But I return to your conclusion, that the excess of the proportion of the increasing quantities above the third part of so many times the greatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the quantity above exceeds the third part of the quantities below by (1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It may be said your meaning is, that the proportion of 1 to the subtriple of 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor proceed further where the terms be but two. Let us therefore take the second row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above is 5, the sum below is 12, the third part whereof is 4; if you mean, that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which is subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity, which is (12)/(12). I do not think you will own such an equation as (12)/(12) = (1)/(12) Therefore I believe you mean (and your next proposition assures me of it), that the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12; if you do so, you are yet deceived. For if the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12 added to the proportion of 1 to 12 must make the proportion of 5 to 4. But if you look on these quantities, 4, 12, 144, you will see, and must not dissemble, that the proportion of 4 to 12 is subtriple, and the proportion of 12 to 144 is the same with that of 1 to 12. Therefore by your assertion it must be as 5 to 4 so 4 to 144, which you must not own. And yet this is manifestly your meaning, as appeareth in these words: “_Ut sit rationis provenientis excessus supra subtriplam ea quam habet unitas ad sextuplum numeri terminorum post 0, adeoque_,” which cannot be rendered in English, nor need to be. For you express yourself in the twentieth proposition very clearly; I noted it only that you may be more merciful hereafter to the stumblings of a hasty pen. For _excessus ea quam_ does not well, nor is to be well excused by _subauditur ratio_. Your twentieth proposition is this: “_Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ termini maximi._”