Ch 1.4.10 — Of Pure Equations of the Third Degree
Summary: Defines pure (incomplete) cubic equations , shows by polynomial division that every cubic has three roots, and derives the two imaginary cube roots alongside the real one.
Sources: chapter-1.4.10
Last updated: 2026-05-03
Definition (§706)
A cubic equation is pure when it contains only the cube of the unknown—no or term:
Basic Solution (§707)
Extract the cube root of both sides:
This gives one value immediately, but Euler asks whether more roots exist.
Three Roots of a Pure Cubic (§708–713)
By analogy with quadratics (which always have two roots), Euler investigates how many roots a cubic has.
Example (§709–711). For , the real root means divides :
Setting the quadratic factor to zero:
These two additional roots are imaginary, but Euler verifies directly that .
General case (§712–713). Let , so . Then:
The quadratic factor gives:
Every pure cubic therefore has three roots:
| # | Root |
|---|---|
| 1 | |
| 2 | |
| 3 |
Only the first is real; the other two are imaginary-numbers. This pattern generalises: an th root has distinct values (§713 remark).
Worked Problems (§714–718)
Five word problems reduce to :
| § | Problem | Equation | Answer |
|---|---|---|---|
| 714 | Square times quarter-part = 432 | ||
| 715 | Fourth power ÷ half + 14¼ = 100 | ||
| 716 | Officers, horsemen, foot-soldiers; total pay 13000 fl | officers | |
| 717 | Merchants’ partnership; profit 2662 sequins | partners | |
| 718 | Cheeses exchanged for egg-laying hens; revenue 72 sous | cheeses |