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...ar|n}}. [[Hans Heilbronn]] proved that the average number of iterations of Euclid's algorithm, for fixed {{mvar|n}} and averaged over all choices of [[coprim ...

This result is found as Proposition 36 in Book 3 of [[Euclid]]'s [[Euclid's Elements|''Elements'']]. ...

...evious example where <math>n = 13</math> and <math>k = 5</math> we perform Euclid's algorithm: == Other uses of Euclid's algorithm in music == ...

The '''exterior angle theorem''' is Proposition 1.16 in [[Euclid's Elements]], which states that the measure of an [[exterior angle]] of a [ ...as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem. ...

{{Short description|Modern formulation of Euclid's parallel postulate}} ...om''' is an [[axiom]] that can be used instead of the fifth postulate of [[Euclid]] (the [[parallel postulate]]): ...

[[File:Brides Chair 01.png|thumb|150px|Bride's Chair: The figure used by Euclid to explain the proof of Pythagorean theorem]] ...November 2023}}</ref> The figure appears in Proposition 47 of Book I of [[Euclid's Elements]].<ref>{{cite book |last1=David Wells |title=The Pengu'inDictio. ...

...numbers and Mersenne primes|the theorem on the infinitude of prime numbers|Euclid's theorem}} ...> − 1}} is a [[prime number]]. The theorem is named after mathematicians [[Euclid]] and [[Leonhard Euler]], who respectively proved the "if" and "only if" as ...

...liam J. |date=1929 |title=Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon |jstor=2300175 |journal=The American Mathematical M ...="Evans">{{citation |last=Evans |first=George W. |date=1927 |title=Some of Euclid's Algebra |jstor=27950916 |jstor-access=free |journal=The Mathematics Teach ...

This is almost the same algorithm as [[Euclidean algorithm|Euclid's]] but the remainder <math>p_{i+2}</math> has negative sign. ...

It is Proposition 35 of Book 3 of [[Euclid]]'s [[Euclid's Elements|''Elements'']]. ...

...n relying on the geometric mean theorem.<ref name="HW"/><ref>[[Euclid]]: [[Euclid's Elements|''Elements'']], book II – prop. 14, book VI – prop. 8, ([http:/ ...

...reflected bottom, and its intersection with the line at eye level, follows Euclid's Book II, Proposition 14, which describes how to construct a square with t ...

...m should not be confused with proposition 48 in book 1 of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', the [[Pythagorean theorem#Converse|converse]] of t ...

...=Heath|first=Thomas L.|author-link=T. L. Heath|title=The Thirteen Books of Euclid's Elements|url=https://archive.org/details/thirteenbooksofe00eucl|url-acces ...

...Little Heath]]: ''A History of Greek Mathematics. Volume 1. From Thales to Euclid''. Clarendon Press 1921 (Nachdruck Elibron Classics 2006), S. 225–230 ({{Go ...

...ared the influence of Euler's ''Introductio'' to that of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', calling the ''Elements'' the foremost textbook of ...

...ay&version=1.0&service=UI&handle=euclid.dmj/1077245037&page=record Project Euclid] ...

...ok|last=Hartshorne|first=Robin|authorlink=Robin Hartshorne|title=Geometry: Euclid and beyond|year=2000|publisher=Springer|location=New York, NY [u.a.]|isbn=9 ...

...of areas.{{r|hoyrup}} This was formalized in Book I of [[Euclid's Elements|Euclid's ''Elements'']], which leads with several common notions including "if equ | title = The Thirteen Books of Euclid's Elements ...

{{term|Euclid's lemma}} {{defn|[[Euclid's lemma]] states that if a prime {{math|''p''}} divides the product of two ...