By Jacques Sesiano

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**Extra resources for An Introduction to the History of Algebra**

**Example text**

But what was the proof, one might ask, that Fermat himself called “truly marvelous”? Since nothing of it remains, we have no idea. However, it is likely that Fermat’s proof was in fact only partial, applying only to certain exponents, and that Fermat hastily generalized his result. Example 2. 5] To divide a given number, which is the sum of two squares, into two other squares. 35 Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, & generaliter nullam in inﬁnitum ultra quadratum potestatem in duos (sic) eiusdem nominis fas est dividere.

The four books extant in Arabic are published in J. ¯ a ibn L¯ uq¯ a , New York 1982. ), Toulouse and Paris 1912–50 (reprint: Paris 1995–). T. , Cambridge 1910 (reprint: New York 1964) is a very complete study of the Greek books that incorporates much of Tannery’s work (often without citation). 34 As mentioned above (page 32), the subtracted terms follow the positive terms. 38 2. ALGEBRA IN ANCIENT GREECE • As was already known in antiquity, the general integral solution of the indeterminate equation u2 + v 2 = w2 is, up to a multiplicative factor, u = 2pq, v = p2 −q 2 , w = p2 +q 2 , with p and q integers, and any rational multiple of such a solution is a solution as well.

Another problem of the Anthologia (number 132) shows that so-called cistern problems already served to trouble school children. It is about the cyclops Polyphemus, one of the numerous obstacles Ulysses found on his way back to Ithaca at the end of the Trojan War. From a statue of Polyphemus, we learn, water ﬂowed through three openings. The stream emerging from the hand alone would ﬁll the basin in three days; the one pouring from his single eye would do this in one day; and the one running from the mouth would need only two-ﬁfths of a day.