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 axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.

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.

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 ).

* Euclid The thirteen books of Euclid's Elements ( Cambridge: Cambridge University Press, 1908 )

Mathematicians and philosophers, such as Bertrand Russell, Alfred North Whitehead, and Baruch Spinoza, have attempted to create their own foundational " Elements " for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced.

Cubism, with its new geometry, its dynamism and multiple view-point perspective, not only represented a departure from Euclid's model, but it achieved, according to Gleizes and Metzinger, a better representation of the real world: one that was mobile and changing in time.

This characterizes Euclid's formulation of geometry, but not Ptolemy's astronomy.

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.

* Euclid's Elements, the mathematical treatise on geometry and number theory

On the surface of a sphere there are no parallel line s. Euclid's Elements contained five postulates that form the basis for Euclidean geometry.

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.

Euclid's construction for proof of the triangle inequality for plane geometry.

The existence of non-Euclidean geometries impacted the " intellectual life " of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements.

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

In the Middle Ages he was known for his rediscovery and teaching of geometry, earning his reputation when he made the first full translation of Euclid's " Elements " and began the process of interpreting the text for a Western audience.

Posidonius was one of the first people to attempt to prove Euclid's fifth postulate of geometry.

* 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.

This method reached its high point with Euclid's Elements ( 300 BC ), monumental treatise on geometry structured with very high standards of rigor: each proposition is justified by a demonstration in the form of chains of syllogisms ( though they do not always conform strictly to Aristotelean templates ).

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.

:" 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.

The mathematician Julius Petersen tried to teach him some of Euclid's theorems, but gave up the task once he realized that their comprehension was beyond Dase's capabilities.

Continuing this procedure ends up in a variant of the Euclid's algorithm.

" The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.

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

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.

Up until that time, Euclid's Elements had remained the standard textbook used in British schools.