Two numbers from a difference and a product
Finding U and V from U − V and U × V is an old problem: square half the difference and add the product, and you get the square of half their sum.
44 / ANTIQUITY TO THE RENAISSANCE
Niccolò Fontana · Tartaglia
Cut a small cube from the corner of a big one. What is left is a smaller cube and three equal slabs, and that picture is the key to Tartaglia’s rule for the cubic.
Explore the mathematics ↓
01 / SOLVING THE CUBIC
A cube, three slabs and a corner
Cut a corner from a big cube and count what is left.
Use the pieces to win Fior’s contest, then follow Tartaglia’s verse when the cubes are not whole.
Big cube:
Problem:
Equation:
=
=
THE IDEAA u-cube without a v-cube corner is a cube of edge x = u − v and three slabs u × v × x, so x³ + 3uv·x = u³ − v³. To solve x³ + px = q, find two numbers with U − V = q and U × V = (p ÷ 3)³; then x = ∛U − ∛V.
The person behind the ideas
- Won a public contest in 1535 by solving all thirty of Antonio Maria Fior’s cubic problems.
- Gave his rule to Cardano in 1539, written as a verse.
- Published the first Italian translation of Euclid’s Elements in 1543.
More of the mathematics
When the rule breaks
For some cubics with three real roots, the rule needs the square root of a negative number. Working through that case led Rafael Bombelli to the first rules for complex numbers (1572).