Steps to Parnassus (Gradus ad Parnassum)

The superquatripartiens fifths proportion superquatripartiens fifths: A ratio of 9:5, where the larger number contains the smaller once, plus four-fifths (4/5) more. in radical terms.

The same Proportion outside of radical terms The "radical terms" are the simplest form of the ratio (9:5). "Outside radical terms" refers to expanded versions of that same ratio, like 18:10 or 27:15..

The superquatripartiens sevenths proportion superquatripartiens sevenths: A ratio of 11:7, where the larger number contains the smaller once, plus four-sevenths (4/7) more. consisting of radical terms.

The same Proportion outside of radical terms.

The superquatripartiens ninths proportion superquatripartiens ninths: A ratio of 13:9, where the larger number contains the smaller once, plus four-ninths (4/9) more. in radical terms.

The same Proportion outside of radical terms.

At this point, we must recall those things which we said above concerning the superbipartiens thirds proportion original: proportionem superbipartientem tertias; the ratio of 5:3. about the golden rule of reduction original: aurea reductionis regula.. This is the rule by which a test is conducted to see whether a certain proportion consisting of terms outside of the radical ones is of the same species as its original proportion. This rule certainly has its place in proportions where both terms consist of rational numbers original: numeris rationalibus., or those which can be measured together by a certain common divisor. However, because it happens that some proportions outside of the radical terms consist of irrational numbers In historical mathematical texts, "irrational" was sometimes used to describe numbers that do not share an obvious common factor, or to describe complex relationships where the underlying ratio is obscured., and for that reason they cannot be reduced to their origin through a common divisor—as in these proportions: 22. 33. / 14. 21. superquatripartiens sevenths—another rule of testing, which is usually called the cross-rule original: regula cruciata; known today as cross-multiplication., will have to be applied.

Let a certain proportion be set down in its radical terms, and the same proportion outside of the radical terms and consisting of irrational numbers; for example, the aforementioned superquatripartiens sevenths.

Multiply the antecedent of the ratio antecedent: The first or leading number in a ratio comparison. consisting of its radical terms with the consequent consequent: The second or following number in a ratio comparison. of the ratio positioned outside the radical terms,