A History of Mathematics From Mesopotamia to Modernity

Geometries andSpace 199

XY

D

B

A

C

Fig. 7Traditional perspective generates the ‘ideal line’ at infinity XY of projective geometry. The lines CA, DB are parallel and meet

at infinity at X in the ‘extended plane’; similarly AB, CD meet at Y.

Indeed, even the study of projective geometry, whichisstill taught as a very abstract subject, was

initially an offshoot of descriptive geometry^5 and was regarded as the study of space enriched by

those ideal points and lines ‘at infinity’ which we find in perspective drawings (Fig. 7). Parallel

lines, in defiance of postulate 5, could meet provided that their meeting point was in that imagined

exclusion zone which was termed the ‘line at infinity’; the Euclidean structure of space was not

challenged, however strange that may seem. Perhaps the strongest expression of the prevailing

orthodoxy was given by Bolzano in 1817 when he attacked the use of geometry to prove results in

analysis:

But it is clear that it is an intolerable offense againstcorrectmethodto derive truths ofpure(or general) mathematics (i.e.

arithmetic, algebra, analysis) from considerations which belong to a merelyapplied(or special) part, namely,geometry.

(Bolzano, in Fauvel and Gray 18.B.1, p. 564)

Geometry was an applied subject, since its truth was derived from our knowledge of the world.
This point would have been almost unquestioned by Bolzano’s readers, even if they did not share
his conclusions. It was not contested by Lobachevsky and Bolyai, and it would not be for some
sixty years. It is only with hindsight that we see non-Euclidean geometry as pointing towards a
democracy of geometries in which all may have equal status and truth-claims are no longer the
issue.

4. Spherical geometry

When God Almighty intended the creation of mankind, He purposely designed the creation of the earth at first, and

gave it the consolidating force to evolve its natural shape, I mean that which is truly spherical. (Al-B ̄ir ̄un ̄i 1967, p. 24)

We are about to confront the major problem of this chapter; that the mathematics which underpins
non-Euclidean geometry is, at times, difficult both conceptually and in terms of sheer calculation.
The ideas of Lobachevsky and Bolyai need their formulae to work, and the formulae are far from
intuitive. Following a common precedent, we shall therefore consider first the ‘geometry’ which
is defined on a sphere—think, as usual, of the Earth—by taking ‘straight lines’ to be lines of
shortest distance, that is great circles. This corresponds to Saccheri’s HOA. By what one might call

5. The founder, Poncelet, was one of Monge’s students.