APPROXIMATION
, a continual approach, still nearer and nearer, to a root or any quantity sought.— Methods of continual approximation for the square roots and cube roots of numbers, have been employed by algebraists and arithmeticians, from Lucas de Burgo down to the present time. And the later writers have given various approximations, not only for the roots of higher powers, or all simple equations, but for the roots of all sorts of compound equations whatever: especially Newton, Wallis, Raphson, Halley, De Lagny, &c, &c; all of them forming a kind of insinite series, either expressed or understood, converging nearer and nearer to the quantity sought, according to the nature of the process.
It is evident that if a number proposed be not a true square, then no exact square root of it can be found, explicable by rational numbers, whether integers or fractions: therefore, in such cases, we must be content with approximations, or coming continually nearer and nearer to the truth. In like manner, for the cube and other roots, when the proposed quantities are not exact cubes, or other powers.
The most easy and general method of approximation, is perhaps by the rule of Double Position, or, what is sometimes called, the Method of Trial-and-error; which method see under its own name. And among all the methods for the roots of pure powers, of which there are many, I believe the best is that which was discovered by myself, and given in the first volume of my Mathematical Tracts, in point of ease, both of execution and for remembering it. The method is this: if N denote any number, out of which is to be extracted the root whose index is denoted by r, and if n be the nearest root first taken; then shall the required root of N very nearly; or as r - 1 times the given number added to r + 1 times the nearest power, is to r + 1 times the given number added to r - 1 times the nearest power, so is the assumed root n, to the required root, very nearly. Then this last value of the root, so found, if one still nearer is wanted, is to be used for n in the same theorem, to repeat the operation with it. And so on, repeating the operation as often as necessary. Which theorem includes all the rational formulæ of Halley and De Lagny.
For example, suppose it were required to double the cube, or to find the cube root of the number 2. Here r=3; consequently , and ; and therefore the general theorem becomes for the cube root of N; or as N + 2n3 : 2N + n3 :: n : the root sought nearly. Now, in this case, N = 2, and therefore the nearest root n is 1, and its cube n3 = 1 also: hence , and ; therefore, as 4 : 5 :: 1 : 5/4 or 1 1/4=1.25 the first approximation. Again, taking r=(5/4), and consequently r3=(125/64); hence ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : (635/504)= 1.259921, which is the cube root of 2, true in all the sigures. And by taking 635/504 for a new value of n, and repeating the process again, a great many more figures may be found.
Of the Roots of Equations by Approximation.— Stevinus and Vieta gave methods for sinding values, always nearer and nearer, of the roots of equations. And Oughtred and others pursued and improved the same. These however were very tedious and imperfect, and required a different process for every degree of equations. But Newton introduced, not only general methods for expressing radical quantities by approximating infinite series, but also for the roots of all sorts of compound equations whatever, which are both easy and expeditious: which will be more particularly described under each respective word or article. His method for approximating of roots, is in substance this: First take a value of the root as near as may be, by trials, either greater or less; then assuming another letter to denote the unknown difference between this and the true value, substitute into the equation the sum or difference of the approximate root and this assumed letter, instead of the unknown letter or root of the equation, which will produce a new equation having only the assumed small difference for its root or unknown letter; and, by any means, find, from this equation, a near value of this small assumed quantity. Assume then another letter for the small difference between this last value and the true one, and substitute the sum or difference of them into the last equation, by which will arise a third equation, involving the second assumed quantity; whose near value is found as before. Proceeding thus as far as we please, all the near values, connected together by their proper signs, will form a series approaching still nearer and nearer to the true value of the root of the first or proposed equation. The approximate values of the several small assumed differences, may be found in different ways: Newton's method is this: As the quan tity sought is small, its higher powers decrease more and more, and therefore neglecting them will not lead to any great error, Newton therefore neglects all the terms having in them the 2d and higher powers, leaving only the 1st power and the absolute known term; from which simple equation he always sinds the value of the assumed unknown letter nearly, in a very simple and easy manner. Halley's method of doing the samething, was to neglect all the terms above the square or 2d power, and then to sind the root of the remaining quadratic equation; which would indeed be a nearer value of the assumed letter than Newton's was, but then it is much more troublesome to perform.—Raphson has another way, which is a little varied from that of New- | ton's again, which is this: having found a near value of the first assumed small quantity or difference, by this he corrects the first approximation to the root of the proposed equation; and then, assuming another letter for the next, or smaller difference, he introduces it into the original equation in the same way as before. And thus he proceeds, from one correction to another, employing always the first proposed equation to find them, instead of the successive new equations used by Newton.
For example, let it be required to find the root of the equation , or :—Here the root x, it is evident, is nearly = 8; for x therefore take 8 + z, and substitute 8 + z for x in the given equation, and the terms will be thus; Hence, then, collecting all the assumed differences, with their signs, it is found that the root of the equation required, by Newton's method.
Example 2. Again, taking the cubic equation the root of the equation . And in the same manner Newton performs the approximation for the roots of literal equations, that is, equations having literal coefficients; so the root of this equation
See also a memoir on this method by the Marquis de Courtivron, in the Memoires de l'Academie for 1744.
Other Methods of Approximation. Besides the foregoing general methods, other particular ways of approximating, for various purposes, have been given by many other persons.—As for example, methods of approximating, by series, to the roots of cubic equations belonging to the irreducible case, by Nicole in the same Memoirs, by M. Clairaut in his Algebra, and by myself in the Philos. Trans. for 1780. See also several parts of Simpson's works, and my Tracts vol. 1. Also the methods of infinite series by Wallis, Newton, Gregory, Mercator, &c, may be considered as approximations, in quadratures, and other branches of the mathematics, many instances of which may be seen in Wallis's Algebra, and other books:—Likewise the method of exhaustions of the ancients, by which Archimedes and others have approximated to the quadrature and rectification of the circle, &c, which was performed by continually bisecting the sides of polygons, both inscribed in a circle and circumscribed about it; by which means the sum of the sides of the like polygons approach continually nearer and nearer together, and the circumference of the circle is nearly a mean between the two sums. See also Equations.
