CH. IX] 



UMCUBSAL PROBLEMS 



171 



will not be affected if we suppose the islands to diminish to 

 points and the bridges to lengthen out. In this way we 



ultimately obtain a geometrical figure or network. In the 

 Kbnigsberg problem this figure is of the shape indicated below, 

 the areas being represented by the points A, B, C, D, and the 

 bridges being represented by the lines I, m, n, p, q, r, a. 



Euler's problem consists therefore in finding whether a 

 given geometrical figure can be described by a point moving 

 so as to traverse every line in it once and only once. A more 

 general question is to determine how many strokes are neces- 

 sary to describe such a figure so that no line is traversed twice: 

 this is covered by the rules hereafter given. The figure may 

 be either in three or in two dimensions, and it may be repre- 

 sented by lines, straight, curved, or tortuous, joining a number 

 of given points, or a model may be constructed by taking a 

 number of rods or pieces of string furnished at each end with 

 a hook so as to allow of any number of them being connected 

 together at one point. 



The theory of such figures is included as a particular case 



