By giving different values to x, §, it is easy to see from this formula, in what manner the visual angle changes when the positions of the object and eye vary.

We notice that this formula is symmetrical with regard to x and έ, so that the positions of the eye and object may be interchanged in every case without altering the visual angle.

When the eye is at a principal focus the apparent magnitude is independent of the position of the object; and similarly when the object is at a principal focus its apparent magnitude is independent of the position of the eye; the apparent magnitude being, in both cases, equal to that under which the object would be seen by the naked eye, when at a distance equal to the focal length of the lens. For if we make either x=ƒ, or § =ƒ, we get tan 0 = B/f.

Again, when either the eye or the object is close to the lens, the apparent magnitude is that under which the object would be seen by the naked eye. For in these cases, we must make or a very small; and therefore tan 0=B/x or B/E.

115. But in all cases the visual angle will necessarily be limited by the aperture of the lens; so that the greatest value of tan will be equal to y/ nearly, where y denotes the semi-aperture of the lens. The greatest linear extent of object, visible through a lens in any position, may be called the field of view. Its magnitude is at once ascertained by equating the value of tan as previously found to y/§; the greatest value of ẞ is accordingly

The linear extent of the field of view, therefore, varies as the aperture of the lens, other things remaining the

same.

When the object is in the principal focus of the lens x=f, and therefore Byf/. When the eye is in the

=

principal focus of the lens, the linear extent is equal to the aperture of the lens, whatever the position of the object. For if §=ƒ, ß=y.

When the object is close to the lens, that is, when x is very small, the value of ẞ becomes very nearly equal to y; so that the extent of the field is in this case independent of the position of the eye. On the other hand, when the eye is close to the lens, that is, when is small, the field becomes very great.

Spectacles and Reading Glasses.

116. The distinctness of objects as seen by the naked eye depends on the accurate convergence of the rays of different pencils to points on the retina. We have seen that the eye is furnished with a mechanism for adapting itself for seeing distinctly objects at different distances. A normal eye when not actively accommodated is adapted for rays coming from a distant object, or for parallel rays; and it must be accommodated for seeing objects which are near, the range of distinct vision extending from five or six inches to infinity. Eyes for which the greatest distance of distinct vision is finite are called short-sighted, or myopic; these eyes can only bring divergent pencils to a focus on the retina. On the other hand, On the other hand, eyes which can bring to a focus on the retina not only parallel rays but convergent pencils are called long-sighted or hypermetropic. The defects in these eyes depend on the length of the axes of the eyes; in a short-sighted eye, the axis is too long, and in a long-sighted eye it is too short. In both short-sighted and long-sighted eyes the accommodating mechanism may be quite perfect. When this is the case, their defects may be entirely remedied and the eyes made normal by the use of spectacles.

117. Let the range of distinct vision by the naked eye extend from points distant a, b from the eye. In a normal eye, b will be infinite; in a short-sighted eye b will be finite and positive, and in a long-sighted eye b

will be finite and negative. Suppose the eye to view an object through a lens of focal length f, placed close to the eye, f being positive for a collective lens, and negative for a dispersive lens. Then if x, x' be the distances from the eye (or from the lens) of an object and its image, respectively, measured outwards,

The rays striking the eye will appear to diverge from the image; and therefore the rays may be brought to a focus provided ' lies between the limits a, b. If we substitute for the values a, b in succession, the corresponding values of x will be the limits of the range of distinct vision through the spectacles.

When the accommodating mechanism is perfect, we have only to choose f, so that the farther limit of distinct vision is at an infinite distance. We must therefore make x infinite when xb, and thus we find the focal length of the spectacle glass, namely, f=-b. The nearer limit of the range of distinct vision becomes

and therefore the range of distinct vision through the spectacles will extend from ab/(b − a) to infinity.

In a short-sighted person b is finite and positive, and therefore ƒ is negative; he must therefore use dispersive lenses, generally double concave lenses, whose focal length is equal to the greatest distance of distinct vision by the naked eye. Thus if the range of distinct vision extends from 3 to 6 inches from the eye, the use of a concave lens whose focal length is 6 inches, will cause the range of distinct vision to extend from 6 inches to infinity.

On the other hand, in a long-sighted eye b is negative and therefore ƒ is positive. For example, if the range of distinct vision extend from 12 inches outwards through infinity to -12 inches, the spectacles chosen must be

collective lenses of 12 inches focal length; substituting these values in the general formula, we find that the range is then from 6 inches to infinity.

Practically, these glasses may be chosen by making the person look at a distant object; then the weakest concave glasses which will enable a short-sighted person to see this object distinctly, and the strongest convex glasses which will enable a long-sighted person to see it distinctly, are the glasses suitable to the eyes.

The limiting points of the range of distinct vision may be measured by making the person look through suitably chosen convex lenses, so that the points in question are brought within 12 inches from the eye, and then their distances can be measured on a divided scale. They are generally not the same for both eyes, so that the two eyes require different glasses.

Short-sighted persons who have to do delicate work, have sometimes to bring things close to the eyes; in this case they should use rather weaker concave glasses, than those prescribed above. For the same purpose, achromatised prismatic glasses, which are thicker towards the sides next the nose, and thinner towards the sides next the temples are used, because the objects can then be seen with less convergence of the axes of the eyes.

118. As the age of a person advances, the eye gradually loses its power of accommodation; it is supposed that the outer layers of the crystalline lens lose their elasticity, so that the lens becomes less capable of changing its form and curvature. This defect is known as presbyopia. It is entirely different from the defect described above, called long-sightedness; though aged persons are sometimes said to be long-sighted. The structure of an eye does not alter with age, so that a person with normal eyes, can still see distant objects when he becomes old; but the range of accommodation of the eye is then less than before, so that it cannot bring to a focus on the retina pencils of rays issuing from points very near to it; in other words, the nearer limit of distinct

vision has receded from the eye. Presbyopic eyes therefore need convex glasses to enable them to see near objects, as in reading or writing; but they must be laid aside to look across a room or at a distant view. Usually the glasses are chosen so as to bring the nearer limit of distinct vision to 10 or 12 inches from the eye. For very aged persons, whose sight has lost its keenness, it is sometimes advisable to use spectacles which will bring this nearer limit to within 8 or even 7 inches from the eye, so that objects may be seen under a greater angle.

From what has been said, it is evident that presbyopia may exist along with the other defects previously mentioned. Both long-sighted eyes and short-sighted eyes can be made normal by the use of spectacles, as we have When presbyopia sets in, these eyes will need two pairs of spectacles, one for walking, and another for reading and writing.

seen.

119. A convex lens of considerable aperture and magnifying power is often used as a reading glass, or for viewing the details of small objects. Such a glass may be used by both short and long-sighted people. For suppose that the glass is placed, so that the object is in the principal focus of the glass, then the rays emerging from the lens are parallel. If the glass be now moved a little nearer to the object, the emergent rays will diverge, and can be brought to a focus on the retina by a short-sighted eye; if on the other hand the glass be moved a little farther away from the object, the emergent rays will converge and will be adapted for distinct vision by a longsighted eye. In each case the magnifying power will be 1+/f, where X is the least distance of distinct vision, and f the focal length of the lens.