[FOM] Understanding Euclid

[Vaughan Pratt]

Euclidean geometry is standardly understood as the geometry of the 
plane, more generally of flat or uncurved space.  To make this stick 
however, Euclid's fifth postulate should be phrased as an equivalence: 
two lines fail to meet if and only if a third line intersecting them 
both meets them at the same angle (or any equivalent phrasing thereof, 
e.g. that the interior angles on the same side of the cutting line sum 
to two right angles, or that a point P and a line L determine a unique 
line through P parallel to L).

Wording the postulate in this way then allows "parallel lines" to be 
defined equivalently as lines that don't meet, or as lines that cut any 
third line if at all in the same angle.  Before contemplating weakening 
the Parallel Postulate in any way, one should first decide which of 
these one is going to take as the definition of "parallel," because the 
respective predicates denoted by the two definitions diverge when it is 
weakened.  Here I'll follow the custom of defining parallel lines to be 
lines that don't meet.

When the Parallel Postulate is omitted altogether, other geometries such 
as hyperbolic and elliptic satisfy the first four postulates.  In 
particular those postulates are true on the sphere, for example the 
Earth's surface so modeled.

Euclid required just the only-if direction of that equivalence, that a 
line cutting parallel lines meet them at the same angle.  His phrasing 
more precisely was of the contrapositive, couched in terms of two 
adjacent angles summing to less than two right angles, which one might 
interpret constructively as the Triangle Postulate, that a base and two 
such angles always determine a (not necessarily unique) triangle, 
analogously to the other two odd-numbered postulates, that two points 
determine both a line and a circle (in those cases uniquely).  Yet 
another phrasing is that *at most* one line can pass through a point P 
parallel to a line L, also constructive but in this case partially and 
uniquely instead of totally and nonuniquely, concomitant with point-line 
and epi-monic duality.

In this weaker form the Parallel Postulate is true not only on the plane 
but on the sphere when lines are understood to be great circles or 
geodesics.  Hence all of Euclid's postulates as originally stated are 
true on the sphere.

The great search for a counterexample to the Parallel Postulate only 
makes sense for Euclid's statement of it.  Its phrasing as an 
equivalence (or even just as the contrapositive, two lines cut by a 
third at the same angle don't meet) admits the sphere as an obvious 
counterexample worth not even brownie points at the level of 
mathematical sophistication at which the search was conducted.  The 
searchers were clearly well aware of the soundness of Euclid's 
postulates for the sphere.  With regard to the preference for the 
one-directional version of the Parallel Postulate, Euclid and the 
searchers were surely on the same page.

I would welcome pointers to sources of information and opinion bearing 
on either of the following questions.

1.  All intuition about Euclidean space per se demands that the Parallel 
Postulate be bidirectional, in order to rule out positive curvature as 
well as negative when proving theorems true on the plane but false on 
the sphere.  Given this, why were Euclid and the searchers in agreement 
about stating the Parallel Postulate in its weakened one-directional 
form?  Was Euclid covertly hoping that his theorems would all hold for 
surveying projects of a scale where the earth's curvature was 
significant, or did it merely not occur to him that he might need the 
converse at some point?

2.  Proposition 47 of Book 1 of the Elements states and proves 
Pythagoras's theorem.  This is clearly false on the sphere, which 
contains an equilateral triangle all three of whose angles are right. 
This is not the earliest counterexample: Proposition 32 shows that the 
angles of a triangle sum to two right angles.  Euclid is evidently not 
playing by his own rules, and is appealing to unstated postulates.  Does 
he appeal somewhere to the converse of his fifth postulate?  If so how 
did he fail to notice he was doing so?  This would make him an early 
user of the Fallacy of the Converse, from P implies Q infer Q implies P, 
and an embarrassingly visible one at that.  Proclus's Commentary raises 
only relatively finicky concerns, I don't know who first pointed out 
this more glaring discrepancy but it's hard to imagine anyone seriously 
working on the independence of the Parallel Postulate without being 
aware of it.  It was surely common knowledge by the time Hilbert 
embarked on his program to make Euclid's system more rigorous by 
formalizing it.

Vaughan Pratt

--I was raised to be rigorous, not formal.