contained in the sector AP S. Suppose we set out from A the perihelion, then will the angle A SP at first increase more rapidly than the mean longitude, and will, therefore, during the whole semi-revolution from A to M, exceed it in amount; or, in other words, the true place will be in advance of the mean: at M, one half the year will have elapsed, and one half the orbit have been described, whether it be circular or elliptic. Here, then, the mean and true places coincide; but in all the other half of the orbit, from M to A, the true place will fall short of the mean, since at M the angular motion is slowest, and the true place from this point begins to lag behind the mean to make up with it, however, as it approaches A, where it once more overtakes it.

(375.) The quantity by which the true longitude of the earth differs from the mean longitude is called the equation of the centre, and is additive during all the half-year, in which the earth passes from A to M, beginning at 0° 0′ 0′′, increasing to a maximum, and again diminishing to zero at M; after which it becomes subtractive, attains a maximum of subtractive magnitude between M and A, and again diminishes to 0 at A. Its maximum, both additive and substractive, is 1° 55′ 33′′-3.

(376.) By applying, then, to the earth's mean longitude, the equation of the centre corresponding to any given time at which we would ascertain its place, the true longitude becomes known; and since the sun is always seen from the earth in 180° more longitude than the earth from the sun, in this way also the sun's true place in the ecliptic becomes known. The calculation of the equation of the centre is performed by a table constructed for that purpose, to be found in all"Solar Tables."

(377.) The maximum value of the equation of the centre depends only on the ellipticity of the orbit, and may be expressed in terms of the excentricity. Vice versa, therefore, if the former quantity can be ascertained by observation, the latter may be derived from it; because, whenever the law, or numerical connection, between two quantities is known, the one can always be determined from the other. Now, by assiduous observation of the sun's transits over the meridian,

we can ascertain, for every day, its exact right ascension, and thence conclude its longitude (art. 309.). After this, it is easy to assign the angle by which this observed longitude exceeds or falls short of the mean; and the greatest amount of this excess or defect which occurs in the whole year, is the maximum equation of the centre. This, as a means of ascertaining the eccentricity of the orbit, is a far more easy and accurate method than that of concluding the sun's distance by measuring its apparent diameter. The results of the two methods coincide, however, perfectly.

(378.) If the ecliptic coincided with the equinoctial, the effect of the equation of the centre, by disturbing the uniformity of the sun's apparent motion in longitude, would cause an inequality in its time of coming on the meridian on successive days. When the sun's centre comes to the meridian, it is apparent noon, and if its motion in longitude were uniform, and the ecliptic coincident with the equinoctial, this would always coincide with mean noon, or the stroke of 12 on a well-regulated solar clock. But, independent of the want of uniformity in its motion, the obliquity of the ecliptic gives rise to another inequality in this respect; in consequence of which, the sun, even supposing its motion in the ecliptic uniform, would yet alternately, in its time of attaining the meridian, anticipate and fall short of the mean noon as shown by the clock. For the right ascension of a celestial object forming a side of a right-angled spherical triangle, of which its longitude is the hypothenuse, it is clear that the uniform increase of the latter must necessitate a deviation from uniformity in the increase of the former.

(379.) These two causes, then, acting conjointly, produce, in fact, a very considerable fluctuation in the time as shown per clock, when the sun really attains the meridian. It amounts, in fact, to upwards of half an hour; apparent noon sometimes taking place as much as 16 min. before mean noon, and at others as much as 14 min. after. This difference between apparent and mean noon is called the equation of time, and is calculated and inserted in ephemerides for every day of the year, under that title: or else, which comes

to the same thing, the moment, in mean time, of the sun's culmination for each day, is set down as an astronomical phanomenon to be observed.

(380.) As the sun, in its apparent annual course, is carried along the ecliptic, its declination is continually varying between the extreme limits of 23° 27′ 30′′ north, and as much south, which it attains at the solstices. It is consequently always vertical over some part or other of that zone or belt of the earth's surface which lies between the north and south parallels of 23° 27' 30". These parallels are called in geography the tropics; the northern one that of Cancer, and the southern, of Capricorn; because the sun, at the respective solstices, is situated in the divisions, or signs of the ecliptic so denominated. Of these signs there are twelve, each occupying 30° of its circumference. They commence at the vernal equinox, and are named in order-Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces. They are denoted also by

the following symbols: -Y, 8, 11, ≈, k, m, ^, m, ↑, W,, X. Longitude itself is also divided into signs, degrees, and minutes, &c. Thus 5 27° 0' corresponds to

177° 0'.

66

(381.) These Signs are purely technical subdivisions of the ecliptic, commencing from the actual equinox, and are not to be confounded with the constellations so called (and sometimes so symbolized). The constellations of the zodiac, as they now stand arranged on the ecliptic, are all a full sign" in advance or anticipation of their symbolic cognomens thereon marked. Thus the constellation Aries actually occupies the sign Taurus 8, the constellation Taurus, the sign Gemini II, and so on, the signs having retreated among the stars (together with the equinox their origin), by the effect of precession. The bright star Spica in the constellation Virgo (a Virginis), by the observations of They may be remembered by the following memorial hexameters :Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo,

Libraque, Scorpius, Arcitenens, Caper, Amphora, Pisces.

Retreated is here used with reference to longitude, not to the apparent diurnal motion.

SIDEREAL, TROPICAL, AND ANOMALISTIC YEARS. 225

Hipparchus, 128 years B.C., preceded, or was westward of the autumnal equinox in longitude by 6°. In 1750 it followed or stood eastward of the same equinox by 20° 21'. Its place then, as referred to the ecliptic at the former epoch, would be in longitude 5o 24° 0′, or in the 24th degree of the sign &, whereas in the latter epoch it stood in the 21st degree of my, the equinox having retreated by 26° 21' in the interval, 1878 years, elapsed. To avoid this source of misunderstanding, the use of "signs" and their symbols in the reckoning of celestial longitudes is now almost entirely abandoned, and the ordinary reckoning (by degrees, &c. from 0 to 360) adopted in its place, and the names Aries, Virgo, &c. are becoming restricted to the constellations so called.*

(382.) When the sun is in either tropic, it enlightens, as we have seen, the pole on that side the equator, and shines over or beyond it to the extent of 23° 27′ 30′′. The parallels of latitude, at this distance from either pole, are called the polar circles, and are distinguished from each other by the names arctic and antarctic. The regions within these circles are sometimes termed frigid zones, while the belt between the tropics is called the torrid zone, and the intermediate belts temperate zones. These last, however, are merely names given for the sake of naming; as, in fact, owing to the different distribution of land and sea in the two hemispheres, zones of climate are not co-terminal with zones of latitude.

(383.) Our seasons are determined by the apparent passages of the sun across the equinoctial, and its alternate arrival in the northern and southern hemisphere. Were the equinox invariable, this would happen at intervals precisely equal to the duration of the sidereal year; but, in fact, owing to the slow conical motion of the earth's axis described in art. 317., the equinox retreats on the ecliptic, and meets the advancing sun somewhat before the whole sidereal circuit is completed. The annual retreat of the equinox is 50"-1, and this arc is

When, however, the place of the sun is spoken of, the old usage prevails. Thus, if we say "the sun is in Aries," it would be interpreted to mean between 0° and 30° of longitude. So, also, "the first point of Aries" is still understood to mean the vernal, and "the first point of Libra," the autumnal equinox; and so in a few other cases.

described by the sun in the ecliptic in 20m 19.9. By so much shorter, then, is the periodical return of our seasons than the true sidereal revolution of the earth round the sun. As the latter period, or sidereal year, is equal to 365a 6h 9m 946, it follows, then, that the former must be only 365d 5h 48m 499.7; and this is what is meant by the tropical year.

(384.) We have already mentioned that the longer axis of the ellipse described by the earth has a slow motion of 11"-8 per annum in advance. From this it results, that when the earth, setting out from the perihelion, has completed one sidereal period, the perihelion will have moved forward by 11"-8, which arc must be described by the earth before it can again reach the perihelion. In so doing, it occupies 4m 39.7 and this must therefore be added to the sidereal period, to give the interval between two consecutive returns to the perihelion. This interval, then, is 365d 6h 13m 498-3*, and is what is called the anomalistic year. All these periods have their uses in astronomy; but that in which mankind in general are most interested is the tropical year, on which the return of the seasons depends, and which we thus perceive to be a compound phenomenon, depending chiefly and directly on the annual revolution of the earth round the sun, but subordinately also, and indirectly, on its rotation round its own axis, which is what occasions the precession of the equinoxes; thus affording an instructive example of the way in which a motion, once admitted in any part of our system, may be traced in its influence on others with which at first sight it could not possibly be supposed to have any thing to do.

(385.) As a rough consideration of the appearance of the earth points out the general roundness of its form, and more exact enquiry has led us first to the discovery of its elliptic figure, and, in the further progress of refinement, to the perception of minuter local deviations from that figure; so, in investigating the solar motions, the first notion we obtain is that of an orbit, generally speaking, round, and not far from

These numbers, as well as all the other numerical data of our system, are taken from Mr. Baily's Astronomical Tables and Formulæ, unless the contrary is expressed.