Cardan’s Rule
Summary: The formula, derived by Euler following Cardano and Scipio Ferreo, that expresses one root of the depressed cubic as a sum of two nested cube roots.
Sources: chapter-1.4.12
Last updated: 2026-05-03
Historical Note
Euler attributes the rule to Gerolamo Cardano but notes that it is more properly due to Scipio Ferreo (§735), both of whom lived “some centuries since.” The rule first appeared in Cardano’s Ars Magna (1545).
The Formula
For the depressed cubic :
Setting , this is more compactly .
Derivation
Start from the identity . Setting :
So is satisfied by if:
where , . These two symmetric conditions determine and via the quadratic , giving the expressions above.
Eliminating the Term
Cardan’s formula applies only to depressed cubics. Given a general cubic , the substitution removes the term and yields:
which is then solvable by the formula.
When the Formula Simplifies
If is a perfect square, the cube roots inside may still be irrational. They collapse to a rational answer only when the cubic has a rational root—and only by recognising that the binomials under the cube roots are themselves perfect cubes. Example: (§744).
When no rational root exists (casus irreducibilis), the nested radicals are irreducible and are the final form of the answer. Example: (§749).
Relation to the Discriminant
The quantity inside the outer square root is the discriminant of the depressed cubic. When it is negative, is imaginary—the casus irreducibilis—which paradoxically occurs precisely when all three roots are real. Compare discriminant for the quadratic analogue.