same motion, the stars must appear to an inhabitant of the earth to move in an opposite direction. In the triangle s p a, Plate VII. Fig. 1, let s p represent the parallactic motion of a star; then, if this star is one that has no real motion, s p will also be its apparent motion; but if the star in the same time, that by its parallactic motion it would have gone from s to p, should have a real motion which would have carried it from s to r, then will it be seen to move along the diagonal s a, of the parallelogram s r p a; and p a, which is parallel and equal to s r, will represent its real motion. Therefore, in the above mentioned triangle s p a, which I suppose to be formed in the concave part of the heavens by three arches of great circles, the eye of the observer being in the center, the three sides will represent, or stand for, the three motions I have named: s p the parallactic, p a the real, and s a the apparent motion of the star. The situation and length of these arches, in seconds of a degree, will express, or rather represent, not only the direction but also the quantity of each motion, such as it must appear to an eye in the above mentioned central situation. And calling the solar motion S, the distance of the star from the sun d, and the sine of the star's distance from the point towards which the sun is moving φ, the parallactic motion, when these are given, will be had by the expression This theorem, and its corollaries, of which frequent use will be made hereafter, it will not be necessary here to demonstrate.
When I call the arch p a the real motion, it should be understood that I only mean its representative; for it must be evident that the absolute motion of a star in space, as well as its intrinsic velocity, will still remain unknown, because the inclination of that motion on which also its real velocity will