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M.A. D.C.L. F.R.S.L. & E. HON. M.R.I.A. F.R.A.S. F.G.S. M.C.U.P.S.

Correspondent or Honorary Member of the Imperial, Royal, and National, Academies of Sciences
of Berlin, Brussels, Copenhagen, Göttingen, Haarlem, Massachussets (U.S.), Modena,
Naples, Paris, Petersburg, Stockholm, Turin, Vienna, and Washington (U.S.);
the Italian and Helvetic Societies;
the Academies, Institutes, &e., of Albany (U.S.), Bologna, Catania, Dijon, Lausanne,
Nantes, Padua, Palermo, Rome, Venice, Utrecht, and Wilna ;

the Philomathic Society of Paris; Asiatic Society of Bengal; South African Lit. and Phil. Society;
Literary and Historical Society of Quebec; Historical Society of New York;
Royal Medico-Chirurgical Soc., and Inst. of Civil Engineers, London;
Geographical Soc. of Berlin; Astronomical and

Meteorological Soc. of British Guiana;

&c. &c. &c.





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The work here offered to the Public is based upon and may be considered as an extension, and, it is hoped, an improvement of a treatise on the same subject, forming Part 43. of the Cabinet Cyclopædia, published in the year 1833. Its object and general character are sufficiently stated in the introductory chapter of that volume, here reprinted with little alteration; but an opportunity having been afforded me by the Proprietors, preparatory to its re-appearance in a form of more pretension, I have gladly availed myself of it, not only to correct some errors which, to my regret, subsisted in the former volume, but to remodel it altogether (though in complete accordance with its original design as a work of explanation); to introduce much new matter in the earlier portions of it; to re-write, upon a far more matured and comprehensive plan, the part relating to the lunar and planetary perturbations, and to bring the subjects of sidereal and nebular astronomy to the level of the present state of our knowledge in those departments.

The chief novelty in the volume, as it now stands, will be found in the manner in which the subject of Perturbations is treated. It is not -- it cannot be made elementary, in the sense in which that word is understood in these days of light reading. The chap


ters devoted to it must, therefore, be considered as addressed to a class of readers in possession of somewhat more mathematical knowledge than those who will find the rest of the work readily and easily accessible; to readers desirous of preparing themselves, by the possession of a sort of carte du pays, for a campaign in the most difficult, but at the same time the most attractive and the most remunerative of all the applications of modern geometry. More especially they may be considered as addressed to students in that university, where the “ Principia” of Newton is not, nor ever will be, put aside as an obsolete book, behind the age; and where the grand though rude outlines of the lunar theory, as delivered in the eleventh section of that immortal work, are studied less for the sake of the theory itself than for the spirit of far-reaching thought, superior to and disencumbered of technical aids, which distinguishes that beyond any other production of the human intellect.

In delivering a rational as distinguished from a technical exposition of this subject, however, the course pursued by Newton in the section of the Principia alluded to, has by no means been servilely followed. As regards the perturbations of the nodes and inclinations, indeed, nothing equally luminous can ever be substituted for his explanation. But as respects the other disturbances, the point of view chosen by Newton has been abandoned for another, which it is somewhat difficult to perceive why he did not, himself, select. By a different resolution of the disturbing forces from that adopted by him, and by the aid of a few obvious conclusions from the laws of elliptic motion which would have found their


place, naturally and consecutively, as corollaries of the seventeenth proposition of his first book (a proposition which seems almost to have been prepared with a special view to this application), the momentary change of place of the upper focus of the disturbed ellipse is brought distinctly under inspection; and a clearness of conception introduced into the perturbations of the excentricities, perihelia, and epochs which the author does not think it presumption to believe can be obtained by no other method, and which certainly is not obtained by that from which it is a departure. It would be out of keeping with the rest of the work to have introduced into this part of it any algebraic investigations ; else it would have been easy to show that the mode of procedure here followed leads direct, and by steps (for the subject) of the most elementary character, to the general formulæ for these perturbations, delivered by Laplace in the Mécanique Céleste.*

The reader will find one class of the lunar and planetary inequalities handled in a very different manner from that in which their explanation is usually presented. It comprehends those which are characterized as incident on the epoch, the principal among them being the annual and secular equations of the moon, and that very delicate and obscure part of the perturbational theory (so little satisfactory in the manner in which it emerges from the analytical treatment of the subject), the constant or permanent effect of the disturbing force in altering the disturbed orbit. I will venture to hope that what is here stated will tend to remove some rather generally

* Livre ï. chap. viii. art. 67.

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