It should also be noted that these five axioms might be combined in one, namely, _If equals are operated on by equals in the same way, the results are equal_. In Axiom 1 this operation is addition, in Axiom 2 it is subtraction, and so on. Indeed, in order to reduce the number of axioms two are already combined in Axiom 5. But there is a good reason for not combining the first four with the fifth, and there is also a good reason for combining two in Axiom 5. The reason is that these are the axioms continually used in equations, and to combine them all in one would be to encourage laxness of thought on the part of the pupil. He would always say "by Axiom 1" whatever he did to an equation, and the teacher would not be certain whether the pupil was thinking definitely of dividing equals by equals, or had a hazy idea that he was manipulating an equation in some other way that led to an answer. On the other hand, Axiom 5 is not used as often as the preceding four, and the interchange of integral and fractional exponents is relatively common, so that the joining of these two axioms in one for the purpose of reducing the total number is justifiable. 6. _If unequals are operated on by positive equals in the same way, the results are unequal in the same order._ This includes in a single statement the six operations mentioned in the preceding axioms; that is, if _a_ > _b_ and if _x_ = _y_, then _a_ + _x_ > _b_ + _y_, _a_ - _x_ > _b_ - _y_, _ax_ > _by_, etc. The reason for thus combining six axioms in one in the case of inequalities is apparent. They are rarely used in geometry, and if a teacher is in doubt as to the pupil's knowledge, he can easily inquire in the few cases that arise, whereas it would consume a great deal of time to do this for the many equations that are met. The axiom is stated in such a way as to exclude multiplying or dividing by negative numbers, this case not being needed. 7. _If unequals are added to unequals in the same order, the sums are unequal in the same order; if unequals are subtracted from equals, the remainders are unequal in the reverse order._ These are the only cases in which unequals are necessarily combined with unequals, or operate upon equals in geometry, and the axiom is easily explained to the class by the use of numbers. 8. _Quantities that are equal to the same quantity or to equal quantities are equal to each other._ In this axiom the word "quantity" is used, in the common manner of the present time, to include number and all geometric magnitudes (length, area, volume).