ch1.3.6-geometrical-ratio

Ch1.3.6 — Of Geometrical Ratio

Summary: Defines the geometrical ratio of two numbers as their quotient; introduces antecedent/consequent notation, the colon symbol, a taxonomy of ratio types, and the scaling invariance of ratios; motivates reduction to lowest terms via the GCD.

Sources: chapter-1.3.6

Last updated: 2026-05-01

---

Definition (§440–442)

The geometrical ratio of two numbers answers the question: how many times is one greater than the other? It is found by dividing the first number (the antecedent) by the second (the consequent), and the quotient is the ratio.

The relation is written $a:b$, read ”$a$ is to $b$“. Because finding the ratio requires the division $a÷b$, Euler reuses the division colon for this purpose.

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

Example: for $18:12$, the ratio is $\frac{18}{12}=\frac{3}{2}=1\frac{1}{2}$, meaning $18$ contains $12$ once and a half.

Three-parameter system (§445)

Given any two of the three quantities $\{a,b,d\}$, the third is determined:

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

Types of ratio (§444)

Name	Condition	Example
Equality	$d=1$	$3:3$, $a:a$
Double	$d=2$	$4:2$
Triple	$d=3$	$12:4$
Quadruple	$d=4$	$24:6$
Subduple	$d=1/2$	$6:12$
Subtriple	$d=1/3$	$5:15$
Rational	$d$ expressible exactly	$11:7$, $8:15$
Irrational (surd)	$d$ not exactly expressible	$\sqrt{5}:8$, $4:\sqrt{3}$

Scaling invariance (§446)

Multiplying or dividing both terms by the same number $n$ leaves the ratio unchanged:

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

Reduction to lowest terms (§447–449)

When $a/b$ reduces to $p/q$ (dividing both by their GCD), we write $a:b=p:q$, or $a:b::p:q$, read ”$a$ is to $b$ as $p$ is to $q$“. This is the clearest way to express the relation.

Example: $57600:25200÷3600=16:7$.

Finding the GCD of two large numbers requires a systematic rule — the subject of Chapter VII.

Related pages

• geometrical-ratio
• ch1.3.7-greatest-common-divisor
• ch1.3.8-geometrical-proportions
• ratios
• fractions
• ch1.3.1-arithmetical-ratio