Page:Popular Science Monthly Volume 19.djvu/765

Rh A S and B S', representing the direction of the star S, as seen from the points A and B, may, in consequence of the immense distance of the star from the earth, be regarded as parallel. On account of the proximity of the mountain G, the plumb-line does not take the direction



A Z, but is deflected toward the mountain, so that it gives the direction A Z' as the apparent vertical, and Z' as the apparent zenith. On this account, the zenith-distance of the star is increased by the angle a, to a degree that is represented by the angle m. The prolonged plumb-lines B Z" and A Z' consequently do not form the angle x at the center of the earth, but another angle, y, which differs from x by the magnitude a, wherefore, a $$=$$ x $$-$$ y. If, now, we imagine the line of direction A S prolonged backward, an equivalent of the angle u is formed at T, and by the lines A C' and A T the angle m, equal to the observed zenith-distance at A. But u being the external angle of a triangle, $$=$$ m $$-$$ y, or y $$=$$ m—u; and since a is equal to x $$-$$ y, if we substitute for y the difference m $$-$$ u, a $$=$$ x $$+$$ u $$-$$ m. The angles u and m have been obtained by observation as zenith-distances of the fixed star S S', and we have only to obtain the value of the angle x, which is deduced from a trigonometrical measurement of the arc A B. The mass of the