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).
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
and
CqL=420000(A+E).
Subtract from 247651713 the sum of the known terms A³ (his Ac) and 420000 A (his CqA). This sum is 232000000 the remainder is 15651713.
“Exemplum II
1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇
Hoc est, Lc+CqL=Dc
2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7
------+-------+-------+------------
4 2 | 0 0 0 | 0 | Cq ------+-------+-------+------------
6 4 | | | Ac 1 6 8 | 0 0 0 | 0 | Cq A ------+-------+-------+------------
2 3 2 | 0 0 0 | 0 | Ablatit.
===================================
R 1 5 | 6 5 1̇ | 7 1 3̣ |
------+-------+-------+------------
4 | 8 | | 3 Aq | 1 2 | | 3 A 4 | 2 0 0 | 0 0 | Cq ------+-------+-------+------------
9 | 1 2 0 | 0 0 | Divisor.
------+-------+-------+------------
4 | 8 | | 3 Aq E | 1 2 | | 3 A Eq | 1 | | Ec 4 | 2 0 0 | 0 0 | Cq E ------+-------+-------+------------
9 | 1 2 1 | 0 0 | Ablatit.
===================================
R 6 | 5 3 0 | 7 1 3̣̇ | 4 | 1 |
------+-------+-------+------------ ----+-----+---
| 5 0 4 | 3 | 3 Aq | |
| 1 | 2 3 | 3 A 1 6 | 8 |
| 4 2 0 | 0 0 0 | Cq | 1 |
------+-------+-------+------------ ----+-----+---
| 9 2 5 | 5 3 0 | Divisor. 1 6 8 1
------+-------+-------+------------
3 | 5 3 0 | 1 | 3 Aq E | 6 0 | 2 7 | 3 A Eq | | 3 4 3 | Ec 2 | 9 4 0 | 0 0 0 | Cq E ------+-------+-------+------------
6 | 5 3 0 | 7 1 3 | Ablatit.”
Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A, the coefficient of E². He obtains 3A²=480000, 3A=1200, Cq=420000. He interprets 3A² and Cq as tens, 3A as hundreds. Accordingly, he obtains as their sum 9120000, which is the divisor for finding the second digit in the approximation. Observe that this divisor is the value of |f(a+s₁)-f(a)|-s₁ⁿ in our general expression, where a=400, s₁=10, n=3, f(x)=x³+420000x.
Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in ten’s place; thus E=10, approximately. He now computes the terms 3A²E, 3AE² and E³ to be, respectively, 4800000, 120000, 1000. Their sum is 9121000. Subtracting it from the previous remainder, 15651713, leaves the new remainder, 6530713.
From here on each step is a repetition of the preceding step. The new A is 410, the new E is to be determined. We have now in closer approximation, L=A+E. This time we do not subtract A³ and CqA, because this subtraction is already affected by the preceding work.
We find the second trial divisor by computing the sum of 3A², 3A and Cq; that is, the sum of 504300, 1230, 420000, which is 925530. Again, this divisor can be computed by our general expression for divisors, by taking a=410, s₁=1, n=3.
Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing 3A²E, 3AE², E³ and subtracting their sum, the remainder is 0. Hence 417 is an exact root of the given equation.
Since the extraction of a cube root is merely the solution of a pure cubic equation, x³=n, the process given above may be utilized in finding cube roots. This is precisely what Oughtred does in chap. xiv of his Clavis. If the foregoing computation is modified by putting Cq=0, the process will yield the approximate cube root of 247651713.
Oughtred solves 16 examples by the process of approximation here explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In all cases he finds only one or two real roots. Of the roots sought, five are irrational, the remaining are rational and are computed to their exact values. Three of the computed roots have 2 figures each, 9 roots have 3 figures each, 4 roots have 4 figures each. While no attempt is made to secure all the roots—methods of computing complex roots were invented much later—he computes roots of equations which involve large coefficients and some of them are of a degree as high as the fifth. In view of the fact that many editions of the Clavis were issued, one impression as late as 1702, it contributed probably more than any other book to the popularization of Vieta’s method in England.
Before Oughtred, Thomas Harriot and William Milbourn are the only Englishmen known to have solved numerical equations of higher degrees. Milbourn published nothing. Harriot slightly modified Vieta’s process by simplifying somewhat the formation of the trial divisor. This method of approximation was the best in existence in Europe until the publication by Wallis in 1685 of Newton’s method of approximation.
It should be stated that, before the time of Newton, the best method of approximation to the roots of numerical equations existed, not in Europe, but in China. As early as the thirteenth century the Chinese possessed a method which is almost identical with what is known today as “Horner’s method.”