172 Unicursal problems [ch. ix 



in the propositions proved by Listing in his Topologie*. I 

 shall, however, adopt here the methods of Euler, and I shall 

 begin by giving some definitions, as it will enable me to put 

 the argument in a more concise form. 



A node (or isle) is a point to or from which lines are 

 drawn. A branch (or bridge or path) is a line connecting two 

 consecutive nodes. An end (or hook) is the point at each 

 termination of a branch. The order of a node is the number 

 of branches which meet at it. A node to which only one 

 branch is drawn is a free node or a free end. A node at which 

 an even number of branches meet is an even node : evidently 

 the presence of a node of the second order is immaterial. A 

 node at which an odd number of branches meet is an odd node. 

 A figure is closed if it has no free end : such a figure is often 

 called a closed network. 



A route consists of a number of branches taken in con- 

 secutive order and so that no branch is traversed twice. A 

 closed route terminates at a point from which it started. 

 A figure is described unicursally when the whole of it is 

 traversed in one route. 



The following are Euler's results, (i) In a closed net- 

 work the number of odd nodes is even, (ii) A figure which 

 has no odd node can be described unicursally, in a re-entrant 

 route, by a moving point which starts from any point on it. 

 (iii) A figure which has two and only two odd nodes can be 

 described unicursally by a moving point which starts from one 

 of the odd nodes and finishes at the other, (iv) A figure 

 which has more than two odd nodes cannot be described com- 

 pletely in one route; to which Listing added the corollary 

 that a figure which has In odd nodes, and no more, can be 

 described completely in n separate routes. I now proceed to 

 prove these theorems. 



* Die Studien, Gottingen, 1847, part x. See also Taiton 'Listing's Topologie,' 

 Philosophical Magazine, London, January, 1884, series 5, vol. xvn, pp. 30 — 46; 

 and Collected Scientific Papers, Cambridge, vol. n, 1900, pp. 85 — 98. The 

 problem was discussed by J. 0. Wilson in his Traversing of Geometrical 

 Figures, Oxford, 1905. 



