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.

This definition of proportion forms the subject of Euclid's Book V.

Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus.

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

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

* 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 ).

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 ).

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

One can argue that Euclid's axioms were arrived upon in this manner.

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.

It can also be proven that none of these factors obeys Euclid's lemma ; e. g.

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

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 ".

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.

Among others these tweaks will have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded.

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.

We can prove the cancellation law easily using Euclid's lemma, which generally states that if an integer b divides a product rs ( where r and s are integers ), and b is relatively prime to r, then b must divide s. Indeed, the equation

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all.

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.

A prime factor can be visualized by understanding Euclid's geometric position.

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.

Lobachevsky replaced Euclid's parallel postulate with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part ; a famous consequence is that the sum of angles in a triangle must be less than 180 degrees.

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.

The classic example is that of Euclid's ( see Euclid's Elements ) geometry ; its hundreds of propositions can be deduced from a set of definitions, postulates, and common notions: all three of which constitute first principles.

An example of using Euclid's algorithm will be shown below.

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.

Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also taken to be unique.

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.

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.

#* 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 >

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.

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.

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 ").

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 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.

* Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the extended Euclidean algorithm to work

Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.

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.

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 case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians.

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.

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.

In geometry, Pasch's theorem, stated in 1882 by a German mathematician Moritz Pasch, is a result of plane geometry which cannot be derived from Euclid's postulates.

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.

It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers.