THE HISTORY OF MATHEMATICS 409 



the number of such changes which can be made. In doing this we 

 naturally make a connection with the theory of invariants. Such a set 

 of changes is called a group. But a more fruitful idea has been ob- 

 tained by considering what combinations amongst the symbols, using 

 a specified law of combination, will always produce another of the 

 symbols, and will produce nothing else. As a simple and familiar 

 example, take the series of numbers 0,1,2, . . . with the law of addition. 

 If we add any two of these numbers we always get another of the same 

 series and we never get any other kind of number. The whole of the 

 series is called a group from this point of view. The even numbers 

 form a sub-group under this definition but the odd numbers do not 

 because the sum of two odd numbers is not an odd number. If we use 

 the same series, omitting the zero, with the law of multiplication instead 

 of that of addition, we again have a group, but now the odd numbers 

 form a sub-group. Again we may consider the operation of turning a 

 straight line through 60° in a plane about one end of the line. 

 There are obviously six positions of the line and in whatever one of 

 the six positions we start, a turn through 60° will always give one of 

 the other positions. The whole set constitutes a group. 



The idea of a group of substitutions enabled Gabois and Abel, 

 about the middle of the nineteenth century, to open up the way to treat 

 algebraic equations of a degree higher than the fourth and in fact to 

 show that the methods used to solve equations of the second, third and 

 fourth degrees could not in general be applied to those of the fifth 

 and higher degrees. The quintic had long been a puzzle to mathe- 

 maticians, all attempts to give a general solution in terms of radicals 

 having failed. Later on, Sophus Lie applied the idea of groups to the 

 solution of differential equations and was able to indicate the nature of 

 the solutions in certain general classes. Before his time the methods 

 for finding them had been disconnected and apparently without any 

 common property. Another form of the group theory has been applied 

 with success to the investigation of curves and surfaces and it is not 

 too much to say that the idea has been one of the most fruitful in pro- 

 ducing progress. "When a problem has been exhibited in group 

 phraseology, the possibility of a solution of a certain character or the 

 exact nature of its inherent difficulties is exhibited by a study of the 

 group of the problem." 1 



One of the most useful efforts of the nineteenth century mathe- 

 maticians has been in the direction of proving the possibility or im- 

 possibility of performing certain operations or "of solving certain prob- 

 lems, that is, in the investigation of existence theroems, as they are 

 often called. The squaring of the circle or the trisection of an angle are 

 two of the oldest of them and later arose the obtaining of a finite 

 numerical expression for the number e which is the base of the Napier- 



