EXAMPLE.

EXAMPLE.
To find the Sun’s true Place April 30th, A. D. 1757, at 18 minutes 40 seconds past 10 in the morning.

EXAMPLE.
To find the Suns true Place May the 28th at 4 hours 3 min. 42 sec. in the afternoon, the year before Christ 585, which was a Leap year[^[82]].

To find the Sun’s Declination.

To find the Angle of the Moon’s visible Path with the Ecliptic.

To find the Moon’s Latitude.

Which is Aries 12° 9ʹ; and this, when taken from three Signs, or the beginning of Cancer, leaves 2 signs 17 deg. 51 min., or 77° 51ʹ for the Sun’s distance from the then nearest Solstice.

360. But because the Sun’s true Place is often wanted when the Moon is neither New nor Full, we shall next shew how it may be found for any given moment of time: though this be digressing from our present purpose.

In [Table XVI] find the nearest lesser year to that in which the Sun’s Place is sought; and take out the Sun’s mean Longitude and Anomaly answering thereto; to which add his mean motion and Anomaly for the compleat residue of the years, with the month, day, hour, and minute, all taken from the same Table, and you have the Sun’s mean Longitude and Anomaly for the given time. Then, from [Table XII] take out the Sun’s Equation by means of his Anomaly (making proportions for the odd minutes of Anomaly) which Equation being added to or subtracted from the Sun’s mean Longitude from Aries, as the titles in the Table direct, gives his true Place, or Longitude from the beginning of Aries, reckoned according to the order of the Signs § [354].

			s	°	ʹ	ʺ	s	°	ʹ
			Sun’s Long.				Sun’s Anom.
The year next less than 1757 in the Table is 1753, at the beginning of which, the Sun’s mean Longitude from the beginning of Aries, and his mean Anomaly, is			9	10	16	52	6	1	38
To which add his mean Mot. and Anom. for four years to make 1757			0	0	1	49	11	29	58
And likewise his mean Mot. and Anom. for	April		2	28	42	30	2	28	42
	days	29	0	28	35	2	0	28	35
	hours	22		0	54	13		0	54
	min.	18			0	44			1
	sec.	49				2			0
Sun’s mean Longitude and Anomaly for the given time is			1	8	31	12	9	29	48
To which add the Equation of the Sun’s mean Place				1	40	14	Sun’s Eq.
And it gives his true Place, viz. ♉ Taurus 10° 11ʹ 26ʺ			1	10	11	26	1°	40ʹ	14ʺ

N. B. In leap-years after February, the Sun’s mean Motion and Anomaly must be taken out for the day next after the given one.

361. To calculate the Sun’s true Place for any time in a given year before the first year of Christ: subtract the mean Motions and Anomalies for the compleat hundreds next above the given year; to the remainder add those for the residue of years, months, &c. and then work in all respects as above taught.

				s	°	ʹ	ʺ	s	°	ʹ
				Sun’s Long.				Sun’s Anom.
From the Sun’s mean Longitude and Anomaly at the beginning of the year Christ 1				9	7	53	10	6	29	54
Subtract his mean Motion and Anomaly for 600 years				0	4	32	0	11	24	2
And the remainder, or radix, is				9	3	21	10	7	5	52
To which add what 585 wants of 600, viz. 15 years				11	29	22	27	11	29	7
And also those of	May			3	28	16	40	3	28	17
	days	28	Bissextile	0	28	35	2	0	28	35
	hours	4			0	9	51		0	10
	min.	3				0	7
	min.	3				0	7
	sec.	42					2	0	2	1
								Sun’s Anom.
								Sun’s Anom.
Sun’s mean Long. May 28th, at 4 hour 3 min. 24 sec. afternoon				1	29	45	19
				1	29	45	19
Equation of the Sun’s mean Place subtract						2	2	2ʹ 22ʺ
								Sun’s Equat.
								Sun’s Equat.
Rem. his true Place for the same time, viz. ♉ Taurus 29° 43ʹ 17ʺ				1	29	43	17	subtract.

N. B. As the Longitudes or Places of all the visible Stars in the Heavens are well known, we have an easy method of finding the Sun’s true Place in the Ecliptic: for the Sun is directly opposite to that Point of the Ecliptic which comes to the Meridian at mid-night.

Fourth Element.

362. Precept. Enter [Table XVII] with the Signs and Degrees of the Sun’s Place; and making proportions, take out his Declination answering thereto. If the Signs are at the head of the Table, the Degrees are at the left hand; but if the Signs are at the foot of the Table, the Degrees are at the right hand. So, the Sun’s Declination answering to his true Place (found by § [359] to be 0^s 12° 9ʹ) is 4 degrees 48 minutes 54 seconds, making allowance for the 9ʹ that his Place exceeds 12°.

Fifth Element.

Precept. This we may state at 5 degrees 38 minutes, as near enough for the purpose; since it is never above 8 minutes of a degree more or less.

Sun’s Long. from Aries.

Sun’s mean Anomaly.

Table I. To the Sun’s mean Place and Anomaly at the mean time of New Moon in March 1764, N. S.

Add the same from Tab. VI. for one Lunation, to carry it to April

Mean Place and Anomaly at the time of New Moon in April

To which place add the Sun’s Equation from Tab. XII.

And it gives the Sun’s true place

Sixth Element.

363. Precept. Having found the Sun’s distance from the Ascending Node by § [357], at the mean time of New Moon, and his Anomaly for that time by § [359], find the Equation of the Node in Table XIII, by the Sun’s Anomaly, and the Equation of the Sun’s mean Place in [Table XII] by his Anomaly: these two Equations applied (as the titles direct) to the Sun’s mean distance from the Ascending Node, give his true distance from it, and also the Moon’s true distance at the time of Change: but when the Moon is Full, this distance must be increased by the addition of 6 Signs, which will then be the Moon’s true distance from the same Node.

The Moon’s true distance from the Ascending Node is called the Argument of the Moon’s Latitude; with the Signs of which, at the head of [Table XIV], and Degrees at the left hand, or with the Signs at the foot of the Table and Degrees at the right hand, take out the Moon’s Latitude: which is North Ascending, North Descending, South Ascending, or South Descending, according to the letters NA, ND, SA or SD, annexed to the Signs of the said Argument.

Plate XII.

The Geometrical Construction of Solar and Lunar Eclipses.

J. Ferguson delin.

J. Mynde Sculp.