At the seventeenth chapter, your first exception is to the definition of proportional proportions, which is this: “_Four proportions are then proportional, when the first is to the second, as the third to the fourth_.” The reader will hardly believe that your exception is in earnest. You say, I mean not by proportionality the “_quantity of the proportions_.” Yes I do. Therefore I say again, that _four proportions are then proportional, when the quantity of the first proportion, is to the quantity of the second proportion, as the quantity of the third proportion, to the quantity of the fourth proportion_. Is not my meaning now plainly enough expressed? Or is it not the same definition with the former. But what do I mean, you will say, by the quantity of a proportion? I mean the determined greatness of it, that is, for example, in these numbers, the quantity of the proportion of two to three, is the same with the quantity of the proportion of four to six, or six to nine; and again, the quantity of the proportion of six to four, is the same with the quantity of the proportion of nine to six, or of three to two. But now what do you mean by the quantity of a proportion? You mean that two and three, are the quantities of the proportion of two to three (for so Euclid calls them) and that six and four are the quantities of the proportion of six to four, which is the same with the proportion of three to two. And by this rule, one and the same proportion shall have an infinite number of quantities; and consequently the quantity of a proportion can never be determined. I call one proportion double to another, when one is equal to twice the other; as the proportion of four to one, is double to the proportion of two to one. You call that proportion double where one number, line, or quantity absolute, is double to the other; so that with you the proportion of two to one is a double proportion. It is easy to understand how the number two is double to one, but to what, I pray you, is double the proportion of two to one, or of one to two? Is not every double proportion double to some proportion? See whether this geometry of yours can be taken by any man of sound mind for sense. “_But it is known_,” you say, “_that in proportions, double is one thing, and duplicate another_;” so that it seems to you, that in talking of proportion men are allowed to speak senselessly. “_It is known_,” you say. To whom? It is indeed in use at this day to call _double duplicate_, and _triple triplicate_. And it is well enough; for they are words that signify the same thing, but that they differ (in what subject soever) I never heard till now. I am sure that Euclid, whom you have undertaken to expound, maketh no such difference. And even there where he putteth these numbers, one, two, four, eight, &c. for numbers in _double_ proportion (which is the last proposition of the ninth element) he meaneth not that one to two, or two