It is rather significant that in this postulate which is really a definition of space should be found grounds for such diverse interpretations as to its nature. Of course, the moment the mind seeks to understand the infinite by interpreting it in the unmodified terms of the apparently unchangeable finite it entangles itself into insurmountable difficulties. As a drowning man grasps after straws so the mind, immersed in endless abysses of infinity, fails to conduct itself in a seemly manner; but gasps, struggles and flounders and is happy if it can, in the depths of its perplexity, discover a way of logical escape. The pure mathematician has a hankering after the logically consistent in all his pursuits; to him it is the "Holy Grail" of his highest aspirations. He seeks it as the devotee seeks immortality. It is to him a philosopher's stone, the elixir of perpetual youth, the eternal criterion of all knowledge.
Failures to demonstrate the celebrated postulate of Euclid led, as a matter of course, to the substitution of various other postulates more or less equivalent to it in that each of them may be deduced from the other without the aid of any new hypothesis.
Among those who sought proof by a restatement of the problem are the following:
1. Ptolemy: The internal angles which two parallels make with a transversal on the same side are supplementary.
2. Clavius: Two parallel straight lines are equidistant.
3. Proclus: If a straight line intersects one of two parallels it also intersects the other.
4. Wallis: A triangle being given another triangle can be constructed similar to the given one and of any size whatever.
5. Bolyai (W.): Through three points not lying on a straight line a sphere can always be drawn.
6. Lorenz: Through a point between the lines bounding an angle a straight line can always be drawn which will intersect these two lines.
7. Saccheri: The sum of the angles of a triangle is equal to two right angles.