FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 95 



and time are transcendental forms of perception which are more exactly 

 determined by the axioms was almost universally accepted among 

 mathematicians and philosophers till Helmholtz brought the matter so 

 far within the scope of his investigations as to raise question with 

 regard to the a priori existence of these axioms. His objections were 

 based, not on the ground of abstract mathematical considerations, as 

 had been the case in part with those of Gauss and Eiemann, but 

 physiological-optical researches had caused him to consider the 

 source of the general perception of space, and he was very soon led to 

 the conviction that only the appearance of space relations causes us to 

 grant as self-evident that which, in reality, is a particular character- 

 istic of our exterior surroundings, and we therefore regard the axioms 

 of geometry as laws given by transcendental perception. 



As early as 1852, in his academic dissertation, delivered at Konigs- 

 berg, "Upon the nature of human sensations," he showed by a thorough 

 physiological-physical comparison of objects and their corresponding 

 sensations, that light and color perceptions are only symbols for the 

 actually existing relations; and thus he paved the way for further 

 progress in the investigation of the nature of sense perceptions. 

 According to what may be termed the "nativistic" theory of form 

 perception, it is assumed that the retina itself discerns impressions 

 with reference to its own surface, particular forms being therefore 

 distinguished by means of an innate mechanism, and thus the special 

 localization of each impression is given by the simple perception. 

 Opposed to this is the empiric theory of which Helmholtz was the 

 originator. By this theory, on the other hand, the sense-perceptions 

 serve only as the symbols to our consciousness of outward things 

 and events, whose significance is referred to our judgment. Follow- 

 ing this theory it would, for example, be unnecessary that in the 

 perception of difference of position there should be any similarity 

 between the local sight-symbol for this interval and the correspond- 

 ing external difference of position; or that in general an exact cor- 

 respondence should exist between the laws of thought and percep- 

 tion and those of the outside world. These physiological- philosophical 

 views were elaborated in a series of researches, interesting also from a 

 mathematical standpoint, which appeared in the years 1862 to 1864 

 under the title " Upon the horopter." The horopter was defined by 

 him as the geometrical position of those points in space whose images 

 are formed upon corresponding parts of the two retinas, and therefore 

 are perceived as single. The general form was found by him to be a 

 curve in space of the third degree. Designating as a line horopter the 

 surface upon which right lines of a given direction must lie in order 

 that during a continuous congruent displacement upon this surface the 

 images of the whole lines shall correspond, without necessary corre- 

 spondence of the separate points, it was shown by Helmholtz that for 

 the vertical and horizontal horopters, or for lines which in both fields 



