[FOM] FOM Understanding Euclid

Vaughan Pratt pratt at cs.stanford.edu
Sat Dec 6 05:35:53 EST 2008

Colin McLarty wrote:
 > Euclid's postulate that a line segment can always be extended was
 > understood to mean "extended to new points," i.e. a segment can always
 > be extended without returning to itself.  That is how he proves
 > Proposition 16, the first of his propositions that fails in the sphere
 > geometry:  "In any triangle, if one of the sides is produced, then the
 > exterior angle is greater than either of the interior and opposite
 > angles."

Colin, thanks very much for that.

By "he proves," do you mean Euclid himself or those looking for some 
justification for his reasoning?

Where does Euclid exploit the no-new-points premise in his proof?

How plausible is it, really, that this premise is what Euclid had in 
mind at the time?

There are just too many missing facts and logical connections to make 
this explanation plausible.

1.  Unless I'm missing something Euclid says nothing about new points 

2.  He proves Proposition 16 by appealing to the diagram, which being 
drawn in a medium satisfying the converse of Postulate 5 makes it 
impossible to see how it could fail to follow from the postulates in the 
absence of that converse.  (I find this the most telling point of all.)

His apparent success with Proposition 16 may have led him to weaken 
Postulate 5 because the other direction *did* follow from 1-4, so he 
thought.  The grand project over the following millennia of eliminating 
Proposition 5 altogether may well have been simply one of somehow 
persuading the other shoe to drop, along similar lines.

3.  The no-new-points premise only rules out spherical geometry under 
the assumption that spherical geometry is the only possible 
counterexample to the converse of Postulate 5, making it a 
model-theoretic argument.  What evidence is there that Euclid intended 
his system to be limited to the models of Postulates 1-5 with 5 
bidirectional, plus spherical geometry?

Only slightly more plausible is that Euclid had just the plane and 
spherical geometry in mind.  But in that case surely he would have said 
*something* by way of explanation of how Postulate 2 was supposed to 
rule out spherical geometry.  This post hoc explanation of Proposition 
16, with nothing to back it up other than the absence of any other 
explanations, might have satisfied those who bought it, but to my 
thinking it sheds little if any light on what Euclid might have had in 
mind here.


A convenient way of organizing Euclid's postulates is according to 
locality: which postulates take their user well outside the scope of 
their data?

Postulate 1 is naturally interpreted as being about line segments rather 
than lines, otherwise there would be no need for Postulate 2 concerning 
extendibility of lines.  With this very reasonable interpretation 
Postulate 1 is entirely local.

Postulate 2 is therefore the first non-local postulate, allowing line 
segments to be extended into foreign countries and beyond.

Postulate 3, two points determine a circle, is intrinsically local: the 
resulting figure fits within twice the separation of the data.

The locality of Postulate 4, all right angles are congruent, is unclear 
for want of any notion of locality of a right angle.  On the one hand 
one might argue that infinitesimal neighborhoods of right angles 
suffice, on the other one might invoke Postulate 2 to argue that right 
angles can grow to arbitrary size, which while not a problem for a 
perfect sphere might be interpreted as ruling out all other 
positive-curvature geometries.  However the precise logic of how 
Postulate 4 as stated would rule those out while leaving the sphere 
intact as a model remains completely unclear to me.  In any event I 
don't see how any such interpretation of Postulate 4 could rule out 
spherical geometry itself, the sphere being perfectly symmetrically and 
therefore admitting no obvious (to me) counterexamples to Postulate 4.

Postulate 5 in the direction stated by Euclid is intrinsically nonlocal 
by virtue of a given base and angles summing to less than 180 degrees 
producing a third point arbitrarily far away.  The (missing) converse of 
Postulate 5 however is easily seen to be local, by consideration of 
Proposition 32 as a suitable surrogate for the converse.  Proposition 32 
asserts that the angles of a triangle add up to 180 degrees, and this is 
a completely local statement.  Taking the earlier Proposition 16 instead 
makes no difference, it being just as local; likewise for Proposition 47 

How is a nonlocal condition such as novelty of a point per Postulate 2 
supposed to bear on a purely local postulate organized along the lines 
of Propositions 16, 32, or 47?  There you are with Toto somewhere out 
near Arcturus and suddenly you stumble upon a triangle whose angles add 
up to 181 degrees.  How natural would it be to say, "Toto, I have a 
feeling of deja vu."  This might make sense if there were some reason 
for the existence of "large" triangles in this sense to imply a return 
to an old haunt, which arguably there is in the very special case of a 
sphere itself.  But this is model-theoretic reasoning based on 
particular models, not logic, and the requisite thought processes seem 
impossibly far from anything suggested by the Elements.

> I know that Johann Heinrich Lambert also took the postulate that way,
> much later, and I believe pretty much everyone did before the 19th
> century.

What was the alternative?  To say that Euclid's argument was unsound? 
Let those who have never argued unsoundly cast the first stone.

While I don't want to stone Euclid I do want to understand him. 
Painting him as an infallible mystic expecting a high degree of 
creativity from his students isn't a hugely satisfying insight into his 
thought processes.


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