: Foundations of Geometry ) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting the traditional axioms of Euclid.

The few historical references to Euclid were written centuries after he lived, by Proclus and Pappus of Alexandria.

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.

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.

* Several works on mechanics are attributed to Euclid by Arabic sources.

These three works complement each other in such a way that it has been suggested that they are remnants of a single treatise on mechanics written by Euclid.

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.

Near the beginning of the first book of the Elements, Euclid gives five postulates ( axioms ) for plane geometry, stated in terms of constructions ( as translated by Thomas Heath ):

Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it.

Euclid often used proof by contradiction.

Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.

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.

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.

With the opening ( and re-opening of previously Jart controlled ) gates, the human migrants gained access to remarkable technologies and abilities, and made possible habitation within the Way, and on the Flaw itself, in the newly created Axis Thoreau and Axis Euclid at 1. 5 ex 7 ( 15 million km ).

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

It brings to mind the famous fifth postulate of Euclid concerning parallels, which is more extensive and complicated in its wording than the other postulates.

The resulting constitutions are prefaced by a history more extensive than any before, again tracing the history of what was now freemasonry back to biblical roots, again forging Euclid into the chain.

Euclid does not go beyond a third measuring and gives no numerical examples.

Euclid ’ s algorithm appears as Proposition II in Book VII (" Elementary Number Theory ") of his Elements.

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 his work Prodromo dell ' Arte Maestra ( 1670 ) he proposes a lighter-than-air vessel based on logical deductions from previous work ranging from Archimedes and Euclid to his contemporaries Robert Boyle and Otto von Guericke.

Here Dürer favours the methods of Ptolemy over Euclid.

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

They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion.

The axioms unify both the plane geometry and solid geometry of Euclid in a single system.

A clear exposition of the " errors " of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry.

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

Euclid ( ; Eukleidēs ), fl.

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

Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

In the only other key reference to Euclid, Pappus briefly mentioned in the fourth century that Apollonius " spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought.

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

The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.

* The Life and Times of Miss Jane Marple -- a biography by Anne Hart

( The only critical edition of Ibn Sina's autobiography, supplemented with material from a biography by his student Abu ' Ubayd al-Juzjani.

According to 20th-century scholar Walther Ludwig, the poems were spuriously inserted into an early biography of Plato sometime between 250 BC and 100 BC and adopted by later writers from this source.

* James S. Easby-Smith The Songs of Alcaeus ( 1901 ) W. H. Lowdermilk and Co., Washington-digitalized by Google: biography, history of criticisms, history of editions / publications, translations of fragments, commentary etc.

According to the biography, Penicillin Man: Alexander Fleming and the Antibiotic Revolution by Kevin Brown, Alexander Fleming, in a letter to his friend and colleague Andre Gratia, described this as " A wondrous fable.

There has only been one biography, written by Paul Allen, and this primarily covers his career in the theatre.

Common among the mislabeled works are all of the reasons identified for misattributing Cuyp ’ s works: the lack of biography and chronology of his works made it difficult to discern when paintings were created ( making it difficult to pinpoint an artist ); contentious signatures added to historians ’ confusion as to who actually painted the works ; and the collaborations and influences by different painters makes it hard to justify that a painting is genuinely that of Aelbert Cuyp ; and finally, accurate identification is made extremely difficult by the fact that this same style was copied ( rather accurately ) by his predecessor.

* Abraham Joshua Heschel: Prophetic Witness & Spiritual Radical: Abraham Joshua Heschel in America, 1940-1972, biography by Edward K. Kaplan ISBN 0-300-11540-7

The biography of Anthony's life by Athanasius of Alexandria helped to spread the concept of monasticism, particularly in Western Europe through Latin translations.

His biography was written by Saint Athanasius and titled Life of Saint Anthony the Great.

Translated to English by John Bester, the book was a biography of one of the last traditional Yakuza bosses in Japan.

Furthermore, John Murray believed that Disraeli had caricatured him and abused his confidence – an accusation denied at the time, and by the official biography, although subsequent biographers ( notably Blake ) have sided with Murray.

Benjamin Franklin ( 1966 ) 228 pp online edition, short biography by scholar

* Benjamin Franklin 1706 – 1790 Text of biography by Rev.

* Karl Benrath's German biography, translated into English by Helen Zimmern, with a preface by the Rev.

Luther's complaints against the book carried past the point of scholarly critique and may reflect Luther's antisemitism, which is disputed, such as in the biography of Luther by Derek Wilson, which points out that Luther's anger at the Jews was not at their race but at their theology.

In his biography of rock legend Elton John, Philip Norman recounted that by his early teens, John ( then known as Reg Dwight ) was wearing glasses " not because he needed them, but in homage to Buddy Holly.

Media reports immediately following his death indicated Haley displayed deranged and erratic behavior in his final weeks, although beyond a biography of Haley by John Swenson, released a year later, which described Haley painting the windows of his home black, there is little information extant about Haley's final days.

* A German-language biography was published soon after Haley's death, written by Peter Cornelsen and Harald D. Kain.

The latest biography of Khmelnytsky by Smoliy and Stepankov, however, challenges the 27 December date and suggests that it is more likely he was born on 9 November ( feast day of St Zenoby ,< ref >

* French translation by Edouard Chavannes of Sima Qian's biography of Confucius ( see pp. 283-435 ) in the Records of the Grand Historian

* Mary Baker Eddy: The Years of Discovery ; Mary Baker Eddy: The Years of Trial ; and Mary Baker Eddy: The Years of Authority comprise the three-volume biography of Eddy favored by Christian Scientists.