Page:Optics.djvu/144

 The apparent lengths with the naked eye, and with the lens, are therefore, as $1⁄120$ and $1⁄8$, or as 8 and 120, of which numbers the latter being fifteen times the former, the object is said to be magnified in length fifteen times.

In order to generalize this, let $c$ be the nearest distance for correct vision, Then since $k=OA$, the linear magnitudes of the object and image are as $∆=AQ$, that is, as $F=AF$.

The angular magnitudes, that is, the angles subtended by the object and image at $δ=Aq$, are as $δ=∆F⁄F−∆$, but the fairer way of stating the matter is to compare the angular magnitudes of the object at the distance $∆:δ$ and the image at the distance $F−∆:F$: these are as

and the magnifying power of the lens is

This of course is increased by diminishing $O$. If we make this $F−∆⁄∆+k:F⁄δ+k$ by placing the eye close to the lens, the magnifying power becomes

and is inversely as the distance $c$, which may consequently be diminished with advantage, as long as $δ+k$ is left greater than $F−∆⁄c:F⁄δ+k$.