rent, if, instead of regarding the equinox, we fix our attention on the pole of the equinoctial, or the vanishing point of the earth's axis. (315.) The place of this point among the stars is easily determined at any epoch, by the most direct of all astronomical observations, those with the meridian or mural circle. By this instrument we are enabled to ascertain at every moment the exact distance of the polar point from any three or more stars, and therefore to lay it down, by triangulating from these stars, with unerring precision, on a chart or globe, without the least reference to the position of the ecliptic, or to any other circle not naturally connected with it. Now, when this is done with proper diligence and exactness, it results that, although for short intervals of time, such as a few days, the place of the pole may be regarded as not sensibly variable, yet in reality it is in a state of constant, although extremely slow motion; and, what is still more remarkable, this motion is not uniform, but compounded of one principal uniform, or nearly uniform, part, and other smaller and subordinate periodical fluctuations: the former giving rise to the phenomena of precession; the latter to another distinct phenomenon called mutation. These two phenomena, it is true, belong, theoretically speaking, to one and the same general head, and are intimately connected together, forming part of a great and complicated chain of consequences flowing from the earth's rotation on its axis: but it will be conducive to clearness at present to consider them separately. (316.) It is found, then, that in virtue of the uniform part of the motion of the pole, it describes a circle in the heavens around the pole of the ecliptic as a centre, keeping constantly at the same distance of 23° 28′ from it in a direction from east to west, and with such a velocity, that the annual angle described by it, in this its imaginary orbit, is 50-10"; so that the whole circle would be described by it in the above-mentioned period of 25,868 years. It is easy to perceive how such a motion of the pole will give rise to the retrograde motion of the equinoxes; for in the figure, art. 308, suppose the pole P in the progress of its motion in the small circle P O Z round K to come to O, then, as the situation of the equinoctial EV Q is determined by that of the pole, this, it is evident, must cause a displacement of the equinoctial, which will take a new situation, EU Q, 90° distant in every part from the new position O of the pole. The point U, therefore, in which the displaced equinoctial will intersect the ecliptic, i. e. the displaced equinox, will lie on that side of V, its original position, towards which the motion of the pole is directed, or to the westward. (317.) The precession of the equinoxes thus conceived, consists, then, in a real but very slow motion of the pole of the heavens among the stars, in a small circle round the pole of the ecliptic. Now this cannot happen without producing corresponding changes in the apparent diurnal motion of the sphere, and the aspect which the heavens must present at very remote periods of history. The pole is nothing more than the vanishing point of the earth's axis. As this point, then, has such a motion as we have described, it necessarily follows that the earth's axis must have a conical motion, in virtue of which it points successively to every part of the small circle in question. We may form the best idea of such a motion by noticing a child's peg-top, when it spins not upright, or that amusing toy the te-to-tum, which, when delicately executed, and nicely balanced, becomes an elegant philosophical instrument, and exhibits, in the most beautiful manner, the whole phenomenon. The reader will take care not to confound the variation of the position of the earth's axis in space with a mere shifting of the imaginary line about which it revolves, in its interior. The whole earth participates in the motion, and goes along with the axis as if it were really a bar of iron driven through it. That such is the case is proved by the two great facts: 1st, that the latitudes of places on the earth, or their geographical situation with respect to the poles, have undergone no perceptible change from the earliest ages. 2dly, that the sea maintains its level, which could not be the case if the motion of the axis were not accompanied with a motion of the whole mass of the earth.1 (318.) The visible effect of precession on the aspect of the heavens consists in the apparent approach of some stars and constellations to the pole and recess of others. The bright star of the Lesser Bear, which we call the pole star, has not always been, nor will always continue to be, our cynosure: at the time of the construction of the earliest catalogues it was 12° from the pole - it is now only 1° 24', and will approach yet nearer, to within half a degree, after which it will again recede, and slowly give place to others, which will succeed in its companionship to the pole. After a lapse of about 12,000 years, the star a Lyræ, the brightest in the northern hemisphere, will occupy the remarkable situation of a pole star approaching within about 5° of the pole. (319.) At the date of the erection of the Great Pyramid of Gizeh, which precedes by 3970 years (say 4000) the present epoch, the longitudes of all the stars were less by 55° 45′ than at present. Calculating 1 Local changes of the sea level, arising from purely geological causes, are easily distinguished from that general and systematic alteration which a shifting of the axis of rotation would give rise to. 'from this datum' the place of the pole of the heavens among the stars, it will be found to fall near a Draconis; its distance from that star being 3° 44' 25". This being the most conspicuous star in the immediate neighbourhood was therefore the pole star at that epoch. And the latitude of Gizeh being just 30° north, and consequently the altitude of the north pole there also 30°, it follows that the star in question must have had at its lower culmination, at Gizeh, an altitude of 26° 15′ 35′′. Now it is a remarkable fact, ascertained by the late researches of Col. Vyse, that of the nine pyramids still existing at Gizeh, six (including all the largest) have the narrow passages by which alone they can be entered, (all which open out on the northern faces of their respective pyramids) inclined to the horizon downwards at angles as follows. Of the two pyramids at Abousseir also, which alone exist in a state of sufficient preservation to admit of the inclinations of their entrance passages being determined, one has the angle 27° 5′, the other 26°. (320.) At the bottom of every one of these passages therefore, the then pole star must have been visible at its lower culmination, a circumstance which can hardly be supposed to have been unintentional, and was doubtless connected (perhaps superstitiously) with the astronomical observation of that star, of whose proximity to the pole at the epoch of the erection of these wonderful structures, we are thus furnished with a monumental record of the most imperishable nature. (321.) The nutation of the earth's axis is a small and slow subordinate gyratory movement, by which, if subsisting alone, the pole would describe among the stars, in a period of about nineteen years, a minute ellipsis, having its longer axis equal to 18′′-5, and its shorter to 13"-74; the longer being directed towards the pole of the ecliptic, and the shorter, of course, at right angles to it. The consequence of this real motion of the pole is 1 On this calculation the diminution of the obliquity of the eliptic in the 4000 years elapsed has no influence. That diminution arises from a change in the plane of the earth's orbit, and has nothing to do with the change in the position of its axis, as referred to the starry sphere. an apparent approach and recess of all the stars in the heavens to the pole in the same period. Since, also, the place of the equinox on the ecliptic is determined by the place of the pole in the heavens, the same cause will give rise to a small alternate advance and recess of the equinoctial points, by which, in the same period, both the longitudes and the right ascensions of the stars will be also alternately increased and diminished. (322.) Both these motions, however, although here considered separately, subsist jointly; and since, while in virtue of the nutation, the pole is describing its little ellipse of 18′′-5 in diameter, it is carried by the greater and regularly progressive motion of precession over so much of its circle round the pole of the ecliptic as corresponds to nineteen years,— that is to say, over an angle of nineteen times 50"-1 round the centre (which, in a small circle of 23° 28' in diameter, corresponds to 6' 20", as seen from the centre of the sphere): the path which it will pursue in virtue of the two motions, subsisting jointly, will be neither an ellipse nor an exact circle, but a gently undulated ring like that in the figure (where, however, the undulations are much exaggerated). (See fig. to art. 325.) (323.) These movements of precession and nutation are common to all the celestial bodies, both fixed and erratic; and this circumstance makes it impossible to attribute them to any other cause than a real motion of the carth's axis such as we have described. Did they only affect the stars, they might, with equal plausibility, be urged to arise from a real rotation of the starry heavens, as a solid shell, round an axis passing through the poles of the ecliptic in 25,868 years, and a real ecliptic gyration of that axis in nineteen years: but since they also affect the sun, moon, and planets, which, having motions independent of the general body of the stars, cannot without extravagance be supposed attached to the celestial concave,1 this idea falls to the ground; and there only remains, then, a real motion in the earth by which they can be accounted for. It will be shown in a subsequent chapter that they are necessary consequences of the rotation of the earth, combined with its elliptical figure, and the unequal attraction of the sun and moon on its polar and equatorial regions. (324.) Uranographically considered, as affecting the apparent places of the stars, they are of the utmost importance in practical astronomy. When we speak of the right ascension and declination of a celestial object, it This argument, cogent as it is, acquires additional and decisive force from the law of nutation, which is dependent on the position, for the time, of the lunar orbit. It we attribute it to a real motion of the celestial sphere, we must then maintain that sphere to be kept in a constant state of tremor by the motion of the moon. - becomes necessary to state what epoch we intend, and whether we mean the mean right ascension — cleared, that is, of the periodical fluctuation in its amount, which arises from nutation, or the apparent right ascension, which, being reckoned from the actual place of the vernal equinox, is affected by the periodical advance and recess of the equinoctial point produced by nutation and so of the other elements. It is the practice of astronomers to reduce, as it is termed, all their observations, both of right ascension and declination, to some common and convenient epoch - such as the beginning of the year for temporary purposes, or of the decade, or the century for more permanent uses, by subtracting from them the whole effect of precession in the interval; and, moreover, to divest them of the influence of nutation by investigating and subducting the amount of change, both in right ascension and declination, due to the displacement of the pole from the centre to the circumference of the little ellipse above mentioned. This last process is technically termed correcting or equating the observation for nutation; by which latter word is always understood, in astronomy, the getting rid of a periodical cause of fluctuation, and presenting a result, not as it was observed, but as it would have been observed, had that cause of fluctuation had no existence. (325.) For these purposes, in the present case, very convenient formulæ have been derived, and tables constructed. They are, however, of too technical a character for this work; we shall, however, point out the manner in which the investigation is conducted. It has been shown in art. 309 by what means the right ascension and declination of an object are derived from its longitude and latitude. Referring to the figure of that article, and supposing the triangle K P X orthographically projected on the plane of the ecliptic as in the annexed figure: in the triangle KPX, KP is the obliquity of the ecliptic, K X the co-latitude (or complement of latitude), and the angle P K X the co-longitude of the object X. These are the data of our question, of which the second is constant, and the other two are varied by the effect of precession and nutation: and their variations (considering the minuteness of the latter effect generally, and the small number of years in comparison of the whole period of 25,868, for which we ever require to estimate the effect of the former,) are of that order which may be regarded as infinitesimal in geometry, and treated as such without fear of error. The whole question, then, is reduced to this: In a spherical triangle K PX, in which one side K X is constant, and an angle K, and adjacent side K P vary by given infinitesimal changes of the position of P: required the changes thence arising in the other side PX, and the angle KPX. This is a very simple and easy problem of spherical geometry, and being resolved, |