Part 12

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. [Illustration: Pl. VI.

_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.]