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.
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,
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.
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 301⁄2 minutes of a degree every hour at a mean rate; but so much of the Moon’s Orbit is equal to 301⁄2 degrees of a great Circle on the Earth, § [320]; and therefore the Moon’s shadow goes 301⁄2 degrees or 1830 geographical miles on the Earth in an hour, or 301⁄2 miles in a minute, which is almost four times as swift as the motion of a cannon-ball.
[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 231⁄2 degrees towards the Sun, falls so many degrees within the Earth’s enlightened Disc, because the Sun is then vertical to D, 231⁄2 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 231⁄2 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 231⁄2 degrees toward the Sun falls 231⁄2 degrees within the enlightened Disc, as seen from the Sun or New Moon which are then vertical to the Tropic of Capricorn t, 231⁄2 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.
[PLATE XI].
Various positions of the Earth’s Axis, as seen from the Sun
at different times of the year.