For in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third part of 12 by a quarter of the square of 4, and 14 exceeds the third part of 36 by 2 quarters of the square of 4, and proceeding on, the sum of the increasing quantities where the terms are 5 (which sum is 30) exceedeth the third part of those below, (those below are 80, and their third part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square of 4, and when the terms are 6, the quantities above will exceed the third part of them below by 5 quarters of the square of 4. Would you have men believe, that the further they go, the excess of the increasing quantities above the third part of those below shall be so much the less? And yet the proportions of those above, to the thirds of those below, shall decrease eternally; and therefore your twenty-first proposition is false, namely this: “_Si proponatur series infinita quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum), continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem totidem maximæ æqualium, ut 1 ad 3._” That is, if an infinite row of quantities be propounded in duplicate proportion of arithmetically-proportionals (or according to the row of quadratic numbers), continually increasing and beginning from a point or 0; that row shall be to the row of as many equals to the greatest, as 1 to 3. This is false, _ut patet ex præcedentibus_; and, consequently, all that you say in proof of the proportion of your _parabola_ to a _parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is in vain. But your spiral puts me in mind of what you have under-written to the diagram of your proposition 5. _The spiral, in both figures, was to be continued whole to the middle, but, by the carelessness of the graver, it is in one figure_ manca, _in the other_ intercisa. Truly, Sir, you will hardly make your reader believe that a graver could commit those faults without the help of your own copy, nor that it had been in your copy, if you had known how to describe a spiral line then as now. This I had not said, though truth, but that you are pleased to say, though not truth, that I attributed to the printer some faults of mine. I come now to the thirty-ninth proposition, which is this: “_Si proponatur series quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum), continue crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.), propositum sit inquirere quam habeat series illa rationem ad seriem totidem maximæ æqualium_: “_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop. 19_): _Eritque_ (0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4) (0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8)