The above shows (very) schematic eyes. When relaxed, the crystalline lens of the normal eye focusses light from an object at infinity on the retina and can expand to bring a nearer object into focus. Some eyes, however, have difficulty in accommodating to near vision, although distant objects may still be clearly seen. The defect increases with age and is termed presbyopia (Gr. presbys, old, ops, opis, the eye). It is further possible that even when relaxed the eye's focus is beyond the retina, a condition termed hyperopia or hypermetropia ( Gr. hyper, over, metron, measure, ops, eye). The opposite can occur, that the eye focusses light from infinity before it reaches the retina, and objects can be clearly seen only when close at hand. This condition is termed myopia (Gr. myops. short-sighted - myein, to shut, ops, the eye ). "Short-sight" is of course the common English term, while "long-sight" (or "far-sight" in American English) is the corresponding term for presbyopia and hyperopia. With reference to the derivation of "myopia" from the Greek, people with short-sight often squint to improve their vision in the absence of glasses. It is well known to photographers that reduction of aperture increases depth of field, that is, the range of distances for which the camera can be said to be in focus - a pinhole camera has infinite depth of field. For the same reason squinting can help long-sighted people too, as can bright illumination, where the pupil is contracted to be very small.
In Europe convex lenses were used to correct long-sightedness from the twelfth century and concave lenses to correct short-sightedness somewhat two centuries later. It would appear that this was done almost purely empirically, with no knowledge of how the improvement to vision came about. A partial explanation of the action of spectacles was provided later still by Franceso Mauricolo, in a work published posthumously in 1611 (Lindberg 1976). He recognised that myopia occurs when the convex crystalline humour (lens) is excessively curved, so that the light passing through converges too much or too soon, while in hyperopia the lens of the eye is insufficiently curved so that convergence is delayed. The function of spectacles is therefore to hasten or delay the convergence, as the case may be.
The diagrams above illustrate the application of the lens formula
\begin{equation} \frac{1}{u}+ \frac{1}{v}=\frac{1}{f} \end{equation}
to the various possibilities of real and virtual objects and images and the corresponding conventions for the signs of $u$, object distance, $v$, image distance, and $f$, focal length. The conventions are
- Focal lengths of converging lenses are positive
- Focal lengths of diverging lenses are negative
- Distances of real objects and images from the optical centre are positive
- Distances of imaginary objects and images from the optical centre are negative.
The results are for a point object O on the optical axis. The angle $D$ is the deviation of a ray meeting the lens at a distance $h$ from the optical axis. Taking the first diagram, we can see that we have taken the angles $\alpha$ and $\beta$ as $h/u$ and $h/v$ respectively, whereas strictly speaking $\tan\alpha=h/u$ and $\tan\beta=h/v$. In practice however, we are restricted to small angles, as otherwise optical aberrations become significant. We are then justified in approximating the tangents of angles by the angles themselves, in which case the deviation may be taken as $D=h/f$, irrespective of the angle at which the ray meets the lens. .
