Matteo Ricci ( left ) and Xu Guangqi ( right ) in the Chinese edition of Euclid's Elements published in 1607.

Euclid's The Elements includes the following " Common Notion 1 ":

Proclus introduces Euclid only briefly in his fifth-century Commentary on the Elements, as the author of Elements, that he was mentioned by Archimedes, and that when King Ptolemy asked if there was a shorter path to learning geometry than Euclid's Elements, " Euclid replied there is no royal road to geometry.

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 ( Papyrus Oxyrhynchus 29 | P. Oxy.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

* Proclus, A commentary on the First Book of Euclid's Elements, translated by Glenn Raymond Morrow, Princeton University Press, 1992.

* Euclid's Elements, with the original Greek and an English translation on facing pages ( includes PDF version for printing ).

* Euclid's Elements, books I-VI, in English pdf, in a Project Gutenberg Victorian textbook edition with diagrams.

* Euclid's Elements, All thirteen books, in several languages as Spanish, Catalan, English, German, Portuguese, Arabic, Italian, Russian and Chinese.

Includes editions and translations of Euclid's Elements, Data, and Optica, Proclus's Commentary on Euclid, and other historical sources.

In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of theorems stated in the Elements.

* Heath's authoritative translation of Euclid's Elements plus his extensive historical research and detailed commentary throughout the text.

* Euclid's Elements, the mathematical treatise on geometry and number theory

The earliest surviving description of the Euclidean algorithm is in Euclid's Elements ( c. 300 BC ), making it one of the oldest numerical algorithms still in common use.

Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem.

The proof uses Euclid's lemma ( Elements VII, 30 ): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b ( or perhaps both ).

Illustration at the beginning of a medieval translation of Euclid's Element ( mathematics ) | Elements, ( c. 1310 )

A proof from Euclid | Euclid's Euclid's Elements | Elements, widely considered the most influential textbook of all time.

( Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.

In a work titled Euclides ab Omni Naevo Vindicatus ( Euclid Freed from All Flaws ), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility ( some others of Euclid's axioms must be modified for elliptic geometry to work ) and set to work proving a great number of results in hyperbolic geometry.

* Euclid's Elements – All thirteen books in one volume, Based on Heath's translation, Green Lion Press ISBN 1-888009-18-7.

* Clay Mathematics Institute Historical Archive – The thirteen books of Euclid's Elements copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD

* Kitāb Taḥrīr uṣūl li-Ūqlīdis Arabic translation of the thirteen books of Euclid's Elements by Nasīr al-Dīn al-Ṭūsī.

* Euclid The thirteen books of Euclid's Elements ( Cambridge: Cambridge University Press, 1908 )

*" About the translator: Thomas L. Heath " in Euclid's Elements: all thirteen books complete in one volume ( 2002 ) Green Lion Press.

* Heath: The thirteen books of Euclid's elements Preface

It considers the pedagogic merit of thirteen contemporary geometry textbooks, demonstrating how each in turn is either inferior to or functionally identical to that of Euclid's Elements.

According to Pappus, " Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics.

Euclid's Elements ( Stoicheia ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC.

It has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's Elements.

A graphical expression on Euclid's algorithm using example with 1599 and 650.

His friend Farkas Wolfgang Bolyai with whom Gauss had sworn " brotherhood and the banner of truth " as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

Pappus believed these results to be important in astronomy and included Euclid's Optics, along with his Phaenomena, in the Little Astronomy, a compendium of smaller works to be studied before the Syntaxis ( Almagest ) of Claudius Ptolemy.

* Porisms might have been an outgrowth of Euclid's work with conic sections, but the exact meaning of the title is controversial.

Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space ( as in elliptic geometry ), and all five axioms are consistent with a variety of topologies ( e. g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry ).

In its simplest form, Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the difference between the larger and smaller numbers.

The simplest form of Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the difference between the larger and smaller numbers.

Thus the division form of Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the remainder obtained by dividing the larger number by the smaller number.

: This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number.

As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors.

An important part of the book is concerned with Euclid's famous parallel postulate, which attracted the interest of Thabit ibn Qurra.

To start with, none of the terms a, 2a, ..., ( p − 1 ) a can be congruent to zero modulo p, since if k is one of the numbers 1, 2, ..., p − 1, then k is relatively prime with p, and so is a, so Euclid's lemma tells us that ka shares no factor with p. Therefore, at least we know that the numbers a, 2a, ..., ( p − 1 ) a, when reduced modulo p, must be found among the numbers 1, 2, 3, ..., p − 1.

He became so obsessed with Euclid's parallel postulate that his father wrote to him: " For God's sake, I beseech you, give it up.

Herman translated Euclid's Elements around 1140, possibly in collaboration with Robert of Ketton.

After Simson's death, restorations of Apollonius's treatise De section determinata and of Euclid's treatise De Porismatibus were printed for private circulation in 1776, at the expense of Earl Stanhope, in a volume with the title Roberti Simson opera quaedam reliqua.