190 ANNUAL KEPORT SMITHSONIAN INSTITUTION, 1912. 



furnish a partial explanation of the fact that it seems impossible to 

 give a complete definition of the term mathematics. If the above 

 view is correct, we have no reason to expect that a complete defini- 

 tion of this term will ever be possible, although it seems possible 

 that a satisfactory definition of the developed parts may be forth- 

 coming.^ 



Among the various fields of research those which surround a 

 standing problem are pferhaps most suitable for a popular exposi- 

 tion, but it should not be inferi'ed that these are necessarily the 

 most important points of attack for the young investigator. On the 

 contrary, one of the chief differences between the great mathemati- 

 cian and the poor one is that the former can direct his students into 

 fields which are likely to become well known in the near future, 

 while the latter can only direct them to the well-known standing 

 problems of the past, whose approaches have been tramped down 

 solid by the feet of the mediocre, who are often even too stupid to 

 realize their Imiitations. The best students can work their way 

 through tliis hard crust, but the paddle of the weaker ones will 

 only serve to increase its thickness if it happens to make any impres- 

 sion whatever. 



It would not be difficult to furnish a long list of standing mathe- 

 matical problems of more or less liistoric interest. Probably all 

 would agree that the most popular one at the present time is Fer- 

 mat's greater theorem. In fact, this theorem has become so popular 

 that it takes courage to mention it before a strictly mathematical 

 audience, but it does not appear to be out of place before a more 

 general audience like tliis. 



The ancient Egyptians knew that 3^ + 4^ = 5^, and the Hindus 

 knew several other such triplets of integers at least as early as the 

 fourth century before the Christian era.^ These triplets constitute 

 positive integi-al solutions of the equation 



x2 + y2 = 23_ 



Pythagoras gave a general rule by means of which one can find 

 any desired number of such solutions, and hence these triplets are 

 often called Pythagorean numbers. Another such rule was given by 

 Plato, while Euclid and Diophantus generalized and extended these 

 rules. 



Fermat, a noted French mathematician of the seventeenth century, 

 wrote on the margin of a page of his copy of Diophantus the theorem 

 that it is impossible to find any positive integral solutioii of the 

 equation 



1 Bficher discussed some of the proposed definitions in the Bulletin of the American Mathematical Society, 

 vol. 2 (1904), p. 115. 

 s Lietzmann, •' Der Pythagoreische Lehrsatz," 1912, p. 52. 



