her revolution round the earth, with respect to some fixed star: this. consists of 27 days, 7 hours, 43 minutes, 5 seconds, mean solar time.

16. The synodical lunation is longer than the periodical; because, while the moon is revolving round the earth from syzygia to syzygia again, she is advancing in the ecliptic at the mean rate of 13:10:35:02 every 24 hours :-for,

As 27:7:435: 360: :: 1 day to 13:10:35:02:-but, during the time that the moon is describing this arc of the ecliptic, the earth describes an arc of 59:8"33:-Hence, 13:10:35:02 - 59′8′′33 = 12:11:26:69, is the diurnal excess of the moon's motion in the ecliptic, over that of the earth. Now, as 12:11:26:69: 1 day :: 360: to 29:12:44:3; which is the correct measure of a synodical lunation, or the true value of the interval of time between new moon and new moon; and therefore it is 2 days, 5 hours, 0 minutes, 58 seconds, longer than the periodical lunation or time that the moon takes to revolve round the earth, with respect to any given fixed star. From what is stated in the four last articles, it will appear manifest to the reader that if the earth had not an annual motion round its orbit, the moon would revolve round it, so as to complete her synodical and her periodical revolutions in the same exact measure of time, viz. in 27 days, 7 hours, 43 minutes, and 5 seconds.

17. A Lunar Year consists of twelve synodical revolutions :-hence, 29:12:44"3: × 12 = 354 days, 8 hours, 48 minutes, 36 seconds, is the correct length of the lunar year; which is 10 days, 21 hours, 0 minutes, 12 seconds shorter than the solar year. This difference is estimated at 11 days in round numbers; and it is upon this that the epact is founded. As the epact and certain subjects connected therewith are prefixed to the Nautical Almanac, it may not be unnecessary to make a few observations relative to "the principal articles of the calendar.”

18. The Lunar Cycle, or golden number, is a revolution of 19 years; in the course of which, the conjunctions, oppositions, and other aspects of the moon, will fall upon the same days that they did nineteen years before..

19. The Solar Cycle is a revolution of 28 years; in the course of which, the days of the month return again to the same days of the week on which they fell 28 years before; and the leap-years begin the same course over again, with respect to the days of the week on which the days of the month fall :-all the variations of the dominical letters will also take place, and then return in the same order that they did twenty-eight years before.

20. The Dominical Letter is the Sunday letter of the year; A, being always taken as the first of January, or the representative of New-year's Day. The first seven letters of the alphabet are placed in the calendar opposite to the seven days of the week, and that which answers to Sun

day is called the dominical letter. If the year consisted of 365 days exactly, a period of the dominical letters would be completed in 7 years; but, because every fourth year is a bissextile, and contains 366 days, the period cannot be completed in a less time than four times seven (7 x 4), or twenty-eight years, agreeably to the revolution of the solar cycle.

21. The Epact signifies the moon's age at the beginning of the year, viz. the interval of time between the first minute of the first of January, and the first minute of the last new moon in the preceding month.

22. The Roman Indiction is a cycle of 15 years. It was used by the Romans, when masters of the then known world, for the purpose of indicating the times for levying a periodical tax upon the inhabitants of the conquered countries.

23. The Julian Period is a cycle of 7980 years, being the product arising from the multiplication of the cycles of the sun, moon, and Roman indiction, viz. 28 × 19 × 15 7980 years.

=

24. The Grand Celestial Period, or the Platonic Day, is a revolution of 25791 years; in which the annual retrograde motion of the colures (see pages between 301 and 305), at the rate of 50. 25 seconds of a degree, will cause the equinoxes to move completely round the ecliptic :-for, As 50251 year:: 360° to 25791 years. At the end of this long period, and not sooner, all the fixed stars in the firmament will return to the same precise places that they now occupy; and again describe the same circles, with respect to the equator and the poles of the earth, that they describe at present in the ethereal vault of heaven.

The respective values of the cycles, &c. mentioned in the above articles, may be easily determined; as thus :

25. To find the Solar Cycle:-Add 9* to the given year of our Lord, and divide the sum by 28: the quotient will be the number of cycles since the epoch of Christianity, and the remainder the solar cycle for the given year :-should nothing remain, the cycle is to be estimated at 28.

26. To find the Golden Number:-Add 1† to the given year of our Lord, and divide the sum by 19; then the quotient is the number of lunar cycles since the birth of Christ, and the remainder is the golden number: if nothing remain, the golden number will be 19.

27. To find the Julian Period:-Increase the given year of our Lord by 1, and add it to 4712; or, since Christ was born in the 4713th year of the Julian period; therefore, to the given year add the common number 4713, and the result will be the Julian period corresponding to the given year of the Christian era.

28. To find the Epact :-Subtract 1 from the golden number found as above, multiply by 11 and divide by 30; the result will be the epact for the year, or the moon's age on the first of January.

*The Solar Cycle was 9 at the birth of Christ.

+ The Golden Number was 1 at the birth of Christ.

29. To find the Dominical Letter:-If the day of the week on which the first day of the year falls be known; let A, be taken as the first of January; B, the 2d; C, the 3d, &c. &c.; then, the letter answering to Sunday will be the dominical letter for the year. But, if the day of the week on which the year commences be not known, proceed as thus:— To the given year add its fourth part, rejecting fractions; then divide the sum by 7, and the remainder will be the number of the dominical letter; calling G, 1; F, 2; E, 3; D, 4; C, 5; B, 6; and A, 0:should there be no remainder, then A is the Sunday letter for the year. As the 52 weeks into which the year is divided contain but 364 days, instead of 365, the dominical letter, therefore, retrogrades or falls back one in every succeeding common year, and two, when the year consists of 366 days :—hence, all leap-years are noted by two Sunday letters, viz. the letter which is peculiar to the intercalated year, and that which precedes it in the order of the first seven letters of the alphabet, as above.

I think it right to observe, that the subjects which are contained in the articles from 18 to the above, are of such minor importance, that they would not be noticed in this work, were it not for the purpose of keeping it in unison with the Nautical Almanac. We must now return to the further consideration of time, from which we broke off at the end of Article 17.

30. Apparent Time signifies the sun's horary distance from the meridian, reckoned westward from the time of transit; or, it simply expresses the hour of the day, shown by a correct sun-dial :-hence, it only relates to the true sun, and not to any other celestial object.

31. Mean Time is the hour which is shown by an equable-going clock or chronometer, adjusted to go 24 hours in an average solar day of 24 hours, 3 minutes, 56. 5554 seconds, measured in sidereal time: it is reckoned westward from the transit of the mean sun's centre over the meridian.

32. Sidereal Time is the hour which is shown by a well-regulated clock, adjusted to go 24 hours in a sidereal day of 23 hours, 56 minutes, 4.0906 seconds, measured in mean solar time. This time is reckoned westward, from the transit of the first point of Aries over the meridian; and since it is always equal to the sum of the right ascension of the mean sun and the mean time at any given place on the earth, it is, therefore, the same as the right ascension of the meridian; (defin. 16, page 299); and thus its value is quite different from the sidereal time, which is given in page II. of the month in the Nautical Almanac, as will be shown presently.

33. The Equation of Time is the difference between the sun's true right ascension and his mean longitude (in time), corrected by the equation of the equinoxes in right ascension. This equation implies a correction which is additive to, or subtractive from, the apparent time, deduced

from an observation of the sun, in order to reduce it to equable or mean time, such as that shown by a perfect chronometer.

34. The diurnal motion of the earth upon its axis being perfectly uniform at all times throughout the year, the sidereal days are therefore always of the same exact length. But, owing to the inequalities of the earth's annual motion in an elliptic orbit, combined with the obliquity of that orbit to the plane of the equator, the length of the solar day is constantly varying. And because the ecliptic is inclined to the equator in an angle of 23:27:38", the equable motion of the earth on its axis brings unequal portions of the ecliptic to the same meridian, in equal portions of time;-and thus it is that the apparent time, deduced from an observation of the sun, never corresponds with the mean time shown by a well-regulated clock or chronometer, except on four days of the year, viz. about the 15th of April, the 15th of June, the 31st of August, and the 24th of December. At all other times of the year, the unequal velocity of the earth in moving round its orbit, will cause the sun to appear upon the same meridian a little earlier or a little later every day, than the time indicated by an equable-going clock or chronometer: and, hence, the sun will appear to be fast, or before a perfect time-keeper on some days of the year, and slow, or behind it, on other days. This may be familiarly explained in the following manner, viz.:—

35. In the annexed diagram, let us suppose that there are two globes,

interior equator, represented by the dotted circle e, e, e, &c., with an uniform motion, according to the order of right ascension indicated by the common numerals 1, 2, 3, &c., on to 24. The black straight lines. answering to the common numerals are the meridians to which the imaginary globe comes every day at mean noon." The dotted straight lines corresponding to the Roman numerals I., II., III., &c., are the apparent meridians to which the real globe comes every day at apparent noon, in its annual revolution round the sun S. The letters A, B, C, D, represent the four days in the year, viz. the 15th April, the 15th June, the 31st August, and the 24th December; on which the right ascensions of the real globe and the imaginary one are equal, and on which the equation of time vanishes.

Let the two globes a and e be on the meridian, in the line S A, on the 15th of April. On this day, the right ascension of each globe will be about 1:33: hence, as there is no difference in the right ascension, the mean noon, shown by a well-regulated clock, that is adjusted to go exactly 24 hours in a mean solar day, will correspond with the apparent noon indicated by a correct sun-dial. Now, whilst the imaginary globe e, moves at an uniform rate round the dotted equator, increasing its right ascension by the diurnal fixed quantity 3"56:5554, the real globe a, will move with an unequal degree of velocity round the plane of the equator, a, a, a, &c., increasing its right ascension by unequal increments, which will be at certain times of the year greater, and at others less, than the above invariable quantity. While the two globes are moving from A to B, the right ascension of a, being less than that of e, the true globe comes to the meridian before the imaginary one; and thus its meridians, marked by the dotted lines S a II., S a III., &c. fall to the left hand of the black meridian lines Se 2, Se 3, &c. belonging to the imaginary globe*: and therefore the equation of time, or the solar angle e S a, contained between the two meridian lines, is subtractive from apparent time, and additive to mean time. The two globes will be in the line S B on the 15th June; on which day, as their right ascensions will be equal, each being about 5:33", the hour shown by a well-regulated clock will correspond with the hour indicated by a correct sun-dial, and therefore there will be no equation of time.

During the time that the two globes are moving from B to C, the right ascension of a, being greater than that of e, the real globe will not come to the meridian until after the imaginary one; and thus its dotted meridians S a VII., S a VIII., &c. fall to the right hand of the black meridian lines, Se 7, Se 8, &c. belonging to the imaginary globe* ; and therefore the equation of time, indicated by the solar angle e Sa, or the difference between the two meridian lines, is to be applied by addition to apparent time, and by subtraction to mean time.

The eye of the reader is to be directed from the centre S to the circumference.