In

course 90° distant from V, the vernal equinox, which is one of its poles so that V R (the right ascension) being given, and also V E, the arc E R, and its measure, the spherical angle EP R, or K P X, is known. the spherical triangle K P X, then, we have given, 1st, The side PK, which, being the distance of the poles of the ecliptic and equinoctial, is equal to the obliquity of the ecliptic; 2d, The side PX, the polar distance, or the complement of the declination R X; and, 3d, the included angle K PX; and therefore, by spherical trigonometry, it is easy to find the other side K X, and the remaining angles. Now K X is the complement of the required latitude X T, and the angle PKX being known, and P K V being a right angle (because S V is 90°), the angle X K V becomes known. Now this is no other than the measure of the longitude VT of the object. The inverse problem is resolved by the same triangle, and by a process exactly similar.

(261.) The same course of observations by which the path of the sun among the fixed stars is traced, and the ecliptic marked out among them, determines, of course, the place of the equinox V upon the starry sphere, at that time a point of great importance in practical astronomy, as it is the origin or zero point of right ascension. Now, when this process is repeated at considerably distant intervals of time, a very remarkable phenomenon is observed; viz. that the equinox does not preserve a constant place among the stars, but shifts its position, travelling continually and regularly, although with extreme lowness, backwards, along the ecliptic, in the direction V W from east to west, or the contrary to that in which the sun appears to move in that circle. As the ecliptic and equinoctial are not very much inclined, this motion of the equinox from east to west along the former, conspires (speaking generally) with the diurnal motion, and carries it, with reference to that motion, continually in advance upon the stars: hence it has acquired the name of the precession of the equinoxes, because the place of the equinox among the stars, at

CHAP. IV.

PRECESSION OF THE EQUINOXES.

169

every subsequent moment, precedes (with reference to the diurnal motion) that which is held the moment before. The amount of this motion by which the equinox travels backward, or retrogrades (as it is called), on the ecliptic, is 0° 0′ 50′′·10 per annum, an extremely minute quantity, but which, by its continual accumulation from year to year, at last makes itself very palpable, and that in a way highly inconvenient to practical astronomers, by destroying, in the lapse of a moderate number of years, the arrangement of their catalogues of stars, and making it necessary to reconstruct them. Since the formation of the earliest catalogue on record, the place of the equinox has retrograded already about 30°. The period in which it performs a complete tour of the ecliptic, is 25,868 years.

(262.) The immediate uranographical effect of the precession of the equinoxes is to produce a uniform increase of longitude in all the heavenly bodies, whether fixed or erratic. For the vernal equinox being the initial point of longitudes, as well as of right ascension, a retreat of this point on the ecliptic tells upon the longitudes of all alike, whether at rest or in motion, and produces, so far as its amount extends, the appearance of a motion in longitude common to all, as if the whole heavens had a slow rotation round the poles of the ecliptic in the long period above mentioned, similar to what they have in twenty-four hours round those of the equinoctial.

(263.) To form a just idea of this curious astronomical phenomenon, however, we must abandon, for a time, the consideration of the ecliptic, as tending to produce confusion in our ideas; for this reason, that the stability of the ecliptic itself among the stars is (as already hinted, art. 257.) only approximate, and that in consequence its intersection with the equinoctial is liable to a certain amount of change, arising from its fluctuation, which mixes itself with what is due to the principal uranographical cause of the phenomenon. This cause will become at once apparent, 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.

(264.) 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 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 nutation. 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 of advantage to present clearness to consider them separately.

(265.) 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. 259., suppose the

CONICAL MOVEMENT OF THE EARTH'S AXIS. 171

pole P in the progress of its motion in the small circle POZ 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. c. 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.

(266.) 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 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, in a way calculated to give at once a clear conception of it as a fact, and a considerable insight into its physical cause as a dynamical effect. 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.

(267.) 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 it 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.

(268.) 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,

(269.) Both these motions, however, although here considered separately, subsist jointly; and since, while in virtue of the nutation, the pole is describing its little