(to demonstrate) a geometrical (problem) by arithmetic, for there are three things in demonstrations, one the demonstrated conclusion, and this is that which is per se inherent in a certain genus. Another are axioms, but axioms are they from which (demonstration is made), the third is the subject genus, whose properties and essential accidents demonstration makes manifest. Now it is possible that the things from which demonstration consists may be the same, but with those whose genus is different, as arithmetic and geometry, we cannot adapt an arithmetical demonstration to the accidents of magnitudes, except magnitudes are numbers, and how this is possible to some shall be told hereafter. But arithmetical demonstration always has the genus about which the demonstration (is conversant), and others in like manner, so that it is either simply necessary that there should be the same genus, or in a certain respect, if demonstration is about to be transferred; but that it is otherwise impossible is evident, for the extremes and the middles must necessarily be of the same genus, since if they are not per se, they will be accidents. On this account we cannot by geometry demonstrate that there is one science of contraries, nor that two cubes make one cube, neither can any science (demonstrate) what belongs to any science, but such as are so related to each other as to be the one under the other, for instance, optics to geometry, and harmonics to arithmetic. Nor if any thing is inherent in lines not so far as they are lines, nor as they are from proper principles, as if a straight line is the most beautiful of lines, or if it is contrary to circumference, for these things are inherent not by reason of their proper genus, but in so far as they have something common.
