reach a conclusion that can claim numerical exactness, we may reach one that will give us a general idea of the subject. The first question at which we aim is that of the number of stars within some limit of distance. It is as if, looking around upon an extensive landscape in an inhabited country, we wished to estimate the average number of houses in a square mile. On the general average, what is the radius of the sphere occupied by a single star? If we divide the number of cubic miles in some immense region of the heavens by the number of stars within that region, what quotient should we get? Of course, cubic miles are not our unit of measure in such a case. It will be more convenient to take as our unit of volume a sphere of such radius that from its center, supposed to be at the sun, the annual parallax of a star on the surface would be 1". The radius of this sphere would be 206,265 times that of the earth's orbit. We may use round numbers, consider it 200,000 of these radii, and designate it by the letter E.
Now, let us conceive drawn around the sun as a center concentric spheres of which the radii are R, 2R, 3R, and so on. At the surfaces of these respective spheres the parallax of a star would be 1", half a second, one-third of a second, and so on. The volumes of spheres being as the cubes of their radii, those of the successive spheres would be proportional to the numbers 1, 8, 27, 64, etc.
If the stars are uniformly scattered through space, the numbers having parallaxes between the corresponding limits will be in the same proportion.
The most obvious method of determining the number of stars within the celestial spaces around us is by measurement of their parallaxes. It is possible to reach a definite conclusion in this way only in the case of parallaxes sufficiently large to be measured with an approach to accuracy. In the case of a small parallax the uncertainty
