Chapter VII: Part 7

+----------------------------------------------------------+ | _Sun faster than the Clock if his Anomaly be_ | +----+--------+-------+-------+-------+-------+-------+----+ | |0 Signs | 1 | 2 | 3 | 4 | 5 | | | D. +--------+-------+-------+-------+-------+-------+ | | | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | | +----+--------+-------+-------+-------+-------+-------+----+ | 0 | 0 0 | 3 48 | 6 39 | 7 45 | 6 47 | 3 57 | 30 | | 1 | 0 8 | 3 55 | 6 43 | 7 45 | 6 43 | 3 50 | 29 | | 2 | 0 16 | 3 2 | 6 47 | 7 45 | 6 39 | 3 43 | 28 | | 3 | 0 24 | 4 9 | 6 51 | 7 45 | 6 35 | 3 35 | 27 | | 4 | 0 32 | 4 16 | 6 54 | 7 45 | 6 30 | 3 28 | 26 | | 5 | 0 40 | 4 22 | 6 58 | 7 44 | 6 26 | 3 20 | 25 | | 6 | 0 48 | 4 29 | 7 1 | 7 44 | 6 21 | 3 13 | 24 | | 7 | 0 56 | 4 35 | 7 5 | 7 43 | 6 16 | 3 5 | 23 | | 8 | 1 3 | 4 42 | 7 8 | 7 42 | 6 11 | 2 58 | 22 | | 9 | 1 11 | 4 48 | 7 11 | 7 41 | 6 6 | 2 50 | 21 | | 10 | 1 19 | 4 54 | 7 14 | 7 40 | 6 1 | 2 42 | 20 | | 11 | 1 27 | 5 0 | 7 17 | 7 38 | 5 56 | 2 35 | 19 | | 12 | 1 35 | 5 6 | 7 20 | 7 37 | 5 51 | 2 27 | 18 | | 13 | 1 43 | 5 12 | 7 22 | 7 35 | 5 45 | 2 19 | 17 | | 14 | 1 50 | 5 18 | 7 25 | 7 34 | 5 40 | 2 11 | 16 | | 15 | 1 58 | 5 24 | 7 27 | 7 32 | 5 34 | 2 3 | 15 | | 16 | 2 6 | 5 30 | 7 29 | 7 30 | 5 28 | 1 55 | 14 | | 17 | 2 13 | 5 35 | 7 31 | 7 28 | 5 22 | 1 47 | 13 | | 18 | 2 21 | 5 41 | 7 33 | 7 25 | 5 16 | 1 39 | 12 | | 19 | 2 28 | 5 46 | 7 35 | 7 23 | 5 10 | 1 31 | 11 | | 20 | 2 36 | 5 52 | 7 36 | 7 20 | 5 4 | 1 22 | 10 | | 21 | 2 43 | 5 57 | 7 38 | 7 18 | 4 58 | 1 14 | 9 | | 22 | 2 51 | 6 2 | 7 39 | 7 15 | 4 51 | 1 6 | 8 | | 23 | 2 58 | 6 7 | 7 41 | 7 12 | 4 45 | 0 58 | 7 | | 24 | 3 6 | 6 12 | 7 42 | 7 9 | 4 38 | 0 50 | 6 | | 25 | 3 13 | 6 16 | 7 43 | 7 5 | 4 31 | 0 41 | 5 | | 26 | 3 20 | 6 21 | 7 43 | 7 2 | 4 25 | 0 33 | 4 | | 27 | 3 27 | 6 26 | 7 44 | 6 58 | 4 18 | 0 25 | 3 | | 28 | 3 34 | 6 30 | 7 44 | 6 55 | 4 11 | 0 17 | 2 | | 29 | 3 41 | 6 34 | 7 45 | 6 51 | 4 4 | 0 8 | 1 | | 30 | 3 48 | 6 39 | 7 45 | 6 47 | 3 57 | 0 0 | 0 | +----+--------+-------+-------+-------+-------+-------+----+ | |11 Signs| 10 | 9 | 8 | 7 | 6 | D. | +----+--------+-------+-------+-------+-------+-------+----+ | _Sun slower than the Clock if his Anomaly be_ | +----------------------------------------------------------+

[Sidenote: A Table of the Equation of Time, depending on the Sun’s
Anomaly.]

241. The annexed Table shews the Variation, or Equation of time depending on the Sun’s Anomaly, and arising from his unequal motion in the Ecliptic; as the former Table § 229 shews the Variation depending on the Sun’s place, and resulting from the obliquity of the Ecliptic: this is to be understood the same way as the other, namely, that when the Signs are at the head of the Table, the Degrees are at the left hand; but when the Signs are at the foot of the Table the respective Degrees are at the right hand; and in both cases the Equation is in the Angle of meeting. When both the above-mentioned Equations are either faster or slower, their sum is the absolute Equation of Time; but when the one is faster, and the other slower, it is their difference. Thus, suppose the Equation depending on the Sun’s place, be 6 minutes 41 seconds too slow, and the Equation depending on the Sun’s Anomaly, be 4 minutes 20 seconds too slow, their Sun is 11 minutes 1 second too slow. But if the one had been 6 minutes 41 seconds too fast, and the other 4 minutes 20 seconds too slow, their difference had been 2 minutes 21 seconds too fast, because the greater quantity is too fast.

242. The obliquity of the Ecliptic to the Equator, which is the first mentioned cause of the Equation of Time, would make the Sun and Clocks agree on four days of the year; which are, when the Sun enters Aries, Cancer, Libra, and Capricorn: but the other cause, now explained, would make the Sun and Clocks equal only twice in a year; that is, when the Sun is in his Apogee and Perigee. Consequently, when these two points fall in the beginnings of Cancer and Capricorn, or of Aries and Libra, they concur in making the Sun and Clocks equal in these points. But the Apogee at present is in the 9th degree of Cancer, and the Perigee in the 9th degree of Capricorn; and therefore the Sun and Clocks cannot be equal about the beginning of these Signs, nor at any time of the year, except when the swiftness or slowness of Equation resulting from one cause just balances the slowness or swiftness arising from the other.

243. The last Table but one, at the end of this Chapter, shews the Sun’s place in the Ecliptic at the noon of every day by the clock, for the second year after leap-year; and also the Sun’s Anomaly to the nearest degree, neglecting the odd minutes of a degree. Their use is only to assist in shewing the method of making a general Equation Table from the two fore-mentioned Tables of Equation depending on the Sun’s Place and Anomaly § 229, 241; concerning which method we shall give a few examples presently. The following Tables are such as might be made from these two; and shew the absolute Equation of Time resulting from the combination of both it’s causes; in which the minutes, as well as degrees, both of the Sun’s Place and Anomaly are considered. The use of these Tables is already explained, § 225; and they serve for every day in leap-year, and the first, second, and third years after: For on most of the same days of all these years the Equation differs, because of the odd six hours more than the 365 days of which the year consists.

[Sidenote: Examples for making Equation Tables.]

EXAMPLE I. On the 15th of _April_ the Sun is in the 25th degree of ♈ Aries, and his Anomaly is 9 Signs 15 Degrees; the Equation resulting from the former is 7 minutes 23 seconds of time too fast § 229; and from the latter, 7 minutes 27 seconds too slow, § 241; the difference is 4 seconds that the Sun is too slow at the noon of that day; taking it in gross for the degrees of the Sun’s Place and Anomaly, without making proportionable allowance for the odd minutes. Hence, at noon the swiftness of the one Equation balancing so nearly the slowness of the other, makes the Sun and Clocks equal on some part of that day.

EXAMPLE II. On the 16th of _June_, the Sun is in the 25th degree of ♊ Gemini, and his Anomaly is 11 Signs 16 Degrees; the Equation arising from the former is 1 minute 48 seconds too fast; and from the latter 1 minute 50 seconds too slow; which balancing one another at noon to 2 seconds, the Sun and Clocks are again equal on that day.

EXAMPLE III. On the 31st of _August_ the Sun’s place is 7 degrees 52 minutes of ♍ Virgo (which we shall call the 8th degree, as it is so near) and his Anomaly is 2 Signs 0 Degrees; the Equation arising from the former is 6 minutes 41 seconds too slow; and from the latter 6 minutes 39 seconds too fast; the difference being only 2 seconds too slow at noon, and decreasing towards an equality will make the Sun and Clocks equal in the afternoon of that day.

EXAMPLE. IV. On the 23d of _December_ the Sun’s place is 1 degree 41 minutes (call it 2 degrees) of ♑ Capricorn, and his Anomaly is 5 Signs 23 Degrees; the Equation for the former is 43 seconds too slow, and for the latter 58 seconds too fast; the difference is 15 seconds too fast at noon; which decreasing will come to an equality, and so make the Sun and Clocks equal in the evening of that day.

And thus we find, that on some part of each of the above-mentioned four days, the Sun and Clocks are equal; but if we work examples for all other days of the year we shall find them different. And,

[Sidenote: Remark.]

244. On those days which are equidistant from any Equinox or Solstice, we do not find that the Equation is as much too fast or too slow, on the one side, as it is too slow or too fast on the other. The reason is, that the line of the Apsides § 238, does not, at present, fall either into the Equinoctial or Solsticial points § 242.

[Sidenote: The reason why Equation Tables are but temporary.]

245. If the line of the Apsides, together with the Equinoctial and Solsticial points, were immoveable, a general Equation Table might be made from the preceding Equation Tables, which would always keep true, because these Tables themselves are permanent. But, with respect to the fixed Stars, the line of the Apsides moves forwards 12 seconds of a degree every year, and the above points 50 seconds backward. So that if in any given year, the Equinoctial points, and line of the Apsides were coincident, in 100 years afterward they would be separated 1 degree 43 minutes 20 seconds; and consequently in 5225.8 years they would be separated 90 degrees, and could not meet again, so that the same Equinoctial point should fall again into the Apogee in less than 20,903 years: and this is the shortest Period in which the Equation of Time can be restored to the same state again, with respect to the same seasons of the year.

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 14-1/2 seconds. The _Sidereal Year_ is therefore 20 minutes 17-1/2 seconds longer than the Solar or Tropical year, and 9 minutes 14-1/2 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.

[Sidenote: 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 17-1/2 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_. | +--------+--------------------------------------++-----------------+ | | Precession of the Equinoctial || Anticipation of | | | Points in the Heavens. || the Equinoxes | | Julian +----------------+---------------------++ on the Earth. | | years. | Motion. | Time. || | | +----------------+---------------------++-----------------+ | | S. ° ʹ ʺ | Days H. M. S. || D. H. M. S. | +--------+----------------+--------------------++------------------+ | 1 | 0 0 0 50 | 0 0 20 17-1/2 || 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 52-1/2 || 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 27-1/2 || 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 2-1/2 || 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 37-1/2 || 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 | +--------+----------------+---------------------++-----------------+

[Sidenote: 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. 17-1/2 sec. of time, annually: this, in so many years, makes 30 days, 10-1/2 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_.

+---------+----------------------------------++----------------+ | | Precession of the Equinoctial || Anticipation | | | Points in the Heavens. || of the | | Julian +-----------------+----------------+| Equinoxes on | | years. | Motion. | Time. || 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 | +---------+-----------------+----------------++----------------+

[Sidenote: 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.

[Sidenote: 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 365-1/4 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.

[Sidenote: 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.

[Sidenote: 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 23-1/2 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 8-1/2 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_.