are very nearly 1461 days, or in one year 365 days. A clock which has indicated 365 days of 24 hours each in one year, has kept mean solar time (97). Twentyfour hours by this clock is a mean solar day. At certain times in the year, the time from noon to noon is about 8 seconds less than 24 hours of mean solar time, and at other times about 24 seconds more. These differences accumulating day by day soon amount to an aggregate which is considerable.

237. Equation of time.-At 12 by the clock the sun may have already passed the meridian, and is said to be fast of the clock; it may not yet have come to the meridian, and is slow of the clock. The difference in time between apparent noon, as shown by the passage of the sun over the meridian, and mean noon, as shown by the clock, on any day of the year, is the equation of time for that day.

CAUSES OF EQUATION.

238. The causes which produce this variation in time

are two:

1. The unequal apparent motion of the sun on the ecliptic, caused by the unequal real motion of the earth in its orbit.

2. The variable inclination of this motion from day to day to the equinoctial.

239. First cause.-The earth's angular motion (194) is fastest when the earth is nearest the sun; that is, from September to March, the greatest rate being on the 1st of January. On that day, the earth makes more than its average angular progress, and therefore the sun makes more than his average apparent day's journey on the ecliptic. Hence, when the meridian is about to pass the

sun on the next day, it finds that the sun has moved to the eastward of his position on the day before by an amount greater than the average, and therefore more time will be required for the meridian to overtake him. Fig. 78.

The sun is accordingly slow of the clock about 8 seconds on this account. The same result occurs on the next day, and the sun is now 16 seconds slow. The difference will continue to increase daily until soon after the vernal equinox, when the earth moves at its average rate.

240. After equinox.-From the vernal equinox until apogee, the rate of the earth's angular motion is less than the average, and is constantly decreasing; the daily easting of the sun diminishes at the same rate. The length of the solar day, although still more than 24 hours of mean solar time, becomes gradually less, until the accumulated difference is entirely lost on the 1st of July, and the sun and clock, so far as this cause is concerned, come together again. After July 1st the sun becomes fast of the clock, as the sun's daily motion is less, and

the meridian comes up with the sun in less than the average time; the action of the preceding half year is reversed.

241. Second cause.-Were the sun's motion in longitude uniform, there would still be an equation of time. Difference in time is caused by difference in right ascension (106), but a uniform amount of motion in celestial lon

A 23°28

Fig. 79.

SUN'S LONG.

SUN'S R.A.

gitude, produces a variable amount of motion in right ascension. Let AC represent part of the equinoctial and BD a part of the ecliptic crossing the equinoctial. at E. Suppose that in one day the sun has moved from

E to B; his difference of right ascension will be EC, less than EB, because the base of a right-angled triangle is less than the hypotenuse. The sun is not so far to the east as his motion would indicate; the meridian overtakes him sooner, and the day is shorter by this cause. The sun is fast of the clock.

23°28

Fig. 80.

SUN'S LONG

\D ECLIPTIC

At the solstices, the path of the sun is nearly parallel to the equinoctial, but is removed from it 2310. The right ascension AB being reckoned on the equinoctial is more than the actual distance traversed by the sun; the sun's relative easting is increased; the day is longer; the sun is slow of the clock.

SUN'S R.A.

BEQUINOCTIAL

242. The equation for the day is found by combining the results obtained for each cause separately. Thus on April 4 the sun is slow from the first cause 7 m. 40 sec.; from the second, fast 4 m. 46 sec.; the equation is, there

fore, (740) + (4 46) (2 54); when the sun is on the meridian, the clock should show 2 minutes 54 seconds past twelve.

243. Morning and afternoon unequal.-Sunrise and sunset are equally distant from apparent noon; hence, if mean noon is, say 7 minutes later than apparent noon, the clock adds 7 minutes to the morning, and subtracts it from the afternoon; the morning is 14 minutes longest. Sunrise and sunset will be as much slow or fast of the clock as midday.

244. Table of equation of time. The values of the equation for each day in the year have been computed, and are arranged in a table at the end of the book. It is more important that a watch should agree with a recognized standard, than that it should be absolutely correct. It is useless, however, to attempt to regulate a watch by a sun-dial, or by a noon mark, without correction for equation of time. To say that a watch runs with the sun is to say that it is a poor time-keeper.

THE CALENDAR.

245. The tropical year.-The sidereal year is the time occupied by the earth in passing once round its orbit, or until it has brought the sun back to the same star in the heavens. Its length is 365 d. 6 h. 9 m. 9.6 sec. But the vernal equinox has a motion backward along the orbit, amounting to 50" of arc per annum; the earth, therefore, comes back to the vernal equinox a little sooner than to the precise place it started from a year before, as shown by the stars. The time required by the earth to return to the vernal equinox is called the tropical year; its length is 365 d. 5 h. 48 m. 46.05 sec. This is

Ast. 11

the year employed in the calendar; it is 20 m. 23.55 sec. less than the sidereal year.

The time required by the earth to return to perihelion is the anomalistic year. It is 365 d. 6 h. 13 m. 49.3 sec. It will be remembered that perihelion moves forward about 12" annually (231); hence, its year must be longer than the sidereal year.

246. The Julian Calendar.-For practical purposes it is convenient to consider some number of whole days a year. The Greek year had at different times 354, 360, and 365 days. The Roman year under Numa had 355 days. There was a continual discordance between the civil year and the astronomical year, which reached such a degree that the autumn festivals were celebrated in the spring, and those of harvest, in midwinter. An extra month, called Mercedonius, was added every second year. The length of this month was not fixed, but was arranged from time to time by the pontiffs, and this gave rise to serious.corruption and fraud, interfering with the duration of office and the collection of debts.

In the year 46, B. C., Julius Cæsar reformed the calendar. To restore the seasons to their proper months he made that year contain 445 days. Assuming the astronomical year to be 365 days, he made each fourth year to contain 366 days; the remainder 365. The added day was placed in the month of February. The 23d of February, called sexto-calendas, being the sixth before the calends, or 1st of March, was celebrated in honor of the expulsion of the kings; the additional day was placed next to this feast, and was called Bis-sexto-calendas, whence our name Bissextile.

247. The Gregorian Calendar.-The astronomical year, as assumed by Cæsar, was too long by 11 m. 13.95 sec., or about 3 days in 400 years. By the year A. D. 1582,