cording as the eastern or western hemisphere is represented by the diagram. Now, when the sun is on the equator in March (Art. 73.) he performs his diurnal motion in the equator EQ, and comes to the hori zon in O the east point; and as the equator is bisected by the horizon, and the sun's apparent diurnal motion is nearly uniform, he will set in the west point, and will be as long below the horizon as above it, making the days and nights equal; this is called the vernal equino.r. As the sun's declination increases to the northward, in his passage from the vernal equinox to the summer solstice, a variation will take place in his diurnal motion; this may be illustrated by imagining the sun's declination to be Ea, his place in the ecliptic to be e, and ab the parallel which he describes that day. Then will he come to the horizon at the point o, between the north and the east; when he arrrives at i it will be six o'clock, being half way between a and b his situations at noon and midnight; at e he will be on the prime vertical, or in the east ; at a he will be on the meridian; afterwards he will proceed in a contrary order along the western hemisphere, and will set between the west and the north. In this situation, the arc oa, occupied by the sun in passing from the horizon to the meridian, called the semidiurnal arc, being greater than ia, the half of a b, requires more than six hours to pass through it, and consequently the day is more than twelve hours long. The days will continue lengthening, and the sun rising and setting nearer the north point, as he approaches towards the point E, where the north declination is the greatest; and then the sun's diurnal motion will be described in the tropic of cancer, the days will be at the longest, the nights shortest, and this will be the summer solstice. From this period the sun's declination will be decreasing, and the days decreasing also, until the sun arrives at the equator, being the first point of libra; then will the length of the day and night be

again equal, and the sun will rise and set in the east and west: this is called the autumnal equinox, being about the 23d of September. After this time the declination of the sun will become southerly, and the days will decrease in length, which, with other attending circumstances, may be thus explained: Imagine m to be the sun's place in the ecliptic, and c d the parallel he describes. Then he arrives at n on the prime vertical before he gets above the horizon, and consequently can never be seen to shine from the east or west while his declination is south; nor can he, since u is below the horizon, for a similar reason, rise till after six o'clock, during the same period: he rises at s between the east and the south, and after proceeding gradually to the meridian, declines as uniformly, and sets between the south and the west. Here the semidiurnal arc c s being less than cu, the half of cd is passed over by the sun in less than six hours, and consequently the day is less than twelve hours in length. The days will continue to decrease until the sun has reached C, the greatest declination south, when his diurnal motion will be described in the tropic of capricorn: he will then rise and set at the farthest from the east and west towards the south, the day will be at the shortest, the night the longest, and this will be the winter solstice. The sun's south declination will be decreasing, and the days lengthening, from this time till he again arrives at the equator, which will be at the vernal equinox after which the seasons will continue gradually following each other, in beautiful harmony, in the manner here described *.

It may be proper to observe, that the different degrees of heat in summer and in winter, do not arise, entirely, from the different times which the sun is above the horizon. The direction and force of the solar rays have also considerable influence. During the short days, the effect of the sun is less, both with respect to the intensity of the rays, their direction, and the time of their conti

53. The change of seasons to an inhabitant of any place in south latitude, might be elucidated in a similar manner; and it would appear that, to such a person, the seasons would be just opposite to ours: that is, his summer would be our winter, his autumn our spring, and vice versa. An inhabitant of the equator has the days and nights all of equal length: but at both the times when the sun is in the equinox he will have summer; for the sun will then be in his zenith at noon and at both the solstices he will have winter; for then the sun at noon declines 23° 28' from his zenith: which must be too evident to need farther illustration.

54. Since the regular return of the seasons is intimately connected with the length of the year, it will be proper, previous to concluding this chapter, to point out a few methods by which the precise length of the year may be learnt. Astronomers consider the year under two distinctions: first, the tropical or solar year, properly so called, is the exact space of time in which the sun moves through the twelve signs of the ecliptic; and secondly, the sidereal or astral year, being the time occupied by the sun in passing from any fixed star till his return to it again. The changes of seasons depending on the first of these, the tropical year, we will first describe the manner in which its length may be determined. The latitude of the

nuance and during the long days it is greater in all these respects. It may be remarked too, that the severest frosts usually take place after the days have begun to lengthen; and the most oppressive heat prevails when the days are decreasing the reason of which is, that during the summer months, the earth having imbibed more heat than it gave out, is not exhausted of its superabundant warmth till towards the close of the year: in like manner, on account of the waste of the earth's heat being greater in winter than its supply, it continues to get colder and colder till a month or longer after the winter solstice. From a similar cause arises the difference between the spring and autumn; though the position of the sun, in respect of the earth, is in both the same.

place being known, and consequently the elevation of the equator (Art. 26.), observe carefully the sun's meridian altitude when he is nearest the equator (at either the vernal or autumnal equinox); and if the observed meridian altitude on any day is exactly equal to the complement of the latitude, that day at noon is the exact time of the equinox. But as it is not very probable that the sun should be on the equator precisely at noon, take the meridian altitude one day when it is less than the co-latitude, and on the following day when it is greater; then say, as the sum of the defect and excess of these two meridian altitudes (when taken from, and added to, the co-latitude): the distance in time between the observations :: the defect the time to be added to the first observation, for the true time of the equinox. After the same manner, may the time of the next vernal or autumnal equinox be found: the difference between these times will be the length of the year. In order to arrive at the greatest accuracy, it will be adviseable to take two such equinoxes as are at the distance of several years from each other: for then, whatever errors may attend the observations, by being divided into many parts, they will become in a manner insensible.

55. I. Cassini has given the following method of finding the length of the tropical year: Observe the meridian altitude (a) of the sun on the day nearest to an equinox: then the next year take its meridian altitude on two succeeding days, one when the altitude (m) is less than a, and the next when the altitude (n) is greater than a, and -m is the change in the sun's declination in twenty-four hours: hence, as the declination near the equinoxes changes uniformly, we have n―m: a-m :: 24 hours: the interval from the first of the two days, till the time when the sun's declination is the same as at the observation the year before. This fourth term, therefore, being added to the number of days between

the first two observations, gives the length of the year required. If there be an interval of several years before the second observation is made, and the interval between the times when the declinations were the same be divided by the number of years, the year's length will be determined more exactly.

For example, on March 20, 1672, M. Cassini's father observed the meridian altitude of the sun's upper limb, at the royal observatory at Paris, to be 41° 43'; and on March 20, 1716, Cassini himself found the meridian altitude of the upper limb to be 41° 27' 10", and on the 21st to be 41° 51'. The dif ference between the two latter altitudes is 23′ 50′′, and between the two former 15' 50". Therefore, by the rule, 23′ 50′′: 15′ 50′′:: 24 hours: 15h 56m 39. Hence, on March 20, 1716, at 15h 56m 39*, the declination of the sun was the same as on March 20, 1672, at noon. Now the interval between the first two observations was 44 common years, of which 34 consisted of 365 days each, and 10 of 366; therefore the interval in days was 16070, and the whole period between the equal declinations was 16070 days, 15h 56m 39: this divided by 44 gives 365 5h 49m 053 for the tropical year. This is called by some authors the apparent solar year: they apply a minute correction, which depends upon the motion of what is called the sun's apogee (Art. 297.), and thus get 3654 5 48 49, for the length of a mean solar year.

56. To find the length of a sidereal year, M. Cassini presents us with this rule: Take the time (t) of a fixed star's transit over the meridian by a clock adjusted to mean solar time; the year following observe the time again on two days, one (m) when the star passes the meridian before, and the other (n) after the time t; then, m-nm-t: twenty-four hours, or more accurately 23h 56 45, the length of a sidereal day (Art. 114.): the time from m till the difference between the star's and sun's right ascension was the