8 
ON THE COLOURING OF MAPS. 
[707 
be sufficient, and taking a third colour for the surrounding area, three colours only 
would be wanted; and so in general according as the number of sectors is even or 
odd, three colours or four colours are wanted. And in any tolerably simple case it can 
be seen that four colours are sufficient. But I have not succeeded in obtaining a 
general proof: and it is worth while to explain wherein the difficulty consists. 
Supposing a system of n areas coloured according to the theorem with four colours 
only, if we add an (n +1 )th area, it by no means follows that we can without 
altering the original colouring colour this with one of the four colours. For instance, 
if the original colouring be such that the four colours all present themselves in the 
exterior boundary of the n areas, and if the new area be an area enclosing the n 
areas, then there is not any one of the four colours available for the new area. 
The theorem, if it is true at all, is true under more stringent conditions. For 
instance, if in any case the figure includes four or more areas meeting in a point 
(such as the sectors of a circle), then if (introducing a new area) we place at the 
point a small circular area, cut out from and attaching itself to each of the original 
sectorial areas, it must according to the theorem be possible with four colours only 
to colour the new figure; and this implies that it must be possible to colour the 
original figure so that only three colours (or it may be two) are used for the 
sectorial areas. And in precisely the same way (the theorem is in fact really the 
same) it must be possible to colour the original figure in such wise that only 
three colours (or it may be two) present themselves in the exterior boundary of the 
figure. 
But now suppose that the theorem under these more stringent conditions is true 
for n areas: say that it is possible with four colours only, to colour the n areas 
in such wise that not more than three colours present themselves in the external 
boundary: then it might be easy to prove that the n +1 areas could be coloured 
with four colours only: but this would be insufficient for the purpose of a general 
proof; it would be necessary to show further that the n +1 areas could be with the 
four colours only coloured in accordance with the foregoing boundary condition; for 
without this we cannot from the case of the n +1 areas pass to the next case of 
n + 2 areas. And so in general, whatever more stringent conditions we import into 
the theorem as regards the n areas, it is necessary to show not only that the n +1 
areas can be coloured with four colours only, but that they can be coloured in 
accordance with the more stringent conditions. As already mentioned, I have failed 
to obtain a proof.