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57-491: Euclid generally refers to the ancient Greek mathematician Euclid of Alexandria (3rd century BC), who wrote a work on geometry called the Elements . Euclid , Euclides , or Eucleides may also refer to: Euclid of Alexandria Euclid ( / ˈ j uː k l ɪ d / ; ‹See Tfd› Greek : Εὐκλείδης ; fl.  300 BC) was an ancient Greek mathematician active as a geometer and logician . Considered

114-405: A corruption of Greek mathematical terms. Euclid is best known for his thirteen-book treatise, the Elements ( ‹See Tfd› Greek : Στοιχεῖα ; Stoicheia ), 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

171-445: A modern axiomatization of the Elements . Proclus The primary source for the life of Proclus is the eulogy Proclus, or On Happiness that was written for him upon his death by his successor, Marinus , Marinus' biography set out to prove that Proclus reached the peak of virtue and attained eudaimonia . There are also a few details about the time in which he lived in the similarly structured Life of Isidore written by

228-516: 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 (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13)—though book 5 (on proportions) and 10 (on irrational lines) do not exactly fit this scheme. The heart of

285-497: A faithful interpretation of Plato, and in this he did not differ from other Neoplatonists, as he considered that "nothing in Plato's corpus is unintended or there by chance", that "Plato's writings were divinely inspired" (ὁ θεῖος Πλάτων ho theios Platon —the divine Plato, inspired by the gods), that "the formal structure and the content of Platonic texts imitated those of the universe", and therefore that they spoke often of things under

342-581: A gifted student, he eventually became dissatisfied with the level of philosophical instruction available in Alexandria , and went to Athens , philosophical center of the day, in 431 to study at the Neoplatonic successor of the New Academy , where he was taught by Plutarch of Athens (not to be confused with Plutarch of Chaeronea ), Syrianus , and Asclepigenia ; he succeeded Syrianus as head of

399-553: 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

456-499: 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 c.  270 BC . Euclid

513-463: A successful practicing lawyer. However, the experience of the practice of law made Proclus realize that he truly preferred philosophy. He returned to Alexandria, and began determinedly studying the works of Aristotle under Olympiodorus the Elder . He also began studying mathematics during this period as well with a teacher named Heron (no relation to Hero of Alexandria , who was also known as Heron). As

570-444: A third yet younger set ( Amyntas , Menaechmus and his brother Dinostratus , Theudius of Magnesia , Hermotimus of Colophon and Philip of Opus ). Some of these mathematicians were influential in arranging the Elements that Euclid later published. Proclus authored a theology of Plato, which is text concerned with the divine hierarchies and their complex ramifications. A number of his Platonic commentaries are lost. In addition to

627-596: A veil, hiding the truth from the philosophically uninitiated. Proclus was however a close reader of Plato, and quite often makes very astute points about his Platonic sources. In his commentary on Plato's Timaeus Proclus explains the role the Soul as a principle has in mediating the Forms in Intellect to the body of the material world as a whole. The Soul is constructed through certain proportions, described mathematically in

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684-428: 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

741-420: 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

798-567: Is his elaboration of a level of individual ones, called henads, between the One which is before being and intelligible divinity. The henads exist "superabundantly", also beyond being, but they stand at the head of chains of causation ( seirai ) and in some manner give to these chains their particular character. He identifies them with the Greek gods, so one henad might be Apollo and be the cause of all things apollonian, while another might be Helios and be

855-628: Is in Apollonius' prefatory letter to the Conics (early 2nd century BC): "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

912-500: Is largely only possible with Plotinus, the only other Neoplatonic writer for whom a significant amount of writings survive. Proclus, like Plotinus and many of the other Neoplatonists , agreed on the three hypostases of Neoplatonism: The One ( hen ), The Intellect ( nous ) and The Soul ( psyche ), and wrote a commentary on the Enneads , of which unfortunately only fragments survive. At other times he criticizes Plotinus' views, such as

969-465: 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 c.  300 BC is called into question by a lack of contemporary references. The earliest original reference to Euclid

1026-418: 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 (4th century BC) as the mathematician to whom Plato sent those asking how to double

1083-660: Is possible because the powers of the gods (the henads ) extend through their series of causation even down to the material world. And by certain power-laden words, acts, and objects, the soul can be drawn back up the series, so to speak. Proclus himself was a devotee of many of the religions in Athens, considering that the power of the gods could be present in these various approaches. The majority of Proclus's works are commentaries on dialogues of Plato ( Alcibiades , Cratylus , Parmenides , Republic , Timaeus ). In these commentaries, he presents his own philosophical system as

1140-654: 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

1197-471: Is that the Neoplatonists of his time did not consider themselves innovators; they believed themselves to be the transmitters of the correct interpretations of Plato himself. Although the neoplatonic doctrines are much different from the doctrines in Plato's dialogues, it's often difficult to distinguish between different Neoplatonic thinkers and determine what is original to each one. For Proclus, this

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1254-532: Is thought to have written many lost works . The English name 'Euclid' is the anglicized version of the Ancient Greek name Eukleídes ( Εὐκλείδης ). It is derived from ' eu- ' ( εὖ ; 'well') and 'klês' ( -κλῆς ; 'fame'), 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 ,

1311-540: 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. Book 1 of the Elements is foundational for the entire text. It begins with a series of 20 definitions for basic geometric concepts such as lines , angles and various regular polygons . Euclid then presents 10 assumptions (see table, right), grouped into five postulates (axioms) and five common notions. These assumptions are intended to provide

1368-483: 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" ("the author of Elements "). In the Middle Ages, some scholars contended Euclid was not a historical personage and that his name arose from

1425-621: The Elements was published in 1570 by Henry Billingsley and John Dee . The mathematician Oliver Byrne published a well-known version of the Elements in 1847 entitled The First Six Books of the Elements of Euclid in Which Coloured Diagrams and Symbols Are Used Instead of Letters for the Greater Ease of Learners , which included colored diagrams intended to increase its pedagogical effect. David Hilbert authored

1482-447: 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

1539-412: The Elements , at least five works of Euclid have survived to the present day. They follow the same logical structure as Elements , with definitions and proved propositions. Four other works are credibly attributed to Euclid, but have been lost. Euclid is generally considered with Archimedes and Apollonius of Perga as among the greatest mathematicians of antiquity. Many commentators cite him as one of

1596-544: The Elements , book 10 is by far the largest and most complex, dealing with irrational numbers in the context of magnitudes. The final three books (11–13) primarily discuss solid geometry . By introducing a list of 37 definitions, Book 11 contextualizes the next two. Although its foundational character resembles Book 1, unlike the latter it features no axiomatic system or postulates. The three sections of Book 11 include content on solid geometry (1–19), solid angles (20–23) and parallelepipedal solids (24–37). In addition to

1653-457: The Timaeus , which allow it to make Body as a divided image of its own arithmetical and geometrical ideas. In addition to his commentaries, Proclus wrote two major systematic works. The Elements of Theology (Στοιχείωσις θεολογική) consists of 211 propositions, each followed by a proof, beginning from the existence of the One (divine Unity) and ending with the descent of individual souls into

1710-552: The area of triangles and parallelograms (35–45); and the Pythagorean theorem (46–48). The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros as "remarkably delicate". Book 2 is traditionally understood as concerning " geometric algebra ", though this interpretation has been heavily debated since the 1970s; critics describe the characterization as anachronistic, since

1767-501: The pentagon . Book 5 is among the work's most important sections and presents what is usually termed as the "general theory of proportion". Book 6 utilizes the "theory of ratios " in the context of plane geometry. It is built almost entirely of its first proposition: "Triangles and parallelograms which are under the same height are to one another as their bases". From Book 7 onwards, the mathematician Benno Artmann  [ de ] notes that "Euclid starts afresh. Nothing from

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1824-454: The prime mover . Unlike Plotinus, Proclus also did not hold that matter was evil, an idea that caused contradictions in the system of Plotinus. It is difficult to determine what, if anything, is different between the doctrines of Proclus and Syrianus: for the latter, only a commentary on Aristotle's Metaphysics survives, and Proclus never criticizes his teacher in any of his preserved writings. The particular characteristic of Proclus's system

1881-529: 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

1938-459: The Academy in 437, and would in turn be succeeded on his death by Marinus of Neapolis . He lived in Athens as a vegetarian bachelor, prosperous and generous to his friends, until the end of his life, except for a one-year exile, to avoid pressure from Christian authorities. Marinus reports that he was writing seven hundred lines each day. One challenge with determining Proclus' specific doctrines

1995-688: The Alcibiades, the Cratylus, the Timaeus, and the Parmenides, he also wrote commentaries on the remainder of the dialogues in the Neoplatonic curriculum. He also wrote a commentary on the Organon , as well as prolegomena to both Plato and Aristotle. Proclus exerted a great deal of influence on Medieval philosophy , though largely indirectly, through the works of the commentator Pseudo-Dionysius

2052-469: The Areopagite . This late-5th- or early-6th-century Christian Greek author wrote under the pseudonym Dionysius the Areopagite , the figure converted by St. Paul in Athens. Because of this fiction, his writings were taken to have almost apostolic authority. He is an original Christian writer, and in his works can be found a great number of Proclus's metaphysical principles. Another important source for

2109-450: The One, and prepare it not only to ascend to the higher levels while still in this life, but to avoid falling immediately back into a new body after death. Because the soul's attention, while inhabiting a body, is turned so far away from its origin in the intelligible world, Proclus thinks that we need to make use of bodily reminders of our spiritual origin. In this he agrees with the doctrines of theurgy put forward by Iamblichus . Theurgy

2166-412: The cause of all sunny things. Each henad participates in every other henad, according to its character. What appears to be multiplicity is not multiplicity at all, because any henad may rightly be considered the center of the polycentric system. According to Proclus, philosophy is the activity which can liberate the soul from a subjection to bodily passions, remind it of its origin in Soul, Intellect, and

2223-441: 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 (now lost), eventually leading Euclid the mathematician to be ascribed details of both men's biographies and described as Megarensis ( lit.   ' of Megara ' ). The Byzantine scholar Theodore Metochites ( c.  1300 ) explicitly conflated

2280-484: 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

2337-514: 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

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2394-442: The early 4th century. According to Proclus, Euclid lived shortly after several of Plato 's ( d.  347 BC) followers and before the mathematician Archimedes ( c.  287  – c.  212 BC); specifically, Proclus placed Euclid during the rule of Ptolemy I ( r.  305/304–282 BC). Euclid's birthdate is unknown; some scholars estimate around 330 or 325 BC, but others refrain from speculating. It

2451-498: The existence of evils ( De malorum subsistentia ). Proclus, the scholiast to Euclid, knew Eudemus of Rhodes ' History of Geometry well, and gave a short sketch of the early history of geometry, which appeared to be founded on the older, lost book of Eudemus. The passage has been referred to as "the Eudemian summary," and determines some approximate dates, which otherwise might have remained unknown. The influential commentary on

2508-490: 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

2565-461: The first book of Euclid 's Elements is one of the most valuable sources we have for the history of ancient mathematics, and its Platonic account of the status of mathematical objects was influential. In this work, Proclus also listed the first mathematicians associated with Plato: a mature set of mathematicians ( Leodamas of Thasos , Archytas of Taras , and Theaetetus ), a second set of younger mathematicians ( Neoclides , Eudoxus of Cnidus ), and

2622-411: The foundations of even nascent algebra occurred many centuries later. The second book has a more focused scope and mostly provides algebraic theorems to accompany various geometric shapes. It focuses on the area of rectangles and squares (see Quadrature ), and leads up to a geometric precursor of the law of cosines . Book 3 focuses on circles, while the 4th discusses regular polygons , especially

2679-462: The influence of Proclus on the Middle Ages is Boethius 's Consolation of Philosophy , which has a number of Proclus principles and motifs. The central poem of Book III is a summary of Proclus's Commentary on the Timaeus , and Book V contains the important principle of Proclus that things are known not according to their own nature, but according to the character of the knowing subject. A summary of Proclus's Elements of Theology circulated under

2736-492: The logical basis for every subsequent theorem, i.e. serve as an axiomatic system . The common notions exclusively concern the comparison of magnitudes . While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book 1 also includes 48 propositions, which can be loosely divided into those concerning basic theorems and constructions of plane geometry and triangle congruence (1–26); parallel lines (27–34);

2793-611: The lunar crater Euclides , and the minor planet 4354 Euclides . The Elements is often considered after the Bible as the most frequently translated, published, and studied book in the Western World 's history. With Aristotle's Metaphysics , the Elements is perhaps the most successful ancient Greek text, and was the dominant mathematical textbook in the Medieval Arab and Latin worlds. The first English edition of

2850-504: The material world. The Platonic Theology (Περὶ τῆς κατὰ Πλάτωνα θεολογίας) is a systematization of material from Platonic dialogues, showing from them the characteristics of the divine orders, the part of the universe which is closest to the One. We also have three essays, extant only in Latin translation: Ten doubts concerning providence ( De decem dubitationibus circa providentiam ); On providence and fate ( De providentia et fato ); On

2907-475: The most influential figures in the history of mathematics . The geometrical system established by the Elements long dominated the field; however, today that system is often referred to as ' Euclidean geometry ' to distinguish it from other non-Euclidean geometries discovered in the early 19th century. Among Euclid's many namesakes are the European Space Agency 's (ESA) Euclid spacecraft,

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2964-470: The name Liber de Causis ( Book of Causes ). This book is of uncertain origin, but circulated in the Arabic world as a work of Aristotle, and was translated into Latin as such. It had great authority because of its supposed Aristotelian origin, and it was only when Proclus's Elements were translated into Latin that Thomas Aquinas realised its true origin. Proclus's works also exercised an influence during

3021-605: The philosopher Damascius in the following century. According to Marinus, Proclus was born in 412 AD in Constantinople to a family of high social status from Lycia , and raised in Xanthus . He studied rhetoric , philosophy and mathematics in Alexandria , with the intent of pursuing a judicial position like his father. Before completing his studies, he returned to Constantinople when his rector, his principal instructor (one Leonas), had business there. Proclus became

3078-533: The preceding books is used". Number theory is covered by books 7 to 10, the former beginning with a set of 22 definitions for parity , prime numbers and other arithmetic-related concepts. Book 7 includes the Euclidean algorithm , a method for finding the greatest common divisor of two numbers. The 8th book discusses geometric progressions , while book 9 includes the proposition, now called Euclid's theorem , that there are infinitely many prime numbers . Of

3135-546: 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" ( ‹See Tfd› Greek : ὅρος or ὁρισμός ), "postulate" ( αἴτημα ), or a "common notion" ( κοινὴ ἔννοια ); 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

3192-769: 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 Bartolomeo Zamberti  [ fr ; de ] 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. 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. Most scholars consider them of dubious authenticity; Heath in particular contends that

3249-556: 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

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