THE SENSE OF SIGHT. 651 



enter the eye from objects situated progressively nearer the eye, a separate 

 calculation is necessitated for each distance for the determination of the 

 size of the image. 



A method by which these difficulties are much reduced was suggested 

 by Gauss and developed by Listing. It was demonstrated by Gauss that 

 in every complicated system of refracting media separated by centered 

 spheric surfaces there may be assumed certain ideal or cardinal points, to 

 which the system may be reduced, and which, if their relative position and 

 properties be known, permit of the determination, either by calculation 

 or geometric construction, of the path of the refracted ray, and the position 

 and size of the image in the last medium, if those of the object in the first 

 medium be known. 



Every dioptric system can be replaced, as Gauss showed, by a single 

 system composed of six cardinal points and six planes perpendicular to the 

 common axis e.g., two focal points, two principle points, two nodal points, 

 two focal planes, two principal planes, and two nodal planes. 



Properties of the Cardinal Points. The first focal point, F v in Fig. 

 309, has the property that every ray which before refraction passes through 

 it, after refraction is parallel to the axis. 



The second focal point, F 2 , has the property that every ray which before 

 refraction is parallel to the axis, passes after refraction through it. 



The second principal point, H 2 , is the image of the first, H v " that is, 

 rays in the first medium which go through the first principal point pass after 



Jf, 



FIG. 309. DIAGRAM SHOWING THE POSITION AND RELATION OF THE CARDINAL POINTS. 



the last refraction though the second. Planes at right angles to the axis at 

 these points are principal planes. The second principal plane is the image 

 of the first. Every point in the first principal plane has its image after 

 refraction at a corresponding point in the second principal plane at the same 

 distance from the axis and on the same side. 



The second nodal point, N 2 , is the image of the first, A7\: a ray which in 

 the first medium is directed to the first nodal point passes after refraction 

 through the second nodal point, and the direction of the rays before and 

 after refraction are paralled to each other. In Fig. 309 let A B represent 

 the axis. The distance of the first focal point, F v from the first principal 

 plane, H v is the anterior focal distance. The distance of the second focal 

 point, F 2 , from the second principal plane, H 2 , is the posterior focal distance. 

 The distance of the first nodal point, N v from the first focal point, F v is equal 

 to the posterior focal distance H 2 F 2 . The distance of the second nodal 

 point, N 2 , from the second focal point, F 2 , is equal to the anterior focal 

 distance, H 1 F 1 . It is evident, therefore, that the distance of the corre- 



