194 A History ofMathematics
2. First problem: the postulate
Let us return to Euclid’s postulate 5.^3 It is worth noting that postulates 1–4 require the reader to
accept what might, in reasonable terms, be considered ‘obvious’: for example, that a circle can be
described with any given centre and any given radius. However, even then there is a disjunction
between geometry and ‘experience’. Would the statement still be so obvious, one might ask, if the
radius were chosen to be a million miles? As Torrelli points out:In the closed Aristotelian world not every straight line can be produced continuously as required by Postulate 2, and
not every point can be the centre of a circle of arbitrary radius as demanded by Postulate 3. (Torrelli 1978, pp. 8–9)This is an example of how the non-specificity of the postulates is useful; one does not have to
consider such questions.
What postulate 5 says is different in kind. However, it is essential in constructing that Euclidean
geometry which most of us would consider sensible. Not only the standard result on the angle-sum
of triangles (two right angles), but the very existence of rectangles and squares (figures with four
right angles at the corners) depend on it. And so, as a consequence, does much of what for the
Greeks’ predecessors was known and used, for example, the Pythagoras theorem.
In its classical form, postulate 5 reads:
If a straight line, falling on two other straight lines, makes the two interior angles on one side less than two right
angles, then the two straight lines, produced indefinitely, will meet on the side on which are the two angles less than
two right angles.
This at first simply appears hard to understand; but what it states is indicated by Fig. 1 above. If we
have such a diagram, andα+β< 180 ◦, then the two lines will meet as stated. Note the two other
points:
- ‘produced indefinitely’; that is, we are allowed to make the lines as long as is necessary;
- ‘on that side’; that is, they will meet on one side, and not on the other.
While now perhaps comprehensible, the postulate—if we think about it—is asking us to accept
quite a strong statement. Once again, if we were allowed to think in terms of numbers, we would
find that it contains an assumption that geometry continues to work at arbitrarily large distances.
Trigonometry tells us, for example:
Claim:If (in radians)α=β=(π/ 2 )− 10 −^10 , and the transversal is of length 1 cm, then the
distance to the meeting point is roughly^12 · 1010 cm.In other words, for postulate 5 to be true of ‘the world’, we must again conclude that the world has
infinite extent. For evidence that the Greeks in general did not think this, it is enough to consider
Archimedes (a very sophisticated Greek) who discusses the size of the universe in hisSand-Reckoner
(see Fauvel and Gray 4.A.2). What figure he came up with is unimportant for our purposes; the
main point is that it was a finite one. Hence, the ideally produced lines of geometry might go beyond
the boundaries of the universe—andconceptually, one can see how an idealized straight line might
continue after the universe had stopped. This points to an interesting disjunction between geometry
(an ideal study) and the study of the real world. In this respect, Plato’s point (see Chapter 2) that- Numbering in Euclid is sometimes problematic, and some authors (Bolyai in particular) call it axiom 11.