_The proportion_, you say, _which is compounded of double and triple proportion, is not_, as I would have it, _quintuple, but sextuple_, as in these numbers, six, three, one; where the proportion of six to three is double, the proportion of three to one triple, and the proportion of six to one sextuple, not quintuple. Tell me, egregious professors, how is six to three double proportion? Is six to three the double of a number, or the double of some proportion? All men know the number six is double to the number three, and the number three triple to an unity. But is the question here of compounding numbers, or of compounding proportions? Euclid, at the last proposition of his ninth Element, says indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι ἀναλογία, in double proportion; yet there is no man that understands it otherwise, than if he had said in proportion of the single quantity to the double quantity; and after the same rate, if he had said three, nine, twenty-seven, &c. had been in triple proportion, all men would have understood it, of the proportion of any quantity to its triple. Your instance, therefore, of six, three, one, is here impertinent, there being in them no doubling, no tripling, no sextupling of proportions, but of numbers. You may observe also, that Euclid never distinguished between double and duplicate, as you do. One word διπλάσιον serves him every where for either. Though, I confess, some curious grammarians take διπλάσιον for duplicate in number, and διπλοῦν for double in quantity; which will not serve your turn. Your geometry is not your own, but you case yourselves with Euclid’s; in which, as I have showed you, there be some few great holes; and you by misunderstanding him, as in this place, have made them greater. Though the beasts that think your railing roaring, have for a time admired you; yet now that through these holes of your case I have showed them your ears, they will be less affrighted. But to exemplify the composition of proportions, take these numbers, thirty-two, eight, one, and then you shall see that the proportion of thirty-two to one is the sum of the proportions of thirty-two to eight, and of eight to one. For the proportion of thirty-two to eight is double the proportion of thirty-two to sixteen; and the proportion of eight to one, is triple the proportion of thirty-two to sixteen; and the proportion of thirty-two to one is quintuple of thirty-two to sixteen; but double and triple added together maketh quintuple. What can be here denied?