Chapter VI. On the third Type of Proportion.

Perhaps it might seem to someone that these proportions—specifically 10:6 with a difference of 4—are not at all equivalent to the previous 5:3 with a difference of 2, since in the former the differential term The "differential term" is the number resulting from subtracting the smaller term from the larger. is two and the parts remaining from division are thirds, whereas in this one (10:6) the differential term is four and the parts remaining from division are sixths. However, this does not at all prevent these proportions existing outside of their radical terms radical terms: The lowest possible whole numbers that express a ratio; what we would today call a "simplified fraction." from being of the same species as the first (5:3), as is demonstrated by the golden rule of reduction original: auream reductionis regulam; a method for simplifying fractions by finding a common divisor..

Let a number be sought which, by dividing each term of the proportion, exhausts the whole with no remainder. For example, in this proportion 10:6, the common divisor is the number 2, because it divides both the number ten and the number six so that nothing remains; from ten it produces 5, and from six it produces 3. This proportion is superbipartiens thirds superbipartiens thirds: A ratio of 5:3, where the larger number (5) contains the smaller (3) once, plus two-thirds of the smaller number (2). consisting of radical numbers, which is the same result found in the others.

In this way, all proportions outside of their radical terms can be reduced to the proportion consisting of radical terms, as if to their own origin, as the following species will further demonstrate.

The superbipartiens fifths proportion consisting of its radical terms.

The radical superbipartiens sevenths proportion.