Geometries andSpace 195
geometers are, or should be studying ‘forms’ rather than things in the world makes more sense
than appears at first sight.
And yet, of course, Euclidean geometry was, and still is vital for surveyors, architects, and
town-planners who care nothing for how far a line can be produced but who very much want to
use the basic results about triangles, rectangles, lengths, and areas. The ‘ideal’ geometry, which in
Greek terms is a fiction, founds the geometry which people need. Worse, a naive reading, at least
of Euclid’s early books, might lead us to think that we were studying on a flat earth, particularly
if we use the variant definition of a straight line as ‘the shortest distance between two points’. The
Greeks knew the earth was round—Eratosthenes measured its radius. Al-B ̄ir ̄un ̄i (see Chapter 5)
used sophisticated spherical geometry, in which the shortest distance between two points is a
great circle, to find routes between cities and determine the qibla. But in this ‘geometry’, all lines
eventually meet, as is shown in Fig. 4, and the standard results referred to above are not true. They
are, however, so nearly true that in (say) town planning, as opposed to long sea or air journeys, any
error in Pythagoras’s theorem could not be detected by our measuring instruments.
These two ways in which Euclidean geometry failed to measure up to the real world are worth
bearing in mind when we consider its problems. For both Greek and Islamic geometers we find that
the question is not strictly: ‘How do parallel lines behave in the world’? Rather, it is how they behave
ingeometry. Here, then, we need to pause and make a historicist evaluation of what, in proving
postulate 5 (or any other Euclidean study) Euclid’s successors were after. The title ‘Ideas of Space’
given to Gray’s book is not, or has not always been a characterization of geometry.
Be that as it may, the influence of Euclidean thinking over subsequent geometers was naturally
enormous, even when they misunderstood him. Consequently, it is not surprising that the terms
in which Proclus stated the problem (in the sixth centuryce—and they were already old by then)
remained constant for so long:
This ought to be struck from the postulates altogether. For it is a theorem—one that raises many questions, which
Ptolemy proposed to resolve in one of his books—and requires for its demonstration a number of definitions as well as
theorems. (Proclus 1970, cited in Fauvel and Gray 3.B1)
In other words: (a) the postulate isnot‘reasonably acceptable’ in the sense that the others are,
and (b) it should be proved to be a consequence of assumptions whichareacceptable. This was
the long-term programme at least up to 1700. As can be seen, it was not, in any sense, a problem
about the coherence or otherwise of Euclidean geometry, which was by universal agreementthe
geometry. Rather, it was a problem about constructing a proof of the postulate. Notice also that,
Sl ABlFig. 4Geometry on the surface of a sphere. Any two ‘straight lines’landl′meet (twice), at A and B.