MATHEMATICS 45 



butter costs twenty cents, we expect 100 pounds to cost 100 

 times as much. Should we expect one million pounds to cost 

 one million times as much? Or should we expect diamonds 

 to follow this same simple rule? If so, surprises await us 

 not only in science, but at every turn in ordinary life. In 

 fact, we find that very few quantities follow ordinary pro- 

 portion, and that if they do, it is only for a very short while. 

 Far more important are other forms of variation which are 

 not linear, that is, not based on ordinary proportion, and 

 these other non-linear forms of variation form the subject- 

 matter of the theory of functions. A highly complicated 

 case known to you is the function which represents the height 

 of a wave of water. Its geometrical picture is indeed pre- 

 cisely of the form of such a wave Its mathematical expres- 

 sion leads to complicated considerations as compared with 

 those of elementary mathematics. Finally the direct interest 

 in functions grows for their own sake with their further 

 study. A grand part of mathematics has to do with elabora- 

 tions of this theory: a striking instance of the possible ex- 

 tension of a theory to render it complete and satisfying to 

 the human mind. It is a theory in which the beauty of mathe- 

 matics stands out as in few other topics, though perhaps to 

 some of you beauty in mathematics is an anomaly indeed. I 

 shall not wonder if it is in view of the fact that elementary 

 mathematics consists so largely of shorthand and deals so little 

 with real mathematics. 



The question of approach of one varying thing to some 

 fixed boundary is one which enters every human mind con- 

 stantly. In nature, changes are occurring in everything about 

 us, and their state, their temperature and length have a goal, 

 are approaching some definite quantity. Again, I might men- 

 tion the famous paradox of Zeno, in which a hare pursuing 

 a tortoise covers successively half the intervening distance 

 and then again half and half of that forever. Clearly the 

 hare is approaching the tortoise; indeed I will confess to 



