Astronomy Explained Upon Sir Isaac Newton's Principles / And made easy to those who have not studied mathematics

by James Ferguson

This Table may be made perpetual, by deducting 6 hours from the time of New Moon in any given year and month from 1724 to 1800, in order to have the mean time of New Moon in any year and month 76 years afterward; or deducting 12 hours for 152 years, 18 hours for 228 years; and 24 hours for 304 years, because in that time the changes of the Moon anticipate almost a complete natural day. And if the like number of hours be added for so many years past, we shall have the mean time of any New Moon already elapsed. Suppose, for example, the mean time of Change was required for January 1802; deduct 76 years and there remains 1726, against which in the following Table under January I find the time of New Moon was on the 21st day at 11 in the evening: from which take 6 hours and there remains the 21st day at 5 in the evening for the mean time of Change in January 1802. Or, if the time be required for May, A. D. 1701, add 76 years and it makes 1777, which I look for in the Table, and against it under May I find the New Moon in that year falls on the 25th day at 9 in the evening; to which add 6 hours, and it gives the 26th day at 3 in the Morning for the time of New Moon in May, A. D. 1701. By this addition for time past, or subtraction for time to come, the Table will not vary 24 hours from the truth in less than 14592 years. And if, instead of 6 hours for every 76 years, we add or subtract only 5 hours 52 minutes, it will not vary a day in 10 millions of years.

Although this Table is calculated for 76 years only, and according to the Old Style, yet by means of two easy Equations it may be made to answer as exactly to the New Style, for any time to come. Thus, because the year 1724 in this Table is the first year of the Cycle for which it is made; if from any year of Christ after 1800 you subtract 1723, and divide the overplus by 76, the Quotient will shew how many entire Cycles of 76 years are elapsed since the beginning of the Cycle here provided for; and the remainder will shew the year of the current Cycle answering to the given year of Christ. Hence, if the remainder be 0, you must instead thereof put 76, and lessen the Quotient by unity.

Then, look in the left hand column of the Table for the number in your remainder, and against it you will find the times of all the mean New Moons in that year of the present Cycle. And whereas in 76 Julian Years the Moon anticipates 5 hours 52 minutes, if therefore these 5 hours 52 minutes be multiplied by the above found Quotient, that is, by the number of entire Cycles past; the product subtracted from the times in the Table will leave the corrected times of the New Moons to the Old Style; which may be reduced to the New Style thus:

Divide the number of entire hundreds in the given year of Christ by 4, multiply this Quotient by 3, to the product add the remainder, and from their sum subtract 2: this last remainder denotes the number of days to be added to the times above corrected, in order to reduce them to the New Style. The reason of this is, that every 400 years of the New Style gains 3 days upon the Old Style: one of which it gains in each of the centurial years succeeding that which is exactly divisible by 4 without remainder; but then, when you have found the days so gained, 2 must be subtracted from their number on account of the rectifications made in the Calendar by the Council of Nice, and since by Pope Gregory. It must also be observed, that the additional days found as above directed do not take place in the centurial Years which are not multiples of 4 till February 29th, O. S. for on that day begins the difference between the Styles; till which day therefore, those that were added in the preceding years must be used. The following Example will make this accommodation plain.

Required the mean time of New Moon in June, A.D. 1909, N.S.

From 1909 take 1723 Years, and there rem.		186
Which divided by 76, gives the Quotient 2 and the remainder		34
Then, against 34 in the Table is June	5d	8h	0m	Afternoon.
And 5h 52m multiplied by 2 make to be subtr.		11	44
Remains the mean time according to the Old Style, June	5d	9h	16m	Morning.
Entire hundred in 1909 are 19, which divided by 4, quotes			4
And leaves a remainder of			3
Which Quotient multiplied by 3 makes 12, and the remainder added makes			15
From which subtract 2, and there remains			13
Which number of days added to the above time Old Style, gives June	18d	9h	16m	Morn.N.S.

So the mean time of New Moon in June 1909 New Style is the 18th day at 16 minutes past 9 in the Morning.

If 11 days be added to the time of any New Moon in this Table, it will give the time thereof according to the New Style till the year 1800. And if 14 days 18 hours 22 minutes be added to the mean time of New Moon in either Style, it will give the mean time of the next Full Moon according to that Style.

A Table shewing the times of all the mean Changes of the Moon, to the nearest Hour, through four Lunar Periods, or 76 years. M signifies morning, A afternoon.

Yrs of the Cyc.	A.D.	January		February		March		April		May		June		July		August		Septemb.		October		Novemb.		Decemb.
		D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.	D.	H.
1	1724	14	5A	13	5M	13	6A	12	7M	11	8A	10	8M	9	9A	8	10M	6	10A	6	11M	4	12A	4	1A
2	1725	3	2M	1	2A	3	3M	1	4A	1	4M	29	6M	28	7A	27	8M	25	8A	25	9M	23	10A	23	11M
										30	5A
3	1726	21	11A	20	11M	21	12A	20	1A	20	1M	18	2A	18	3M	16	4A	15	5M	14	5A	13	6M	12	7A
4	1727	11	8M	9	9A	11	9M	9	10A	9	11M	7	12A	7	0A	6	1M	4	1A	4	2M	2	3A	2	4M
																								31	5A
5	1728	30	6M	28	7A	29	7M	27	8A	27	8M	25	9A	25	10M	23	11A	22	11M	21	12A	20	1A	20	2M
6	1729	18	2A	17	3M	18	4A	17	4M	16	5A	15	6M	14	7A	12	7M	11	8A	11	9M	9	0A	9	11M
7	1730	7	11A	6	0A	8	1M	6	1A	6	2M	4	3A	4	3M	2	4A	2	5M	30	7M	28	8A	28	9M
																		30	6A
8	1731	26	9A	25	10M	26	10A	25	11M	24	11A	23	0A	23	1M	21	2A	20	2M	19	3A	18	4M	17	5A
9	1732	16	5M	14	6A	15	7M	13	8A	13	8M	11	9A	11	10M	9	11A	8	11M	7	12A	6	1A	6	2M
10	1733	4	2A	3	3M	4	4A	3	4M	2	5A	1	6M	30	8M	28	8A	27	9M	26	10A	25	11M	24	11A
												30	7A
11	1734	23	0A	22	1M	23	1A	22	2M	21	2A	20	3M	19	4A	18	5M	16	5A	16	6M	14	7A	14	8M
12	1735	12	9A	11	9M	12	10A	11	11M	10	11A	9	0A	9	1M	7	2A	6	2M	5	3A	4	4M	3	5A
13	1736	2	5M	----		1	7M	29	9M	28	9A	27	10M	26	11A	25	0A	23	12A	23	1A	22	2M	21	3A
		31	6A			30	8A
14	1737	20	3M	18	4A	20	4M	18	5A	18	5M	16	6A	16	7M	14	8A	13	8M	12	9A	11	10M	10	11A
15	1738	9	11M	7	12A	9	1A	8	1M	7	2A	6	3M	5	4A	4	5M	2	5A	2	6M	30	8M	29	8A
																				31	7A
16	1739	28	9M	26	10A	28	11M	26	12A	26	0A	25	1M	24	2A	23	3M	21	3A	21	4M	19	5A	19	6M
17	1740	17	6A	16	7M	16	8A	15	9M	14	9A	13	10M	12	11A	11	0A	9	12A	9	1A	8	2M	7	3A
18	1741	6	3M	4	4A	6	4M	4	5A	4	5M	2	6A	2	7M	30	8M	28	9A	28	10M	26	11A	26	11M
														31	7A
19	1742	24	12A	23	1A	25	2M	23	3A	23	3M	21	4A	21	5M	19	6A	18	6M	17	7A	16	8M	15	9A
20	1743	14	9M	12	10A	14	11M	12	12A	12	0A	11	1M	10	2A	9	3M	7	3A	7	4M	5	5A	5	6M
21	1744	3	6A	2	7M	2	8A	1	9M	30	10M	28	11A	28	0A	26	12A	25	1A	25	2M	23	3A	23	3M
								30	9A
22	1745	21	4A	20	5M	21	5A	20	6M	19	6A	18	7M	17	8A	16	8M	14	9A	14	10M	12	11A	12	0A
23	1746	10	12A	9	1A	11	2M	9	3A	9	3M	7	4A	7	5M	5	6A	4	6M	3	7A	2	8M	1	9A
																								31	10M
24	1747	29	10A	28	11M	29	11A	28	0A	27	12A	26	1A	26	2M	24	3A	23	3M	22	4A	21	5M	20	6A
25	1748	19	6M	17	7A	18	8M	16	9A	16	9M	14	10A	14	11M	12	12A	11	0A	11	1M	9	2A	9	3M
26	1749	7	3A	6	4M	7	5A	6	6M	5	6A	4	7M	3	8A	2	9M	30	10M	29	11A	28	0A	27	12A
																31	9A
27	1750	26	1A	25	2M	26	3A	25	4M	24	4A	23	5M	22	6A	21	7M	19	7A	19	8M	17	9A	17	10M
28	1751	15	10A	14	11M	15	11A	14	0A	13	12A	12	1A	12	2M	10	3A	9	3M	8	4A	7	5M	6	6A
29	1752	5	6M	3	7A	4	8M	2	9A	2	9M	30	11M	29	12A	28	0A	27	1M	26	2A	25	3M	24	3A
										31	10A
30	1753	23	4M	21	5A	23	6M	21	7A	21	7M	19	8A	19	9M	17	10A	16	10M	15	11A	14	0A	14	1M
31	1754	12	1A	11	2M	12	3A	11	4M	10	4A	9	5M	8	6A	7	7M	5	7A	5	8M	3	9A	3	10M
32	1755	1	10A	----		1	11A	29	12A	29	1A	28	2M	27	3A	25	3M	24	4A	24	5M	22	6A	22	6M
		31	11M			31	0A
33	1756	20	7A	19	8M	19	9A	18	9M	17	10A	16	11M	15	12A	14	1A	13	1M	12	2A	11	3M	10	4A
34	1757	9	4M	7	5A	9	6M	7	7A	7	7M	5	8A	5	9M	3	10A	2	10M	1	14A	30	1M	29	1A
																				31	0A
35	1758	28	2M	26	3A	28	3M	26	4A	26	4M	24	5A	24	6M	22	7A	21	7M	20	8A	19	9M	18	10A
36	1759	17	10M	15	11A	17	0A	16	1M	15	1A	14	2M	13	3A	12	2M	10	4A	10	5M	8	6A	8	7M
37	1760	6	7A	5	8M	5	9A	4	10M	3	10A	2	11M	1	12A	30	1M	28	2A	28	3M	26	4A	26	4M
														31	1A
38	1761	24	5A	23	6M	24	7A	23	8M	22	9A	21	10M	20	10A	19	11M	17	11A	17	0A	16	1M	15	2A
39	1762	14	2M	12	3A	14	3M	12	4A	12	4M	10	5A	10	6M	8	7A	7	7M	6	8A	5	9M	4	10A
40	1763	3	11M	1	12A	3	0A	2	1M	1	1A	29	3A	29	4M	27	4M	26	5M	25	6A	24	7M	23	7A
										31	2M
41	1764	22	8M	20	9A	21	10M	19	11A	19	11M	17	12A	17	1A	16	2M	14	2A	14	3M	12	4A	12	5M
42	1765	10	5A	9	6M	10	6A	9	7M	8	7A	7	8M	6	9A	5	10M	3	10A	3	11M	1	12A	1	1A
																								31	1M
43	1766	29	2A	28	3M	29	4A	28	5M	27	5A	26	6M	25	7A	24	8M	22	8A	22	9M	20	10A	20	11M
44	1767	18	11A	17	0A	19	1M	17	2A	17	2M	15	3A	15	4M	13	5A	12	6M	11	6A	10	7M	9	8A
45	1768	8	8M	6	9A	7	10M	5	11A	5	11M	3	12A	3	1A	2	2M	30	3M	29	4A	28	5M	27	5A
																31	2A
46	1769	26	6M	24	7A	26	7M	24	8A	24	8M	22	9A	22	10M	20	11A	19	11M	18	12A	17	1A	17	2M
47	1770	15	2A	14	3M	15	4A	14	5M	13	5A	12	4M	11	7A	10	8M	8	8A	8	9M	6	10A	6	11M
48	1771	4	11M	3	0A	5	1M	3	2A	3	2M	1	3A	1	4M	29	5M	27	6A	27	7M	25	8A	25	9M
														30	5A
49	1772	23	9A	22	10M	22	10A	21	11M	20	11A	19	0A	19	1M	17	2A	16	2M	15	3A	14	4M	13	5A
50	1773	12	5M	10	6A	12	7M	10	8A	10	8M	8	9A	8	9M	6	10A	5	11M	4	12A	3	1A	3	2M
51	1774	1	2A	----		1	4A	29	5A	29	6M	27	7A	27	8M	25	8A	24	9M	23	10A	22	11M	21	11A
		31	3M			31	5M
52	1775	20	0A	19	1M	20	2A	19	3M	18	3A	17	4M	16	5A	15	6M	13	6A	13	7M	11	8A	11	9M
53	1776	9	9A	8	10M	8	10A	7	11M	6	12A	5	0A	5	1M	3	2A	2	2M	1	3A	29	5A	29	5M
																				31	4M
54	1777	27	6A	26	7M	27	8A	26	9M	25	9A	24	10M	23	11A	22	0A	20	12A	20	1A	19	2M	18	3A
55	1778	17	3M	15	4A	17	5M	15	6A	15	6M	13	7A	13	8M	11	9A	10	9M	9	10A	8	11M	7	12A
56	1779	6	0A	5	1M	6	2A	5	3M	4	3A	3	4M	2	5A	1	6M	29	7M	28	8A	27	9M	26	9A
																30	6A
57	1780	25	10M	23	11A	24	11M	22	12A	22	0A	21	1M	20	2A	19	3M	17	3A	17	4M	15	5A	15	6M
58	1781	13	6A	12	7M	13	8A	12	9M	11	9A	10	10M	9	11A	8	0A	6	12A	6	1A	5	2M	4	3A
59	1782	3	3M	1	4A	3	5M	1	6A	1	6M	29	8M	28	9A	27	9M	25	10A	25	11M	23	12A	23	0A
										30	7A
60	1783	22	1M	20	2A	22	2M	20	3A	20	3M	18	4A	18	5M	16	6A	15	6M	14	7A	13	8M	12	9A
61	1784	11	9M	9	10A	10	11M	8	12A	8	0A	7	1M	6	2A	5	3M	3	3A	3	4M	1	5A	1	6M
																								30	6A
62	1785	29	7M	27	8A	29	9M	27	10A	27	10M	25	11A	25	0A	24	1M	22	1A	22	2M	20	3A	20	3M
63	1786	18	4A	17	5M	18	5A	17	6M	16	6A	15	7M	14	8A	13	9M	11	9A	11	10M	9	11A	9	0A
64	1787	7	12A	6	1A	8	2M	6	3A	6	3M	4	4A	4	5M	2	6A	1	6M	30	8M	28	9A	28	9M
																		30	7A
65	1788	26	10A	25	11M	25	12A	24	1A	24	1M	22	2A	22	3M	20	4M	19	4M	18	5A	17	6M	16	7A
66	1789	15	7M	13	8A	15	9M	13	10A	13	10M	11	11A	11	0A	10	1M	8	1A	8	2M	6	3A	6	4M
67	1790	4	4A	3	5M	4	5A	3	6M	2	6A	1	7M	30	9M	28	9A	27	10M	26	11A	25	0A	24	12A
												30	8A
68	1791	23	1A	22	2M	23	3A	22	4M	21	4A	20	5M	19	6A	18	7M	16	7A	16	8M	14	9A	14	10M
69	1792	12	10A	11	11M	11	12A	10	1A	10	1M	8	2A	8	3M	6	4A	5	4A	4	5A	3	6M	2	7A
70	1793	1	7M	----		1	9M	29	10M	28	11A	27	0A	27	1M	25	1A	24	2M	23	3A	22	4M	21	4A
		30	8A			30	10A
71	1794	20	5M	18	6A	20	6M	18	7A	18	7M	16	8A	16	9M	14	10A	13	10M	12	11A	11	0A	11	1M
72	1795	9	1A	8	2M	9	3A	8	4M	7	4A	6	5M	5	6A	4	7M	2	7A	2	8M	30	10M	29	10A
																				31	9A
73	1796	28	11M	26	12A	27	0A	26	1M	25	1A	24	2M	23	3A	22	4M	20	4A	20	5M	18	6A	18	7M
74	1797	16	7A	15	8M	16	9A	15	10M	14	10A	13	11M	12	12A	11	1A	10	1M	9	2A	8	3M	7	4A
75	1798	6	4M	4	5A	6	6M	4	7A	4	7M	2	8A	2	9M	30	10M	28	11A	28	0A	27	1M	26	1A
														31	10A
76	1799	25	2M	23	3A	25	4M	23	5A	23	5M	21	6A	21	6M	19	8A	18	8M	17	9A	16	10M	15	11A
1	1800	14	11A	12	12A	13	0A	12	1M	11	1A	10	2M	9	3A	8	4M	6	4A	6	5M	4	6A	4	7M