For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers ' common measure is in fact the greatest.

In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms.

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 the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.

" It is further believed that Euclid may have studied at Plato's Academy in Athens.

There is no mention of Euclid in the earliest remaining copies of the Elements, and most of the copies say they are " from the edition of Theon " or the " lectures of Theon ", while the text considered to be primary, held by the Vatican, mentions no author.

In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye.

It is likely that the first four books of Apollonius's work come directly from Euclid.

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, e. g., a 45-degree angle would be referred to as half of a right angle.

Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as " if the line is extended to a sufficient length ," although he occasionally referred to " infinite lines.

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite ( see below ) and what its topology is.

212 BCE ), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians.

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

The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle ; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex.

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

In particular, it is thought that Euclid felt the parallel postulate was forced upon him, as indicated by his reluctance to make use of it, and his arrival upon it by the method of contradiction.

The term “ Euclidean ” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.

This work, which contained only the first six and the eleventh and twelfth books, and to which, in its English version, he added the Data in 1762, was for long the standard text of Euclid in England.

Frankie Yankovic, a Slovenian from Cleveland Ohio traveled the world playing his version (" Cleveland Style " as per Polka Hall of Fame, Euclid Ohio ) of the Viennese Waltzes.

The Arabs received the Elements from the Byzantines in approximately 760 ; this version, by a pupil of Euclid called Proclo, was translated into Arabic under Harun al Rashid c. 800.

Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem.

" At the very end we learn that the child's name is Euclid and he pulls out a small version of the platohedron showing that the destruction of the world was metaphysical in nature and that the new world of Euclid and Ptolemy I Soter has replaced the old world of Aristotle and Plato.

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

The city hosted such leading lights as the mathematician Euclid and anatomist Herophilus ; constructed the great Library of Alexandria ; and translated the Hebrew Bible into Greek ( called the Septuagint for it was the work of 70 translators ).

300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the " Father of Geometry ".

It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.

Long before the first photographs were made, Chinese philosopher Mo Di and Greek mathematicians Aristotle and Euclid described a pinhole camera in the 5th and 4th centuries BC.

* Proclus ' Commentary on Euclid, Book I. PDF scans of Friedlein's Greek edition, now in the public domain ( Classical Greek )

" The phrase was used by many early Greek mathematicians, including Euclid and Archimedes.

Catholics charge that modalistic monarchianism has its origin by means of influence in Greek pagan philosophy, including pagan philosophers like Euclid and Aristotle, who based their logic on Monism and Aristotle's arguments around his concept energeia ( i. e. energy ) called metaphysics.

Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Euclid replacing such proofs with ones using other techniques such as the method of exhaustion.

* 300 Euclid, Greek mathematician, publishes Elements, treating both geometry and number theory ( see also Euclidean algorithm ).

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

The discoveries of several Greek mathematicians, including Pythagoras, Euclid, and Archimedes, are still used in mathematical teaching today.

* Euclid, Greek mathematician who will come to live in Alexandria ( d. c. 275 BC )

He edited works of Euclid, Apollonius, Archimedes, and other Greek mathematicians.

Mathematical proofs were revolutionized by Euclid ( 300 BCE ), who introduced the axiomatic method still in use today, starting with undefined terms and axioms ( propositions regarding the undefined terms assumed to be self-evidently true from the Greek " axios " meaning " something worthy "), and used these to prove theorems using deductive logic.

Euclid uses the Greek ἀναλόγον ( analogon ), this has the same root as λόγος and is related to the English word " analog ".

Euclidean ( or, less commonly, Euclidian ) relates to Euclid ( an ancient Greek mathematician ), a town or others.

In 1797, Moses Cleaveland named the area east of the Cuyahoga River Euclid, after the Greek mathematician and patron saint of surveyors.

Mentor is named after the Greek figure Mentor, in keeping with the Connecticut Western Reserve settlers ' tradition, as well as that of most other Americans at the time, of celebrating aspects of Greek classicism ( nearby Solon, Macedonia, Euclid, and Akron also were named using that principle ).