96 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. , 



of vision appear normal or vertical to the retinal horizon, these sur- 

 faces were of the second degree. An exact mathematical discussion of 

 them appeared in his Physiological Optics, with the addition of the 

 assumption of the asymmetry of the retina and a somewhat modified 

 definition of the identical points in the two retinas. Thus there was 

 growing up within him the conviction, based on incontrovertible mathe- 

 matical-physical reasoning, that since in the act of seeing there are two 

 channels through which the sensations come separately to the brain, 

 there to be blended to form a single perception of the material world, 

 through an act of intelligence dictated by experience, it is altogether 

 impossible to separate that part of our perception which corresponds to 

 the simple sensation from that which is the result of experience. It 

 appeared also that only in the relations of space and of time and of the 

 function derived from them, number — that is, only in mathematics — is 

 the outer and inner world the same, and that, therefore, here alone can 

 a complete correspondence between the images and the things per- 

 ceived be expected. The questions now arose, In what manner is this 

 correspondence of space and time perceptions with the things which 

 give rise to them brought about; what in these perceptions is a priori; 

 what the result of experience, and what is the origin of space percep- 

 tions in general? 



An investigation of these questions appeared in a memoir published 

 in 1868, " On the data which form the basis of geometry." Helmholtz 

 guarded himself perhaps from raising objections to Kant's conception 

 of space as a transcendental form of perception; but he made himself 

 clear as regards perception by means of the senses. Thus, for example, 

 it results from the very organization of our eyes that all which we see 

 must consist in a distribution of colors upon a surface, and this dis- 

 tribution of colors does not necessarily condition any particular series 

 of phenomena of time or space. The question might then be raised 

 whether for this form of transcendental space perception the assump- 

 tion is involved that after, or at the same time with a given space per- 

 ception, another determined by it must appear; or, in other words, 

 whether the assumption of certain axioms is necessarily implied. In 

 the endeavor to distinguish between the logical development of geom- 

 etry and the results derived from experience, which are apparently 

 necessary to the processes of thought, he recognized as the basis of 

 all the demonstrations of the geometry of Euclid the proof of the con- 

 gruence of figures in space, and therewith as a postulate the supposi- 

 tion that these figures can be brought together without alteration in 

 their form or dimensions. He was then confronted with the question 

 whether the assumption of the possibility of free movement, which 

 we have experienced from our earliest youth, contains no logically 

 unproved hypothesis, and by profound investigation of the question he 

 was able to show with a very great degree of certainty that it did not. 



Helmholtz depicted geometry limited to two dimensions, as it might 



