224. The Earth’s motion on it’s Axis being perfectly uniform, and equal at all times of the year, the sidereal days are always precisely of the same length; and so would the solar or natural days be, if the Earth’s orbit were a perfect Circle, and it’s Axis perpendicular to it’s orbit. But the Earth’s diurnal motion on an inclined Axis, and it’s annual motion in an elliptic orbit, cause the Sun’s apparent motion in the Heavens to be unequal: for sometimes he revolves from the Meridian to the Meridian again in somewhat less than 24 hours, shewn by a well regulated clock; and at other times in somewhat more: so that the time shewn by an equal going clock and a true Sun-dial is never the same but on the 15th of April, the 16th of June, the 31st of August, and the 24th of December. The clock, if it goes equally and true all the year round, will be before the Sun from the 24th of December till the 15th of April; from that time till the 16th of June the Sun will be before the clock; from the 16th of June till the 31st of August the clock will be again before the Sun; and from thence to the 24th of December the Sun will be faster than the clock.
225. The Tables of the Equation of natural days, at the end of the next Chapter, shew the time that ought to be pointed out by a well regulated clock or watch every day of the year at the precise moment of solar noon; that is, when the Sun’s centre is on the Meridian, or when a true Sun-dial shews it to be precisely Twelve. Thus, on the 5th of January in Leap-year, when the Sun is on the Meridian, it ought to be 5 minutes 51 seconds past twelve by the clock; and on the 15th of May, when the Sun is on the Meridian, the time by the clock should be but 55 minutes 57 seconds past eleven; in the former case, the clock is 5 minutes 51 seconds beforehand with the Sun; and in the latter case, the Sun is 4 minutes 3 seconds faster than the clock. The column at the right hand of each month shews the daily difference of this equation, as it increases or decreases. But without a Meridian Line, or a Transit-Instrument fixed in the plane of the Meridian, we cannot set a Sun-dial true.
226. The easiest and most expeditious way of drawing a Meridian Line is this: Make four or five concentric Circles, about a quarter of an inch from one another, on a flat board about a foot in breadth; and let the outmost Circle be but little less than the board will contain. Fix a pin perpendicularly in the center, and of such a length that it’s whole shadow may fall within the innermost Circle for at least four hours in the middle of the day. The pin ought to be about an eighth part of an inch thick, with a round blunt point. The board being set exactly level in a place where the Sun shines, suppose from eight in the morning till four in the afternoon, about which hours the end of the shadow should fall without all the Circles; watch the times in the forenoon, when the extremity of the shortening shadow just touches the several Circles, and there make marks. Then, in the afternoon of the same day, watch the lengthening shadow, and where it’s end touches the several Circles in going over them, make marks also. Lastly, with a pair of compasses, find exactly the middle point between the two marks on any Circle, and draw a straight line from the center to that point; which Line will be covered at noon by the shadow of a small upright wire, which should be put in the place of the pin. The reason for drawing several Circles is, that in case one part of the day should prove clear, and the other part somewhat cloudy, if you miss the time when the point of the shadow should touch one Circle, you may perhaps catch it in touching another. The best time for drawing a Meridian Line in this manner is about the middle of summer; because the Sun changes his Declination slowest and his Altitude fastest in the longest days.
If the casement of a window on which the Sun shines at noon be quite upright, you may draw a line along the edge of it’s shadow on the floor, when the shadow of the pin is exactly on the Meridian Line of the board: and as the motion of the shadow of the casement will be much more sensible on the Floor, than that of the shadow of the pin on the board, you may know to a few seconds when it touches the Meridian Line on the floor, and so regulate your clock for the day of observation by that line and the Equation Tables above-mentioned § 225.
227. As the Equation of time, or difference between the time shewn by a well regulated Clock and a true Sun-dial, depends upon two causes, namely, the obliquity of the Ecliptic, and the unequal motion of the Earth in it, we shall first explain the effects of these causes separately considered, and then the united effects resulting from their combination.
228. The Earth’s motion on it’s Axis being perfectly equable, or always at the same rate, and the [55]plane of the Equator being perpendicular to it’s Axis, ’tis evident that in equal times equal portions of the Equator pass over the Meridian; and so would equal portions of the Ecliptic if it were parallel to or coincident with the Equator. But, as the Ecliptic is oblique to the Equator, the equable motion of the Earth carries unequal portions of the Ecliptic over the Meridian in equal times, the difference being proportionate to the obliquity; and as some parts of the Ecliptic are much more oblique than others, those differences are unequal among themselves. Therefore, if two Suns should start either from the beginning of Aries or Libra, and continue to move through equal arcs in equal times, one in the Equator, and the other in the Ecliptic, the equatoreal Sun would always return to the Meridian in 24 hours time, as measured by a well regulated clock; but the Sun in the Ecliptic would return to the Meridian sometimes sooner, and sometimes later than the equatoreal Sun; and only at the same moments with him on four days of the year; namely, the 20th of March, when the Sun enters Aries; the 21st of June, when he enters Cancer; the 23d of September, when he enters Libra; and the 21st of December, when he enters Capricorn. But, as there is only one Sun, and his apparent motion is always in the Ecliptic, let us henceforth call him the real Sun, and the other which is supposed to move in the Equator the fictitious; to which last, the motion of a well regulated clock always answers.
Let Z♈z♎ be the Earth, ZFRz it’s Axis, abcde &c. the Equator, ABCDE &c. the northern half of the Ecliptic from ♈ to ♎ on the side of the Globe next the eye, and MNOP &c. the southern half on the opposite side from ♎ to ♈. Let the points at A, B, C, D, E, F, &c. quite round from ♈ to ♈ again bound equal portions of the Ecliptic, gone through in equal times by the real Sun; and those at a, b, c, d, e, f, &c. equal portions of the Equator described in equal times by the fictitious Sun; and let Z♈z be the Meridian.
As the real Sun moves obliquely in the Ecliptic, and the fictitious Sun directly in the Equator, with respect to the Meridian, a degree, or any number of degrees, between ♈ and F on the Ecliptic, must be nearer the Meridian Z♈z, than a degree, or any corresponding number of degrees on the Equator from ♈ to f; and the more so, as they are the more oblique: and therefore the true Sun comes sooner to the Meridian whilst he is in the quadrant ♈ F, than the fictitious Sun does in the quadrant ♈ f; for which reason, the solar noon precedes noon by the Clock, until the real Sun comes to F, and the fictitious to f; which two points, being equidistant from the Meridian, both Suns will come to it precisely at noon by the Clock.
Whilst the real Sun describes the second quadrant of the Ecliptic FGHIKL from ♋ to ♎; he comes later to the Meridian every day, than the fictitious Sun moving through the second quadrant of the Equator from f to ♎; for the points at G, H, I, K, and L being farther from the Meridian than their corresponding points at g, h, i, k, and l, they must be later of coming to it: and as both Suns come at the same moment to the point ♎, they come to the Meridian at the moment of noon by the Clock.
In departing from Libra, through the third quadrant, the real Sun going through MNOPQ towards ♑ at R, and the fictitious Sun through mnopq towards r, the former comes to the Meridian every day sooner than the latter, until the real Sun comes to ♑, and the fictitious to r, and then they both come to the Meridian at the same time.
Lastly, as the real Sun moves equably through STUVW, from ♑ towards ♈; and the fictitious Sun through stuvw, from r towards ♈, the former comes later every day to the Meridian than the latter, until they both arrive at the point ♈, and then they make noon at the same time with the clock.
229. The annexed Table shews how much the Sun is faster or slower than the clock ought to be, so far as the difference depends upon the obliquity of the Ecliptic; of which the Signs of the first and third quadrants are at the head of the Table, and their Degrees at the left hand; and in these the Sun is faster than the Clock: the Signs of the second and fourth quadrants are at the foot of the Table, and their degrees at the right hand; in all which the Sun is slower than the Clock: so that entering the Table with the given Sign of the Sun’s place at the head of the Table, and the Degree of his place in that Sign at the left hand; or with the given Sign at the foot of the Table, and Degree at the right hand; in the Angle of meeting is the number of minutes and seconds that the Sun is faster or slower than the clock: or in other words, the quantity of time in which the real Sun, when in that part of the Ecliptic, comes sooner or later to the Meridian than the fictitious Sun in the Equator. Thus, when the Sun’s place is ♉ Taurus 12 degrees, he is 9 minutes 49 seconds faster than the clock; and when his place is ♋ Cancer 18 degrees, he is 6 minutes 2 seconds slower.
| Sun faster than the Clock in | |||||||
|---|---|---|---|---|---|---|---|
| Degrees | ♈ | ♉ | ♊ | 1st Q. | |||
| ♎ | ♏ | ♐ | 3d Q. | ||||
| ʹ | ʺ | ʹ | ʺ | ʹ | ʺ | Deg. | |
| 0 | 0 | 0 | 8 | 24 | 8 | 46 | 30 |
| 1 | 0 | 20 | 8 | 35 | 8 | 36 | 29 |
| 2 | 0 | 40 | 8 | 45 | 8 | 25 | 28 |
| 3 | 1 | 0 | 8 | 54 | 8 | 14 | 27 |
| 4 | 1 | 19 | 9 | 3 | 8 | 1 | 26 |
| 5 | 1 | 39 | 9 | 11 | 7 | 49 | 25 |
| 6 | 1 | 59 | 9 | 18 | 7 | 35 | 24 |
| 7 | 2 | 18 | 9 | 24 | 7 | 21 | 23 |
| 8 | 2 | 37 | 9 | 31 | 7 | 6 | 22 |
| 9 | 2 | 56 | 9 | 36 | 6 | 51 | 21 |
| 10 | 3 | 16 | 9 | 41 | 6 | 35 | 20 |
| 11 | 3 | 34 | 9 | 45 | 6 | 19 | 19 |
| 12 | 3 | 53 | 9 | 49 | 6 | 2 | 18 |
| 13 | 4 | 11 | 9 | 51 | 5 | 45 | 17 |
| 14 | 4 | 29 | 9 | 53 | 5 | 27 | 16 |
| 15 | 4 | 47 | 9 | 54 | 5 | 9 | 15 |
| 16 | 5 | 4 | 9 | 55 | 4 | 50 | 14 |
| 17 | 5 | 21 | 9 | 55 | 4 | 31 | 13 |
| 18 | 5 | 38 | 9 | 54 | 4 | 12 | 12 |
| 19 | 5 | 54 | 9 | 52 | 3 | 52 | 11 |
| 20 | 6 | 10 | 9 | 50 | 3 | 32 | 10 |
| 21 | 6 | 26 | 9 | 47 | 3 | 12 | 9 |
| 22 | 6 | 41 | 9 | 43 | 2 | 51 | 8 |
| 23 | 6 | 55 | 9 | 38 | 2 | 30 | 7 |
| 24 | 7 | 9 | 9 | 33 | 2 | 9 | 6 |
| 25 | 7 | 23 | 9 | 27 | 1 | 48 | 5 |
| 26 | 7 | 36 | 9 | 20 | 1 | 27 | 4 |
| 27 | 7 | 49 | 9 | 13 | 1 | 5 | 3 |
| 28 | 8 | 1 | 9 | 5 | 0 | 43 | 2 |
| 29 | 8 | 13 | 8 | 56 | 0 | 22 | 1 |
| 30 | 8 | 24 | 8 | 46 | 0 | 0 | 0 |
| 2d Q. | ♍ | ♌ | ♋ | Deg. | |||
| 4th Q. | ♓ | ♒ | ♑ | ||||
| Sun slower than the Clock in | |||||||
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.
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 No 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.
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.
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.
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.
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.
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.
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 | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| D. | 0 Signs | 1 | 2 | 3 | 4 | 5 | |||||||
| 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 | |||||||||||||
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.
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,