Distance given by Binocular Vision. 311 



be strained so as to unite the points A, B. Tlie united image 

 of these points will be seen at the binocular centre D'", and 

 the united lines AC, BC will have the position D"'C. In like 

 manner, when the eye descends to E'", E', E, the united image 

 D"'C will rise and diminish, taking the positions D"C, D'C, 

 DC till it disappears on the line CM, when the eyes reach 

 M. If the eye deviates from the vertical plane GMN the 

 united image will also deviate from it, and is always in a plane 

 passing through the eye and the line GM. 



If at any altitude EM the eye advances towards ACB in 

 the line EG, the binocular centre D will also advance towards 

 ACB in the line EG, and the image DC will rise and become 

 shorter as its extremity D moves along DG, and after passing 

 the perpendicular to GE it will increase in length. If the 

 eye, on the other hand, recedes from ACB in the line GE, 

 the binocular centre D will also recede, and the image DC 

 will descend to the plane CM and increase in length. 



The preceding diagram is, for the purpose of illustration, 

 drawn in a sort of perspective, and therefore does not give 

 the true positions and lengths of the united images. This 

 defect however is remedied in fig. 3, where E, E', E", E'" is 

 the middle point between the two eyes, the plane GMN being, 

 as before, perpendicular to the plane passing through ACB. 

 Now, as the distance of the eye from G is supposed to be the 

 same, and as A B is invariable as well as the distance between 

 the eyes, the distance of the binocular centres O, D, D', D", 

 D'", P, from G will also be invariable, and lie in a circle ODP 

 whose centre is G, and whose radius is GO, the point O being 



determined by the formula G O = GD = XBTTITL ' ^^"'^®» 

 in order to find the binocular centres D, D', D", D'", &c. at 

 any altitude E, E', &c., we have only to join EG, E'G, &c. 

 and the points of intersection D, D', &c. will be the binocular 

 centres, and the lines DC, D'C, &c. drawn to C, will be the 

 real lengths and inclinations of the united images of the lines 

 AC, BC. 



When GO is greater than GC there is obviously some 



angle A, or E"GM at which D"C is perpendicular to GC. 



/-I /-I 



This takes place when cos A= ^^i. When O coincides with 



C, the images CD, CD', &c. will have the same positions 

 and magnitudes as the chords of the altitudes A of the eyes 

 above the plane GC. In this case, the raised or united images 

 will just reach the perpendicular when the eye is in the plane 

 GCM, for since GC = GO, cos A = l, and A = 0°. 



When the eye at any |)osition, E" for example, sees the. 



