[61]. The Ecliptic, together with the fixed Stars, make 3661⁄4 apparent diurnal revolutions about the Earth in a year; the Sun only 3651⁄4. Therefore the Stars gain 3 minutes 56 seconds upon the Sun every day: so that a Sidereal day contains only 23 hours 56 minutes of mean Solar time; and a natural or Solar day 24 hours. Hence 12 Sidereal hours are 1 minute 58 seconds shorter than 12 Solar.
[62]. The Sun advances almost a degree in the Ecliptic in 24 hours, the same way that the Moon moves: and therefore, the Moon by advancing 131⁄6 degrees in that time goes little more than 12 degrees farther from the Sun than she was on the day before.
[63]. This center is as much nearer the Earth’s center than the Moon’s as the Earth is heavier, or contains a greater quantity of matter than the Moon, namely about 40 times. If both bodies were suspended on it they would hang in æquilibria. So that dividing 240,000 miles, the Moon’s distance from the Earth’s center, by 40 the excess of the Earth’s weight above the Moon’s, the quotient will be 6000 miles, which is the distance of the common center of gravity of the Earth and Moon from the Earth’s center.
[64]. The Penumbra is a faint kind of shadow all around the perfect shadow of the Planet or Satellite; and will be more fully explained by and by.
[65]. Which is the time that the Eclipse would be at the greatest obscuration, if the motions of the Sun and Moon were equable, or the same in all parts of their Orbits.
[66]. The above period of 18 years 11 days 7 hours 43 minutes, which was found out by the Chaldeans, and by them called Saros.
[67]. A Digit is a twelfth part of the diameter of the Sun or Moon.
[68]. There are two antient Eclipses of the Moon, recorded by Ptolemy from Hipparchus, which afford an undeniable proof of the Moon’s acceleration. The first of these was observed at Babylon, December the 22d, in the year before Christ 383: when the Moon began to be eclipsed about half an hour before the Sun rose, and the Eclipse was not over before the Moon set: but by our best Astronomical Tables, the Moon was set at Babylon half an hour before the Eclipse began; in which case, there could have been no possibility of observing it. The second Eclipse was observed at Alexandria, September the 22d, the year before Christ 201; where the Moon rose so much eclipsed, that the Eclipse must have begun about half an hour before she rose: whereas by our Tables the beginning of this Eclipse was not till about 10 minutes after the Moon rose at Alexandria. Had these Eclipses begun and ended while the Sun was below the Horizon, we might have imagined, that as the antients had no certain way of measuring time, they might have been so far mistaken in the hours, that we could not have laid any stress on the accounts given by them. But, as in the first Eclipse the Moon was set, and consequently the Sun risen, before it was over; and in the second Eclipse the Sun was set, and the Moon not risen, till some time after it began; these are such circumstances as the observers could not possibly be mistaken in. Mr. Struyk in the following Catalogue, notwithstanding the express words of Ptolemy, puts down these two Eclipses as observed at Athens; where they might have been seen as above, without any acceleration of the Moon’s motion: Athens being 20 degrees West of Babylon, and 7 degrees West of Alexandria.
[69]. Each Olympiad began at the time of Full Moon next after the Summer Solstice, and lasted four years, which were of unequal lengths because the time of Full Moon differs 11 days every year: so that they might sometimes begin on the next day after the Solstice, and at other times not till four weeks after it. The first Olympiad began in the year of the Julian Period 3938, which was 776 years before the first year of Christ, or 775 before the year of his birth; and the last Olympiad, which was the 293d, began A. D. 393. At the expiration of each Olympiad, the Olympic Games were celebrated in the Elean fields, near the river Alpheus in the Peloponnesus (now Morea) in honour of Jupiter Olympus. See Strauchius’s Breviarium Chronologium, p. 247-251.
[70]. The reader may probably find it difficult to understand why Mr. Smith should reckon this Eclipse to have been in the 4th year of the 48th Olympiad; as it was only in the end of the third year: and also why the 28th of May, in the 585th year before Christ should answer to the present 10th of that month. But we hope the following explanation will remove these difficulties.