Different cubics, different curves
Khayyam used a suitable pair of conics for each type of cubic. This lesson explores just one type.
28 / ANTIQUITY TO THE RENAISSANCE
Omar Khayyam
A cube and a beam must hold a set amount. Often no neat length works. Two curves can find the length anyway.
Explore the mathematics ↓
01 / CUBICS FROM CONICS
Where two curves cross, a cubic is solved.
A cube and a 2 × 2 beam share one length x. Together they must hold a target number of small cubes: x³ + 4x = B.
Khayyam drew a parabola and a circle built from the problem. Where they cross, both of their rules hold at once, and that point’s x solves the cubic.
Target volume:
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THE IDEAFor x³ + 4x = B, the parabola gives x² = 2y and the circle on a diameter of B ÷ 4 gives y² = x(B ÷ 4 − x). At their crossing both hold, and together they give x³ + 4x = B.
The person behind the ideas
- Wrote an influential treatise on algebra.
- Studied geometry and ratios as well as equations.
- Worked on astronomy and calendar reform.
More of the mathematics
Why not the origin?
Both curves also pass through (0, 0), but x = 0 means no cube at all. Only the other crossing solves the problem.