Arithmetic proportion likewise divides into continuous and discontinuous. And the continuous is that which falls at the least among 3 terms, of which the first is to the second as the second is to the third, that is, in the excesses 1 2 3. See here the proportionality of equiparance made, that is, that unity is that by which 2 exceeds 1 and 3 exceeds 2. The discontinuous arithmetic proportion is this: 1-2, 5-6, 8-9, whose parts are interpolated and discontinuous; nonetheless their excesses are in the proportion of equiparance.

Geometric quantities commensurable among themselves are those in which one and the same line measures both exactly, without any excess, as is the line of one foot, which measures exactly a line of 2 and of 4 feet; and so in arithmetic the 2 measures exactly the 4 and the 8.

Incommensurable quantities are those between which falls no common measure that renders each of them precisely, as happens between the diagonal of the square and its side, between which will never be found a measure that measures each with precision, as is demonstrated in the third part of the seventh [proposition] of the tenth [book] of Euclid. And therefore surd roots are incommensurable and incommunicable.

And by this it is made plain that geometric proportion is of greater abstraction and consideration than the arithmetic. See, for it considers the rational and the irrational, as has been said, whereas the arithmetic considers only the rational, which can be named by some number. Of the other it is wholly deprived, as is shown in that which is found between two surd roots, or one surd root and the other known, like that of the diagonal of the square and its side: which, if the side be 10, the diagonal will be said to be the root of 200.

Equality and inequality.

Rational proportion divides into 2 parts, because all the quantities in the world are either equal or unequal. If they are equal among themselves, there arises the proportion of equality; and if they are unequal, there is created between them the proportion of inequality.

Greater and lesser inequality.

When you make comparison from the greater to the lesser inequality, there will arise the species of the proportion of greater inequality; and if you make comparison from the lesser to the greater inequality, there will arise the species of the proportion of lesser inequality.

Simple and composite.

Greater inequality divides into 2 members, that is, simple and composite; and the like is understood of lesser inequality, because whatever is said, or shall be said, of the one is said also of the other.