Those magnitudes are said to be *commensurable* which are measured by the same measure, and those *incommensurable* which cannot have any common measure.

Two magnitudes *A* and *B* of the same kind are *commensurable* if there is another magnitude *C* of the same kind such that both are multiples of *C,* that is, there are numbers *m* and *n* such that *nC* = *A* and *mC* = *B.* See definition V.Def.5 for the definition of equality of ratios (also known as a *proportion*). If the two magnitudes are not commensurable, then they’re called *incommensurable.*

Propositions X.2 through X.8 and several later ones deal with commensurable and incommensurable magnitudes. In particular X.5 and X.6 state that two magnitudes are commensurable if and only if their ratio is the ratio of a number to a number. For example, if *nC* = *A* and *mC* = *B,* then the ratio of magnitudes *A* : *B* is the same as the ratio of numbers *n* : *m.* And conversely, if *A* : *B* = *n* : *m,* then the 1/*n*^{th} part of *A* equals the 1/*m*^{th} part of *B.*

Ratios of numbers are known to modern mathematicians as *rational numbers* while other ratios are known as *irrational numbers.* Unfortunately, Euclid used the words rational and irrational in a different way in Definition 3, see below.

Straight lines are *commensurable in square* when the squares on them are measured by the same area, and *incommensurable in square* when the squares on them cannot possibly have any area as a common measure.

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called *rational,* and those straight lines which are commensurable with it, whether in length and in square, or in square only, *rational,* but those that are incommensurable with it *irrational.*

The proof referred to at the beginning of this definition is that of X.10 which finds lines commensurable in square only, and lines incommensurable in square.

Euclid uses the words rational and irrational differently than mathematicians both before and after him. The usual uses of these words correspond to commensurable and incommensurable, respectively. But when applied to lines, Euclid makes them correspond to commensurable in square and incommensurable in square. First, one line is chosen as a standard, then another line is called *rational* if it is commensurable in square, and *irrational* if not. Thus, the diagonal on the square on the standard line is rational, even though it’s incommensurable with the standard line, since it’s commensurable in square with it.

This confusing terminology makes Book X difficult to understand.

And the let the square on the assigned straight line be called *rational,* and those areas which are commensurable with it *rational,* but those which are incommensurable with it *irrational,* and the straight lines which produce them *irrational,* that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.