Chapter 1.3.2 — Of Arithmetical Proportion
Summary: Euler defines an arithmetical proportion as the equality of two arithmetical ratios (differences), identifies its key property — sum of means equals sum of extremes — and introduces continued arithmetical proportions (arithmetical progressions).
Sources: chapter-1.3.2
Last updated: 2026-04-30
Definition (§390)
When two arithmetical ratios are equal — that is, when — the four numbers are said to form an arithmetical proportion, written:
Example: (both differences equal ).
Permissible transpositions (§392–393)
Given :
- The second and third terms may be swapped: .
- The proportion may be reversed: .
Example: since , we also have and .
The principal property: sum of means = sum of extremes (§394–395)
The most important property of an arithmetical proportion is:
Proof: from , add to both sides to get . Conversely, if then subtracting recovers .
Example: because .
Finding the fourth term (§396)
Given three terms of an arithmetical proportion, the fourth term is:
Example: first three terms give , yielding .
Continued arithmetical proportion and progressions (§397–401)
When the second and third terms coincide (say ), we have three numbers whose successive differences are equal. These form a continued arithmetical proportion, also called an arithmetical progression, written .
Such a progression may be increasing (e.g. ) or decreasing (e.g. ).
From it follows that , so . Using this rule, if the first term is and the second is , the successive terms are:
The general -th term is , i.e. where is the common difference. (This is further developed in ch1.3.3-arithmetical-progressions.)