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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 diffused misapprehensions as to the true bearings of Newton's explanation of the annual equation.

If proof were wanted of the inexhaustible fertility of astronomical science in points of novelty and interest, it would suffice to adduce the addition to the list of members of our system of no less than eight now planets and satellites during the preparation of these sheets for the press. Among them is one whose discovery must ever be regarded as one of the noblest triumphs of theory. In the account here given of this discovery, I trust to have expressed myself with complete impartiality; and in the exposition of the perturbative action on Uranus, by which the existence and situation of the disturbing planet became revealed to us, I have endeavoured, in pursuance of the general plan of this work, rather to exhibit a rational view of the dynamical action, than to convey the slightest idea of the conduct of those masterpieces of analytical skill which the researches of Messrs. Leverrier and Adams exhibit.

To the latter of these eminent geometers, as well as to my excellent and esteemed friend the Astronomer Royal, I have to

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return my best thanks for communications which would have effectually relieved some doubts I at one period entertained, had I not succeeded in the interim in getting clear of them, as to the compatibility of my views on the subject of the annual equation already alluded to, with the tenor of Newton's account of it. To my valued friend, Professor De Morgan, I am indebted for some most ingenious suggestions on the subject of the mistakes committed in the early working of the Julian reformation of the calendar, of which I should have availed myself, had it not appeared preferable, on mature consideration, to present the subject in its simplest form, avoiding altogether entering into minutiæ of chronological discussion.


Collingwood, April 12, 1849.




Terminology and elementary geometrical conceptions and relations. Termino-

logy relating to the globe of the Earth — to the celestial sphere. Celestial


... 62


Of the nature of astronomical instruments and observations in general. Of

sidereal and solar time. Of the measurements of time. Clocks, chronome-

ters. Of astronomical measurements. Principle of telescopic sights to

increase the accuracy of pointing. Simplest application of this principle.

The transit instrument. Of the measurement of angular intervals. Methods

of increasing the accuracy of reading. The vernier. The microscope. Of

the mural circle. The Meridian circle. Fixation of polar and horizontal

points. The level, plumb-line, artificial horizon. Principle of collimation.

Collimators of Rittenhouse, Kater, and Benzenberg. Of compound instru-

ments with co-ordinate circles. The equatorial, altitude, and azimuth instru-

ment. Theodolite. Of the sextant and reflecting circle. Principle of repe-

tition. Of micrometers. Parallel wire micrometer. Principle of the dupli-

cation of images. The heliometer. Double refracting eye-piece. Variable

prison micrometer. Of the position micrometer

... 76

Of the Moon. Its sidereal period. Its apparent diameter. Its parallax, dis-

tance, and real diameter. First approximation to its orbit. An ellipse about

the Earth in the focus. Its excentricity and inclination. Motion of its nodes

and apsides. Of occultations and solar eclipses generally. Limits within

which they are possible. They prove the Moon to be an opaque solid. Its

light derived from the Sun. Its phases. Synodic revolution or lunar month.

Of eclipses more particularly. Their phenomena. Their periodical recur-

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