Archimedes's cattle problem

Smallest solution to Archimedes's cattle problem with each icon representing around 10206543 cattle

Archimedes's cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773.[1]

The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by Carl Ernst August Amthor [de] (1845–1916), headmaster of the Gymnasium zum Heiligen Kreuz (Gymnasium of the Holy Cross) in Dresden, Germany.[2][3][4] Using logarithmic tables, he calculated the first digits of the smallest solution, showing that it is about 7.76×10206544 cattle, far more than could fit in the observable universe.[5] The decimal form is too long for humans to calculate exactly, but multiple-precision arithmetic packages on computers can write it out explicitly.

  1. ^ Lessing, Gotthold Ephraim (1773). Zur Geschichte und Litteratur: aus den Schätzen der Herzoglichen Bibliothek zu Wolfenbüttel, Zweyter Beytrag [On History and Literature: from the treasures of the ducal library at Wolfenbüttel, second article] (in German and Greek). Braunschweig, (Germany): Fürstlicher Waysenhaus. pp. 421–425. From pp. 422–423: "Denn, wie gesagt, das Problem soll, wenn es nicht von dem Archimedes selbst abgefaßt worden, doch von ihm für werth erkannt seyn, daß er es den Eratosthenes geschicket hätte, um es den Meßkünstern zu Alexandria zur Auflösung vorzulegen. Dieses besagt die Aufschrift; ..." (For, as said [above], the problem [Greek: ΠΡΟΒΛΗΜΑ] shall, if it had not been composed by Archimedes [Greek: Α'ΡΧΙΜΗΔΗΣ] himself, yet have been recognized by him [as so] worthy that he would have sent it to Eratosthenes [Greek: ΕΡΑΤΟΣΘΕΝΗΝ], in order to submit it to the surveyor at Alexandria for solution. The title says this; ...) See pages 423–424 (in Greek).
  2. ^ Krumbiegel, B.; Amthor, A. (1880). "Das Problema bovinum des Archimedes" [The cattle problem of Archimedes]. Zeitschrift für Mathematik und Physik: Historisch-literarische Abtheilung [Journal for Mathematics and Physics: Historical-literary section] (in German, Greek, and Latin). 25: 121–136, 153–171.
  3. ^ Biographical information about August Amthor:
    • Amthor's full name appears in: (School administration) (1876). Programm des Gymnasiums zum Heiligen Kreuz in Dresden [Program of the Gymnasium of the Holy Cross in Dresden] (in German). Dresden, Germany: K. Blochmann und Sohn. p. 31.
    • Brief biography about Amthor appears in: Singer, Isadore; de Leon, Edward Warren, eds. (1910). "Amthor, August (Ph.D.)". International Insurance Encyclopedia. Vol. 1. New York, New York, USA: American Encyclopedic Library Association. p. 18.
  4. ^ The problem was solved independently in 1895 by Adam Henry Bell, a surveyor and civil engineer of Hillsboro, Illinois, USA. See:
    • Bell, A.H. (1895). "On the celebrated 'Cattle Problem' of Archimedes". The Mathematical Magazine. 2: 163–164.
    • Bell, A.H. (1895). "The 'Cattle Problem' by Archimedes 251 B.C". American Mathematical Monthly. 2: 140–141.
    • Bell's full name appears in: Bateman, Newton; Selby, Paul, eds. (1918). "Fish, Albert E.". Historical Encyclopedia of Illinois. Vol. 2. Chicago, Illinois, USA: Munsell Publishing Co. pp. 1049–1050.; see p. 1050.
    • Bell's occupations appear in: Merriman, Mansfield (November 1905). "The cattle problem of Archimedes". Popular Science Monthly. 67: 660–665.; see p. 664.
  5. ^ Lenstra, H. W. Jr. (2002), "Solving the Pell Equation" (PDF), Notices of the American Mathematical Society, 49 (2): 182–192, MR 1875156