[Math] Why the name “square root”

Question

Why do we say that $\sqrt{a}$ is a square root of $a$?

Is this because $\sqrt{a}$ is a root of the function $f(x)=x^2-a$?

Cubic root similarly?

Thanks in advance

Answer 1

The "root" of "square root" is from latin radix.

From Florian Cajori, A history of mathematical notations (1928), page 361 of I vol of Dover reprint :

The principal symbolisms for the designation of roots, which have been developed since the influx of Arabic learning into Europe in the twelfth century, fall under four groups having for their basic symbols, respectively, $R$ (radix), $l$ (latus), the sign $\surd$, and the fractional exponent.

The sign $R$; first appearance.-In a translationa from the Arabic into Latin of a commentary of the tenth book of the Elements of Euclid, the word radix is used for "square root." The sign $R$ came to be used very extensively for "root," but occasionally it stood also for the first power of the unknown quantity, $x$. The word radix was used for $x$ in translations from Arabic into Latin by John of Seville and Gerard of Cremona. [...]

With the close of the seventeenth century [the symbol $R$] practically passed away as a radical sign; the symbol $\surd$ gained general ascendancy.

See page 366 :

Origin of $\surd$.-This symbol originated in Germany. L.Euler guessed that it was a deformed letter $r$, the first letter in radix [see page 213 of II vol : L.Euler says in his Institutiones calculi differentialis (Petrograd, 1755), p.100 : "in place of the letter $r$ which first stood for $radix$, there has now passed into common usage this distorted form of it $\surd$."]

This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has convinced Germans that the old explanation is hardly tenable. [...] The oldest of these is in the Dresden Library, in a volume of manuscripts which contains different algebraic treatises in Latin and one in German. [...] They [the main facts found in the four manuscripts] show conclusively that the dot was associated as a symbol with root extraction.

Christoff Rudolff was familiar with the Vienna manuscript which uses the dot with a tail. In his Coss of 1525 he speaks of the Punkt in connection with root symbolism, but uses a mark with a very short heavy downward stroke (almost a point), followed by a straight line or stroke, slanting upward. As late as 1551, Scheubel, in his printed Algebra, speaks of points.

See page 144 :

In 1553 Stifel brought out a revised edition of Rudolff's Coss. Interesting is Stifel's comparison of Rudolff's notation of radicals with his own, and his declaration of superiority of his own symbols. We read: "How much more convenient my own signs are than those of Rudolff, no doubt everyone who deals with these algorithms will notice for himself. But I too shall often use the sign $\surd$ [...]."

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Added

From John Fauvel & Jeremy Gray (editors), The History of Mathematics: A Reader (1987).

Page 229 :

Al-Khwarizmi (ca.780 – ca.850) : "A square is the whole amount of the root multiplied by itself."

Page 250 :

Luca Pacioli (1445–1517) regarding quadratic equations : "Now we must see in how many ways they can be made equal, one to the other, and the other to the one, and two of them to one of them, and one to two of them. On this I say that they can be made equal to each other in six ways. First, the square to the things. Second, the square to the numbers. Third, thing or things to numbers. [...]"

Page 260 :

Gerolamo Cardano (1501 – 1576), Ars Magna, on the cubic equations : "Cube the third part of the number of 'things', to which you add the square of half the number of the equation, and take the root of the whole, that is, the square root, which you will use, in the one case adding the half of the number which you just multiplied by itself, in the other case [...]."

Finally :

René Descartes (1596 – 1650), Geometry, page 299 of 1637 edition (see Dover reprint) : "if I wish to extract the square root [racine quarrée] of $a^2+b^2$, I write $\sqrt{a^2+b^2}$; if I wish to extract the cube root [racine cubique] of $a^3 - b^3 +abb$, I write $\sqrt{C.a^3 - b^3 +abb}$ [...]"