728 
Proceedings of the Royal Society 
Some thirty years ago, when I was attending Professor De 
Morgan’s class, my brother, Francis Guthrie, who had recently 
ceased to attend them (and who is now professor of mathematics at 
the South African University, Cape Town), showed me the fact 
that the greatest necessary number of colours 
to be used in colouring a map so as to avoid 
identity of colour in lineally contiguous dis- 
tricts is four. I should not be justified, after 
this lapse of time, in trying to give his proof, 
but the critical diagram was as in the margin. 
With my brother’s permission I sub- 
mitted the theorem to Professor De Morgan, who expressed himself 
very pleased with it ; accepted it as new ; and, as I am informed by 
those who subsequently at- 
tended his classes, was in the 
habit of acknowledging whence 
he had got his information. 
If I remember rightly, the 
proof which my brother gave did not seem alto- 
gether satisfactory to himself ; but I must refer 
to him those interested in the subject. I have 
at various intervals urged my brother to com- 
plete the theorem in three dimensions, but with 
little success. 
It is clear that, at all events when unrestricted 
by continuity of curvature, the maximum number 
of solids having superficial contact each with all is infinite. Thus, 
to take only one case, n straight rods, one edge of whose projections 
forms the tangent to successive points of a curve of one curvature, 
may so overlap one another that, when pressed and flattened at 
their points of contact, they give n — 1 surfaces of contact. 
How far the number is restricted when only one kind of super- 
ficial curvature is permitted must be left to be considered by those 
more apt than myself to think in three dimensions and knots. 
