Elements of Algebra

292 ELEMENTS SECT. IV. Now, the assumed value of $x=3$ not being far from the true one, we shall suppose $n=3$, which gives us $x=\frac{62}{21}$; and if we were to substitute this new value instead of $n$, we should find another still more exact. 791. We shall give only the following example, for equations of higher dimensions than the third. Let $x^5 = 6x + 10$, or $x^5 - 6x - 10 = 0$, where we readily perceive that $1$ is too small, and that $2$ is too great. Now, if $x=n$ be a value not far from the true one, and we make $x = n + p$, we shall have $x^5 = n^5 + 5n^4p$; and, consequently, $n^5 + 5n^4p = 6n + 6p + 10$; or $p(5n^4 - 6) = 6n + 10 - n^5$. Wherefore $p = \frac{6n+10-n^5}{5n^4-6}$, and $x(=n+p) = \frac{4n^5+10}{5n^4-6}$. If we suppose $n=1$, we shall have $x = \frac{14}{-1} = -14$; this value is altogether inapplicable, a circumstance which arises from the approximated value of $n$ having been taken by much too small. We shall therefore make $n=2$, and shall thus obtain $x = \frac{138}{74} = \frac{69}{37}$, a value which is much nearer the truth. And if we were now to substitute for $n$, the fraction $\frac{69}{37}$, we should obtain a still more exact value of the root $x$. 792. Such is the most usual method of finding the roots of an equation by approximation, and it applies successfully to all cases. We shall however explain another method*, which deserves attention, on account of the facility of the calculation. The foundation of this method consists in determining for each equation a series of numbers, as $a, b, c$, &c. such, that each term of the series, divided by the preceding one, may express the value of the root with so much the more exactness, according as this series of numbers is carried to a greater length. Suppose we have already got the terms $p, q, r, s, t$, &c. * The theory of approximation here given, is founded on the theory of what are called *recurring series*, invented by M. de Moivre. This method was given by Daniel Bernoulli, in vol. iii. of the Ancient Commentaries of Petersburg. But Euler has here presented it in rather a different point of view. Those who wish to investigate these matters, may consult chapters 13 and 17 of vol. i. of our author's Introd. in Anal. Infin.; an excellent work, in which several subjects treated of in this first Part, beside others equally connected with pure mathematics, are profoundly analysed and clearly explained. F. T.