CH. Ill] GEOMETRICAL RECREATIONS 57 



meets any other line except at one of the vertices, which is all 

 that we require for the map theorem ; but it has not been proved. 

 Without this limitation it is not correct. For instance the 

 accompanying figure, representing a closed network in three 

 dimensions of 15 lines formed by the sides of two pentagons 

 and the lines joining their corresponding angular points, 

 cannot be coloured as described by Tait. If the figure is in 

 three dimensions, the lines intersect only at the ten vertices 

 of the network. If it is regarded as being in two dimensions, 

 only the ten angular points of the pentagons are treated as 

 vertices of the network, and any other point of intersection of 



the lines is not regarded as such a vertex. Expressed in tech- 

 nical language the difficulty is this. Petersen* has shown that 

 a graph (or network) of the 2nth order and third degree and 

 without offshoots (ovfeuilles) can be resolved into three graphs 

 of the 2wth order and each of the first degree, or into two graphs 

 of the 2»th order one being of the first degree and one of the 

 second degree. Tait assumed that the former resolution was 

 the only one possible. The question is whether the limitations 

 mentioned above exclude the second resolution. 



Assuming that the theorem as thus limited can be estab- 

 lished, Tait's argument that four colours will suffice for a map 

 is divided into two parts and is as follows. 



* See J. Petersen of Copenhagen, L' Intermedial™ des MatMmaticiens, vol. v, 

 1898, pp. 225-227 ; and vol. vi, 1899, pp. 36—38. Also Acta Mathematica, 

 Stookholm, vol. xv, 1891, pp. 193—220. 



