ON THE NATURE OF MATHEMATICAL KNOWLKDCE. 29 



their mathematical successors, required qualification. But this did 

 not invalidate at all the correctness of the method of differentiation, 

 nor its application in all practical cases ; the theoretical specula- 

 tions pursued on this subject simply clarified ideas and sifted out 

 the conditions upon which differentiability depended. Happily the 

 gifted minds who invent the new methods and open up the new 

 paths of research in mathematics, are not deterred by the fear that 

 a subsequent generation gifted with unusual acumen will spy out 

 isolated cases in which their methods fail. Happily the creators 

 of the differential calculus pushed onward without a thought that a 

 critical posterity would discover exceptions to their results. In 

 every great advance that mathematics makes, the clarification and 

 scrutinisation of the results reached are reserved necessarily for a 

 subsequent period, but with it the demonstration of those results is 

 more rigorously established. Despite all this, however, in no sci- 

 ence does cognition bear so unmistakably the imprint of truth as in 

 pure mathematics. And this fact bestows on mathematics its con- 

 servative character. 



This conservative character again is displayed in the objects of 

 mathematical research. The physician, the historian, the geogra- 

 pher, and the philologist have to-day quite different fields of inves- 

 tigation from what they had centuries ago. In mathematics, too, 

 every new age gives-birth to new problems, arising partly from the 

 advance of the science itself, and partly also from the advance of 

 civilisation, where improvements in the other sciences bring in their 

 train new problems that are constantly taxing afresh the resources 

 of mathematics. But despite all this, in mathematics more than in 

 any other science problems exist that have played a role for hun- 

 dreds, nay, for thousands of years. 



In the oldest mathematical manuscript which we possess, the 

 Rhind Papyrus of the British Museum, which dates back to the 

 eighteenth century before Christ, and whose decipherment we owe 

 to the industry of Eisenlohr, we find an attempt to solve the prob- 

 lem of converting a circle into a square of equal area, a problem 

 whose history covers a period of three and a half thousand years. 

 For it was not until 1882 that a rigorous proof was given of the im- 



