Archimedes, Axiom
المؤلف:
Boyer, C. B. and Merzbach, U. C
المصدر:
A History of Mathematics, 2nd ed. New York: Wiley
الجزء والصفحة:
...
21-10-2019
3352
Archimedes' Axiom
Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.
Symbolically, the axiom states that
iff the appropriate one of following conditions is satisfied for integers 
and 
:
1. If 
, then 
.
2. If 
, then 
.
3. If 
, then 
.
Formally, Archimedes' axiom states that if 
and 
are two line segments, then there exist a finite number of points 
, 
, ..., 
on 
such that
and 
is between 
and 
(Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry.
REFERENCES:
Boyer, C. B. and Merzbach, U. C. A History of Mathematics, 2nd ed. New York: Wiley, pp. 89 and 129, 1991.
Itô, K. (Ed.). §155B and 155D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 611, 1986.
Stolz, O. "Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes." Math. Ann. 22, 504-520, 1883.
Stolz, O. "Über das Axiom des Archimedes." Math. Ann. 39, 107-112, 1891.
Veronese, G. "Il continuo rettilineo e l'assioma cinque d'Archimede." Atti della Reale Accademia dei Lincei Ser. 4, No. 6, 603-624, 1890.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
