7. FURTHER SIMPLIFICATIONS OF NEWTON'S METHOD

In modern text-books we find the Newtonian method of approximation stated thus: if f(x) = o, and a is a first approximation, then

had used these values, but obtained them by transforming the equation. Halley used the quadratic part of the transformed equation; Newton did the same at times. Brooks Taylor also used the quadratic, but obtained his coefficient by differentiation.

writer to use the linear differential formula x = a ·

f (a)

f' (a)

The first

was Thom

0; x =

as Simpson in his essay, "A new Method for the Solution of Equations in Numbers" (1740).20 He states the formula rhetorically and illustrates it by this example: f(x) = 300x x3 1000 = a + A; by trial a is found to lie between 3 and 4; try a = 3.5; f'(x) = 300 3x2; f(a) ÷ f'(a) = A; ƒ(3.5) = 7.125; ƒ'(3.5) = 263.25; 0.027 = 3.473.

A

= 0.027; 3.5 Proceeding as before, letting a = 3.473, we have x = 3.47296351. Simpson uses Newton's method in solving simultaneous equations. His fifth example is a set of exponential equations: x + = 0 and x" + y2

--

100 = 0.

1000

De Courtivron (1744), Euler (1755), and Lambert (c. 1770) also applied the calculus to the solution of higher equations.21 In his algebra 22 (1770), Euler discards the higher powers in both dividend and divisor during the substitution instead of carrying them along until the division takes place. Thus in solving the equation x3+ ax2 + bx + c = 0, he sets x = n − p, x2 = n2

3n2p; from these four expressions he finds p

=

2np, x3 = n3. n3 + an2 + bn + c

3n2 + 2an + b

Edward Waring (1770) extended the Newtonian approximation process into the realm of complex roots. The functional expression

20 Simpson, Essays on Mathematics, London, 1740, p. 81.

21 Euler, Institutiones calculi differentialis, sections 224, 234, 235; Histoire de l'Académie, 1744, p. 405-14.

22 Euler, Algebra, I (1770), Chap. 16.

23 Lagrange, Traité de la Résolution des Équations Numériques (1798), Note XI.

8. SUMMARY

I. The method of solving equations of higher degree by successive substitutions in derived equations was originated by Newton (1669) incidental to his getting integral expressions for his work in areas. He performed a transformation for every new supplement to the root. His method was first made public in the algebra of Wallis (1685).

2. Raphson (1690) worked Newton's method into a system. Following the example of Harriot he derived canonical forms to simplify calculation. In his plan only one transformation and one divisor are needed for each equation.

3. DeLagny (1691) derived notable approximations in the solution of pure equations by a special adaptation of Newton's method, and worked out an elaborate set of tables for the solution of numerical equations.

4. Halley (1694) generalized DeLagny's method for pure equations and extended it to affected equations; in his table of powers he gave a general formula for the coefficients of transformed equations. This made obsolete Raphson's canonical forms and DeLagny's tables.

5. Taylor (1717) discovered that Halley's coefficients were the same as the coefficients in Taylor's series. This made possible the direct use of the calculus in finding roots.

6. Our present standard linear approximation formula x = a

f(a)

was first used by Simpson (1740); it received its conventional f' (a)

form through the writings of Lagrange (1798).

VIII

CERTAIN EFFECTIVE BUT NON

PRACTICABLE METHODS

I. THE ASCENDANCY OF NEWTON'S METHOD

Newton's method of approximating roots by substituting values in equations of an infinite number of terms did not immediately upon its publication in 1685 supersede Vieta's method of finding roots by evolving the successive digits. The latter was still used by Dechâles in 1690 and appears as late as 1702 in a new edition of Oughtred's Clavis. It seems as if the transition from the one to the other must have been facilitated by the eclectic character of Raphson's arrangement. Halley's general formula increased the popularity of the new method. Its ascendancy is clearly shown in an article written by Wallis in 1694. After Taylor's discovery (1717) it had no serious rival old or new, until the publication of Horner's method in 1819. But new methods sprang up to challenge it in this period, and we shall describe the three that gave the most promise.

2. ROLLE'S METHOD OF CASCADES

A considerable contribution to the location of roots was made by Michel Rolle in his two works Traité d'Algèbre (1690) and Démonstratio d'une Méthode pour Résoudre les Égalitez de toutes Degrez (1691). He demonstrated that there cannot be more than one real root of f(x) = o between two successive real roots of f'(x) = 0 (Rolle's Theorem). He invented a method of finding roots, called the "Method of Cascades"; it consisted in locating the roots by first finding the limits ("hypotheses") of the roots of the different cascades. Between the roots of each cascade lie the roots of the next higher cascade, and so on.1 It was an anticipation of the method of using successive derivatives (Rolle's cascades) now used in calculus for locating roots.

xn

With Rolle an equation is "prepared" when it has the form ɑ2x2-3 + . . . . . = o where the a's are posi

n-2

n-3

1 Traité d'Algèbre, p. 124, p. 103; p. 118; Démonstratio, article VI (New York Public Library); Cajori in Bibl. Math. XI (1910-11), p. 300–313.

tive integers. Then the upper limit of the root is found by dividing the largest negative coefficient by the highest exponent, and adding unity and as much more as necessary to make the result an integer. His Method of Cascades he illustrates thus:2

Let x3-57x2 +936x - 3780 = 0 (third cascade).

Multiply the terms of the cascade by parts of the progression 3, 2, I, 0, respectively; divide by x. Then 3x2 - 114x + 936 = 0 (second cascade).

Multiply by 2, 1, 0; divide by 2x. Then 3x cascade).

57 =

o (first

The "hypotheses" for the second cascade are 0, 19 and 39 (or

The "hypotheses" for the first cascade are o and 19.

Between 0 and 19, the second cascade is found to have 12 as a root; and another root, 21, between 19 and 39.

Finally for the third cascade the extreme "hypotheses" are o and 3781

; 0, 12, 26, and 3781 are its four "hypotheses."

The 100ts of the third cascade (that is, the given equation) can then be found to be: 6 between 0 and 12; 21 between 12 and 26; 30 between 26 and 3781.

3. THE METHOD OF RECURRING SERIES

De Moivre (1667–1754) first conceived of series in which each coefficient bears a given relation to certain of the preceding coefficients; such series he called 1ecurring series (1720). Daniel Bernoulli (1700-1782) first used them for approximating roots (1728). Following is his process:

Take such an equation as I = ax + bx2 + cx3 + dx4. Select arbitrarily the four terms A, B, C, D. Then E, F, G, H, etc., constitute a recurring series if

2 Traité d'Algèbre, pp. 127-28.

3 Phil. Trans., Vol. 32 (1722), pp. 162-78; DeMoivre, Miscellanea Analytica, London, 1730, Bk. I, II.

Cantor. III (1898), pp. 621-22.

EaD + bC + cB + dA,
FaEbD+cC + dB,

G = aFbE + cD + dC,

E F

F'G'

HaG+bF+cE + aD;

and the approximate values of x are etc.

In solving the equation = -2x+5x2 - 4x3 + x, Bernoulli sets

A = B =

C

= D =

I, and obtains the series 0, 2, -7, 25, -93, 341, -1254.

approximations of the root. The last value, when substituted in the given equation gives I = 0.999487, a very small error.

Bernoulli gave no analytical proof for the validity of his process. That was supplied by Euler (1748) and later by Lagrange (1798). A considerable improvement was made by Euler in his algebra (1770). He shows the general relation that must exist for the different powers, and thus immediately gets a recursion formula for any equation under consideration. The recursion formula performs a function similar to that of Raphson's canonical form or of Halley's general formula.

Suppose our series p, q, r, s, t, etc., is such that

Р

= x; hence =x3; similarly

Р

have a substitution table for powers, but we have one with the first term as the common denominator.

x3

As an illustration, let us substitute these values in the equation x2 + 2x + 1, This gives s = r + 29 + p. Using this as a re

=

'Euler's Introductio in Analysin Infinitorum, Chap. 17; Lagrange, Traité de la Résolution des Équations Numériques, Note VI.

• Euler's Algebra, Vol. I, Chap. 16.