That is, if there be propounded a row of quantities in duplicate proportion of arithmetically-proportionals (or according to the row of square numbers) continually increasing, and beginning with a point or O. The proportion of that row to a row of so many equals to the greatest, shall be greater than subtriple proportion, and the excess shall be that proportion which unity hath to the sextuple of the number of terms after 0, or the same which the square root of the first number after 0, hath to the sextuple of the square root of the greatest. For proof whereof you have no more here than _patet ex præcedentibus_; and no more before but _adeoque_. You do not well to pass over such curious propositions so slightly; none of the ancients did so, nor, that I remember, any man before yourself. The proposition is false, as you shall presently see. Take, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By this proportion of yours 1 + 4, which makes 5, is to 12 in more than subtriple proportion; by the proportion of 1 to the sextuple of 2 which is 12. Put in order these three quantities 5, 4, 12, and you must see the proportion of 5 to 12 is greater than the proportion of 4 to 12, that is, subtriple proportion, by the proportion of 5 to 4. But by your account the proportion of 5 to 4 is greater than that of 4 to 12 by the proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a very strange paradox. After this you bring in this consectary: “_Cum autem crescente numero terminorum excessus ille supra rationem subtriplam continue minuatur, ut tandem quovis assignabili minor evadat (ut patet) si in infinitum producatur, prorsus evaniturus est. Adeoque._” That is, seeing as the number of terms increaseth, that excess above subtriple proportion continually decreaseth, so as at length it becomes less than any assignable (as is manifest) if it be produced infinitely, it shall utterly vanish, and so. And so what? Sir, this consequence of yours is false. For two quantities being given, and the excess of the greater above the less, that excess may continually be decreased, and yet never quite vanish. Suppose any two unequal quantities differing by more than an unit, as 3 and 6, the excess 3, let 3 be diminished, first by an unit, and the excess will be 2, and the quantities will be 3 and 5; 5 is greater than 4, the excess 1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the quantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again diminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4), yet still 4(1)/(4) is greater than 4. In the same manner you may proceed to (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never come within an unit (though your unit stand for 100 miles) of the lesser quantity propounded 3, if that 3 stands for 300 miles. The excesses above subtriple proportion do not decrease in the manner you say it does, but in the manner which I now shall show you.