Chapter 1.3.1 — Of Arithmetical Ratio, or of the Difference between Two Numbers
Summary: Euler introduces the two ways to compare unequal quantities — by difference (arithmetical ratio) and by quotient (geometrical ratio) — and derives the three-way relationship among a greater number, a lesser number, and their difference.
Sources: chapter-1.3.1
Last updated: 2026-04-30
Two kinds of ratio (§378)
When two quantities are unequal we can ask either how much one exceeds the other, or how many times one contains the other. Both answers are called ratios or relations.
- The answer to “how much” is the arithmetical ratio — i.e. the difference.
- The answer to “how many times” is the geometrical ratio — i.e. the quotient.
Euler notes that the names “arithmetical” and “geometrical” are entirely arbitrary conventions. In what follows he reserves the words ratio and relation for the geometrical case, and uses difference for the arithmetical case.
Comparing only like quantities (§379)
Comparison only makes sense between quantities of the same kind. Asking whether two pounds equals three ells is absurd. Because all such quantities can be expressed as numbers, the discussion reduces to comparing numbers.
The difference as arithmetical ratio (§380–382)
The arithmetical ratio of and is simply (assuming ). If the numbers are equal their difference is ; otherwise the difference is found by subtraction.
Examples: exceeds by ; exceeds by .
Three connected quantities (§383–386)
Given:
- = the greater number
- = the lesser number
- = the difference
any two of these determine the third:
These three equalities are all equivalent: if any one holds, so do the other two. In general, implies and .
Invariance under translation and scaling (§387–389)
Translation invariance: Adding or subtracting the same constant to both numbers leaves the difference unchanged:
Scaling: Multiplying both numbers by scales the difference by :