is called a solar, or, as above, an astronomical day ; it is measured by the sum of the whole equator 360°), and an arc of it equal to the daily motion in right ascension. For at the end of a diurnal rotation, which is known by observation to be uniform, the meridian is against the same star, or point of the ecliptic, it was against at the preceding noon (setting aside the very minute difference arising from the precession of the equinoxes); but the sun, during this period, has removed from that star to another, which has a greater right ascension: therefore, before the sun can be again on the same meridian, such an additional arch must be described as is equal to the daily motion in right ascension.

112. A sidereal day is the interval between two successive returns of the same fixed star to the same meridian it is less than the solar day, for it is measured by 360°, whereas the mean solar day is measured by 360° 59′ 8′′ nearly.

113. True or mean time, is that shewn by a clock whose 24 hours measure the time which the sun takes to describe an equatorial arc equal to 360° 59' 8": apparent time, is that shewn by the sun or a dial, where 24 hours are measured by the sum of 360°, and that day's motion in right ascension. The equation of time, is the difference between mean and apparent time, or between the mean and apparent noons, or between the times shewn by a clock and a sun-aial. If this equation can be accurately known at all periods, it will always enable us to reduce apparent time to that which is absolute or true.

114. The difference between the measures of a mean solar day, and a sidereal day, viz. 59′ 8′′, reduced to time, at the rate of 24 hours to 360°, gives 3m 56; from which we learn that a star which was on the meridian with the sun on one noon, will return to that meridian 3 56 previous to the next noon therefore a clock, which measures mean days

by 24 hours, will give 23h 56m 4s for the length of a sidereal day.

115. The mean and apparent solar days are never equal, except when the sun's daily motion in rightascension is 59' 8"; this is nearly the case about April 15th, June 15th, September ist, and December 24th on these days the equation is nothing, or nearly so; it is at the greatest about November 1st, when it is 16m 14.

116. The equation of time is calculated by tracing out the effects of three combined causes; the obli quity of the ecliptic, the sun's unequal apparent motion therein, and the precession of the equinoctial points in consequence of the first of these, in the first and third quadrants of the ecliptic from aries, that is, between aries and cancer, and between libra and capricorn, the right ascension being less than the mean longitude, the point of right ascension is to the west, and therefore the apparent noon precedes the mean noon; but in the second and fourth quadrants, namely between cancer and libra, or capricorn and aries, the right ascension being greater than the longitude, or the mean motion taken in the equator, the mean noon is westward, and therefore precedes the apparent noon. But, even if the plane of the ecliptic coincided with that of the equator, there would be a correction necessary; for the apparent annual motion of the sun being not quite uniform, a longer arc would be described in some days than others; that is, since the right-ascension and longitude would in this case be the same, the daily increments of right ascension would be unequal.

117. Dr. Maskelyne has invented a rule for computing the equation of time, in which all the three causes are considered; it was investigated in the following manner: Let APLQ (fig. 3, Pl. II.) be the ecliptic, ALQ the equator, A the first point of

aries, P the point where the sun's apparent motion is slowest, S any place of the sun; draw Sv perpendicular to the equator, and take An AP. When the sun begins to move from P, suppose a star to begin to move from with the sun's mean motion in right-ascension or longitude, viz. (Art.110.) at the rate of 59' 8" in a day, and when n passes the meridian let the clock be adjusted to 12. Take nmPs, and when the star comes to m, if the sun moved uniformly with his mean motion, he would be found at s; but at that time let S be the place of the sun. Let the sun S, and consequently v, be on the meridian; and then as m is the place of the imaginary star at that instant, my must be the equation of time. The sun's mean place is at s, and as An AP, and nm=Ps, we have Am=APs, consequently mo― Ac-Am-Av-APs. Let a be the mean equinox, or the point where it would have been if it had moved with its mean velocity, and draw a z perpendicular to AQ; then Am = A ≈ + x m = Aax co-sines Au+m: or because the co-sine of≈ A a the obliquity of the ecliptic, 23°

=

true right ascension;m the mean right ascension,

11

or mean longitude; and Aa (viz. A) is the

ascension..

12

equation of the equinoxes in right ascension; therefore the equation of time is equal to the difference of the sun's true right ascension and his mean longitude corrected by the equation of the equinoxes in right When Am is less than Av, mean or true time precedes apparent; when it is greater, apparent time precedes mean. That is, when the sun's true right ascension is greater than his mean longitude corrected as above shewn, we must add the

equation of time to the apparent to obtain the mean time; and when it is less, we must subtract. To convert mean time into apparent, we must subtract in the former case, and add in the latter.

Tables of the equation of time are computed by this rule, for the use of astronomers: they are either calculated for the noon of each day, as given in the Nautical and some other almanacs; or for every degree of the sun's place in the ecliptic, as is done in Table VI. at the end of the volume. But a table of this kind will not answer accurately for many years, on account of the precession and other causes, which render a frequent revisal of the calculations necessary.

118. It must be evident from the nature of the chief subjects treated on in this chapter, viz. Parallax, Refraction, and the Equation of Time, that the respective corrections for each of them must be carefully attended to in our observations upon the heavenly bodies, in order that the conclusions resulting from them may be relied upon in point of accuracy. The ancients, though not entirely ignorant of the nature of, these corrections, conceived the necessity of attending to them to be much les than it really is, and consequently but seldom regarded them on this account we cannot place so much dependence upon some of their observations as might be wished; though, on the whole, we have much greater reason to admire them for their skill, than to complain of their want of exactness.

CHAPTER VI.

On determining the Times of the Rising, Culminating, Setting, &c. of the Sun and fixed Stars.

ART. 119. WHEN we enquire on what part of the day the sun is in different positions, with respect to the horizon, meridian, prime vertical, &c. we always consider either the sun's declination, longitude, or right ascension, as known: for the nature of the sun's apparent motion is now so well ascertained, that his declination for every day at noon is determined with great precision; and this being known, together with the obliquity of the ecliptic, we readily find either the longitude or right ascension, by a common and easy rule in right-angled spherical triangles. Nay, in the Nautical Almanac (which, together with the Requisite Tables, cannot be too strongly recommended to the young astronomer), not only the declination, but the longitude and right ascension of the sun are given for every day at noon; and either of them may be found for any intermediate time, by proportion. Thus, suppose the declination on April 15th at noon is 9° 41′ 50′′ N., and on the 16th at noon 10° 3′ 14′′ N.; and the declination for 8 o'clock P.M. be required; say, as 24h: 8h 21′ 24′′ (the difference between the two declinations): 7' 8"; which added to 9° 41′ 50′′, gives 9° 48′ 58" N. for the declination sought.

120. It will be proper to remark in this place, that the declinations, longitudes, or right ascensions, as given in the Nautical Almanac, will not answer exactly for any other meridian than that of Green