9. _A quantity may be substituted for its equal in an equation or in an inequality._ This axiom is tacitly assumed by all writers, and is very useful in the proofs of geometry. It is really the basis of several other axioms, and if we were seeking the "irreducible minimum," it would replace them. Since, however, we are seeking only a reasonably abridged list of convenient assumptions that beginners will understand and use, this axiom has much to commend it. If we consider the equations (1) _a_ = _x_ and (2) _b_ = _x_, we see that for _x_ in equation (1) we may substitute _b_ from equation (2) and have _a_ = _b_; in other words, that Axiom 8 is included in Axiom 9. Furthermore, if (1) _a_ = _b_ and (2) _x_ = _y_, then since _a_ + _x_ is the same as _a_ + _x_, we may, by substituting, say that _a_ + _x_ = _a_ + _x_ = _b_ + _x_ = _b_ + _y_. In other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes others has a legitimate place, because a beginner would be too much confused by seeing its entire scope, and because he will make frequent use of it in his mathematical work. 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 [Greek: aitemata](_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._