ch1.2.7-extraction-of-roots-compound

Chapter 1.2.7 – Of the Extraction of Roots applied to Compound Quantities

Summary: Euler develops a systematic algorithm for extracting square roots of compound algebraic expressions and numbers, analogous to long division, and introduces radical notation for non-perfect squares.

Sources: chapter-1.2.7

Last updated: 2026-04-28

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The Algorithm

The method exploits the factored form of the remainder when squaring $a+b$ (§317–320): (source: chapter-1.2.7)

$$a^2+2ab+b^2-a^2=2ab+b^2=(2a+b)b.$$

Procedure:

1. Arrange the square in decreasing powers of one variable.
2. The first term of the root is the square root of the leading term.
3. Subtract its square, leaving a remainder.
4. Divide the remainder by twice the first term found (leaving a blank for the next term); the quotient is the next term $b$ of the root.
5. Complete the divisor to $2a+b$, confirm the remainder is zero.
6. If a remainder persists, treat the two terms found so far as a single new “first part” and repeat.

Euler shows the algorithm in tableau form (§321): (source: chapter-1.2.7)

$$\sqrt{a^2+2ab+b^2}=a+b.$$

Two-Term Examples

Worked examples from §322: (source: chapter-1.2.7)

Square	Root
$a^2+6ab+9b^2$	$a+3b$
$4a^2-4ab+b^2$	$2a-b$
$9p^2+24pq+16q^2$	$3p+4q$
$25x^2-60x+36$	$5x-6$

Three- and Four-Term Examples

When a remainder is left after the first step, the algorithm continues (§323): (source: chapter-1.2.7)

$$\sqrt{a^2+2ab-2ac-2bc+b^2+c^2}=a+b-c,$$ $$\sqrt{a^4+2a^3+3a^2+2a+1}=a^2+a+1.$$

A notable six-term example: (source: chapter-1.2.7)

$$\sqrt{a^6-6a^{5}b+15a^4{b}^2-20a^3{b}^3+15a^2{b}^4-6a{b}^5+b^6}=a^3-3a^{2}b+3a{b}^2-b^3.$$

Numerical Square Root Extraction

The same algorithm applies to integers, reproducing the standard arithmetic procedure (§324): (source: chapter-1.2.7)

$$\sqrt{529}=23,\sqrt{2304}=48,\sqrt{15625}=125,\sqrt{998001}=999.$$

Non-Perfect Squares and Radical Notation

If a remainder persists after all digits are used, the number has no exact square root (§325). Euler uses the radical sign: (source: chapter-1.2.7)

$$\sqrt{a^2+b^2},\sqrt{1-x^2},$$

or equivalently the fractional exponent $\frac{1}{2}$:

$$(a^2+b^2{)}^{1/2}.$$

Related pages

• ch1.2.6-squares-of-compound-quantities
• ch1.2.8-calculation-irrational-quantities
• square-roots-and-irrational-numbers
• irrational-numbers
• algebraic-identities
• roots