The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry ; Euclid himself wrote eight more advanced books on geometry.

It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.

Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight ; he related that he said to himself, " You never can make a lawyer if you do not understand what demonstrate means ; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight ".

He decides he should flee to the town of Euclid, where his uncle lives ; however, Chester refuses because he wants to stay behind and bury the dead first, and Cress finally decides to go by himself.

Euclid himself wrote six dialogues — the Lamprias, the Aeschines, the Phoenix, the Crito, the Alcibiades, and the Amatory dialogue — but none survive.

" However, Euclid himself taught logic, and his pupil, Eubulides, who was famous for employing celebrated paradoxes, was the teacher of several later dialecticians.

He is known to have studied Euclid and to have translated the work of Al Battani and Avicenna, and it seems that he would not have made the translation for which he is famous, that of the Qur ' an, without the encouragement of the French Abbot Peter the Venerable, who wished to have access to Islamic texts.

The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention.

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

In addition to the Elements, at least five works of Euclid have survived to the present day.

Other works are credibly attributed to Euclid, but have been lost.

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

Under the instruction of Cosmas, who also taught John's orphan friend ( the future St. Cosmas of Maiuma ), John is said to have made great advances in music, astronomy and theology, soon rivalling Pythagoras in arithmetic and Euclid in geometry.

Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of the quotients of incommensurables, so such a definition would have been meaningless to him.

Zoning laws are, of course, the classic example, see Hadacheck v. Sebastian, 239 U. S. 394 ( 1915 ) ( prohibition of brickyard operations within certain neighborhoods ); Village of Euclid, Ohio v. Ambler Realty Co., 272 U. S. 365 ( 1926 ) ( prohibition of industrial use ); Gorieb v. Fox, 274 U. S. 603, 608 ( 1927 ) ( requirement that portions of parcels be left unbuilt ); Welch v. Swasey, 214 U. S. 91 ( 1909 ) ( height restriction ), which have been viewed as permissible governmental action even when prohibiting the most beneficial use of the property.

While Euclid was the originator of what we now understand as the published geometric proof, Pythagoras created a closed community and suppressed results ; he is even said to have drowned a student in a barrel for revealing the existence of irrational numbers.

Propositions 30 and 32 together are essentially equivalent to the fundamental theorem of arithmetic stating that every positive integer can be written as a product of primes in an essentially unique way, though Euclid would have had trouble stating it in this modern form as he did not use the product of more than 3 numbers.

It has been argued that, given some differences between the two models, it is more likely that Copernicus could have taken the ideas found in the Tusi couple from Proclus's Commentary on the First Book of Euclid.

Euclid is known to have assumed the commutative property of multiplication in his book Elements.

First, some have objected to applying the name " mathematics " to subject matter that is not developed abstractly and logically, with proofs, as in the academic tradition descended from Hellenistic Greeks like Pythagoras, Euclid, and Archimedes and comparable traditions in China, Japan, and India.

With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem.

Functions in Euclid are closed scopes, may not have side effects, and must explicitly declare imports.

Socrates is also supposed to have reproved Euclid for his fondness for eristic disputes.

Euclid described a line as " breadthless length ", and introduced several postulates as basic unprovable properties from which he constructed the geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenth century ( such as non-Euclidean geometry, projective geometry, and affine geometry ).

Euclid references some of Autolycus ' work, and Autolycus is known to have taught Arcesilaus.

In fact, the only relation that the historians have been able to trace it to is with Euclid, who again came many centuries after Pythagoras!

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.

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

" Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn't realize was equivalent to his own postulate.

The Saccheri quadrilateral was first considered by Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.

Khayyám then considered the three cases ( right, obtuse, and acute ) that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he ( correctly ) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.

Cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property.

Laruelle's non-philosophy, he claims, should be considered to philosophy what non-Euclidean geometry is to the work of Euclid.

The ancient peoples who are considered the first scientists may have thought of themselves as natural philosophers, as practitioners of a skilled profession ( for example, physicians ), or as followers of a religious tradition ( for example, temple healers ). The encyclopedic works of Aristotle, Archimedes, Hippocrates, Galen, Ptolemy, Euclid, and others spread throughout the world.

It was considered essential for the theaters ' marquees to face Euclid Avenue, but because of space constraints the State Theatre was built at the back of the lot, although its lobby shares the Euclid frontage with the Ohio Theatre.

Residents of South Euclid eventually wanted autonomy from the larger Euclid Township, and voted on October 13, 1917, to be incorporated as a village, with Edward C. Foote being elected the first mayor a few weeks on November 6.

A large earth mover which was being widely used in the construction of the M1, the UK's first motorway, the Euclid factory was only two miles from Corgi headquarters which allowed easy access to all the data required to produce a very accurate model.

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.

Mixing these two ideas, Euclid claimed that good is the knowledge of this being.

When geometry was first formalised by Euclid in the Elements, he defined lines to be " breadthless length " with a straight line being a line " which lies evenly with the points on itself ".

In fact, Euclid did not use these definitions in work and probably included them just to make it clear to the reader what was being discussed.

In 1893, the neighborhood began being served by electric streetcars, which ran along Westcott Street and Euclid Avenue towards downtown Syracuse.

Prior to introduction of the MetroCard, the Euclid Avenue-bound platform had a same-level fare control and a paper bus-style transfer was handed out at either IND fare control areas ( the only place in the subway where this was still being done ) for access to the shuttle station only.

Provides service between the north and south sides of the CWRU campus from 5: 15pm – 12: 30am Provides service from University Circle to Coventry Village from 5: 15pm – 12: 30am Provides service around the CWRU campus and the University Hospitals of Cleveland Connects the Urban Child Research Center with the main Case campus The new BRT HealthLine, which opened on October 24, 2008, is the newest option to the neighborhood, being a major destination on the line along Euclid Avenue that connects Public Square to Louis Stokes Station at Windermere in East Cleveland.