At the thirteenth article you find fault with, that I say _that the proportion of inequality, whether it be of excess or of defect, is quantity, but the proportion of equality is not quantity_. Whether that which you say, or that which I say, be the truth, is a question worthy of a very strict examination. The first time I heard it argued, was in Mersennus’ chamber at Paris, at such time as the first volume of his _Cogitata Physico-Mathematica_ was almost printed; in which, because he had not said all he would say of proportion, he was forced to put the rest into a general preface, which, as was his custom, he did read to his friends before he sent it to the press. In that general preface, under the title _De Rationibus atque Proportionibus_, at the numbers twelve, thirteen, fourteen, he maintaineth against Clavius, _that the composition of proportion is_ (as of all other things) _a composition of the parts to make a total_, and _that the proportion of equality answereth in quantity to_ non-ens, _or nothing; the proportion of excess, to_ ens, _or quantity; and the proportion of defect, to less than nothing; because equality_ (he says) _is a term of middle signification between excess and defect_. And there also he refuteth the arguments which Clavius, at the end of the ninth Element of Euclid, bringeth to the contrary. And though this were approved by divers good geometricians then present, and never gainsaid by any since, yet do not I say it upon the credit of them, but upon sufficient grounds. For it hath been demonstrated by Eutocius, that _if there be three magnitudes, the proportion of the first to the third is compounded of the proportions of the first to the second, and of the second to the third_; which also I demonstrate in this article. And if there were never so many magnitudes ranked, it might be likewise demonstrated, that the proportion of the first to the last is compounded of the proportions of the first to the second, and of the second to the third, and of the third to the fourth, and so on to the last. If, therefore, we put in order any three numbers, whereof the two last be equal, as four, seven, seven, the proportion of four the first to seven the last, will be compounded of the proportions of four the first to seven the second, and of seven the second to seven the third. Wherefore the proportion of seven to seven (which is of equality) addeth nothing to the proportion of four the first, to seven the second; and consequently the proportion of seven to seven hath no quantity; but that the proportion of inequality hath quantity, I prove it from this, that one inequality may be greater than another.