CHAPTER VI.

On the third Type of Proportion.

...the difference will be two, and the proportion must be called superbipartiens superbipartiens: A ratio where the larger number contains the smaller once, plus two additional parts (e.g., 5:3)., as in this proportion $5:3$, where the difference is two.

If the difference is three, as in this proportion $8:5$, it is called a supertripartiens proportion original: proportio supertripartiens; literally "three parts over.".

If the difference is four—for example, $9:4$ The author identifies the difference here as four, though $9-4$ is five; he likely intended the ratio $9:5$ to fit the "superquatripartiens" definition.—the proportion will be called superquatripartiens, and so on for other numbers.

In forming this type of proportion, the following rule should be observed: take the third number from the Pythagorean table Pythagorean table: A grid used to study multiplication and the relationships between numbers., which is 3, to serve as the smaller term of the proportion. Then, skipping the number 4 (the next in the natural order), take the number 5 for the larger term of the proportion, in this manner: $5:3$. This will be called superbipartiens thirds, which is the first species of this class.

The prefix super indicates that the smaller number is contained once within the larger, plus some additional parts of it. What is meant by the prefix bi meaning "two" is evident from what was said just above. Furthermore, for a fuller understanding of the different species and names of this class, it is helpful to divide the larger term of the proportion by the smaller. For example: $5 \div 3 = 1 \frac{2}{3}$. Here, the numerator of the fraction (the two) indicates the difference original: terminum differentialem of the proportion—showing that the larger number contains the smaller once, plus two parts. The denominator of the fraction (the three) shows what those extra parts are, namely two thirds. Therefore, the ratio just cited, $5:3$, is correctly called "superbipartiens thirds." From this, it is clearly seen that the numerator of the fraction indicates the difference of the proportion (how many parts of the smaller term the larger contains in addition to the whole), and the denominator indicates the quality of those parts remaining from the division. Let us now look at the various species and names of this class, so that the truth of what has been said may shine forth more clearly.