A ratio grows by enlarging its greater term or by diminishing the lesser term. Example of a double, which is 4 and 8: now if you add a 4 to the 8 you will have made a triple, that is 12 against 4; and if you leave the 8 alone and take 3 from the 4, you will have 1 against 8, which is an octuple. And so the denominations of the first ratio have grown — which at first was double and is now octuple, the greater name. A ratio grows if to the greater term a number greater than itself is added and to the lesser term a number smaller than itself is added. Example, 4 and 8: add 12 to the 8 and you will have 20 for one term; then increase the lesser, which is 4, by 2, so 6 remains for the lesser term, and you will have 6 against 20, which will have made a triple-and-one-third (tripla sesquiterza), which has become a greater denomination. A ratio decreases if between its terms a number is interposed that is less than the greater extreme and greater than the lesser extreme. Example: let the terms of the first ratio be 2 and 4, which is a double, between which 3 is interposed, which is more than 2 and less than 4; and so from one ratio you have made two, each of which is less than the whole, which was a double — because from 2 to 3 is a sesquialtera, and from 3 to 4 is a sesquiterza. If, to the quantities between which an arithmetic proportionality falls ... If to an arithmetic proportionality equal quantities are added, the proportions will remain equal arithmetically. Suppose that to 4 and 8, and 8 and 12, 4 is added to each: where before the differences were 4 and are now made 8, they are equal as before, even though the number has been increased. If to the greater extremes of the quantities between which an arithmetic proportionality falls equal quantities are added, the arithmetic proportionality will not change. If to the lesser extremes of the arithmetic proportionality equal quantities are added, the arithmetic proportionality will not change. Thus: let there be 4 and 8, 8 and 12; and if 6 is added to the 4, the lesser term of the lesser pair, it makes 10, and if 6 is added to the 8, the lesser term of the greater pair, it makes 14; whence the same arithmetic ratio will hold between 10 and 8 as between 14 and 12 — although it is not the same as before, because here the differences are by 2 and before they were by 4. The arithmetic proportionality is likewise maintained by equally diminishing the greater and the lesser term of both pairs. Thus: let the terms be 4 and 8, 8 and 12, whose differences are 4 and 4. Now let 2 be taken from each term: there will remain 2 and 6, 6 and 10, whose differences are 4 and 4, as at first.