When the moon, or any heavenly body, is seen in the horizon, as at E, the change of place is called the horizontal parallax. Thus, the angle A E C, measures the horizontal parallax of the moon. Were a spectator to view the earth from the centre of the moon, he would see the semidiameter of the earth under this same angle; hence, the horizontal parallax of any body is the angle subtended by the semidiameter of the earth, as seen from the body. Please to remember this fact.
It is evident from the figure, that the effect of parallax upon the place of a celestial body is to depress it. Thus, in consequence of parallax, E is depressed by the arc H h; F, by the arc P p; G, by the arc R r; while O sustains no change. Hence, in all calculations respecting the altitude of the sun, moon, or planets, the amount of parallax is to be added: the stars, as we shall see hereafter, have no sensible parallax.
It is now very easy to see how, when the parallax of a body is known, we may find its distance from the centre of the earth. Thus, in the triangle A C E, Fig. 19, the side A C is known, being the semidiameter of the earth; the angle C A E, being a right angle, is also known; and the parallactic angle, A E C, is found from observation; and it is a well-known principle of trigonometry, that when we have any two angles of a triangle, we may find the remaining angle by subtracting the sum of these two from one hundred and eighty degrees. Consequently, in the triangle A E C, we know all the angles and one side, namely, the side A C; hence, we have the means of finding the side C E, which is the distance from the centre of the earth to the centre of the moon.
Fig. 20.
When the distance of a heavenly body is known, and we can measure, with instruments, its angular breadth, we can easily determine its magnitude. Thus, if we have the distance of the moon, E S, Fig. 20, and half the breadth of its disk S C, (which is measured by the angle S E C,) we can find the length of the line, S C, in miles. Twice this line is the diameter of the body; and when we know the diameter of a sphere, we can, by well-known rules, find the contents of the surface, and its solidity.
You will perhaps be curious to know, how the moon's horizontal parallax is found; for it must have been previously ascertained, before we could apply this method to finding the distance of the moon from the earth. Suppose that two astronomers take their stations on the same meridian, but one south of the equator, as at the Cape of Good Hope, and another north of the equator, as at Berlin, in Prussia, which two places lie nearly on the same meridian. The observers would severally refer the moon to different points on the face of the sky,—the southern observer carrying it further north, and the northern observer further south, than its true place, as seen from the centre of the earth. This will be plain from the diagram, Fig. 21. If A and B represent the positions of the spectators, M the moon, and C D an arc of the sky, then it is evident, that C D would be the parallactic arc.
Fig. 21.
These observations furnish materials for calculating, by the aid of trigonometry, the moon's horizontal parallax, and we have before seen how, when we know the parallax of a heavenly body, we can find both its distance from the earth and its magnitude.