Part 13
230. This part of the Equation of time may perhaps be somewhat difficult to understand by a Figure, because both halves of the Ecliptic seem to be on the same side of the Globe; but it may be made very easy to any person who has a real Globe before him, by putting small patches on every tenth or fifteenth degree both of the Equator and Ecliptic; and then, turning the ball slowly round westward, he will see all the patches from Aries to Cancer come to the brazen Meridian sooner than the corresponding patches on the Equator; all those from Cancer to Libra will come later to the Meridian than their corresponding patches on the Equator; those from Libra to Capricorn sooner, and those from Capricorn to Aries later: and the patches at the beginnings of Aries, Cancer, Libra, and Capricorn, being also on the Equator, shew that the two Suns meet there, and come to the Meridian together.
[Sidenote: A machine for shewing the sidereal, the equal, and the solar Time.
PLATE VI.]
231. Let us suppose that there are two little balls moving equably round a celestial Globe by clock-work, one always keeping in the Ecliptic, and gilt with gold, to represent the real Sun; and the other keeping in the Equator, and silvered, to represent the fictitious Sun: and that whilst these balls move once, round the Globe according to the order of Signs, the Clock turns the Globe 366 times round it’s Axis westward. The Stars will make 366 diurnal revolutions from the brasen Meridian to it again; and the two balls representing the real and fictitious Sun always going farther eastward from any given Star, will come later than it to the Meridian every following day; and each ball will make 365 revolutions to the Meridian; coming equally to it at the beginnings of Aries, Cancer, Libra, and Capricorn: but in every other point of the Ecliptic, the gilt ball will come either sooner or later to the Meridian than the silvered ball, like the patches above-mentioned. This would be a pretty-enough way of shewing the reason why any given Star, which, on a certain day of the year, comes to the Meridian with the Sun, passes over it so much sooner every following day, as on that day twelvemonth to come to the Meridian with the Sun again; and also to shew the reason why the real Sun comes to the Meridian sometimes sooner, sometimes later, than it is noon by the clock; and, on four days of the year, at the same time; whilst the fictitious Sun always comes to the Meridian when it is twelve at noon by the clock. This would be no difficult task for an artist to perform; for the gold ball might be carried round the Ecliptic by a wire from it’s north Pole, and the silver ball round the Equator by a wire from it’s south Pole, with a few wheels to each; which might be easily added to my improvement of the celestial Globe, described in N^o 483 of the _Philosophical Transactions_; and of which I shall give a description in the latter part of this Book, from the 3d Figure of the 3d plate.
[Sidenote: Fig. III.]
232. ’Tis plain that if the Ecliptic were more obliquely posited to the Equator, as the dotted Circle ♈_x_♎, the equal divisions from ♈ to _x_ would come still sooner to the Meridian _Z0_♈ than those marked _A_, _B_, _C_, _D_, and _E_ do: for two divisions containing 30 degrees, from ♈ to the second dott, a little short of the figure 1, come sooner to the Meridian than one division containing only 15 degrees from ♈ to _A_ does, as the Ecliptic now stands; and those of the second quadrant from _x_ to ♎ would be so much later. The third quadrant would be as the first, and the fourth as the second. And it is likewise plain, that where the Ecliptic is most oblique, namely about Aries and Libra, the difference would be greatest: and least about Cancer and Capricorn, where the obliquity is least.
[Sidenote: The second part of the Equation of Time.
PLATE VI.]
234. Having explained one cause of the difference of time shewn by a well-regulated Clock and a true Sun-dial; and considered the Sun, not the Earth, as moving in the Ecliptic; we now proceed to explain the other cause of this difference, namely, the inequality of the Sun’s apparent motion § 205, which is slowest in summer, when the Sun is farthest from the Earth, and swiftest in winter when he is nearest to it. But the Earth’s motion on it’s Axis is equable all the year round, and is performed from west to east; which is the way that the Sun appears to change his place in the Ecliptic.
235. If the Sun’s motion were equable in the Ecliptic, the whole difference between the equal time as shewn by a Clock, and the unequal time as shewn by the Sun, would arise from the obliquity of the Ecliptic. But the Sun’s motion sometimes exceeds a degree in 24 hours, though generally it is less: and when his motion is slowest any particular Meridian will revolve sooner to him than when his motion is quickest; for it will overtake him in less time when he advances a less space than when he moves through a larger.
236. Now, if there were two Suns moving in the plane of the Ecliptic, so as to go round it in a year; the one describing an equal arc every 24 hours, and the other describing sometimes a less arc in 24 hours, and at other times a larger; gaining at one time of the year what it lost at the opposite; ’tis evident that either of these Suns would come sooner or later to the Meridian than the other as it happened to be behind or before the other: and when they were both in conjunction they would come to the Meridian at the same moment.
[Sidenote: Fig. IV.]
237. As the real Sun moves unequably in the Ecliptic, let us suppose a fictitious Sun to move equably in it. Let _ABCD_ be the Ecliptic or Orbit in which the real Sun moves, and the dotted Circle _abcd_ the imaginary Orbit of the fictitious Sun; each going round in a year according to the order of letters, or from west to east. Let _HIKL_ be the Earth turning round it’s Axis the same way every 24 hours; and suppose both Suns to start from _A_ and _a_, in a right line with the plane of the Meridian _EH_, at the same moment: the real Sun at _A_, being then at his greatest distance from the Earth, at which time his motion is slowest; and the fictitious Sun at _a_, whose motion is always equable because his distance from the Earth is supposed to be always the same. In the time that the Meridian revolves from _H_ to _H_ again, according to the order of the letters _HIKL_, the real Sun has moved from _A_ to _F_; and the fictitious with a quicker motion from _a_ to _f_, through a larger arc: therefore, the Meridian _EH_ will revolve sooner from _H_ to _h_ under the real Sun at _F_, than from _H_ to _k_ under the fictitious Sun at _f_; and consequently it will be noon by the Sun-dial sooner than by the Clock.
[Sidenote: PLATE VI.]
As the real Sun moves from _A_ towards _C_, the swiftness of his motion increases all the way to _C_, where it is at the quickest. But notwithstanding this, the fictitious Sun gains so much upon the real, soon after his departing from _A_, that the increasing velocity of the real Sun does not bring him up with the equally moving fictitious Sun till the former comes to _C_, and the latter to _c_, when each has gone half round it’s respective orbit; and then being in conjunction, the Meridian _EH_ revolving to _EK_ comes to both Suns at the same time, and therefore it is noon by them both at the same moment.
But the increased velocity of the real Sun, now being at the quickest, carries him before the fictitious; and therefore, the same Meridian will come to the fictitious Sun sooner than to the real: for whilst the fictitious Sun moves from _c_ to _g_, the real Sun moves through a greater arc from _C_ to _G_: consequently the point _K_ has it’s fictitious noon when it comes to _k_, but not it’s real noon till it comes to _l_. And although the velocity of the real Sun diminishes all the way from _C_ to _A_, and the fictitious Sun by an equable motion is still coming nearer to the real Sun, yet they are not in conjunction till the one comes to _A_ and the other to _a_; and then it is noon by them both at the same moment.
And thus it appears, that the real noon by the Sun is always later than the fictitious noon by the clock whilst the Sun goes from _C_ to _A_, sooner whilst he goes from _A_ to _C_, and at these two points the Sun and Clock being equal, it is noon by them both at the same moment.
[Sidenote: Apogee, Perigee, and Apsides, what.
Fig. IV.]
238. The point _A_ is called _the Sun’s Apogee_, because when he is there he is at his greatest distance from the Earth; the point _C_ his _Perigee_, because when in it he is at his least distance from the Earth: and a right line, as _AEC_, drawn through the Earth’s center, from one of these points to the other, is called _the line of the Apsides_.
[Sidenote: Mean Anomaly, what.]
239. The distance that the Sun has gone in any time from his Apogee (not the distance he has to go to it though ever so little) is called _his mean Anomaly_, and is reckoned in Signs and Degrees, allowing 30 Degrees to a Sign. Thus, when the Sun has gone suppose 174 degrees from his Apogee at _A_, he is said to be 5 Signs 24 Degrees from it, which is his mean Anomaly: and when he is gone suppose 355 degrees from his Apogee, he is said to be 11 Signs 25 Degrees from it, although he be but 5 Degrees short of _A_ in coming round to it again.
240. From what was said above it appears, that when the Sun’s Anomaly is less than 6 Signs, that is, when he is any where between _A_ and _C_, in the half _ABC_ of his orbit, the true noon precedes the fictitious; but when his Anomaly is more than 6 Signs, that is, when he is any where between _C_ and _A_, in the half _CDA_ of his Orbit, the fictitious noon precedes the true. When his Anomaly is 0 Signs 0 Degrees, that is, when he is in his Apogee at _A_; or 6 Signs 0 Degrees, which is when he is in his Perigee at _C_; he comes to the Meridian at the moment that the fictitious Sun does, and then it is noon by them both at the same instant.
+----------------------------------------------------------+ | _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.