Why does Hadamard claim that Pascal could have discovered non-Euclidean geometry In The psychology of invention in the mathematical field, p. 53, Hadamard makes the following claim:

Is his point that there must, in any axiomatic theory, be undefined terms, and if you write the parallel postulate entirely expanded, i.e., with definiens substituted for definiendum , then you will see that its not a truth of logic? But that seems completely off. You may still "see" that its a truth for perceptual space (let us grant, for the sake of the argument), and moreover, you may still fail to see that it isn't implied by the other axioms+logic. Thus, while you may arguably conceive of the possiblity of non-Euclidean axiom systems, you might still not yet perceive that of non-Euclidean geometries (qua sciences of space); and in fact, without a proof of relative consistency of these systems, e.g. by an inner model in Euclidean geometry, or by a model in R, you aren't technically allowed to even assume so much as the possibility of these systems, since (lacking such proof), the PP might -- for all you know -- after all be a consequence of the other axioms.
Am I completely misunderstanding Hadamard's point here?
 A: 
Why, according to Hadamard, Pascal could have discovered non-Euclidean geometry?

Let start from Euclid's definition:

Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

The issue is not about substitution of definiens in place of definiendum in the statement of the postulate.
The original text of Euclid's Postulate 5 does not use the definiendum "parallel lines" but refers to the "meeting condition" of the definies:

Postulate 5. That, if a straight line falling on two straight lines makes the interior angles 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 the two right angles.

The issue is to note that in Post.5 Euclid uses a property of lines: to meet that is not mentioned in any other postulates.
Thus, we cannot derive anything about "lines meeting" from them.
Obviously, it is not easy at all to note this fact: the modern concept of model (the intuitive one) dates from late XIX Century and was very far away from Ancient and Early Modern mathematics.
Hilbert's proof is a mathematical proof, and not an intuition (made with hindsight).
The question posed by Hadamard can be rewritten as: can Pascal have had that intuition without a development of mathematics comparable to that of Hilbert?
