Bauhaus-Universität Weimar

Helmholtz's treatise on physiological optics. Volume 1. Edited by James P. C. Southall. Translated from the 3rd German edition
Helmholtz, Hermann von
Dioptries of the Eye 
[231, 232. G. 
ished until the line in question becomes so short that it is indistinguish¬ 
able from a piece of any other curved line. The general constitution 
of a bundle of optical rays may be defined also by the fact that the 
two focal lines with respect to any given ray are intersected by all the 
contiguous rays (disregarding infinitesimal aberrations of order higher 
than the first); but this statement requires an explicit definition of 
what is meant by a contiguous ray. Draw the ray corresponding to a 
point on the ruled surface at a finite distance from the given ray, 
and do the same thing for points nearer and nearer the given ray ; then 
ultimately the distance between the point and the given ray will 
vanish entirely, and the ray corresponding to this point will be con¬ 
tiguous to the given ray. But this definition will be fundamentally 
misunderstood if the conclusion is deduced that all the rays of an 
infinitely narrow bundle go approximately through two Sturm focal 
Since (neglecting infinitesimals of order higher than the first) all 
the contiguous rays of a bundle of rays which is anastigmatic along a 
certain ray meet this ray in the focal point, there occurs in this case a 
perfect ray-convergence of the first order ; and the focal point is the optical 
reproduction or image of the point where the light originates. Hence, 
in case of a stop of finite aperture, the optical imagery cannot be said 
to be due to the fact that all the rays coming from an object-point go 
approximately through the image-point, but merely to the fact that 
contiguous rays intersect at this point, so that the concentration of the 
light here is infinitely great as compared with that at a point at a finite 
distance from this spot. The imagery supposed in the first case above 
is indeed mathematically correct for an infinitely narrow bundle of 
rays, but since it is physically impossible on account of diffraction 
effects to produce an optical image by such means, this imagery repre¬ 
sents merely an ideal dreamed of in olden times; whereas the latter 
case mentioned above describes exactly the actual process as it takes 
place. If the reality here appears to be little different from the ideal, 
the explanation is to be found in the circumstance that both in the 
eye and likewise on the photographic plate shades of brightness are of 
more importance than absolute brightness. 
The criterion of actual optical imagery is just this perfect ray- 
convergence of the first order. As is evident from the foregoing, this 
convergence of the rays is susceptible of mathematical investigation 
only along definite rays. The rays chosen for this purpose are the chief 
rays defined above. Assuming that the chief ray corresponding 
to each point of the surface of the object has been constructed 
and traced through the optical system by trigonometrical calcula¬ 
tion, and that somewhere in the path of the light a screen is


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