21. Greciaheroica2 It can also be used to square the circle although dinostratus gave a cleardemonstration of this in the following century. http://descartes.cnice.mecd.es/ingles/maths_workshop/A_history_of_Mathematics/Gr

THE GREEK HEROIC AGE II History HIPPIAS OF ELIS Unlike the Pythagoreans, Hippias de Elis (460 B.C.) was a Sophist ; in other words he earned his living by teaching his disciples. This is mentioned in Plato's Dialogues , where he is described as having little substance, earning more money than his peers and somewhat proud in character. Proclus ascribed to him the invention of the first curve, which is different to the circumference , known as the trisectrix or quadratrix of Hippias, which allows the angle to be divided into three equal parts. It can also be used to square the circle although Dinostratus gave a clear demonstration of this in the following century. Hippias' trisectrix Whilst a moves around the circle at constant velocity b moves along the segment at constant velocity too. Each point on the curve represents the point where the arc and segment coincide as we move along them at the same time. In this window you can see how Hippias' trisectrix is used to trisect the angle in three equal parts.

22. Dinostratus turnbull.mcs.stand.ac.uk/history/References/Dinos More results from turnbull.mcs.st-and.ac.uk Footnotesdinostratus dinostratus showed how to square the circle using the trisectrix .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . http://turnbull.mcs.st-and.ac.uk/history/Mathematicians/Dinostratus.html

Dinostratus Born: about 390 BC in Greece Died: about 320 BC Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing Dinostratus is mentioned by Proclus who says (see for example [1] or [3]):- Amyclas of Heraclea, one of the associates of Plato , and Menaechmus , a pupil of Eudoxus who had studied with Plato , and his brother Dinostratus made the whole of geometry still more perfect. It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias , to solve the problem of squaring the circle Pappus tells us (see for example [1] or [3]):- For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix It appears from this quote that Hippias discovered the curve but that it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Proclus , who claims to be quoting from Eudemus , writes (see [1]):- Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of

23. Mathematics Magazine: News And Letters dinostratus some 80 years later realized that the same curve could be used tosquare the circle, and hence renamed it the quadratrix (cf. http://www.findarticles.com/p/articles/mi_qa3789/is_199810/ai_n8818629

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Advanced Search Home Help IN free articles only all articles this publication Automotive Sports 10,000,000 articles - not found on any other search engine. FindArticles Mathematics Magazine Oct 1998 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports News and letters Mathematics Magazine Oct 1998 Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Letters to the Editor Dear Editor: In the charming article "Functions with compact preimages of compact sets" in the December 1997 issue of Mathematics Magazine, two topology students and their instructor discuss functions from the real line into itself with the property that the preimage of every compact set is compact. They show that such a "preimagecompact" function need not be continuous, but its set of discontinuities must be a closed, nowhere dense set, and they give some examples to show that this discontinuity set can be rather large. A slight adaptation of the authors' introductory example shows that every closed, nowhere dense set F is the set of discontinuities of some preimage-compact function. Indeed, define a function f via f (z) = z if x E F and f (z) = z + dist(z, F) -1 when x V F. Evidently f is continuous on the complement of F and unbounded in a neighborhood of every point of F. Hence F is the set of discontinuities of f. Since F is closed, the restriction of f to F is preimage-compact. On the other hand, if x V F, then f (x) blows up when z approaches either F or infinity, and consequently the restriction of f to the complement of F is also preimage-compact.

24. References For Dinostratus References for dinostratus. Biography in Dictionary of http//wwwhistory.mcs.st-andrews.ac.uk/history/References/dinostratus.html. http://202.38.126.65/mirror/www-history.mcs.st-and.ac.uk/history/References/Dino

25. Squaring The Circle in this archive. Hippias and dinostratus are associated with the methodof squaring the circle using a quadratrix. The curve it http://202.38.126.65/mirror/www-history.mcs.st-and.ac.uk/history/HistTopics/Squa

26. Quadratrice De Dinostrate dinostratus (oder des Hippias). Courbe étudiée par Hippias d Elis en 430 http://www.mathcurve.com/courbes2d/dinostrate/dinostrate.shtml

courbe suivante courbes 2D courbes 3D surfaces ... fractals QUADRATRICE DE DINOSTRATE Dinostratus' (or Hippias') quadratrix, Quadratrix des Dinostratus (oder des Hippias) Autre nom : sectrice d'Hippias. La quadratrice de Dinostrate sectrice de Maclaurin O d'une Comme son nom l'indique, cette courbe est une quadratrice ; en effet : n -sectrice ; en effet q p courbe suivante courbes 2D courbes 3D surfaces ... fractals

27. INTERNATIONAL COMPUTER HIGH SCHOOL MATH DEPARTMENT - Squaring The Circle Hippias and dinostratus are associated with the method of squaring thecircle using a quadratrix. The curve it thought to be the http://math.ichb.ro/modules.php?name=News&file=print&sid=19

28. Archimedes (287 B.C. - 212 B.C.) dinostratus discovered the quadratrix resulting from the intersection of a rotatingline with another moving parallel to itself. http://www.usefultrivia.com/biographies/archimedes_001.html

ARCHIMEDES ARCHIMEDES was a native of Syracuse, one of the greatest cities of the West Grecian world. His letters to Dositheus of Alexandria show him to have been in constant communication with the students of geometry in that city. Plutarch, in his life of Marcellus, speaks of his intimate friendship with King Hiero of Syracuse, who induced him to apply his mechanical principles to the construction of military engines; though the time thus withdrawn from his theoretical researches was most unwillingly given. During the second Punic War Hiero had been in close alliance with Rome. But after his death, Hippocrates, an ambitious general, enlisted the city on the side of Carthage, and a Roman force, under the command of Marcellus, besieged it by sea and land. The fleet, equipped with the usual engines of war, especially the sambuca Cicero . It bore the image of a cylinder circumscribing a sphere, with a verse indicating, what Archimedes had held to be his greatest achievement, the measurement and mutual proportion of these two bodies. Dramatic, surely, was the contrast offered by the siege of Syracuse between the scientific intellect of Greece and the disciplined force of Rome; and not less remarkable is the admiration of the conqueror for the conquered, which, in a few generations, would weld the Greco-Roman world into one. Geometry, when Archimedes began his career, had made more progress than is shown by the thirteen books of Euclid's

29. Pappus (4th Century A.D.) There are special studies on various curves; as the spirals of Archimedes, thequadratix of dinostratus, and the conchoid of Nicomedes. http://www.usefultrivia.com/biographies/pappus_001.html

PAPPUS PAPPUS was a contemporary of Theon of Alexandria, and taught mathematics in that city during the reign of Theodosius I. He wrote a commentary on the Great Syntaxis Almagest ] of Ptolemy, which has not come down to us. He is known to us by the work entitled Synagoge or Assemblage ; a collection in eight books of mathematical papers having no very distinct connection, and consisting of commentaries on the geometrical work of the previous six centuries, enriched by very fruitful additions of his own. For the history of ancient mathematics this work, of which the last six books and part of the second have been preserved, is invaluable. Already in the third century B.C. the filiation of discovery, so evident in this science, had been traced by Eudemus, a pupil of Aristotle, parts of whose work have been preserved by Proclus. Pappus supplies many details of Apollonius and of later writers who would otherwise have been unknown to us. Special studies on isolated problems occupy the greater part of his attention. Various modes of inserting two mean geometrical proportionals are discussed; new methods of inscribing the five regular solids in a sphere are put forward, (book iii). There are special studies on various curves; as the spirals of Archimedes , the quadratix of Dinostratus, and the conchoid of Nicomedes. Much attention is given to the work of Zenodorus on isoperimetry; and new problems on this subject are solved (book v). In the 6th book the earlier astronomers are spoken of.

30. Τετραγωνισμός Ï„Î¿Ï ÎºÏÎºÎ The summary for this English page contains characters that cannot be correctly displayed in this language/character set. http://users.ira.sch.gr/thafounar/Genika/problemGeometry/SquaringTheCircle/Dinos

q q q r q x

31. Quadratrix Of Hippias -- From MathWorld Elias in 430 BC Eric Weisstein s World of Astronomy , and later studied bydinostratus in 350 BC Eric Weisstein s World of Astronomy (MacTutor Archive). http://mathworld.wolfram.com/QuadratrixofHippias.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Curves Plane Curves ... Geometric Construction Quadratrix of Hippias The quadratrix was discovered by Hippias of Elias in 430 BC , and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal parts, and circle squaring It has polar equation with corresponding parametric equation and Cartesian equation Using the parametric representation, the curvature and tangential angle are given by for SEE ALSO: Angle trisection Cochleoid [Pages Linking Here] REFERENCES: Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 223, 1987. Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972. Loomis, E. S. "The Quadratrix." §2.1 in

32. CHRONOLOGY OF MATHEMATICIANS -1100 CHOU-PEI -585 THALES OF MILETUS 350 dinostratus QUADRATRIX. -335 EUDEMUS HISTORY OF GEOMETRY. -330 AUTOLYCUSON THE MOVING SPHERE. -320 ARISTAEUS CONICS. -300 EUCLID THE ELEMENTS http://users.adelphia.net/~mathhomeworkhelp/timeline.html

CHRONOLOGY OF MATHEMATICIANS -1100 CHOU-PEI -585 THALES OF MILETUS: DEDUCTIVE GEOMETRY PYTHAGORAS : ARITHMETIC AND GEOMETRY -450 PARMENIDES: SPHERICAL EARTH -430 DEMOCRITUS -430 PHILOLAUS: ASTRONOMY -430 HIPPOCRATES OF CHIOS: ELEMENTS -428 ARCHYTAS -420 HIPPIAS: TRISECTRIX -360 EUDOXUS: PROPORTION AND EXHAUSTION -350 MENAECHMUS: CONIC SECTIONS -350 DINOSTRATUS: QUADRATRIX -335 EUDEMUS: HISTORY OF GEOMETRY -330 AUTOLYCUS: ON THE MOVING SPHERE -320 ARISTAEUS: CONICS EUCLID : THE ELEMENTS -260 ARISTARCHUS: HELIOCENTRIC ASTRONOMY -230 ERATOSTHENES: SIEVE -225 APOLLONIUS: CONICS -212 DEATH OF ARCHIMEDES -180 DIOCLES: CISSOID -180 NICOMEDES: CONCHOID -180 HYPSICLES: 360 DEGREE CIRCLE -150 PERSEUS: SPIRES -140 HIPPARCHUS: TRIGONOMETRY -60 GEMINUS: ON THE PARALLEL POSTULATE +75 HERON OF ALEXANDRIA 100 NICOMACHUS: ARITHMETICA 100 MENELAUS: SPHERICS 125 THEON OF SMYRNA: PLATONIC MATHEMATICS PTOLEMY : THE ALMAGEST 250 DIOPHANTUS: ARITHMETICA 320 PAPPUS: MATHEMATICAL COLLECTIONS 390 THEON OF ALEXANDRIA 415 DEATH OF HYPATIA 470 TSU CH'UNG-CHI: VALUE OF PI 476 ARYABHATA 485 DEATH OF PROCLUS 520 ANTHEMIUS OF TRALLES AND ISIDORE OF MILETUS 524 DEATH OF BOETHIUS 560 EUTOCIUS: COMMENTARIES ON ARCHIMEDES 628 BRAHMA-SPHUTA-SIDDHANTA 662 BISHOP SEBOKHT: HINDU NUMERALS 735 DEATH OF BEDE 775 HINDU WORKS TRANSLATED INTO ARABIC 830 AL-KHWARIZMI: ALGEBRA 901 DEATH OF THABIT IBN - QURRA 998 DEATH OF ABU'L - WEFA 1037 DEATH OF AVICENNA 1039 DEATH OF ALHAZEN

33. New Page 0 dinostratus Diocles Dionysodorus Diophantus Domninus Eratosthenes Euclid Eudemus of Rhodes Eudoxus Eutocius Geminus Heraclides of http://www.edfiles.com/top/GR6B/geometryP.htm

SCIENCE / GEOMETRY / MATH EDFILES SOCIAL STUDIES ANCIENT GREECE science math geometry index Ancient Greek mathematics greek contributions to science greek contributions to science ii ancient greek medicine ... euclids elements Ancient Greek mathematics Greek mathematics Anaxagoras Anthemius Antiphon ... Zenodorus greek contributions to science Ancient Greek Agriculture Botany Ancient Greek Astronomy Ancient Greek Earth Science Origins of Greek ScienCE ... go to index greek contributions to science ii (from the vatican) Vatican Exhibit Main Hall Greek Astronomy Greek Mathematics and Modern Heirs Mathematics Ancient Science Modern Fates ... go to index ancient greek medicine Ancient drugs BBC Medicine Asclepius (1200BC - 500AD) BBC Medicine - Greek Medicine BBC Medicine Hippocrates ... go to index on ancient medicine Part 1 Part 2 Part 3 Part 4 ... go to index on air waters and places Part 1 Part 2 Part 3 Part 4 ... go to index Articles about Greek mathematics Squaring the circle Doubling the cube Trisecting an angle Greek Astronomy ... Greek mathematics?

34. Sfabel.tripod.com/mathematik/database/Dinostratus.html Quadratrix definition of Quadratrix by the Free Online (Geom.) A curve made use of in the quadrature of other curves; as the quadratrix,of dinostratus, or of Tschirnhausen . http://sfabel.tripod.com/mathematik/database/Dinostratus.html

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Tripod Star Wars Share This Page Report Abuse Edit your Site ... Next Dinostratus Born: about 390 BC in Greece Died: about 320 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Dinostratus is mentioned by Proclus who says Amyclas of Heraclea, one of the associates of Plato , and Menaechmus , a pupil of Eudoxus who had studied with Plato , and his brother Dinostratus made the whole of geometry still more perfect. Dinostratus used the quadratrix, discovered by Hippias , to solve the problem of squaring the circle. Pappus tells us For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix. It appears that Hippias discovered the curve but it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Dinostratus probably did much more work on geometry but nothing is known of it.

35. History Of Mathematics: Greece 350); dinostratus (c. 350); Speusippus (d. 339); Aristotle (384322); Aristaeus theElder (fl. c. 350-330); Eudemus of Rhodes (the Peripatetic) (c. http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

36. History Of Mathematics: Chronology Of Mathematicians 350); dinostratus (fl. c. 350) *SB; Speusippus (d. 339); Aristotle (384322) *SB *MT;Aristaeus the Elder (fl. c. 350-330) *SB *MT; Eudemus of Rhodes (the http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan Table of Contents 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below List of Mathematicians 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *MT 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) 500 B.C.E. Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *MT Zeno of Elea (c. 490-c. 430) *MT Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *MT Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *MT Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB

37. Assignment 19 Great mathematicians whose works were revived by Pappus include Euclid, Archimedes,Apollonius, Nicomedes, and dinostratus. Herkimer s Corner http://www.herkimershideaway.org/algebra2/doc_page27.html

Assignment 19 Obvious is the most dangerous word in mathematics." (Eric Temple Bell ) PAPPUS (ca 300): An excellent mathematician who lived in Alexandria, Pappus attempted to rekindle interest in the mathematical works of the Greeks. This was not an easy task, since Christians had destroyed much of the ancient Greek documents, but Pappus managed to create his Mathematical Collection which cites or references over thirty different ancient mathematicians. Much of our knowledge of Greek mathematics has been derived from the works of Pappus. His work is often called the requiem of Greek mathematics. Great mathematicians whose works were revived by Pappus include Euclid Archimedes Apollonius Nicomedes , and Dinostratus Why did Herkimer have trouble making a phone call to the zoo? Answer : Because the lion was busy. Herky s friends JUSTIN TIME ...this guy always arrives at the very last minute. LEE KEEROOF ... a repairman who prevents rain water from dripping into your house. Reading : Section 3.4, pages 163-166. Written: Page 166-167/9, 13, 15, 17

38. Imperial Gloriensburg Presents: Webster's Unabridged Dictionary (1913) :: Letter A curve made use of in the quadrature of other curves; as the quadratrix, ofdinostratus, or of Tschirnhausen. Quadrature (a.) The act of squaring; http://www.gloriensburg.org.uk/library/dictionary/Q.htm

loading... Imperial Gloriensburg... writing a legend on the net! Skip to main content. The Imperial Gazetteer Article date: Modern and Vintage News updated as for Wednesday, Aug 31, 2005 Navigation: Editorial Front Page Reviews Archives ... Search... Dictionary: A B C D ... Z Translations: es de fr it ... nl by FreeTranslation.com Font Size Font Size Q ) the seventeenth letter of the English alphabet, has but one sound (that of k), and is always followed by u, the two letters together being sounded like kw, except in some words in which the u is silent. See Guide to Pronunciation, / 249. Q is not found in Anglo-Saxon, cw being used instead of qu; as in cwic, quick; cwen, queen. The name (k/) is from the French ku, which is from the Latin name of the same letter; its form is from the Latin, which derived it, through a Greek alphabet, from the Ph/nician, the ultimate origin being Egyptian. Qua conj. ) In so far as; in the capacity or character of; as. Quab n. ) An unfledged bird; hence, something immature or unfinished. Quab v. i.

39. ALC III,2: The Science Of Magnitudes Menaechmus, dinostratus, Athenaeus, Helicon, and especially Eudoxs made veryimportant mathematical discoveries. Their names today seem strange to us, http://www.domcentral.org/study/ashley/arts/arts302.htm

BENEDICT M. ASHLEY, O.P.: THE ARTS OF LEARNING AND COMMUNICATION CHAPTER II The Science of Magnitudes THE BEGINNINGS THE GREEKS, SCIENTISTS AND ARTISTS In the last chapter we indicated that, while mathematical calculation was developed in a practical way by the people of Mesopotamia and Egypt, and carried still further by the Hindus and Chinese, it was the Greeks who made it a theoretical study. They transformed it into a true science, rigorously logical in structure, and a model for all other sciences. It was these same scientifically minded Greeks who first arrived at a perfect conception of the fine arts. The art of Mesopotamia was strong and grandiose, but without grace or subtlety. The art of Egypt was subtle and mysterious, but strangely static and without inner thought or feeling. Only in the art of Greece is there achieved a living balance of all the elements of beauty. Their art was classical (from Latin classicus ,meaning "first class"), and became a standard for all later art. Not, indeed, that art of later ages need confine itself to copying the style and subject-matter of Greek art, as some people have thought but that we can learn from Greek literature, sculpture, and architecture a true conception of the elements that go into a work of art and of the harmony with which they should be united. Today we are inclined to think of science and art as unrelated fields. The artist seems to be all imagination and emotion, living in a subjective world of free fancy. The scientist seems to be all facts and abstract theories, living in the objective world of experiment and measurement. Yet the Greeks excelled both in art and science. In order to learn something of this lesson from the Greeks in this chapter we are going to try to get clearer notions of two questions:

40. Hippias2.html dinostratus (circa 350 BC) was the first to use it for this purpose, accordingto Pappus (circa 300 AD). Maple Plot http://www.ms.uky.edu/~carl/ma330/hippias/hippias21.html

Hippias and his quadratrix Hippias of Elis (430 BC) was a sophist who invented the quadratrix curve to trisect an angle. The problem of trisecting a given angle was one of the problems that generated a lot of mathematics during this period, and several mathematicians devised methods for solving this problem. Like many other sophists, Hippias was an itinerant teacher who made his living wowing the locals with his knowledge. Apparently, he did alright, but didn't leave much of a legacy except for the quadratrix. Definition of the curve The curve can be described in a few sentences. Let ABCD denote a square. Over a unit time period, allow the top segment of the square to fall at a uniform speed to the bottom of the square. During the same time, allow the left side of the square to rotate clockwise at a uniform speed to the bottom of the square. At each time, the two segments will intersect in a point P. The totality of all these points P is defined as the quadratrix. Drawing the quadratrix One can imagine how Hippias might have sketched the quadratrix in the sand, but one can hardly image how he would have made an accurate sketch of it.

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

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