Elements of Algebra

216 ELEMENTS SECT. IV. so that $s = 1537$; which is the whole number of soldiers. By this means we find, $x = 1802 - 1537 = 265$; $2y = 2703 - 1537 = 1166$, or $y = 583$; $3z = 3604 - 1537 = 2067$, or $z = 689$. The company of Swiss therefore has 265 men; that of Swabians 583; and that of Saxons 689. *** CHAP. V. *Of the Resolution of Pure Quadratic Equations.* 623. An equation is said to be of the second degree, when it contains the square, or the second power, of the unknown quantity, without any of its higher powers; and an equation, containing likewise the third power of the unknown quantity, belongs to cubic equations, and its resolution requires particular rules. 624. There are, therefore, only three kinds of terms in an equation of the second degree: 1. The terms in which the unknown quantity is not found at all, or which is composed only of known numbers. 2. The terms in which we find only the first power of the unknown quantity. 3. The terms which contain the square, or the second power, of the unknown quantity. So that $x$ representing an unknown quantity, and the letters $a$, $b$, $c$, $d$, &c. the known quantities, the terms of the first kind will have the form $a$, the terms of the second kind will have the form $bx$, and the terms of the third kind will have the form $cx^2$. 625. We have already seen, how two or more terms of the same kind may be united together, and considered as a single term. For example, we may consider the formula $ax^2 - bx^2 + cx^2$ as a single term, representing it thus, $(a - b + c)x^2$; since, in fact, $(a - b + c)$ is a known quantity. And also, when such terms are found on both sides of the sign $=$, we have seen how they may be brought to one side,