Geometrical Proportion
Summary: A geometrical proportion holds if and only if (product of extremes equals product of means); this single condition governs all transpositions, derived proportions, the Rule of Three, and the combination of multiple proportions.
Sources: chapter-1.3.8, chapter-1.3.9
Last updated: 2026-05-01
Definition
Four numbers form a geometrical proportion when
and are the extremes; and are the means.
This is the geometrical analogue of arithmetical-proportion (equality of two differences).
Product rule
The criterion (“product of extremes equals product of means”) is proved by multiplying through by . The converse follows by dividing by .
Transpositions
From , any of the following are valid:
Derived proportions
Rule of Three (fourth-term formula)
Given with unknown:
This is the basis of the Rule of Three: three given quantities determine a fourth in proportion.
Combining proportions
- If two proportions share the same first and third terms: and , then .
- If two proportions share the same mean terms: and , then .
- Two proportions can always be multiplied term-by-term: and yield .
Connection to compound relations
Compound relations (§488 ff.) extend this by taking the product of several ratios simultaneously — the foundation of the Rule of Five. See compound-relations.
Practical use
Currency exchange chains (Chapter IX) reduce to repeated application of the Rule of Three; the Rule of Reduction collapses the chain into a single fraction. See ch1.3.9-observations-proportion-utility.