* D. E. Joyce's presentation of Euclid's Elements

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.

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.

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.

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

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.

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

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.

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.

The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work Elements was written.

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.

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.

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.

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

The first recorded use of the Greek word translated trapezoid ( τραπέζοειδη, trapézoeide, " table-like ") was by Marinus Proclus ( 412 to 485 AD ) in his Commentary on the first book of Euclid's Elements.

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 ).

The word was first anglicized and used in the magical sense in John Dee's book Mathematicall Praeface to Euclid's Elements ( 1570 ).

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.

For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students.

Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that were the most important.

This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

Newton was convinced to read the Elements again with care, and formed a more favourable estimate of Euclid's merit.

Copernicus, to an extent unachieved by Ptolemy, approximated to Euclid's vision.

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.

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.

* Proclus, A commentary on the First Book of Euclid's Elements, translated by Glenn Raymond Morrow, Princeton University Press, 1992.

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

Thus the division form of Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the remainder obtained by dividing the larger number by the smaller number.

: 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.

( Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.

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.

Roger Ascham thought that his pupil Robert had an uncommon talent for languages and writing, " exceed almost all other by nature ", and regretted that he had done himself harm by preferring " Euclid's pricks and lines " ( mathematics ).

) More formally, the only permissible constructions are those granted by Euclid's first three postulates.

However, by Proposition 2 of Book 1 of Euclid's Elements, no computational power is lost by using such a collapsing compass ; there is no need to transfer a distance from one location to another.

The frontispiece of an Adelard of Bath Latin translation of Euclid's Elements, c. 1309 – 1316 ; the oldest surviving Latin translation of the Elements is a 12th-century translation by Adelard from an Arabic version.

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

Euclid's idea of a line is perhaps clarified by the statement " The extremities of a line are points ," ( Def.

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.

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

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

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.

" 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 ( Stoicheia ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC.

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.