The early part of his exposition shows how an equation may be transformed so as to make its roots 10, 100, 1000, or 10^m 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 L_c+C_qL=D_c, where L_c is our x, C_q=420000, D_c=247651713. In the first step of approximation, let L=A+E, where A=400 and E is, as yet, undetermined. We have L_c=(A+E)³=A³+3A²E+3AE²+E³ and C_qL=420000(A+E). Subtract from 247651713 the sum of the known terms A³ (his A_c) and 420000 A (his C_qA). This sum is 232000000 the remainder is 15651713. “Exemplum II 1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇ Hoc est, L_c+C_qL=D_c