10. If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third. This axiom is needed several times in geometry. The case in which a > b and b = c, therefore a > c, is provided for in Axiom 9.
11. The whole is greater than any of its parts and is equal to the sum of all its parts. The latter part of this axiom is really only the definition of "whole," and it would be legitimate to state a definition accordingly and refer to it where the word is employed. Where, however, we wish to speak of a polygon, for example, and wish to say that the area is equal to the combined areas of the triangles composing it, it is more satisfactory to have this axiom to which to refer. It will be noticed that two related axioms are here combined in one, for a reason similar to the one stated under Axiom 5.
In the case of the postulates we are met by a problem similar to the one confronting us in connection with the axioms,—the problem of the "irreducible minimum" as related to the question of teaching. Manifestly Euclid used postulates that he did not state, and proved some statements that he might have postulated.[47]
The postulates given by Euclid under the name αἰτήματα (aitemata) were requests made by the teacher to his pupil that certain things be conceded. They were five in number, as follows:
1. Let the following be conceded: to draw a straight line from any point to any point.
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.