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, 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 prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers ; an example ( there are others ) is described by the flow chart above and as an example in a later section.

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.

It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner ( much as we can take Euclid's parallel postulate as either true or false ).

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.

Little is known about Euclid's life, as there are only a handful of references to him.

Therefore, Euclid's depiction in works of art is the product of the artist's imagination.

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

Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original.

In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

It is possible to object to this interpretation of general relativity on the grounds that light rays might be improper physical models of Euclid's lines, or that relativity could be rephrased so as to avoid the geometrical interpretations.

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor ( GCD ) of two integers, also known as the greatest common factor ( GCF ) or highest common factor ( HCF ).

Euclid's classical lemma can be rephrased as " in the ring of integers every irreducible is prime ".

: 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.

The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite and in Bhaskara's " cyclic method ".

While Greek astronomy — thanks to Alexander's conquests — probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition ; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.

* Codex Nitriensis, a volume containing a work of Severus of Antioch of the beginning of the 9th century is written on palimpsest leaves taken from 6th century manuscripts of the Iliad and the Gospel of St Luke, both of the 6th century, and the Euclid's Elements of the seventh or 8th century, British Museum

But p is coprime to q and therefore to q < sup > n </ sup >, so by ( the generalized form of ) Euclid's lemma it must divide the remaining factor a < sub > 0 </ sub > of the product.

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.

It is frequently noted that Aristotle's logic is unable to represent even the most elementary inferences in Euclid's geometry, but Frege's " conceptual notation " can represent inferences involving indefinitely complex mathematical statements.

" His magnum opus, Ethics, contains unresolved obscurities and has a forbidding mathematical structure modeled on Euclid's geometry.

It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.

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

Also, he stated that Euclid's system of proving geometry theorems was unique to the classical Greeks and did not evolve similarly in other mathematical cultures in China, India, and Arabia.

Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places.

A formal system need not be mathematical as such, Spinoza's Ethics for example imitates the form of Euclid's Elements.

Barozzi translated many works of the ancients, including Proclus ’ s edition of Euclid's Elements ( published in Venice in 1560 ), as well as mathematical works by Hero, Pappus of Alexandria, and Archimedes.