Geometrical Ratio

Summary: A geometrical ratio $a:b$ is the quotient $a/b$, answering “how many times is $a$ greater than $b$?”; it is invariant under common scaling and is reduced to lowest terms by dividing both terms by their GCD.

Sources: chapter-1.3.6

Last updated: 2026-05-01

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Definition

Given two numbers $a$ (the antecedent) and $b$ (the consequent), their geometrical ratio is

$d=\frac{a}{b}.$

The notation $a:b$ is read ”$a$ is to $b$“. Euler notes that the colon already denotes division, which is precisely what computing the ratio requires.

This is the second of the two ways to compare quantities (§378): the first is the arithmetical ratio (difference $a-b$); the geometrical ratio (quotient $a/b$) is the more powerful of the two and dominates Sections III–IV.

Scaling invariance

$na:nb=a:b,\frac{a}{n}:\frac{b}{n}=a:b.$

Two ratios that reduce to the same fraction $p/q$ in lowest terms are considered equal, so $18:12=3:2$.

Types

Ratios are named by the value of $d$: equality ($d=1$), double ($d=2$), triple, quadruple; subduple ($d=1/2$), subtriple ($d=1/3$), etc. A ratio is rational when $d$ is expressible as a fraction of integers; otherwise it is irrational (surd), e.g. $\sqrt{5}:8$.

Connection to GCD

To reduce $a:b$ to lowest terms, divide both by $\gcd (a,b)$. This is why Chapter VI motivates Chapter VII (the Euclidean algorithm for GCD). See greatest-common-divisor for the algorithm and proof.

Related pages

• ch1.3.6-geometrical-ratio
• ch1.3.7-greatest-common-divisor
• greatest-common-divisor
• geometrical-proportion
• ratios
• fractions