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ASV, the proportion between the sides AS, AV, is known by the periodical times of the earth and Venus; as also the ratio between AV and VS. Hence the horizontal parallax of the sun is found by saying, As VS: AV: the horizontal parallax of Venus from the sun the horizontal parallax of the sun on the day of the transit; then as the sun's distance from the earth on the day of the transit is to his mean distance, so is his horizontal parallax on the day of the transit, found as above, to his horizontal parallax at his mean distance.

In this method of determining the parallax of the sun, the observations should be made in two places 90 degrees apart; one of the observers having the sun in his meridian or near to it, and the other, in the horizon. But as this may be difficult, the observations may be made in two places at any other considerable distance from each other; where the difference of times between the observations will be great, occasioned by the parallax of Venus from the sun; from which his parallax may be ascertained. And when his parallax is once determined, we have sufficient data to determine the distances of all the planets of the solar system, from the known proportion of their periodical times: and then, from their apparent diameters at these known distances, their real diameters and bulks may be found; their bulks being as the cubes of their diameters; the diameter of the earth being found from different measures of a degree of the meridian, in different latitudes, compared together.

The parallax of the sun being found by observations of this kind, the distances of all the planets are found to be, what we have already given in a preceding lecture. And to this important observation we

are indebted for all the precision we have now acquired in the distances and dimensions of the solar system.

The diurnal parallax of a heavenly body always depresses it in a vertical circle; and consequently will produce a parallax of right ascension and of declination, excepting when it is in the meridian, when the whole parallax is in declination: it will also produce a parallax of longitude and latitude; excepting when the vertical circle, in which it appears, is a secondary of the ecliptic, when the whole parallax will be in latitude. The stars, from their immense distance, have no parallax.

ELLIPTIC ORBITS OF THE PLANETS.

THE orbits of the planets differ but little from circles, though all in reality are ellipses, having the sun in the lower focus. If the velocity of a planet in its orbit were such as it would acquire by falling through one fourth of its diameter, that velocity would be just sufficient to retain it in a circle; and it would revolve in it, without approaching towards or receding from the center of force. This balance between the central and projectile forces, must soon however have been destroyed by the mutual attractions of the planets, had it even been so constituted at first. The projectile and centripetal forces conspiring together in some degree during one half of its elliptical revolution, the planet will thereby be brought within the circle, and approach nearer to the central body; whence its velocity will at last become more than sufficient to retain it in the circle, and consequently it will then recede from the center of force: but the projectile and centripetal forces now acting in some measure against each other, the velocity will gradually decrease, until it become less than what would be necessary to retain it in a

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circle, when it arrives at its aphelion. In that point, having the same velocity, that it had at the beginning, it will describe the same ellipsis over again, in the next revolution.

Hence all the planets are observed to move with different velocities in different parts of their orbits; sometimes slower and sometimes faster, than what would be necessary to describe their whole orbits in their periodical times, which velocity is therefore called their mean motion. And hence our summers are eight days longer than our winters; or, the sun takes eight days more to go from the vernal to the autumnal equinox, than to go from the autumnal to the vernal equinox. Because in the summer season, the earth passes through her aphelion, where her motion is slowest, about the 30th of June; as it is also quickest on the 30th of December, when the earth passes by her perihelion. Since we are nearer to the sun in winter than in summer, his diameter must then, of course, appear larger. Accordingly we find that his diameter in summer is 31' 33", whereas in winter it is=32′ 38′′. His daily motion in aphelion is 57' 12'', in perihelion, 61' 12', and at the mean distance 59' 8".

As the elliptic orbits of the planets are nearly circles, the distance between the centers and foci is but very small in any of them, being largest in Mercury. If the mean distance of any planet from the sun be divided into 1000 parts, the eccentricities of the several planets will be as follows; Mercury's 210, Venus's 7, the Earth's 17, Mars's 93, Jupiter's 48, Saturn's 55, Herschel's 82,034.

All the planets are regulated in their motions by this fundamental law; that however variable their velocities, yet in their orbits they describe areas proportional

to the times of description. In order to find their places in their orbits, astronomers have contrived the following method. They have supposed a body, revolving in a circle with an equable motion, to have commenced its motion from the aphelion of the planet at the same time with it, and to complete its revolution in the same time with the planet. The motion of this body is called the mean anomaly of the planet; and as it would be sometimes before and sometimes behind the planet, the difference between them being added to the mean anomaly or subtracted from it, as occasion may require, will give the true place of the planet in its orbit. As the real place of the planet in its orbit is the same with the mean, both at the aphelion and perihelion, there will be no equation to be added or subtracted at these points: but the mean motion will be greater than the true motion from the place of the aphelion to the place of the mean distance, or near to it, when the equation to be subtracted from the mean anomaly will be the greatest possible, in order to give the true place of the planet. From that place to the perihelion the mean place will be still before the true place of the planet in its orbit; but the difference will gradually grow less and less, till it vanish at the perihelion, where the mean and true places will be the same. So that the equation of the center will be subtractive from the mean anomaly to give the true place of the planet, from the aphelion to the perihelion, or while the mean anomaly is less than six signs; and for the same reason it will be additive for the other six signs, or from the perihelion to the aphelion.

When the time of the passage of the planet through its aphelion is previously known by observation, and the mean anomaly is determined from the periodical

time, the planet's place is easily known, by applying the equation of the center, according as it is additive or subtractive. This will give the planet's distance from its aphelion. But its distance from the first point of Aries is also found, by first knowing the distance of the aphelion from the first point of Aries, and the motion of the aphelion, if it have any. Astronomers have disposed these motions for all the planets in proper tables, to facilitate the calculation of their places. The table of its mean motion shows its distance from the first point of Aries, and of its mean anomaly from the aphelion; and the equation of the center is the difference between the mean and true anomaly.

It is found that the place of the earth's aphelion is not stationary, but moves forward from west to east at the rate of 16 seconds of a degree in a year, in respect to the fixed stars, or 50" +16" 66", with respect to the equinoxes. The place of the sun's apogee on the first of January, in the year 1760, was 3' 8° 47′ 25".

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It is found, from observation, that the orbits of the other planets are quiescent, and their aphelion points are found to be in the following places, viz.

The aphelion of Mercury in

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12° 44' 0"

ww 4 19 54

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8 47 25 variable.

0 31 54

9 9 54

Saturn... ↑ 27 49 54

Herschel.. ↑ 17 13 17

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