Critique of Pure Reason [Max Müller] (New York, 1900), pp. 18-20 and Supplement 8.

[2003]. Schopenhauer’s Predicabilia a priori.[11]

OF TIME OF SPACE
1. There is but one time, all different times are parts of it. There is but one space, all different spaces are parts of it.
2. Different times are not simultaneous but successive. Different spaces are not successive but simultaneous.
3. Everything in time may be thought of as non-existent, but not time. Everything in space may be thought of as non-existent, but not space.
4. Time has three divisions: past, present and future, which form two directions with a point of indifference. Space has three dimensions: height, breadth, and length.
5. Time is infinitely divisible. Space is infinitely divisible.
6. Time is homogeneous and a continuum: i.e. no part is different from another, nor separated by something which is not time. Space is homogeneous and a continuum: i.e. no part is different from another, nor separated by something which is not space.
7. Time has no beginning nor end, but all beginning and end is in time. Space has no limits [Gränzen], but all limits are in space.
8. Time makes counting possible. Space makes measurement possible.
9. Rhythm exists only in time. Symmetry exists only in space.
10. The laws of time are a priori conceptions. The laws of space are a priori conceptions.
11. Time is perceptible a priori, but only by means of a line-image. Space is immediately perceptible a priori.
12. Time has no permanence but passes the moment it is present. Space never passes but is permanent throughout all time.
13. Time never rests. Space never moves.
14. Everything in time has duration. Everything in space has position.
15. Time has no duration, but all duration is in time; time is the persistence of what is permanent in contrast with its restless course. Space has no motion, but all motion is in space; space is the change in position of that which moves in contrast to its imperturbable rest.
16. Motion is only possible in time. Motion is only possible in space.
17. Velocity, the space being the same, is in the inverse ratio of the time. Velocity, the time being the same, is in the direct ratio of the space.
18. Time is not directly measurable by means of itself but only by means of motion which takes place in both space and time.... Space is measurable directly through itself and indirectly through motion which takes place in both time and space....
19. Time is omnipresent: each part of it is everywhere. Space is eternal: each part of it exists always.
20. In time alone all things are successive. In space alone all things are simultaneous.
21. Time makes possible the change of accidents. Space makes possible the endurance of substance.
22. Each part of time contains all substance. No part of space contains the same substance as another.
23. Time is the principium individuationis. Space is the principium individuationis.
24. The now is without duration. The point is without extension.
25. Time of itself is empty and indeterminate. Space is of itself empty and indeterminate.
26. Each moment is conditioned by the one which precedes it, and only so far as this one has ceased to exist. (Principle of sufficient reason of being in time.) The relation of each boundary in space to every other is determined by its relation to any one. (Principle of sufficient reason of being in space.)
27. Time makes Arithmetic possible. Space makes Geometry possible.
28. The simple element of Arithmetic is unity. The element of Geometry is the point.

—Schopenhauer, A.

Die Welt als Vorstellung und Wille; Werke (Frauenstädt) (Leipzig, 1877), Bd. 2, p. 55.

[2004]. The clear possession of the Idea of Space is the first requisite for all geometrical reasoning; and this clearness of idea may be tested by examining whether the axioms offer themselves to the mind as evident.—Whewell, William.

The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. 4, sect. 4 (London, 1858).

[2005]. Geometrical axioms are neither synthetic a priori conclusions nor experimental facts. They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction.... In other words, axioms of geometry are only definitions in disguise.

That being so what ought one to think of this question: Is the Euclidean Geometry true?