ch1.3.10-compound-relations

Ch1.3.10 — Of Compound Relations

Summary: Defines compound relations as the product of two or more geometrical ratios; introduces duplicate and triplicate ratios (ratios of squares and cubes); and applies these ideas to areas, volumes, falling bodies, diamond prices, and the Rule of Five.

Sources: chapter-1.3.10

Last updated: 2026-05-01

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Definition (§488–489)

The compound relation of $a:b$, $c:d$, $e:f$ is obtained by multiplying antecedents together and consequents together:

$\text{compound}=(ace):(bdf).$

The ratio $ace/bdf$ can be simplified by cancelling any common factors between the combined numerators and denominators before multiplying. This often reduces dramatically.

Example: relations $12:25$, $28:33$, $55:56$ compound to $(12\times 28\times 55):(25\times 33\times 56)=2:5$.

Telescoping chains (§490)

If each antecedent equals the consequent of the preceding relation:

$a:b,b:c,c:d,d:e,e:a$

the compound is $abcde:abcde=1:1$.

Areas as compound ratios (§491–492)

The ratio of two rectangular areas equals the compound of their length ratio and breadth ratio:

$\frac{{\text{Area}}_A}{{\text{Area}}_B}=\frac{l_A}{l_B}\cdot \frac{w_A}{w_B}.$

Example 1: field $A$ is $500\times 60$, field $B$ is $360\times 100$. Compound: $(500\times 60):(360\times 100)=30000:36000=5:6$.

Example 2: $720\times 88$ vs $660\times 90$ → $16:15$.

Volumes as compound ratios (§493)

Three-dimensional contents involve three relations (length, breadth, height):

$\frac{{\text{Vol}}_A}{{\text{Vol}}_B}=\frac{l_A}{l_B}\cdot \frac{w_A}{w_B}\cdot \frac{h_A}{h_B}.$

Example: room $A$ is $36\times 16\times 14$, room $B$ is $42\times 24\times 10$. Compound: $4:5$.

Duplicate and triplicate ratios (§494)

When two equal ratios are compounded, the result is the duplicate ratio (ratio of squares):

$a:b\text{ compounded with }a:b\Rightarrow a^2:b^2.$

Three equal ratios give the triplicate ratio (ratio of cubes): $a^3:b^3$.

Euler’s phrasing: “squares are in the duplicate ratio of their sides”; “cubes are in the triplicate ratio of their sides.”

Geometric applications (§495–497)

• Circular areas are in the duplicate ratio of their diameters (Euclid): $A_1:A_2=d_1^2:d_2^2$.
  Example: diameters 45 and 30 → areas in ratio $9:4$.

• Sphere volumes are in the triplicate ratio of their diameters: $V_1:V_2=d_1^3:d_2^3$.
  Diameter 1 ft vs 2 ft → volumes $1:8$.

• Cannon ball weights (same material): if ball $A$ has diameter 2 in and weighs 5 lb, then ball $B$ with diameter 8 in weighs $2^3:8^3::5:320$ lb; ball $C$ with diameter 15 in weighs $\approx 2109\frac{3}{8}$ lb.

Ratios of fractions (§498–499)

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

Multiply both fractions by $bd$ to convert to integer ratio. Special cases:

• $\frac{1}{a}:\frac{1}{b}::b:a$ (unit-numerator fractions are in the inverse ratio of their denominators).
• $\frac{a}{c}:\frac{b}{c}::a:b$ (equal-denominator fractions are in the direct ratio of their numerators).

Falling bodies (§500–501)

Euler cites the empirical law: a freely falling body covers 16 English feet in 1 second; heights are in the duplicate ratio of elapsed times.

$h_1:h_2::t_1^2:t_2^2.$

• Stone falling from 2304 ft: $16:2304::1:144$, so time $=12$ s.
• Falling for 3600 s (1 hour): height $=207360000$ ft $\approx 39272$ miles (nearly $5\times$ Earth’s diameter).

Diamond prices (§502)

Diamond prices follow the duplicate ratio of weight (in carats): $P_1:P_2::w_1^2:w_2^2$. If a 1-carat diamond is worth 10 livres:

• 100-carat diamond: $1^2:100^2::10:100000$ livres.
• 1680-carat Portuguese diamond: $1^2:1680^2::10:28224000$ livres.

Rule of Five / Double Rule of Three (§503–504)

When a price depends on two independent quantities (e.g. number of horses and number of stages), the price is governed by a compound ratio. Compound the two ratios, then apply the Rule of Three.

Example: 1 horse costs 20 sous per post; cost for 28 horses for $4\frac{1}{2}$ posts:

$1:28\text{ (horses)}\times 2:9\text{ (stages)}=2:252=1:126;\text{cost}=126\text{ fr.}$

This generalisation of the Rule of Three to five given quantities is called the Rule of Five (or Double Rule of Three).

Related pages

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