40 SCIENCE AND METHOD. 



become aware of the similarity of their form in spite 

 of the dissimilarity of their matter, when they have 

 modelled themselves upon each other in such a way 

 that each could profit by the triumphs of the other. 

 At the same time we should look to concurrences of 

 a similar nature for progress in the future. 



Arithmetic. 



The progress of arithmetic has been much slower 

 than that of algebra and analysis, and it is easy to 

 understand the reason. The feeling of continuity is 

 a precious guide which fails the arithmetician. 

 Every whole number is separated from the rest, and 

 has, so to speak, its own individuality ; each of them 

 is a sort of exception, and that is the reason why 

 general theorems will always be less common in 

 the theory of numbers, and also why those that do 

 exist will be more hidden and will longer escape 

 detection. 



If arithmetic is backward as compared with algebra 

 and analysis, the best thing for it to do is to try to 

 model itself on these sciences, in order to profit by 

 their advance. The arithmetician then should be 

 guided by the analogies with algebra. These analo- 

 gies are numerous, and if in many cases they have 

 not yet been studied sufficiently closely to become 

 serviceable, they have at least been long foreshadowed, 

 and the very language of the two sciences shows 

 that they have been perceived. Thus we speak of 

 transcendental numbers, and so become aware of 

 the fact that the future classification of these numbers 

 has already a model in the classification of transcen- 

 dental functions. However, it is not yet very clear 



