| and put | α = ½ (−1 + √−3), |
| β = ½ (−1 − √−3). |
Then, from what has been shown (§ 1), it is evident that v and z have each these three values,
v = 3√A, v = α3√A, v = β3√A;
z = 3√B, z = α3√B, z = β3√B.
To determine the corresponding values of v and z, we must consider that vz = −1⁄3 q = 3√(AB). Now if we observe that αβ = 1, it will immediately appear that v + z has these three values,
v + z = 3√A + 3√B,
v + z = α3√A + β3√B,
v + z = β3√A + α3√B,
which are therefore the three values of y.
The first of these formulae is commonly known by the name of Cardan’s rule (see [Algebra]: History).
The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows:—Because vz = −1⁄3 q, therefore z = −1⁄3q × 1/v = −1⁄3 q / 3√A; hence v + z = 3√A − 1⁄3 q / 3√A: thus it appears that the three values of y may also be expressed thus:
y = 3√A − 1⁄3 q / 3√A
y = α3√A − 1⁄3 qβ / 3√A
y = β3√A − 1⁄3 qα / 3√A.