I. A Direct Translation of Euclid’s Postulates:
Let the following be postulated
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a
straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines
make the interior angle on the same side less than two
right angles, the two straight lines, if produced
indefinitely, meet on that side on which are the angles
less than two right angles.
Or, in straightforward declarative sentences:
1. A straight line segment can be drawn joining any
two points.
2. Any straight line segment can be extended
indefinitely in a straight line.
3. Given any straight line segment, a circle can be
drawn having the segment as radius and one endpoint
as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in
such a way that the sum of the inner angles on one
side is less than two right angles, then the two lines
inevitably must intersect each other on that side if
extended far enough.
II. Euclid's Postulates, in modern guise:
1. For every point P and every point Q not equal to P
there exists a unique line that passes through P and Q.
2. For every segment AB and for every segment CD
there exists a unique point E such that B is between A
and E and segment CD is congruent to segment BE.
3. For every point O and every point A not equal to O
there exists a circle with center O and radius OA.
4. All right angles are congruent to each other.
5. For every line l and for every point P that does not lie
on l there exists a unique line m through P that is
parallel to l.
III. Flaws in Euclid's Postulates
1. Euclid takes existence of points for granted, never
stating such existence as a postulate. In Hilbert's
system these assumptions are stated in axiom I-2 and
I-3.
2. Euclid takes betweenness and line separation for
granted, never stating the properties he uses in any
axioms or postulates. In Hilbert's system these
properties are stated in axioms B-1, B-2, B-3, and B-4.
3. Euclid has a faulty proof of SAS where he assumes
that certain motions are possible without stating in
postulates or axioms that such motions are possible.
Some modern treatments of geometry do assume
motions in their axioms. However, Hilbert's does not,
so in Hilbert's system SAS has to be taken as an axiom
(axiom C-6).
4. Euclid takes continuity properties for granted. For
example, he assumes (without stating it as an axiom)
that if two circles are sufficiently close together then
they intersect in two points.
Geometry BolyaiLobachevski
(hyperbolic)
Parallels Many
C/D for a > 
circle
Sum of
< 180o
interior
angles for
a triangle
2-D
Saddle
surfaces
surface
Or Pseudosphere
Euclid
Riemann
(ellipitical)
One
=
None
<
= 1800
> 180o
Plane
Surface
of a
Sphere