ch1.3.9-observations-proportion-utility

Ch1.3.9 — Observations on the Rules of Proportion and their Utility

Summary: Applies the Rule of Three to practical problems of currency conversion, introduces the Rule of Reduction (Double Rule of Three) for multi-step chain proportions, and illustrates the method with extended examples involving several European currencies.

Sources: chapter-1.3.9

Last updated: 2026-05-01

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Proportion in commerce (§477)

Euler opens with the observation that proportion is indispensable in everyday life: prices and commodities are always related proportionally, and currency exchange amounts to finding the ratio between different monetary units.

Currency exchange: single step (§478)

An old louis d’or at Berlin is worth $5\frac{1}{3}$ rixdollars; a ducat is worth $3$ rixdollars. Reducing to a common denominator (or converting both to drachms: $1L=128$ drachms, $1D=72$ drachms) gives:

$1L:1D::16:9,\text{i.e. }9\text{ louis}=16\text{ ducats.}$

From this single proportion, any conversion is immediate:

• 1000 louis → $\frac{16\times 1000}{9}=1777\frac{7}{9}$ ducats.
• 1000 ducats → $\frac{9\times 1000}{16}=562\frac{1}{2}$ louis.

Ruble–ducat exchange rates (§479)

At St. Petersburg the exchange rate (stivers per ruble) is variable; 105 stivers = 1 ducat.

Exchange rate	Proportion	Result
45 stivers/ruble	$45:105::3:7$	1 ducat $=2\frac{1}{3}$ rubles
50 stivers/ruble	$50:105::10:21$	1 ducat $=2\frac{1}{10}$ rubles
44 stivers/ruble	$44:105::1:2\frac{17}{44}$	1 ducat $=2$ rubles $38\frac{7}{11}$ copecks

Multi-step chain (§480–482)

Problem: convert 1000 rubles (St. Petersburg) into ducats payable at Berlin, given:

1. 1 ruble = $47\frac{1}{2}$ stivers (i.e. 2 rubles = 95 stivers)
2. 20 stivers = 1 Dutch florin
3. $2\frac{1}{2}$ Dutch florins = 1 Dutch dollar (i.e. 5 florins = 2 Dutch dollars)
4. 100 Dutch dollars → 142 Berlin dollars (exchange rate)
5. 3 Berlin dollars = 1 ducat

Applying five separate Rule-of-Three steps in sequence gives $449\frac{2}{3}$ ducats. With a 5% banker’s discount (pay 100 instead of 105), the result is $428\frac{16}{63}$ ducats.

Rule of Reduction / Double Rule of Three (§483–485)

Instead of six separate operations, list the antecedents and consequents of each step:

Step	Antecedent	Consequent
1	2 rubles	95 stivers
2	20 stivers	1 Dutch florin
3	5 Dutch florins	2 Dutch dollars
4	100 Dutch dollars	142 Berlin dollars
5	3 Berlin dollars	1 ducat
6	105 ducats	100 ducats (discount)

The answer is:

$\frac{1000\times 95\times 1\times 2\times 142\times 1\times 100}{2\times 20\times 5\times 100\times 3\times 105}=\frac{10\times 19\times 142}{3\times 21}=\frac{26980}{63}=428\frac{16}{63}.$

Method: each relation in the chain starts with the same unit the previous relation ended with, so the units cancel across the chain. After writing all antecedents and consequents, cancel common factors before multiplying (§484).

Further examples (§486–487)

• Ducats to Polish florins via Hamburg: 1000 ducats → $8012\frac{2}{3}$ Polish florins.
• Ducats to Leipzig dollars via Amsterdam: 1000 ducats → $2639\frac{13}{21}$ Leipzig dollars.

Related pages

• geometrical-proportion
• ch1.3.8-geometrical-proportions
• ch1.3.10-compound-relations
• ratios
• fractions