Arithmetical Proportion

Summary: An arithmetical proportion is the equality of two differences; its central property is that the sum of the means equals the sum of the extremes, from which the fourth term can always be recovered.

Sources: chapter-1.3.2

Last updated: 2026-04-30

---

Definition

Four numbers $a,b,p,q$ are in arithmetical proportion when $a-b=p-q$, i.e. the difference between the first pair equals the difference between the second pair.

Standard notation: $a-b=p-q$.

Principal property (§394–395)

$b+p=a+q(\text{sum of means}=\text{sum of extremes}).$

This bidirectional: $a-b=p-q⟺b+p=a+q$.

Finding the fourth term (§396)

Given $a,b,p$:

$q=b+p-a.$

Allowed transpositions (§392–393)

From $a-b=p-q$ one may freely derive:

• $a-p=b-q$ (swap middle terms)
• $b-a=q-p$ (reverse both differences)

Continued proportion → progression

When $b=p$ (the middle two terms coincide), the proportion $a-b=b-q$ describes three numbers in arithmetical progression. See arithmetical-progressions and ch1.3.2-arithmetical-proportion.

Related pages

• ratios
• ch1.3.2-arithmetical-proportion
• arithmetical-progressions
• ch1.3.1-arithmetical-ratio