ch1.1.14-cubic-numbers

Chapter XIV – Of Cubic Numbers

Summary: Defines cubes as third powers, extends cubing to fractions and mixed numbers, and shows that unlike squares, cubes preserve the sign of the root. (source: chapter-1.1.14)

Sources: chapter-1.1.14

Last updated: 2026-04-24

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§152–153: Definition and Table of Cubes

A cube or cubic number is produced by multiplying a number twice by itself: $a^3=a\cdot a\cdot a.$ (source: chapter-1.1.14)

Euler lists the cubes of the natural numbers from $1$ to $10$, such as $2^3=8$, $3^3=27$, $4^3=64$, and $10^3=1000$. (source: chapter-1.1.14)

He observes that the first differences of consecutive cubes are irregular, but their second differences increase by $6$. (source: chapter-1.1.14)

§154–156: Cubes of Fractions, Mixed Numbers, and Products

Fractions are cubed by cubing numerator and denominator separately: ${\left(\frac{a}{b}\right)}^3=\frac{a^3}{b^3}.$ (source: chapter-1.1.14)

Mixed numbers must first be reduced to improper fractions. Euler works examples such as $1\frac{1}{2}$ and $1\frac{1}{4}$. (source: chapter-1.1.14)

If a number is factored, its cube is the product of the cubes of the factors: ${\left(ab\right)}^3=a^3{b}^3.$ (source: chapter-1.1.14)

He uses this to compute $12^3$ from $12=3\cdot 4$, so $12^3=3^3\cdot 4^3=27\cdot 64=1728$. (source: chapter-1.1.14)

§157: Signs of Cubes

Positive numbers have positive cubes, but negative numbers have negative cubes: ${\left(+a\right)}^3=+a^3,{\left(-a\right)}^3=-a^3.$ (source: chapter-1.1.14)

This is a key contrast with squares, which are always positive in Euler’s treatment. (source: chapter-1.1.14)

Related pages

• cube-roots
• powers-and-exponents
• roots
• positive-negative-numbers