will cause the equinoctial colures to be in the plane of the dotted line eSb-Hence, this actual retrograde motion of the equinoxes gives an F D B E E apparent progressive motion to the fixed star at a in the firmament, and causes it to appear at the point c, a little to the eastward of where it was seen the year before: and therefore the earth must travel farther, or move on from tox, before it can bring the sun and star in the same right line, viz. in the dotted line x S dc, so as to complete the sidereal year. The arc of the ecliptic, which is intercepted between the points and xb, is the actual recession of the equinoctial points, Aries and Libra:-and the arc d, which is similar tox, and equal to the arc a c, in the starry firmament, is what is called "the precession of the equinoxes." Precession is evidently an incorrect denomination; but, it is in perfect harmony with that optical illusion, under which, to an observer on the earth at, the fixed star, a, will appear to have advanced to the eastward, or to have described the arc a c-The annual value of this arc, deduced from numberless observations, is 501 seconds of a degree. Now, because the recession of the equinoxes has no reference to the sun, S, but to the fixed star, a, which is at an infinite distance beyond it; therefore the earth will revolve from the point to the point again, and thus go through the orbital circle of 360 degrees in its an nual revolution round the sun, the same as if the equinoctial points and were immovable, or had no retrograde motion. Hence, the solar year will be completed in every return of the earth to the same relative point of the ecliptic; at which point it will always arrive, the instant that it has performed 365. 24222 diurnal revolutions round its axis. But, with respect to the fixed star, a; while the earth is moving round the ecliptic from west to east, the line of the equinoctial colures, marked Sva, is moving in an opposite direction, or from east to west; and therefore the earth and the equinoctial point will meet at e, before the sidereal year is completed.-The fixed star, a, will then be, apparently, at the point C, in the firmament; and thus the earth will have to move from e to æ, so as to be in the same right line (the dotted line x 8 d c) with the sun and the fixed star, at c; which completes the sidereal year. The earth, in moving round its orbit, so as to accomplish the sidereal year, must always perform something more than one diurnal revolution on its axis, beyond the number of diurnal turns which it takes to complete the solar year. This, though seemingly paradoxical, will be easily comprehended, by reflecting that, since the equinoctial point is moving from east to west, or advancing, as it were, to meet the earth; this retrograde motion of the equinox shortens the measure of the earth's daily advance in the ecliptic by a certain portion of a degree, which bears the same proportion to the annual value of the equinoctial recessione, that the earth's daily motion in its orbit does to the great circle of 360 degrees :-and, hence, by the time that the earth has arrived at the point, it will have made 366. 25222 revolutions upon its axis; that is, it will reckon one complete revolution and a small fraction more upon its axis, than it did in performing the same circuit with respect to the sun. But, as it must move from to x, before the sidereal year is completed; therefore, it will reckon the fractional part .00420 beyond the above expression: thus making in the whole. 366.25642 revolutions. The earth, in this respect, may be likened to a ship, which, by sailing round the globe in an easterly direction, would gain one complete day by the time that she returned to the port from which she set out: -for, when a ship sails easterly, she advances towards the sun, and therefore shortens the interval of time between every two returns of the sun to her noon, or 12 o'clock meridian line, in proportion to the meridional distance made good during that interval. Hence, being farther advanced towards the east every evening than in the preceding morning, she will cause the sun to set below her western horizon, something sooner than if she had not so advanced :-and, therefore, by curtailing, each diurnal arc in proportion to the east longitude made good, she will register one day more in her log-book at her return (let the period of her circumnavigation be ever so short or ever so long), than will be reckoned by the inhabitants of the port or place from whence she sailed. 6. Although the illustration of the diagram is sufficient to show that it is the recession of the equinoctial points which causes the sidereal year to be longer than the solar; yet, with the view of elucidating this curious subject in the most ample manner, I shall arrange a few simple proportions, which will not only confirm the above fact, but also prove that it is owing to the same cause, that the earth takes a little beyond one diurnal revolution more upon its axis to complete the sidereal year, than it does to finish the solar year. The earth moves round the ecliptic, and completes the tropical year, in 365 days, 5 hours, 48 minutes, 48 seconds, or in 31556928 seconds of mean solar time. The ecliptic circle contains 360 degrees, or 1296000 seconds of motion. A natural day, viz. while the earth is turning once round upon its axis, consists of 24 hours, or 86400 seconds, mean solar time. Then-As 31556928: 1296000%::86400! to 3548" 33018 (=59:8:33;) which, therefore, is the arc of the ecliptic, that the earth describes every day, or the mean rate at which it moves round the ecliptic, during the period of making one complete revolution round its axis. Now, as the arc of the ecliptic, thus described, is to one diurnal revolution, so is the great circle of the ecliptic, to the number of diurnal revolutions which the earth takes to complete the solar year :-Hence, As 3548 330181 Rev. :: 1296000? to 365.24222 revolutions; which is the correct number of times that the earth must revolve round its axis, in performing its annual circuit round the sun. It has been determined by numberless observations, as stated at bottom of page 302, that the recession of the equinoxes amounts to 50 seconds of a degree every year; and we have seen, as above, that the earth advances 3548733018 in the ecliptic, while it is describing the great circle of its diurnal revolution; and since this circle, like all others, contains 360 degrees or 1296000 seconds, we are thus fúrnished with the necessary data for determining the absolute length of the sidereal year:-Hence, As 3548733018: 1296000"::50:25, to 18353:42; which being converted into time, gives 20 minutes 23 seconds for the excess of the sidereal year above the solar :-then, this excess being added to the solar year, gives 365 days, 6 hours, 9 minutes, 11 seconds, or 31558151 seconds; which, therefore, is the absolute length of the sidereal year. Now, observation shows that the earth performs one diurnal revolution round its axis, viz. that it turns completely round from any fixed star to the same star again, in 23 hours, 56 minutes, 4. 0966 seconds, or in 86164. 0906 seconds of mean solar time.-Hence, As 86164:0906: 1 Rev.::31558151.5, to 366. 25642; which, therefore, is the actual number of revolutions that the earth must make round its axis, to complete the sidereal year; which is one complete revolution, and the fraction .01420 of another, more than it takes to finish the solar year. Having thus shown that the excess of the sidereal year above the solar, is entirely owing to the retrograde motion of the equinoxes, we shall now resume the subject relating to the consideration of time, from which we broke off at the end of Article 4, page 301. 7. An Apparent Solar Day is the interval of time between two consecutive transits of the sun over the same meridian, as shown by a correct sun-dial:—this species of day is subject to an incessant variation, arising from the obliquity of the ecliptic, and the unequal motion of the earth round its orbit. 8. A Natural Day.-This consists of 24 hours, as shown by a wellregulated clock, or chronometer; being exactly equal to the time that the earth takes to turn once round upon its axis. 9. A Mean Solar Day is equal to the average length of all the days in a tropical year, and consists of 24 hours, 3 minutes, 56. 5554 seconds, in sidereal time; but of 24 hours, exactly, in mean or equable time. 10. A Sidereal Day is the interval of time between two consecutive returns of any fixed star to the same meridian; or, rather, it is the absolute time in which the earth performs one revolution round its axis, in relation to a fixed star this consists of 23 hours, 56 minutes, 4.0906 seconds, in mean solar time, or 24 hours in sidereal time. 11. If the earth had not an annual motion round the sun, the solar day and the sidereal day would be precisely of the same length; but, while the earth revolves once round its axis, it advances 59:8:33 in its orbit (paragraph 2, Article 6); and, therefore, should the sun and a fixed star be on the meridian of a place on any given day, the star will come to the same meridian the next day, when the sun is 59:8:33 short of it-Hence, the earth must perform something more than one complete turn on its axis; or go through an arc of its diurnal circle, equal to the measure of its daily advance in the ecliptic, before it can bring the sun to the same meridian again. The value of this diurnal rotatory arc of excess, may be determined in the following manner, viz.: As 360 degrees: 1 Rev. :: 59:8:33 to .00273;* which is the absolute value of the arc that the earth must describe beyond one diurnal rotatory turn upon its axis, before it can cause the sun to be upon the same meridian that it was the day preceding. But, as to the fixed stars; they will always return to the same meridian at the end of every 23 hours, 56 minutes, 4.0906 seconds, mean solar time, or 24 hours Or,-As 24h. 1 diurnal turn: 3m. 56s. 5554 to .00273 of a diurnal turn. X sidereal time :-for the distance at which they are placed in the firmament of heaven, beyond the sun, is so immeasurably great, that the diameter of the earth's orbit, compared therewith (though upwards of 190 millions of miles in extent), dwindles into a mere dimensionless point; and therefore the earth will bring the meridian of any given place to the same star again, at the end of every complete revolution round its axis, the same as if it had no annual motion round the ecliptic. 12. The equable motion of the earth, with respect to the fixed stars, will cause any given star in the heavens to return to the same meridian again, at the end of every complete diurnal revolution round its axis ; -and since this is performed in 23:56"4:0906, mean solar time, or 24 hours sidereal time; the fixed stars, therefore, anticipate 3:55:9094 in mean solar time, upon the sun, every day, or 356-5554 in sidereal time. The never-failing uniformity of this measure of time, furnishes us with an infallible standard for proving the correctness of clocks and watches, as pointed out in the explanation of Table XLV. in pages 117 and 118. 13. A Lunar Day is the interval of time between two consecutive returns of the moon to the same meridian: this, upon an average, is about 24 hours, 48 minutes, 46 seconds. The lunar day being so much longer than the solar, is owing to the combined motions of the earth and moon for, while the earth turns once round upon its axis, in 24 hours, it advances 59:8:33 in its orbit; but, during that time, the moon, in going through a portion (betwixt the 29th and 30th part) of her periodical revolution round the earth, advances, at a mean rate, 13:10:35:02 in the ecliptic: hence, she gains, on an average, 12:11:26:69 upon the earth every day; and therefore the earth cannot bring the moon upon the same meridian that she passed the day before, until it has described an arc of 12:11:26:69 over and above the great circle of 360 degrees, which measures the diurnal circuit round its axis. This arc, reduced to time, is 48:45 77, or, rejecting fractions, 48 minutes and 46 seconds; and, as the arc of excess, thus described, bears the same proportion to a great circle, that the fraction over a diurnal turn of the earth does to one complete revolution round its axis; therefore, As 360 degrees: 12:11:26769 :: 1 Rev. to.03386; which is the value of the fraction of excess in relation to one diurnal turn :—hence, the earth must make one complete revolution round its axis and . 03386, or rather more than the third part of another turn, before it can cause the moon to transit over the same meridian again. 14. A Synodical Lunation is the interval of time between two consecutive new, or full moons: this consists of 29 days, 12 hours, 44 minutes, and 3 seconds, mean solar time. 15. A Periodical Lunation is the time which the moon takes to finish |