FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 99 



and hue for which the sum of the perceptible differences is a mini- 

 mum. This conception of the field of color sensation led him, in the 

 memoir on "The extinction of the application of the law of Fechner 

 in the color system," to extend this law, which contemplated only 

 changes of brightness in a light of a constant color, to embrace a 

 diversity of cases of more than one dimension, and to take into con- 

 sideration the greatness of successive gradations of the tone and of 

 the saturation of colors corresponding to change in the brightness. 



In one of his last investigations u On the cause of the correct inter- 

 pretation of sensory impressions," he returned again to the question of 

 space perception, and was led to very acute and significant philosoph- 

 ical considerations upon the subject. The perception of tbe stereo- 

 metric form of a material object played for him the role of one of a 

 great number of cases of impressions derived through the senses which, 

 quite independently of the geometrical definition, can only be brought 

 together by an understanding of the law in accordance with which the 

 perspective impressions follow each other. Starting with this view, he 

 recognized our unconscious mental activity as the cause of the merging 

 together of separate impressions with results essentially like those of 

 our conscious thinking. According to Helmholtz, the conclusions of 

 induction are nothing more nor less than the expectation that the phe- 

 nomena observed in their beginning will proceed in a way correspond- 

 ing with our previous observations, and false induction is identical 

 with deceptions of our senses; so that our science is only the expres- 

 sion in words of such knowledge as, with our natural organization and 

 with the help of the conclusions of induction depending on the uncon- 

 scious activity of our minds, we are able to collect. 



Long after the publication of his memoirs on the axioms of geometry 

 he returned again to a similar subject in his investigation of the theory 

 of the conception of number and measure, in a paper dedicated to 

 Edward Zeller, in 1887, on the fiftieth anniversary of taking his doc- 

 tor's degree. In this paper he opposed the view of Kant, that the 

 axioms of arithmetic are laws given a priori, which determine the 

 transcendental perception of time in the same sense that the axioms of 

 geometry govern that of space. He investigated the significance and 

 correctness of calculation with pure numbers and the possibility of 

 tbeir application to physical quantities. As he derived from consid- 

 ering numeration that we are able to retain in mind the order of suc- 

 cession in which acts of consciousness are performed, the science of 

 pure numbers was for him essentially a method built up on physio- 

 logical facts for the consistent application of a system of signs of 

 unbounded capacities for extension and improvement, with the purpose 

 of representing the different methods of combining these signs to reach 

 the same final conclusion. After deducing from this conception a defi- 

 nition of the regular series of positive whole numbers and the signifi- 

 cance of their succession, he proceeded to establish the conception of 



