Elements of Algebra

CHAP. XVI. OF ALGEBRA. 289 CHAP. XVI. Of the Resolution of Equations by Approximation. 784. When the roots of an equation are not rational, and can only be expressed by radical quantities, or when we have not even that resource, as is the case with equations which exceed the fourth degree, we must be satisfied with determining their values by approximation; that is to say, by methods which are continually bringing us nearer to the true value, till at last the error being very small, it may be neglected. Different methods of this kind have been proposed, the chief of which we shall explain. 785. The first method which we shall mention, supposes that we have already determined, with tolerable exactness, the value of one root *; that we know, for example, that such a value exceeds 4, and that it is less than 5. In this case, if we suppose this value $= 4 + p$, we are certain that $p$ expresses a fraction. Now, as $p$ is a fraction, and consequently less than unity, the square of $p$, its cube, and, in general, all the higher powers of $p$, will be much less with respect to unity; and, for this reason, since we require only an approximation, they may be neglected in the calculation. When we have, therefore, nearly determined the fraction $p$, we shall know more exactly the root $4 + p$; from that we proceed to determine a new value still more exact, and continue the same process till we come as near the truth as we desire. 786. We shall illustrate this method first by an easy example, requiring by approximation the root of the equation $x^2 = 20$. Here we perceive, that $x$ is greater than 4, and less than 5; making, therefore, $x = 4 + p$, we shall have $x^2 = 16 + 8p + p^2 = 20$; but as $p^2$ must be very small, we shall neglect it, in order that we may have only the equation 16 + * This is the method given by Sir Is. Newton at the beginning of his Method of Fluxions. When investigated, it is found subject to different imperfections; for which reason we may with advantage substitute the method given by M. de la Grange, in the Memoirs of Berlin for 1767 and 1768. F. T. This method has since been published by De la Grange, in a separate Treatise, where the subject is discussed in the usual masterly style of this author. U