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

This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

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.

Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions ( theorems ) from these.

Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called Euclidean motions, which include translations and rotations of figures.

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

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.

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.

* Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time.

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

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.

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.

Hilbert's system consisting of 20 axioms < ref > a 21 < sup > st </ sup > axiom appeared in the French translation of Hilbert's Grundlagen der Geometrie according to most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs.

However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2, 000 years.

* 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 clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed entirely implausible: it was one of Tycho Brahe's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere ( the fixed stars ).

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.

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.

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.

A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

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.

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.

Playfair's axiom is fundamental since comparative measures of angle size are foreign to affine geometry so that that Euclid's parallel postulate is beyond the scope of pure affine 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.

This sum contains a term a < sub > r </ sub > b < sub > s </ sub > which is not divisible by p ( because p is prime, by Euclid's lemma ), yet all the remaining ones are ( because either or ), so the entire sum is not divisible by p. But by assumption all coefficients in the product are divisible by p, leading to a contradiction.

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.

Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ.

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.

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.

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.

Euclid's fifth is a rule in Euclidean geometry which states ( in John Playfair's reformulation ) that for any given line and point not on the line, there is one parallel line through the point not intersecting the line.

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.

* Niccolò Fontana Tartaglia publishes a translation of Euclid's Elements into Italian, the first into any modern European language.

:" Because the proofs which we shall use in almost the entire work deal with straight lines and arcs, with plane and spherical triangles and because Euclid's Elements, although they clear up much of this, do not have what is here most required, namely, how to find the sides from angles and the angles from the sides ... there has accordingly been found a method whereby the lines subtending any arc may be known.

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.

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

The theorem extends Euclid's theorem that there are infinitely many prime numbers.

The direct theorem was Proposition 22 in Book 3 of Euclid's Elements.

# REDIRECT Euclid's theorem

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.

In number theory, Euclid's lemma ( also called Euclid's first theorem ) is a lemma that captures one of the fundamental properties of prime numbers.

Euclid's proof of the fundamental theorem of arithmetic is a simple proof using a minimal counterexample.

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.