Chapter VI: Part 6

217. To illustrate this progressive motion of light, let _A_ and _B_ be the Earth in two different parts of it’s Orbit, whose distance is 81 millions of miles, equal to the Earth’s distance from the Sun _S_. It is plain, that if the motion of light were instantaneous, the Satellite 1 would appear to enter into Jupiter’s shadow _FF_ at the same moment of time to a spectator in _A_ as to another in _B_. But by many years observations it has been found, that the immersion of the Satellite into the shadow is seen 8-1/4 minutes sooner when the Earth is at _B_, than when it is at _A_. And so, as Mr. ROMER first discovered, the motion of light is thereby proved to be progressive, and not instantaneous, as was formerly believed. It is easy to compute in what time the Earth moves from _A_ to _B_; for the chord of 60 degrees of any Circle is equal to the Semidiameter of that Circle; and as the Earth goes through all the 360 degrees of it’s Orbit in a year, it goes through 60 of those degrees in about 61 days. Therefore, if on any given day, suppose the first of _June_, the Earth is at _A_, on the first of _August_ it will be at _B_: the chord, or straight line _AB_, being equal to _DS_ the Radius of the Earth’s Orbit, the same with _AS_ it’s distance from the Sun.

218. As the Earth moves from _D_ to _C_, through the side _AB_ of it’s Orbit, it is constantly meeting the light of Jupiter’s Satellites sooner, which occasions an apparent acceleration of their Eclipses: and as it moves through the other half _H_ of it’s Orbit, from _C_ to _D_, it is receding from their light, which occasions an apparent retardation of their Eclipses, because their light is then longer ere it overtakes the Earth.

219. That these accelerations of the immersions of Jupiter’s Satellites into his shadow, as the Earth approaches towards Jupiter, and the retardations of their emersions out of his shadow, as the Earth is going from him, are not occasioned by any inequality arising from the motions of the Satellites in excentric Orbits, is plain, because it affects them all alike, in whatever parts of their Orbits they are eclipsed. Besides, they go often round their Orbits every year, and their motions are no way commensurate to the Earth’s. Therefore, a Phenomenon not to be accounted for from the real motions of the Satellites, but so easily deducible from the Earth’s motion, and so answerable thereto, must be allowed to result from it. This affords one very good proof of the Earth’s annual motion.

220. TABLES for converting mean solar TIME into Degrees and Parts of the
terrestrial EQUATOR; and also for converting Degrees and Parts of the
EQUATOR into mean solar Time.

+---------------------------------------------+ | TABLE I. For converting Time into | | Degrees and Parts of the Equator. | +-----+-------+-----+---------+-----+---------+ | | | *M. | D. M. | *M. | D. M. | |Hours|Degrees| S. | M. S. | S. | M. S. | | | | T. | S. T. | T. | S. T. | +-----+-------+-----+---------+-----+---------+ | 1 | 15 | 1 | 0 15 | 31 | 7 45 | | 2 | 30 | 2 | 0 30 | 32 | 8 0 | | 3 | 45 | 3 | 0 45 | 33 | 8 15 | | 4 | 60 | 4 | 1 0 | 34 | 8 30 | | 5 | 75 | 5 | 1 15 | 35 | 8 45 | +-----+-------+-----+---------+-----+---------+ | 6 | 90 | 6 | 1 30 | 36 | 9 0 | | 7 | 105 | 7 | 1 45 | 37 | 9 15 | | 8 | 120 | 8 | 2 0 | 38 | 9 30 | | 9 | 135 | 9 | 2 15 | 39 | 9 45 | | 10 | 150 | 10 | 2 30 | 40 | 10 0 | +-----+-------+-----+---------+-----+---------+ | 11 | 165 | 11 | 2 45 | 41 | 10 15 | | 12 | 180 | 12 | 3 0 | 42 | 10 30 | | 13 | 195 | 13 | 3 15 | 43 | 10 45 | | 14 | 210 | 14 | 3 30 | 44 | 11 0 | | 15 | 225 | 15 | 3 45 | 45 | 11 15 | +-----+-------+-----+---------+-----+---------+ | 16 | 240 | 16 | 4 0 | 46 | 11 30 | | 17 | 255 | 17 | 4 15 | 47 | 11 45 | | 18 | 270 | 18 | 4 30 | 48 | 12 0 | | 19 | 285 | 19 | 4 45 | 49 | 12 15 | | 20 | 300 | 20 | 5 0 | 50 | 12 30 | +-----+-------+-----+---------+-----+---------+ | 21 | 315 | 21 | 5 15 | 51 | 12 45 | | 22 | 330 | 22 | 5 30 | 52 | 13 0 | | 23 | 345 | 23 | 5 45 | 53 | 13 15 | | 24 | 360 | 24 | 6 0 | 54 | 13 30 | | 25 | 375 | 25 | 6 15 | 55 | 13 45 | +-----+-------+-----+---------+-----+---------+ | 26 | 390 | 26 | 6 30 | 56 | 14 0 | | 27 | 405 | 27 | 6 45 | 57 | 14 15 | | 28 | 420 | 28 | 7 0 | 58 | 14 30 | | 29 | 435 | 29 | 7 15 | 59 | 14 45 | | 30 | 450 | 30 | 7 30 | 60 | 15 0 | +-----+-------+-----+---------+-----+---------+

+---------------------------------------------------+ | TABLE II. For converting Degrees and | | Parts of the Equator into Time. | +-----+--------+-----+--------+-------+-----+-------+ | *D. | H. M. | *D. | H. M. | | | | | M. | M. S. | M. | M. S. |Degrees|Hours|Minutes| | S. | S. T. | S. | S. T. | | | | +-----+--------+-----+--------+-------+-----+-------+ | 1 | 0 4 | 31 | 2 4 | 70 | 4 | 40 | | 2 | 0 8 | 32 | 2 8 | 80 | 5 | 20 | | 3 | 0 12 | 33 | 2 12 | 90 | 6 | 0 | | 4 | 0 16 | 34 | 2 16 | 100 | 6 | 40 | | 5 | 0 20 | 35 | 2 20 | 110 | 7 | 20 | +-----+--------+-----+--------+-------+-----+-------+ | 6 | 0 24 | 36 | 2 24 | 120 | 8 | 0 | | 7 | 0 28 | 37 | 2 28 | 130 | 8 | 40 | | 8 | 0 32 | 38 | 2 32 | 140 | 9 | 20 | | 9 | 0 36 | 39 | 2 36 | 150 | 10 | 0 | | 10 | 0 40 | 40 | 2 40 | 160 | 10 | 40 | +-----+--------+-----+--------+-------+-----+-------+ | 11 | 0 44 | 41 | 2 44 | 170 | 11 | 20 | | 12 | 0 48 | 42 | 2 48 | 180 | 12 | 0 | | 13 | 0 52 | 43 | 2 52 | 190 | 12 | 40 | | 14 | 0 56 | 44 | 2 56 | 200 | 13 | 20 | | 15 | 1 0 | 45 | 3 0 | 210 | 14 | 0 | +-----+--------+-----+--------+-------+-----+-------+ | 16 | 1 4 | 46 | 3 4 | 220 | 14 | 40 | | 17 | 1 8 | 47 | 3 8 | 230 | 15 | 20 | | 18 | 1 12 | 48 | 3 12 | 240 | 16 | 0 | | 19 | 1 16 | 49 | 3 16 | 250 | 16 | 40 | | 20 | 1 20 | 50 | 3 20 | 260 | 17 | 20 | +-----+--------+-----+--------+-------+-----+-------+ | 21 | 1 24 | 51 | 3 24 | 270 | 18 | 0 | | 22 | 1 28 | 52 | 3 28 | 280 | 18 | 40 | | 23 | 1 32 | 53 | 3 32 | 290 | 19 | 20 | | 24 | 1 36 | 54 | 3 36 | 300 | 20 | 0 | | 25 | 1 40 | 55 | 3 40 | 310 | 20 | 40 | +-----+--------+-----+--------+-------+-----+-------+ | 26 | 1 44 | 56 | 3 44 | 320 | 21 | 20 | | 27 | 1 48 | 57 | 3 48 | 330 | 22 | 0 | | 28 | 1 52 | 58 | 3 52 | 340 | 22 | 40 | | 29 | 1 56 | 59 | 3 56 | 350 | 23 | 20 | | 30 | 2 0 | 60 | 4 0 | 360 | 24 | 0 | +-----+--------+-----+--------+-------+-----+-------+

These are the Tables mentioned in the 208th Article, and are so easy that they scarce require any farther explanation than to inform the reader, that if, in Table I. he reckons the columns marked with Asterisks to be minutes of time, the other columns give the equatoreal parts or motion in degrees and minutes; if he reckons the Asterisk columns to be seconds, the others give the motion in minutes and seconds of the Equator; if thirds, in seconds and thirds: And if in Table II. he reckons the Asterisk columns to be degrees of motion, the others give the time answering thereto in hours and minutes; if minutes of motion, the time is minutes and seconds; if seconds of motion, the corresponding time is given in seconds and thirds. An example in each case will make the whole very plain.

EXAMPLE I. | EXAMPLE II. | In 10 hours 15 minutes 24 | In what time will 153 degrees seconds 20 thirds, _Qu._ How | 51 minutes 5 seconds of the much of the Equator revolves | Equator revolve through the through the Meridian? | Meridian? | | Deg. M. S. | H. M. S. T. Hours 10 150 0 0 | Deg. { 150 10 0 0 0 Min. 15 3 45 0 | { 3 12 0 0 Sec. 24 6 0 | Min. 51 3 24 0 Thirds 20 5 | Sec. 5 20 ------------ | ------------ _Answer_ 153 51 5 | _Answer_ 10 15 24 20

CHAP. XII.

_Of Solar and Sidereal Time._

[Sidenote: Sidereal days shorter than solar days, and why.]

221. The fixed Stars appear to go round the Earth in 23 hours 56 minutes 4 seconds, and the Sun in 24 hours: so that the Stars gain three minutes 56 seconds upon the Sun every day, which amounts to one diurnal revolution in a year; and therefore, in 365 days as measured by the returns of the Sun to the Meridian, there are 366 days as measured by the Stars returning to it: the former are called _Solar Days_, and the latter _Sidereal_.

[Sidenote: PLATE III.]

The diameter of the Earth’s Orbit is but a physical point in proportion to the distance of the Stars; for which reason, and the Earth’s uniform motion on it’s Axis, any given Meridian will revolve from any Star to the same Star again in every absolute turn of the Earth on it’s Axis, without the least perceptible difference of time shewn by a clock which goes exactly true.

If the Earth had only a diurnal motion, without an annual, any given Meridian would revolve from the Sun to the Sun again in the same quantity of time as from any Star to the same Star again; because the Sun would never change his place with respect to the Stars. But, as the Earth advances almost a degree eastward in it’s Orbit in the time that it turns eastward round its Axis, whatever Star passes over the Meridian on any day with the Sun, will pass over the same Meridian on the next day when the Sun is almost a degree short of it; that is, 3 minutes 56 seconds sooner. If the year contained only 360 days as the Ecliptic does 360 degrees, the Sun’s apparent place, so far as his motion is equable, would change a degree every day; and then the sidereal days would be just four minutes shorter than the solar.

[Sidenote: Fig. II.]

Let _ABCDEFGHIKLM_ be the Earth’s Orbit, in which it goes round the Sun every year, according to the order of the letters, that is, from west to east, and turns round it’s Axis the same way from the Sun to the Sun again every 24 hours. Let _S_ be the Sun, and _R_ a fixed Star at such an immense distance that the diameter of the Earth’s Orbit bears no sensible proportion to that distance. Let _Nm_ be any particular Meridian of the Earth, and _N_ a given point or place upon that Meridian. When the Earth is at _A_, the Sun _S_ hides the Star _R_, which would always be hid if the Earth never removed from _A_; and consequently, as the Earth turns round it’s Axis, the point _N_ would always come round to the Sun and Star at the same time. But when the Earth has advanced, suppose a twelfth part of it’s Orbit from _A_ to _B_, it’s motion round it’s Axis will bring the point _N_ a twelfth part of a day or two hours sooner to the Star than to the Sun; for the Angle _NBn_ is equal to the Angle _ASB_: and therefore, any Star which comes to the Meridian at noon with the Sun when the Earth is at _A_, will come to the Meridian at 10 in the forenoon when the Earth is at _B_. When the Earth comes to _C_ the point _N_ will have the Star on it’s Meridian at 8 in the morning, or four hours sooner than it comes round to the Sun; for it must revolve from _N_ to _n_, before it has the Sun in it’s Meridian. When the Earth comes to _D_, the point _N_ will have the Star on it’s Meridian at six in the morning, but that point must revolve six hours more from _N_ to _n_, before it has mid-day by the Sun: for now the Angle _ASD_ is a right Angle, and so is _NDn_; that is, the Earth has advanced 90 degrees in it’s Orbit, and must turn 90 degrees on its Axis to carry the point _N_ from the Star to the Sun: for the Star always comes to the Meridian when _Nm_ is parallel to _RSA_; because _DS_ is but a point in respect of _RS_. When the Earth is at _E_, the Star comes to the Meridian at 4 in the morning; at _F_, at two in the morning; and at _G_, the Earth having gone half round it’s Orbit, _N_ points to the Star _R_ at midnight, being then directly opposite to the Sun; and therefore, by the Earth’s diurnal motion the Star comes to the Meridian 12 hours before the Sun. When the Earth is at _H_, the Star comes to the Meridian at 10 in the evening; at _I_ it comes to the Meridian at 8, that is, 16 hours before the Sun; at _K_ 18 hours before him; at _L_ 20 hours; at _M_ 22; and at _A_ equally with the Sun again.

A TABLE, shewing how much of the Celestial Equator passes over the
Meridian in any part of a mean SOLAR DAY; and how much the FIXED STARS
gain upon the mean SOLAR TIME every Day, for a Month.

+-----+-----------+-----+------------+-----+------------+ | Time| Motion. | Time| Motion. |Time | Motion. | | | | | | | | +-----+-----------+-----+------------+-----+------------+ |Hours| D. M. S. | *M. | D. M. S. | *M. | D. M. S. | | | | S. | M. S. T. | S. | M. S. T. + | | | T. | S. T. ʺʺ | T. | S. T. ʺʺ | +-----+-----------+-----+------------+-----+------------+ | 1 | 15 2 28 | 1 | 0 15 2 | 31 | 7 46 16 | | 2 | 30 4 56 | 2 | 0 30 5 | 32 | 8 1 19 | | 3 | 45 7 24 | 3 | 0 45 7 | 33 | 8 16 21 | | 4 | 60 9 51 | 4 | 1 0 10 | 34 | 8 31 24 | | 5 | 75 12 19 | 5 | 1 15 12 | 35 | 8 46 26 | +-----+-----------+-----+------------+-----+------------+ | 6 | 90 14 47 | 6 | 1 30 15 | 36 | 9 1 29 | | 7 | 105 17 15 | 7 | 1 45 17 | 37 | 9 16 31 | | 8 | 120 19 43 | 8 | 2 0 20 | 38 | 9 31 34 | | 9 | 135 22 11 | 9 | 2 15 22 | 39 | 9 46 36 | | 10 | 150 24 38 | 10 | 2 30 25 | 40 | 10 1 39 | +-----+-----------+-----+------------+-----+------------+ | 11 | 165 27 6 | 11 | 2 45 27 | 41 | 10 16 41 | | 12 | 180 29 34 | 12 | 3 0 30 | 42 | 10 31 43 | | 13 | 195 32 2 | 13 | 3 15 32 | 43 | 10 46 46 | | 14 | 210 34 30 | 14 | 3 30 34 | 44 | 11 1 48 | | 15 | 225 36 58 | 15 | 3 45 37 | 45 | 11 16 51 | +-----+-----------+-----+------------+-----+------------+ | 16 | 240 39 26 | 16 | 4 0 39 | 46 | 11 31 53 | | 17 | 255 41 53 | 17 | 4 15 41 | 47 | 11 46 56 | | 18 | 270 44 21 | 18 | 4 30 44 | 48 | 12 1 58 | | 19 | 285 46 49 | 19 | 4 45 47 | 49 | 12 17 1 | | 20 | 300 49 17 | 20 | 5 0 49 | 50 | 12 32 3 | +-----+-----------+-----+------------+-----+------------+ | 21 | 315 51 45 | 21 | 5 15 52 | 51 | 12 47 6 | | 22 | 330 54 13 | 22 | 5 30 54 | 52 | 13 2 8 | | 23 | 345 56 40 | 23 | 5 45 57 | 53 | 13 17 11 | | 24 | 360 59 8 | 24 | 6 0 59 | 54 | 13 32 13 | | 25 | 376 1 36 | 25 | 6 16 2 | 55 | 13 47 16 | +-----+-----------+-----+------------+-----+------------+ | 26 | 391 4 4 | 26 | 6 31 4 | 56 | 14 2 18 | | 27 | 406 6 32 | 27 | 6 46 7 | 57 | 14 17 21 | | 28 | 421 9 0 | 28 | 7 1 9 | 58 | 14 32 23 | | 29 | 436 11 28 | 29 | 7 16 11 | 59 | 14 47 26 | | 30 | 451 13 56 | 30 | 7 31 14 | 60 | 15 2 28 | +-----+-----------+-----+------------+-----+------------+

Accelerations of the Fixed Stars. +----+----------+ | D. | H. M. S. | +----+----------+ | 1 | 0 3 56 | | 2 | 0 7 52 | | 3 | 0 11 48 | | 4 | 0 15 44 | | 5 | 0 19 39 | +----+----------+ | 6 | 0 23 35 | | 7 | 0 27 31 | | 8 | 0 31 27 | | 9 | 0 35 23 | | 10 | 0 39 19 | +----+----------+ | 11 | 0 43 15 | | 12 | 0 47 11 | | 13 | 0 51 7 | | 14 | 0 55 3 | | 15 | 0 58 58 | +----+----------+ | 16 | 1 2 54 | | 17 | 1 6 50 | | 18 | 1 10 46 | | 19 | 1 14 42 | | 20 | 1 18 38 | +----+----------+ | 21 | 1 22 34 | | 22 | 1 26 30 | | 23 | 1 30 26 | | 24 | 1 34 22 | | 25 | 1 38 17 | +----+----------+ | 26 | 1 42 13 | | 27 | 1 46 9 | | 28 | 1 50 5 | | 29 | 1 54 1 | | 30 | 1 57 57 | +----+----------+

[Sidenote: PLATE III.

An absolute Turn of the Earth on it’s Axis never finishes a
solar day.

Fig. II.]

222. Thus it is plain, that an absolute turn of the Earth on it’s Axis (which is always completed when the same Meridian comes to be parallel to it’s situation at any time of the day before) never brings the same Meridian round from the Sun to the Sun again; but that the Earth requires as much more than one turn on it’s Axis to finish a natural day, as it has gone forward in that time; which, at a mean state is a 365th part of a Circle. Hence, in 365 days the Earth turns 366 times round it’s Axis; and therefore, as a turn of the Earth on it’s Axis compleats a sidereal day, there must be one sidereal day more in a year than the number of solar days, be the number what it will, on the Earth, or any other Planet. One turn being lost with respect to the number of solar days in a year, by the Planet’s going round the Sun; just as it would be lost to a traveller, who, in going round the Earth, would lose one day by following the apparent diurnal motion of the Sun: and consequently, would reckon one day less at his return (let him take what time he would to go round the Earth) than those who remained all the while at the place from which he set out. So, if there were two Earths revolving equably on their Axes, and if one remained at _A_ until the other travelled round the Sun from _A_ to _A_ again, _that_ Earth which kept it’s place at _A_ would have it’s solar and sidereal days always of the same length; and so, would have one solar day more than the other at it’s return. Hence, if the Earth turned but once round it’s Axis in a year, and if _that_ turn was made the same way as the Earth goes round the Sun, there would be continual day on one side of the Earth, and continual night on the other.

[Sidenote: To know by the Stars whether a Clock goes true or not.]

223. The first part of the preceding Table shews how much of the celestial Equator passes over the Meridian in any given part of a mean solar day, and is to be understood the same way as the Table in the 220th article. The latter part, intitled, _Accelerations of the fixed Stars_, affords us an easy method of knowing whether or no our clocks and watches go true: For if, through a small hole in a window-shutter, or in a thin plate of metal fixed to a window, we observe at what time any Star disappears behind a chimney, or corner of a house, at a little distance; and if the same Star disappears the next night 3 minutes 56 seconds sooner by the clock or watch; and on the second night, 7 minutes 52 seconds sooner; the third night 11 minutes 48 seconds sooner; and so on, every night, as in the Table, which shews this difference for 30 natural days, it is an infallible Sign that the machine goes true; otherwise it does not go true; and must be regulated accordingly: and as the disappearing of a Star is instantaneous, we may depend on this information to half a second.

_J. Ferguson inv. et delin._ _J. Mynde Sc._]

CHAP. XIII.

_Of the Equation of Time._

[Sidenote: The Sun and Clocks equal only on four days of the year.]

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.

[Sidenote: Use of the Equation Table.]

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.

[Sidenote: How to draw a Meridian Line.]

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.

[Sidenote: Equation of natural days explained.]

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.

[Sidenote: PLATE VI.

The first part of the Equation of time.]

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.

[Sidenote: Fig. III.]

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.

[Sidenote: A Table of the Equation of Time depending on the Sun’s place
in the Ecliptic.

PLATE VI.]

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_ | +---------+--------+--------+--------+--------+ | | ♈ | ♉ | ♊ | 1st Q. | | | ♎ | ♏ | ♐ | 3d Q. | + +--------+--------+--------+--------+ | Degrees | 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_ | +---------------------------------------------+

[Sidenote: Fig. III.]

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.