Euclid
Euclid was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.
Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.
In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour. In addition to the Elements, Euclid wrote a central early text in the optics field, Optics, and lesser-known works including Data and Phaenomena. Euclid's authorship of On Divisions of Figures and Catoptrics has been questioned. He is thought to have written many [|lost works].
Life
Traditional narrative
The English name 'Euclid' is the anglicized version of the Ancient Greek name . It is derived from 'eu-' and 'klês', meaning "renowned, glorious". In English, by metonymy, 'Euclid' can mean his most well-known work, Euclid's Elements, or a copy thereof, and is sometimes synonymous with 'geometry'.As with many ancient Greek mathematicians, the details of Euclid's life are mostly unknown. He is accepted as the author of four mostly extant treatises—the Elements, Optics, Data, Phaenomena—but besides this, there is nothing known for certain of him. The traditional narrative mainly follows the 5th century AD account by Proclus in his Commentary on the First Book of Euclid's Elements, as well as a few anecdotes from Pappus of Alexandria in the early 4th century.
According to Proclus, Euclid lived shortly after several of Plato's followers and before the mathematician Archimedes ; specifically, Proclus placed Euclid during the rule of Ptolemy I. Euclid's birthdate is unknown; some scholars estimate around 330 or 325 BC, but others refrain from speculating. It is presumed that he was of Greek descent, but his birthplace is unknown. Proclus held that Euclid followed the Platonic tradition, but there is no definitive confirmation for this. It is unlikely he was a contemporary of Plato, so it is often presumed that he was educated by Plato's disciples at the Platonic Academy in Athens. Historian Thomas Heath supported this theory, noting that most capable geometers lived in Athens, including many of those whose work Euclid built on; historian Michalis Sialaros considers this a mere conjecture. In any event, the contents of Euclid's work demonstrate familiarity with the Platonic geometry tradition.
In his Collection, Pappus mentions that Apollonius studied with Euclid's students in Alexandria, and this has been taken to imply that Euclid worked and founded a mathematical tradition there. The city was founded by Alexander the Great in 331 BC, and the rule of Ptolemy I from 306 BC onwards gave it a stability which was relatively unique amid the chaotic wars over dividing Alexander's empire. Ptolemy began a process of hellenization and commissioned numerous constructions, building the massive Musaeum institution, which was a leading center of education. Euclid is speculated to have been among the Musaeum's first scholars. Euclid's date of death is unknown; it has been speculated that he died.
Medieval Islamic Narrative
Islamic biographical sources contain several detailed accounts of Euclid’s life; however, these reports are widely considered to be late and unsubstantiated. One such account is preserved by Ali Ibn Yusuf al-Qifti, who writes:Identity and historicity
Euclid is often referred to as 'Euclid of Alexandria' to differentiate him from the earlier philosopher Euclid of Megara, a pupil of Socrates included in dialogues of Plato with whom he was historically conflated. Valerius Maximus, the 1st century AD Roman compiler of anecdotes, mistakenly substituted Euclid's name for Eudoxus as the mathematician to whom Plato sent those asking how to double the cube. Perhaps on the basis of this mention of a mathematical Euclid roughly a century early, Euclid became mixed up with Euclid of Megara in medieval Byzantine sources, eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as Megarensis. The Byzantine scholar Theodore Metochites explicitly conflated the two Euclids, as did printer Erhard Ratdolt's 1482 editio princeps of Campanus of Novara's Latin translation of the Elements. After the mathematician appended most of the extant biographical fragments about either Euclid to the preface of his 1505 translation of the Elements, subsequent publications passed on this identification. A further confusion, which gives the birthplace of Euclid as Gela, Sicily, arises from the fact that Euclid of Megara is sometimes said to have been born in Gela. Later Renaissance scholars, particularly Peter Ramus, reevaluated this claim, proving it false via issues in chronology and contradiction in early sources.Medieval Arabic sources give vast amounts of information concerning Euclid's life, but are completely unverifiable. Euclid, who was supposedly a Tyre-born Greek domiciled at Damascus, was claimed to have been the son of Naucrates. Most scholars consider them of dubious authenticity. Heath in particular contends that the fictionalization was done to strengthen the connection between a revered mathematician and the Arab world. There are also numerous anecdotal stories concerning to Euclid, all of uncertain historicity, which "picture him as a kindly and gentle old man". The best known of these is Proclus' story about Ptolemy asking Euclid if there was a quicker path to learning geometry than reading his Elements, which Euclid replied with "there is no royal road to geometry". This anecdote is questionable since a very similar interaction between Menaechmus and Alexander the Great is recorded from Stobaeus. Both accounts were written in the 5th century AD, neither indicates its source, and neither appears in ancient Greek literature.
Any firm dating of Euclid's activity is called into question by a lack of contemporary references. The earliest original reference to Euclid is in Apollonius' prefatory letter to the Conics : "The third book of the Conics contains many astonishing theorems that are useful for both the syntheses and the determinations of number of solutions of solid loci. Most of these, and the finest of them, are novel. And when we discovered them we realized that Euclid had not made the synthesis of the locus on three and four lines but only an accidental fragment of it, and even that was not felicitously done." The Elements is speculated to have been at least partly in circulation by the 3rd century BC, as Archimedes and Apollonius take several of its propositions for granted; however, Archimedes employs an older variant of the theory of proportions than the one found in the Elements. The oldest physical copies of material included in the Elements, dating from roughly 100 AD, can be found on papyrus fragments unearthed in an ancient rubbish heap from Oxyrhynchus, Roman Egypt. The oldest extant direct citations to the Elements in works whose dates are firmly known are not until the 2nd century AD, by Galen and Alexander of Aphrodisias; by this time it was a standard school text. Some ancient Greek mathematicians mention Euclid by name, but he is usually referred to as "ὁ στοιχειώτης". In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from a corruption of Greek mathematical terms.
Works
''Elements''
Euclid is best known for his thirteen-book treatise, the Elements, considered his magnum opus. Much of its content originates from earlier mathematicians, including Eudoxus, Hippocrates of Chios, Thales and Theaetetus, while other theorems are mentioned by Plato and Aristotle. It is difficult to differentiate the work of Euclid from that of his predecessors, especially because the Elements essentially superseded much earlier and now-lost Greek mathematics. The classicist Markus Asper concludes that "apparently Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps" and the historian Serafina Cuomo described it as a "reservoir of results". Despite this, Sialaros furthers that "the remarkably tight structure of the Elements reveals authorial control beyond the limits of a mere editor".The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry, basic number theory and solid geometry —though book 5 and 10 do not exactly fit this scheme. The heart of the text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition", "postulate", or a "common notion" ; only the first book includes postulates—later known as axioms—and common notions. The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it a natural fit. As a whole, the authorial voice remains general and impersonal.