ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 31 



whose solution has been sought for a long time, but not yet reached, 

 and in the case of which there is no reason for supposing that they 

 are insoluble. Of such problems the two following perhaps have 

 found their way out of the isolated circles of mathematicians and 

 have become more or less known to other scholars. I refer to the 

 astronomical Problem of Three Bodies and to the problem of the 

 frequency of prime numbers. The first of these two problems as- 

 sumes three or more heavenly bodies whose movements are mutually 

 influenced by one another according to Newton's law of gravitation, 

 and requires the exact determination of the path which each body 

 describes. The second problem requires the construction of a for- 

 mula which shall tell how many prime numbers there are below a 

 certain given number. So far these two problems have been solved 

 only approximately, and not with absolute mathematical exactness. 

 If the eternal and inviolable correctness of its truths lends to 

 mathematical research, and therefore also to mathematical knowl- 

 edge, a conservative character, on the other hand, by the continuous 

 outgrowth of new truihs and methods from the old, progressivcness 

 is also one of its characteristics. In marvellous profusion old knowl- 

 edge is augmented by new, which has the old as its necessary con- 

 dition, and, therefore, could not have arisen had not the old pre- 

 ceded it. The indestructibility of the edifice of mathematics renders 

 it possible that the work can be carried to ever loftier and loftier 

 heights without fear that the highest stories shall be less solid and 

 safe than the foundations, which are the axioms, or the lower sto- 

 ries, which are the elementary propositions. But it is necessary for 

 this that all the stones should be properly fitted together ; and it 

 would be idle labor to attempt to lay a stone that belonged above in 

 a place below. A good example of a stone of this character belong- 

 ing in what is now the uppermost layer of the edifice, is Linde- 

 mann's demonstration of the insolubility of the quadrature of the 

 circle, a demonstration of which interesting simplifications have 

 been given by several mathematicians, including Weierstrass and 

 Felix Klein. Lindemann's demonstration could not have been pro- 

 duced in the preceding century, because it rests necessarily on theo- 

 ries whose development falls in the present century. It is true, 



