Let us now try what better success we shall have where the places are three, as here: (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12): If your symbols be fractions, the compound of them by addition is (5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently (because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be allowed, and therefore that was not your meaning. If you meant that the proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal to the proportion of 5 to 12, you will fall again into no less an inconvenience. For the proportion arising out of that composition will be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely little. Then to compound the other two, set them in this order 1. 4. 4. and you have a proportion compounded of 1 to 4 and of 4 to 4, namely, the proportion of the first to the last, which is of 1 to 4, which must be equal, by this your meaning, to the proportion of 5 to 12, and consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly, if you mean that the uppermost quantities to the uppermost, and the lowermost to the lowermost in the first equation are equal, it is granted, but then again in the second equation it is false. It concerns your fame in the mathematics to look about how to justify these equations which are the premises to your conclusion following, namely, that the proportion arising is every where greater than sub-triple, or a third; and that the excess (that is, the excess above subtriple) perpetually decreaseth as the number of terms is augmented, as here (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you plainly is false. But first I wonder why you were so angry with me for saying you made proportion to consist in the quotient, as to tell me it was abominably false, and to justify it, cite your own words _penes quotientem_; do not you say here, the proportion is everywhere greater than subtriple, or (1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot say in this place that _penes_ is understood; for if it were expressed you would not be able to proceed.