ss. [706 
707] 
7 
707. 
e P denotes 
—, and for 
cx 
comes when 
cal problem, 
image of X 
yard to the 
>m A. And 
Lg the image 
n regard to 
age from B. 
yes of X in 
mages of X 
3 is in fact 
don between 
ON THE COLOURING OF MAPS. 
[From the Proceedings of the Royal Geographical Society, vol. I., no. 4 (1879), 
pp. 259—261.] 
The theorem that four colours are sufficient for any map, is mentioned somewhere 
by the late Professor De Morgan, who refers to it as a theorem known to map-makers. 
To state the theorem in a precise form, let the term “area” be understood to mean 
a simply or multiply connected* area: and let two areas, if they touch along a line, 
be said to be “ attached ” to each other; but if they touch only at a point or points, 
let them be said to be “appointed” to each other. For instance, if a circular area 
be divided by radii into sectors, then each sector is attached to the two contiguous 
sectors, but it is appointed to the several other sectors. The theorem then is, that 
if an area be partitioned in any manner into areas, these can be, with four colours 
only, coloured in such wise that in every case two attached areas have distinct 
colours; appointed areas may have the same colour. Detached areas may in a map 
represent parts of the same country, but this relation is not in anywise attended 
to: the colours of such detached areas will be the same, or different, as the theorem 
may require. 
It is easy to see that four colours are wanted; for instance, we have a circle 
divided into three sectors, the whole circle forming an enclave in another area; then 
we require three colours for the three sectors, and a fourth colour for the surrounding 
area: if the circle were divided into four sectors, then for these two colours would 
* An area is “connected” when every two points of the area can be joined by a continuous line lying 
wholly within the area; the area within a non-intersecting closed curve, or say an area having a single 
boundary, is “simply connected”; but if besides the exterior boundary there is one or more than one 
interior boundary (that is, if there is within the exterior boundary one or more than one enclave not 
belonging to the area), then the area is “multiply connected.” The theorem extends to multiply connected 
areas, but there is no real loss of generality in taking, and we may for convenience take the areas of the 
theorem to be each of them a simply connected area.