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tions of every degree. He restricts the rule to equations where the coefficients are positive or zero.

The following is his derivation of the rule, presented in modern analytical form:

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Let f(x) k, where f(x) is a polynomial in x arranged in descending order. Let a and (a + 1) be two positive integers such that f(a) kb and f(a + 1) = k + b'. Then the root x lies between a and a + 1. That is, x = a + t(a + 1 a) or x = (å + 1) e(a + I - a), where the corrections t and e are positive proper fractions. The first may be called the additive, the second the subtractive form. Since f(a + 1) > f(x) > ƒ(a), we have f(a + 1) f(a) > f(x) − f(a) > o and consequently

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b

b + b'

b

as an approximate value of t. Then x = a +

(a

< I our next additive form will then be ƒ ( a +

b+b'

Placing this new additive value in the subtractive form we have

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In the subtractive form e is less than unity. For an approximate

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This gives the third approximation x = a + 1 –

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b + b'

b'

=

b' + b"

=r, say. According as k lies between f(r) and f(a) or be

tween f(r) and f(a + 1) one must, in continuing, use the additive or the subtractive process.

His first illustrative example is the solution of the equation

x2 + 3x3 = 100, the root of which he finds to be 2

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lowing is his solution and arrangement:

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Contrary to his own restrictions on the signs of the coefficients Cardan uses the rule in the equation x3 = 6x+20, with disastrous results.

4. STEVIN'S CONTRIBUTION

The approximation work in Stevin's algebra (1685) might with equal propriety be discussed in connection with Vieta or Newton. For Stevin (1548-1620) anticipated Newton's analytical form and treatment as well as Vieta's idea of evolving the digits of the root

of a complete equation order by order. But since his scheme of deriving the digits involved the successive choice of two positions, we discuss his method in this chapter. We can best explain his process by recording one of his illustrations.5

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Hence x lies between 100 and 1000. By trying x = 200, x = 300, x = 400, he finds the root to lie between 300 and 400. Similarly trying 310, then 320, then 330, he finds x to lie between 320 and 330. A similar procedure for the units gives 324 as the exact value of the root. Stevin observes that an irrational root can be approximated to within any desired degree of accuracy by using this method and his scheme of computing decimal fractions. This method was advocated and used considerably by Albert Girard in his Invention Nouvelle en l'Algèbre (1629). Though laborious, the method is general. "Stevin's rule" was used by later algebraists, like Oughtred (1631), Kersey (1673), and Saunderson, in connection with other methods."

5. THE METHODS OF BÜRGI AND PITISCUS

Pacioli's Summa (1494), the earliest printed work on arithmetic and algebra, gave to Renaissance scholars the method of double false position and interpolation by proportional parts used by Leonardo for finding roots of numbers. Leonardo's method now came to be employed frequently also in the solution of higher numerical equations, and Pacioli even used it in an exponential equation. This method, which has always been basic in finding trigonometric functions, was used by Jost Bürgi in solving equations which rose out of his work in trigonometry. We give his solu

' Stevin, Les Œuvres Mathématiques (editor Girard, 1634), I, problem 77; Cantor, II (1900), p. 628.

• N. Saunderson, Elements of Algebra, II, Cambridge, 1790, p. 728.

7 Pacioli, Summa, fol. 186 recto to 188 recto; Cantor, II (1900), pp. 325-26.

8 Cantor, II (1900), pp. 626, 645-46, 648; C. I. Gerhardt, Geschichte der Math. in Deutschland, Munich, 1877, pp. 76-82; 84-87.

tion of such an equation, namely, 9-30x2 + 27x4

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The root has been found to lie between 0.68 and 0.69; let x = 0.68 +h, and f(x) = 0. Then

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Hence 0.1397 : 0.0569 = 0.01: h; or h = 0.004, and x = 0.684. Repeating the process with f(0.6840) and f(0.6841) he gets 0.00140012 0.00056410 = 0.0001 : h; or h

Hence x = 0.68414029.

= 0.00004029.

A more conventional method of double false position is used by Pitiscus (1612), a pupil of Bürgi. In solving the equation9 5,176,380 3x − x3 he sets x = 1,725,460 +3. Manifestly x >

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3

1,725,460. By trial he finds x to lie between 1,730,000 and 1,740,000. Let us call these two values a1 and a2 and represent the equation by f(x)=k. Then f(a1) kd2. Then

38,158

=

k

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k

di; f(a2)

=

k

9,061 =

di - d2

= 1,743,114.

6. SUMMARY

I. The first successful general method of solving complete numerical equations by approximation was given by Chuquet in his Triparty (1484). It was not printed until 1880 and remained comparatively little known.

2. The first printed method was Cardan's Regula Aurea (1545). It was restricted as to the signs of the coefficients but not as to the degree of the equation. It never came into general use.

3. In 1585 Stevin explained a method of evolving the digits of the roots by order.

4. Pacioli and Bürgi extended Leonardo's method of finding cube roots to the solution of complete equations of higher degree. This method and the conventional method of double false position exemplified by Pitiscus were the ones most generally used before the year 1600.

• Pitiscus, Trigonometriae, pp. 50-53.

VI

VIETA'S EXTENSION OF THE HINDU METHOD
OF APPROXIMATION TO COMPLETE EQUATIONS

I. VIETA'S METHOD OF APPROXIMATION

A new epoch in the history of the solution of numerical equations begins with the year 1600, the date of the first publication of Vieta's De Numerosa Potestatum Purarum atque Adfectarum ad Exegesin Resolutione Tractatus. In it was explained a systematic and concise way of solving any equation, complete or incomplete, by methodically finding the successive digits of the root, beginning with the highest order. Why no one before Vieta should have thought of applying to the solution of complete equations the Hindu method of finding roots of large numbers may seem strange to us unless we reflect that useful inventions usually seem very natural after they are invented. At any rate, this method had been applied to large numbers for a thousand years, had appeared in definite forms for four hundred years, and had been perfected in its finest details, before any one thought of extending it to complete equations. It was the first comprehensive method of solving such equations that had been attempted, and it involved no restrictions as to terms, signs, or degree, for it applied to a form into which any equation can be changed. Vieta calls an equation "duly prepared" if it has the form x" + ax + bx"2 + = K where

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K is positive and all the coefficients are integers.

De Numerosa Potestatum was first printed in Paris in 1600 for private circulation. In 1646 Franz van Schooten, of Leyden, again printed it in a one-volume collection of Vieta's works. It contains sixty-six pages, of which the first ten deal with incomplete equations. He arranges the solution in a way slightly different from the conventional one in order the better to show the analogy between the pure and the affected powers.2

1 Vietae Opera Mathematica, Leyden, 1646; hereafter referred to as Vieta.

? Vieta classified powers and equations as pure and affected ("ad-fected"). The names are geometrical in origin; for Vieta considered all the terms in an equation homodimensional. Thus in the "affected" equation x3 + 30x K, the "resolvend" K is considered three-dimensional; and 30 is the "plane coefficient" in the "adjoined solid"

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