In the next place you say, when I had defined arithmetical proportions to be the same when the difference is the same; it was to be expected I should define geometrical proportions to be then the same, when the antecedents are of their consequents _totuple_ or _tantuple_, that is, equimultiple (for _tantuplum_ signifies nothing). In plain words, you expected, that as I defined one by subtraction, I should define the other by the quotient in division. But why should you expect a definition of the same proportion by the quotient? Neither reason nor the authority of Euclid could move you to expect it. Or why should you say _it was to be expected_? But it seems you have the vanity to place the measure of truth in your own learning. In lines incommensurable there may be the same proportion, when, nevertheless, there is no quotient; for setting their symbols one above another doth not make a quotient: for quotient there is none, but in _aliquot parts_. It is therefore impossible to define proportion universally, by comparing quotients. This incommensurability of magnitudes was it that confounded Euclid in the framing of his definition of proportion at the fifth Element. For when he came to numbers, he defined the _same proportion_ irreprehensibly thus: _numbers are then proportional, when the first of the second and the third of the fourth are equimultiple, or the same part, or the same parts_; and yet there is in this definition no mention at all of a quotient. For though it be true, that if in dividing two numbers you make the same quotient, the dividends and the divisors are proportional, yet that is not the definition of the same proportion, but a theorem demonstrable from it. But this definition Euclid could not accommodate to proportion in general, because of incommensurability.