19 20

30 40

"

5° 60

"

3 1 16 0 4 58
119 11 5 4 46

9 56

14 55

19 54 24 51 29 51

9 33

14 21

19 11

24 2 28 54

10 8 23:10 4 8

8 17

12 23

16 42

2) 58 25 16

TABLE 44.-Astronomical time at Greenwich of all the mean New Moons for TABLE 43.-Equation for funding the De76 years, or four Lunar Cycles; Old Style. With this Table and two easy equa-clination of a celestial body. Arguments. tons, by Problem 42, the time of mean New Moo.. in any month of any year for Latitude of the body at the head of one of the ten thousand years after the Creation may readily be fortid, for either O. or N. S. divisions of the table, and Ecliptic declination ZYrs Jan. Feb. Mar. Apr. May June July Aug. Sep. | Oct | Nov. Dec.in the left hand column. SA.CD.H.D.H. D.H. D.H. D.H. D.H. D.H. D.H. D.II. D.H. D.H. D. H. Latitude and Declination of the same name. 11724 14 512 17 13 611 1911 8 9 20 9 9 7 22 6 10 5 23 4 12 41 21725 2 14 1 2 2 15 1 4 0 16 28 18 23 7 26 20 25 8 21 21 23 10 22 23 31726 21 11 19 23 21 12 20 119 1318 217 15 16 4 14 17 14 5 12 18 12 7 o 41727 10 20 9 9 10 21 9 10 8 23 7 12 7 0 5 13 4 13 14 2 5 1728 29 18 23 723 1927 826 20 25 9 24 22 23 11 21 23 21 12 20 61729 18 216 15 18 416 16 16 511 18 14 711 1911 810 21 9 71730 7 11 6 0 13 6 15 14 4 3 3 15 2 430 629 19 23 827 2115 3 13 81731 26 924 22 26 1024 23 24 11 23 022 1321 219 14 19 317 16 17 51 1 2 9 91732 5 17 14 614 19 13 812 2011 910 22 9 1 7 23 7 12 6 1 5 1420 1 26 10 1733 4 2 2 15 4 4 2 16 2 5 0 18 29 20 28 8 26 21 25 10 24 23 24 1122 0 39 11 1734 23 21 13 23 12 14 21 219 15 19 417 17 16 515 1914 713 20:23 0 13 12 1735 12 9 10 21 12 10 10 23 10 11 9 0 8 13 7 25 14 5 3 4 16 3 52328 0 13 1736 1 17 0 19 28 2123 926 2226 1125 023 1223 121 1421 3 141737 19 15 18 419 16 18 517 1716 615 1914 8 12 20 12 9 10 22 10 11 151733 8 23 7 12 9 1 7 13 7 2 6 3 5 4 3 17 2 5 1 1829 2029 8 16 1739 27 21 26 10 22 13 26 12 26 021 1324 222 15 21 10 20 16 19 5 18 18 17 1740 17 615 1916 814 21 14 912 22 12 11 11 0 9 12 9 1 7 14 7 32 18 1741 5 15 4 4 5 16 4 5 3 17 2 6 1 19 29 2023 927 22 26 11 25 23 19 1742 24 12/23 124 1423 322 1521 420 1719 617 18 17 7 15 20 15 9 0 20 1743 13 21 12 10 13 23 12 12 12 010 13 10 2 8 15 7 3 6 16 5 5 4 18 0 4 58 21 174 3 6 1 19 2 8 29 2423 1128 026 12 25 124 1423 322 15 5 4 45 221745 21 419 17 21 519 18 19 6 17 19 17 815 2014 913 22 12 11 12 010 4 6 23 1746 10 12 9 110 14 9 3 8 15 7 4 6 17 5 6 3 18 3 7 1 20 1 915 3 1 24 1747 29 10 27 23 29 11 23 027 126 125 1124 322 15 22 420 1720 6 18 28 25 1748 18 18 17 717 2016 9 15 21 14 10 13 23 12 12 11 010 13 9 2 8 15 20 1 25 25 1740 7 3 5 16 7 5 5 18 5 6 3 19 3 8 1 21 29 22 29 11 8 027 1222 0 33 271750 26 124 1426 3 24 1624 422 17 22 6 20 19 19 7 18 20 17 916 22 23 0 13 23 1751 15 10 13 23 15 11 14 013 12 12 111 1410 3 8 15 8 4 6 17 6 6 23 29 0 291752 4 18 3 7 3 20 2 9 1 21 29 23 29 12 28 025 13 26 224 154 3 30 1753 22 16 21 522 18 21 720 19 19 8 18 21 17 10 15 22 15 11 14 013 13 31175112 110 14 12 3 10 16 10 48 17 8 6 6 19 5 7 4 20 3 9 2 22 32 1755 1 10 1 11 29 12 29 127 14 27 3 24 1524 423 17 22 621 18 33 1756 20 7 18 20 19 9 17 21 17 10 15 23 15 12 11 12 13 12 210 15 10 4 311757 8 16 7 5 8 18 7 7 6 19 5 8 4 21 3 10 1 22 1 11 29 13 29 1 35 1758 27 14 26 327 1526 425 16 24 523 1822 7 20 19 20 318 21 18 10 36 1759 16 22 15 1117 0 15 13 15 113 14 13 311 1410 4 9 17 8 6 7 19 37 1760 6 7 4 20 5 9 3 22 3 10 1 23 1 12 29 13 28 227 1526 425 16 38 1761 24 522 1824 722 2022 920 22 20 10 18 23 17 11 17 0 15 13 15 2 39 1762 13 14 12 313 15 12 411 16 10 5 9 18 8 7 5 19 6 8 4 21 4 10 40 1763 2 23 1 12 3 0 1 13 1 129 323 16 27 4 25 1725 623 1923 7 41 1764 41 20 20 920 2219 11 18 23 17 12 17 115 14 14 2 13 15 12 411 17 421765 10 5 8 18 10 6 8 19 8 17 6 20 6 9 4 22 3 10 2 23 1 12 1 1 43 1766 29 227 15 29 427 17 27 5 25 18 25 7 23 20 32 821 21 20 10 19 23 411757 18 1117 018 13 17 216 14 15 3 14 16 13 511 18 11 6 9 19 9 8 45 1763 7 20 6 9 6 22 5 11 4 23 3 12 3 1 1 14 29 15 29 427 1727 5 46 1769 25 1921 725 1924 823 20 22 921 1920 11 13 13 18 12 17 116 14 47 1770 15 213 15 15 4 13-17 13 511 16 11 7 9 20 8 8 7 21 6 10 5 23 48 1771 3 23 3 0 4 13 3 2 2 14 1 3 0 16 28 17 27 626 1925 824 21 491772 23 921 22 22 10 20 23 20 1119 018 13 17 215 14 15 3 13 16 13 5 50 1773 11 17 10 611 19 10 89 26 8 9 7 21 6 10 4 23 4 12 3 12 14 51 177 1 2 1 429 528 18 27 726 2025 8 23 21 23 1021 2321 11 52 1775 20 0 18 13 20 218 17 18 316 16 16 514 18 13 612 1911 810 21 53 1776 9 9 7 22 8 10 6 23 6 12 5 0 4 13 3 2 1 14 1 39 5 8 17 51 1777 27 625 1927 825 21 25 923 22 23 11 22 020 12 20 113 14 18 3 55 1775 16 15 15 4 16 17 15 614 1813 712 2011 9 9 21 9 10 7 23 7 12 561779 6 0 4 13 6 2 4 15 4 3 2 16 2 5 0 18 23 19 23 826 21 26 9 57 1780 21 22 23 11 23 23 22 1222 020 13 20 2 18 15 17 316 16 15 514 18 581751 13 611 1913 811 21 11 9 9 22 9 11 8 0 6 12 6 14 14 4 3 591782 2 15 1 4 2 17 1 6 0 18 28 2028 926 21 25 1021 23 23 1223 0 50 1733 21 13 20 221 14 20 319 1518 417 1716 614 18 14 712 20 12 9 61 1781 10 21 9 10 9 23 8 12 8 0 6 13 6 2 4 15 3 3 2 15 1 5 0 18 61785 28 19 27 828 21 27 10 26 22 25 11 25 023 13 22 121 1420 319 15 53 1786 18 4 16 17 18 5 16 1816 611 1914 812 2111 910 22 9 119 0 64 1757 7 12 6 1 7 14 6 3 5 15 4 4 3 17 2 6 0 18 29 2028 927 21 65 1788 26 10 24 23 25 12 24 123 1322 221 15/20 413 16 18 5 16 1816 7 66 1789 14 19 13 814 21 13 10 12 2211 11 0 9 13 8 17 14 6 3 5 16 671790 3 16 2 17 4 5 2 18 2 6 0 19 29 21 23 9 26 22 26 1125 024 14 681791 23 121 123 321 16 21 419 17 19 617 1916 715 2014 913 22 69 1792 12 10 10 2311 12 10 1 9 13 8 2 7 15 6 4 5 4 4 5 2 18 2 72 70 1793 0 19 0 21 28 24 23 11 27 026 13 25 123 14 23 321 1621 43 71 174 19 17 18 619 18 18 7 17 19 16 8 15 21 14 10 12 22 12 11 11 010 13 721795 9 1 7 14 9 3 7 16 7 4 5 17 5 6 3 19 2 7 1 20 29 22 29 1625 73 1795 27 23 26 12 27 025 13 25 123 1423 321 16 20 419 17 13 617 15 71 1797 16 7 14 2016 914 22 14 10 12 23 12 12 11 1 9 13 9 2 7 15 7 4: 75 1793 5 16 4 5 5 18 4 7 3 19 2 8 1 21 29 22 28 11 28 026 13 26 1799 24 14 23 3/24 1623 522 1721-620 18 7 1617 2017, 915 215 11

Diff between the
O. S. and N. S.

26 5 1

27 5 7

31 42

28 5 13

41 55

29 5 18

52 12

30 5 24

62 25

40 6 14

72 36

50 6 53

TABLE 52,-Miscellaneous Particulars of the Sun, Moon, al Planets.

distance Year, English

from the Sun, in

Medusa's head

the Goose

Northern Fly

Little florse
Lizard

Andromeda

Arrow

3818 27.

1 18 51.0333
2 29 36 .0099
0 46 28.0025
Greatest

Mean Lon.

Mo. in Longitude, in 365 Days.

34 62 90

Co-Si. Sine: Co-ta. Tang. Co-Se. Secant.] Note. To detect an error, should there be one, in the above Table, or in a similar table; or to supply one or more figures that may be effaced in any column; let it be observed that the sum of the logarithmic Sine and Co-secant of any arc or angle is 20.00000; so also is the sum of the Co-Sine and Secant 20.00000; and the sum of the Tangent and Co-Tangent is likewise 20,00000, So that if one of these logarithms is blurred, or illegible, subtract the other from 20.00000, and the remainder is the illegible one.

CHAPTER III.-DIVISION OF TIME.

ONE of the first cares of every nation or community, must undoubtedly always have been to establish some uniform method of reckoning time. Without such a standard to refer to, as occasion may require, the administration of public affairs, and the common concerns of life, would be subject to perpetual confusion. A well regulated calendar, therefore, is of the utmost importance. By means of it we are enabled to ascertain the return of the seasons, and the proper times for cultivating the earth; to adjust the observance of civil and ecclesiastical institutions, and to transmit to posterity the dates of such events as are worthy of remembrance.

Time, as measured by the heavenly bodies, is divided into Periods, Cycles, Years, Months, Weeks, Days, Hours, Minutes, and Seconds.

The most common measure for time is the year and day. There are several kinds of years; but the one most in use, or indeed used at all in ordinary cases, is the Solar, or Tropical year, on which the seasons depend. This is the time in which the earth moves once round the sun, and all the variety of the seasons occur, and afterwards begin anew. All nations do not begin their year at the same time. The Romans, whom we and most European nations follow, began theirs at the first of January; the Mahometans begin theirs at the Vernal Equinox; the Persians at the Summer Solstice; and the Chinese and East Indians at the first New Moon in March.

A Day is the time in which the earth turns once round on its axis; or it is the time from one noon to the next, or from one midnight to the next. About 365 1-4 days make one solar year. Different nations begin their day at different times. The Babylonians began theirs at sunrise; the Jews and Athenians at sunset; which the Italians, Austrians, and Bohemians, do at this time; the Egyptians, whom we and most European nations follow, began theirs at midnight; but Astronomers and Navigators begin their day at noon; the Astronomical day beginning where the Nautical day ends.

An Hour is the 24th part of a day; that is, there are 24 hours in one day. A Minute is the 60th part of an hour; and a Second is the 60th part of a minute. An exact tropical year is 365 days, 5 hours, 48 minutes, 49 seconds.

A Week is seven days. This is the most ancient collection of days that ever was. There are several kinds of Months, as the civil, solar, and lunar. The Lunar month is of two kinds, the Tropical or Sidereal, and the Synodic. The tropical or sidereal lunar month is the time in which the moon meves once round the earth through the signs of the zodiac, from any point in the zodiac to the same again; or from any fixed star to the same again; which is 27 days, 7 hours, 43 minutes, nearly. The synodic lunar month is one lunation,the time in which the moon moves round the earth, from the sun to the sun again; from one new moon to the next; or from one full moon to the next, which is 29 days, 12 hours, 44 minutes, 2.8283 seconds. The solar month is the time in which the sun or earth moves through one sign. The solar months are not exactly equal; but each is about the twelfth part of a solar year, there being 12 solar months in a solar year. The civil month is of two kinds, the weekly or legal month, and the calendar month. The weekly or legal month is 4 weeks, or 28 days. There are 12 calendar months in the solar year: the 2d, in common years, has 28 days, and in leap-years 29 days; the 4th, 6th, 9th, and 11th, have each 30 days; the others have 31 days each.

The names of the calendar months of the year, as well as of the days of the week, were mostly derived from heathen names of kings, gods, and idols, many of which were worshipped by the ancient Romans, and other eastern idolatrous nations. For this reason these names have been rejected by some; but in general they have been continued, to avoid that confusion in the calendar which would result from the introduction of new names. The first month of the year was called January, from the god Janus; the 2d, February, from Februa, a feast held in this month; the 3d, March, from the god Mars; the 4th, April, from aperio, a Latin word signifying to open, because of the opening of buds and flowers this month; the 5th, May, from Maia, a goddess to whom this month was dedicated; the 6th, June, from the goddess Juno; the 7th, July, frem Julius Cesar, a Roman general; the 8th, August, from Augustus Cesar, the first Roman emperor; the 9th, September, from septem, the Latin word for seven, this being then the 7th month of the year-March was then the first month of the year; the 10th, 11th, and 12th, October, November, and December, from octem, novem, and decem, Latin words for eight, nine, and ten, these being the 8th, 9th, and 10th months of the year-the year then began on the first of March. The 6th month of the ancient year, until the time of Augustus, was called Sextilis, from sex, the Latin word for six; and when it was changed in honour of that emperor, a day was taken from February and added to it. This month anciently had but 30 days, and February, in common years, 29 days, and in leap-years 30.

The first day of the week was called Sunday, from the Sun; the 2d, Monday, from the Moon; the 3d, Tuesday, from Tuisco, or Tew, the Saxon name of an idol; the 4th, Wednesday, from the Saxon idol Woden; the 5th, Thursday, from Thor, the Saxon name of Jupiter; the 6th, Friday, from Friga, supposed to be the Venus of the Saxons; the 7th, Saturday, from Sateor, signifying, in the Saxon, the day of Saturn.

The sidereal year, the time in which the earth moves round the sun from a fixed star to the same again, is 365 days, 6 hours, 9 minutes, 11.5 seconds, which is 20 minutes 225 seconds longer than a mean solar year.

The anomalistic year, the time in which the earth moves from its apogee to its apogee again, is 365 days, 6 hours, 13 minutes, 58.8 seconds, which is 25 minutes 9.8 seconds longer than a mean solar year.

Cycles and periods are explained in the Glossary.

AN EXPLANATION OF LEAP-YEAR.

While the earth moves once round the sun, it revolves round its own axis 365 1-4 times nearly, making about 365 1-4 days in a solar year. As every solar year is about 6 hours, or one fourth pait of a day, more than 365 days, three of our common civil years have 365 days each, and every fourth has 366 days. The exact time in which the earth moves round the sun is 365d. 5h 48m. 49s., which is the length of the true solar year, on which the seasons depend. Four solar years, therefore, amount to 1460 days, 23 hours, 15 minutes, 16 seconds, nearly one day more than 4 common years of 365 days each, which amount only to 1460 days. Therefore, to make the civil and solar years agree, every fourth year is a leap-year; but as this makes 182 civil years contain 8,213 days, or about one day more than 132 solar years, which contain only 48,211 days, 23 hours, 23 minutes, 48 seconds, it is necessary to abate one leap-year in every 132 years; or 3 leap-years in 4 centuries: therefore, the years 1700 and 1800, by the New Style, were not leap-years, and 1900 will not be a leap-year.

Were it not for the leap-years, the longest day of the year, which is now June 21st, would in 4 years be a day later; in 100 years it would be 25 days later; and in about 800 years, the longest day of the year would happen about January 1st, or near the middle of winter. And were every fourth year a leap-year, in 13 years the longest day would happen a day earlier, and in 264 years 2 days earlier, and so on: but by the present arrangement, (the New Style,) the solar and civil years will keep pace, or not differ a day in more than 12,000 years.

By the Old Style, every fourth year was a leap-year. When reckoning by the Old Style was adopted, in the year of our Lord 325, the longest day of the year happened June 21st, and the vernal equinox March 21st: and before the New Style was adopted by Great Britain, and the United States, then British colonies, in 1752, the longest day happened June 10th, and the vernal equinox March 10th. To restore these to the original days, 11 days were struck out of the month of September, 1752, by calling the 3d day the 14th; and it was ordered, that hereafter one leap-year in 132 years, or 3 leap-years in 400 years, should be omitted; that is, that the years 1700, 1800, and 1900, which by the Old Style would have been leap-years, should be common years of 365 days each; and that every other year which can be divided by 4, without a remainder, should be a leap-year. This arrangement is called the NEW STYLE."

To keep the equinoxes and the solstices to the same days of the same months; and to keep the time of celebrating Easter, and the other feasts, fasts, and holydays of the episcopal church, to the same season of the year, it was, that the style was altered. Reckoning by the Old Style is still continued in Russia, and some other eastern nations. As the year 1800 was not a leap-year by the New Style, and would have been by the Old Style, the difference between the styles is now 12 days. To reduce the New to the Old Style, subtract 12 days: after February 28th, 1900, subtract 13 days, until Feb. 28th, 2100; then subtract 14 days, until Feb. 28th, 2200; then 15, until 2300; then 16, until 2500, and so on. To change the New to the Old, or the Old to the New Style, for any date, see Table 45, page 24.

If any century of the Christian era can be divided by 4 without a remainder, the last year of that century is a leap-year; otherwise, it is not a leap-year. Thus, the year 2000, the last year of the 20th century, will be a leap-year, because 20 can be divided by 4 without a remainder; but the year 1900, the last year of the 19th century, will not be a leap-year, because 19 cannot be divided by 4 without a remainder. So the years 1800 and 1700, by the New Style, were not leap-years: every other year, divisible by 4 without a remainder, until the year 2100, is a leap-year.

OF THE DOMINICAL LETTER.

As there are 7 days in a week, it was formerly customary to mark them, or the days of the year, in Calendars, or Almanacs, with the first 7 letters of the alphabet, repeated continually; always beginning the year with A, that is, always calling the first day of January A, the 2d B, the 3d C, the 4th D, the 5th E, the 6th F, the 7th G; and then the same over again, calling the 8th A, the 9th B, the 10th C, the 11th D, the 12th E, the 13th F, the 14th G, the 15th A, the 16th B, and so on throughout the year. The letter that marks the Sundays, or Lord's days, is called the Sunday Letter, or the Dominical Letter, from Dominus, the Latin word for Lord. Thus, the 1st, 8th, 15th, 22d, and 29th days of any month, always have the same letter. By this arrangement, the first day of January and October are always A; of February, March, and November,