Oresme's proof of the density of rotations of a circle through an irrational angle

Abstract

One of the theorems of Nicole Oresme's (ca. 1320-1382) says that, for two points moving uniformly but incommensurably along a circle, “no sector of a circle is so small that two such mobiles could not conjunct in it at some future time, and could not have conjuncted in it at some time.” A detailed study of his proof of this and related theorems shows that he was in the possession of all the arguments needed for the proof to be conclusive. Nicole Oresme (ca. 1320-1382) gibt folgenden Satz. Für zwei Punkte, die sich gleichförmig, aber inkommensurabel auf einmm Kreis bewegen, gilt, daβ “kein Sektor des Kreises so klein ist, daβ nicht zwei solche bewegliche Punkte einmal in der Zukunft in Konjunktion stehen könnten, und einmal in Konjunktion hätten stehen können.” Eine genaue Untersuchung von Oresmes Beweis dieses Satzes and einiger damit zusammenhängender Sätze zeigt, daβ er alle Argumente zur Verfügung hatte, die man für einen schlüssigen Beweis braucht. Nicole Oresme (ca. 1320-1382) enuncia il teorema secondo il quale, dati due punti in movimento rotatorio uniforme ma incommensurabile, “non vi e alcun settore del cerchio tanto piccolo, the tali due mobili non possano congiungersi in futuro, e non avrebbero potuto farlo.” Lo studio dettagliato della prova di questo teorema e di alcuni altri connessi dimostra the Oresme possedeva tutti gli argumenti necessari per una prova conclusiva.