute atoms as the satellites of Jupiter?
Such is the work of La Place, affording an example, which is yet solitary in the history of human knowledge, of a theory entirely complete; one that has not only accounted for all the phenomena that were known, but that has discovered many before unknown, which observation has since recognized. In this theory, not only the elliptic motion of the planets, relatively to the sun, but the irregularities produced by their mutual action, whether of the primary on the primary, of the primary on the secondary, or of the secondary on one another, are all deduced from the principle of gravitation, that mysterious power, which unites the most distant regions of space, and the most remote periods of duration. To this we must add the great truths brought in view and fully demonstrated, by tracing the action of the same power through all its mazes: That all the inequalities in our system are periodical; that, by a fixt appointment in nature, they are each destined to revolve in the same order, and between the same limits; that the mean distances of the planets from the sun, and the time of their revolutions round that body, are susceptible of no change whatsoever; that our system is thus secured against natural decay; order and regularity preserved in the midst of so many disturbing causes ;-and anarchy and misrule eternally proscribed.
The work where this sublime picture is delineated, does honour, not to the author only, but to the human race; and marks, undoubtedly, the highest point to which man has yet ascended in the scale of intellectual attainment. The glory, therefore, of having produced this work, belongs not to the author