The method, an extremely rough one, which was in use among the ancients, was something like the following. In an eclipse of the moon, that body passes through the earth's shadow in about four hours. If, then, the moon travels along its orbit in four hours a distance equal to the diameter of the earth, in twenty-four hours it would pass over six times, and in a lunar month (about thirty days) one hundred and eighty times, that distance. The circumference of the lunar orbit must be then one hundred and eighty times the diameter of the earth. The ancients supposed the heavenly orbits to be circles, and as the diameter of a circle is about of the circumference, they deduced directly the diameter of the moon's orbit as 120 times, and the distance of the the moon from the earth as 60 times the semi-diameter of the earth.

(2.) Modern method by the lunar parallax.-Under the head of parallax we saw how, in common life, we obtain a correct idea of the distance of an object by means of our two eyes. We proved that one eye alone gives no notion of distance. Just, then, as we use two eyes to find how far from us an object is, so the astronomer uses two astronomical eyes or observatories, located as far apart as possible, to find the parallax of a heavenly body. In the figure, M represents the moon, G an observatory at Greenwich, and C another at the Cape of Good Hope. At the former, the distance from the north pole to the centre of the moon, measured on a meridian of the

celestial sphere, is found to be 108°. At the latter station, the distance from the south pole to the moon's centre is measured in the same way, and found to be 731°. The sum of these angles is 18110. Now, the entire distance from the north pole around to the south pole, measured on a meridian, can be only half a great circle, or 180°. This difference of

110 must be the difference in the position of the moon, as seen from the two observatories. For the observer at the former ştation will see the moon projected on the celestial sphere at G', and in measuring its distance from the north pole will measure an arc bG' further than if he were located at E, the centre of the earth. The observer at the latter station will see the moon projected on the celestial sphere at C', and in measuring its distance from the

=

The sum

south pole will measure an arc bC' more than if he were located at E, the centre of the earth. of bG' and bC' = G'C' is the difference in the position of the moon as seen from the two stations. In other words, it is the moon's parallax. The arc G'C' measures the angle C'MG'; that angle is equal to the opposite angle GMC 1°. Now, in the foursided figure GECM, the sides GE and CE are each equal radii of the earth 3956 miles; while the distance from G to C is the difference in the latitude of the two places. The angles ZGM and Z'CM, being the zenith distances of the moon, are known, and so the angles MGE and MCE are easily found. EM, the moon's distance from the centre of the earth, is thus readily computed by a simple trigonometrical formula.

(3.) The horizontal parallax of the moon is most commonly found by estimating its distance, not from the north and south poles, as just explained under the general meaning of the term parallax, but from a fixed star. The moon's horizontal parallax is now estimated at 57', which makes its distance about sixty times the earth's semi-diameter.*

TO FIND THE SUN'S DISTANCE FROM THE EARTH.This might be estimated by obtaining the solar

* In figure 95, let S represent the moon, sun, or any other heavenly body, AB the semi-diameter of the earth, and ASB the "horizontal parallax" of the body. Then, by the following trig onometrical formula, the distance from the earth may be easily calculated

AS AB:: Radius: Sin of ASB.

parallax in the same manner as the lunar parallax. It would be only necessary to take the sun's distance from the north and south poles respectively at Greenwich and the Cape of Good Hope, and then subtracting 180° from the sum of the two angular distances, the remainder would be the solar parallax. The difficulty in this method lies in the fact that when the sun shines the air is full of tremulous motion. This increases refraction-that plague of all astronomical calculations-to such an extent that it becomes impossible to calculate so small an angle with any accuracy. Neither can the parallax be estimated, as in the case of the moon, by measuring

the distance from a fixed star, since when the sun shines the stars near by are invisible even in a telescope. Astronomers have therefore been compelled to resort to other methods.

(1.) Calculation of solar parallax by observation of the planet Mars.-We have already seen that the distance of Mars from the sun is that of the earth

from the sun. If, therefore, we

can find Mars' dis

tance from the earth, we can multiply it by three,

=

and so obtain the distance of the sun from the earth. In 1862, when Mars was in opposition, it came very near us, for it was in perihelion while the earth was in aphelion, so that its distance (as since ascertained) was only 126,300,000-93,000,000 33,300,000 miles. Observers at Greenwich and the Cape, and at various American and European observatories, calculated the distance of the planet from the north and south poles, as well as from several fixed stars, in precisely the manner just explained for obtaining the lunar parallax. The result of these observations fixed the solar parallax at 8.94".*

(2.) Calculation of solar parallax by observation of the transit of Venus. In the figure, let A and B represent the positions of two observers stationed at

opposite sides of the earth. At the time of the transit, the one at A will see the planet Venus as a round black spot at V" on the sun's disk, while the one at B will see it at V'. The distance V'V" is the

* By the formula on page 302, we have―

AS AB:: Radius: Sin 8.94".