ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
Ill 
The proper numbering of lenses in dioptrics has been an enormous gain 
in one branch of the optical industries — that of the ophthalmists — and 
it is much to be wished that other lenses than those of spectacles should 
also be henceforth described in the same way. 
Another important step has been the introduction of the conception 
of focal planes — a conception to which the use of the photographic 
camera has doubtless contributed its share. Less fully recognized, but 
none the less important, is the conception introduced by Gauss as the 
result of his studies in geometrical optics, that the properties of any 
centered combination of lenses might be represented by a system of 
planes and points which are, so to speak, the characteristics of the 
equivalent lens. A lens can be considered as having a single optical 
centre only when it is infinitely thin, or at least of negligible thickness. 
All thick lenses and combinations of lenses have two optical centres, 
described by Gauss as the two “principal” points (“ Hauptpunkte ”), 
which are considered as the places where the axis intersects two “ princi- 
pal planes.” These principal planes are at a certain distance apart, and 
equidistant between the two principal foci at the back and front of the 
lens. They possess certain properties most useful in the geometrical 
treatment of lens problems, and act as though the light, however 
obliquely it may be crossing the lens, were transferred straight from one 
to the other. The two “ principal ” points, or optical centres, possess 
the property that light pro- 
ceeding from any direction 
towards the one of these 
points passes out from tho 
lens combination as though 
it had passed through the 
other. Fig. 8 shows a thick 
lens (in diagram), in which 
the two principal points of 
Gauss are marked Hj and 
IF, with the two principal 
planes drawn through them. 
These two points, together 
with the two principal foci, 
completely determine the action of the lens. When the positions of 
these four * cardinal points are known for any lens or lens-system, then 
all is known that, is necessary for a complete discussion of the formation 
of images for all objects lying near the principal axis. The true focal 
length is the distance from either of the two principal planes to the 
corresponding principal focus ; the back focus and the front focus being 
for other purposes than focal powers — for example, to express the curvatures of 
surfaces. Unit curvature is taken as the curvature of a circle of one metre radius. 
Or a lens having a metre focal length is described in modern language as having 
a power of one dioptrie. A lens having half a metre as focal length has a power of 
two dioptries. 
* If the lens system is not bounded by similar media on its two faces — for ex- 
ample, if one side is bounded by air and the other by water — the two optical ceutres 
are shifted away from the two principal planes towards the denser medium, and aro 
kuown as “ nodal ” points. In this case there are six cardinal points to consider. 
