Ch1.3.8 — Of Geometrical Proportions
Summary: Defines a geometrical proportion as the equality of two geometrical ratios, proves that the product of extremes equals the product of means, derives the Rule of Three, and establishes how proportions may be transposed, combined, and scaled.
Sources: chapter-1.3.8
Last updated: 2026-05-01
Definition (§461–462)
A geometrical proportion is an equality of two geometrical ratios:
Read: ” is to as is to “. Here are the extremes and the means.
Example: , since both ratios equal .
Product of extremes equals product of means (§463–465)
Theorem: .
Proof (→): From , multiply both sides by : . Multiply by : .
Proof (←): From , divide by : .
This is the central computational tool for all work with proportions.
Transpositions (§466)
From any of the following proportions are equally valid:
| invert both ratios | |
| swap inner pair | |
| swap outer pair | |
| fully reverse |
Derived proportions (§467–468)
Starting from , i.e. :
These generalise to: for any integers ,
Proof: the product of the extremes and of the means both simplify to after substituting .
Finding the fourth term — Rule of Three (§470–471)
Given with unknown:
This is the basis of the Rule of Three: given three quantities in proportion, compute the fourth by multiplying the second by the third and dividing by the first.
Example: gives .
Properties when proportions share terms (§472–473)
- If and (same first and third terms), then .
- If and (same mean terms), then .
Multiplying two proportions (§474–476)
If and , then multiplying term-by-term:
Proof: and imply , which is exactly .
Conversely, any product equality yields valid proportions such as or .