Page:A History of Mathematics (1893).djvu/166

could be found, there remained nothing else to be done than the devising of rules by which at least the numerical values of the roots could be ascertained. Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta, a French mathematician, whose transcendent genius enriched mathematics with several important innovations. Taking the equation $\scriptstyle {f(x)=Q}$, wherein $\scriptstyle {f(x)}$ is a polynomial containing different powers of x, with numerical coefficients, and Q is a given number, Vieta first substitutes in $\scriptstyle {f(x)}$ a known approximate value of the root, and then shows that another figure of the root can be obtained by division. A repetition of the same process gives the next figure of the root, and so on. Thus, in $\scriptstyle {x^{2}+14x=7929}$, taking 80 for the approximate root, and placing $\scriptstyle {x=80+b}$, we get

or

${\begin{aligned}&\scriptstyle {(80+b)^{2}+14(80+b)=7929},\\&\scriptstyle {174b+b^{2}=409}\end{aligned}}$.

Since $\scriptstyle {174b}$ is much greater than $\scriptstyle {b^{2}}$, we place $\scriptstyle {174b=409}$, and obtain thereby $\scriptstyle {b=2}$. Hence the second approximation is 82. Put $\scriptstyle {x=82+c}$, then $\scriptstyle {(82+c)^{2}+14(82+c)=7929}$, or $\scriptstyle {178c+c^{2}=57}$. As before, place $\scriptstyle {178c=57}$, then $\scriptstyle {c=.3}$, and the third approximation gives 82.3. Assuming $\scriptstyle {x=82.3+d}$, and substituting, gives $\scriptstyle {178.6d+d^{2}=3.51}$, and $\scriptstyle {178.6d=3.51}$, $\scriptstyle {\therefore ~d=.01}$; giving for the fourth approximation 82.31. In the same way, $\scriptstyle {e=.009}$, and the value for the root of the given equation is 82.319… For this process, Vieta was greatly admired by his contemporaries. It was employed by Harriot, Oughtred, Pell, and others. Its principle is identical with the main principle involved in the methods of approximation of Newton and Homer. The only change lies in the arrangement of the