56 GEOMETEICAL RECREATIONS [CH. Ill 



of the above areas may diminish to a point and finally disappear 

 without affecting the argument. 



That we may require at least four colours is obvious from 

 the above diagram, since in that case the areas A, B, C, and X 

 would have to be coloured differently. 



A proof of the proposition involves difficulties of a high 

 order, which as yet have baffled all attempts to surmount them. 

 This is partly due to the fact that if, using only four colours, we 

 build up our map, district by district, and assign definite colours 

 to the districts as we insert them, we can always contrive the 

 addition of two or three fresh districts which cannot be coloured 

 differently from those next to them, and which accordingly 

 upset our scheme of colouring. But by starting afresh, it 

 would seem that we can always re-arrange the colours so as 

 to allow of the addition of such extra districts. 



The argument by which the truth of the proposition was 

 formerly supposed to be demonstrated was given by A. B. Kempe * 

 in 1879, but there is a flaw in it. 



In 1880, Tait published a solution f depending on the 

 theorem that if a closed network of lines joining an even 

 number of points is such that three and only three lines meet 

 at each point then three colours are sufficient to colour the 

 lines in such a way that no two lines meeting at a point are of 

 the same colour; a closed network being supposed to exclude 

 the case where the lines can be divided into two groups 

 between which there is but one connecting line. 



This theorem may be true, if we understand it with the 

 limitation that the network is in one plane and that no line 



* He sent his first demonstration across the Atlantic to the American Journal 

 of Mathematics, 1879, vol. n, pp. 193 — 200 ; but subsequently he communicated 

 it in simplified forms to the London Mathematical Society, Transactions, 1879, 

 vol. x, pp. 229—231, and to Nature, Feb. 26, 1880, vol. xxi, pp. 399—400. The 

 flaw in the argument was indicated in articles by P. J. Heawood in the Quarterly 

 Journal of Mathematics, London, 1890, vol. xxiv, pp. 332 — 338 ; and 1897, 

 vol. xxxi, pp. 270—285. 



+ Proceedings of the Royal Society of Edinburgh, July 19, 1880, vol. x, 

 p. 729 ; Philosophical Magazine, January, 1884, series 5, vol. xvn, p. 41 ; and 

 Collected Scientific Papers, Cambridge, vol. n, 1890, p. 93. 



