proof is one in which you assume the negation of a conclusion, show that
this leads to a contradiction, and consequently the only option left is that
the sought-after conclusion must be correct. A large number of indirect
proofs in geometry are the result of Euclid’s infamous fifth postulate—
the parallel postulate.
Noncontroversial Geometry
Noncontroversial geometry is everything up to, but not including, the
parallel postulate. It includes basic objects that we can’t really define but
everybody knows what they are, some definitions involving basic objects,
some obvious arithmetic and geometric facts, and the four postulates that
precede the parallel postulate.
Basic objects are things like points. Euclid defined a point as that which
has no part.^1 Works for me—I’m not philosopher enough to say exactly
what these abstract constructs are, but I (and you) know what Euclid was
getting at, so we can move on. Obvious arithmetic facts were such state-
ments as equals added to equals are equal. Euclid’s one obvious geometric
fact was that things that coincided with each other were equal—if line seg-
ments AB and CD can both be positioned to coincide, then ABCD.
We come now to the four noncontroversial postulates. I’ll assume that
line segments have endpoints, but straight lines don’t. Using this termi-
nology, the postulates are
Postulate 1: Any two points can be connected by a unique line seg-
ment.
Postulate 2: Any line segment can be extended to a straight line.
Postulate 3: There is a unique circle with given center and radius.
Postulate 4: All right angles are equal.^2There is an amazing amount of geometry that can be done using only
those four postulates—but that doesn’t concern us here.
The Parallel Postulate
Euclid’s initial version of the parallel postulate was, to say the least, un-
wieldy.
Postulate 5 (Euclid): If a straight line falling on two straight lines makes
the interior angles on the same side less than two right angles, the two
straight lines, if extended indefinitely, meet on that side on which the
angles are less than two right angles.^3
To understand what is happening here, think of a triangle with all its
102 How Math Explains the World