sent epoch, the longitudes of all the stars were less by 55° 45' than at present. Calculating 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 erec

On this calculation the diminution of the obliquity of the ecliptic 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.

tion 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 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 earth'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 elliptic 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*, 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 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

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. If 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!

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 KPX orthographically projected on the plane of the ecliptic as in the annexed figure: in the triangle KPX, KP is the obliquity of the ecliptic, KX the co-latitude (or complement of latitude), and the angle PKX 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

quired 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, it gives at once the reductions we are seeking; for PX being the polar distance of the object, and the angle K PX its right ascension plus 90°, their variations are the very quantities we scek. It only remains, then, to express in proper form the amount of the precession and nutation in longitude and latitude, when their amount in right ascension and declination will immediately be obtained.

(326.) The precession in latitude is zero, since the latitudes of objects are not changed by it: that in longitude is a quantity proportional to the time at the rate of 50"-10 per annum. With regard to the nutation in longitude and latitude, these are no other than the abscissa and ordinate of the little ellipse in which the pole moves. The law of its motion, however, therein, cannot be understood till the reader has been made acquainted with the principal features of the moon's motion on which it depends.

(327.) Another consequence of what has been shown respecting precession and nutation is, that sidereal time, as astronomers use it, i. e. as reckoned from the transit of the equinoctial point, is not a mean or uniformly flowing quantity, being affected by nutation; and, moreover, that so reckoned, even when cleared of the periodical fluctuation of nutation, it does not strictly correspond to the earth's diurnal rotation. As the sun loses one day in the year on the stars, by its direct motion in longitude; so the equinox gains one day in 25,868 years on them by its retrogradation. We ought, therefore, as carefully to distinguish between mean and apparent sidereal as between mean and apparent solar time.

(328.) Neither precession nor nutation change the apparent places of celestial objects inter se. We see them, so far as these causes go, as they are, though from a station more or less unstable, as we see distant land objects correctly formed, though appearing to rise and fall when viewed from the heaving deck of a ship in the act of pitching and rolling. But there is an optical cause, independent of refraction or of perspective, which displaces them one among the other, and causes us to view the heavens under an aspect always to a