184 Prof. J. LeConte on some Phenomena 



tancc the verticals approach but do not attain parallelism ; 

 within the proper distance they cross in a direction contrary to 

 that in the diagram. When the circles are ten inches apart, the 

 proper distance is nearly three feet, and the image therefore 

 about seven inches from the eyes. 



Helmholtz has a diagram similar in all respects to my own, 

 except turned upside down, in which, he states, both verticals 

 and horizontals coincide perfectly when the circles are combined. 

 Our own figure (fig. 7) turned upside down will answer for Pro- 

 fessor Helmholtz's. We quote his own words : — " The horizontal 

 lines arc parts of the same straight line ; the vertical lines are 

 not perfectly vertical. The upper end of those of the right 

 figure are inclined to the right, and those of the left figure to the 

 left, by about 1%°" But his experience differs from our own in 

 a most unaccountable manner. He says : " Now combine the 

 two sides stercoscopically, either by squinting or by a stereoscope, 

 and you will see that the white lines of the one coincide with the 

 black lines of the other as soon as the centres of both figures 

 coincide, although the vertical lines of the two figures are not 

 parallel to each other." He accounts for this, not by rotation 

 of the eyes, but by the principle of the difference between real and 

 apparent verticality. The ignorance of this principle he believes 

 has vitiated the results of all previous observers. He illustrates 

 this principle thus : " When you draw on paper a horizontal 

 line, and another line crossing it exactly at right angles, the 

 right superior angle will appear to your right eye too great and 

 to your left eye too small ; the other angles show corresponding 

 deviations. To have an apparently right angle, you must make 

 the vertical line incline by an angle of about \\° for it to appear 

 really vertical. We must distinguish, therefore, between the 

 really vertical lines and the apparently vertical lines in the field 



of view Now look alternately with the right and the left 



eye at these figures (fig. 7 turned upside down). You will find 

 that the angles of the right figure appear to the right eye equal 

 to right angles, and those of the left figure so appear to the left 

 eye ; but the angles of the left figure appear to the right eye to 

 deviate much from a right angle, as also do those of the right 

 figure to the left eye." Professor Helmholtz therefore believes 

 that the perfect stereoscopic coincidence of the vertical lines of 

 his diagram is the result of this principle. "Therefore," he 

 says, " not the really vertical meridians of the two fields corre- 

 spond as has been hitherto supposed, but the apparently vertical 

 meridians. On the contrary, the horizontal meridians really cor- 

 respond, at least for normal eyes which are not fatigued." 



On this principle Professor Helmholtz builds his whole theory 

 of the horopter. But that this principle cannot account for the 



