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Pappus Of Alexandria: more detail

Pappus of Alexandria - Book 7 of the Collection by Alexander Jones, 1986-12-31

Pappus of Alexandria and the Mathematics of Late Antiquity (Cambridge Classical Studies) by Serafina Cuomo, 2007-06-21

Selections Illustrating The History Of Greek Mathematical Works..2 Volume Set..Vol. 1:Thales To Euclid:Vol.2:Aristarchus To Pappus Of Alexandria...Loeb Classical Library

Pappus of Alexandria: Book 7 of the Collection by Pappus, 1985

Pappus of Alexandria Book 7 Part 2 Only by Alexander Jones, 1986

On the duplication of the cube in Pappus of Alexandria (IIIrd century A.D.) (Rapport / SeÌminaires de matheÌmatique pure) by E EÌtienne, 1978

The Commentary of Pappus on Book X of Euclid's Elements: Arabic Text and Translation by Pappus of Alexandria & William Thomson, 1930

lists with details

Info-history
For example, pappus of alexandria (about the end of the 4th century AD has written It (Taprobane) is one of the largest islands in the world, being 1,000
http://www.parahol.com/maldives/history1.htm

Extractions: HISTORY of Maldives The History of the Maldives is lost in antiquity. Very little information is available on the ancient people and their way of life. The late H.C.P. Bell, a British archaeologist states: "Indeed it may be preferable to assign to the original colonization of the group of dates synchronic with that of Ceylon itself (viz., several centuries before the Christian era)". Looking among the scanty information available it is suggested that people might have settled from Asia, Africa, and the Indian subcontinent, Sri Lanka, the Mediterranean, and the Far East and from many other parts of the world. Historical records and ancient stone carvings similar to those found in Sri Lanka support the theory of scholars that Sri Lankan migrated to the islands in the 12th century A.D. Yet, there are indications of the Maldives being populated as early as the 4th century BC. The Maldives was certainly known among some of the classical writers. For example, Pappus of Alexandria (about the end of the 4th century A.D. has written: "It (Taprobane) is one of the largest islands in the world, being 1,000 miles in length by 1,500 miles broad and encompasses 1,370 adjacent islands among its dependencies" This is a clear reference to Sri Lanka and Maldives.
Aust. Math. Soc. Gazette Vol 22 No 4
(See 1, 2.). I believe that pappus of alexandria (c. 300 AD) deservesmore credit than is commonly attributed to him in this matter.
http://www.austms.org.au/Publ/Gazette/Oct95/letters.html

Extractions: Australian Math Society Web Site - the Gazette There is a common belief that the method of mathematical induction is a product of Western civilisation, discovered by Maurolyco (1495-1575) and that the first clear account of the method was given by Pascal. The term `mathematical induction' was first used by the English mathematician Auguste de Morgan in 1838. In his `Justification Theorem', Dedekind proved that this procedure does not lead to logical complications. (See [1], [2].) I believe that Pappus of Alexandria (c. 300 A.D.) deserves more credit than is commonly attributed to him in this matter. Here is a short account of Pappus' proof of one of his theorems. On the same line segment , three semicircles are drawn so that they are tangent in pairs. In the region bounded by the three semicircles, a chain of tangent circles is inscribed in the manner shown below. The diameters of these tangent circles are denoted by and the distances of their centres to the baseline by respectively. Pappus' Theorem states that
On Wisconsin
Described as the father of algebra. Influenced alKhwarizmi in his work. 320 ADpappus of alexandria (Greek) Summarizes knowledge of Greek mathematicians.
http://www.uwalumni.com/onwisconsin/summer02/laska.html

Extractions: Letters On Wisconsin Magazine welcomes letters from our readers. The editors reserve the right to edit letters for length or clarity. Please mail comments to On Wisconsin, 650 North Lake Street, Madison WI 53706; fax them to (608) 265-8771; or e-mail them to WAA@uwalumni.com In the article titled "A Muslim's Jihad" in the Winter 2001 edition of On Wisconsin , some statements are made which are not entirely correct. In particular, on page 37, it states that in the last part of the first millennium and the first part of the second, "Islam produced the world's leading scientists, mathematicians, architects, and artists." It may be considered only a minor discrepancy, but this implies that all the leading scientists, etc., were produced by Islam. The words "many of" should be inserted between "produced" and "the" to make the statement true. Another statement is completely inaccurate. Muslims did not
Lecture Notes 2 - Math 3210
Lecture Notes 2. pappus of alexandria (340 AD) Pappus' Theorem Ifpoints A,B and C are on one line and A', B' and C' are on another
http://www-math.cudenver.edu/~wcherowi/courses/m3210/hg3lc2.html

Extractions: Lecture Notes 2 Pappus of Alexandria (340 A.D.) Pappus' Theorem: If points A,B and C are on one line and A', B' and C' are on another line then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and CB' lie on a common line called the Pappus line of the configuration. Axioms for the Finite Geometry of Pappus There exists at least one line. Every line has exactly three points. Not all lines are on the same point. [N.B. Change from the text] If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line. If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'. With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them. Theorem 1.10 Each point in the geometry of Pappus lies on exactly three lines. Pf . Let X be any point. By corrected axiom 3, there is a line not containing X. This line contains points A,B,C [Axiom 2]. X lies on lines meeting two of these points, say B and C [Axiom 5]. There is exactly one line through X parallel to BC [Axiom 4]. There can be no other line through X since by Axiom 4 it would have to meet BC at a point other than A, B or C [Axioms 6 and 5], and this would contradict Axiom 2. Pappus geometry has 9 points and 9 lines.
The Beginnings Of Trigonometry
pappus of alexandria, who was a teacher of mathema tics in the fourth century, observedthat Hipparchus in his book on the rising of the twelve signs of the
http://www.math.rutgers.edu/courses/436/436-s00/Papers2000/hunt.html

Extractions: Rutgers, Spring 2000 The ancient Greeks transformed trigonometry into an ordered science. Astronomy was the driving force behind advancements in trigonometry. Most of the early advancements in trigonometry were in spherical trigonometry mostly because of its application to astronomy. The three main figures that we know of in the development of Greek trigonometry are Hipparchus, Menelaus, and Ptolomy. There were likely other contributors but over time their works have been loss and their names have been forgotten. "Even if he did not invent it, Hipparchus is the first person of whose systematic use of trigonometry we have documentary evidence." (Heath 257) Some historians go as far as to say that he invented trigonometry. Not much is known about the life of Hipp archus. It is believed that he was born at Nicaea in Bithynia. (Sarton 285) The town of Nicaea is now called Iznik and is situated in northwestern Turkey. Founded in the 4th century BC, Nicaea lies on the eastern shore of Lake Iznik. He is one of the g reatest astronomers of all time. We know from Ptolemy's references that he made astronomical observations from 161 to 127 BC. (Sarton 285) Unfortunately, nearly all of his works are lost, and all that remains is his commentary on the Phainomena of Eudoxos of Cnidos, and a commentary on an astronomical poem by Aratos of Soloi. (Sarton 285) Most of what we know about Hipparchus comes from Ptolemy's
Girard Desargues (1591-1661) Desargues Applet Desargues With
Girard Desargues (15911661) Desargues applet Desargues with Cinderella,pappus of alexandria (~290-~350) Cut the knot (Pappus applet).
http://www.maths.uwa.edu.au/~2DG/papdes.html
EUCRATIDES
geometry more easily than by studying the Elements There is no royal road togeometry. pappus of alexandria, in his Mathematical Collection, says that
http://24.1911encyclopedia.org/E/EU/EUCRATIDES.htm

Extractions: certain kinds of beryl (aquamarine) and topaz, from which it may be distinguished by its specific gravity (3.1). Its hardness (7.5) is rather less than that of topaz. Euclase occurs with topaz at Boa Vista, near Ouro Preto (Villa Rica) in. the province of Minas Geraes, Brazil. It is found also with topaz and chrysoberyl in the gold-bearing gravels of the R. Sanarka in the South Urals; and is met with as a rarity in the mica-schist of the Rauris in the Austrian Alps. EUCLID OF MEGARA founder of the Megarian (also called the eristic or dialectic) school of philosophy, was born c. 450 B.c., probably at Megara, though Gela in Sicily has also been named as his birthplace (Diogenes Laertius i~. 106), and died in 374. He was one of the most devoted of the disciples of Socrates. Aulus Gellius (vi. 10) states that, when a decree was passed forbidding the Megarians to enter Athens, he regularly visited his master by night in the disguise of a woman; and he was one of the little band of intimate friends who listened to the last discourse. He withdrew subsequently with a number of fellow disciples to Megara, and it has been conjectured, though there is no direct evidence, that this was the period of Platos residence in Megara, of which indications appear in the Theaetetus. He is said to have written six dialogues, of which only the titles have been preserved. For his doctrine (a combination of the principles of Parmenides and Socrates) see MEGARIAN SCHOOL.
Untitled Document
pappus of alexandria was a Greek Mathematician. In 320 AD he composeda work with the title Collection (Synagoge). This work was
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Simmons/Essay3/Essay3.html

Extractions: EMAT 6690 Essay 3 Pappus Areas by Kimberly Burrell, Brad Simmons, and Doug Westmoreland Pappus of Alexandria was a Greek Mathematician. In 320 A.D. he composed a work with the title Collection (Synagoge) . This work was very important because on several reasons. It is the most valuable historical record of Greek Mathematics that would otherwise be unknown to us. We are able to learn that Archimedes' discovered the 13 semiregular polyhedra, which are today known as "Archimedian solids." He also include alternate proofs and supplementary lemmas for propositions from Euclid, Archimedes, Apollonius, and Ptolemy. Pappus' treatise includes new discoveries and generalizations not found in early work. The Collection contained eight books. The first book and the beginning to book two have been lost. In Book IV, Pappus included an elementary generalization of the Pythagorean theorem. He also included the following problem, which has came to be known as the Pappus areas theorem. It is not known whether or not the problem originated with Pappus, but it has been suggested that possibly it was known earlier to Heron. Consider any triangle ABC.
Adventures In CyberSound: Euclid
Renditions of the Elements. In ancient times, Hero and pappus of alexandriaand Proclus and Simplicius all wrote commentaries. Theon
http://www.acmi.net.au/AIC/EUCLID_BIO.html

Extractions: Euclid (alt: Euklid, Eucleides) : 365 - 300 BC Euclid's The Optics is the earliest surviving work on geometrical optics, and is generally found in Greek manuscripts along with elementary works on spherical astronomy. There were a number of medieval Latin translations, which became of new importance in the fifteenth century for the theory of linear perspective. This technique is beautifully illustrated in the miniature of a street scene in this elegant manuscript from the library of the Duke of Urbino. It may once have been in the possession of Piero della Francesca, who wrote one of the principal treatises on perspective in painting.
Zeal.com - United States - New - Library - Sciences - Mathematics - Mathematicia
Add a Site Profile. 1. History of Mathematics pappus of alexandria http//www.math.tamu.edu/~don.allen/history/pappus/pappus.ht
http://www.zeal.com/category/preview.jhtml?cid=554590
Egypt Math Web Sites
Hasan ibn al'Haitam; 12 pappus of alexandria; 13 Abu'lHasan ibnYunus; 14 Jean Baptiste Joseph Fourier; 15 Hypsicles of Alexandria;
http://showcase.netins.net/web/rmozzer/Egypt.html

Extractions: Born: 835 in Baghdad (now in Iraq) Died: 912 in Cairo, Egypt. Ahmed ibn Yusuf wrote on ratio and proportion and it was translated into Latin by Gherard of Cremona. The book is largely a commentary on, and expansion of, Book 5 of Euclid's Elements . Ahmed ibn Yusuf also gave methods to solve tax problems which appear in Fibonacci's Liber Abaci . He was also quoted by Bradwardine, Jordanus and Pacioli. Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja Born: about 850 in (possibly) Egypt. Died: about 930. Abu Kamil Shuja is sometimes known as al'Hasib and he worked on integer solutions of equations. He also gave the solution of a fourth degree equation and of a quadratic equation with irrational coefficients. Abu Kamil's work was the basis of Fibonacci's books. He lived later than al'Khwarizmi and his biggest advance was in the use of irrational coefficients. Theon of Alexandria Born: about 335 in (possibly) Alexandria, Egypt. Died: about 395. Theon was the father of Hypatia and worked in Alexandria as a professor of mathematics and astronomy. He produced commentaries on many works such as Ptolemy's Almagest and works of Euclid. Theon was a competent but unoriginal mathematician. Theon's version of Euclid's Elements (with textual changes and some additions) was the only Greek text of the Elements known, until an earlier one was discovered in the Vatican in the late 19
3960. Archimedes. The Columbia World Of Quotations. 1996
Quoted in Mathematical Collection, book VIII, proposition 10, section 11,pappus of alexandria (date unknown); translated into Latin (1588).
http://www.bartleby.com/66/60/3960.html

Extractions: Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference Quotations The Columbia World of Quotations PREVIOUS ... AUTHOR INDEX The Columbia World of Quotations. NUMBER: QUOTATION: Give me where to stand, and I will move the earth.
Mathematicians
Xiahou Yang (c. 350?). 300 CE. pappus of alexandria (fl. c. 300c. 350)*SB *mt. Serenus of Antinopolis (c. 350). Theon of Alexandria (c. 390).
http://www.chill.org/csss/mathcsss/mathematicians.html

Extractions: List of Mathematicians printed from: http://aleph0.clarku.edu:80/~djoyce/mathhist/mathhist.html 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 Hippias of Elis (fl. c. 425) *SB *mt Theodorus of Cyrene (c. 425) Socrates (469-399) Philolaus of Croton (d. c. 390) *SB Democritus of Abdera (c. 460-370) *SB *mt 400 B.C.E. Hippasus of Metapontum (or of Sybaris or Croton) (c. 400?) Archytas of Tarentum (of Taras) (c. 428-c. 347) *SB *mt Plato (427-347) *SB *MT Theaetetus of Athens (c. 415-c. 369) *mt Leodamas of Thasos (fl. c. 380) *SB
Directory :: Look.com
pappus of alexandria (c. 320) Wrote treatise, the Mathematical Collection, as aguide to Greek geometry, discusses theorems and constructions of more than
http://www.look.com/searchroute/directorysearch.asp?p=166424
History Of Geometry
compute detailed trigonometric tables. pappus of alexandria (290350AD) was the last of the great Greek geometers. His major work
http://geometryalgorithms.com/history.htm

Extractions: A Short History of Geometry Ancient This page gives a short outline of geometry's history, exemplified by major geometers responsible for it's evolution. Click on a person's picture or name for an expanded biography at the excellent: History of Mathematics Archive (Univ of St Andrews, Scotland). Also, Click these links for our recommended: Greek Medieval Modern History Books ... History Web Sites The geometry of Babylon (in Mesopotamia) and Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They knew how to compute areas, and even knew the "Pythagorian Theorem" 1000 years before the Greeks (see: Pythagoras's theorem in Babylonian mathematics ). But there is no evidence that they logically deduced geometric facts from basic principles. Nevertheless, they established the framework that inspired Greek geometry. A detailed analysis of Egyptian mathematics is given in the book: Mathematics in the Time of the Pharaohs India (1500 BC - 200 BC)
Lever - Wikipedia
levers are provided by Archimedes ( Give me a place to stand, and I will move theEarth. , a remark of Archimedes quoted by pappus of alexandria) who formally
http://www.wikipedia.org/wiki/Lever

Extractions: Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles Interlanguage links All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help From Wikipedia, the free encyclopedia. A lever is a rigid object that is used with an appropriate fulcrum or pivot point to multiply the mechanical force that can be applied to another object. This is also termed mechanical advantage The earliest remaining writings regarding levers are provided by Archimedes "Give me a place to stand, and I will move the Earth." , a remark of Archimedes quoted by Pappus of Alexandria ) who formally stated the correct mathematical principle of levers. The force applied (at end points of the lever) is proportional to the ratio of the lever arms measured between the fulcrum and application point of the force applied at each end of the lever. Insert a better freebody diagram here when one is available.
Reflections Vol20, No3, Aug 95
(A History of Mathematics, Carl B. Boyer, 1968, p. 209). pappus of alexandria(c. 300 AD) composed the work Collection (Synagogue) in eight books.
http://hsc.csu.edu.au/pta/mansw/reflections/vol23no2grant.htm

Extractions: A line may be regarded as the path in which a point moves, or more strictly as the 'locus' of a point, locus meaning the path which is formed by the motion of anything. So again, a line by its motion will 'generate' a surface - a moving surface generates a solid; and this surface is the locus of a line, and a solid the locus of a surface. (Outlines of Geometry, or The Motion of a Point, by Walter M. Adams, B.A., 1866, p.13) Locus The only means recognized by the ancients for defining plane curves were (1) kinematic definitions in which a point moves subject to two superimposed motions, and (2) the section by a plane of a geometrical surface, such as a cone or sphere or cylinder. (A History of Mathematics, Carl B. Boyer, 1968, p. 209) Pappus of Alexandria (c. 300 AD) composed the work Collection (Synagogue) in eight books. Book VII, Treasury of Analysis , contained works by Euclid, Apollonius, Aristaeus and Eratosthenes, which Pappus advised as most suitable for an advanced course in the method of analysis and synthesis (The Treasury of Mathematics: 1 Henrietta Midonick, 1965, p.401). This contained what is known as the Problem of Pappus, which is a generalization of 'the locus to three or four lines'. About 500 years earlier this locus to three or four lines was completely solved by Apollonius of Perga in Book III of his Conics Some fundamental loci and useful hints The locus of a point is the path traced by that point as it moves according to a given condition or conditions.
All Titles - Cambridge University Press
pappus of alexandria and the Mathematics of Late Antiquity SerafinaCuomo Hardback Published March 2000. Passions and Perceptions
http://publishing.cambridge.org/hss/classical/philosophy/all/page4
Historical Note
The problem was solved geometrically by pappus of alexandria (3rdcentury AD), and analytically by Jakob Bernoulli (1654 1705).
http://blueox.uoregon.edu/~belitz/ph612/problems/hist_note1.html

Extractions: Dido was a mythical queen of the ancient city of Carthage. According to the myth, she bought as much land as `could be covered by a cow's hide'. (This was ambiguous in Greek, where `covered' can also mean `enclosed'). She cut the hide into thin stripes and made these into a long rope. Then she faced the problem of which shape of the rope would maximize the enclosed area. The problem was solved geometrically by Pappus of Alexandria (3rd century AD), and analytically by Jakob Bernoulli
ALEXANDRIA DOCTA : BIBLIOGRAPHIE GÉNÉRALE
Translate this page Cuomo (2000) = S. Cuomo, pappus of alexandria and the Mathematics of LateAntiquity, Cambridge-New York, Cambridge University Press, 2000.
http://www.ulg.ac.be/facphl/services/cedopal/ALEXDOCT.htm

Extractions: Alexandria docta : bibliographie générale par Nathaël Istasse Avertissement: Au seuil de ce travail, nous tenons à préciser que cette bibliographie, non exhaustive, et que nous projetons de compléter régulièrement, a pour but essentiel de proposer au chercheur, comme à l'amateur, un matériau brut sur la vie intellectuelle et scientifique à Alexandrie ad Aegyptum ) durant les périodes ptolémaïque, romaine et byzantine. Ainsi, on trouvera, à côté d'ouvrages et d'articles scientifiques, des travaux de vulgarisation, portant sur des thèmes aussi divers que la vie intellectuelle et scientifique, mais également, en moindre proportion, sur l'histoire, la religion, la vie quotidienne, les realia , etc. Toutes ces études sont fondées sur des sources tant littéraires que papyrologiques, épigraphiques ou archéologiques. Y sont inclus tant des ouvrages de synthèse que des recherches particulières, par exemple sur tel ou tel représentant de ce foyer intellectuel que fut Alexandrie, sur la place de celle-ci dans le monde culturel antique, sur l'influence aristotélicienne, etc. En fin de bibliographie, on trouvera en outre plusieurs adresses URL de

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