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

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 Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms.

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

The most notorious of the postulates is often referred to as " Euclid's Fifth Postulate ," or simply the " parallel postulate ", which in Euclid's original formulation is:

Unfortunately, Euclid's original system of five postulates ( axioms ) is not one of these as his proofs relied on several unstated assumptions which should also have been taken as axioms.

( Euclid's original definition and some English dictionaries ' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.

However, Euclid's original proof of this proposition is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.

The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.

In an axiomatic formulation of Euclidean geometry, such as that of Hilbert ( Euclid's original axioms contained various flaws which have been corrected by modern mathematicians ), a line is stated to have certain properties which relate it to other lines and points.

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

The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e. g., the proof of the infinitude of primes.

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 ).

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

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 ".

#* 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 construction for proof of the triangle inequality for plane geometry.

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

Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions.

Because if not, then an elementary proof of Euclid's result is also impossible.

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

His proof is in Euclid's Elements Book X Proposition 9.

In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture and participated in the Cunningham project.

It is widely believed, but false, that the idea of primorial primes appears in Euclid's proof of the infinitude of the prime numbers: First, assume that the first n primes are the only primes that exist.

In fact, Euclid's proof did not assume that a finite set contains all primes that exist.

The pons asinorum in Byrne's edition of the Elements showing part of Euclid's proof.

The easiest proof of Euclid's lemma uses another lemma called Bézout's identity.

The logic of this proof is basically Euclid's, but the notation and some of the concepts ( zero, negative ) would be foreign to him.

Much like Euclid's first and third definitions and Plato's ' beginning of a line ', the Mo Jing stated that " a point may stand at the end ( of a line ) or at its beginning like a head-presentation in childbirth.

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.

Absolute geometry assumes the first four of Euclid's Axioms ( or their equivalents ), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms.

The usage primarily comes to us from translations of Euclid's Elements, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another.

Much like Euclid's first and third definitions and Plato's ' beginning of a line ', the Mo Jing stated that " a point may stand at the end ( of a line ) or at its beginning like a head-presentation in childbirth.

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.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Euclid's method for finding the greatest common divisor ( GCD ) of two starting lengths BA and DC, both defined to be multiples of a common " unit " length.

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 ).

Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers.

Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates ( which include, for example, " Between any two points a straight line may be drawn ").

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.

* Euclid's lemma: if a prime number divides a product of two numbers, then it divides at least one of those two numbers.

Spherical geometry obeys two of Euclid's postulates: the second postulate (" to produce a finite straight line continuously in a straight line ") and the fourth postulate (" that all right angles are equal to one another ").

With this identification of antipodal points, the model satisfies Euclid's first postulate, which states that two points uniquely determine a line.

For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry.

He therefore bought an English edition of Euclid's Elements which included an index of propositions, and, having turned to two or three which he thought might be helpful, found them so obvious that he dismissed it " as a trifling book ", and applied himself to the study of René Descartes ' Geometry.

Euclid's algorithm can be used to determine whether two integers are coprime without knowing their prime factors ; the algorithm runs in a time that is polynomial in the number of digits involved.

See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers.

His works include Rekhaganita, a translation of Euclid's Elements made from Nasir al-Din al-Tusi's Arabic recension of same ; Siddhantasarakaustubha, a translation of the Almagest from Arabic ; and two works on astronomical instruments such as the astrolabe, Siddhanta-samrat and Yantraprakara, which also record astronomical observations made by Jagannatha.

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