and al-Qalasâdî (c. 1475).1o This rule was also used by the Jewish writer Johannes Hispalensis (c. 1140). Al-Karchî (c. 1020) used a different formula, namely va2 + b = a + b 2a + I How he de rived this formula is not known; possibly he used a method of double false position and interpolation by proportional parts similar to the one used later by Leonardo of Pisa for cube roots. Ibn Albannâ (c. 1300) used al-Karchî's formula for b > a, but Theon's formula for ba. Leonardo used Theon's formula with a corrective sup Renaissance period Bombelli (1572), Cataldi (1613), and Schwenter (1618) approximated irrational square roots by the use of continued fractions. A formula for approximating irrational cube roots was invented by Leonardo of Pisa." In deriving this he used double false position and interpolation by proportional parts. He lets a3 and (a + 1)3 be the two cubes nearest to a3 + b. Then, since a3 < a3 + b < (a+1), we have o<b< 3a (a + 1) + 1. This relation gives a criterion for the proper choice of a. Also, if we increase a by unity we increase the resulting cube by зa(a + 1) + 1. What increase in a will increase the resulting cube by b? By proportion 10 Günther, Quadratische Irrationaliteten der Alten, pp. 45-46; Cantor, I (1908), pp. 633, 641; Hankel, Zur Geschichte der Mathematik im Alterthum und Mittelalter (hereafter referred to as Hankel), Leipzig, 1874, p. 185. 11 Boncompagni, Leonardo Pisano, I, pp. 370, 378, 380-81; Cantor, II (1900), pp. 31-32; Bibl. Math. (3) II (1902), pp. 350-54. 12 Juan de Ortega (1512) used a modification of this formula, viz., vos +b = To find the roots of large irrational numbers two plans were followed, the Hindu arrangement being used in both cases. Leonardo illustrates them both.13 In one of his solutions, √927435 = the derivation of 963 is shown as 587 9274 35 963 We find this method used as late as the seventeenth century in the Kholasat-al-Hisab 14 of Behâ Eddîn (c. 1600). Another plan is followed by Leonardo in this solution: This plan became increasingly popular and when in 1539 a man of Cardan's prestige adopted it systematically in his arithmetic,15 accompanied by clear and concise rules, it became the standard method in Europe. It had taken a thousand years, from Aryabhata to Cardan, to perfect this approximation method. But it is the most effective and most widely used method known. The methods of Vieta and of Horner are its lineal descendants. 18 Boncompagni, Leonardo Pisano, I, p. 355; Cantor, II (1900), p. 30; Curtze, "Ueber eine Algorismus Schrift des XII. Jahrhunderts" in Abhandl. Geschicht. Math., VIII (1898), pp. 1-28. 14 Taylor's Lilawati, Preface, p. 14; Hutton's Tracts II, London, 1812, p. 180. A manuscript copy of the Kholasat-al-Hisâb is found in Professor D. E. Smith's private collection. 15 Cardan's Arithmetic, chap. 23. 3. SUMMARY 1. The superior symbolism and notation of the Hindus enabled them to invent and use with ease formulas based on the principle of inversion. 2. For finding the rational roots of numbers too large for handling by inspection they used a method of exhaustion. It was based on the inversion of the expressions a2+2ab + b2 = (a + b)2 and a2 + 3a2b+3ab2 + b3 = (a + b)3, and on the decimal notation with its place values. Its essence was the evolving of the digits of the root by orders (Evolution). This method was also used for irrational roots later on, and European scholars standardized an adaptation of it for approximating irrational roots by decimal fractions. 3. The five principal approximation formulas invented and used in the Middle Ages were: IV APPROXIMATIONS FOR SPECIFIC PURPOSES I. THE RISE OF NUMERICAL HIGHER EQUATIONS AMONG THE ARABS Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations. Archimedes's problem of the section of the sphere, which led to a cubic, was first attempted by al-Mâhânî (c. 860), and was later solved successfully by al-Khâzin (c. 950) with the aid of conic sections. The trisection of angles and the computation of sides of regular polygons led to other cubics,' which were also solved by conic sections; for they soon came empirically to the conclusion that cubics could not be solved algebraically. The solution of numerical cubic equations by intersecting conics was the greatest original contribution to algebra made by the Arabs. These solutions remained unknown to the Western world, and were rediscovered in the seventeenth century by Descartes, Thomas Baker and Edmund Halley. The success of the Arab scholars in this field may have deterred them from trying methods of approximation. What they might have done in this field may be inferred rather than judged from the solitary example left us in Saracen writings. 2. A SOLUTION BY AN UNKNOWN ARAB SCHOLAR In Miram Chelebi's annotations of Ulugh Beg's astronomical tables (1498) there is explained a method of solving the equations Px = x2+Q, where P and Q are positive and x3 is small in comparison with Q. Chelebî explains that this equation was used for a specific purpose, namely, to find the sine of 1°. With the Persian astronomers, as with us, the ordinary trigonometric interpolation 1 L'algèbre d'Omar Alkayyami (translated and edited by Woepcke), Paris, 1851, pp. 54-57; p. 82; Hankel, p. 277. 2 Cantor I (1907), pp. 781-82; Hankel, pp. 289–93; Braunmühl, Geschichte der Vorlesungen der Trigonometrie I, Leipzig, 1900-3, pp. 72-74. would fail for sin 1°. The solution is commonly attributed to Giyât Eddîn al-Kâschi. Braunmühl, however, thinks it is due to al-Zarkâli (c. 1080). In this particular equation P = 45' and Q = 47′ 6o 8′ 29′′, where the radius length 1' is the sexagesimal unit and is equal to 60o. The solution is as follows: b, with remainder S; that gives R = bP + Sa3. S − a3 + (a + b + s)3 = a+b+ S+ (a + b)3 — a3 P Suppose the last fraction equals c, with a remainder T. Then x = a + b + c + s, and so on. = P The actual problem is to evaluate sin 1° from the known trigonometric relation 60 sin 3° 3.60 sin 1° 4 sin3 1°. Letting x = sin 1° and simplifying, the equation takes the form 45 x = x+460 sin 3°. The Arabs knew the last term to equal 47" 6' 8' 29" = Q. The following3 is Chelebî's problem, carried out to seconds; in the Arab text it is carried out to fourths. Hankel, p. 292. |