This section is from the book "Complete Self-Instructing Library Of Practical Photography", by J. B. Schriever. Also available from Amazon: Complete Self-Instructing Library Of Practical Photography.
Forms Of Lenses. Lenses are divided into two classes: positive or converging lenses which bend the rays toward the axis; negative or diverging lenses which bend the rays away from the axis. The various forms of lenses are shown in Fig 7. Positive lenses are, I., double convex; II., piano convex; III., convex meniscus. Negative lenses are, IV., double concave; V., piano concave; VI., concave meniscus.
587. A lens surface is either plane or curved, and the curved surface is a perfect portion of a sphere. The principal axis (often called only the axis) of a lens is a straight line passing through the centers of curvature if both surfaces are curved, or the center of curvature and the center of the plane surface. A ray of light coinciding with the principal axis will pass straight through the lens, without deviation. When two or more lenses are used together, either in one combination or in different combinations, their principal axes must coincide - there must be one common axis for all of them. To so adjust them - center them - is a very delicate operation.
Astigmatism - Anastigmat. Astigmatism is the inability of a lens to render at the same time a sharp image of horizontal and vertical lines situated in the same plane. It exists only in oblique pencils, - pencils of light passing through the lens obliquely, not parallel with the axis - and affects the marginal definition. It is thus possible for a lens which is astigmatic to render both horizontal and vertical lines equally sharp at the same time in the center of the plate, but only one set at a time, either the horizontal or the vertical, at the margin of the plate. When a lens is free from astigmatism it is termed an anastigmat.
589. The following is a simple test for astigmatism. Draw a black cross on a white card and set it up in front of the lens so that one bar is horizontal and the other one vertical. Focus it on the center of the ground-glass, and both bars will be equally sharp at the same time. Then move either the camera or the card to one side, until the image of the cross is near one corner of the ground-glass. If both bars show equally sharp at the same time, the lens gives freedom from astigmatism over an area at least as large as the ground-glass. If astigmatism is present, only one bar can be focused sharply at a time. The distance which the ground-glass must be moved to bring the other bar in focus, measures the degree of astigmatism, the astigmatic difference.
590. A set of rays emanating from one point should be brought together in one point and form one image point. However, if they pass obliquely through the lens, they are not brought to one point, but to two lines (or properly speaking, two very attenuated ellipses) some distance apart and at right angles to each other. The image point is extended into a vertical line (or ellipse) nearer the lens, and into a horizontal line (or ellipse) farther from the lens. Somewhere between the two, at a point where a cross section of the rays is most nearly circular (the "circle of least confusion") is the best mean focus. The smaller this cross section is, the more nearly it becomes a point and the less apparent is the effect of astigmatism.
591. Being confined to oblique pencils, unless the lens is very poorly made, astigmatism is not present in the center of the field. It increases toward the margin, and, as with spherical aberration, the difficulty of correcting it increases as we approach nearer to the margin and as the lens is given a larger relative opening.