Strictly speaking, Euclid might have been required to postulate that points and straight lines exist, but he evidently considered this statement sufficient. Aristotle had, however, already called attention to the fact that a mere definition was sufficient only to show what a concept is, and that this must be followed by a proof that the thing exists. We might, for example, define _x_ as a line that bisects an angle without meeting the vertex, but this would not show that an _x_ exists, and indeed it does not exist. Euclid evidently intended the postulate to assert that this line joining two points is unique, which is only another way of saying that two points determine a straight line, and really includes the idea that two straight lines cannot inclose space. For purposes of instruction, the postulate would be clearer if it read, _One straight line, and only one, can be drawn through two given points_. 2. _To produce a finite straight line continuously in a straight line._ In this postulate Euclid practically assumes that a straight line can be produced only in a straight line; in other words, that two different straight lines cannot have a common segment. Several attempts have been made to prove this fact, but without any marked success. 3. _To describe a circle with any center and radius._ 4. _That all right angles are equal to one another._ While this postulate asserts the essential truth that a right angle is a _determinate magnitude_ so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be _proved_, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the _invariability of figures_, which would have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be taken as equivalent to the principle of _invariability of figures_, or its equivalent, the _homogeneity_ of space.[48]