ch1.1.20-methods-of-calculation

Chapter XX – Of the different Methods of Calculation, and of their mutual Connection

Summary: Reviews the chain of algebraic operations from addition to roots, introduces the equality sign $=$, and shows that logarithms arise when one asks for the unknown exponent in $a^b=c$. (source: chapter-1.1.20)

Sources: chapter-1.1.20

Last updated: 2026-04-24

---

§206: Equality Sign

Euler explicitly introduces the sign $=$ as shorthand for “is equal to.” (source: chapter-1.1.20)

§207–212: From Addition to Fractions

Starting from $a+b=c$, reversing the question gives subtraction: $b=c-a.$ (source: chapter-1.1.20)

When the number to be subtracted is larger than the starting quantity, negative numbers arise. Euler gives $5-9=-4$. (source: chapter-1.1.20)

Starting from multiplication $ab=c$, reversing the question gives division: $b=\frac{c}{a}.$ (source: chapter-1.1.20)

When the division is not exact, fractions arise, as in $4b=3$, so $b=\frac{3}{4}$. (source: chapter-1.1.20)

§213–218: From Powers to Irrational and Imaginary Numbers

Euler next considers powers through $a^b=c$. If the root $a$ and exponent $b$ are given, one computes the power $c$. (source: chapter-1.1.20)

If the power $c$ and exponent $b$ are given, one extracts a root: $a=\sqrt[b]{c}.$ (source: chapter-1.1.20)

This leads to irrational numbers when the root is not exact, and to imaginary numbers in the cases discussed earlier. (source: chapter-1.1.20)

§219: Unknown Exponent and the Birth of Logarithms

The remaining reversed question is to determine the exponent $b$ from the root $a$ and the power $c$. Euler identifies this as the foundation of logarithms. (source: chapter-1.1.20)

Related pages

• logarithms
• powers-and-exponents
• roots
• imaginary-numbers