220 GREAT MEN OF SCIENCE 



calculus invented by Newton and Leibniz and since still 

 further developed. 



This calculus had already been of extensive service to 

 scientific research. Even before the time of Scheele and 

 Watt (from 1759 onwards), the development of the differential 

 equations of hydrodynamics, for example, was begun. 

 Here the answer to all questions whatever concerning the 

 motion of liquids is referred to the mathematical treatment 

 of these equations, which themselves, however, contain only 

 the fundamental laws of Galileo and Newton concerning all 

 motion, applied to the single space element of the liquid. 

 All that is required in addition is certain equations which 

 contain special and simple facts of experience, such as the 

 small compressibility of liquids and the like, and also special 

 statements relating to the case considered. Though these 

 equations, when skilfully handled, answer questions which 

 might otherwise appear insoluble, for example those relating 

 to all details of waves upon liquids, and though they go in 

 their generality far beyond the consideration of characteristic 

 and hence important cases already dealt with by Newton, 

 they nevertheless contain only old knowledge; nothing funda- 

 mentally new is added or made evident by their treatment. 

 Nothing else can possibly be the case; for mathematics is 

 throughout scientific research, simply a tool, enabling us to 

 apply knowledge derived from the observation of accessible 

 and generally simple cases, in a completely logical manner 

 and without danger of mental error, to all other cases whatso- 

 ever, no matter how complicated; but knowledge of nature 

 can only be derived from observation. 



It is the special virtue of the art of mathematics that, when 

 correctly applied without arbitrariness, it keeps away all 

 foreign matter, and only allows that to take effect which was 

 originally derived from observation and put into the equa- 

 tions. If the final result then leads to matters which excite 

 astonishment, as when for example Laplace concludes that 



