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52-559: The book that talks about porisms first is Euclid 's Porisms . What is known of it is in Pappus of Alexandria 's Collection , who mentions it along with other geometrical treatises, and gives several lemmas necessary for understanding it. Pappus states: Pappus said that the last definition was changed by certain later geometers, who defined a porism as an accidental characteristic as τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος ( to leîpon hypothései topikoû theōrḗmatos ), that which falls short of

104-460: 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

156-621: A modern axiomatization of the Elements . Michel Chasles Michel Floréal Chasles ( French pronunciation: [miʃɛl flɔʁeal ʃal] ; 15 November 1793 – 18 December 1880) was a French mathematician . He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson . In the War of the Sixth Coalition he was drafted to fight in

208-516: 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

260-592: A Euclidean motion in space as screw displacement was seminal to the development of the theories of dynamics of rigid bodies . Chasles was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1864. In 1865 he was awarded the Copley Medal . As described in A Treasury of Deception , by Michael Farquhar (Peguin Books, 2005), between 1861 and 1869 Chasles purchased some of

312-496: A controversy between Breton and A. J. H. Vincent, who disputed the interpretation given by the former of Pappus's text, and declared himself in favour of Frans van Schooten 's idea, put forward in his Mathematicae exercitationes (1657). According to Schooten, if the various relations between straight lines in a figure are written down in the form of equations or proportions, then the combination of these equations in all possible ways, and of new equations thus derived from them leads to

364-567: A given ratio X to AM. The lemmas which Pappus gives in connection with the porisms are: Robert Simson explained the only three propositions which Pappus indicates with any completeness, which was published in the Philosophical Transactions in 1723. Later he investigated the subject of porisms generally in a work entitled De porismatibus traclatus; quo doctrinam porisrnatum satis explicatam, et in posterum ab oblivion tutam fore sperat auctor , and published after his death in

416-450: A locus is a species of porism. Then follows a Latin translation of Pappus's note on the porisms, and the propositions which form the bulk of the treatise. John Playfair 's memoir ( Trans. Roy. Soc. Edin. , 1794, vol. iii.), a sort of sequel to Simson's treatise, explored the probable origin of porisms, or the steps that led ancient geometers to discover them. Playfair remarked that the careful investigation of all possible particular cases of

468-556: A locus-theorem by a (or in its) hypothesis. Proclus pointed out that the word porism was used in two senses: one sense is that of "corollary", as a result unsought but seen to follow from a theorem. In the other sense, he added nothing to the definition of "the older geometers", except to say that the finding of the center of a circle and the finding of the greatest common measure are porisms. Pappus rejected Euclid's definition of porism . A porism, expressed in modern language, asserts that given four straight lines, of which three turn about

520-611: 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

572-548: A proposition would show that These cases could be defined separately, were in a manner intermediate between theorems and problems, and were called "porisms." Playfair defined a porism as "[a] proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." Although Playfair's definition of a porism appears to be most favoured in England, Simson's view has been most generally accepted abroad, and had

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624-499: 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

676-434: A straight line. The above can be expressed as: If about two fixed points, P and Q, one makes the turn two straight lines meeting on a given straight line, L, and if one of them cuts off a segment, AM, from a fixed straight line, AX, given in position, another fixed straight line BY, and a point B fixed on it can be determined, such that the segment BM' made by the second moving line on this second fixed-line measured from B has

728-465: A triangular number whose side is n  − 1. If they are made to turn about the n fixed points so that any n  − 1 of their 1 ⁄ 2 n ( n  − 1) points of intersection, chosen subject to a certain limitation, lie on n  − 1 given fixed straight lines, then each of the remaining points of intersection, 1 ⁄ 2 n ( n  − 1)( n  − 2) in number, describes

780-704: A volume, Roberti Simson opera quaedam reliqua (Glasgow, 1776). Simson's treatise, De porismatibus , begins with the definitions for theorem, problem, datum, porism, and locus. Simon wrote that Pappus's definition is too general, and that he substituted it as: Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent rationem, convenire ostendendum est affectionem quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda sunt, invenienda proponantur. Simson said that

832-428: 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

884-420: 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

936-628: 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

988-465: 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

1040-418: 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

1092-707: 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

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1144-532: 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 ,

1196-540: 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

1248-483: 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

1300-621: 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

1352-447: 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

1404-412: 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

1456-544: 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

1508-552: 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

1560-501: 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

1612-509: The 27,000 forged letters from Frenchman Denis Vrain-Lucas . Included in this trove were letters from Alexander the Great to Aristotle , from Cleopatra to Julius Caesar , and from Mary Magdalene to a revived Lazarus , all in a fake medieval French. In 2004, the journal Critical Inquiry published a recently "discovered" 1871 letter written by Vrain-Lucas (from prison) to Chasles, conveying Vrain-Lucas's perspective on these events, itself an invention. In 1986, Alexander Jones published

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1664-764: The General Properties of Cones of the Second Degree and on the Spherical Conics , adding a significant amount of original material. Jakob Steiner had proposed Steiner's conic problem of enumerating the number of conic sections tangent to each of five given conics and had answered it incorrectly. Chasles developed a theory of characteristics that enabled the correct enumeration of the conics (there are 3264) (see enumerative geometry ). He established several important theorems (all called Chasles's theorem ). In kinematics , Chasles's description of

1716-441: 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

1768-456: The defence of Paris in 1814. After the war, he gave up on a career as an engineer or stockbroker in order to pursue his mathematical studies. In 1837 he published the book Aperçu historique sur l'origine et le développement des méthodes en géométrie ("Historical view of the origin and development of methods in geometry"), a study of the method of reciprocal polars in projective geometry . The work gained him considerable fame and respect and he

1820-484: 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

1872-564: The discovery of innumerable new properties of the figure. The discussions between Breton and Vincent, which C. Housel joined, did not carry forward the work of restoring Euclid's Porisms , which was left for Chasles. His work ( Les Trois livres de porismes d'Euclide , Paris, 1860) makes full use of all the material found in Pappus. An interesting hypothesis about porisms was put forward by H. G. Zeuthen ( Die Lehre von den Kegelschnitten im Altertum , 1886, ch. viii.). Zeuthen observed, for example

1924-514: 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

1976-442: 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

2028-490: 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

2080-411: 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

2132-427: The intercept-porism is still true if the two fixed points are points on a conic, and the straight lines drawn through them intersect on the conic instead of on a fixed straight line. He conjectured that the porisms were a by-product of a fully developed projective geometry of conics. Attribution: Euclid Euclid ( / ˈ j uː k l ɪ d / ; ‹See Tfd› Greek : Εὐκλείδης ; fl.  300 BC)

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2184-492: 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);

2236-611: 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

2288-475: 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,

2340-502: The points in which they meet the fourth if two of the points of intersection of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line. The general definition applies to any number, n , of straight lines, of which n can turn about as many points fixed on the ( n  + 1)th. These n straight lines cut two and two into 1 ⁄ 2 n ( n  − 1) points, 1 ⁄ 2 n ( n  − 1) being

2392-533: 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

2444-536: The support of Michel Chasles . However, in Liouville 's Journal de mathematiques pures et appliquées (vol. xx., July, 1855), P. Breton published Recherches nouvelles sur les porismes d'Euclide , in which he gave a new translation of the text of Pappus, and sought to base a view of the nature of a porism that conforms more closely to Pappus's definition. This was followed in the same journal and in La Science by

2496-546: 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

2548-769: 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

2600-619: 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

2652-514: Was appointed Professor at the École Polytechnique in 1841, then he was awarded a chair at the Sorbonne in 1846. A second edition of this book was published in 1875. In 1839, Ludwig Adolph Sohncke (the father of Leonhard Sohncke ) translated the original into German as Geschichte der geometrie, hauptsachlich mit bezug auf die neueren methoden . Shortly thereafter, in 1841, Charles Graves published an English version as Two Geometrical Memoirs on

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2704-556: 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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