200 A History ofMathematics
coincidence, it has always been dismissed as the HAA has not, as clearly contrary to what is obvious.
In Section 1, I gave it a quick dismissal; a more interesting one rests on Euclid’s proposition I.16.In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite
angles.Euclid’s proof, together with a picture which illustrates how it fails to work on a sphere, is given as
Appendix A. In his discussion of the ‘standard account’, Gray raises the question of why spherical
geometry (which was, as he says, ‘well known throughout the entire period’) was not seen as an
answer to the question (i.e. whether the fifth postulate could be deduced from other self-evident
facts). It may be the mere fact that obvious Euclidean postulates, like the existence of circles of
arbitrary radius, were untrue in spherical geometry; or it may be that other simple defects such as
are brought to light in the failure of proposition I.16 were responsible. In any case, the aim of this
section is not so much to discuss this ‘what if ’ question (why did geometers not see the sphere as an
answer?), but to look at what was known of that geometry and how it influenced later thinking.
Already in the time of the Greeks, as has been mentioned, it was recognized that a line of shortest
distance on a sphere (let us call the sphere S) was an arc of a ‘great circle’—the intersection of S
with a plane through the centre (Fig. 4). Because of their importance in astronomy, the Greeks, in
particular Ptolemy gave attention to understanding spherical triangles (triangles whose sides are
shortest lines on a sphere), and the Islamic mathematicians who had (roughly) our trigonometric
functions to help them were able to derive the key formulae for ‘solving’ them, which are given in
Appendix B.
The formulae are essential if we are to find our way about on a sphere; they have been used
by geographers ancient and modern. Of much less interest to geographers, but more to mathem-
aticians, is something about theangle-sumof spherical triangles which was discovered by Albert
Girard in the seventeenth century: that- the angle-sum A+B+C is always greater thanπ(so much is obvious);
- the ‘excess’ A+B+C−πincreases with area; in fact, it is precisely equal to( 1 /R^2 )×area(ABC).
This is easy to see for the triangle all of whose angles areπ/2, which makes up an eighth of the
sphere (why?). To see that the excess is simply a multiple of the area is a subtler argument, but
acceptable if you are prepared to take a little time thinking about pasting triangles together.
The way in which Girard’s formula might lead to a better understanding of what it means to deny
postulate 5 seems first to have occurred to Johann Heinrich Lambert, whose posthumousTheory
of parallelsappeared in 1786. (Indeed, Lambert is singled out as a key transitional figure by Gray
(1979, ch. 5) for this and other reasons.) It was Lambert who, by reasoning with quadrilaterals
again, arrived at a key consequence of the HAA:[I]t is not only the case that in every triangle the angle sum is less than 180◦, as we have already seen, but also that
the difference from 180◦increases directly with the area of the triangle. (Lambert, in Fauvel and Gray p. 518)Second, he saw that in consequence an HAA geometry must, like a spherical geometry, have an
absolute measure of length. This comes from reasoning with a quadrilateral ADGB (Fig. 8) in which
AB=AD and the three angles A,B,D are right angles, but the fourth (G) may not be:Since the angle has a measure intelligible in itself [i.e. as a fraction of 360◦], if one took e.g.AB=ADas a Paris foot
and then the angleGwas 80◦this is only to say that if one should make the quadrilateralADGBso big that the angleG