An Introduction to the History of Algebra by Jacques Sesiano

By Jacques Sesiano

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By adding to each equation the factor appearing in parentheses, we obtain ⎧ ⎪ ⎨x1 + x2 + x3 + x4 = (a + 1)(x3 + x4 ) x1 + x2 + x3 + x4 = (b + 1)(x2 + x4 ) ⎪ ⎩ x1 + x2 + x3 + x4 = (c + 1)(x2 + x3 ). Let S be the sum of the four unknowns. Then the first of these equations becomes 1 S. x3 + x4 = a+1 Substituting the right side for x3 + x4 in the first given equation yields a S, x1 + x2 = a+1 while proceeding similarly for the other two equations yields b S b+1 c S. x1 + x4 = c+1 This brings us to the situation of Thymaridas’s rule, which we can apply to obtain a b c S a+1 + b+1 + c+1 −S .

In the second problem, we are given u + v = k and w = l. As u − v = √ 2w2 − (u + v)2 = 2l2 − k 2 by identity (4 ), we can calculate v= 1 1 [(u + v) − (u − v)] = k− 2 2 2l2 − k 2 , and then u = (u + v) − v = k − v. This is indeed how v and u are determined; with k√= 17, l = 13, we √ have the 2 , k 2 , 2l2 , 2l2 − k 2 , 2 − k2 , k − 2l 2l2 − k 2 , successive computations of l √ 1 2 2 2 [k − 2l − k ] = v, whence u. 21 This use of the article in the Greek text indicates that the number in question has already been encountered, either as a given value or as the result of a calculation.

Anbouba, L’Algèbre al-Bad¯ı‘ d’al-Karag¯ı, Beirut 1964. For a study of the indeterminate algebra in the latter work, see J. Sesiano, “Le Traitement des équations indéterminées dans le Bad¯ı‘ f¯ı’l-H ab d’Ab¯ u Bakr al-Karaj¯ı,” . is¯ Archive for history of exact sciences, 17 (1977), pp. 297–379. 42 See J. Sesiano, “Les Méthodes d’analyse indéterminée chez Ab¯ u K¯ amil,” Centaurus, 21 (1977), pp. 89–105. 3. DIOPHANTINE ALGEBRA 47 terms was not immediately possible (page 34). This is exactly the case that some of the problems below address.

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