Therefore, as A M, the tangent of the circle stands at right angles to the radius, A C, the angle at A is a right angle, and the magnitude of the angle at C is found by means of the arc, A B, the distance of the two observers from each other. As soon, however, as we are acquainted with the magnitude of two angles of a triangle, we arrive at that of the third, because we know that all the angles of a triangle together equal two right angles (180°). The angle at M, generally called the moon’s parallax, is thus found to be fifty-six minutes and fifty-eight seconds. We know that in the right-angled triangle M C A, the measure of the angle, M = 56´ 58″, and also that A C, the semi-diameter of the earth = 3,964 miles. This is sufficient, in order by trigonometry, to obtain the length of the side, M C; that is, to find the moon’s distance from the earth. A C is the sine of the angle, M, and by the table the sine of an angle of 56´ 58″ is equal to 1652/100000; or, in other words, if we divide the constant, M C, the distance of the moon, into 100,000 equal parts, the sine, A C, the earth’s semi-diameter = 1,652 of these parts. And this last quantity being contained 60 times in 100,000, the distance of the moon from the earth is equal to 60 semi-diameters of the earth, or 60 × 3964 = 237,840 miles.
Fig. 524.—Parallax explained.
In a similar way the parallax of the sun has been found = 8″·6, and the distance of the sun from the earth to be 91,000,000 miles.
Let us first see how we can obtain the distance of any inaccessible or distant object. We have already mentioned an experiment, but this method is by a calculation of angles. The three angles of a triangle, we know, are equal to two right angles; that is an axiom which cannot be explained away. We first establish a base line; that is, we plant a pole at one point, A, and take up our position at another point, B, at some distance in a straight line, and measure that distance very carefully. By means of the theodolite we can calculate the angles which our eye, or a supposed line drawn from our eye to the top of the object (C) we wish to find the distance of, makes with that object. We now have an imaginary triangle with the length of one side, A B, known, and all the angles known; for if all three angles are equal to 180°, and we have calculated the angles at the base, we can easily find the other. We can then complete our triangle on paper to scale, and find out the length of the side of the triangle by measurement; that is the distance between our first position, A, and the object, C. It is of course necessary that all measurements should be exact, and the line we adopt for a base should bear some relative proportion to the distance at which we may guess the object to be.
In celestial measurements two observers go to different points of the earth, and their distance in a straight line is known, and the difference of the latitudes. By calling the line between the observers a base line, a figure may be constructed and angles measured; then by some abstruse calculation the distance between the centre of the earth and the centre of the moon may be ascertained. The mean distance is sixty times the radius of the earth. The measurement of the sun’s distance is calculated by the observations of the transit of Venus across his disc, a phenomenon which will again occur on 6th December, 1882, and on 8th June, 2004, the next transit will take place; there will be no others for a long time after 2004.
All astronomical observations are referred to the centre of the earth, but of course can only be viewed from the surface, and correction is made. In the cut above, let E be the earth and B a point on the surface. From B the stars, a b c d, will be seen in the direction of the dotted lines, and be projected to e i k l respectively. But from the centre of the earth they would appear at e f g h correctly. The angles formed by the lines at b c d are the parallactic angles, f i g h and h l show the parallax. An object on the zenith thus has no parallax. (See fig. 524.)
Fig. 525.—Halo Nebulæ.