The History of Mathematics: A Brief Course

422 14. EQUATIONS AND ALGORITHMS or quartic equations, it must be found by the solver's own ingenuity" (Colebrooke, 1817, pp. 20 ...

THE MUSLIMS 423 Colebrooke also notes (p. Ixxi) that Mohammed Abu'l-Wafa al-Buzjani (940- 998) wrote a translation or commenta ...

424 14. EQUATIONS AND ALGORITHMS Ç— 2 i—Ç J FIGURE 3. Al-Khwarizmi's solution of "square plus 10 roots equals 39 dirhems." such ...

THE MUSLIMS 425 explains the solution as follows (we put the legal principle that provides the equation in capital letters): C ...

426 14. EQUATIONS AND ALGORITHMS FIGURE 4. Omar Khayyam's solution of x^3 + ax^2 + b^2 x = b^2 c. we would phrase it, x^3 + ax^2 ...

EUROPE 427 He then turned to the five types that could have no (positive) solutions for some values of the data. As an example ...

428 14. EQUATIONS AND ALGORITHMS 6.1. Leonardo of Pisa (Fibonacci). Many of the problems in the Liber abaci re- flect the routin ...

EUROPE^429 given by Euclid in Book 10 of the Elements, that is, to construct a line of this length using ruler and compass. In ...

430 14. EQUATIONS AND ALGORITHMS (see Parshall, 1988), the work was largely a compilation of the works of Leonardo of Pisa, but ...

EUROPE 431 6.5. Solution of cubic and quartic equations. In Europe algebra was confined to linear and quadratic equations for ...

432 14. EQUATIONS AND ALGORITHMS This is a cubic equation called the resolvent cubic. Once it is solved, the original quartic br ...

EUROPE 433 we obtain 3 Ì © D sec © D ter ©. Although notation still had far to go, from the modern point of view, at least it ...

434 14. EQUATIONS AND ALGORITHMS Questions and problems 14.1. Problem 6 of Book 1 of the Arithmetica is to separate a given numb ...

QUESTIONS AND PROBLEMS^435 14.9. Consider the cubic equation of Sharaf al-Tusi's third type, which we write as -x^3 - ax^2 + bx ...

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Chapter 15. Modern Algebra By the mid-seventeenth century, the relation between the coefficients and roots of a general equation ...

438 15. MODERN ALGEBRA factions as there are numbers given." He noted that the number of terms in each faction could be found by ...

THEORY OF EQUATIONS 439 Tschirnhaus claimed that the the middle terms (the á& above) would be elimi- nated by a polynomial ...

440 15. MODERN ALGEBRA To explain the most important of these new insights, let us consider what Girard's result means when appl ...

THEORY OF EQUATIONS 441 Euler. In his 1749 paper "Recherches sur les racines imaginaires des equations" ("Investigations into ...