...composed; likewise, let the first term original: antecedens of this proportion be multiplied by the second term original: sequente of the ratio that consists of its simplest "radical" terms. If both sums produced by this multiplication are equal in number, it will be an evident sign that the two proportions are of the same species. Here is the trial:

A cross-multiplication diagram. The number 11 is at the top left and 21 at the bottom right, joined by a diagonal line. The number 33 is at the top right and 7 at the bottom left, joined by a diagonal line forming an X. Below this, the product 231 is written twice, once under each side of the diagram.

This should be understood to apply to all other proportions consisting of irrational numbers In early modern mathematics, "irrational" often referred to ratios that were complex or not yet reduced to their simplest whole-number form, rather than the modern definition of numbers like pi..

CHAPTER VII.

On the fourth Type of Proportion, called the multiple superparticular.

This composite category contains as many species as its simple superparticular counterpart The "superparticular" category involves ratios like 3:2 or 4:3. The "multiple" version adds a whole number to that, resulting in ratios like 5:2 (2 and 1/2).. It is a Proportion where the larger term includes the smaller twice, three times, four times, etc., and simultaneously one aliquot part aliquot part: a part that, when multiplied, exactly equals the whole (e.g., 2 is an aliquot part of 4). of it. It is constructed from the species of its simple superparticular category in this manner:

Let us take, for example, the first species of the superparticular category in its radical terms, namely the sesquialter 3/2 sesquialter: A ratio of 3:2, where the larger number is 1.5 times the smaller.. Add the smaller number to the larger, and you will have 5; then place the smaller number of that same sesquialter (which is 2) beneath the larger number. The resulting proportion is 5/2, which is the first species of this category. It is called the double sesquialter because its larger number contains the smaller twice, plus one aliquot part, as is clear from the division: 5 / 2 = 2 1/2. See the examples of this species: