[FOM] Understanding Euclid
pratt at cs.stanford.edu
Sat Dec 13 00:31:13 EST 2008
--Original question answered--
After mulling over the very helpful input from everybody for a few days,
I concluded that my original question (was Euclid considering spherical
geometry in Book I?) had been answered definitively.
Answer: No. Euclid did not intend to accommodate spherical geometry in
Book I or he would not have allowed Proposition 16, that an exterior
angle of a triangle exceeds each of its two opposite interior angles.
He did not notice that his postulates allowed spherical geometry, and so
failed to realize that in that model of his axioms the relevant median
(the line segment BE in the first figure at
could be greater than a right angle.
In that case vertex F completing ABC to the parallelogram ABCF passes
through (namely below in the figure) the plane of the great circle
formed by BCD, simultaneously with its antipode reaching B. This
results in the line segment CF suddenly and very violently (from the
standpoint of the diagram) flipping around 180 degrees at the instant of
passage of F through BCD (thinking of F being pushed out continuously
until EF equals BE rather than all at once) and splitting BCG instead of
An easily visualized such triangle is the thick octant, consisting of
two points on the Equator separated by a right angle, together with the
North Pole, thickened by pulling each vertex distance epsilon away from
the centroid of the triangle. All internal angles are slightly over a
right angle while all external angles are slightly under.
An equivalent coordinatization making the North Pole the centroid of the
octant (for better symmetry) places the vertices of the triangle on
three equally spaced meridians (i.e. 120 degrees apart), all at latitude
epsilon less than asin(sqrt(1/3)), 35 degrees works fine.
This answer generated for me a slew of questions, starting with the
An interesting feature of this failure of I.16 is that this breakdown of
the argument depended on the surface continuing to curve around to the
back of the sphere well outside the triangle. Taking the thick octant
as a case in point, if one were to chop it and a neighborhood out of the
sphere and continue its boundary with a different curvature aimed at
making the octant one of a host of bumps in a large dimpled but
otherwise flat sheet, the breakdown shouldn't be able to happen in the
way it does for the sphere because there is no longer a back of the
sphere for EF to go around; instead EF goes off to the upper right
consistently with the diagram, rolling up hill and down but always away
from B and remaining in a more or less flat (but dimpled) plane on average.
(The reason for making it multiple bumps rather than one isolated bump
is that two points on opposite sides of the bump equally and
sufficiently far removed from it will clearly find it shorter to go
round the bump than over its top, giving the same ambiguity as arises
for antipodes of the sphere. Maybe this still happens with multiple
bumps but I find this much harder to see.)
However the theorem remains false in this model because the thick-octant
bump is a counterexample. Why is it false? The basic backwards flip of
CF shouldn't happen. Is it false because even a dimpled surface is
problematic, and the only smooth surface, even without homogeneity, that
has no pairs of points having two geodesics go through them is the
hyperbolic plane (including the Euclidean plane as its limiting
degenerate case)? That's still not convincing because even if it does
the flip is nowhere near as bad as for the sphere. Because of this I
have trouble understanding how I.32 can be true even taking I.16 as a
postulate! Could Euclid be smuggling yet more stuff in somewhere to get
from I.16 to I.32?
I have several more questions, which divide into model-theoretic,
concerning possible models of both the postulates and the propositions,
and proof-theoretic, concerning the domain-independent aspects of
Euclid's apparent reasoning. I'll ask some of the former in this post
and defer the others to later in the interests of not making this
already overlong message (for which my apologies) impossibly long.
My model-theoretic questions all fall out as special cases of the
following master question for the model theory of Book I.
What is a model of Euclidean geometry?
More precisely, what surfaces does Euclidean geometry describe? (We can
take two-dimensionality for granted.) And of those, which are
reasonable in some sense?
One might judge the reasonableness of a space by some combination of the
following criteria (P2 denotes Postulate 2).
C1. Flat (zero Gaussian curvature, it's ok to roll up architectural
plans for a pyramid when stowing them in the overhead bin because the
Gaussian curvature is zero in that situation).
C2. Continuous (every Cauchy sequence has a limit).
C3. Orientable (clockwise and counterclockwise are distinct notions).
C4. Unbounded (infinite in extent) (entailed by C1 and P2)
C5. Homogeneous (a maze of points all alike).
C6. Boundaryless (not a half-plane and no holes) (entailed by C5 and P2).
--Three refinements of the master question--
These criteria allow the following refinements of the master question.
R1. Which of these criteria might Euclid have had in mind?
R2. Taking a model of Euclid's axioms as any surface for which the
propositions of Book I hold, which criteria are satisfied by every such
surface, and what logical relationships then hold between those criteria?
R3. As for R2 with "postulates" in place of "propositions."
Refinement R1 is intrinsically speculative, while R2 and R3 replace the
speculative element with more mathematically defined questions.
Although R1 is a nice question I don't know enough about pre-Euclidean
mathematics, or Euclid's environment, to have any thoughts on it myself,
but would be very interested to hear from those who know something about it.
Regarding the subject line of "understanding Euclid," R2 is really the
right question because we depend on the propositions to clarify and fill
in the gaps in Euclid's postulates making the latter a poor indicator of
his intentions. The answers to R2 should be a proper superset of those
By Proposition 32, that triangles contain exactly two right angles,
there is no doubt that Euclid intended total flatness. So Euclid's
propositions, if not postulates, would appear to axiomatize the plane in
the sense of zero Gaussian curvature if not e.g. continuity.
R3 is more in the nature of the game genies play when you make three
wishes and they take them literally (don't you hate when that happens,
like when the genie is your computer, or your two-year-old?). In this
case Euclid made the postulates and we're now proposing to take them
literally. As such it is more like pure mathematics than a serious
effort to understand Euclid: a quite different question, more or less
interesting than R2 depending on whether one is more interested in
mathematics or mathematicians respectively. I primed the inter-criteria
relationships pump with two easy ones, namely C1 -> C4 and C5 -> C6,
both solely on account of P2, to illustrate what I have in mind here.
--Consideration of each criterion--
If nonzero curvature is deemed unreasonable then I would be inclined to
stop there and not pursue the remaining criteria. Since nonflat models
of Euclid's postulates have long been of interest and therefore are
presumably not entirely unreasonable, let's push on.
Continuity doesn't seem high on people's list given that I've never
heard anyone complain about iterated quadratic irrationals, the closure
of the rationals under square root, as not being a model of Euclid's
axioms on that ground. This may be the result of wanting to consider
Euclidean geometry a first order theory, which Tarski if not Hilbert
adhered to. (That ZF is first order should be an orthogonal issue.)
In the absence of continuity, how do you prove that a spherical model of
Euclid's postulates contains any antipodal pairs of points? Every point
has its antipode when you work with degrees, or with ideal metres
(40,000,000 of which will take you round a spherical Earth of radius of
curvature that of the (nonspherical) WGS-84 ellipsoid at Lèves, Alsace
or any other point at latitude 48.46791 degrees, north or south), or
with cosines of angles instead of the angles themselves so as to retain
the simplicity of iterated quadratic irrationals in relating sides and
angles of spherical triangles, because cos(pi) = -1. But if you make
life computationally hard for yourself by using radians instead of
cosines, taking as the basis for a coordinate system the vertices of an
equilateral triangle one radian on each side, then it's not at all clear
to me whether any great circles ever intersect in 2 points, nor whether
they always intersect in 1 (though clearly infinitely many do). (There
exist antipodal points iff there exist great circles intersecting in two
points iff there exist circles with center that are great circles.)
The absence of antipodal points has its good features and its bad. On
the good side, geodesics intersect in at most one point. On the bad
side, can we be sure that geodesics always intersect even once, let
alone twice? And how can we live with the thought of two geodesics
coming closer to each other than any positive epsilon without actually
intersecting? Did any ancient Greek geometer contemplate such a
travesty of common sense?
Homogeneity would imply no boundaries. However even in the absence of
homogeneity, boundarylessness is a no-brainer consequence of Postulate
2: a line normal to a boundary is blocked by it. Hence we can safely
assume no boundaries.
Orientability seems a very natural requirement. One tends to lose track
of the time when your analog watch runs backward at the same time as
it's running forward, a consequence of nonorientability. This happens
on the Bloch sphere modeling electron spin, being the projective sphere,
but physicists are too busy worrying about Schroedinger's cat (curious)
and Schroedinger's fish (delicious) to find nonorientability
disorienting. The designers of the Acropolis surely would have rejected
a nonorientable surface---how would you indicate the orientation of a
On that basis therefore the projective sphere (antipodal points
identified, with the metric adjusted accordingly) seems a non-starter
for a model of Euclidean geometry suitable for the typical applications
of geometry contemplated by Euclid's contemporaries, or for modern
architects for that matter.
Summarizing thus far, we've suggested that discontinuities are of
debatable reasonableness (and we have the associated question of whether
any discontinuous sphere can model the postulates without introducing
antipodal points). We expressed a sufficiently strong distaste for
nonorientable surfaces that those can be ruled out.
(One approach to dealing with antipodal points would be to strengthen
the requirement in Postulate 1 that the two points be distinct to the
further requirement that they also not be antipodal when using them to
determine a line. But that does nothing to prevent the counterexample
in Proposition 16 (violent flipping); it's not so much the singularity
at an antipodal pair itself as the discontinuity it creates in its
neighborhood that makes the proof of Proposition 16 unsound.)
--Down to just boundedness and homogenity.
This leaves boundedness and homogeneity. I haven't heard anyone say
that Euclid's postulates imply his domain must be unbounded. If that
follows from any reasonable interpretation of the postulates that
doesn't assume homogeneity (currently also on the table) I would very
much like to see the proof.
There does seem to be a strong sense that Euclid meant his space to be
homogeneous, but that's not at issue because we know furthermore that he
meant it to be Euclidean space, in particular flat. The question here
is not what he meant, only what Euclid's postulates might reasonably be
taken to mean when others decide what they should mean.
There are a great many useful nonhomogeneous yet reasonably regular
surfaces in the world, gently dimpled plastic sheets for example, which
are often encountered, though more vigorously dimpled sheets containing
thick octants might be much less common. How about the Fermi surface of
a monolayer crystal?
If Euclid's postulates taken as they stand, without Propositions 16, 17,
32, etc. shoring them up, have dimpled sheets as a model, that could be
--Onset of misgivings about the organization of Book I--
After contemplating all these possibilities, I came away with a feeling
similar to that of Proclus as cited by Alasdair Urquhart: "[Postulate 5]
ought even to be struck out of the Postulates altogether; for it is a
theorem involving many difficulties..."
What I object to is not really Postulate 5 itself, however patched, but
the whole protocol that it is part of. I don't believe Euclid has quite
the right axioms for Euclidean geometry. Postulate 3 is a major part of
the problem, and needs to be replaced with a set of postulates better
adapted to precisely capturing the Euclidean plane, without using any
form of Postulate 5. Postulates 1, 2, and 4 are ok but a bit feeble,
which Euclid wishfully compensates for by making Postulate 3 Euclid's
hammer. The problem with the hammer is that it is too universal for the
purposes of capturing the geometry of the flat plane: it tries to do too
much with a single axiom yet doesn't accomplish enough, and Postulate 5
is the wrong way to take up that slack. On the other hand I don't
believe the right axioms need to differ from Euclid's as much as
Tarski's did. I'll say more about this (in part in response to George
McNulty's post mentioning Borsuk and Szmielew) in another message after
my promised proof theory message.
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