# An Introduction to the History of Algebra by Jacques Sesiano

By Jacques Sesiano

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Indeed, this is how the text calculates u and v from the known values k = 25 and l = 150. 23 See F. Blume, K. Lachmann, and A. ), Berlin 1848–52 (reprint: Hildesheim 1967), I, pp. 297–98, or (a better text) N. Bubnov, Gerberti postea Silvestri II papae Opera mathematica, Berlin 1899, pp. 511–12 (or the second reference mentioned in note 20). 24 “foot” is used indistinctly for length and area measurement. 25 In these texts (see also the following example), “always” means that the rule is general and independent of the values given in the problem.

We have seen the reduction for a = 0 and c = 0 (ﬁrst two cases above). Now suppose that b = 0, in which case the equation is = m2 or = m2 x2 , we obtain ax2 + c = . Whether we set 2 an equality between x and a constant. The rationality of the solution then depends on the coeﬃcients a and c. 14, Diophantus obtains the equation 15x2 − 36 = , which he declares impossible ( ); indeed, as he says, a rational solution would imply that 15 is representable as a sum of two squares, which it is not. We will see the reason for this later, in our last example of a Diophantine problem.

Second, the ﬁnal form of the equation is obtained by means of two operations. One of them is to add to both sides the (positive) amount of the subtracted terms, thus obtaining an equation involving only positive terms. The other is to remove from both sides the common quantities, whereby the equation will contain a single term for each power of the unknown. Thus, as a modern illustration, 5x2 + 18 − 11x = 16 would ﬁrst be changed by addition of 11x to both sides to 5x2 + 18 = 16 + 11x, and then, by removal of 16, to 5x2 + 2 = 11x.