Part 27

339. Thus it appears, that from the vernal equinox to the autumnal, the North Pole is enlightened; and the Equator and all its parallels appear Semi-ellipses as seen from the Sun, more or less curved as the time is nearer to or farther from the Summer Solstice; and bending downwards or towards the South Pole; the reverse of which happens from the autumnal Equinox to the vernal. A little consideration will be sufficient to convince the reader, that the Earth’s Axis inclines towards the Sun at the Summer Solstice; from the Sun at the Winter Solstice; and sidewise to the Sun at the Equinoxes; but towards the right hand, as seen from the Sun at the vernal Equinox; and towards the left hand at the autumnal. From the Winter to the Summer Solstice, the Earth’s Axis inclines more or less to the right hand, as seen from the Sun; and the contrary from the Summer to the Winter Solstice.

[Sidenote: How these positions affect solar Eclipses.]

340. The different positions of the Earth’s Axis, as seen from the Sun at different times of the year, affect solar Eclipses greatly with regard to particular places; yea so far as would make central Eclipses which fall at one time of the year invisible if they fell at another, even though the Moon should always change in the Nodes and at the same hour of the day: of which indefinitely various affections, we shall only give Examples for the times of the Equinoxes and Solstices.

[Sidenote: Fig. IV.]

In the same Diagram, let _FG_ be part of the Ecliptic, and _IK_ _ik_ _ik_ _ik_ part of the Moon’s Orbit; both seen edgewise, and therefore projected into right lines; and let the intersections _N_, _O_, _D_, _E_ be one and the same Node at the above times, when the Earth has the forementioned different positions; and let the spaces included by the Circles _P_, _p_, _p_, _p_ be the Penumbra at these times, as its center is passing over the center of the Earth’s Disc. At the Winter Solstice, when the Earth’s Axis has the position _NNS_, the center of the Penumbra _P_ touches the Tropic of Capricorn _t_ in _N_ at the middle of the general Eclipse; but no part of the Penumbra touches the Tropic of Cancer _T_. At the Summer Solstice, when the Earth’s Axis has the position _NDS_ (_iDk_ being then part of the Moon’s Orbit whose Node is at _D_) the Penumbra _p_ has its center on the Tropic of Cancer _T_ at the middle of the general Eclipse, and then no part of it touches the Tropic of Capricorn _t_. At the autumnal Equinox the Earth’s Axis has the position _NOS_ (_iOk_ being then part of the Moon’s Orbit) and the Penumbra equally includes part of both Tropics _T_ and _t_ at the middle of the general Eclipse: at the vernal Equinox it does the same, because the Earth’s Axis has the position _NES_: But, in the former of these two last cases, the Penumbra enters the Earth at _A_, north of the Tropic of Cancer _T_, and leaves it at _m_, south of the Tropic of Capricorn _t_; having gone over the Earth obliquely southward, as its center described the line _AOm_: whereas in the latter case the Penumbra touches the Earth at _n_, south of the Equator _ÆQ_, and describing the line _nEq_ (similar to the former line _AOm_ in open space) goes obliquely northward over the Earth, and leaves it at _q_, north of the Equator.

In all these circumstances, the Moon has been supposed to change at noon in her descending Node: had she changed in her ascending Node, the Phenomena would have been as various the contrary way, with respect to the Penumbra’s going northward or southward over the Earth. But because the Moon changes at all hours, as often in one Node as the other, and at all distances from them both at different times as it happens, the variety of the Phases of Eclipses are almost innumerable, even at the same places, considering also how variously the same places are situated on the enlightened Disc of the Earth, with respect to the Penumbra’s motion, at the different hours that Eclipses happen.

[Sidenote: How much of the Penumbra falls on the Earth at different distances from the Nodes.]

341. When the Moon changes 17 degrees short of her descending Node, the Penumbra _P_ 18 just touches the northern part of the Earth’s Disc, near the North Pole _N_; and, as seen from that place the Moon appears to touch the Sun, but hides no part of him from sight. Had the Change been as far short of the ascending Node, the Penumbra would have touched the southern part of the Disc near the South Pole _S_. When the Moon changes 12 degrees short of the descending Node, more than a third part of the Penumbra _P 12_ falls on the northern parts of the Earth at the middle of the general Eclipse: had she changed as far past the same Node, as much of the other side of the Penumbra about _P_ would have fallen on the southern part of the Earth; all the rest in the _expansum_, or open space. When the Moon changes 6 degrees from the Node, almost the whole Penumbra _P6_ falls on the Earth at the middle of the general Eclipse. And lastly, when the Moon changes in the Node, the Penumbra _PN_ takes the longest course possible on the Earth’s Disc; its center falling on the middle thereof, at the middle of the general Eclipse. The farther the Moon changes from either Node within 17 degrees of it, the shorter is the Penumbra’s continuance on the Earth, because it goes over a less portion of the Disc, as is evident by the Figure.

[Sidenote: The Earth’s diurnal motion lengthens the duration of solar Eclipses, which fall without the polar Circles.]

342. The nearer that the Penumbra’s center is to the Equator at the middle of the general Eclipse, the longer is the duration of the Eclipse at all those places where it is central; because, the nearer that any place is to the Equator, the greater is the Circle it describes by the Earth’s motion on its Axis: and so, the place moving quicker keeps longer in the Penumbra whose motion is the same way with that of the place, tho’ faster as has been already mentioned § 337. Thus, (see the Earth at _D_ and the Penumbra at _12_) whilst the point _b_ in the polar Circle _abcd_ is carried from _b_ to _c_ by the Earth’s diurnal motion, the point _d_ on the Tropick of Cancer _T_ is carried a much greater length from _d_ to _D_: and therefore, if the Penumbra’s center goes one time over _c_ and another time over _D_, the Penumbra will be longer in passing over the moving place _d_ than it was in passing over the moving place _b_. Consequently, central Eclipses about the Poles are of the shortest duration; and about the Equator of the longest.

[Sidenote: And shortens the duration of some which fall within these Circles.]

343. In the middle of Summer the whole frigid Zone included by the polar Circle _abcd_ is enlightened; and if it then happens that the Penumbra’s center goes over the north Pole, the Sun will be eclipsed much the same number of Digits at _a_ as at _c_; but whilst the Penumbra moves eastward over _c_ it moves westward over _a_, because with respect to the Penumbra, the motions of _a_ and _c_ are contrary: for _c_ moves the same way with the Penumbra towards _d_, but _a_ moves the contrary way towards _b_; and therefore the Eclipse will be of longer duration at _c_ than at _a_. At _a_ the Eclipse begins on the Sun’s eastern limb, but at _c_ on his western: at all places lying without the polar Circles, the Sun’s Eclipses begin on his western limb, or near it, and end on or near his eastern. At those places where the Penumbra touches the Earth, the Eclipse begins with the rising Sun, on the top of his western or uppermost edge; and at those places where the Penumbra leaves the Earth, the Eclipse ends with the setting Sun, on the top of his eastern edge which is then the uppermost, just at its disappearing in the Horizon.

[Sidenote: The Moon has no Atmosphere.]

344. If the Moon were surrounded by an Atmosphere of any considerable Density, it would seem to touch the Sun a little before the Moon made her appulse to his edge, and we should see a little faintness on that edge before it were eclipsed by the Moon: But as no such faintness has been observed, at least so far as I ever heard, it seems plain, that the Moon has no such Atmosphere as that of the Earth. The faint ring of light surrounding the Sun in total Eclipses, called by CASSINI _la Chevelure du Soleil_, seems to be the Atmosphere of the Sun; because it has been observed to move equally with the Sun, not with the Moon.

[Sidenote: PLATE XI.]

345. Having been so prolix concerning Eclipses of the Sun, we shall drop that subject at present, and proceed to the doctrine of lunar Eclipses; which, being more simple, may be explained in less time.

[Sidenote: Eclipses of the Moon.

Fig. II.]

That the Moon can never be eclipsed but at the time of her being Full, and the reason why she is not eclipsed at every Full, have been shewn already § 316, 317. Let _S_ be the Sun, _E_ the Earth, _RR_ the Earth’s shadow, and _B_ the Moon in opposition to the Sun: in this situation the Earth intercepts the Sun’s light in its way to the Moon; and when the Moon touches the Earth’s shadow at _v_ she begins to be eclipsed on her eastern limb _x_, and continues eclipsed until her western limb _y_ leaves the shadow at _w_: at _B_ she is in the middle of the shadow, and consequently in the middle of the Eclipse.

[Sidenote: Why the Moon is visible in a total Eclipse.]

346. The Moon when totally eclipsed, is not invisible if she be above the Horizon and the Sky be clear; but appears generally of a dusky colour like tarnished copper, which some have thought to be the Moon’s native light. But the true cause of her being visible is the scattered beams of the Sun, bent into the Earth’s shadow by going through the Atmosphere; which, being more dense near the Earth than at considerable heights above it, refracts or bends the Sun’s rays more inward § 179, the nearer they are passing by the Earth’s surface, than those rays which go through higher parts of the Atmosphere, where it is less dense according to its height, until it be so thin or rare as to lose its refractive power. Let the Circle _fghi_, concentric to the Earth, include the Atmosphere whose refractive power vanishes at the heights _f_ and _i_; so that the rays _Wfw_ and _Viv_ go on straight without suffering the least refraction: But all those rays which enter the Atmosphere between _f_ and _k_, and between _i_ and _l_, on opposite sides of the Earth, are gradually more bent inward as they go through a greater portion of the Atmosphere, until the rays _Wk_ and _Vl_, touching the Earth at _m_ and _n_, are bent so as to meet at _q_, a little short of the Moon; and therefore the dark shadow of the Earth is contained in the space _moqpn_ where none of the Sun’s rays can enter: all the rest _RR_, being mixed by the scattered rays which are refracted as above, is in some measure enlightened by them; and some of those rays falling on the Moon give her the colour of tarnished copper, or of iron almost red hot. So that if the Earth had no Atmosphere, the Moon would be as invisible in total Eclipses as she is when New. If the Moon were so near the Earth as to go into its dark shadow, suppose about _po_, she would be invisible during her stay in it; but visible before and after in the fainter shadow _RR_.

[Sidenote: PLATE XI.

Why the Sun and Moon are sometimes visible when the Moon is totally eclipsed.]

347. When the Moon goes through the center of the Earth’s shadow she is directly opposite to the Sun: yet the Moon has been often seen totally eclipsed in the Horizon when the Sun was also visible in the opposite part of it: for, the horizontal refraction being almost 34 minutes of a degree § 181, and the diameter of the Sun and Moon being each at a mean state but 32 minutes, the refraction causes both Luminaries to appear above the Horizon when they are really below it § 179.

[Sidenote: Fig. V.

Duration of central Eclipses of the Moon.]

348. When the Moon is Full at 12 degrees from either of her Nodes, she just touches the Earth’s shadow but enters not into it. Let _GH_ be the Ecliptic, _ef_ the Moon’s Orbit where she is 12 degrees from the Node at her Full; _cd_ her Orbit where she is 6 degrees from the Node, _ab_ her Orbit where she is Full in the Node, _AB_ the Earth’s shadow, and _M_ the Moon. When the Moon describes the line _ef_ she just touches the shadow but does not enter into it; when she describes the line _cd_ she is totally though not centrally immersed in the shadow; and when she describes the line _ab_ she passes by the Node at _M_ in the center of the shadow, and takes the longest line possible, which is a diameter, through it: and such an Eclipse being both total and central is of the longest duration, namely, 3 hours 57 minutes 6 seconds from the beginning to the end, if the Moon be at her greatest distance from the Earth: and 3 hours 37 minutes 26 seconds, if she be at her least distance. The reason of this difference is, that when the Moon is farthest from the Earth she moves slowest; and when nearest to it, quickest.

[Sidenote: Digits.]

349. The Moon’s diameter, as well as the Sun’s, is supposed to be divided into twelve equal parts called _Digits_; and so many of these parts as are darkened by the Earth’s shadow, so many Digits is the Moon eclipsed. All that the Moon is eclipsed above 12 Digits, shew how far the shadow of the Earth is over the body of the Moon, on that edge to which she is nearest at the middle of the Eclipse.

[Sidenote: Why the beginning and end of a lunar Eclipse is so difficult to be determined by observation.]

350. It is difficult to observe exactly either the beginning or ending of a lunar Eclipse, even with a good Telescope; because the Earth’s shadow is so faint, and ill defined about the edges, that when the Moon is either just touching or leaving it, the obscuration of her limb is scarce sensible; and therefore the nicest observers can hardly be certain to four or five seconds of time. But both the beginning and ending of solar Eclipses are visibly instantaneous; for the moment that the edge of the Moon’s Disc touches the Sun’s, his roundness seems a little broke on that part; and the moment she leaves it he appears perfectly round again.

[Sidenote: The use of Eclipses in Astronomy, Geography, and Chronology.]

351. In Astronomy, Eclipses of the Moon are of great use for ascertaining the periods of her motions; especially such Eclipses as are observed to be alike in all circumstances, and have long intervals of time between them. In Geography, the Longitudes of places are found by Eclipses, as already shewn in the eleventh chapter: but for this purpose Eclipses of the Moon are more useful than those of the Sun, because they are more frequently visible, and the same lunar Eclipse is of equal largeness and duration at all places where it is seen. In Chronology, both solar and lunar Eclipses serve to determine exactly the time of any past event: for there are so many particulars observable in every Eclipse, with respect to its quantity, the places where it is visible (if of the Sun) and the time of the day or night; that ’tis impossible there can be two Eclipses in the course of many ages which are alike in all circumstances.

[Sidenote: The darkness at our SAVIOUR’s crucifixion supernatural.]

352. From the above explanation of the doctrine of Eclipses it is evident, that the darkness at our SAVIOUR’s crucifixion was supernatural. For he suffered on the next day after eating his last Passover-Supper, on which day it was impossible that the Moon’s shadow could fall on the Earth, for the _Jews_ kept the Passover at the time of Full Moon: nor does the darkness in total Eclipses of the Sun last four minutes in any place § 333, whereas the darkness at the crucifixion lasted three hours, _Matt._ xxviii. 15. and overspread at least all the land of _Judea_.

CHAP. XIX.

_The Calculation of New and Full Moons and Eclipses. The geometrical Construction of Solar and Lunar Eclipses. The examination of antient Eclipses._

353. To construct an Eclipse of the Sun, we must collect these ten Elements or Requisites from the following Astronomical Tables.

[Sidenote: Requisites for a solar Eclipse.]

I. The true time of conjunction of the Sun and Moon: to know at what conjunctions the Sun must be eclipsed; and to the times of those conjunctions,

II. The Moon’s horizontal parallax, or angle which the semi-diameter of the Earth subtends as seen from the Moon.

III. The Sun’s true place, and distance from the solstitial colure to which he is then nearest, either in coming to it or going from it.

IV. The Sun’s declination.

V. The angle of the Moon’s visible path with the Ecliptic.

VI. The Moon’s Latitude or Declination from the Ecliptic.

VII. The Moon’s true hourly motion from the Sun.

VIII. The Angle of the Sun’s semi-diameter as seen from the Earth.

IX. The Angle of the Moon’s semi-diameter as seen from the Earth.

X. The semi-diameter of the Penumbra.

And for an Eclipse of the Moon, the following Elements.

[Sidenote: Requisites for a lunar Eclipse.]

I. The true time of opposition of the Sun and Moon; and for that time,

II. The Moon’s horizontal parallax.

III. The Sun’s semi-diameter.

IV. The semi-diameter of the Earth’s shadow.

V. The Moon’s semi-diameter.

VI. The Moon’s Latitude.

VII. The Moon’s true hourly motion from the Sun.

VIII. The Angle of the Moon’s visible path with the Ecliptic.

These Elements are easily found from the following Tables and Precepts, by the common Rules of Arithmetic.

_Note_, 60 minutes make a Degree, 30 degrees a Sign, and 12 Signs a Circle. A Sign is marked thus ^s, a Degree thus °, and a Minute thus ʹ.

When you exceed 12 Signs, always reject them and set down the remainder. When the number of Signs to be subtracted is greater than the number you subtract from, add 12 Signs to that which you subtract from; and then you will have a remainder to set down.

[Sidenote: How the Signs are reckoned.]

354. As we fix arbitrarily upon the beginning of the Sign _Aries_ to reckon from, when we speak of the places of the Sun, Moon, and Nodes; we call _Aries_ 0 Signs, _Taurus_ 1 Sign, _Gemini_ 2 Signs, _Cancer_ 3 Signs, _&c._ So, when the Sun is in the 10th degree of Aries, we say his Place or Longitude is 0 Signs 10 Degrees, because he is only 10 Degrees from the beginning of Aries: if he is in the 5th, 10th, _&c._ Degree of Taurus, we say his Place or Longitude is 1 Sign, 5, 10, _&c._ Degrees: and so on, till he comes quite round again. But in reckoning the Anomalies of the Sun and Moon, and their distance from the Nodes, we only consider the number of Signs and Degrees the Luminaries are gone past their Apogee or Nodes; not how far they have to go to these points, were the distance ever so little. The Sun, Moon, and Apogee move according to the order of Signs, but the Nodes contrary. We shall now give the Precepts and Examples for the above Requisites in their due order.

_To calculate the time of New and Full Moon._

[Sidenote: First Element or Requisite.]

355. PRECEPT I. For any proposed year in the 18th Century, take out the mean time of the New Moon in _March_ from Table I., and the mean time of Full Moon from Table III., for the _Old Stile_; or from Tables II and IV for _New Stile_; with the mean Anomalies of the Sun and Moon for these times, and set them by themselves. Then, from Table VI, take out as many Lunations as the proposed Month is after _March_, with the days, hours, and minutes belonging to them; and also the mean Anomalies of the Sun and Moon for these Lunations.

II. Add the days, hours, and minutes of these Lunations to the time of New or Full Moon in _March_, and the Anomalies for the Lunations to the Anomalies for _March_: the sums give the hours and minutes of the mean New or Full Moon required, and the mean Anomalies of the Sun and Moon for that time.

III. Then, with the number of days enter Table VII, under the given Month, and right against this number, in the left hand column you have the day of New or Full Moon; which set before the hours and minutes above-mentioned.

IV. But, (as it will sometimes happen) if the number of days fall short of all those under the given Month, add one Lunation with its Anomalies from Table VI to the foresaid sums; so you will have a new sum of days wherewith to enter the 7th Table under the given Month, where you are sure to find that sum the second time, if the first falls short.

V. With the Signs and Degrees of the Sun’s Anomaly enter Table VIII, _The Moon’s annual Equation_, and take out the minutes of time of that Equation by the Anomaly; remembring, that if the Signs are at the head of the Table, the degrees are at the left hand, in which case the Equation found in the Angle of meeting must be subtracted from the mean time of New or Full Moon, as the title _Subtract_, at the head of the Table directs: but if the Signs are at the foot of the Table their degrees are in the right-hand column, and the Equation where the Signs and Degrees meet in the Table is to be added to the mean time, as the title _Add_, at the foot of the Table directs; which Equation, so applied, gives the mean time of New or Full Moon corrected.

VI. With the Signs and Degrees of the Sun’s Anomaly enter Table IX, _Equation of the Moon’s mean Anomaly_, and take out the Equation thereof; adding it to the mean Anomaly or subtracting it therefrom, as the titles at the head or foot of the Table direct; and it gives the mean Anomaly corrected. Then, with the Sun’s Anomaly enter Table XII, _Equation of the Sun’s mean Place_, and take out that Equation, applying it to the Moon’s corrected Anomaly as the titles direct; and it will give the Moon’s Anomaly equated[77]. _N. B._ In all these Equations, care must be taken to make proper allowance for the odd minutes of Anomaly; the Tables having the Equations only for compleat Degrees.

VII. With the Moon’s equated Anomaly enter Table X, _The Moon’s elliptic Equation_, and take out that Equation in the same manner as the preceding: adding it to the former corrected time if the Signs be at the head of the Table, or subtracting it if they be at the foot, as the Table directs; and this gives the mean time equated.

VIII. Lastly, enter Table XI, _The Sun’s Equation at New and Full Moon_, with the Sun’s Anomaly, and take out the Sun’s Equation in the same manner as the others; adding it to, or subtracting it from the former equated time, as the titles direct: and by this last Equation you have the true time of New or Full Moon, agreeing with well regulated Clocks and Watches. But to make it agree with true Sun-Dials, the Equation of time must be applied as taught § 225.

EXAMPLE I.

_To find the time of New Moon in_ April 1764, _N. S._