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landscape lens, an inverted meniscus is used. It generally turns out, 
however, that when an advantage is gained in one direction something 
is lost in another. In this instance, spherical aberration is introduced 
by this form of lens, so the correction of the third error is accom- 
plished by introducing the second. In a lens which has a flat field, 
a want of sharpness may be noticed in images formed by even direct 
axial pencils. 
The fourth error is distortion ; this, which is commonly called pin- 
cushion and barrel distortion, is caused by the too great refraction of 
oblique centrical pencils. It is corrected by opposing two lenses of 
symmetrical form to one another. It should be noted, however, that 
these lenses may be, and often are, asymmetrical both as to their foci 
and as to chromatic corrections ; all that is necessary is that they should 
be symmetrical in their outward form or nearly so. 
The fifth, the astigmatic error — one most difficult to correct — is 
caused by the focal lines of oblique and excentrical pencils. The 
mathematical analysis of this subject is both complicated and prolix. 
The corrections are obtained by combining glasses possessing rela- 
tively high dispersion and low refraction with those of low dispersion 
and high refraction. This is now rendered possible, owing to the 
production by the Jena firm of glasses having their ratios of refraction 
to dispersion different from any hitherto made. 
To very briefly explain astigmatism : — When a small oblique pencil 
is refracted by a lens, the rays are not brought to a focal point, but 
to a line, and afterwards to another line ; these lines are called the 
primary and secondary focal lines. When a lens has negative aberra- 
tion, that is, when the lens brings aberrated rays to a focal point 
situated between the lens and its geometrical focus, then the primary 
focal line lies between the lens and the secondary focal line ; but 
when the lens has positive aberration, that is, brings aberrated rays 
to a focal point beyond its geometrical focus, then the secondary focal 
line lies between the lens and the primary focal line. In general, 
the circle of least confusion lies half-way between the focal lines. 
Now, a plane passing through the axis of the lens and the axis 
of the small oblique incident pencil, is called the primary plane, and 
the primary focal line is always perpendicular to this plane ; but the 
secondary focal line lies in it ; therefore the focal lines are at right 
angles to one another. If now points are imaged by lines, it is very 
easy to see how the image of a lined object in a certain direction 
would be blotted out. By means of a hole in a card, a small oblique 
beam of sunlight may be passed excentrically through a hand magni- 
fying glass, and the focal lines be conveniently studied by receiving 
their images on a piece of paper. 
These focal lines have nothing whatever to do with spherical 
aberration, as some have supposed, for they can be originated by 
pencils falling obliquely on a plane refracting surface. It is here 
that we meet one of the differences between reflection and refraction ; 
