32 ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 



Lambert succeeded in 1761 in demonstrating the irrationality of the 

 ratio of the circumference of a circle to its diameter, or, which is 

 the same thing, the irrationality of the ratio of the area of a circle 

 to the area of the square on its radius. Afterwards, Lambert also 

 supplied a proof that it was impossible for this ratio to be the square 

 root of a rational number. But this was the first step only in a long 

 journey. The attempt to prove that the old problem is insoluble 

 was still destined to fail. An astounding mass of mathematical in- 

 vestigations were necessary before the demonstration could be suc- 

 cessfully accomplished. 



As we see, the majority of the mathematical truths now pos- 

 sessed by us presuppose the intellectual toil of many centuries. A 

 mathematician, therefore, who wishes to-day to acquire a thorough 

 understanding of modern research in this department, must think 

 over again in quickened tempo the mathematical labors of several 

 centuries. This constant dependence of new results on old ones 

 stamps mathematics as a science of uncommon exclusiveness and 

 renders it generally impossible to lay open to uninitiated readers a 

 speedy path to the apprehension of the higher mathematical truths. 

 For this reason, too, the theories and results of mathematics are 

 rarely adapted for popular presentation. There is no royal road to 

 the knowledge of mathematics, as Euclid once said to the first 

 Egyptian Ptolemy. This same inaccessibility of mathematics, al- 

 though it secures for it a lofty and aristocratic place among the sci- 

 ences, also renders it odious to those who have never learned it, and 

 who dread the great labor involved in acquiring an understanding 

 of the questions of modern mathematics. Neither in the languages 

 nor in the natural sciences are the investigations and results so 

 closely interdependent as to make it impossible to acquaint the un- 

 initiated student with single branches or with particular results of 

 these sciences, without causing him to go through a long course of 

 preliminary study. 



The third trait which distinguishes mathematical research is its 

 self-sufficiency. In philology the field of inquiry is the organic one 

 of languages, and philology, therefore, is dependent in its investiga- 

 tions on the mo'de of development of languages, which is more or 



