Astronomy Explained Upon Sir Isaac Newton's Principles

Chapter XIV: Part 14

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+------+-----+----------+------------+ | Aft. | | Months | Time of | | Chr. | | and | the Day | | | | Days. | or Night. | +------+-----+----------+------------+ | 1701 | 🌑︎ | Feb. 22 | 11 A. | | 1703 | 🌑︎ | Jan. 3 | 7 M. | | 1703 | 🌑︎ | June 29 | 1 M. _T._ | | 1703 | 🌑︎ | Dec. 23 | 7 M. _T._ | | 1704 | 🌑︎ | Dec. 11 | 7 M. | | 1706 | 🌑︎ | Apr. 28 | 2 M. | | 1706 | ☉ | May 12 | 10 M. | | 1706 | 🌑︎ | Oct. 21 | 7 A. | | 1707 | 🌑︎ | Apr. 17 | 2 M. _T._ | | 1708 | 🌑︎ | April 5 | 6 M. | | 1708 | ☉ | Dec. 14 | 8 M. | | 1708 | 🌑︎ | Sept. 29 | 9 A. | | 1709 | ☉ | Mar. 11 | 2 A. | | 1710 | 🌑︎ | Feb. 13 | 11 A. | | 1710 | ☉ | Feb. 28 | 1 A. | | 1711 | ☉ | July 15 | 8 A. | | 1711 | 🌑︎ | July 29 | 6 A. _T._ | | 1712 | 🌑︎ | Jan. 23 | 8 A. | | 1713 | 🌑︎ | June 8 | 6 A. | | 1713 | 🌑︎ | Dec. 2 | 4 M. | | 1715 | ☉ | May 3 | 9 M. _T._ | | 1715 | 🌑︎ | Nov. 11 | 5 M. | | 1717 | 🌑︎ | Mar. 27 | 3 M. | | 1717 | 🌑︎ | May 20 | 6 A. | | 1718 | 🌑︎ | Sept. 9 | 8 A. _T._ | | 1719 | 🌑︎ | Aug. 29 | 9 A. | | 1721 | 🌑︎ | Jan. 13 | 3 A. | | 1722 | 🌑︎ | June 29 | 3 M. | | 1722 | ☉ | Dec. 8 | 3 A. | | 1722 | 🌑︎ | Dec. 22 | 4 A. | | 1724 | ☉ | May 22 | 7 A. _T._ | | 1724 | 🌑︎ | Nov. 1 | 4 M. | | 1725 | 🌑︎ | Oct. 21 | 7 A. | | 1726 | ☉ | Sept. 25 | 6 A. | | 1726 | 🌑︎ | Oct. 11 | 5 M. | | 1727 | ☉ | Sept. 15 | 7 M. | | 1729 | 🌑︎ | Feb. 13 | 9 A. _T._ | | 1729 | 🌑︎ | Aug. 9 | 1 M. | | 1730 | 🌑︎ | Feb. 4 | 4 M. | | 1731 | 🌑︎ | June 20 | 2 M. | | 1732 | 🌑︎ | Dec. 1 | 10 A. _T._ | | 1733 | ☉ | May 13 | 7 A. | | 1733 | 🌑︎ | May 28 | 7 A. | | 1735 | 🌑︎ | Oct. 2 | 1 M. | | 1736 | 🌑︎ | Mar. 26 | 12 A. _T._ | | 1736 | 🌑︎ | Sept. 20 | 3 M. _T._ | | 1736 | ☉ | Oct. 4 | 6 A. | | 1737 | ☉ | Mar. 1 | 4 A. | | 1737 | 🌑︎ | Sept. 9 | 4 M. | | 1738 | ☉ | Aug. 15 | 11 M. | | 1739 | 🌑︎ | Jan. 24 | 11 A. | | 1739 | ☉ | Aug. 4 | 5 A. | | 1739 | ☉ | Dec. 30 | 9 M. | | 1740 | 🌑︎ | Jan. 13 | 11 A. _T._ | | 1741 | 🌑︎ | Jan. 1 | 12 A. | | 1743 | 🌑︎ | Nov. 2 | 3 M. _T._ | | 1744 | 🌑︎ | Aug. 26 | 9 A. | | 1746 | 🌑︎ | Aug. 30 | 12 A. | | 1747 | 🌑︎ | Feb. 14 | 5 M. _T._ | | 1748 | ☉ | July 25 | 11 M. | | 1748 | 🌑︎ | Aug. 8 | 12 A. | | 1749 | 🌑︎ | Dec. 23 | 8 A. | | 1750 | ☉ | Jan. 8 | 9 M. | | 1750 | 🌑︎ | June 19 | 9 A. _T._ | | 1750 | 🌑︎ | Dec. 13 | 7 M. | | 1751 | 🌑︎ | June 9 | 2 M. | | 1751 | 🌑︎ | Dec. 2 | 10 A. | | 1752 | ☉ | May 13 | 8 A. | | 1753 | 🌑︎ | Apr. 17 | 7 A. | | 1753 | ☉ | Oct. 26 | 10 M. | | 1755 | 🌑︎ | Mar. 28 | 1 M. | | 1757 | 🌑︎ | Feb. 4 | 6 M. | | 1757 | 🌑︎ | July 30 | 12 A. | | 1758 | 🌑︎ | Jan. 24 | 7 M. _T._ | | 1758 | ☉ | Dec. 30 | 7 M. | | 1759 | ☉ | June 24 | 7 A. | | 1759 | ☉ | Dec. 19 | 2 A. | | 1760 | 🌑︎ | May 29 | 9 A. | | 1760 | ☉ | June 13 | 7 M. | | 1760 | 🌑︎ | Nov. 22 | 9 A. | | 1761 | 🌑︎ | May 18 | 11 A. _T._ | | 1762 | 🌑︎ | May 8 | 4 M. | | 1762 | ☉ | Oct. 17 | 8 M. | | 1762 | 🌑︎ | Nov. 1 | 8 A. | | 1763 | ☉ | Apr. 13 | 8 M. | | 1764 | ☉ | Apr. 1 | 10 M. | | 1764 | 🌑︎ | Apr. 16 | 1 M. | | 1765 | ☉ | Mar. 21 | 2 A. | | 1765 | ☉ | Aug. 16 | 5 A. | | 1766 | 🌑︎ | Feb. 24 | 7 A. | | 1766 | ☉ | Aug. 5 | 7 A. | | 1768 | 🌑︎ | Jan. 4 | 5 M. | | 1768 | 🌑︎ | June 30 | 4 M. _T._ | | 1768 | 🌑︎ | Dec. 23 | 4 A. _T._ | | 1769 | ☉ | June 4 | 8 M. | | 1769 | 🌑︎ | Dec. 13 | 7 M. | | 1770 | ☉ | Nov. 17 | 10 M. | | 1771 | 🌑︎ | Apr. 28 | 2 M. | | 1771 | 🌑︎ | Oct. 23 | 5 A. | | 1772 | 🌑︎ | Oct. 11 | 6 A. _T._ | | 1772 | ☉ | Oct. 26 | 10 M. | | 1773 | ☉ | Mar. 23 | 5 M. | | 1773 | 🌑︎ | Sept. 30 | 7 A. | | 1774 | ☉ | Mar. 12 | 10 M. | | 1776 | 🌑︎ | July 31 | 1 M. _T._ | | 1776 | ☉ | Aug. 14 | 5 M. | | 1777 | ☉ | Jan. 9 | 5 A. | | 1778 | ☉ | June 24 | 4 A. | | 1778 | 🌑︎ | Dec. 4 | 6 M. | | 1779 | 🌑︎ | May 30 | 5 M. _T._ | | 1779 | ☉ | June 14 | 8 M. | | 1779 | 🌑︎ | Nov. 23 | 8 A. | | 1780 | ☉ | Oct. 27 | 6 A. | | 1780 | 🌑︎ | Nov. 12 | 4 M. | | 1781 | ☉ | Apr. 23 | 6 A. | | 1781 | ☉ | Oct. 17 | 8 M. | | 1782 | 🌑︎ | Apr. 12 | 7 A. | | 1783 | 🌑︎ | Mar. 18 | 9 A. _T._ | | 1783 | 🌑︎ | Sept. 10 | 11 A. _T._ | | 1784 | 🌑︎ | Mar. 7 | 3 M. | | 1785 | ☉ | Feb. 9 | 1 A. | | 1787 | 🌑︎ | Jan. 3 | 12 A. _T._ | | 1787 | ☉ | Jan. 19 | 10 M. | | 1787 | ☉ | June 15 | 5 A. | | 1787 | 🌑︎ | Dec. 24 | 3 A. | | 1788 | ☉ | June 4 | 9 M. | | 1789 | 🌑︎ | Nov. 2 | 12 A. | | 1790 | 🌑︎ | Apr. 28 | 12 A. _T._ | | 1790 | 🌑︎ | Oct. 23 | 1 M. _T._ | | 1791 | ☉ | April 3 | 1 A. | | 1791 | 🌑︎ | Oct. 12 | 3 M. | | 1792 | ☉ | Sept. 16 | 11 M. | | 1793 | 🌑︎ | Feb. 25 | 10 A. | | 1793 | ☉ | Sept. 5 | 3 A. | | 1794 | ☉ | Jan. 31 | 4 A. | | 1794 | 🌑︎ | Feb. 14 | 11 A. _T._ | | 1794 | ☉ | Aug. 25 | 5 A. | | 1795 | 🌑︎ | Feb. 4 | 1 M. | | 1795 | ☉ | July 16 | 9 M. | | 1795 | 🌑︎ | July 31 | 8 A. | | 1797 | ☉ | June 25 | 8 A. | | 1797 | 🌑︎ | Dec. 4 | 6 M. | | 1798 | 🌑︎ | May 27 | 7 A. _T._ | | 1800 | 🌑︎ | Oct. 2 | 11 A. | +------+-----+----------+------------+

328. _A List of Eclipses, and historical Events, which happened about
the same Times, from_ RICCIOLUS.

[Sidenote: Historical Eclipses.]

After CHRIST.
| |
59 | _April_ 30 | An Eclipse of the Sun. This is reckoned among
| | the prodigies, on account of the murther of
| | _Agrippinus_ by _Nero_.
| |
237 | _April_ 12 | A total Eclipse of the Sun. A sign that the reign
| | of the _Gordiani_ would not continue long. A sixth
| | persecution of the Christians.
| |
306 | _July_ 27 | An Eclipse of the Sun. The Stars were seen,
| | and the Emperor _Constantius_ died.
| |
840 | _May_ 4 | A dreadful Eclipse of the Sun. And _Lewis_ the
| | Pious died within six months after it.
| |
1009 | ---- | An Eclipse of the Sun. And _Jerusalem_ taken by
| | the _Saracens_.
| |
1133 | _Aug._ 2 | A terrible Eclipse of the Sun. The Stars were
| | seen. A schism in the church, occasioned by there
| | being three Popes at once.

_J. Ferguson delin._ _J. Mynde Sculp._]

[Sidenote: The superstitious notions of the antients with regard to
Eclipses.

PLATE XI.]

329. I have not cited one half of RICCIOLUS’s list of potentous Eclipses; and for the same reason that he declines giving any more of them than what that list contains: namely, that ’tis most disagreeable to dwell any longer on such nonsense, and as much as possible to avoid tiring the reader: the superstition of the antients may be seen by the few here copied. My author farther says, that there were treatises written to shew against what regions the malevolent effects of any particular Eclipse was aimed: and the writers affirmed, that the effects of an Eclipse of the Sun continued as many years as the Eclipse lasted hours; and that of the Moon as many months.

[Sidenote: Very fortunate once for CHRISTOPHER COLUMBUS.]

330. Yet such idle notions were once of no small advantage to CHRISTOPHER COLUMBUS; who, in the year 1493, was driven on the island of _Jamaica_, where he was in the greatest distress for want of provisions, and was moreover refused any assistance from the inhabitants; on which he threatened them with a plague, and that in token of it there should be an Eclipse: which accordingly fell on the day he had foretold, and so terrified the Barbarians, that they strove who should be first in bringing him all sorts of provisions; throwing them at his feet, and imploring his forgiveness. RICCIOLUS’s _Almagest_, Vol. I. 1. v. c. ii.

[Sidenote: Why there are more visible Eclipses of the Moon than of the
Sun.]

331. Eclipses of the Sun are more frequent than of the Moon, because the Sun’s ecliptic limits are greater than the Moon’s § 317: yet we have more visible Eclipses of the Moon than of the Sun, because Eclipses of the Moon are seen from all parts of that Hemisphere of the Earth which is next her, and equally great to each of these parts; but the Sun’s Eclipses are visible only to that small portion of the Hemisphere next him whereon the Moon’s shadow falls; as shall be explained by and by at large.

[Sidenote: Fig. I.

Total and annular Eclipses of the Sun.

PLATE XI.]

332. The Moon’s Orbit being elliptical, and the Earth in one of its focuses, she is once at her least distance from the Earth, and once at her greatest in every Lunation. When the Moon changes at her least distance from the Earth, and so near the Node that her dark shadow falls on the Earth, she appears big enough to cover the whole [75]Disc of the Sun from that part on which her shadow falls; and the Sun appears totally eclipsed there, as at _A_, for some minutes: But when the Moon changes at her greatest distance from the Earth, and so near the Node that her dark shadow is directed towards the Earth, her diameter subtends a less angle than the Sun’s; and therefore she cannot hide his whole Disc from any part of the Earth, nor does her shadow reach it at that time; and to the place over which the point of her shadow hangs, the Eclipse is annular as at _B_; the Sun’s edge appearing like a luminous ring all around the body of the Moon. When the Change happens within 17 degrees of the Node, and the Moon at her mean distance from the Earth, the point of her shadow just touches the Earth, and she eclipseth the Sun totally to that small spot whereon her shadow falls; but the darkness is not of a moment’s continuance.

[Sidenote: The longest duration of total Eclipses of the Sun.]

333. The Moon’s apparent diameter when largest exceeds the Sun’s when least only 1 minute 38 seconds of a degree: And in the greatest Eclipse of the Sun that can happen at any time and place, the total darkness continues no longer than whilst the Moon is going 1 minute 38 seconds from the Sun in her Orbit; which is about 3 minutes and 13 seconds of an hour.

[Sidenote: To how much of the Earth the Sun may be totally or partially
eclipsed at once.]

334. The Moon’s dark shadow covers only a spot on the Earth’s surface, about 180 _English_ miles broad, when the Moon’s diameter appears largest and the Sun’s least; and the total darkness can extend no farther than the dark shadow covers. Yet the Moon’s partial Shadow or Penumbra may then cover a circular space 4900 miles in diameter, within all which the Sun is more or less eclipsed as the places are less or more distant from the Center of the Penumbra. When the Moon changes exactly in the Node, the Penumbra is circular on the Earth at the middle of the general Eclipse; because at that time it falls perpendicularly on the Earth’s surface: But at every other moment it falls obliquely, and will therefore be elliptical; and the more so, as the time is longer before or after the middle of the general Eclipse; and then, much greater portions of the Earth’s surface are involved in the Penumbra.

[Sidenote: Duration of general and particular Eclipses.

The Moon’s dark shadow.

And Penumbra.]

335. When the Penumbra first touches the Earth the general Eclipse begins: when it leaves the Earth the general Eclipse ends: from the beginning to the end the Sun appears eclipsed in some part of the Earth or other. When the Penumbra touches any place the Eclipse begins at that place, and ends when the Penumbra leaves it. When the Moon changes in the Node, the Penumbra goes over the center of the Earth’s Disc as seen from the Moon; and consequently, by describing the longest line possible on the Earth, continues the longest upon it; namely, at a mean rate, 5 hours 50 minutes: more, if the Moon be at her greatest distance from the Earth, because she then moves slowest; less, if she be at her least distance, because of her quicker motion.

[Sidenote: Fig. II.]

336. To make the last five articles and several other Phenomena plainer, let _S_ be the Sun, _E_ the Earth, _M_ the Moon, and _AMP_ the Moon’s Orbit. Draw the right line _Wc 12_ from the western edge of the Sun at _W_, touching the western edge of the Moon at _c_ and the Earth at _12_: draw also the right line _Vd 12_ from the eastern edge of the Sun at _V_, touching the eastern edge of the Moon at _d_ and the Earth at _12_: the dark space _ce 12 d_ included between those lines is the Moon’s shadow, ending in a point at _12_ where it touches the Earth; because in this case the Moon is supposed to change at _M_ in the middle between _A_ the Apogee, or farthest point of her Orbit from the Earth, and _P_ the Perigee, or nearest point to it. For, had the point _P_ been at _M_, the Moon had been nearer the Earth; and her dark shadow at _e_ would have covered a space upon it about 180 miles broad, and the Sun would have been totally darkened as at _A_ (Fig I) with some continuance: but had the point _A_ (Fig. II) been at _M_, the Moon would have been farther from the Earth, and her shadow would have ended in a point about _e_, and therefore the Sun would have appeared as at _B_ (Fig. I) like a luminous ring all around the Moon. Draw the right lines _WXdh_ and _VXcg_, touching the contrary sides of the Sun and Moon, and ending on the Earth at _a_ and _b_: draw also the right line _SXM 12_, from the center of the Sun’s Disc, through the Moon’s center, to the Earth at _12_; and suppose the two former lines _WXdh_ and _VXcg_ to revolve on the line _SXM 12_ as an Axis, and their points _a_ and _b_ will describe the limits of the Penumbra _TT_ on the Earth’s surface, including the large space _a0b12a_; within which the Sun appears more or less eclipsed as the places are more or less distant from the verge of the Penumbra _a0b_.

[Sidenote: Digits, what.]

Draw the right line _y 12_ across the Sun’s Disc, and parallel to the plane of the Moon’s Orbit; divide this line into twelve equal parts, as in the Figure, for the twelve [76]Digits of the Sun’s diameter: and at equal distances from the center of the Penumbra _TT_ to its edge on the Earth, or from _12_ to _0_, draw twelve concentric Circles, as marked with the numeral Figures _1_ _2_ _3_ _4_ &c. and remember that the Moon’s motion in her Orbit _AMP_ is from west to east, as from _s_ to _t_. Then,

[Sidenote: The different phases of a solar Eclipse.

PLATE XI.

Fig. III.]

To an observer on the Earth at _b_, the eastern limb of the Moon at _d_ seems to touch the western limb of the Sun at _W_, when the Moon is at _M_; and the Sun’s Eclipse begins at _b_; appearing as at _A_ in Fig. III at the left hand; but at the same moment of absolute time to an observer at _a_ in Fig. II the western edge of the Moon at _c_ leaves the eastern edge of the Sun at _V_, and the Eclipse ends, as at the right hand _C_ of Fig. III. At the very same instant, to all those who live on the Circle marked _1_ on the Earth _E_ in Fig. II, the Moon _M_ cuts off or darkens a twelfth part of the Sun _S_, and eclipses him one Digit, as at _1_ in Fig. III: to those who live on the Circle marked _2_ in Fig. II the Moon cuts off two twelfth parts of the Sun, as at _2_ in Fig. III: to those on the Circle _3_, three parts; and so on to the center at _12_ in Fig. II, where the Sun is centrally eclipsed as at _B_ in the middle of Fig. III: under which Figure there is a scale of hours and minutes, to shew at a mean state how long it is from the beginning to the end of a central Eclipse of the Sun on the parallel of _London_; and how many Digits are eclipsed at any particular time from the beginning at _A_ to the middle at _B_, or the end at _C_. Thus in 16 minutes from the beginning, the Sun is two Digits eclipsed; in an hour and five minutes, 8 Digits; and in an hour and thirty-seven minutes, 12 Digits.

[Sidenote: Fig. II.

The Velocity of the Moon’s shadow on the Earth.

Fig. IV.]

337. By Fig. II it is plain, that the Sun is totally or centrally eclipsed but to a small part of the Earth at any time; because the dark conical shadow _e_ of the Moon _M_ falls but on a small part of the Earth: and that the partial Eclipse is confined at that time to the space included by the Circle _a 0 b_, of which only one half can be projected in the Figure, the other half being supposed to be hid by the convexity of the Earth _E_: and likewise, that no part of the Sun is eclipsed to the large space _YY_ of the Earth, because the Moon is not between the Sun and that part of the Earth: and therefore to all that part the Eclipse is invisible. The Earth turns eastward on its Axis, as from _g_ to _h_, which is the same way that the Moon’s shadow moves; but the Moon’s motion is much swifter in her Orbit from _s_ to _t_: and therefore, altho’ Eclipses of the Sun are of longer duration on account of the Earth’s motion on its Axis, than they would be if that motion was stopt, yet in 3 minutes and 13 seconds of time, the Moon’s swifter motion carries her dark shadow quite over any place that its center touches at the time of greatest obscuration. The motion of the shadow on the Earth’s Disc is equal to the Moon’s motion from the Sun, which is about 30-1/2 minutes of a degree every hour at a mean rate; but so much of the Moon’s Orbit is equal to 30-1/2 degrees of a great Circle on the Earth, § 320; and therefore the Moon’s shadow goes 30-1/2 degrees or 1830 geographical miles on the Earth in an hour, or 30-1/2 miles in a minute, which is almost four times as swift as the motion of a cannon-ball.

[Sidenote: PLATE XI.

Fig. IV.

Phenomena of the Earth as seen from the Sun or New Moon
at different times of the year.]

338. As seen from the Sun or Moon, the Earth’s Axis appears differently inclined every day of the year, on account of keeping its parallelism throughout its annual course. Let _E_, _D_, _O_, _N_, be the Earth at the two Equinoxes and the two Solstices; _N S_ its Axis, _N_ the North Pole, _S_ the South Pole, _Æ Q_ the Equator, _T_ the Tropic of Cancer, _t_ the Tropick of Capricorn, and _ABC_ the Circumference of the Earth’s enlightened Disc as seen from the Sun or New Moon at these times. The Earth’s Axis has the position _NES_ at the vernal Equinox, lying towards the right hand, as seen from the Sun or New Moon; its Poles _N_ and _S_ being then in the Circumference of the Disc; and the Equator and all its parallels seem to be straight lines, because their planes pass through the observer’s eye looking down upon the Earth from the Sun or Moon directly over _E_, where the Ecliptic _FG_ intersects the Equator _Æ_. At the Summer Solstice, the Earth’s Axis has the position _NDS_; and that part of the Ecliptic _FG_ in which the Moon is then New, touches the Tropic of Cancer _T_ at _D_. The North Pole _N_ at that time inclining 23-1/2 degrees towards the Sun, falls so many degrees within the Earth’s enlightened Disc, because the Sun is then vertical to _D_, 23-1/2 degrees north of the Equator _ÆQ_; and the Equator with all its parallels seem elliptic curves bending downward, or towards the South Pole as seen from the Sun: which Pole, together with 23-1/2 degrees all round it, is hid behind the Disc in the dark Hemisphere of the Earth. At the autumnal Equinox the Earth’s Axis has the position _NOS_, lying to the left hand as seen from the Sun or New Moon, which are then vertical to _O_, where the Ecliptic cuts the Equator _ÆQ_. Both Poles now lie in the circumference of the Disc, the North Pole just going to disappear behind it, and the South Pole just entering into it; and the Equator with all its parallels seem to be straight lines, because their planes pass through the observer’s eye, as seen from the Sun, and very nearly so as seen from the Moon. At the Winter Solstice the Earth’s Axis has the position _NNS_; when its South Pole _S_ inclining 23-1/2 degrees toward the Sun falls 23-1/2 degrees within the enlightened Disc, as seen from the Sun or New Moon which are then vertical to the Tropic of Capricorn _t_, 23-1/2 degrees south of the Equator _ÆQ_; and the Equator with all its parallels seem elliptic curves bending upward; the North Pole being as far hid behind the Disc in the dark Hemisphere, as the South Pole is come into the light. The nearer that any time of the year is to the Equinoxes or Solstices, the more it partakes of the Phenomena relating to them.

[Sidenote: PLATE XI.

Various positions of the Earth’s Axis, as seen from the Sun
at different times of the year.]

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._

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Astronomy Explained Upon Sir Isaac Newton's PrinciplesChapter XIV: Part 14

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