which, taken two and two, have the same ratio;

which, taken two and two, have the same ratio, that is to say,

which, taken two and two, have the same ratio;

and so on, whatever the number of magnitudes be.

PROP. XXIII. THEO.

If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last of the same.

N.B. This is usually cited by the words "ex æquali in proportione perturbatâ ;" or "ex æquo perturbato."

First, let there be three magnitudes,

and other three,

3

3

which, taken two and two in a cross order, have the same ratio ;

Let these magnitudes and their respective equimultiples be arranged as follows:

then, because there are three magnitudes, M, M, m

3

which, taken two and two in a cross order, have the same ratio;

which, when taken two and two in a cross order, have the same