Ratios

Summary: Euler distinguishes two ways of comparing unequal quantities: the arithmetical ratio (difference) and the geometrical ratio (quotient), and notes that the names are conventional rather than descriptive.

Sources: chapter-1.3.1, chapter-1.3.6

Last updated: 2026-05-01

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Two notions of ratio (§378)

Given two unequal quantities we can ask:

1. How much is one greater than the other? → Arithmetical ratio = the difference.
2. How many times is one greater than the other? → Geometrical ratio = the quotient.

Euler explicitly states that calling the first “arithmetical” and the second “geometrical” is an arbitrary convention with no deeper meaning. In Elements of Algebra he reserves the words ratio and relation for the geometrical case and uses difference for the arithmetical case.

Restriction to like quantities (§379)

Comparing quantities from different domains (e.g. pounds and ells) is meaningless. All quantities under discussion must be expressible as numbers of the same kind.

Arithmetical ratio in practice

The arithmetical ratio of $a$ and $b$ (with $a\ge b$) is $d=a-b$. The three quantities $a$, $b$, $d$ satisfy:

$d=a-b,a=b+d,b=a-d.$

Key properties:

• Translation invariant: $(a+c)-(b+c)=d$.
• Scales with multiplication: $na-nb=nd$.

See ch1.3.1-arithmetical-ratio for the full derivation.

Geometrical ratio

The geometrical ratio $a:b=a/b$ answers how many times one quantity contains the other. The three quantities $a$ (antecedent), $b$ (consequent), and $d$ (ratio) satisfy:

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

The ratio is invariant under common scaling ($na:nb=a:b$) and is reduced to lowest terms by dividing both terms by their $\gcd (a,b)$.

Types include equality ($d=1$), double, triple, subduple, subtriple, rational, and irrational (surd) ratios. See geometrical-ratio and ch1.3.6-geometrical-ratio for full details.

Related pages

• ch1.3.1-arithmetical-ratio
• ch1.3.2-arithmetical-proportion
• ch1.3.6-geometrical-ratio
• arithmetical-proportion
• geometrical-ratio
• geometrical-proportion
• greatest-common-divisor
• fractions