376 HERMANN VON HELMHOLTZ 



activity, by the essay on Natural Philosophy entitled ' Numbers 

 and Measurements treated from the Epistemological Point of 

 View ', and dedicated to his friend Eduard Zeller for the fiftieth 

 year of his doctorate. This paper was an important amplifica- 

 tion of the empiricist theory which he had previously put 

 forward, that the axioms of geometry can no longer be regarded 

 as propositions incapable of and not requiring proof, and also 

 established this theory with respect to the origin of the arith- 

 metical axioms, which stand in the corresponding relation to 

 the conceptional forms of time. 



The five well-known axioms of arithmetic (i) when two 

 magnitudes are equal to a third they are equal to each other ; 

 (2) the associative law of addition, (a + b) + c = a + (b + c) ; (3) the 

 commutative law of addition, a + b = b + a ; (4) if equals be added 

 to equals, the wholes are equal ; (5) if equals be added to unequals, 

 the wholes are unequal were then examined in regard to their 

 independence of each other, and in their relation to experience. 

 While he derives numbers from the fact that we are able to 

 carry in our memory the sequences in which the acts of 

 consciousness have followed one another in time, the theory of 

 pure numeration is for him merely a method built up on psy- 

 chological facts, for the logical application of a system of signs, 

 unlimited in its extension and possibility of refinement, with 

 the object of representing the different modes of association of 

 these signs, all of which tend to the same ultimate result. After 

 the definition thus obtained of the ordered series of the positive 

 whole numbers and of the unequivocal character of their 

 succession, he established the concept of the addition of pure 

 numbers, in the first place explaining the signs by saying that 

 if any number be designated by the letter a, the next following in 

 the normal series shall be (a -f- 1), while the definition of (a + b) is 

 that number of the principal series which is reached by counting 

 One for (fl + i), Two for [(a + i)+i)], and so on, until one has 

 counted up to b. And then he shows how this idea of the 

 addition of pure numbers confirms the arithmetical axioms of 

 the equality of two numbers in relation to a third, the associ- 

 ative law of addition, and the commutative law merely by 

 the agreement of the result with that which can be derived 

 from the numbers of external enumerable objects. Figures 

 are to him arbitrary signs, to establish the time order of 



