Chapter 1.2.6 – Of the Squares of Compound Quantities
Summary: Euler derives the binomial square identity and extends it to trinomials and expressions with negative terms, with numerical applications.
Sources: chapter-1.2.6
Last updated: 2026-04-28
The Binomial Square
Squaring by direct multiplication gives (source: chapter-1.2.6)
The result contains the square of each term plus twice the product of the two terms.
Euler illustrates with numbers: , and (§308). (source: chapter-1.2.6)
Successive Squares
From the binomial identity it follows immediately that
So the square of is obtained by adding — the sum of the two consecutive integers and — to (§309). This provides a fast way to step from one perfect square to the next: (source: chapter-1.2.6)
More instances of the pattern (§310): (source: chapter-1.2.6)
Square of a Difference
When the root is the middle term becomes negative (§311): (source: chapter-1.2.6)
Equivalently, : subtract the sum of the two consecutive integers and from (§312). (source: chapter-1.2.6)
Trinomial Square
For a root with three terms (§314): (source: chapter-1.2.6)
The square contains the square of every term and twice the product of every pair of terms.
Euler verifies with : the partial squares and the cross-products total (§315). (source: chapter-1.2.6)
Negative Terms
When some terms are negative, the cross-products involving an odd number of negative factors become negative (§316): (source: chapter-1.2.6)
Verified with : positive parts ; negative parts ; result . (source: chapter-1.2.6)