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{{r|Euclid's lemma}} {{r|Euclid}}

{{r|Euclid's Elements}}

A result about unique factorization appeared in book IX of [[Euclid]]'s [[Euclid's Elements|Elements]]. However, it did not apply to all whole numbers, but ...and proof this property to [[Carl Friedrich Gauss]] about 2000 years after Euclid. A precise statement and proof may be found on the "advanced" subpage.

{{r|Euclid's lemma}}

...the study and teaching of mathematics in the West. Also in other fields [[Euclid]]'s work led the way. The philosopher [[Baruch Spinoza|Spinoza]] wrote his {{Image|Euclids parallel axiom.png|right|250px|Euclid's parallel axiom. In the upper figure the angles α and β are equ

{{r|Euclid}}

{{r|Euclid}}

{{r|Euclid}}

{{r|Euclid}}

...first described by [[Euclid]] more than two thousand years ago in his ''[[Euclid's Elements|The Elements]]'' ...artesian coordinates]], to describe points in the plane and in space. In Euclid's geometry there is no origin, all points are equal.

...jor street surveyed, [[Euclid Avenue]], was named for the Greek geometer [[Euclid]]. Cleaveland and his crew of surveyors departed after completing their wo

...Suppose it is desired to find the smallest common multiple of 63 and 77. Euclid's algorithm tells us that the greatest common divisor of 63 and 77 is 7. T

...e plane and the geometry are named after the ancient-Greek mathematician [[Euclid]].

...algorithm''', named after the ancient Greek geometer and number-theorist [[Euclid]], is an [[algorithm]] for finding the [[greatest common divisor]] (gcd) of ..., rather than only the remainders, in the divisions we did while executing Euclid's algorithm, we can find ''x'' and ''y''. Here is how:

[[Euclid]] states it — somewhat disguised — in his ''[[Elements]]'' as h Since this statement is much less natural or evident than Euclid's other axioms and postulates,

...factors of a number are large, the algorithm above may be inefficient. [[Euclid's algorithm]] does not involve prime factorizations and runs fast in such c

T. Heath, A history of Greek mathematics, Vol. I: From Thales to Euclid, Dover, New York, 1981.

...hing between [[axiom]]s (and postulates), definitions, and [[theorem]]s. [[Euclid]], a Greek mathematician living in [[Alexandria]] about 300 BC wrote a 13-v

{{r|Euclid}}

...'s elements]]; it greatly rearranged and simplified the propositions of [[Euclid]]'s classical work. Legendre's ''Elements of Geometry'' was to become the