210 A History ofMathematics
Appendix C. From Helmholtz’s 1876 paper
[Reproduced in (1979, pp. 249–50. Helmholtz, to simplify his argument, considers two-
dimensional ‘beings’ constructing geometry from observation of the world in which they live.]But intelligent beings...might also live on the surface of a sphere. Their shortest or straightest line
between two points would then be an arc of the great circle passing through them. Every great
circle passing through two points is divided by them into two parts. If the parts are unequal, the
shorter is certainly the shortest line on the sphere between the two points, but the other, or larger,
arc of the same great circle is also a geodesic, or straightest, line; that is, every smallest part of it is
the shortest line between its ends. Thus the notion of the geodesic, or straightest, line is not quite
identical with that of the shortest line...
Of parallel lines the sphere-dwellers would know nothing. They would declare that any two
straightest lines, if sufficiently extended, must finally intersect not only in one but in two points.
The sum of the angles of a triangle would be always greater than two right angles, increasing as the
surface of the triangle grew greater. They could thus have no conception of geometric similarity
between greater and smaller figures of the same kind, for with them a greater triangle must have
greater angles than a smaller one. Their space would be unlimited, but would be found to be finite
or at least represented as such.
It is clear, then, that such beings must set up a very different system of geometric axioms from
that of the inhabitants of a plane or from ours, with our space of three dimensions, though the
logical processes of all were the same; nor are more examples necessary to show that geometric
axioms must vary according to the kind of space inhabited. But let us proceed still further.
Let us think of reasoning beings existing on the surface of an egg-shaped body. Shortest lines
could be drawn between three points of such a surface and a triangle constructed. But if the attempt
were made to construct congruent triangles at different points of the surface, it would be found that
two triangles with three pairs of equal sides would not have equal angles. The sum of the angles of
a triangle drawn at the sharper pole of the body would depart further from two right angles than
if the body were drawn at the blunter pole or at the equator. Hence it appears that not even such a
simple figure as a triangle could be moved on such a surface without change of form.
Solutions to exercises
- (a) The ‘Aristotle proof ’ (see Fauvel and Gray 3.B.4 (b) and (c)) is the nicest. Let ABC be a triangle;
draw PAQ through A parallel to BC. (See Fig. 14.) Then∠PA C+∠ACB are two right angles;
and∠PA C+∠CAQ are also two right angles (angles on a straight line). So∠ACB=∠CAQ.
(This is just the alternate angle theorem.) Similarly,∠ABC=∠PAB. But the sum of the three
angles∠PAB,∠BAC,∠CAQ is two right angles (straight line again); so the same is true of the
three angles in the triangle, which we have proved equal to them.
(The key point is that the line PAQ has the ‘alternate angles property’ with respect toboththe
transversals AB,AC, which will only be true if the property is equivalent to being the unique
parallel.)
(b) (See Fig. 15.) This is best thought of as a result about so-called ‘Saccheri quadrilaterals’,
see later. To stay strictly in the framework of what Euclid would have done (one imagines): the