THE EVIDENCE OF GEOMETRICAL PROPOSITIONS
It has often been maintained that the twentieth proposition of the first
book of Euclid--that two sides of a triangle are together greater than
the third side--is evident even to asses. This does not, however, seem
to me generally true. I once asked a coastguardsman the distance from A
to B; he replied: "Eight miles." On further inquiry I elicited the fact
that the distance from A to C was two miles and the distance from C to B
was twenty-two miles. Now the paths from A to B and from C to B were by
sea; while the path from A to C was by land. Hence if the path by land
was rugged and the distance along the road was two miles, it would
appear that the coastguardsman believed that not only could one side of
a triangle be greater than the other two, but that one straight side of
a triangle might be greater than one straight side and any curvilinear
side of the same triangle. The only escape from part of this astonishing
creed would be by assuming that the distance of two miles from A to C
was measured "as the crow flies," while the road A to C was so hilly
that a pedestrian would traverse more than fourteen miles when
proceeding from A to C. Then indeed the coastguardsman could maintain
the true proposition that there is at least one triangle ABC, with the
side AC curvilinear, such that the sum of the lengths of AB and AC is
greater than the length of BC, and only deny the twentieth proposition
of the first book of Euclid.
Reasoning with the coastguardsman only had the effect of his adducing
the authority of one Captain Jones in support of the accuracy of his
data. Possibly Captain Jones held strange views as to the influence of
temperature or other physical circumstances, or even the nature of space
itself, on the lengths of lines in the neighbourhood of the triangle
ABC.