When a focussing screen intersects an oblique pencil in a certain place a point may be represented by an oval pointing towards the principal axis of the lens. Moving the screen slightly, another oval is then found with an axis at right angles to that of the first one, and somewhere between these two ovals the point is represented by a disc. These appearances can usually be observed with any cheap so-called "rectilinear" doublet lens, and they are a sign that spherical aberration is complicated by the effect known as astigmatism, which, with photographic lenses, can be produced in oblique pencils only.

The two ovals are the astigmatic foci of the oblique pencil, and the disc is the mean focus. The oval pointing towards the principal axis is the " secondary " or " radial' focus; the other is the " primary " or " tangent " focus. If the primary focus is nearest the lens, then the aberration is under-corrected or " positive," but if the foci are reversed it is over-corrected or " negative," this being the usual state of affairs.

As there are three foci in each oblique astigmatic pencil there must be three image fields. One is formed by radial foci, and in it an object line radiating from the principal axis will be fairly well defined, because each point of the line is extended mainly in a longitudinal direction. A line at right angles to a radial line, or a tangent line, will, however, be blurred, as each point is extended laterally. In the image field of the tangent foci radial lines are blurred laterally, while tangent lines are comparatively sharp. In the field of the mean foci all lines are equally blurred to a small extent, as every point is represented by a disc. Only one of the three fields can be flat, and if that is either the radial or tangent field, then only lines running in one particular direction can be in focus on a flat plate; it is, therefore, preferable that the mean field should be flat if the astigmatism is not corrected, and a flat mean field is the condition usually aimed at by the optician.

When astigmatism is perfectly corrected there is, of course, only one image field, which may be either curved or flat, and good definition will be possible in any part of that field if the removal of the astigmatism leaves no residue of spherical aberration. A corrected lens is now generally styled an " anastigmat," but the name only is new, not the condition that it expresses. The modern anastigmats are distinguished by the fact that they are rapid and have a flat field, whereas the older ones when rapid had a violently curved field that rendered them practically useless, and a flat field anastigmat was extremely slow. The only well known specimen of the old type of anastigmat belongs to the second class, and is called the Concentric lens. It is a very fine lens so far as correction is concerned, but its aperture is only about a quarter of the diameter of that of a modern anastigmat. New methods and new materials have enabled opticians to overcome the defects that once appeared to be inevitable, and the modern anastigmat is not only of extreme rapidity, but well corrected for all errors.

By Edgar H. Carpenter.