Ch 1.4.12 — Of the Rule of Cardan, or of Scipio Ferreo
Summary: Derives the Cardan–Ferreo formula for solving the depressed cubic , shows how to eliminate the term from a general cubic by a linear substitution, and applies both techniques with examples including the casus irreducibilis.
Sources: chapter-1.4.12
Last updated: 2026-05-03
When Rational Roots Fail (§734–735)
If an integer-coefficient cubic has no divisor of its constant term as a root, it has no rational root at all—not even a fraction. The argument: substituting (in lowest terms) into forces the first term to have denominator while all others have smaller denominators, so they cannot sum to zero.
In this case the roots are irrational (or imaginary), and a different method is needed: Cardan’s Rule, attributed to Gerolamo Cardano but credited more accurately to Scipio Ferreo (§735).
Structure of a Binomial Cube (§736–737)
The key identity:
Setting yields:
So any equation of the form has the known root .
Reducing to Two Conditions (§738–740)
Let and , so . The equation matches this form when:
From the first condition: . From the second, squaring: . Subtracting :
Combined with :
Cardan’s Formula (§741)
For the depressed cubic , one root is:
The expression involves nested square and cube roots. See cardanos-rule for the general concept page.
Worked Examples (§742–744)
§742. : , , . Then .
§743. : . Then .
§744. (known root ): the formula gives , which looks irrational but simplifies because , so .
This illustrates a key subtlety: the formula always gives the correct root, but recognising that the nested radicals collapse to a rational number requires further work—and is only possible when a rational root exists.
Eliminating the Term (§745–746)
Cardan’s formula applies directly only to the depressed cubic (no term). For a complete cubic , substitute:
This kills the quadratic term. Working through the algebra (§746):
which is now of the form and can be attacked with Cardan’s formula.
Full Example (§747–748)
. Substitute :
Apply Cardan with , : after simplification, , so . Dividing by gives , with roots (imaginary).
The step of recognising is possible only because a rational root exists (§748).
Casus Irreducibilis (§749)
When the equation has no rational root, the cube roots in Cardan’s formula cannot be simplified further. Example: gives , which cannot be reduced. (Euler does not use the term casus irreducibilis, but this is the phenomenon.) In such cases the preceding chapter’s divisor-trial method fails and Cardan’s formula is the only recourse.
Questions for Practice (§749, end)
Eleven practice problems, including (), (), and (, treated as a cubic in ).