SOLUTION OF NUMERICAL EQUATIONS In the solution of numerical equations Oughtred does not mention the sources from which he drew, but the method is substantially that of the great French algebraist Vieta, as explained in a publication which appeared in 1600 in Paris under the title, De numerosa potestatum purarum atque adfectarum ad exegesin resolutione tractatus. In view of the fact that Vieta’s process has been described inaccurately by leading modern historians including H. Hankel[45] and M. Cantor,[46] it may be worth while to go into some detail.[47] By them it is made to appear as identical with the procedure given later by Newton. The two are not the same. The difference lies in the divisor used. What is now called “Newton’s method” is Newton’s method as modified by Joseph Raphson.[48] The Newton-Raphson method of approximation to the roots of an equation f(x)=0 is usually given the form a-[f(a)/f´(a)], where a is an approximate value of the required root. It will be seen that the divisor is f´(a). Vieta’s divisor is different; it is |f(a+s₁)-f(a)|-s₁ⁿ, where f(x) is the left of the equation f(x)=k, n is the degree of equation, and s₁ is a unit of the denomination of the digit next to be found. Thus in x³+420000x=247651713, it can be shown that 417 is approximately a root; suppose that a has been taken to be 400, then s₁=10; but if, at the next step of approximation, a is taken to be 410, then s₁=1. In this example, taking a=400, Vieta’s divisor would have been 9120000; Newton’s divisor would have been 900000. A comparison of Vieta’s method with the Newton-Raphson method reveals the fact that Vieta’s divisor is more reliable, but labors under the very great disadvantage of requiring a much larger amount of computation. The latter divisor is accurate enough and easier to compute. Altogether the Newton-Raphson process marks a decided advance over that of Vieta. As already stated, it is the method of Vieta that Oughtred explains. The Englishman’s exposition is an improvement on that of Vieta, printed forty years earlier. Nevertheless, Oughtred’s explanation is far from easy to follow. The theory of equations was at that time still in its primitive stage of development. Algebraic notation was not sufficiently developed to enable the argument to be condensed into a form easily surveyed. So complicated does Vieta’s process of approximation appear that M. Cantor failed to recognize that Vieta possessed a uniform mode of procedure. But when one has in mind the general expression for Vieta’s divisor which we gave above, one will recognize that there was marked uniformity in Vieta’s approximations. Oughtred allows himself twenty-eight sections in which to explain the process and at the close cannot forbear remarking that 28 is a “perfect” number (being equal to the sum of its divisors, 1, 2, 4, 7, 14).