Compound Relations

Summary: A compound relation is formed by multiplying two or more geometrical ratios (antecedents by antecedents, consequents by consequents); duplicate and triplicate ratios (squares and cubes) arise when equal ratios are compounded, and appear in the geometry of areas, volumes, and physical laws.

Sources: chapter-1.3.10

Last updated: 2026-05-01

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Definition

The compound relation of $a:b$, $c:d$, $e:f$, … is

$(a\cdot c\cdot e\cdots ):(b\cdot d\cdot f\cdots ).$

Common factors across all antecedents and consequents should be cancelled before multiplying. This is just multiplication of fractions: $\frac{a}{b}\cdot \frac{c}{d}\cdot \frac{e}{f}=\frac{ace}{bdf}$.

Telescoping chains

If each antecedent equals the previous consequent — $a:b$, $b:c$, $c:d$, $d:e$, $e:a$ — the compound is $1:1$. This is the mechanism behind the Rule of Reduction in chain-proportion problems.

Duplicate and triplicate ratios

Name	Construction	Result
Duplicate ratio	$a:b$ compounded with $a:b$	$a^2:b^2$
Triplicate ratio	$a:b$ compounded three times	$a^3:b^3$

Euler states: “squares are in the duplicate ratio of their roots” and “cubes are in the triplicate ratio of their roots.”

Geometric and physical laws

Quantity	Law	Ratio
Rectangular area	compound of lengths and breadths	$l_A{w}_A:l_B{w}_B$
Box/room volume	compound of lengths, breadths, heights	$l_A{w}_A{h}_A:l_B{w}_B{h}_B$
Circular area	duplicate ratio of diameters	$d_A^2:d_B^2$
Sphere volume	triplicate ratio of diameters	$d_A^3:d_B^3$
Falling body heights	duplicate ratio of times	$t_A^2:t_B^2$
Diamond prices	duplicate ratio of carat weights	$w_A^2:w_B^2$
Cannon ball weights	triplicate ratio of diameters	$d_A^3:d_B^3$

Ratio of fractions

$\frac{a}{b}:\frac{c}{d}=ad:bc.$

Fractions with the same numerator are in the inverse ratio of their denominators ($\frac{c}{a}:\frac{c}{b}=b:a$); fractions with the same denominator are in the direct ratio of their numerators.

Rule of Five (Double Rule of Three)

When an unknown depends on two variable quantities (e.g. horses and distance), the relevant ratio is compounded from both, and the Rule of Three is applied to the compound. Five given quantities determine the sixth. This is what Euler calls the Rule of Five or Double Rule of Three.

Related pages

• ch1.3.10-compound-relations
• geometrical-proportion
• ch1.3.8-geometrical-proportions
• ch1.3.9-observations-proportion-utility
• ratios
• powers-and-exponents