Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

archimedes and aristotle 59Italicus (Geymonat 2006, p. 110): “Non illum mundi numerasse capaci harenas vanafides,” which may be approximately translated as “It was not an empty belief that hehad counted all the grains of sands that the world contains.”A second problem, 3 attributed to Archimedes, appeared in a Greek epigram oftwenty-four verses, published by Gotthold Ephraim Lessing in 1773, and is referredto as the cattle problem of Archimedes. The problem is to find the composition of theHerd of the Sun, namely, the number of the white, black, spotted, and brown bulls, andthe number of cows of corresponding colors, with the numbers in question satisfyingnine relations. There are eight unknowns, and seven relations are linear (as an example,it is required that the number of white bulls is five-sixths of the number of black bulls,plus the number of brown bulls). The remaining two relations are arithmetic in natureand ask that the white bulls, together with the black bulls, can be aligned to form aperfect square with the same number of bulls in each line. In a similar way, the spottedbulls, together with the brown bulls, can be corralled in a perfect triangle made of rowsof lines, with the number of bulls increasing by one from a line to the next.The problem reduces to finding the smallest solution, satisfying certain divisibilityconditions, of a certain Pell equation 4 Ax 2 + 1 = y 2 . The Lessing version leads toA = 4, 729, 494 and, in the end, to the truly gigantic smallest solutionx = 50, 549, 485, 234, 315, 033, 074, 477, 819, 735, 540, 408, 986, 340with a corresponding integer y that I will not write down. From this solution we cango back to the smallest solution of the original problem, and it turns out that the totalnumber of cattle has 206, 545 digits, as proved by Amthor in 1880 (see Vardi [1998]for a modern treatment and a discussion of the history of the problem). Nygrén (2001)proposed a more elementary approach to the problem that, at least in principle, wouldgive a procedure for finding the solution using an abacus.Was this really the original Archimedes’ problem? Probably not. Some scholarsmaintain that the epigram text is corrupted and proposed alternative versions leadingto solutions of a few hundred digits that, in view of Archimedes’s known feat with thesandreckoner, were within his range. (See Bartocci and Vipera [2004] for a discussion ofalternative versions and solutions.) An amusing alternative explanation can be found inArchimedes’ writings (see http://www.groups.dcs.st.ac.uk/∼history/Mathematicians).In his treatise On Spirals, Archimedes tells us that he was in the habit of sendingstatements of his latest theorems to his friends in Alexandria, but without givingproofs. Since some of the Alexandrine mathematicians there had claimed his resultsas their own, Archimedes says that on the last occasion when he sent them theoremshe included two that were false: “ . . . so that those who claim to discover everything,but produce no proofs of the same, may be confuted as having pretended to discoverthe impossible.” Could it be that Archimedes formulated such a problem, apparently3 See the account of this problem in Dickson (1952, XII 5, p. 342). A precursor is in Homer’s The Odyssey, bookXII, lines 194–98.4 This classical equation was well known in antiquity. The attribution to Pell is a misnomer originating withEuler. See Dickson (1952, p. 354).