The ratio of the light entering the eye when directed toward a star, to the whole light given out by the star, would be as the area of the pupil of the eye to the area of the whole sphere having the star as a centre and our distance from the star as a radius. If the eye is assisted by a telescope, the ratio is quite different. In that case the ratio of the light which enters the eye to the whole light, would be as the area of the mirror or object-glass to the area of the whole sphere having the star as a centre and its distance as a radius. Thus the light received by the _eye_ in the two cases would be as the area of the pupil is to the area of the object-glass. For instance, if the pupil has a diameter of two-fifths of an inch, and the mirror a diameter of four inches, then a hundred times as much light would enter the eye when assisted by the telescope as when unarmed, since the _area_ of the pupil is one-hundredth the _area_ of the objective.
If a particular star is just visible to the naked eye, it will be quite bright if viewed with this special telescope, which makes it one hundred times more brilliant in appearance. If we could move the star bodily away from us to a distance ten times its present distance, we could thus reduce its brightness, as seen with the telescope, to what it was at first, as seen with the eye alone, _i. e._, to bare visibility. Moving the star to ten times its present distance would increase the surface of the sphere which it illuminates a hundred-fold. We cannot move any special star, but we can examine stars of all brightnesses, and thus (presumably) of all distances.
HERSCHEL'S argument was, then, as follows: Since with such a telescope one can see a star ten times as far off as is possible to the naked eye, this telescope has the power of penetrating into space ten times farther than the eye alone. But this number ten, also, expresses the ratio of the diameter of the objective to that of the pupil of the eye, consequently the general law is that the _space-penetrating power_ of a telescope is found by dividing the diameter of the mirror in inches by two-fifths. The diameter of the pupil of the eye (two-fifths of an inch) HERSCHEL determined by many measures.