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).
The early part of his exposition shows how an equation may be transformed so as to make its roots 10, 100, 1000, or 10m times smaller. This simplifies the task of “locating a root”; that is, of finding between what integers the root lies.
Taking one of Oughtred’s equations, x⁴-72x³+238600x=8725815, upon dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we obtain x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4.
But 4)872·5(218·1; 4 is too small. Next let x=5, we have 125+238·6-180=183·6.
But 5)872·5(174·5; 5 is too large. We take the lesser value, x=4, or in the original equation, x=40. This method may be used to find the second digit in the root. Oughtred divides both sides of the equation by x², and obtains x²+x)238600-72x=x²)8725815. He tries x=47 and x=48, and finds that x=47.
He explains also how the last computation may be done by logarithms. Thereby he established for himself the record of being the first to use logarithms in the solution of affected equations.
As an illustration of Oughtred’s method of approximation after the root sought has been located, we have chosen for brevity a cubic in preference to a quartic. We selected the equation x³+420000x=247651713. By the process explained above a root is found to lie between x=400 and x=500. From this point on, the approximation as given by Oughtred is as shown on [p. 43].
In further explanation of this process, observe that the given equation is of the form Lc+CqL=Dc, where Lc is our x, Cq=420000, Dc=247651713. In the first step of approximation, let L=A+E, where A=400 and E is, as yet, undetermined. We have
Lc=(A+E)³=A³+3A²E+3AE²+E³