Chap. XIV — Of the Precession of the Equinoxes — Astronomy Explained Upon Sir Isaac Newton’s Principles by James Ferguson

Chap. XIV Of the Precession of the Equinoxes.

246. It has been already observed, § 116, that by the Earth’s motion on it’s Axis, there is more matter accumulated all round the equatoreal parts than any where else on the Earth.

The Sun and Moon, by attracting this redundancy of matter, bring the Equator sooner under them in every return towards it than if there was no such accumulation. Therefore, if the Sun sets out, as from any Star, or other fixed point in the Heavens, the moment he is departing from the Equinoctial or either Tropic, he will come to the same again before he compleats his annual course, so as to arrive at the same fixed Star or Point from whence he set out.

When the Sun arrives at the same ^[56]Equinoctial or Solstitial Point, he finishes what we call the Tropical Year, which, by long observation, is found to contain 365 days 5 hours 48 minutes 57 seconds: and when he arrives at the same fixed Star again, as seen from the Earth, he compleats the Sidereal Year; which is found to contain 365 days 6 hours 9 minutes 14¹⁄₂ seconds. The Sidereal Year is therefore 20 minutes 17¹⁄₂ seconds longer than the Solar or Tropical year, and 9 minutes 14¹⁄₂ seconds longer than the Julian or Civil year, which we state at 365 days 6 hours: so that the Civil year is almost a mean betwixt the Sidereal and Tropical.

PLATE VI.

247. As the Sun describes the whole Ecliptic, or 360 degrees, in a Tropical year, he moves 59ʹ 8ʺ of a degree every day; and consequently 50ʺ of a degree in 20 minutes 17¹⁄₂ seconds of time: therefore, he will arrive at the same Equinox or Solstice when he is 50ʺ of a degree short of the same Star or fixed point in the Heavens from which he set out in the year before. So that, with respect to the fixed Stars, the Sun and Equinoctial points fall back (as it were) 30 degrees in 2160 years; which will make the Stars appear to have gone 30 deg. forward, with respect to the Signs of the Ecliptic in that time: for the same Signs always keep in the same points of the Ecliptic, without regard to the constellations.

A Table shewing the Precession of the Equinoctial Points in the Heavens, both in Motion and Time; and the Anticipation of the Equinoxes on Earth.
Julian years.	Precession of the Equinoctial Points in the Heavens.								Anticipation of the Equinoxes on the Earth.
	Motion.				Time.				Anticipation of the Equinoxes on the Earth.
	S.	°	ʹ	ʺ	Days	H.	M.	S.	D.	H.	M.	S.
1	0	0	0	50	0	0	20	17¹⁄₂	0	0	11	3
2	0	0	1	40	0	0	40	35	0	0	22	6
3	0	0	2	30	0	1	0	52¹⁄₂	0	0	33	9
4	0	0	3	20	0	1	21	10	0	0	44	12
5	0	0	4	10	0	1	41	27¹⁄₂	0	0	55	15
6	0	0	5	0	0	2	1	45	0	1	6	18
7	0	0	5	50	0	2	22	2¹⁄₂	0	1	17	21
8	0	0	6	40	0	2	42	20	0	1	28	24
9	0	0	7	30	0	3	2	37¹⁄₂	0	1	39	27
10	0	0	8	20	0	3	22	55	0	1	50	30
20	0	0	16	40	0	6	45	50	0	3	41	0
30	0	0	25	0	0	10	8	45	0	5	31	30
40	0	0	33	20	0	13	31	40	0	7	22	0
50	0	0	41	40	0	16	54	35	0	9	12	30
60	0	0	50	0	0	20	17	30	0	11	3	0
70	0	0	58	20	0	23	40	25	0	12	53	30
80	0	1	6	40	1	3	3	20	0	14	44	0
90	0	1	15	0	1	6	26	15	0	16	34	30
100	0	1	23	20	1	9	49	10	0	18	25	0
200	0	2	46	40	2	19	38	20	1	12	50	0
300	0	4	10	0	4	5	27	30	2	7	15	0
400	0	5	33	20	5	15	16	40	3	1	40	0
500	0	6	56	40	7	1	5	50	3	20	5	0
600	0	8	20	0	8	10	55	0	4	14	30	0
700	0	9	43	20	9	20	44	10	5	8	55	0
800	0	11	6	40	11	6	33	20	6	3	20	0
900	0	12	29	0	12	16	22	30	6	21	45	0
1000	0	13	53	20	14	2	11	40	7	16	10	0
2000	0	27	46	40	28	4	23	20	15	8	20	0
3000	1	11	40	0	42	6	35	0	23	0	30	0
4000	1	25	33	20	56	8	46	40	30	16	40	0
5000	2	9	26	40	70	10	58	20	38	8	50	0
6000	2	23	20	0	84	13	10	0	46	1	0	0
7000	3	7	13	20	98	15	21	40	53	17	10	0
8000	3	21	6	40	112	17	33	20	61	9	20	0
9000	4	5	0	0	126	19	45	0	69	1	30	0
10000	4	18	53	20	140	21	56	40	76	17	40	0
20000	9	7	46	40	281	19	53	20	153	11	20	0
25920	12	0	0	0	365	6	0	0	198	21	36	0

Fig. IV.

To explain this by a Figure, let the Sun be in conjunction with a fixed Star at S, suppose in the 30th degree of ♉, on the 20th day of May 1756. Then, making 2160 revolutions through the Ecliptic VWX, at the end of so many Sidereal years, he will be found again at S: but at the end of so many Julian years, he will be found at M, short of S: and at the end of so many Tropical years, he will be found short of M, in the 30th deg. of Taurus at T, which has receded back from S to T in that time, by the Precession of the Equinoctial points ♈ Aries and ♎ Libra. The Arc ST will be equal to the amount of the Precession of the Equinox in 2160 years, at the rate of 50ʺ of a degree, or 20 min. 17¹⁄₂ sec. of time, annually: this, in so many years, makes 30 days, 10¹⁄₂ hours; which is the difference between 2160 Sidereal and Tropical years: And the Arc MT will be equal to the space moved through by the Sun in 2160 times 11 min. 3 sec. or 16 days, 13 hours 48 minutes, which is the difference between 2160 Julian and Tropical years.

248. From the shifting of the Equinoctial points, and with them all the Signs of the Ecliptic, it follows that those Stars which in the infancy of astronomy were in Aries are now got into Taurus; those of Taurus into Gemini, &c. Hence likewise it is, that the Stars which rose or set at any particular season of the year, in the time of Hesiod, Eudoxus, Virgil, Pliny, &c. by no means answer at this time to their descriptions. The preceding table shews the quantity of this shifting both in the heavens and on the earth, for any number of years to 25,920; which compleats the grand celestial period: within which any number and its quantity is easily found; as in the following example, for 5763 years; which at the Autumnal Equinox, A. D. 1756, is thought to be the age of the world. So that with regard to the fixed Stars, the Equinoctial points in the heavens, have receded 2^s 20° 2ʹ 30ʺ since the creation; which is as much as the Sun moves in 81^d 5^h 0^m 52^s. And since that time, or in 5763 years, the Equinoxes with us have fallen back 44^d 5^h 21^m 9^s; hence, reckoning from the time of the Julian Equinox, A. D. 1756, viz. Sept. 12th, it appears that the Autumnal Equinox at the creation was on the 26th of October.

	Motion.				Time.				Anticipation of the Equinoxes on the Earth.
Julian years.	Precession of the Equinoctial Points in the Heavens.								Anticipation of the Equinoxes on the Earth.
	S.	°	ʹ	ʺ	D.	H.	M.	S.	D.	H.	M.	S.
5000	2	9	26	40	70	10	58	20	38	8	50	0
700	0	9	43	20	9	20	44	10	5	8	55	0
60	0	0	50	0	0	20	17	30	0	11	3	0
3	0	0	2	30	0	1	0	52	0	0	33	9
5763	2	20	2	30	81	5	0	52	44	5	21	9

The anticipation of the Equinoxes and Seasons.

PLATE VI.

249. The anticipation of the Equinoxes, and consequently of the seasons, is by no means owing to the Precession of the Equinoctial and Solsticial points in the Heavens, (which can only affect the apparent motions, places and declinations of the fixed Stars) but to the difference between the Civil and Solar year, which is 11 minutes 3 seconds; the Civil year containing 365 days 6 hours, and the Solar year 365 days 5 hours 48 minutes 57 seconds. The following table shews the length, and consequently the difference of any number of Sidereal, Civil, and Solar years from 1 to 10,000.

The reason for altering the Style.

250. The above 11 minutes 3 seconds, by which the Civil or Julian year exceeds the Solar, amounts to 11 days in 1433 years: and so much our seasons have fallen back with respect to the days of the months, since the time of the Nicene Council in A.D. 325, and therefore in order to bring back all the Fasts and Festivals to the days then settled, it was requisite to suppress 11 nominal days. And that the same seasons might be kept to the same times of the year for the future, to leave out the Bissextile day in February at the end of every century of years not divisible by 4; reckoning them only common years, as the 17th, 18th and 19th centuries, viz. the years 1700, 1800, 1900, &c. because a day intercalated every fourth year was too much, and retaining the Bissextile-day at the end of those Centuries of years which are divisible by 4, as the 16th, 20th and 24th Centuries; viz. the years 1600, 2000, 2400, &c. Otherwise, in length of time the seasons would have been quite reversed with regard to the months of the years; though it would have required near 23,783 years to have brought about such a total change. If the Earth had made exactly 365¹⁄₄ diurnal rotations on its axis, whilst it revolved from any Equinoctial or Solstitial point to the same again, the Civil and Solar years would always have kept pace together; and the style would never have needed any alteration.

The Precession of the Equinoctial Points.

251. Having already mentioned the cause of the Precession of the Equinoctial points in the heavens, § 246, which occasions a flow deviation of the earth’s axis from its parallelism, and thereby a change of the declination of the Stars from the Equator, together with a slow apparent motion of the Stars forward with respect to the Signs of the Ecliptic; we shall now describe the Phenomena by a Diagram.

Fig. V.

Let NZSVL be the Earth, SONA its Axis produced to the starry Heavens, and terminating in A, the present north Pole of the Heavens, which is vertical to N the north Pole of the Earth. Let EOQ be the Equator, T♋Z the Tropic of Cancer, and VT♑ the Tropic of Capricorn: VOZ the Ecliptic, and BO its Axis, both which are immoveable among the Stars. But, as ^[57]the Equinoctial points recede in the Ecliptic, the Earth’s Axis SON is in motion upon the Earth’s center O, in such a manner as to describe the double Cone NOn and SOs, round the Axis of the Ecliptic BO, in the time that the Equinoctial points move quite round the Ecliptic, which is 25,920 years; and in that length of time, the north Pole of the Earth’s Axis produced, describes the Circle ABCDA in the starry Heavens, round the Pole of the Ecliptic, which keeps immoveable in the center of that Circle. The Earth’s Axis being 23¹⁄₂ degrees inclined to the Axis of the Ecliptic, the Circle ABCDA, described by the north Pole of the Earth’s Axis produced to A, is 47 degrees in diameter, or double the inclination of the Earth’s Axis. In consequence of this, the point A, which at present is the North Pole of the Heavens, and near to a Star of the second magnitude in the tail of the constellation called the Little Bear, must be deserted by the Earth’s Axis; which moving backwards a degree every 72 years, will be directed towards the Star or Point B in 6480 years hence: and in double of that time, or 12,960 years, it will be directed towards the Star or Point C; which will then be the North Pole of the Heavens, although it is at present 8¹⁄₂ degrees south of the Zenith of London L. The present position of the Equator EOQ will then be changed into eOq, the Tropic of Cancer T♋Z into Vt♋, and the Tropic of Capricorn VT♑ into t♑Z; as is evident by the Figure. And the Sun, in the same part of the Heavens where he is now over the earthly Tropic of Capricorn, and makes the shortest days and longest nights in the Northern Hemisphere, will then be over the earthly Tropic of Cancer, and make the days longest, and nights shortest. So that it will require 12,960 years yet more, or 25,920 from the present time, to bring the North Pole N quite round, so as to be directed toward that point of the Heavens which is vertical to it at present. And then, and not till then, the same Stars which at present describe the Equator, Tropics, polar Circles, and Poles, by the Earth’s diurnal motion, will describe them over again.

A TABLE shewing the Time contained in any number of Sidereal, Julian, and Solar Years, from 1 to 10000.

Years	Days	H.	M.	S.	Days	H.	Days	H.	M.	S.
Sidereal Years.					Julian Years.		Solar Years.
1	Contain 365	6	9	14¹⁄₂	Contain 365	6	Contain 365	5	48	57
2	730	12	18	29	730	12	370	11	37	54
3	1095	18	27	43¹⁄₂	1095	18	1095	17	26	51
4	1461	0	36	58	1461	0	1460	23	15	48
5	1826	6	46	12¹⁄₂	1826	6	1826	5	4	45
6	2191	12	55	27	2191	12	2191	10	53	42
7	2556	19	5	41¹⁄₂	2556	18	2556	16	42	39
8	2922	1	13	56	2922	0	2921	22	31	36
9	3287	7	23	10¹⁄₂	3287	6	3287	4	20	33
10	3652	13	32	25	3652	12	3652	10	9	30
20	7305	3	4	50	7305	0	7304	20	19	0
30	10957	16	37	15	10957	12	10957	6	28	30
40	14610	6	9	40	14610	0	14609	16	38	0
50	18262	19	42	5	18262	12	18262	2	47	30
60	21915	9	14	30	21915	0	21914	12	57	0
70	25567	22	46	55	25567	12	25566	23	6	30
80	29220	12	19	20	25220	0	29219	9	16	0
90	32873	1	51	45	32872	12	32871	19	25	30
100	36525	15	24	10	36525		36524	5	35
200	73051	6	48	20	73050		73048	11	10
300	109576	22	12	30	109575		109572	16	45
400	146102	13	36	40	146100		146096	22	20
500	182628	5	0	50	182625		182621	3	55
600	219153	20	25		219150		219145	9	30
700	255679	11	49	10	255675		255669	15	5
800	292205	3	13	20	292200		292193	20	10
900	328730	18	37	30	328725		328718	2	15
1000	365256	10	1	40	365250		365242	7	50
2000	730512	20	3	20	730500		730484	15	40
3000	1095769	6	5		1095750		1095726	23	30
4000	1461025	16	6	40	1461000		1460969	7	20
5000	1826282	2	8	20	1826250		1826211	15	10
6000	2191538	12	10		2191500		2191453	14	40
7000	2556794	22	11	40	2556750		2556696	6	50
8000	2922051	8	13	20	2922000		2921938	14	40
9000	3287037	18	15		3287250		3287180	22	30
10000	3652564	4	16	40	3652500		3652423	6	20

A TABLE shewing the Sun’s true Place, and Distance from his Apogee, for the second Year after Leap-year.
Days	January				February				March				April				May				June
	Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.
	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.
1	11♑	7	6	2	12♒	39	7	3	10♓	53	8	0	11♈	40	9	1	10♉	57	10	0	10♊	46	11	1
2	12	8	6	3	13	40	7	4	11	53	8	1	12	39	9	2	11	55	10	1	11	44	11	2
3	13	9	6	4	14	41	7	5	12	53	8	2	13	38	9	3	12	53	10	2	12	41	11	3
4	14	10	6	5	15	42	7	6	13	53	8	3	14	37	9	4	13	51	10	3	13	38	11	4
5	15	11	6	6	16	43	7	7	14	53	8	4	15	36	9	5	14	49	10	4	14	35	11	5
6	16	12	6	7	17	43	7	8	5	53	8	5	16	35	9	6	15	47	10	5	15	33	11	6
7	17	14	6	8	18	44	7	9	16	53	8	6	17	34	9	7	16	45	10	6	16	30	11	7
8	18	15	6	9	19	45	7	10	17	53	8	7	18	33	9	8	17	43	10	7	17	28	11	8
9	19	16	6	10	20	46	7	11	18	53	8	8	19	32	9	9	18	41	10	8	18	25	11	9
10	20	17	6	11	21	46	7	12	19	53	8	9	20	30	9	10	19	39	10	9	19	22	11	10
11	21	18	6	12	22	47	7	13	20	52	8	10	21	29	9	11	20	37	10	10	20	20	11	11
12	22	19	6	13	23	47	7	14	21	52	8	11	22	28	9	12	21	34	10	11	21	17	11	12
13	23	21	6	14	24	48	7	15	22	52	8	12	23	26	9	13	22	32	10	12	22	14	11	13
14	24	22	6	15	25	48	7	16	23	52	8	13	24	25	9	14	23	30	10	13	23	11	11	14
15	25	23	6	16	26	49	7	17	24	51	8	14	25	24	9	15	24	28	10	14	24	8	11	15
16	26	24	6	17	27	49	7	18	25	51	8	15	26	22	9	16	25	26	10	15	25	6	11	16
17	27	25	6	18	28	50	7	19	26	51	8	16	27	21	9	17	26	23	10	16	26	3	11	17
18	28	26	6	19	29	50	7	20	27	50	8	17	28	19	9	18	27	21	10	17	27	0	11	18
19	29	27	6	20	♓	51	7	21	28	50	8	18	29	18	9	19	28	19	10	18	27	58	11	18
20	♒	28	6	21	1	51	7	22	29	49	8	19	♉	16	9	20	29	16	10	19	28	55	11	19
21	1	29	6	22	2	51	7	23	♈	49	8	20	1	15	9	21	♊	15	10	20	29	52	11	20
22	2	30	6	23	3	52	7	24	1	48	8	21	2	13	9	22	1	11	10	21	♋	49	11	21
23	3	31	6	24	4	52	7	25	2	47	8	22	3	11	9	23	2	9	10	22	1	46	11	22
24	4	32	6	25	5	52	7	26	3	47	8	23	4	10	9	24	3	6	10	23	2	44	11	23
25	5	33	6	26	6	52	7	27	4	46	8	24	5	8	9	25	4	4	10	24	3	41	11	24
26	6	34	6	27	7	53	7	28	5	45	8	25	6	6	9	26	5	2	10	25	4	38	11	25
27	7	35	6	28	8	53	7	29	6	45	8	26	7	4	9	27	5	59	10	26	5	35	11	26
28	8	36	6	29	9	53	8	0	7	44	8	27	8	3	9	28	6	56	10	27	6	32	11	27
29	9	37	7	0					8	43	8	28	9	1	9	29	7	54	10	28	7	30	11	28
30	10	38	7	1					9	42	8	29	9	59	9	29	8	51	10	29	8	27	11	29
31	11	39	7	2					10	41	9	0					9	48	11	0

A TABLE shewing the Sun’s true Place, and Distance from his Apogee, for the second Year after Leap-year.
Days	July				August				September				October				November				December
	Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.		Sun’s Place.		Sun’s Anom.
	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.	D.	M.	S.	D.
1	9♋	24	0	0	8♌	59	1	0	8♍	51	2	1	8♎	10	3	1	9♏	0	4	2	9♐	18	5	1
2	10	21	0	1	9	57	1	1	9	49	2	2	9	9	3	2	10	0	4	3	10	19	5	2
3	11	18	0	2	10	54	1	2	10	47	2	3	10	8	3	3	11	0	4	4	11	20	5	3
4	12	15	0	3	11	52	1	3	11	45	2	4	11	8	3	4	12	1	4	5	12	21	5	4
5	13	13	0	4	12	49	1	4	12	43	2	5	12	7	3	5	13	1	4	6	13	22	5	5
6	14	10	0	5	13	47	1	5	13	42	2	6	13	6	3	6	14	1	4	7	14	23	5	6
7	15	7	0	6	14	44	1	6	14	40	2	7	14	6	3	7	15	2	4	8	15	24	5	7
8	16	4	0	7	15	42	1	7	15	39	2	8	15	5	3	8	16	2	4	9	16	25	5	8
9	17	1	0	8	16	39	1	8	16	37	2	9	16	4	3	9	17	2	4	10	17	26	5	9
10	17	59	0	8	17	37	1	9	17	35	2	10	17	4	3	10	18	3	4	11	18	27	5	10
11	18	56	0	9	18	35	1	10	18	34	2	11	18	3	3	11	19	3	4	12	19	28	5	11
12	19	53	0	10	19	32	1	11	19	32	2	12	19	3	3	12	20	4	4	13	20	29	5	12
13	20	50	0	11	20	30	1	12	20	31	2	13	20	2	3	13	21	4	4	14	21	30	5	13
14	21	47	0	12	21	28	1	13	21	29	2	14	21	2	3	14	22	5	4	15	22	31	5	14
15	22	45	0	13	22	25	1	14	22	28	2	15	22	2	3	15	23	5	4	16	23	32	5	15
16	23	42	0	14	23	23	1	15	23	27	2	16	23	1	3	16	24	6	4	17	24	33	5	16
17	24	39	0	15	24	21	1	16	24	25	2	17	24	1	3	17	25	7	4	18	25	34	5	17
18	25	36	0	16	25	19	1	17	25	24	2	18	25	1	3	18	26	7	4	19	26	35	5	18
19	26	34	0	17	26	17	1	18	26	23	2	19	26	0	3	19	27	8	4	20	27	36	5	19
20	27	31	0	18	27	14	1	19	27	21	2	20	27	0	3	20	28	9	4	21	28	38	5	20
21	28	28	0	19	28	12	1	20	28	20	2	21	28	0	3	21	29	9	4	22	29	39	5	21
22	29	26	0	20	29	10	1	21	29	19	2	22	29	0	3	22	♐	10	4	23	♑	40	5	22
23	♌	23	0	21	♍	8	1	22	♎	18	2	23	♏	0	3	23	1	11	4	24	1	41	5	23
24	1	20	0	22	1	6	1	23	1	17	2	24	1	0	3	24	2	12	4	25	2	42	5	24
25	2	18	0	23	2	4	1	24	2	16	2	25	2	0	3	25	3	12	4	26	3	44	5	25
26	3	15	0	24	3	2	1	25	3	15	2	26	3	0	3	26	4	13	4	27	4	45	5	26
27	4	12	0	25	4	0	1	26	4	14	2	27	4	0	3	27	5	14	4	28	5	46	5	27
28	5	10	0	26	4	58	1	27	5	13	2	28	5	0	3	28	6	15	4	29	6	47	5	28
29	6	7	0	27	5	56	1	28	6	12	2	29	6	0	3	29	7	16	4	29	7	48	5	29
30	7	5	0	28	6	54	1	29	7	11	3	0	7	0	4	0	8	17	5	0	8	49	6	0
31	8	2	0	29	7	52	2	0					8	0	4	1					9	51	6	1

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The Bissextile, or Leap-year.
Days	January			Dif.	February			Dif.	March			Dif.	April			Dif.	May			Dif.	June			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Inc.				Dec.				Dec.				Dec.				Inc.
1	12	4	0		12	14	5		12	12	36		12	3	48		11	56	47		11	57	22
				28				7				13				18				7				9
2	12	4	28		12	14	12		12	12	23		12	3	30		11	56	40		11	57	31
				28				7				13				19				7				9
3	12	4	56		12	14	19		12	12	10		12	3	11		11	56	33		11	57	40
				28				6				14				18				6				10
4	12	5	24		12	14	25		12	11	56		12	2	53		11	56	27		11	57	50
				27				5				14				18				6				10
5	12	5	51		12	14	30		12	11	42		12	2	35		11	56	21		11	58	0
				27				4				14				18				5				11
6	12	6	18	27		12	14	4	34		12	14	11	28		18	12	2	17	5		11	56	11	16	11	58	11
				26				3				15				17				4				11
7	12	6	44		12	14	37		12	11	13		12	2	0		11	56	12		11	58	22
				26				3				15				17				4				11
8	12	7	10		12	14	40		12	10	58		12	1	43		11	56	8		11	58	33
				25				2				16				17				4				11
9	12	7	35		12	14	42		12	10	42		12	1	26		11	56	4		11	58	44
				25				1				16				17				3				12
10	12	8	0		12	14	43		12	10	46		12	1	9		11	56	1		11	58	56
				24				Dec.				16				16				2				12
11	12	8	24	24		12	14	Dec.	44		12	16	10	10		16	12	0	53	2		11	55	12	59	11	59	8
				23				1				17				16				1				12
12	12	8	47		12	14	43		12	9	53		12	0	37		11	55	58		11	59	20
				23				1				17				16				1				12
13	12	9	10		12	14	42		12	9	36		12	0	21		11	55	57		11	59	32
				22				2				17				15				Inc.				12
14	12	9	32		12	14	40		12	9	19		12	0	6		11	55	56		11	59	44
				22				3				17				15				1				13
15	12	9	54		12	14	37		12	9	2		11	59	51		11	55	57		11	59	57
				21				4				18				15				1				13
16	12	10	15	21		12	14	4	33		12	18	8	44		15	11	59	36	1		11	55	13	58	12	0	10
				20				4				18				15				1				13
17	12	10	35		12	14	29		12	8	26		11	59	21		11	55	59		12	0	23
				19				5				18				14				2				12
18	12	10	54		12	14	24		12	8	8		11	59	7		11	56	1		12	0	35
				19				5				18				13				2				13
19	12	10	13		12	14	19		12	7	50		11	58	54		11	56	3		12	0	48
				18				6				18				13				3				13
20	12	10	31		12	14	13		12	7	32		11	58	41		11	56	6		12	1	1
				17				7				18				13				3				13
21	12	11	48	17		12	14	7	6		12	18	7	14		13	11	58	28	3		11	56	13	9	12	1	14
				17				8				19				12				4				13
22	12	12	5		12	13	58		12	6	55		11	58	16		11	56	13		12	1	27
				16				8				19				12				5				13
23	12	12	21		12	13	50		12	6	36		11	58	4		11	56	18		12	1	40
				15				9				19				12				5				13
24	12	12	36		12	13	41		12	6	17		11	57	52		11	56	23		12	1	53
				14				9				19				11				6				13
25	12	12	50		12	13	32		12	5	58		11	57	41		11	56	29		12	2	6
				13				10				18				10				6				12
26	12	13	3	13		12	13	10	22		12	18	5	40		10	11	57	31	6		11	56	12	35	12	2	18
				12				11				19				10				7				13
27	12	13	15		12	13	11		12	5	21		11	57	21		11	56	42		12	2	31
				12				11				19				9				7				12
28	12	13	27		12	13	0		12	5	2		11	57	12		11	56	49		12	2	43
				11				12				18				9				7				12
29	12	13	38	11		12	12	12	48		12	18	4	44		9	11	57	3	7		11	56	12	56	12	2	55
				10				12				19				8				8				12
30	12	13	48						12	4	25		11	56	55		11	57	4		12	3	7
				9								19				8				9				11
31	12	13	57	9						12	4	19	6			8				9	11	57	13	11
				8								18								9
	Incr. 9ʹ 57ʺ				Incr. 0ʹ 39ʺ				Decr. 8ʹ 30ʺ				Decr. 6ʹ 53ʺ				Decr. 0ʹ 50ʺ				Incr. 5ʹ 45ʺ
					Decr. 1 56												Incr. 1 17

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The first after Leap-year.
Days	January			Dif.	February			Dif.	March			Dif.	April			Dif.	May			Dif.	June			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Inc.				Dec.				Dec.				Dec.				Inc.
1	12	4	21		12	14	11		12	12	39		12	3	52		11	56	49		11	57	18
				28				7				13				18				8				9
2	12	4	49		12	14	18		12	12	26		12	3	34		11	56	41		11	57	27
				27				6				13				18				7				9
3	12	5	16		12	14	24		12	12	13		12	3	16		11	56	34		11	57	36
				27				5				13				18				6				10
4	12	5	43		12	14	29		12	12	0		12	2	58		11	56	28		11	57	46
				27				5				14				18				6				10
5	12	6	10		12	14	34		12	11	46		12	2	40		11	56	22		11	57	56
				27				4				14				18				5				10
6	12	6	37	27		12	14	4	38		12	14	11	32		18	12	2	22	5		11	56	10	17	11	58	6
				26				3				15				18				4				11
7	12	7	3		12	14	41		12	11	17		12	2	4		11	56	13		11	58	17
				25				2				15				17				4				11
8	12	7	28		12	14	43		12	11	2		12	1	47		11	56	9		11	58	28
				25				1				16				17				4				12
9	12	7	53		12	14	44		12	10	46		12	1	30		11	56	5		11	58	40
				24				0				16				17				3				12
10	12	8	17		12	14	44		12	10	30		12	1	13		11	56	2		11	58	52
				24				Dec.				16				16				2				12
11	12	8	41		12	14	44		12	10	14		12	0	57		11	56	0		11	59	4
				23				1				17				16				2				12
12	12	9	4		12	14	43		12	9	57		12	0	41		11	55	58		11	59	16
				22				2				17				16				1				12
13	12	9	26		12	14	41		12	9	40		12	0	25		11	55	57		11	59	28
				22				3				17				16				1				12
14	12	9	48		12	14	38		12	9	23		12	0	9		11	55	56		11	59	40
				21				3				17				15				Inc.				13
15	12	10	9		12	14	35		12	9	6		11	59	54		11	55	56		11	59	53
				21				4				18				15				1				13
16	12	10	30	21		12	14	4	31		12	18	8	48		15	11	59	39	1		11	55	13	57	12	0	6
				20				5				18				14				1				13
17	12	10	50		12	14	26		12	8	30		11	59	25		11	55	58		12	0	19
				19				6				18				14				2				13
18	12	11	9		12	14	20		12	8	12		11	59	11		11	56	0		12	0	32
				18				6				18				14				2				13
19	12	11	27		12	14	14		12	7	54		11	58	57		11	56	2		12	0	45
				17				7				18				13				3				13
20	12	11	44		12	14	7		12	7	36		11	58	44		11	56	5		12	0	58
				17				7				18				13				3				13
21	12	12	1		12	14	0		12	7	18		11	58	31		11	56	8		12	1	11
				16				8				19				12				4				13
22	12	12	17		12	13	52		12	6	59		11	58	19		11	56	12		12	1	24
				15				9				19				12				4				13
23	12	12	32		12	13	43		12	6	40		11	58	7		11	56	16		12	1	37
				14				9				19				12				5				13
24	12	12	46		12	13	34		12	6	21		11	57	55		11	56	21		12	1	50
				13				10				19				11				6				13
25	12	12	59		12	13	24		12	6	2		11	57	44		11	56	27		12	2	3
				13				10				19				10				6				13
26	12	13	12	13		12	13	10	14		12	19	5	43		10	11	57	34	6		11	56	13	33	12	2	16
				12				11				19				10				6				12
27	12	13	24		12	13	3		12	5	24		11	57	24		11	56	39		12	2	28
				11				12				19				10				7				12
28	12	13	35		12	12	51		12	5	5		11	57	14		11	56	46		12	2	40
				10				12				19				9				7				12
29	12	13	45						12	4	46		11	57	5		11	56	53		12	2	52
								9				18				8				8				12
30	12	13	54						12	4	28		11	56	57		11	57	1		12	3	4
								9				18				8				8				11
31	12	14	3					9		12	4	18	10			8				8	11	57	9	11
				8								18								9
	Incr. 9ʹ 42ʺ				Incr. 0ʹ 33ʺ				Decr. 8ʹ 29ʺ				Decr. 6ʹ 55ʺ				Decr. 0ʹ 53ʺ				Incr. 5ʹ 46ʺ
					Decr. 1 53												Incr. 1 12

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The first after Leap-year.
Days	July			Dif.	August			Dif.	September			Dif.	October			Dif.	November			Dif.	December			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Dec.				Dec.				Dec.				Dec.				Inc.
1	12	3	15		12	5	48		11	59	38		11	49	33		11	43	49		11	49	36
				12				4				19				19				0				24
2	12	3	27		12	5	44		11	59	19		11	49	14		11	43	49		11	50	0
				11				5				19				19				Inc.				24
3	12	3	38		12	5	39		11	59	0		11	48	55		11	43	49		11	50	24
				11				5				19				18				1				25
4	12	3	49		12	5	34		11	58	41		11	48	37		11	43	50		11	50	49
				10				6				20				17				2				25
5	12	3	59		12	5	28		11	58	21		11	48	20		11	43	52		11	51	14
				10				6				20				17				3				26
6	12	4	9	10		12	5	6	22		11	20	58	1		17	11	48	3	3		11	43	26	55	11	51	40
				10				7				20				17				3				26
7	12	4	19		12	5	15		11	57	41		11	47	46		11	43	58		11	52	6
				10				7				20				17				4				27
8	12	4	29		12	5	8		11	57	21		11	47	29		11	44	2		11	52	33
				9				8				20				16				5				27
9	12	4	38		12	5	0		11	57	1		11	47	13		11	44	7		11	53	0
				8				9				20				15				6				27
10	12	4	46		12	4	51		11	56	41		11	46	58		11	44	13		11	53	27
				8				9				20				15				7				28
11	12	4	54	8		12	4	9	42		11	20	56	21		15	11	46	43	7		11	44	28	20	11	53	35
				8				10				21				14				8				28
12	12	5	2		12	4	32		11	56	0		11	46	29		11	44	28		11	54	23
				8				10				21				13				9				29
13	12	5	10		12	4	22		11	55	39		11	46	16		11	44	37		11	54	52
				7				11				21				13				10				29
14	12	5	17		12	4	11		11	55	18		11	46	3		11	44	47		11	55	21
				6				11				21				13				11				29
15	12	5	23		12	4	0		11	54	57		11	45	50		11	44	58		11	55	50
				6				12				21				13				12				30
16	12	5	29	6		12	3	12	48		11	21	54	36		13	11	45	37	12		11	45	30	10	11	56	19
				5				12				21				12				13				30
17	12	5	34		12	3	36		11	54	15		11	45	25		11	45	23		11	56	49
				5				13				21				11				13				30
18	12	5	39		12	3	23		11	53	54		11	45	14		11	45	36		11	57	19
				4				13				21				11				14				30
19	12	5	43		12	3	10		11	53	33		11	45	3		33	45	50		11	57	49
				4				14				21				10				14				30
20	12	5	47		12	2	56		11	53	12		11	44	53		11	46	4		11	58	19
				4				14				21				10				15				30
21	12	5	51	4		12	2	14	42		11	21	52	51		10	11	44	43	15		11	46	30	19	11	58	49
				3				15				20				9				16				30
22	12	5	54		12	2	17		11	52	31		11	44	34		11	46	35		11	59	19
				2				15				20				8				17				30
23	12	5	56		12	2	12		11	52	11		11	44	26		11	46	52		11	59	49
				1				16				21				7				18				30
24	12	5	57		12	1	56		11	51	50		11	44	19		11	47	10		12	0	19
				1				16				21				6				19				30
25	12	5	58		12	1	40		11	51	29		11	44	13		11	47	29		12	0	49
				Dec.				16				20				6				19				30
26	12	5	59	Dec.		12	1	16	24		11	20	51	9		6	11	44	7	19		11	47	30	48	12	1	19
				1				17				20				5				20				30
27	12	5	58		12	1	7		11	50	40		11	44	2		11	48	8		12	1	49
				1				17				19				4				21				29
28	12	5	57		12	1	50		11	50	30		11	43	58		11	48	29		12	2	18
				2				18				19				3				22				29
29	12	5	55		12	1	32		11	50	11		11	43	55		11	48	51		12	2	47
				2				18				19				3				22				29
30	12	5	53		12	0	14		11	49	52		11	43	52		11	49	13		12	3	16
				2				18				19				2				23				29
31	12	5	51	2		11	59	18	56			19				2	11	43	50	23				29		12	3	45
				3				18								1								29
	Incr. 2ʹ 43ʺ				Decr. 5ʹ 52ʺ				Decr. 9ʹ 46ʺ				Decr. 5ʹ 43ʺ				Decr. 0ʹ 0ʺ				Incr. 14ʹ 9ʺ
	Decr. 0 8																Incr. 5 24

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The second after Leap-year.
Days	January			Dif.	February			Dif.	March			Dif.	April			Dif.	May			Dif.	June			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Dec.				Inc.				Dec.				Dec.				Inc.
1	12	4	14		12	14	9		12	12	42		12	3	56		11	56	50		11	57	16
				28				7				13				18				8				9
2	12	4	42		12	14	16		12	12	20		12	3	38		11	56	42		11	57	25
				28				6				13				18				7				10
3	12	5	10		12	14	22		12	12	16		12	3	20		11	56	35		11	57	35
				27				5				13				18				6				10
4	12	5	37		12	14	27		12	12	3		12	3	2		11	56	29		11	57	45
				27				5				14				18				6				10
5	12	6	4		12	14	32		12	11	49		12	2	44		11	56	23		11	57	55
				27				4				14				18				5				10
6	12	6	30	27		12	14	4	36		12	14	11	35		18	12	2	26	5		11	56	10	18	11	58	5
				26				3				15				17				5				11
7	12	6	56		12	14	39		12	11	20		12	2	9		11	56	23		11	58	16
				26				2				15				17				4				11
8	12	7	22		12	14	41		12	11	5		12	1	52		11	56	9		11	58	27
				25				2				15				17				3				11
9	12	7	47		12	14	43		12	10	50		12	1	35		11	56	6		11	58	38
				24				1				16				17				3				12
10	12	8	11		12	14	44		12	10	34		12	1	18		11	56	3		11	58	50
				24				Dec.				16				17				2				12
11	12	8	35	24		12	14	Dec.	44		12	16	10	18		17	12	1	1	2		11	56	12	1	11	59	2
				23				1				17				16				2				12
12	12	8	58		12	14	43		12	10	1		12	0	45		11	55	59		11	59	14
				22				2				17				16				2				12
13	12	9	20		12	14	41		12	9	44		12	0	29		11	55	57		11	59	26
				22				3				17				16				1				12
14	12	9	42		12	14	38		12	9	27		12	0	13		11	55	56		11	59	38
				21				3				17				15				Inc.				12
15	12	10	3		12	14	35		12	9	10		11	59	58		11	55	56		11	59	50
				21				4				18				15				1				13
16	12	10	24	21		12	14	4	31		12	18	8	52		15	11	59	43	1		11	55	13	57	12	0	3
				20				4				18				14				1				13
17	12	10	44		12	14	27		12	8	34		11	59	29		11	55	58		12	0	16
				19				5				18				14				2				13
18	12	11	3		12	14	22		12	8	16		11	59	15		11	56	0		12	0	29
				18				6				18				14				2				13
19	12	11	21		12	14	16		12	7	58		11	59	1		11	56	2		12	0	42
				18				7				18				14				3				13
20	12	11	39		12	14	9		12	7	40		11	58	47		11	56	5		12	0	55
				17				7				18				13				3				13
21	12	11	56	17		12	14	7	2		12	18	7	22		13	11	58	34	3		11	56	13	8	12	1	8
				16				8				18				12				3				13
22	12	12	12		12	13	54		12	7	4		11	58	22		11	56	11		12	1	21
				15				9				19				12				4				13
23	12	12	27		12	13	45		12	6	45		11	58	10		11	56	15		12	1	34
				15				9				19				12				5				13
24	12	12	42		12	13	36		12	6	26		11	57	58		11	56	20		12	1	47
				14				10				19				11				6				12
25	12	12	56		12	13	26		12	6	7		11	57	47		11	56	26		12	1	59
				13				10				19				11				6				13
26	12	13	9	13		12	13	10	16		12	19	5	48		11	11	57	36	6		11	56	13	32	12	2	12
				12				11				19				10				6				13
27	12	13	21		12	13	5		12	5	29		11	57	26		11	56	38		12	2	25
				11				11				19				10				7				12
28	12	13	32		12	12	54		12	5	10		11	57	16		11	56	45		12	2	37
				10				12				19				9				7				12
29	12	13	42						12	4	51		11	57	7		11	56	52		12	2	49
				10								18				9				8				12
30	12	13	52						12	4	33		11	56	58		11	57	0		12	3	1
				9								18				8				8				11
31	12	14	1	9						12	4	18	15			8				8	11	57	8	11
				8								18								8
	Incr. 9ʹ 47ʺ				Incr. 0ʹ 35ʺ				Decr. 8ʹ 27ʺ				Decr. 6ʹ 58ʺ				Decr. 0ʹ 54ʺ				Incr. 5ʹ 45ʺ
					Decr. 1 50												Incr. 1 12

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The second after Leap-year.
Days	July			Dif.	August			Dif.	September			Dif.	October			Dif.	November			Dif.	December			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Dec.				Dec.				Dec.				Dec.				Inc.
1	12	3	12		12	5	48		11	59	43		11	49	37		11	43	49		11	49	30
				12				4				19				19				1				23
2	12	3	24		12	5	44		11	59	24		11	49	18		11	43	48		11	49	53
				11				4				19				18				Inc.				24
3	12	3	35		12	5	40		11	59	5		11	49	0		11	43	49		11	50	17
				11				5				19				18				1				25
4	12	3	46		12	5	35		11	58	46		11	48	42		11	43	50		11	50	42
				11				5				20				18				2				25
5	12	3	57		12	5	30		11	58	26		11	48	24		11	43	52		11	51	7
				10				6				20				17				3				26
6	12	4	7	10		12	5	6	24		11	20	58	6		17	11	48	7	3		11	43	26	55	11	51	33
				10				7				20				17				3				26
7	12	4	17		12	5	17		11	57	46		11	47	50		11	43	58		11	51	59
				9				7				20				17				4				26
8	12	4	26		12	5	10		11	57	26		11	47	33		11	44	2		11	52	25
				9				8				21				16				5				27
9	12	4	35		12	5	2		11	57	5		11	47	17		11	44	7		11	52	52
				9				9				20				16				6				28
10	12	4	44		12	4	53		11	56	45		11	47	1		11	44	13		11	53	20
				8				9				21				15				7				28
11	12	4	52	8		12	4	9	44		11	21	56	24		15	11	46	46	7		11	44	28	20	11	53	48
				8				9				21				14				8				28
12	12	5	0		12	4	35		11	56	3		11	46	32		11	44	28		11	54	16
				8				10				21				14				9				28
13	12	5	8		12	4	25		11	55	42		11	46	18		11	44	37		11	54	44
				7				11				20				13				9				29
14	12	5	15		12	4	13		11	55	22		11	46	5		11	44	46		11	54	13
				6				11				20				13				10				29
15	12	5	21		12	4	3		11	55	2		11	45	52		11	44	56		11	55	42
				6				12				21				13				11				29
16	12	5	27	6		12	3	12	51		\|11	21	54	41		13	11	45	39	11		11	45	29	7	11	56	11
				6				12				21				12				12				30
17	12	5	33		12	3	39		11	54	20		11	45	27		11	45	19		11	56	41
				5				12				21				11				13				30
18	12	5	38		12	3	27		11	53	59		11	45	16		11	45	32		11	57	11
				4				13				20				10				14				30
19	12	5	42		12	3	14		11	53	39		11	45	6		11	45	46		11	57	41
				4				14				21				10				15				30
20	12	5	46		12	3	0		11	53	18		11	44	56		11	46	1		11	58	11
				3				14				21				10				15				30
21	12	5	49	3		12	2	14	46		11	21	52	57		10	11	44	46	15		11	46	30	16	11	58	41
				3				15				20				9				16				30
22	12	5	52		12	2	31		11	52	37		11	44	37		11	46	32		11	59	11
				2				15				21				8				17				30
23	12	5	54		12	2	16		11	52	16		11	44	29		11	46	49		11	59	41
				2				15				21				7				18				30
24	12	5	56		12	2	1		11	51	55		11	44	22		11	47	7		12	0	11
				2				16				21				7				18				30
25	12	5	58		12	1	45		11	51	34		11	44	15		11	47	25		12	0	41
				1				16				20				6				19				30
26	12	5	59	1		12	1	16	29		11	20	51	14		6	11	44	9	19		11	47	30	44	12	1	11
				Dec.				17				20				5				20				30
27	12	5	58		12	1	12		11	50	54		11	44	4		11	48	4		12	1	41
				1				17				20				5				21				30
28	12	5	57		12	0	55		11	50	34		11	43	59		11	48	25		12	2	11
				1				18				19				4				21				29
29	12	5	56		12	0	37		11	50	15		11	43	55		11	48	46		12	2	40
				2				18				19				3				22				29
30	12	5	54		12	0	19		11	49	56		11	43	52		11	49	8		12	3	9
				3				18				19				2				22				29
31	12	5	51	3		12	0	18	1			19				2	11	43	50	22				29		12	3	38
				3				18								1								29
	Incr. 2ʹ 46ʺ				Decr. 5ʹ 47ʺ				Decr. 9ʹ 47ʺ				Decr. 5ʹ 47ʺ				Decr. 0ʹ 1ʺ				Incr. 14ʹ 8ʺ
	Decr. 0 8																Incr. 5 19

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The third after Leap-year.
Days	January			Dif.	February			Dif.	March			Dif.	April			Dif.	May			Dif.	June			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
				Inc.				Inc.				Dec.				Dec.				Dec.				Inc.
1	12	4	7		12	14	6		12	12	44		12	4	1		11	56	52		11	57	15
				28				7				12				18				8				9
2	12	4	35		12	14	13		12	12	32		12	3	43		11	56	44		11	57	24
				28				7				13				18				7				9
3	12	5	3		12	14	20		12	12	19		12	3	25		11	56	37		11	57	33
				27				6				13				18				7				9
4	12	5	30		12	14	26		12	12	6		12	3	7		11	56	30		11	57	42
				27				5				14				18				6				10
5	12	5	57		12	14	31		12	11	52		12	2	49		11	56	24		11	57	52
				27				4				14				18				5				10
6	12	6	24	27		12	14	4	35		12	14	11	38		18	12	2	31	5		11	56	10	19	11	58	2
				26				3				14				18				5				11
7	12	6	50		12	14	38		12	11	24		12	2	13		11	56	14		11	58	13
				25				2				15				18				4				11
8	12	7	15		12	14	41		12	11	9		12	1	55		11	56	10		11	58	24
				25				1				16				17				4				11
9	12	7	40		12	14	43		12	10	53		12	1	38		11	56	6		11	58	35
				25				1				16				17				3				11
10	12	8	5		12	14	44		12	10	37		12	1	21		11	56	3		11	58	46
				24				Dec.				16				16				2				12
11	12	8	29	24		12	14	Dec.	44		12	16	10	21		16	12	1	5	2		11	56	12	1	11	58	58
				23				1				16				16				2				12
12	12	8	52		12	14	43		12	10	5		12	0	49		11	55	59		11	59	10
				23				2				17				16				2				12
13	12	9	15		12	14	41		12	10	48		12	0	33		11	55	57		11	59	22
				22				2				17				16				1				12
14	12	9	37		12	14	39		12	9	31		12	0	17		11	55	56		11	59	34
				21				3				17				15				Inc.				13
15	12	9	58		12	14	36		12	9	14		12	0	2		11	55	56		11	59	47
				21				4				17				15				1				13
16	12	10	19	21		12	14	4	32		12	17	8	57		15	11	59	47	1		11	55	13	57	12	0	0
				20				4				18				15				1				13
17	12	10	39		12	14	28		12	8	39		11	59	32		11	55	58		12	0	13
				19				5				18				14				1				13
18	12	10	58		12	14	23		12	8	21		11	59	18		11	55	59		12	0	26
				18				6				18				14				2				13
19	12	11	16		12	14	17		12	8	3		11	59	4		11	56	1		12	0	39
				18				7				18				14				2				13
20	12	11	34		12	14	10		12	7	45		11	58	50		11	56	3		12	0	52
				17				7				18				13				3				13
21	12	11	51	17		12	14	7	3		12	18	7	27		13	11	58	37	3		11	56	13	6	12	1	5
				16				8				19				13				4				12
22	12	12	7		12	13	55		12	7	8		11	58	24		11	56	10		12	1	17
				16				8				19				12				4				13
23	12	12	23		12	13	47		12	6	49		11	58	12		11	56	14		12	1	30
				15				9				19				12				5				13
24	12	12	38		12	13	38		12	6	30		11	58	0		11	56	19		12	1	43
				14				9				19				11				5				13
25	12	12	52		12	13	29		12	6	11		11	57	49		11	56	24		12	1	56
				13				10				18				11				6				13
26	12	13	5	13		12	13	10	19		12	18	5	53		11	11	57	38	6		11	56	13	30	12	2	9
				12				11				19				10				6				13
27	12	13	17		12	13	8		12	5	34		11	57	28		11	56	36		12	2	22
				11				12				19				10				7				12
28	12	13	28		12	12	56		12	5	15		11	57	18		11	56	43		12	2	34
				11				12				18				9				7				12
29	12	13	39						12	4	57		11	57	9		11	56	50		12	2	46
				10								19				9				8				12
30	12	13	49						12	4	38		11	57	0		11	56	58		12	2	58
				9								19				8				8				12
31	12	13	58	9						12	4	19	19			8				8	11	57	6	12
				8								18								9
	Incr. 9ʹ 51ʺ				Incr. 0ʹ 38ʺ				Decr. 8ʹ 25ʺ				Decr. 7ʹ 1ʺ				Decr. 0ʹ 56ʺ				Incr. 5ʹ 43ʺ
					Decr. 1 48												Incr. 1 10

A TABLE of the Equation of natural Days, shewing what Time it ought to be by the Clock when the Sun is on the Meridian.
The third after Leap-year.
Days	July			Dif.	August			Dif.	September			Dif.	October			Dif.	November			Dif.	December			Dif.
Days	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.	H.	M.	S.	S.
	&nbps;			Inc.	&nbps;			Dec.	&nbps;			Dec.	&nbps;			Dec.	&nbps;			Dec.	&nbps;			Inc.
1	12	3	10		12	5	49		11	59	47		11	49	42		11	43	49		11	49	25
				11				4				19				18				1				23
2	12	3	21		12	5	45		11	59	28		11	49	24		11	43	48		11	49	48
				11				4				19				18				Inc.				24
3	12	3	32		12	5	41		11	59	9		11	49	6		11	43	48		11	50	12
				11				5				19				18				1				24
4	12	3	43		12	5	36		11	58	50		11	48	48		11	43	49		11	50	36
				11				5				19				18				2				25
5	12	3	54		12	5	31		11	58	31		11	48	30		11	43	51		11	51	1
				10				6				20				18				2				25
6	12	4	4	10		12	5	6	25		11	20	58	11		18	11	48	12	2		11	43	25	53	11	51	26
				10				7				20				17				3				26
7	12	4	14		12	5	18		11	57	51		11	47	55		11	43	56		11	52	52
				10				7				20				16				4				27
8	12	4	24		12	5	11		11	57	31		11	47	39		11	44	0		11	52	19
				9				7				20				16				5				27
9	12	4	33		12	5	4		11	57	11		11	47	23		11	44	5		11	52	46
				9				8				20				16				6				27
10	12	4	42		12	4	56		11	56	51		11	47	7		11	44	11		11	53	13
				8				8				20				15				7				28
11	12	4	50	8		12	4	8	48		11	20	56	31		15	11	46	52	7		11	44	28	18	11	53	41
				8				9				21				15				8				28
12	12	4	58		12	4	37		11	56	10		11	46	37		11	44	26		11	54	9
				8				10				21				14				8				28
13	12	5	6		12	4	27		11	55	49		11	46	23		11	44	34		11	54	37
				7				10				21				14				9				29
14	12	5	13		12	4	17		11	55	28		11	46	9		11	44	43		11	55	6
				6				11				21				13				10				30
15	12	5	19		12	4	6		11	55	7		11	45	56		11	44	53		11	55	36
				6				12				20				12				11				29
16	12	5	25	6		12	3	12	54		11	20	54	47		12	11	45	44	11		11	45	29	4	11	56	6
				6				12				21				12				12				30
17	12	5	31		12	3	42		11	54	26		11	45	32		11	45	16		11	56	36
				5				13				21				12				13				30
18	12	5	36		12	3	29		11	54	5		11	45	20		11	45	29		11	57	6
				5				13				21				11				14				29
19	12	5	41		12	3	16		11	53	44		11	45	9		11	45	43		11	57	35
				4				13				21				10				14				30
20	12	5	45		12	3	3		11	53	23		11	44	59		11	45	57		11	58	5
				4				14				20				9				15				30
21	12	5	49	4		12	2	14	49		11	20	53	3		9	11	44	50	15		11	46	30	12	11	58	34
				3				15				21				9				16				30
22	12	5	52		13	2	34		11	52	42		11	44	41		11	46	28		11	59	4
				3				15				21				9				17				30
23	12	5	55		12	2	19		11	52	21		11	44	32		11	46	45		11	59	34
				2				15				20				8				17				30
24	12	5	57		12	2	4		11	52	1		11	44	24		11	47	2		12	0	4
				1				16				21				7				18				30
25	12	5	58		12	1	48		11	51	40		11	44	17		11	47	20		12	0	34
				1				16				20				6				19				30
26	12	5	59	1		12	1	16	32		11	20	51	20		6	11	44	11	19		11	47	30	39	12	1	4
				Dec.				16				20				5				20				30
27	12	5	58		12	1	16		11	51	0		11	44	6		11	47	59		12	1	34
				1				17				20				5				20				30
28	12	5	57		12	0	59		11	50	40		11	44	1		11	48	19		12	2	4
				1				17				20				4				21				29
29	12	5	56		12	0	42		11	50	20		11	43	57		11	48	40		12	2	33
				2				18				19				3				22				29
30	12	5	54		12	0	24		11	50	1		11	43	54		11	49	2		12	3	2
				2				18				19				3				23				29
31	12	5	52	2		12	0	18	6			19				3	11	43	51	23				29		12	3	31
				3				19								2								29
	Incr. 2ʹ 48ʺ				Decr. 5ʹ 43ʺ				Decr. 9ʹ 46ʺ				Decr. 5ʹ 51ʺ				Decr. 0ʹ 1ʺ				Incr. 14ʹ 6ʺ
	Decr. 0 7																Incr. 5 14

CHAP. XV.

The Moon’s surface mountainous: Her Phases described: Her path, and the paths of Jupiter’s Moons delineated: The proportions of the Diameters of their Orbits, and those of Saturn’s Moons, to each other; and to the Diameter of the Sun.

PL. VII.

The Moon’s surface mountainous.

252. By looking at the Moon with an ordinary telescope we perceive that her surface is diversified with long tracts of prodigious high mountains and deep cavities. Some of her mountains, by comparing their height with her diameter (which is 2180 miles) are found to be three times higher than the highest hills on our Earth. This ruggedness of the Moon’s surface is of great use to us, by reflecting the Sun’s light to all sides: for if the Moon were smooth and polished like a looking-glass, or covered with water, she could never distribute the Sun’s light all round; only in some positions she would shew us his image, no bigger than a point, but with such a lustre as would be hurtful to our eyes.

Why no hills appear on her edge.

253. The Moon’s surface being so uneven, many have wondered why her edge appears not jagged, as well as the curve bounding the light and dark places. But if we consider, that what we call the edge of the Moon’s Disc is not a single line set round with mountains, in which case it would appear irregularly indented, but a large Zone having many mountains lying behind one another from the observer’s eye, we shall find that the mountains in some rows will be opposite to the vales in others; and so fill up the inequalities as to make her appear quite round: just as when one looks at an orange, although it’s roughness be very discernible on the side next the eye, especially if the Sun or a Candle shines obliquely on that side, yet the line terminating the visible part still appears smooth and even.

Plate VII.

J. Ferguson delin.

J. Mynde Sculp.

The Moon has no twilight.

Fig. I.

254. As the Sun can only enlighten that half of the Earth which is at any moment turned towards him, and being withdrawn from the opposite half leaves it in darkness; so he likewise doth to the Moon: only with this difference, that the Earth being surrounded by an Atmosphere, and the Moon having none, we have twilight after the Sun sets; but the Lunar Inhabitants have an immediate transition from the brightest Sun-shine to the blackest darkness § 177. For, let tkrsw be the Earth, and A, B, C, D, E, F, G, H the Moon in eight different parts of her Orbit. As the Earth turns round its Axis, from west to east, when any place comes to t the twilight begins there, and when it revolves from thence to r the Sun S rises; when the place comes to s the Sun sets, and when it comes to w the twilight ends. But as the Moon turns round her Axis, which is only once a month, the moment that any point of her surface comes to r (see the Moon at G) the Sun rises there without any previous warning by twilight; and when the same point comes to s the Sun sets, and that point goes into darkness as black as at midnight.

The Moon’s Phases.

255. The Moon being an opaque spherical body, (for her hills take off no more from her roundness than the inequalities on the surface of an orange takes off from its roundness) we can only see that part of the enlightened half of her which is towards the Earth. And therefore, when the Moon is at A, in conjunction with the Sun S, her dark half is towards the Earth, and she disappears as at a, there being no light on that half to render it visible. When she comes to her first Octant at B, or has gone an eighth part of her orbit from her Conjunction, a quarter of her enlightened side is towards the Earth, and she appears horned as at b. When she has gone a quarter of her orbit from between the Earth and Sun to C, she shews us one half of her enlightened side as at c, and we say, she is a quarter old. At D she is in her second Octant, and by shewing us more of her enlightened side she appears gibbous as at d. At E her whole enlightened side is towards the Earth, and therefore she appears round as at e, when we say, it is Full Moon. In her third Octant at F, part of her dark side being towards the Earth, she again appears gibbous, and is on the decrease, as at f. At G we see just one half of her enlightened side, and she appears half decreased, or in her third Quarter, as at g. At H we only see a quarter of her enlightened side, being in her fourth Octant, where she appears horned as at h. And at A, having compleated her course from the Sun to the Sun again, she disappears; and we say, it is New Moon. Thus in going from A to E the Moon seems continually to increase; and in going from E to A, to decrease in the same proportion; having like Phases at equal distances from A or E, but as seen from the Sun S, she is always Full.

The Moon’s Disc not always quite round when full.

256. The Moon appears not perfectly round when she is Full in the highest or lowest part of her Orbit, because we have not a direct view of her enlightened side at that time. When Full in the highest part of her orbit, a small deficiency appears on her lower edge; and the contrary when Full in the lowest part of her Orbit.

The Phases of the Earth and Moon contrary.

257. ’Tis plain by the Figure, that when the Moon changes to the Earth, the Earth appears Full to the Moon; and vice versâ. For when the Moon is at A, New to the Earth, the whole enlightened side of the Earth is towards the Moon: and when the Moon is at E, Full to the Earth, it’s dark side is towards her. Hence a New Moon answers to a Full Earth, and a Full Moon to a New Earth. The Quarters are also reversed to each other.

An agreeable Phenomenon.

258. Between the third Quarter and Change, the Moon is frequently visible in the forenoon, even when the Sun shines; and then she affords us an opportunity of seeing a very agreeable appearance, wherever we find a globular stone above the level of the eye, as suppose on the top of a gate. For, if the Sun shines on the stone, and we place ourselves so as the upper part of the stone may just seem to touch the point of the Moon’s lowermost horn, we shall then see the enlightened part of the stone exactly of the same shape with the Moon; horned as she is, and inclining the same way to the Horizon. The reason is plain; for the Sun enlightens the stone the same way as he does the Moon: and both being Globes, when we put ourselves into the above situation, the Moon and stone have the same position to our eyes; and therefore we must see as much of the illuminated part of the one as of the other.

The nonagesimal Degree, what.

259. The position of the Moon’s Cusps, or a right line touching the points of her horns, is very differently inclined to the Horizon at different hours of the same days of her age. Sometimes she stands, as it were, upright on her lower horn, and then such a line is perpendicular to the Horizon: when this, happens, she is in what the Astronomers call the Nonagesimal Degree; which is the highest point of the Ecliptic above the Horizon at that time, and is 90 degrees from both sides of the Horizon where it is then cut by the Ecliptic. But this never happens when the Moon is on the Meridian, except when she is at the very beginning of Cancer or Capricorn.

How the inclination of the Ecliptic may be found by the
position of the Moon horns.

PL. VII.

260. The inclination of that part of the Ecliptic to the Horizon in which the Moon is at any time when horned, may be known by the position of her horns; for a right line touching their points is perpendicular to the Ecliptic. And as the Angle that the Moon’s orbit makes with the Ecliptic can never raise her above, nor depress her below the Ecliptic, more than two minutes of a degree, as seen from the Sun; it can have no sensible effect upon the position of her horns. Therefore, if a Quadrant be held up, so as one of it’s edges may seem to touch the Moon’s horns, the graduated side being kept towards the eye, and as far from the eye as it can be conveniently held, the arc between the Plumb-line and that edge of the Quadrant which seems to touch the Moon’s horns will shew the inclination of that part of the Ecliptic to the Horizon. And the arc between the other edge of the Quadrant and Plumb-line will shew the inclination of the Moon’s horns to the Horizon at that time also.

Fig. I.

Why the Moon appears as big as the Sun.

261. The Moon generally appears as large as the Sun; for the Angle vkA, under which the Moon is seen from the Earth, is the same with the Angle LkM, under which the Sun is seen from it. And therefore the Moon may hide the Sun’s whole Disc from us, as she sometimes does in solar Eclipses. The reason why she does not eclipse the Sun at every Change shall be explained afterwards. If the Moon were farther from the Earth as at a, she could never hide the whole of the Sun from us; for then she would appear under the Angle NkO, eclipsing only that part of the Sun which lies between N and O: were she still further from the Earth, as at X, she would appear under the small Angle TkW, like a spot on the Sun, hiding only the part TW from our sight.

A proof of the Moon’s turning round her Axis.

262. The Moon turns round her Axis in the time that she goes round her orbit; which is evident from hence, that a spectator at rest, without the periphery of the Moon’s orbit, would see all her sides turned regularly towards him in that time. She turns round her Axis from any Star to the same Star again in 27 days 8 hours; from the Sun to the Sun again in 29¹⁄₂ days: the former is the length of her sidereal day, and the latter the length of her solar day. A body moving round the Sun would have a solar day in every revolution, without turning on it’s Axis; the same as if it had kept all the while at rest, and the Sun moved round it: but without turning round it’s Axis it could never have one sidereal day, because it would always keep the same side towards any given Star.

Her periodical and synodical Revolution.

263. If the Earth had no annual motion, the Moon would go round it so as to compleat a Lunation, a sidereal, and a solar day, all in the same time. But, because the Earth goes forward in it’s orbit while the Moon goes round the Earth in her orbit, the Moon must go as much more than round her orbit from Change to Change in compleating a solar day as the Earth has gone forward in it’s orbit during that time, i. e. almost a twelfth part of a Circle.

Familiarly represented.

A Table shewing the times that the hour and minute hands of a
watch are in conjunction.

A machine for shewing the motions of the Sun and Moon.

PL. VII.

264. The Moon’s periodical and synodical revolution may be familiarly represented by the motions of the hour and minute hands of a watch round it’s dial-plate, which is divided into 12 equal parts or hours, as the Ecliptic is divided into 12 Signs, and the year into 12 months. Let us suppose these 12 hours to be 12 months, the hour hand the Sun, and the minute hand the Moon; then will the former go round once in a year, and the latter once in a month; but the Moon, or minute hand must go more than round from any point of the Circle where it was last conjoined with the Sun, or hour hand, to overtake it again: for the hour hand being in motion, can never be overtaken by the minute hand at that point from which they started at their last conjunction. The first column of the annexed Table shews the number of conjunctions which the hour and minute hand make whilst the hour hand goes once round the dial-plate; and the other columns shew the times when the two hands meet at every conjunction. Thus, suppose the two hands to be in conjunction at XII, as they always are; then, at the first following conjunction it is 5 minutes 27 seconds 16 thirds 21 fourths 49¹⁄₁₁ fifths past I where they meet; at the second conjunction it is 10 minutes 54 seconds 32 thirds 43 fourths 38¹⁄₂ fifths past II; and so on. This, though an easy illustration of the motions of the Sun and Moon, is not precise as to the times of their conjunctions; because, while the Sun goes round the Ecliptic, the Moon makes 12¹⁄₃ conjunctions with him; but the minute hand of a watch or clock makes only 11 conjunctions with the hour hand in one period round the dial-plate. But if, instead of the common wheel-work at the back of the dial-plate, the Axis of the minute hand had a pinion of 6 leaves turning a wheel of 40, and this last turning the hour hand, in every revolution it makes round the dial-plate the minute hand would make 12¹⁄₃ conjunctions with it; and so would be a pretty device for shewing the motions of the Sun and Moon; especially, as the slowest moving hand might have a little Sun fixed on it’s point, and the quickest a little Moon. Besides, the plate, instead of hours and quarters, might have a Circle of months, with the 12 Signs and their Degrees; and if a plate of 29¹⁄₂ equal parts for the days of the Moon’s age were fixed to the Axis of the Sun-hand, and below it, so as the Sun always kept at the ¹⁄₂ day of that plate, the Moon-hand would shew the Moon’s age upon that plate for every day pointed out by the Sun-hand in the Circle of months; and both Sun and Moon would shew their places in the Ecliptic: for the Sun would go round the Ecliptic in 365 Days and the Moon in 27¹⁄₃ days, which is her periodical revolution; but from the Sun to the Sun again, or from Change to Change, in 29¹⁄₂ days, which is her synodical revolution.

Conj.	H.	M.	S.	ʺʹ	ʺʺ	v p^ts.
1	I	5	27	16	21	49¹⁄₁₁
2	II	10	54	32	43	38²⁄₁₁
3	III	16	21	49	5	27³⁄₁₁
4	IIII	21	49	5	27	16⁴⁄₁₁
5	V	27	16	21	49	5⁵⁄₁₁
6	VI	32	43	38	10	54⁶⁄₁₁
7	VII	38	10	54	32	43⁷⁄₁₁
8	VIII	43	38	10	54	32⁸⁄₁₁
9	IX	49	5	27	16	21⁹⁄₁₁
10	X	54	32	43	38	10¹⁰⁄₁₁
11	XII	0	0	0	0	0

The Moon’s motion thro’ open space described.

265. If the Earth had no annual motion, the Moon’s motion round the Earth, and her track in absolute space, would be always the same^[58]. But as the Earth and Moon move round the Sun, the Moon’s real path in the Heavens is very different from her path round the Earth: the latter being in a progressive Circle, and the former in a curve of different degrees of concavity, which would always be the same in the same parts of the Heavens, if the Moon performed a compleat number of Lunations in a year.

An idea of the Earth’s path and the Moon’s.

266. Let a nail in the end of the axle of a chariot-wheel represent the Earth, and a pin in the nave the Moon; if the body of the chariot be propped up so as to keep that wheel from touching the ground, and the wheel be then turned round by hand, the pin will describe a Circle both round the nail and in the space it moves through. But if the props be taken away, the horses put to, and the chariot driven over a piece of ground which is circularly convex; the nail in the axle will describe a circular curve, and the pin in the nave will still describe a circle round the progressive nail in the axle, but not in the space through which it moves. In this case, the curve described by the nail will resemble in miniature as much of the Earth’s annual path round the Sun, as it describes whilst the Moon goes as often round the Earth as the pin does round the nail: and the curve described by the nail will have some resemblance of the Moon’s path during so many Lunations.

Fig. II.

PL. VII.

Let us now suppose that the Radius of the circular curve described by the nail in the axle is to the Radius of the Circle which the pin in the nave describes round the axle as 337¹⁄₂ to 1; which is the proportion of the Radius or Semidiameter of the Earth’s Orbit to that of the Moon’s; or of the circular curve A 1 2 3 4 5 6 7 B &c. to the little Circle a; and then, whilst the progressive nail describes the said curve from A to E, the pin will go once round the nail with regard to the center of it’s path, and in doing so, will describe the curve abcde. The former will be a true representation of the Earth’s path for one Lunation, and the latter of the Moon’s for that time. Here we may set aside the inequalities of the Moon’s Moon, and also the Earth’s moving round it’s common center of gravity and the Moon’s: all which, if they were truly copied in this experiment, would not sensibly alter the figure of the paths described by the nail and pin, even though they should rub against a plain upright surface all the way, and leave their tracks visible. And if the chariot should be driven forward on such a convex piece of ground, so as to turn the wheel several times round, the track of the pin in the nave would still be concave toward the center of the circular curve described by the pin in the Axle; as the Moon’s path is always concave to the Sun in the center of the Earth’s annual Orbit.

Proportion of the Moon’s Orbit to the Earth’s.

In this Diagram, the thickest curve line ABCD, with the numeral figures set to it, represents as much of the Earth’s annual Orbit as it describes in 32 days from west to east; the little Circles at a, b, c, d, e shew the Moon’s Orbit in due proportion to the Earth’s; and the smallest curve abcdef represents the line of the Moon’s path in the Heavens for 32 days, accounted from any particular New Moon at a. The machine, Fig. 5th is for delineating the Moon’s path, and will be described, with the rest of my Astronomical machinery, in the last Chapter. The Sun is supposed to be in the center of the curve A 1 2 3 4 5 6 7 B &c. and the small dotted Circles upon it represent the Moon’s Orbit, of which the Radius is in the same proportion to the Earth’s path in this scheme, that the Radius of the Moon’s Orbit in the Heavens bears to the Radius of the Earth’s annual path round the Sun; that is, as 240,000 to 81,000,000, or as 1 to 337¹⁄₂.

Fig. II.

When the Earth is at A the New Moon is at a; and in the seven days that the Earth describes the curve 1 2 3 4 5 6 7, the Moon in accompanying the Earth describes the curve ab; and is in her first Quarter at b when the Earth is at B. As the Earth describes the curve B 8 9 10 11 12 13 14 the Moon describes the curve bc; and is opposite to the Sun at c, when the Earth is at C. Whilst the Earth describes the curve C 15 16 17 18 19 20 21 22 the Moon describes the curve cd; and is in her third Quarter at d when the Earth is at D. Once more, whilst the Earth describes the curve D 23 24 25 26 27 28 29 the Moon describes the curve de; and is again in conjunction at e with the Sun when the Earth is at E, between the 29th and 30th day of the Moon’s age, accounted by the numeral Figures from the New Moon at A. In describing the curve abcde, the Moon goes round the progressive Earth as really as if she had kept in the dotted Circle A, and the Earth continued immoveable in the center of that Circle.

The Moon’s motion always concave towards the Sun.

And thus we see, that although the Moon goes round the Earth in a Circle, with respect to the Earth’s center, her real path in the Heavens is not very different in appearance from the Earth’s path. To shew that the Moon’s path is concave to the Sun, even at the time of Change, it is carried on a little farther into a second Lunation, as to f.

How her motion is alternately retarded and accelerated.

267. The Moon’s absolute motion from her Change to her first Quarter, or from a to b, is so much slower than the Earth’s, that she falls 240 thousand miles (equal to the Semidiameter of her Orbit) behind the Earth at her first Quarter in b, when the Earth is in B; that is, she falls back a space equal to her distance from the Earth. From that time her motion is gradually accelerated to her Opposition or Full at c, and then she is come up as far as the Earth, having regained what she lost in her first Quarter from a to b. From the Full to the last Quarter at d her motion continues accelerated, so as to be just as far before the Earth at D, as she was behind it at her first Quarter in b. But, from d to e her motion is retarded so, that she loses as much with respect to the Earth as is equal to her distance from it, or to the Semidiameter of her Orbit; and by that means she comes to e, and is then in conjunction with the Sun as seen from the Earth at E. Hence we find, that the Moon’s absolute motion is slower than the Earth’s from her third Quarter to her first; and swifter than the Earth’s from her first Quarter to her third: her path being less curved than the Earth’s in the former case, and more in the latter. Yet it is still bent the same way towards the Sun; for if we imagine the concavity of the Earth’s Orbit to be measured by the length of a perpendicular line Cg, let down from the Earth’s place upon the straight line bgd at the Full of the Moon, and connecting the places of the Earth at the end of the Moon’s first and third Quarters, that length will be about 640 thousand miles; and the Moon when New only approaching nearer to the Sun by 240 thousand miles than the Earth is, the length of the perpendicular let down from her place at that time upon the same straight line, and which shews the concavity of that part of her path, will be about 400 thousand miles.

A difficulty removed.

PL. VII.

268. The Moon’s path being concave to the Sun throughout, demonstrates that her gravity towards the Sun, at her conjunction, exceeds her gravity towards the Earth. And if we consider that the quantity of matter in the Sun is almost 230 thousand times as great as the quantity of matter in the Earth, and that the attraction of each body diminishes as the square of the distance from it increases, we shall soon find, that the point of equal attraction where these two powers would be equally strong, is about 70 thousand miles nearer the Earth than the Moon is at her Change. It may now appear surprising that the Moon does not abandon the Earth when she is between it and the Sun, because she is considerably more attracted by the Sun than by the Earth at that time. But this difficulty vanishes when we consider, that the Moon is so near the Earth in proportion to the Earth’s distance from the Sun, that she is but very little more attracted by the Sun at that time than the Earth is; and whilst the Earth’s attraction is greater upon the Moon than the difference of the Sun’s attraction upon the Earth and her (and that it is always much greater is demonstrable) there is no danger of the Moon’s leaving the Earth; for if she should fall towards the Sun, the Earth would follow her almost with equal speed. The absolute attraction of the Earth upon a drop of falling rain is much greater than the absolute attraction of the particles of that drop upon each other, or of it’s center upon all parts of it’s circumference; but then the side of the drop next the Earth is attracted with so very little more force than it’s center, or even it’s opposite side; that the attraction of the center of the drop upon it’s side next the Earth is much greater than the difference of force by which the Earth attracts it’s nearer surface and center: on which account the drop preserves it’s round figure, and might be projected about the Earth by a strong circulating wind so as to be kept from falling to the Earth. It is much the same with the Earth and Moon in respect to the Sun; for if we should suppose the Moon’s Orbit to be filled with a fluid Globe, of which all the parts would be attracted towards the Earth in it’s center, but the whole of it much more attracted by the Sun; one part of it could not fall to the Sun without the other, and a sufficient projectile force would carry the whole fluid Globe round the Sun. A ship, at the distance of the Moon, sailing round the Earth on the surface of the fluid Globe, could no more be taken away by the Sun when it is on the side next him, than the Earth could be taken away from it when it is on the opposite side; which could never happen unless the Earth’s projectile motion were stopt; and if it were stopt, the Ship with the whole fluid Globe, Earth and all together, would as naturally fall to the Sun as a drop of rain in calm air falls to the Earth. Hence we may see, that the Earth is in no more danger of being left by the Moon at the Change, than the Moon is of being left by the Earth at the Full: the diameter of the Moon’s Orbit being so small in comparison of the Sun’s distance, that the Moon is but little more or less attracted than the Earth at any time. And as the Moon’s projectile force keeps her from falling to the Earth, so the Earth’s projectile force keeps it from falling to the Sun.

Fig. III.

269. All the curves which Jupiter’s Satellites describe, are different from the path described by our Moon, although these Satellites go round Jupiter, as the Moon goes round the Earth. Let ABCDE &c. be as much of Jupiter’s Orbit as he describes in 18 days from A to T; and the curves a, b, c, d will be the paths of his four Moons going round him in his progressive motion.

The absolute Path of Jupiter and his Satellites delineated.

Fig. III.

Now let us suppose all these Moons to set out from a conjunction with the Sun, as seen from Jupiter. When Jupiter is at A his first or nearest Moon will be at a, his second at b, his third at c, and his fourth at d. At the end of 24 terrestrial hours after this conjunction, Jupiter has moved to B, his first Moon or Satellite has described the curve a1, his second the curve b1, his third c1, and his fourth d1. The next day when Jupiter is at C, his first Satellite has described the curve a2 from its conjunction, his second the curve b2, his third the curve c2, and his fourth the curve d2, and so on. The numeral Figures under the capital letters shew Jupiter’s place in his path every day for 18 days, accounted from A to T; and the like Figures set to the paths of his Satellites, shew where they are at the like times. The first Satellite, almost under C, is stationary at + as seen from the Sun; and retrograde from + to 2: at 2 it appears stationary again, and thence it moves forward until it has past 3, being twice stationary, and once retrograde between 3 and 4. The path of this Satellite intersects itself every 42¹⁄₂ hours of our time, making such loops as in the Diagram at 2. 3. 5. 7. 9. 10. 12. 14. 16. 18, a little after every Conjunction. The second Satellite b, moving slower, barely crosses it’s path every 3 days 13 hours; as at 4. 7. 11. 14. 18, making only five loops and as many conjunctions in the time that the first makes ten. The third Satellite c moving still slower, and having described the curve c 1. 2. 3. 4. 5. 6. 7, comes to an Angle at 7 in conjunction with the Sun at the end of 7 days 4 hours; and so goes on to describe such another curve 7. 8. 9. 10. 11. 12. 13. 14, and is at 14 in it’s next conjunction. The fourth Satellite d is always progressive, making neither loops nor angles in the Heavens; but comes to it’s next conjunction at e between the numeral figures 16 and 17, or in 16 days 18 hours. In order to have a tolerably good figure of the paths of these Satellites, I took the following method.

Fig. IV.

PL. VII.

How to delineate the paths of Jupiter’s Moons.

And Saturn’s.

Having drawn their Orbits on a Card, in proportion to their relative distances from Jupiter, I measured the radius of the Orbit of the fourth Satellite, which was an inch and a tenth part; then multiplied this by 424 for the radius of Jupiter’s Orbit, because Jupiter is 424 times as far from the Sun’s center as his fourth Satellite is from his center; and the product thence arising was 466⁴⁄₁₀ inches. Then taking a small cord of this length, and fixing one end of it to the floor of a long room by a nail, with a black lead pencil at the other end I drew the curve ABCD &c. and set off a degree and an half thereon, from A to T; because Jupiter moves only so much, whilst his outermost Satellite goes once round him, and somewhat more; so that this small portion of so large a circle differs but very little from a straight line. This done, I divided the space AT into 18 equal parts, as AB, BC, &c. for the daily progress of Jupiter; and each part into 24 for his hourly progress. The Orbit of each Satellite was also divided into as many equal parts as the Satellite is hours in finishing it’s synodical period round Jupiter. Then drawing a right line through the center of the Card, as a diameter to all the 4 Orbits upon it, I put the card upon the line of Jupiter’s motion, and transferred it to every horary division thereon, keeping always the said diameter-line on the line of Jupiter’s path; and running a pin through each horary division in the Orbit of each Satellite as the card was gradually transferred along the Line ABCD etc. of Jupiter’s motion, I marked points for every hour through the Card for the Curves described by the Satellites as the primary planet in the center of the Card was carried forward on the line: and so finished the Figure, by drawing the lines of each Satellite’s motion, through those (almost innumerable) points: by which means, this is perhaps as true a Figure of the paths of these Satellites as can be desired. And in the same manner might those for Saturn’s Satellites be delineated.

The grand Period of Jupiter’s Moons.

270. It appears by the scheme, that the three first Satellites come almost into the same line or position every seventh day; the first being only a little behind with the second, and the second behind with the third. But the period of the fourth Satellite is so incommensurate to the periods of the other three, that it cannot be guessed at by the diagram when it would fall again into a line of conjunction with them, between Jupiter and the Sun. And no wonder; for supposing them all to have been once in conjunction, it will require 3,087,043,493,260 years to bring them in a conjunction again: See § 73.

Fig. IV. The proportions of the Orbits of the Planets and Satellites.

271. In Fig. 4th we have the proportions of the Orbits of Saturn’s five Satellites, and of Jupiter’s four, to one another, to our Moon’s Orbit, and to the Disc of the Sun. S is the Sun; M m the Moon’s Orbit (the Earth supposed to be at E;) J Jupiter; 1. 2. 3. 4 the Orbits of his four Moons or Satellites; Sat Saturn; and 1. 2. 3. 4. 5 the Orbits of his five Moons. Hence it appears, that the Sun would much more than fill the whole Orbit of the Moon; for the Sun’s diameter is 763,000 miles, and the diameter of the Moon’s Orbit only 480,000. In proportion to all these Orbits of the Satellites, the Radius of Saturn’s annual Orbit would be 21¹⁄₄ yards, of Jupiter’s orbit 11²⁄₃, and of the Earth’s 2¹⁄₄, taking them in round numbers.

272. The annexed table shews at once what proportion the Orbits, Revolutions, and Velocities, of all the Satellites bear to those of their primary Planets, and what sort of curves the several Satellites describe. For, those Satellites whose velocities round their primaries are greater than the velocities of their primaries in open space, make loops at their conjunctions § 269; appearing retrograde as seen from the Sun whilst they describe the inferior parts of their Orbits, and direct whilst they describe the superior. This is the case with Jupiter’s first and second Satellites, and with Saturn’s first. But those Satellites whose velocities are less than the velocities of their primary planets move direct in their whole circumvolutions; which is the case of the third and fourth Satellites of Jupiter, and of the second, third, fourth, and fifth Satellites of Saturn, as well as of our Satellite the Moon: But the Moon is the only Satellite whose motion is always concave to the Sun. There is a table of this sort in De la Caile’s Astronomy, but it is very different from the above, which I have computed from our English accounts of the periods and distances of these Planets and Satellites.

The Satellites		Proportion of the Radius of the Planet’s Orbit to the Radius of the Orbit of each Satellite.		Proportion of the Time of the Planet’s Revolution to the Revolution of each Satellite.		Proportion of the Velocity of each Satellite to the Velocity of its primary Planet.
of Saturn	1	As 5322	to 1	As 5738	to 1	As 5738	to 5322
	2	4155	1	3912	1	3912	4155
	3	2954	1	2347	1	2347	2954
	4	1295	1	674	1	674	1295
	5	432	1	134	1	134	432
of Jupiter	1	As 1851	to 1	As 2445	to 1	As 2445	to 1851
	2	1165	1	1219	1	1219	1165
	3	731	1	604	1	604	731
	4	424	1	258	1	258	424
The Moon		As 337¹⁄₂	to 1	As 12¹⁄₃	to 1	As 12¹⁄₃	to 337¹⁄₂

CHAP. XVI.

The Phenomena of the Harvest-Moon explained by a common Globe: The years in which the Harvest-Moons are least and most beneficial from 1751, to 1861. The long duration of Moon-light at the Poles in winter.

No Harvest-Moon at the Equator.

273. It is generally believed that the Moon rises about 48 minutes later every day than on the preceding; but this is true only with regard to places on the Equator. In places of considerable Latitude there is a remarkable difference, especially in the harvest time; with which Farmers were better acquainted than Astronomers till of late; and gratefully ascribed the early rising of the Full Moon at that time of the year to the goodness of God, not doubting that he had ordered it so on purpose to give them an immediate supply of moon-light after sun-set for their greater conveniency in reaping the fruits of the earth.

But remarkable according to the distance of places from it.

In this instance of the harvest-moon, as in many others discoverable by Astronomy, the wisdom and beneficence of the Deity is conspicuous, who really ordered the course of the Moon so, as to bestow more or less light on all parts of the earth as their several circumstances and seasons render it more or less serviceable. About the Equator, where there is no variety of seasons, and the weather changes seldom, and at stated times, Moon-light is not necessary for gathering in the produce of the ground; and there the moon rises about 48 minutes later every day or night than on the former. At considerable distances from the Equator, where the weather and seasons are more uncertain, the autumnal Full Moons rise very soon after sun-set for several evenings together. At the polar circles, where the mild season is of very short duration, the autumnal Full Moon rises at Sun-set from the first to the third quarter. And at the Poles, where the Sun is for half a year absent, the winter Full moons shine constantly without setting from the first to the third quarter.

The reason of this.

It is soon said that all these Phenomena are owing to the different Angles made by the Horizon and different parts of the Moon’s orbit; and that the Moon can be full but once or twice in a year in those parts of her orbit which rise with the least angles. But to explain this subject intelligibly we must dwell much longer upon it.

PLATE III.

274. The ^[59]plane of the Equinoctial is perpendicular to the Earth’s Axis: and therefore, as the Earth turns round its Axis, all parts of the Equinoctial make equal Angles with the Horizon both at rising and setting; so that equal portions of it always rise or set in equal times. Consequently, if the Moon’s motion were equable, and in the Equinoctial, at the rate of 12 degrees from the Sun every day, as it is in her orbit, she would rise and set 48 minutes later every day than on the preceding: for 12 degrees of the Equinoctial rise or set in 48 minutes of time in all Latitudes.

Fig. III.

275. But the Moon’s motion is so nearly in the Ecliptic that we may consider her at present as moving in it. Now the different parts of the Ecliptic, on account of its obliquity to the Earth’s Axis, make very different Angles with the Horizon as they rise or set. Those parts or Signs which rise with the smallest Angles set with the greatest, and vice versâ. In equal times, whenever this Angle is least, a greater portion of the Ecliptic rises than when the Angle is larger; as may be seen by elevating the pole of a Globe to any considerable Latitude, and then turning it round its Axis in the Horizon. Consequently, when the Moon is in those Signs which rise or set with the smallest Angles, she rises or sets with the least difference of time; and with the greatest difference in those Signs which rise or set with the greatest Angles.

Fig. III.

The different Angles made by the Ecliptic and Horizon.

But, because all who read this Treatise may not be provided with Globes, though in this case it is requisite to know how to use them, we shall substitute the Figure of a Globe; in which FUP is the Axis, ♋TR the Tropic of Cancer, LT♑ the Tropic of Capricorn, ♋EU♑ the Ecliptic touching both the Tropics which are 47 degrees from each other, and AB the Horizon. The Equator, being in the middle between the Tropics, is cut by the Ecliptic in two opposite points, which are the beginnings of ♈ Aries and ♎ Libra. K is the Hour circle with its Index, F the North pole of the Globe elevated to the Latitude of London^[60], namely 51¹⁄₂ degrees above the Horizon; and P the South Pole depressed as much below it. Because of the oblique position of the Sphere in this Latitude, the Ecliptic has the high elevation N♋ above the Horizon, making the Angle NU♋ of 62 degrees with it when ♋ Cancer is on the Meridian, at which time ♎ Libra rises in the East. But let the Globe be turned half round its Axis, till ♑ Capricorn comes to the Meridian and ♈ Aries rises in the East, and then the Ecliptic will have the low elevation NL above the Horizon making only an Angle NUL of 15 degrees, with it; which is 47 degrees less than the former Angle, equal to the distance between the Tropics.

Least and greatest, when.

276. The smallest Angle made by the Ecliptic and Horizon is when Aries rises, at which time Libra sets: the greatest when Libra rises, at which time Aries sets. From the rising of Aries to the rising of Libra (which is twelve ^[61]Sidereal hours) the angle increases; and from the rising of Libra to the rising of Aries it decreases in the same proportion. By this article and the preceding, it appears that the Ecliptic rises fastest about Aries and slowest about Libra.

Days	Signs	Degrees	Rising Diff.		Setting Diff.
Days	Signs	Degrees	H.	M.	H.	M.
1	♋	13	1	5	0	50
2		26	1	10	0	43
3	♌	10	1	14	0	37
4		23	1	17	0	32
5	♍	6	1	16	0	28
6		19	1	15	0	24
7	♎	2	1	15	0	20
8		15	1	15	0	18
9		28	1	15	0	17
10	♏	12	1	15	0	22
11		25	1	14	0	30
12	♐	8	1	13	0	39
13		21	1	10	0	47
14	♑	4	1	4	0	56
15		17	0	46	1	5
16	♒	1	0	40	1	8
17		14	0	35	1	12
18		27	0	30	1	15
19	♓	10	0	25	1	16
20		23	0	20	1	17
21	♈	7	0	17	1	16
22		20	0	17	1	15
23	♉	3	0	20	1	15
24		16	0	24	1	15
25		29	0	30	1	14
26	♊	13	0	40	1	13
27		26	0	50	1	7
28	♋	9	1	0	1	58

Quantity of this Angle at London.

277. On the Parallel of London, as much of the Ecliptic rises about Pisces and Aries in two hours as the Moon goes through in six days: and therefore whilst the Moon is in these Signs, she differs but two hours in rising for six days together; that is, 20 minutes later every day or night than on the preceding. But in fourteen days afterwards, the Moon comes to Virgo and Libra; which are the opposite Signs to Pisces and Aries; and then she differs almost four times as much in rising; namely, one hour and about fifteen minutes later every day or night than the former, whilst she is in these Signs; for by § 275 their rising Angle is at least four times as great as that of Pisces and Aries. The annexed Table shews the daily mean difference of the Moon’s rising and setting on the Parallel of London, for 28 days; in which time the Moon finishes her period round the Ecliptic, and gets 9 degrees into the same Sign from the beginning of which she set out. So it appears by the Table, that while the Moon is in ♍ and ♎ she rises an hour and a quarter later every day than the former; and differs only 24, 20, 18 or 17 minutes in setting. But, when she comes to ♓ and ♈, she is only 20 or 17 minutes later of rising; and an hour and a quarter later in setting.

278. All these things will be made plain by putting small patches on the Ecliptic of a Globe, as far from one another as the Moon moves from any Point of the celestial Ecliptic in 24 hours, which at a mean rate is ^[62] 13¹⁄₆ degrees; and then in turning the globe round, observe the rising and setting of the patches in the Horizon, as the Index points out the different times in the hour circle. A few of these patches are represented by dots at 0 1 2 3 &c. on the Ecliptic, which has the position LUI when Aries rises in the East; and by the dots 0 1 2 3, &c. when Libra rises in the East, at which time the Ecliptic has the position EU♑: making an angle of 62 degrees with the Horizon in the latter case, and an angle of no more than 15 degrees with it in the former; supposing the Globe rectified to the Latitude of London.

279. Having rectified the Globe, turn it until the patch at 0, about the beginning of ♓ Pisces on the half LUI of the Ecliptic, comes to the Eastern side of the Horizon; and then keeping the ball steady, set the hour Index to XII, because that hour may perhaps be more easily remembred than any other. Then, turn the Globe round westward, and in that time, suppose the patch 0 to have moved thence to 1, 13¹⁄₆ degrees, whilst the Earth turns once round its Axis, and you will see that 1 rises only about 20 minutes later than 0 did on the day before. Turn the Globe round again, and in that time suppose the same patch to have moved from 1 to 2; and it will rise only 20 minutes later by the hour-index than it did at 1 on the day or turn before. At the end of the next turn, suppose the patch to have gone from 2 to 3 at U, and it will rise 20 minutes later than it did at 2. And so on for six turns, in which time there will scarce be two hours difference: Nor would there have been so much if the 6 degrees of the Sun’s motion in that time had been allowed for. At the first Turn the patch rises south of the East, at the middle Turn due East, and at the last Turn north of the East. But these patches will be 9 hours of setting on the western side of the Horizon, which shews that the Moon will be so much later of setting in that week in which she moves through these two Signs. The cause of this difference is evident; for Pisces and Aries make only an Angle of 15 degrees with the Horizon when they rise; but they make an Angle of 62 degrees with it when they set § 275. As the Signs Taurus, Gemini, Cancer, Leo, Virgo, and Libra rise successively, the Angle increases gradually which they make with the Horizon; and decreases in the same proportion as they set. And for that reason, the Moon differs gradually more in the time of her rising every day whilst she is in these Signs, and less in her setting: After which, through the other six Signs, viz. Scorpio, Sagittary, Capricorn, Aquarius, Pisces, and Aries, the rising difference becomes less every day, until it be at the least of all, namely, in Pisces and Aries.

280. The Moon goes round the Ecliptic in 27 days 8 hours; but not from Change to Change in less than 29 days 12 hours: so that she is in Pisces and Aries at least once in every Lunation, and in some Lunations twice.

Why the Moon is always Full in different Signs.

Her periodical and synodical Revolution exemplified.

281. If the Earth had no annual motion, the Sun would never appear to shift his place in the Ecliptic. And then every New Moon would fall in the same Sign and degree of the Ecliptic, and every Full Moon in the opposite: for the Moon would go precisely round the Ecliptic from Change to Change. So that if the Moon was once Full in Pisces, or Aries, she would always be Full when she came round to the same Sign and Degree again. And as the Full Moon rises at Sun-set (because when any point of the Ecliptic sets the opposite point rises) she would constantly rise within two hours of Sun-set during the week in which she were Full. But in the time that the Moon goes round the Ecliptic from any conjunction or opposition, the Earth goes almost a Sign forward; and therefore the Sun will seem to go as far forward in that time, namely 27¹⁄₂ degrees: so that the Moon must go 27¹⁄₂ degrees more than round; and as much farther as the Sun advances in that interval, which is 2¹⁄₁₅ degrees, before she can be in conjunction with, or opposite to the Sun again. Hence it is evident, that there can be but one conjunction or opposition of the Sun and Moon in a year in any particular part of the Ecliptic. This may be familiarly exemplified by the hour and minute hands of a watch, which are never in conjunction or opposition in that part of the dial-plate where they were so last before. And indeed if we compare the twelve hours on the dial-plate to the twelve Signs of the Ecliptic, the hour-hand to the Sun and the minute-hand to the Moon, we shall have a tolerably near resemblance in miniature to the motions of our great celestial Luminaries. The only difference is, that whilst the Sun goes once round the Ecliptic the Moon makes 12¹⁄₃ conjunctions with him: but whilst the hour-hand goes round the dial-plate the minute-hand makes only 11 conjunctions with it; because the minute hand moves slower in respect of the hour-hand than the Moon does with regard to the Sun.

The Harvest and Hunter’s Moon.

282. As the Moon can never be full but when she is opposite to the Sun, and the Sun is never in Virgo and Libra but in our autumnal months, ’tis plain that the Moon is never full in the opposite Signs, Pisces and Aries, but in these two months. And therefore we can have only two Full Moons in the year, which rise so near the time of Sun-set for a week together as above-mentioned. The former of these is called the Harvest Moon, and the latter the Hunter’s Moon.

Why the Moon’s regular rising is never perceived but in Harvest.

283. Here it will probably be asked, why we never observe this remarkable rising of the Moon but in harvest, since she is in Pisces and Aries at least twelve times in the year besides; and must then rise with as little difference of time as in harvest? The answer is plain: for in winter these Signs rise at noon; and being then only a Quarter of a Circle distant from the Sun, the Moon in them is in her first Quarter: but when the Sun is above the Horizon the Moon’s rising is neither regarded nor perceived. In spring these Signs rise with the Sun because he is then in them; and as the Moon changeth in them at that time of the year, she is quite invisible. In summer they rise about mid-night, and the Sun being then three Signs, or a Quarter of a Circle before them, the Moon is in them about her third Quarter; when rising so late, and giving but very little light, her rising passes unobserved. And in autumn, these Signs being opposite to the Sun, rise when he sets, with the Moon in opposition, or at the Full, which makes her rising very conspicuous.

284. At the Equator, the North and South Poles lie in the Horizon; and therefore the Ecliptic makes the same Angle southward with the Horizon when Aries rises as it does northward when Libra rises. Consequently, as the Moon at all the fore-mentioned patches rises and sets nearly at equal Angles with the Horizon all the year round; and about 48 minutes later every day or night than on the preceding, there can be no particular Harvest Moon at the Equator.

285. The farther that any place is from the Equator, if it be not beyond the Polar Circle, the Angle gradually diminishes which the Ecliptic and Horizon make when Pisces and Aries rise; and therefore when the Moon is in these Signs she rises with a nearly proportionable difference later every day than on the former; and is for that reason the more remarkable about the Full, until we come to the Polar Circles, or 66 degrees from the Equator; in which Latitude the Ecliptic and Horizon become coincident, every day for a moment, at the same sidereal hour (or 3 minutes 56 seconds sooner every day than the former) and the very next moment one half of the Ecliptic containing Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini rises, and the opposite half sets. Therefore, whilst the Moon is going from the beginning of Capricorn to the beginning of Cancer, which is almost 14 days, she rises at the same sidereal hour; and in autumn just at Sun-set, because all that half of the Ecliptic in which the Sun is at that time sets at the same sidereal hour, and the opposite half rises: that is, 3 minutes 56 seconds, of mean solar time, sooner every day than on the day before. So whilst the Moon is going from Capricorn to Cancer she rises earlier every day than on the preceding; contrary to what she does at all places between the polar Circles. But during the above fourteen days, the Moon is 24 sidereal hours later in setting; for the six Signs which rise all at once on the eastern side of the Horizon are 24 hours in setting on the western side of it: as any one may see by making chalk-marks at the beginning of Capricorn and of Cancer, and then, having elevated the Pole 66¹⁄₂ degrees, turn the Globe slowly round it’s Axis, and observe the rising and setting of the Ecliptic. As the beginning of Aries is equally distant from the beginning of Cancer and of Capricorn, it is in the middle of that half of the Ecliptic which rises all at once. And when the Sun is at the beginning of Libra, he is in the middle of the other half. Therefore, when the Sun is in Libra and the Moon in Capricorn, the Moon is a Quarter of a Circle before the Sun; opposite to him, and consequently full in Aries, and a Quarter of a Circle behind him when in Cancer. But when Libra rises Aries sets, and all that half of the Ecliptic of which Aries is the middle. And therefore, at that time of the year the Moon rises at Sun-set from her first to her third Quarter.

The Harvest Moons regular on both sides of the Equator.

286. In northern Latitudes, the autumnal Full Moons are in Pisces and Aries; and the vernal Full Moons in Virgo and Libra: in southern Latitudes just the reverse because the seasons are contrary. But Virgo and Libra rise at as small Angles with the Horizon in southern Latitudes as Pisces and Aries do in the northern; and therefore the Harvest Moons are just as regular on one side of the Equator as on the other.

287. As these Signs which rise with the least Angles set with the greatest, the vernal Full Moons differ as much in their times of rising every night as the autumnal Full Moons differ in their times of setting; and set with as little difference as the autumnal Full Moons rise: the one being in all cases the reverse of the other.

The Moon’s Nodes.

288. Hitherto, for the sake of plainness, we have supposed the Moon to move in the Ecliptic, from which the Sun never deviates. But the orbit in which the Moon really moves is different from the Ecliptic: one half being elevated 5¹⁄₃ degrees above it, and the other half as much depressed below it. The Moon’s orbit therefore intersects the Ecliptic in two points diametrically opposite to each other: and these intersections are called the Moon’s Nodes. So the Moon can never be in the Ecliptic but when she is in either of her Nodes, which is at least twice in every course from Change to Change, and sometimes thrice. For, as the Moon goes almost a whole Sign more than round her Orbit from Change to Change; if she passes by either Node about the time of Change, she will pass by the other in about fourteen days after, and come round to the former Node two days again before the next Change. That Node from which the Moon begins to ascend northward, or above the Ecliptic, in northern Latitudes, is called the Ascending Node; and the other the Descending Node, because the Moon, when she passes by it, descends below the Ecliptic southward.

289. The Moon’s oblique motion with regard to the Ecliptic causes some difference in the times of her rising and setting from what is already mentioned. For whilst she is northward of the Ecliptic, she rises sooner and sets later than if she moved in the Ecliptic: and when she is southward of the Ecliptic she rises later and sets sooner. This difference is variable even in the same Signs, because the Nodes shift backward about 19²⁄₃ degrees in the Ecliptic every year; and so go round it contrary to the order of Signs in 18 years 225 days.

290. When the Ascending Node is in Aries, the southern half of the Moon’s Orbit makes an Angle of 5¹⁄₃ degrees less with the Horizon than the Ecliptic does, when Aries rises in northern Latitudes: for which reason the Moon rises with less difference of time whilst she is in Pisces and Aries than there would be if she kept in the Ecliptic. But in 9 years and 112 days afterward, the Descending Node comes to Aries; and then the Moon’s Orbit makes an Angle 5¹⁄₃ degrees greater with the Horizon when Aries rises, than the Ecliptic does at that time; which causes the Moon to rise with greater difference of time in Pisces and Aries than if she moved in the Ecliptic.

291. To be a little more particular, when the Ascending Node is in Aries, the Angle is only 9²⁄₃ degrees on the parallel of London when Aries rises. But when the Descending Node comes to Aries, the Angle is 20¹⁄₃ degrees; this occasions as great a difference of the Moon’s rising in the same Signs every 9 years, on the parallel of London, as there would be on two parallels 10²⁄₃ degrees from one another, if the Moon’s course were in the Ecliptic. The following Table shews how much the obliquity of the Moon’s Orbit affects her rising and setting on the parallel of London from the 12th to the 18th day of her age; supposing her to be Full at the autumnal Equinox; and then, either in the Ascending Node, highest part of her Orbit, Descending Node, or lowest part of her Orbit. M signifies morning, A afternoon; and the line at the foot of the Table shews a week’s difference in rising and setting.

Moon’s Age	Full in her Ascending node.				In the highest part of her Orbit.				Full in her Descending node.				In the lowest part of her Orbit.
	Rises at		Sets at		Rises at		Sets at		Rises at		Sets at		Rises at		Sets at
	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.
12	5	A15	3	M20	4	A30	3	M15	4	A32	3	M40	5	A16	3	M0
13	5	32	4	25	4	50	4	45	5	15	4	20	6	0	4	15
14	5	48	5	30	5	15	6	0	5	45	5	40	6	20	5	28
15	6	5	7	0	5	42	7	20	6	15	6	56	6	45	6	32
16	6	20	8	15	6	2	8	35	6	46	8	0	7	8	7	45
17	6	36	9	12	6	26	9	45	7	18	9	15	7	30	9	15
18	6	54	10	30	7	0	10	40	8	0	10	20	7	52	10	0
Dif.	1	39	7	10	2	30	7	25	3	28	6	40	2	36	7	0

This Table was not computed, but only estimated as near as could be done from a common Globe, on which the Moon’s Orbit was delineated with a black lead pencil. It may at first sight appear erroneous; since as we have supposed the Moon to be full in either Node at the autumnal Equinox, she ought by the Table to rise just at six o’clock, or at Sun-set, on the 15th day of her age; being in the Ecliptic at that time. But it must be considered, that the Moon is only 14¹⁄₄ days old when she is Full; and therefore in both cases she is a little past the Node on the 15th day, being above it at one time, and below it at the other.

The period of the Harvest Moons.

292. As there is a compleat revolution of the Nodes in 18²⁄₃ years, there must be a regular period of all the Varieties which can happen in the rising and setting of the Moon during that time. But this shifting of the Nodes never affects the Moon’s rising so much, even in her quickest descending Latitude, as not to allow us still the benefit of her rising nearer the time of Sun-set for a few days together about the Full in Harvest, than when she is Full at any other time of the year. The following Table shews in what years the Harvest-Moons are least beneficial as to the times of their rising, and in what years most, from 1751 to 1861. The column of years under the letter L are those in which the Harvest-Moons are least of all beneficial, because they fall about the Descending Node: and those under M are the most of all beneficial, because they fall about the Ascending Node. In all the columns from N to S the Harvest-Moons descend gradually in the Lunar Orbit, and rise to less heights above the Horizon. From S to N they ascend in the same proportion, and rise to greater heights above the Horizon. In both the columns under S the Harvest-Moons are in the lowest part of the Moon’s Orbit, that is, farthest South of the Ecliptic; and therefore stay shortest of all above the Horizon: in the columns under N just the reverse. And in both cases, their rising, though not at the same times, are nearly the same with regard to difference of time, as if the Moon’s Orbit were coincident with the Ecliptic.

Years in which the Harvest-Moons are least beneficial.

N				L				S
1751	1752	1753	1754	1755	1756	1757	1758	1759
1770	1771	1772	1773	1774	1775	1776	1777	1778
1788	1789	1790	1791	1792	1793	1794	1795	1796	1797
1807	1808	1809	1810	1811	1812	1813	1814	1815
1826	1827	1828	1829	1830	1831	1832	1833	1834
1844	1845	1846	1847	1848	1849	1850	1851	1852

Years in which they are most beneficial.

S				M				N
1760	1761	1762	1763	1764	1765	1766	1767	1768	1769
1779	1780	1781	1782	1783	1784	1785	1786	1787
1798	1799	1800	1801	1802	1803	1804	1805	1806
1816	1817	1818	1819	1820	1821	1822	1823	1824	1825
1835	1836	1837	1838	1839	1840	1841	1842	1843
1853	1854	1855	1856	1857	1858	1859	1860	1861

PL. VIII.

293. At the Polar Circles, when the Sun touches the Summer Tropic, he continues 24 hours above the Horizon; and 24 hours below it when he touches the Winter Tropic. For the same reason the Full Moon neither rises in Summer, nor sets in Winter, considering her as moving in the Ecliptic. For the Winter Full Moon being as high in the Ecliptic as the Summer Sun, must therefore continue as long above the Horizon; and the Summer Full Moon being as low in the Ecliptic as the Winter Sun, can no more rise than he does. But these are only the two Full Moons which happen about the Tropics, for all the others rise and set. In Summer the Full Moons are low, and their stay is short above the Horizon, when the nights are short, and we have least occasion for Moon-light: in Winter they go high, and stay long, above the Horizon when the nights are long, and we want the greatest quantity of Moon-light.

The long continuance of Moon-light at the Poles.

Fig. V.

294. At the Poles, one half of the Ecliptic never sets, and the other half never rises: and therefore, as the Sun is always half a year in describing one half of the Ecliptic, and as long in going through the other half, ’tis natural to imagine that the Sun continues half a year together above the Horizon of each Pole in it’s turn, and as long below it; rising to one Pole when he sets to the other. This would be exactly the case if there were no refraction: but by the Atmosphere’s refracting the Sun’s rays, he becomes visible some days sooner § 183, and continues some days longer in sight than he would otherwise do: so that he appears above the Horizon of either Pole before he has got below the Horizon of the other. And, as he never goes more than 23¹⁄₂ degrees below the Horizon of the Poles, they have very little dark night: it being twilight there as well as at all other places till the Sun be 18 degrees below the Horizon, § 177. The Full Moon being always opposite to the Sun, can never be seen while the Sun is above the Horizon, except when the Moon falls in the northern half of her Orbit; for whenever any point of the Ecliptic rises the opposite point sets. Therefore, as the Sun is above the Horizon of the north Pole from the 20th of March till the 23d of September, it is plain that the Moon, when Full, being opposite to the Sun, must be below the Horizon during that half of the year. But when the Sun is in the southern half of the Ecliptic he never rises to the north Pole, during which half of the year, every Full Moon happens in some part of the northern half of the Ecliptic, which never sets. Consequently, as the polar Inhabitants never see the Full Moon in Summer, they have her always in the Winter, before, at, and after the Full, shining for 14 of our days and nights. And when the Sun is at his greatest depression below the Horizon, being then in Capricorn, the Moon is at her First Quarter in Aries, Full in Cancer, and at her Third Quarter in Libra. And as the beginning of Aries is the rising point of the Ecliptic, Cancer the highest, and Libra the setting point, the Moon rises at her First Quarter in Aries, is most elevated above the Horizon, and Full in Cancer, and sets at the beginning of Libra in her Third Quarter, having continued visible for 14 diurnal rotations of the Earth. Thus the Poles are supplied one half of the winter time with constant Moon-light in the Sun’s absence; and only lose sight of the Moon from her Third to her First Quarter, while she gives but very little light; and could be but of little, and sometimes of no service to them. A bare view of the Figure will make this plain; in which let S be the Sun, e the Earth in Summer when it’s north Pole n inclines toward the Sun, and E the Earth in Winter, when it’s north Pole declines from him. SEN and NWS is the Horizon of the north Pole, which is coincident with the Equator; and, in both these positions of the Earth, ♈♋♎♑ is the Moon’s Orbit, in which she goes round the Earth, according to the order of the letters abcd, ABCD. When the Moon is at a she is in her Third Quarter to the Earth at e, and just rising to the north Pole n; at b she changes, and is at the greatest height above the Horizon, as the Sun likewise is; at c she is in her First Quarter, setting below the Horizon; and is lowest of all under it at d, when opposite to the Sun, and her enlightened side toward the Earth. But then she is full in view to the south Pole p; which is as much turned from the Sun as the north Pole inclines towards him. Thus in our Summer, the Moon is above the Horizon of the north Pole whilst she describes the northern half of the Ecliptic ♈♋♎, or from her Third Quarter to her First; and below the Horizon during the progress through the southern half ♎♑♈; highest at the Change, most depressed at the Full. But in winter, when the Earth is at E, and it’s north Pole declines from the Sun, the New Moon at D is at her greatest depression below the Horizon NWS, and the Full Moon at B at her greatest height above it; rising at her First Quarter A, and keeping above the Horizon till she comes to her Third Quarter C. At a mean state she is 23¹⁄₂ degrees above the Horizon at B and b, and as much below it at D and d, equal to the inclination of the Earth’s Axis F. S♋ and S♑ are, as it were, a ray of light proceeding from the Sun to the Earth; and shews that when the Earth is at e, the Sun is above the Horizon, vertical to the Tropic of Cancer; and when the Earth is at E, he is below the Horizon, vertical to the Tropic of Capricorn.

Plate VIII.

J. Ferguson delin.

J. Mynde Sculp.

Plate IX.

J. Ferguson delin.

J. Mynde Sculp.

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