Elements of Algebra

36 ELEMENTS SECT. I. CHAP. XI. Of Square Numbers. 115. The product of a number, when multiplied by itself, is called a square; and, for this reason, the number, considered in relation to such a product, is called a square root. For example, when we multiply $12$ by $12$, the product $144$ is a square, of which the root is $12$. The origin of this term is borrowed from geometry, which teaches us that the contents of a square are found by multiplying its side by itself. 116. Square numbers are found therefore by multiplication; that is to say, by multiplying the root by itself: thus, $1$ is the square of $1$, since $1$ multiplied by $1$ makes $1$; likewise, $4$ is the square of $2$; and $9$ the square of $3$; $2$ also is the root of $4$, and $3$ is the root of $9$. We shall begin by considering the squares of natural numbers; and for this purpose shall give the following small Table, on the first line of which several numbers, or roots, are ranged, and on the second their squares *. Numbers. $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ --|---|---|---|---|---|---|---|---|---|---|---|--- Squares. $1$ | $4$ | $9$ | $16$ | $25$ | $36$ | $49$ | $64$ | $81$ | $100$ | $121$ | $144$ | $169$ 117. Here it will be readily perceived that the series of square numbers thus arranged has a singular property; namely, that if each of them be subtracted from that which immediately follows, the remainders always increase by $2$, and form this series; $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, &c. which is that of the odd numbers. 118. The squares of fractions are found in the same manner, by multiplying any given fraction by itself. For example, the square of $\frac{1}{2}$ is $\frac{1}{4}$. * We have very complete tables for the squares of natural numbers, published under the title "Tetragonometria Tabularia, &c. Auct. J. Jobo Ludolfo, Amstelodami, $1690$, in $4$to." These Tables are continued from $1$ to $100000$, not only for finding those squares, but also the products of any two numbers less than $100000$; not to mention several other uses, which are explained in the introduction to the work. F. T.