To differ in number, magnitude, species, and genus, what.
2. And, first of all, it is manifest that no two bodies are the same; for seeing they are two, they are in two places at the same time; as that, which is the same, is at the same time in one and the same place. All bodies therefore differ from one another in number, namely, as one and another; so that the same and different in number, are names opposed to one another by contradiction.
In magnitude bodies differ when one is greater than another, as a cubit long, and two cubits long, of two pound weight, and of three pound weight. And to these, equals are opposed.
Bodies, which differ more than in magnitude, are called unlike; and those, which differ only in magnitude, like. Also, of unlike bodies, some are said to differ in the species, others in the genus; in the species, when their difference is perceived by one and the same sense, as white and black; and in the genus, when their difference is not perceived but by divers senses, as white and hot.
What is relation, proportion, and relatives.
3. And the likeness, or unlikeness, equality, or inequality of one body to another, is called their RELATION; and the bodies themselves relatives or correlatives; Aristotle calls them τὰ πρὸς τί; the first whereof is usually named the antecedent, and the second the consequent; and the relation of the antecedent to the consequent, according to magnitude, namely, the equality, the excess or defect thereof, is called the PROPORTION of the antecedent to the consequent; so that proportion is nothing but the equality or inequality of the magnitude of the antecedent compared to the magnitude of the consequent by their difference only, or compared also with their difference. For example, the proportion of three to two consists only in this, that three exceeds two by unity; and the proportion of two to five in this, that two, compared with five, is deficient of it by three, either simply, or compared with the numbers different; and therefore in the proportion of unequals, the proportion of the less to the greater, is called DEFECT; and that of the greater to the less, EXCESS.
Proportionals, what.
4. Besides, of unequals, some are more, some less, and some equally unequal; so that there is proportion of proportions, as well as of magnitudes; namely, where two unequals have relation to two other unequals; as, when the inequality which is between 2 and 3, is compared with the inequality which is between 4 and 5. In which comparison there are always four magnitudes; or, which is all one, if there be but three, the middlemost is twice numbered; and if the proportion of the first to the second, be equal to the proportion of the third to the fourth, then the four are said to be proportionals; otherwise they are not proportionals.
The proportion of magnitudes to one another, wherein it consists.
5. The proportion of the antecedent to the consequent consists in their difference, not only simply taken, but also as compared with one of the relatives; that is, either in that part of the greater, by which it exceeds the less, or in the remainder, after the less is taken out of the greater; as the proportion of two to five consists in the three by which five exceeds two, not in three simply only, but also as compared with five or two. For though there be the same difference between two and five, which is between nine and twelve, namely three, yet there is not the same inequality; and therefore the proportion of two to five is not in all relation the same with that of nine to twelve, but only in that which is called arithmetical.