410 THE SCIENTIFIC MONTHLY 



ian system of logarithms and which arises in numerous mathe- 

 matical and physical investigations. The failure of attempts to solve 

 these problems finally led mathematicians to consider whether they 

 could not be proved to be insoluble under the conditions laid down. 

 Complete success has rewarded them. We now know that with the use 

 of the ruler and compasses alone, it is not possible to find a square 

 which shall be exactly equal in area to that of a given circle, nor given 

 any angle is it possible to construct the lines which shall divide it into 

 three equal parts. The number e too cannot be exactly expressed by 

 fractions or square roots or any other such simple numerical repre- 

 sentations, though it can be approximated to as closely as we wish by 

 decimals or in other ways. The labor of useless effort on the part of 

 the mathematician is thus avoided, though we shall still probably con- 

 tinue to hear of those who claim to have performed the impossible. In 

 our time it is quite usual as a part of an investigation to find included 

 in the construction of some new function or in a new representation of 

 a known function, a proof of its existence; especially in those cases 

 where the possibility may be called into question. In celestial me- 

 chanics an important part of Poincare's work consisted in proofs of the 

 existence or non-existence of different kinds of motion and of different 

 kinds of integrals. Indeed we have a considerable class of literature 

 which consists solely in demonstrating the existence of functions or 

 curves with little indication of the methods by which they may be con- 

 structed. The stimulating value of such researches in suggesting prob- 

 lems is often forgotten by those who, with some justice, complain of 

 their dullness. 



It is strange in connection with existence theorems that some of the 

 problems, most simple in statement, are still unproved. Long ago 

 Fermat stated that there are no whole numbers which will satisfy the 

 statement that the sum of the n A powers of two whole numbers is equal 

 to the n th power of a third whole number, except when n is equal to 2. 

 This impossibility has been proved for all values of n up to 100 and 

 for a few beyond, but no general proof has yet been given that it is 

 universally true. Again, there is no general method which will enable 

 us to pick out the prime numbers, that is, those which are not divisible 

 by any other number except unity. In geometry we have the famous 

 four-color problem in which it is desired to prove that a map consisting 

 of countries of any shape and arrangement can always be painted with 

 four colors so that no two adjoining countries will receive the same 

 color. In these and similar cases, no exceptions to the statements have 

 been found and there exist no complete demonstrations of their pos- 

 sibility or impossibility. It is of course assumed that if they are true 



1 h. E. Dickson, Paris International Congress, Vol. 2, p. 225. 



