These Elements are easily found from the following Tables and Precepts,
by the common Rules of Arithmetic.
_Note_, 60 minutes make a Degree, 30 degrees a Sign, and 12 Signs a
Circle. A Sign is marked thus ^s, a Degree thus °, and a Minute thus ʹ.
When you exceed 12 Signs, always reject them and set down the remainder.
When the number of Signs to be subtracted is greater than the number you
subtract from, add 12 Signs to that which you subtract from; and then
you will have a remainder to set down.
[Sidenote: How the Signs are reckoned.]
354. As we fix arbitrarily upon the beginning of the Sign _Aries_ to
reckon from, when we speak of the places of the Sun, Moon, and Nodes; we
call _Aries_ 0 Signs, _Taurus_ 1 Sign, _Gemini_ 2 Signs, _Cancer_ 3
Signs, _&c._ So, when the Sun is in the 10th degree of Aries, we say his
Place or Longitude is 0 Signs 10 Degrees, because he is only 10 Degrees
from the beginning of Aries: if he is in the 5th, 10th, _&c._ Degree of
Taurus, we say his Place or Longitude is 1 Sign, 5, 10, _&c._ Degrees:
and so on, till he comes quite round again. But in reckoning the
Anomalies of the Sun and Moon, and their distance from the Nodes, we
only consider the number of Signs and Degrees the Luminaries are gone
past their Apogee or Nodes; not how far they have to go to these points,
were the distance ever so little. The Sun, Moon, and Apogee move
according to the order of Signs, but the Nodes contrary. We shall now
give the Precepts and Examples for the above Requisites in their due
order.
_To calculate the time of New and Full Moon._
[Sidenote: First Element or Requisite.]
355. PRECEPT I. For any proposed year in the 18th Century, take out the
mean time of the New Moon in _March_ from Table I., and the mean time of
Full Moon from Table III., for the _Old Stile_; or from Tables II and IV
for _New Stile_; with the mean Anomalies of the Sun and Moon for these
times, and set them by themselves. Then, from Table VI, take out as many
Lunations as the proposed Month is after _March_, with the days, hours,
and minutes belonging to them; and also the mean Anomalies of the Sun
and Moon for these Lunations.
II. Add the days, hours, and minutes of these Lunations to the time of
New or Full Moon in _March_, and the Anomalies for the Lunations to the
Anomalies for _March_: the sums give the hours and minutes of the mean
New or Full Moon required, and the mean Anomalies of the Sun and Moon
for that time.
III. Then, with the number of days enter Table VII, under the given
Month, and right against this number, in the left hand column you have
the day of New or Full Moon; which set before the hours and minutes
above-mentioned.
IV. But, (as it will sometimes happen) if the number of days fall short
of all those under the given Month, add one Lunation with its Anomalies
from Table VI to the foresaid sums; so you will have a new sum of days
wherewith to enter the 7th Table under the given Month, where you are
sure to find that sum the second time, if the first falls short.
V. With the Signs and Degrees of the Sun’s Anomaly enter Table VIII,
_The Moon’s annual Equation_, and take out the minutes of time of that
Equation by the Anomaly; remembring, that if the Signs are at the head
of the Table, the degrees are at the left hand, in which case the
Equation found in the Angle of meeting must be subtracted from the mean
time of New or Full Moon, as the title _Subtract_, at the head of the
Table directs: but if the Signs are at the foot of the Table their
degrees are in the right-hand column, and the Equation where the Signs
and Degrees meet in the Table is to be added to the mean time, as the
title _Add_, at the foot of the Table directs; which Equation, so
applied, gives the mean time of New or Full Moon corrected.
VI. With the Signs and Degrees of the Sun’s Anomaly enter Table IX,
_Equation of the Moon’s mean Anomaly_, and take out the Equation
thereof; adding it to the mean Anomaly or subtracting it therefrom, as
the titles at the head or foot of the Table direct; and it gives the
mean Anomaly corrected. Then, with the Sun’s Anomaly enter Table XII,
_Equation of the Sun’s mean Place_, and take out that Equation, applying
it to the Moon’s corrected Anomaly as the titles direct; and it will
give the Moon’s Anomaly equated[77]. _N. B._ In all these Equations,
care must be taken to make proper allowance for the odd minutes of
Anomaly; the Tables having the Equations only for compleat Degrees.
VII. With the Moon’s equated Anomaly enter Table X, _The Moon’s elliptic
Equation_, and take out that Equation in the same manner as the
preceding: adding it to the former corrected time if the Signs be at the
head of the Table, or subtracting it if they be at the foot, as the
Table directs; and this gives the mean time equated.
VIII. Lastly, enter Table XI, _The Sun’s Equation at New and Full Moon_,
with the Sun’s Anomaly, and take out the Sun’s Equation in the same
manner as the others; adding it to, or subtracting it from the former
equated time, as the titles direct: and by this last Equation you have
the true time of New or Full Moon, agreeing with well regulated Clocks
and Watches. But to make it agree with true Sun-Dials, the Equation of
time must be applied as taught § 225.