At the fourth article, you allow the demonstration all the way (except the faults of the third, which I have already proved to be none) till I come to say, “_that because the proportion of F K to B I is double to the proportion of A F to A B, therefore the proportion of A B to A F is double to the proportion of B I to F K_.” This you deny, and wonder at as strange, (for it is indeed strange to you), and in many places you exclaim against it as extreme ignorance in geometry. In this place you only say, “_no such matter; for though one proportion be double to another, yet it does not follow that the converse is the double of the converse_.” So that this is the issue to which the question is reduced, whether you have any or no geometry. I say, if there be three quantities in continual proportion, and the first be the least, the proportion of the first to the second is double to the proportion of the first to the third; and you deny it. The reason of our dissent consisteth in this, that you think the doubling of a proportion to be the doubling of the quantity of the proportion, as well in proportions of defect, as in proportions of excess; and I think that the doubling of a proportion of defect, is the doubling of the defect of the quantity of the same. As for example in these three numbers, 1, 2, 4, which are in continual proportion, I say the quantity of the proportion of one to two, is double the quantity of the proportion of one to four. And the quantity of the proportion of one to four, is half the quantity of the proportion of one to two. And yet deny not but that the quantity of the defect in the proportion of one to two is doubled in the proportion of one to four. But because the doubling of defect makes greater defect, it maketh the quantity of the proportion less. And as for the part which I hold in this question, first, there is thus much demonstrated by Euclid, El. v. prop. 8; that the proportion of one to two, is greater than the proportion of one to four, though how much it is greater be not there demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that it is twice as great, that is to say, (to a man that speaks English), double. The introducing of _duplicate_, _triplicate_, &c. instead of _double_, _triple_, &c. (though now they be words well understood by such as understand what proportion is), proceeded at first from such as durst not for fear of absurdity, call the half of any thing double to the whole, though it be manifest that the half of any defect is a double quantity to the whole defect; for want added to want maketh greater want, that is, a less positive quantity. This difference between _double_ and _duplicate_, lighting upon weak understandings, has put men out of the way of true reasoning in very many questions of geometry. Euclid never used but one word both for _double_ and _duplicate_. It is the same fault when men call half a quantity _subduplicate_, and a third