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

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, All thirteen books, with interactive diagrams using Java.

* 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.

A proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides.

* Euclid's Elements, a 13-book mathematical treatise written by Euclid, that includes both geometry and number theory

A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides.

E. T. Bell in his book Men of Mathematics wrote about Lobachevsky's influence on the following development of mathematics: The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other ' axioms ' or accepted ' truths ', for example the ' law ' of causality which, for centuries, have seemed as necessary to straight thinking as Euclid's postulate appeared till Lobatchewsky discarded it.

While Nicomachus ' algorithm is the same as Euclid's, when the numbers are prime to one another it yields the number " 1 " for their common measure.

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane 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.

By Euclid's lemma p < sub > 1 </ sub > must divide one of the q < sub > j </ sub >; relabeling the q < sub > j </ sub > if necessary, say that p < sub > 1 </ sub > divides q < sub > 1 </ sub >.

#* 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 >

In classical geometry, a proposition may be a construction that satisfies given requirements ; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.

This proof appears in Euclid's Elements, Book 1, Proposition 20.

However, by Proposition 2 of Book 1 of Euclid's Elements, no computational power is lost by using such a collapsing compass ; there is no need to transfer a distance from one location to another.

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.

By Euclid's lemma it divides one of the factors or, implying that x is congruent to either 1 or − 1 modulo p.

The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica ( 1797 ) to describe two numbers whose product is 1 ; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements.

For an example of homogeneity, note that Euclid's proposition I. 1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way.

* Book 1 contains Euclid's 10 axioms ( 5 named postulates — including the parallel postulate — and 5 named axioms ) and the basic propositions of geometry: the pons asinorum ( proposition 5 ), the Pythagorean theorem ( Proposition 47 ), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are " equal " ( have the same area ).

Pons asinorum ( Latin for " bridge of asses ") is the name given to Euclid's fifth proposition in Book 1 of his Elements of geometry, also known as the theorem on isosceles triangles.

Although best known for its geometric results, the Elements also includes number theory.

In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

Later in his career he went to the United States of America and wrote Elements of Civil Engineering which includes much information on canal building.

The David Adler Collection includes photographs, research files and materials collected and produced by the museum's Department of Architecture for the 2001 exhibition " David Adler, Architect: The Elements of Style.

The lowest tier or layer of the EKMS architecture which includes the AN / CYZ-10 ( Data Transfer Device ( DTD ), the SKL ( Simple Key Loader ) AN / PYQ-10, and all other means used to fill keys to End Cryptographic Units ( ECUs ); hard copy material holdings only ; and STU-III / STE material only using Key Management Entities ( KMEs ) ( i. e., Local Elements ( LEs )).

Unicode includes 128 such characters: The adjacent Block Elements table contains 32 block element, shade and terminal graphics characters.

His considerable body of chamber music, often written expressly for a performer's individual abilities, includes solos for clarinet, drumkit, cello, flute, piccolo, guitar, violin, viola, and accordion, alongside six string quartets ( 1983, 1991, 1998, 2004, 2008, 2010 ), the five-part Book of Elements for piano ( 1997-2002 ) and the soadie waste for piano and string quartet ( 2002 / 3 ).