In view of these efforts it is well to go back to Euclid and see what this great teacher of university men[46] had to suggest. The following are the five "common notions" that Euclid deemed sufficient for the purposes of elementary geometry. 1. _Things equal to the same thing are also equal to each other._ This axiom has persisted in all elementary textbooks. Of course it is a simple matter to attempt criticism,--to say that -2 is the square root of 4, and +2 is also the square root of 4, whence -2 = +2; but it is evident that the argument is not sound, and that it does not invalidate the axiom. Proclus tells us that Apollonius attempted to prove the axiom by saying, "Let _a_ equal _b_, and _b_ equal _c_. I say that _a_ equals _c_. For, since _a_ equals _b_, _a_ occupies the same space as _b_. Therefore _a_ occupies the same space as _c_. Therefore _a_ equals _c_." The proof is of no value, however, save as a curiosity. 2. _And if to equals equals are added, the wholes are equal._ 3. _If equals are subtracted from equals, the remainders are equal._ Axioms 2 and 3 are older than Euclid's time, and are the only ones given by him relating to the solution of the equation. Certain other axioms were added by later writers, as, "Things which are double of the same thing are equal to one another," and "Things which are halves of the same thing are equal to one another." These two illustrate the ancient use of _duplatio_ (doubling) and _mediatio_ (halving), the primitive forms of multiplication and division. Euclid would not admit the multiplication axiom, since to him this meant merely repeated addition. The partition (halving) axiom he did not need, and if needed, he would have inferred its truth. There are also the axioms, "If equals are added to unequals, the wholes are unequal," and "If equals are subtracted from unequals, the remainders are unequal," neither of which Euclid would have used because he did not define "unequals." The modern arrangement of axioms, covering addition, subtraction, multiplication, division, powers, and roots, sometimes of unequals as well as equals, comes from the development of algebra. They are not all needed for geometry, but in so far as they show the relation of arithmetic, algebra, and geometry, they serve a useful purpose. There are also other axioms concerning unequals that are of advantage to beginners, even though unnecessary from the standpoint of strict logic.