Chapter XV – Of Cube Roots, and of Irrational Numbers resulting from them
Summary: Defines cube roots, distinguishes exact cube roots from irrational cube roots, and shows that cube roots of negative numbers remain real and negative rather than imaginary. (source: chapter-1.1.15)
Sources: chapter-1.1.15
Last updated: 2026-04-24
§158–160: Definition and Exact Cases
The cube root of a number is the number whose cube equals the given number. (source: chapter-1.1.15)
Exact cube roots are easy to identify for perfect cubes, fractions, mixed numbers, and negative cubes: (source: chapter-1.1.15)
§160–163: Irrational Cube Roots
When the proposed number is not a perfect cube, its cube root cannot be expressed by integers or fractions. Euler’s example is , whose cube root lies between and . (source: chapter-1.1.15)
He therefore introduces the sign to represent such quantities, and treats them as a distinct class of irrational numbers not reducible to square roots. (source: chapter-1.1.15)
§164–166: Arithmetic with Cube Roots
Perfect cubes simplify in the expected way: (source: chapter-1.1.15)
Cube roots multiply and divide by combining the radicands: (source: chapter-1.1.15)
Euler also simplifies factors outside the radical when a cube factor is present, for example: (source: chapter-1.1.15)
§167: Negative Cube Roots
Negative numbers do not create imaginary cube roots because odd powers preserve sign: (source: chapter-1.1.15)
Euler explicitly contrasts this with square roots, where negative radicands lead to imaginary quantities. (source: chapter-1.1.15)