Curvature of field results from the tendency of oblique rays in objectives, otherwise well corrected, to come to shorter focus than axial rays, from their more considerable refraction resulting from greatly increased angles of incidence. This applies to both the astigmatic image surfaces, which are concave toward the objective in all ordinary cases. Fortunately both these faults are negligible near the axis. They are both proportional to tan²{u}/f where u is the obliquity to the axis and f the focal length; turn up with serious effect in wide angled lenses such as are used in photography, but may generally be forgotten in telescopes of the ordinary _F_ ratios, like _F_/12 to _F_/16. So also one may commonly forget a group of residual aberrations of higher orders, but below about _F_/8 look out for trouble. Objectives of wider aperture require a very careful choice of special glasses or the sub-division of the curvatures by the use of three or more lenses instead of two. Fig. 57 shows a cemented triplet of Steinheil’s design, with a crown lens between two flints. Such triplets are made up to about 4 inches diameter and of apertures ranging from _F_/4 to _F_/5. [Illustration: FIG. 57.—Steinheil Triple Objective.] [Illustration: FIG. 58.—Tolles Quadruple Objective.] In cases of demand for extreme relative aperture, objectives composed of four cemented elements have now and then been produced. An example is shown in Fig. 58, a four-part objective of 1 inch aperture made by Tolles years ago for a small hand telescope. Its performance, although it worked at _F_/4, was reported to be excellent even up to 75 diameters. The main difficulty with these objectives of high aperture is the relatively great curvature of field due to short focal length which prevents full utilization of the improved corrections off the axis. Distortion is similarly due to the fact that magnification is not quite the same for rays passing at different distances from the axis. It varies in general with the cube of the distance from the axis, and is usually negligible save in photographic telescopes, ordinary visual fields being too small to show it conspicuously. Distortion is most readily avoided by adopting the form of a symmetrical doublet of at least four lenses as in common photographic use. No simple achromatic pair gives a field wholly free of distortion and also of the ordinary aberrations, except very near the axis, and in measuring plates taken with such simple objectives corrections for distortion are generally required. At times it becomes necessary to depart somewhat from the objective form which theoretically gives the least aberrations in order to meet some specific requirement. Luckily one may modify the ratios of the curves very perceptibly without serious results. The aberrations produced come on gradually and not by jumps. [Illustration: FIG. 59.—“Bent” Objective.]