3 ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



tention. All the notions of mechanics, and consequently of all 

 the other departments of physics, are composed, by multiplication 

 or division, of three fundamental notions length, time, and mass. 

 That is to say, to the notions of geometry resting on length and its 

 powers, two other fundamental notions, time and mass, are added, 

 which, joined to that of length, lead to the notions of force, work, 

 horse-power, atmospheric pressure, etc. The knowledge of me- 

 chanics, thus, highly certain though it be, is rendered less certain 

 than that of geometry and a fortiori than that of arithmetic. The 

 uncertainty of knowledge continues to increase in branches which 

 are still more remote from mathematics, owing to the increasing com- 

 plexity of the observational material which must here be put to the 

 test. 



Still, although mathematical knowledge does not lead to abso- 

 lutely certain results, it yet invests known results with incomparably 

 greater trustworthiness than does the knowledge of the other sci- 

 ences. But after all, it remains a useless accumulation of capital 

 so long as it is not turned to practical account in other sciences, 

 such as metaphysics, physics, chemistry, biology, political economy, 

 etc. Hence also arises an obligation on the part of the other sci- 

 ences, so to shape their problems and investigations that they can 

 be made susceptible of mathematical treatment. Then will mathe- 

 matics gladly perform her duty. The moment a science has ad- 

 vanced far enough to permit of the mathematical formulation of its 

 problems, mathematics will not be slow to treat and to solve these 

 problems. Mathematical knowledge, aristocratic as it may' appear 

 by the greater certainty of its results, will, so far as the advance- 

 ment of human kind is concerned, never be more than a useless 

 mass of self-evident truths, unless it constantly places itself in the 

 service of the other sciences. 



