Chapter III – Of the Multiplication of Compound Quantities
Summary: Extends multiplication to compound expressions by distributing each term of one factor across every term of the other, while preserving the usual sign rules, and derives several important identities from that rule. (source: chapter-1.2.3)
Sources: chapter-1.2.3
Last updated: 2026-04-26
§270–274: Distributive Multiplication
Euler writes the product of two compound quantities as
(source: chapter-1.2.3)
He first notes that multiplying a compound quantity by a simple one means multiplying each term separately:
(source: chapter-1.2.3)
From there he obtains the general rule: each term of one expression must be multiplied by each term of the other, using the ordinary sign rules for products. (source: chapter-1.2.3)
For example,
(source: chapter-1.2.3)
§275–276: Difference of Two Squares
Euler computes
(source: chapter-1.2.3)
He treats this as a theorem: the sum of two numbers multiplied by their difference equals the difference of their squares. (source: chapter-1.2.3)
He also concludes that any difference of two squares is composite, because it is divisible by both the sum and the difference of the roots. (source: chapter-1.2.3)
§277–281: Further Identities and Rearrangement of Factors
The chapter develops more elaborate products, including
and
(source: chapter-1.2.3)
Euler then shows that the same final product may be reached by grouping several factors in different orders, illustrating the flexibility of multiplication of compound quantities. (source: chapter-1.2.3)
One major example ends with
(source: chapter-1.2.3)