(0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12) = (1)/(4) + (1)/(12) _Et sic deinceps._ “_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4). Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur, puta (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero quaternatio, ut patet, ut sit rationis provenientis excessus supra subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post 0 adeoque._” That is, if a row of quantities be propounded in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning from a point or 0, as 0, 1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that row hath to a row of as many equals to the greatest. Be it sought by way of induction, as in proposition 1 and 19. The proposition arising is everywhere greater than subquadruple, or (1)/(4), and the excess perpetually decreaseth as the number of terms increaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The denominator of the fraction, or consequent of the proportion, being in every place augmented by the number 4, as is manifest, so that the excess of the arising proportion above subquadruple is the same with that which an unit hath to the quadruple of the number of the terms after 0, and so. Here are just the same faults which are in proposition 19. For, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be another fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false. For this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1) = (1)/(2), that is, the whole equal to half. But perhaps you do not mean them fractions, but proportions; and, consequently, that the proportion of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not that I, but that you think there is a composition of proportions by multiplication, which I shall show you anon is false), must be equal to the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1 is infinitely little, that is, none at all; and, consequently, the proportion of 1 to 1 is equal to the proportion of 1 to 2, which is again absurd. There is no doubt but the whole number of 0 + 1 is equal to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as you do, not for whole numbers, but for fractions or proportions, the equations are false.