The proportion arising is everywhere greater than subtriple, or (1)/(3), and the excess perpetually decreaseth as the number of terms is augmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. denominator of the fraction being in every place augmented by the number six, as is manifest; so that the excess of the rising proportion above subtriple is the same which unity hath to six times the number of terms after 0; and so. Sir, in these your characters I understand by the cross + that the quantities on each side of it are to be added together and make one aggregate; and I understand by the two parallel lines = that the quantities between which they are placed are one to another equal; this is your meaning, or you should have told us what you meant else; I understand also, that in the first row 0 + 1 is equal to 1, and 1 + 1 equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 + 4 + 4 equal to 12; but (which you are too apt to grant) I understand your symbols no further; but must confer with yourself about the rest. And first I ask you (because fractions are commonly written in that manner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2) be a fraction, that is to say, a part of an unit, and if you will, for the cypher’s sake, whether (0)/(1), be an infinitely little part of 1; and whether (1)/(1) or 1 divided by 1 signify an unity? if that be your meaning, then the fraction (0)/(1) added to the fraction (1)/(1) is equal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O; therefore the fraction (0)/(1) + (1)/(1) is equal to the fraction (1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an absurd conclusion, and cannot own that meaning. I ask you therefore again, if by (0)/(1) you mean the proportion of 0 to 1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2) the proportion of 1 to 2: if so, then it will follow, that if the proportions of 0 to 1 and of 1 to 1 be compounded by addition, the proportion arising will be the proportion of 1 to 2. But the proportion of 0 to 1 is infinitely little, that is, none. Therefore the proposition arising by composition will be that of 1 to 1, and equal (because of the symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so absurd that I dare say that you will not own it. There may be another meaning yet: perhaps you mean that the uppermost quantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how then in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can this be your meaning. Unless you make your symbols more significant, you must not blame me for want of understanding them.