Root of a number, in Mathematics, means a number that when multiplied by itself results in the number which is its root ; or in general, the word root means a quantity regarded as the base and foundation of a higher power. See Power, etc.

In general the root takes the name of the power of which it is the root; for instance, it is called a square root if the power is square; cube root if the power is cube, etc. Thus the square root of 4 is 2, because 2 multiplied by 2 gives 4. The product 4 is called the square of 2, and 2 is the square root or simply the root .

It is clear that one [1] is to the square root , as the square root is to the square. Therefore the square root is the mean proportional [2] of the square and unity. Thus 1:2::2:4.

If a square number such as 4 is multiplied by its root 2, the product 8 is called the cube or the third power of 2; and the number 2, with regards to the number 8, is its cube root .

Since one is to the root as the root is to the square, & that one is to the root as the square is to the cube, it follows that one, the root , the square, & the cube are in continuous proportion, that is to say 1 : 2 :: 2 : 4 :: 4 : 8. Consequently the cube root is the first of the two proportional means between unity and the cube.

Extracting the root of a number of a given power, such as 8, is the same thing as finding a number such as 2, which when multiplied by itself, even a certain number of times, for example, twice, produces the number 8. See Extraction.

Any root , be it a square, or cube, or of a higher power, is called the binomial root , or simply, binomial if it consists of two parts; such 20+4 or a+b. See Binomial.

If the root is composed of three parts, it is called trinomial, such as 200+40+5 or a+b+c. See Trinomial. If the root has more than three parts, it is called multinomial, like 2000+400+50+6, or a+b+c+d. See Multinomial.

M. l’abbé de Gua, in a mémoire published on p. 455 of the same volume [3], further provided a method on the number of imaginary, positive real, or negative real roots . As we are unable to go into any detail on the subject, it suffices to say along with the author that we find in this method some general approaches, but quite vaguely expressed in a letter from Collins to Dr. Wallis. Thereafter M. Stirling pushed these approaches a bit further in his enumeration of lines of the third order; but they are so lacking that this geometer’s method leaves nothing else to be desired. We believe the same can be said about M. l’abbé de Gua’s method, since this method, by his own admission, supposes solving equations that are not even absolutely determined for the third degree. At the end of the article on Equation, we spoke of the work of M. Fontaine on the same subject.

1. In contemporary mathematics it is also common to refer to the number “one” as “unity” in certain contexts, including the subject of roots.

2. More commonly referred to as “geometric mean.”

3. Mémoires de l'Académie Royale des Sciences de Paris, Année 1741 ,