Having had a look at Krumbeigel:
1. Who wrote the text?
The text is probably not by Archimedes, language of the original is Ionic rather than Doric Greek.
2. Who posed the problem?
The problem itself is in two parts, 1st possibly by Eratosthenes, 2nd possibly by Arch., (Hultsch quoted in Vardi). In his Charmides p. 324 Plato refers to a problem “called by Archimedes the Cattle–problem.” It could be even earlier.
3. Could Archimedes have solved the problem?
Probably not, due to the huge numbers, but he probably knew how to do it with continued fractions.
These questions were reviewed by Krumbiegel 1880 (published along with the first published solution to the second section, by A. Amthor.)
[25] B. Krumbiegel, Das problema bovinum des Archimedes, Zeitschrift f¨ur Math. u. Physik (Hist. litt.
Abtheilung) 25 (1880), 121–136. Online here:
"Zeitschrift für Mathematik u. Physik" (1880) Vol XXV (25) Part 2, July-December 1880
Vardi cites Thomas Heath (author of Heath (1921) in the Bibliography) and also E.J. Dijksterhuis, in support of Krumbiegel.
[3] Archimedes, The Works of Archimedes, edited in modern notation with introductory chapters by T.L. Heath, Dover, New York, 1953. Reprinted (translation only) in Great Books of the Western World, Vol. 11, R.M. Hutchins, editor, Encyclopaedia Britannica, Inc., Chicago, 1952.
E.J. Dijksterhuis, Archimedes, Princeton University Press, Princeton, 1987. [1934]
From Tom Rike, Archimedes:
... [the solution] is detailed in Albert H. Beiler’s Recreations in the Theory of Numbers [9]. However, there is a misprint in the book. The value printed for the variable t is actually the value of t2. A truncated version of the problem in prose is also given. The discussion of the problem appears in the chapter entitled The Pellian where you will find out why Pell, who had almost nothing to do with solving this type of equation has his name gloriously attached to it. There is also a fairly clear presentation of the method of solution via continued fractions.
9. Albert H. Beiler. Recreations in the Theory of Numbers. Dover Publications,Inc., 1964.
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