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degrees and 8 minutes, which she passed on the 26th of June. The earth's longitude on that day, was 274 degrees and 44 minutes. The longitude of the descending node of Venus, was 258 degrees and 8 minutes, which she passed on the 5th of March. The earth's longitude on that day, was one hundred and sixty-four degrees and fifty-five minutes, consequently there was no transit of Venus in 1824.

PROBLEM XI.

To find when any two given planets shall have a given heliocentric aspect, taking their longitudes as stated in the Table for 1823.

Rule. Add the degrees in the aspect given to the heliocentric longitude of either given planet. Find the difference between that sum and the heliocentric longitude of the other given planet: then say, as the difference in the daily motions of the two given planets, is to one day, so is the difference in their longitude found as above to the answer required.

EXAMPLES.

1. At what time in the year 1824, did the earth and Venus have a trine aspect ?

The longitude of the earth for January 1st, for that year, was 100 degrees and 6 minutes; to the earth's longitude, add 120 degrees, (the given aspect,) and the sum is 220 degrees and 6 minutes.

The longitude of Venus on the first day of January, 1824, was 150 degrees and 2 minutes; the difference was 70 degrees and 4 minutes; then 1,6021 degrees,9856=,6165, difference of daily motion. Then, 6165:1 day:: 70 degrees 4 minutes: one hundred and thirteen days, or the 22d of April.

2. On what day were the earth and Jupiter in conjunction in the year 1826?

3. When in 1835, will the earth and Venus be in conjunction ?

NOTE.-The preceding PROBLEMS would be correct, if the Planets moved in perfect circular orbits, which, however, is not the fact: yet they approach so near to circles, that deductions founded upon their figures as circles, are sufficiently accurate for ordinary calculations.

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SECTION TWELFTH.

ON ECLIPSES.

In the solar system, the Sun is the great fountain of light, and every planet and satellite is illuminated by him, receive the distribution of his rays, and are irradiated by his beams. The rays of light are seen in direct lines, and consequently are frequently intercepted by the dark and opaque body of the Moon, passing directly between the earth and the Sun, and hiding a portion or the whole of his disk from the view of those parts of the earth where the penumbra, or the shadow of the Moon, happens to fall. This is called an ECLIPSE OF THE SUN.

It is only at the time of new Moon that an eclipse of this kind can possibly take place, and then only when the Sun is within seventeen degrees of either the ascending or descending nodes; for if his distance at the time of new Moon be greater than seventeen degrees from either node, no part of the Moon's shadow will touch the earth, and consequently there will be no eclipse.

The orbit in which the Moon really moves is different from the ecliptic, one half being elevated five and onethird degrees above it, and the other half as much depressed below. The Moon's orbit therefore intersects the ecliptic in two points diametrically opposite to each other, and these intersections are called the Moon's nodes.The Moon, therefore, can never be in the ecliptic but when she is in either of her nodes, which is at least twice in every lunation, or course from change to change, and sometimes thrice. That node from which the Moon begins to ascend northward, or above the ecliptic in northern latitudes, is called the ascending node; and the other the descending node; because the Moon when she passes by it descends below the Ecliptic southward.The Ecliptic is the great circle which the earth describes in its annual revolution around the Sun, and is divided into twelve equal parts, of 30 degrees each, called signs. Six of these, namely Aries, Taurus, Gemini, Cancer, Leo, and Virgo, are north; and the other six, to wit, Libra, Scorpio, Sagitarius, Capricornus, Aquarius, and Pisces, south of the equator.*

When the earth comes between the Sun and Moon, the Moon passes through the earth's shadow, and having no light of her own, she suffers a real eclipse, the rays of the Sun being intercepted by the earth. This can only happen at the time of full Moon, and when the Sun is within twelve degrees of the Moon's ascending or descending nodes. Should the Sun's distance from the node exceed twelve degrees, the shadow of the earth would nowhere touch the surface of the Moon, and consequently she could not suffer an eclipse.

When the Sun is eclipsed to us, the inhabitants of the Moon on the side next the earth, see her shadow like a dark spot travelling over the earth about twice as fast as its equatorial parts move, and the same way.

When the earth passes between the Sun and Moon, the Sun appears in every part of the Moon where the earth's shadow falls totally eclipsed; and the duration is as long as she remains in the earth's shadow.

If the earth and Sun were of equal sizes, the shadow of the earth would be infinitely extended, and wholly of the same breadth, and the planet Mars when in either of her nodes, and in opposition to the Sun, (although fortytwo millions of miles from the earth,) would be eclipsed by the shadow. If the earth were larger than the Sun, her shadow would be sufficient to eclipse the larger planets, Jupiter and Saturn, with all their satellites, when they were opposite to him; but the shadow of the earth terminates in a point long before it reaches any of the

• The Equator is an imaginary circle passing round the earth from east to west, dividing it into equal parts, called hemispheres.

primary planets. (Plate 6th, fig. Sth. S the Sun, AE the earth, ABE earth's shadow terminating at B.] It is therefore evident, that the earth is much less than the Sun, or its shadow could not end in a point at so short a distance.

If the Sun and Moon were of equal sizes, she would cast a shadow on the earth's surface of more than two thousand miles in breadth, even if it fell directly against its centre. But the shadow of the Moon is seldom more than one hundred and fifty miles in breadth at the earth, unless in total eclipses of the Sun, her shadow strikes on the earth in a very oblique direction.

In annular eclipses, the Moon's shadow terminates in a point at some distance before it reaches the earth; and consequently the Moon is much less than the Sun. If the Moon were actually thrice its present size, it would still in many instances be totally eclipsed. A sufficient proof of this is given by her long continuance in the earth's shadow during any of her total eclipses. Therefore the diameter of the earth is more than three times the diameter of the Moon.

Though all opaque bodies on which the Sun shines, have their shadows; yet such is the magnitude of the Sun, and the distances of the planets, that the primaries can never eclipse each other. A primary can only eclipse its secondary, or be eclipsed by it, and never by those except when they are in opposition or conjunction with the Sun, as before stated. The primary planets are very seldom in such positions, but the Sun and Moon are in every month.

If the Moon's orbit were coincident with the plane of the ecliptic, in which the earth wheels its stated courses, the Moon's shadow would fall on the earth at every change, and the Sun be eclipsed to every part of the earth where the penumbra happened to fall. In the same manner the Moon would have to travel through the middle of the earth's shadow, and be totally eclipsed at every full. The duration of total darkness in every instance, exceeding an hour and a half.

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