- Euclid (circa 300 BCE): Euclid's original "Postulate 5" is the cumbersome statement you see on p. 360 of Stillwell. (In telegraph style: Interior angles less than 180 degrees implies intersection.)
- Proclus (circa 450 CE), Playfair (1795): Parallels exist and are unique. That is, for any straight line L and any point P not on L, there is exactly one line through P that does not intersect L. (This became the standard formulation of the parallel postulate.)
- al-Haytham (circa 1000), Clavius (1574): Parallel lines are the same thing as equidistant lines. (The trouble with this one is that a set of points equidistant from a given straight line is not necessarily itself a straight line. For example, on a sphere the closest equivalent of a line is a "great circle"; the equator is a great circle, but the other parallels of longitude are not. Any other great circle, in fact, intersects the equator in two places.)
- Wallis (1693): Euclidean geometry is scale-invariant; in particular, similar triangles of all sizes exist. (This is my favorite, since it states a fundamental symmetry property of the space.)
- Clairaut (1741): Rectangles exist. (If a quadrilateral has 3 right angles, then the fourth angle is also right.)
- Legendre (circa 1800): For any acute angle A and any point D interior to angle A, there exists a line through D and not through the vertex A that intersects both sides of angle A.

The first four of these models were discovered by E. Beltrami, although only the fourth one is traditionally identified by his name.

- Klein's disk model
- Poincaré's disk model
- Poincaré's upper half plane model
- The pseudosphere or revolved tractrix model
- Hyperboloid model.

The first three models are more elementary from the point of view of axiomatic geometry. The other two live in modern differential geometry.

The pseudosphere provides us with a tangible trumpet-shaped surface
in Euclidean 3-space that *locally* has a geometry of constant
negative curvature. Unfortunately, only a small part of the full
hyperbolic space can be embedded in this way, so the straight lines
cannot be "indefinitely extended" and hence the *global* hyperbolic
parallel property is not really demonstrated. The hyperboloid is a
genuine, complete realization of the full hyperbolic geometry, but the
3-dimensional space in which it lives has an *indefinite metric*,
like the space-time of special relativity. [punt]

From the point of view of differential geometry, there is a close
symmetry between elliptic and hyperbolic geometry. More precisely, we
are dealing with *Riemannian* geometry, in which lengths are
defined by an arc-length formula to be integrated along a curve. The
metric also makes it possible to define **geodesics** as curves that
are the closest possible analogues of straight lines in a curved space.
In a hyperbolic (negative curvature) space, geodesic lines that
initially look parallel eventually veer away from each other (cf.
alHaytham-Clavius, above). In an elliptic (positive curvature) space,
geodesic lines that initially look parallel veer toward each other and
eventually cross (as do meridians on the earth). It follows that in the
hyperbolic space, triangles have angle sums less than 180 degrees (and
are increasingly skinny as their dimensions grow), while in the elliptic
space, triangles have angle sums greater than 180 degrees. (Consider a
triangle with one vertex at the north pole and the other two on the
equator.)

This nice picture deteriorates in axiomatic geometry ("Euclidean" geometry as carefully reaxiomatized by David Hilbert at the end of the 19th century, building on important work by Moritz Pasch (1843-1930)). The axioms imply that two lines intersect in at most one point, which is violated by the great circles on a sphere. (This problem can be eliminated by replacing the sphere by the projective plane, but then there's a problem with another axiom that requires each line to divide the space into two separate halves.) See Greenberg's book for details. This is the reason why Saccheri (Sec. 18.1) was able to prove existence of parallels but not uniqueness.

One more technical point: Somewhere Stillwell says that the
parallel postulate is *equivalent* to "Angle sum of any triangle =
180 degrees." In fact, all that can be proved from the elementary Hilbert
axioms
is that "Euclid 5" *implies* the angle sum formula, not the converse.
The problem is that the elementary axioms uniquely characterize
the standard Euclidean and hyperbolic planes
(depending upon whether Euclid 5 is affirmed or denied, respectively) only
when supplemented by a more technical "continuity axiom" stating that
every line segment has the topology of the real numbers (as established
by Dedekind in the 19th century). If that assumption is removed,
other models of the axioms become possible, notably the **Dehn models**
(after Max Dehn (1878-1952),
a student of Hilbert who later became a famous topologist).
In these spaces parallel lines are nonunique (the usual definition
of "hyperbolic") but nevertheless the angle sums of triangles
can be always equal to two right angles, or always greater,
as well as always less. Roughly speaking, these models are created
by replacing the real numbers by a *non-Archimedean field*
containing "infinitesimal elements" and then restricting attention
to an infinitesimal neighborhood of a point.
[time to punt again]

An ironic consequence of the foregoing is that the parallel
postulate *can't be proved* within the Hilbert axioms, unless
Euclidean geometry itself is *inconsistent*.
In other words, if Saccheri et al. had *succeeded* in
"vindicating" Euclid by proving his fifth postulate, they would have
destroyed Euclidean geometry by proving it inconsistent.

Note that in these models the primitive terms (especially "straight
line" and "congruent") are reinterpreted to mean something
rather different than they mean in the embodying Euclidean spaces.
This may leave the impression that the "true" geometry of a plane
really is Euclidean, and hyperbolic geometry describes something
different, so why the big deal? If you *define* Euclidean
geometry as the geometry of the vector space **R**^{2}
with its standard inner product, no one can argue with you.

On the other hand, now that we know that the parallel postulate is not
inevitable, or built into the framework of the human mind so that
denying it is inconceivable, as so many mathematicians and
philosophers used to say,
whether the geometry of *physical space* is Euclidean
becomes an experimental question. Logically, it is entirely
possible that space is hyperbolic, this time with the primitives
meaning (almost) exactly what they always have in Euclidean
thinking.
In fact, in modern physical theory (general relativity, cosmology)
space is something even more general, not homogeneous (i.e., may be
different in different regions) and possibly changing in time.
It can be "bumpy" as well as curved.
In other words, nature is not *exactly* described by the
**R**^{3} geometry given by linear algebra.
On the small scale, planets etc. create bumps in the geometry,
and the influence of the bumps on moving bodies constitutes the
gravitational force of the planets. Gravity = nonhomogeneous
geometry.
On the large scale, however, current observational evidence
indicates that the average curvature of the observable universe
is
very close to flat; but there is no good theoretical reason why
that must be true, and elliptic (usually spherical) and
hyperbolic 3-spaces are frequently studied by serious physicists.
(Because the universe is expanding, the geometry of
four-dimensional space-*time* is not flat.)