61. Pappus' Theorem pappus of alexandria was a Greek mathematician who lived around the end of thethird century AD, although the exact date is uncertain. http://www.mathpages.com/home/kmath542/kmath542.htm

Pappus' Theorem Pappus of Alexandria was a Greek mathematician who lived around the end of the third century AD, although the exact date is uncertain. Theon made a marginal note in one of his manuscripts stating that Pappus wrote during the reign of Roman emperor Diocletian, which places him in the period from 284 to 305 AD, but it also seems from Pappus' commentary on Ptolemy's Almagest that he witnessed a particular solar eclipse that occurred in 320 AD In any case, the "Collection" (Synagoge) of Pappus is an invaluable source of information on Greek mathematics, since it contains summaries and attributions of results based on ancient works that have subsequently been lost. In addition, Pappus gave some apparently original results, such as the proposition that is commonly called "Pappus' Theorem" involving a hexagon inscribed between two lines. Consider two straight lines emanating from point O and containing the points P through P as shown in the figure below. Letting L ij denote the line through points P i and P j , and letting a,b,c denote the points of intersection between the pairs of lines [L ,L ], [L

62. On Wisconsin - Summer 2002 pappus of alexandria (Greek) Summarizes knowledge of Greek mathematicians.Fourth century Author unknown Bakshali manuscript, written in Brahmi (a script in http://www.uwalumni.com/onwisconsin/summer02/laska.html

Travel Lifelong Learning UW Libraries Resources On Wisconsin ... UW-MADISON 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

63. From Euclid To Newton pappus of alexandria (circa 300350) Mathematicarum collectionum libri V quiextant cum commentariis Federici Commandini Peasro Hieronymous Concordia http://www.brown.edu/Facilities/University_Library/exhibits/math/nofr.html

From Euclid to Newton: An Exhibition in Honor of the 1999 Conference of the Mathematical Association of America Math Exhibit Home Exhibits Home Euclid (ca. 326-265 BC) Archimedes (ca. 287-212 BC) Apollonius of Perga (ca. 260-200 BC) Nichomachus of Gerasa (ca. 100) Claudius Ptolemy (ca. 85-165) Diophantus of Alexandria (ca. 200-284) Pappus of Alexandria (ca. 300-350) Proclus (ca. 410-485) Boethius (ca. 480-524) Thomas Bradwardine (ca. 1290-1349) Girolamo Cardano Robert Recorde Johann MŸller of Kšnigsberg called Regiomontanus Franois Vite John Napier Henry Briggs Adriaan Vlacq ... Bonaventura Cavalieri (ca. 1598-1647) Christiaan Huygens RenŽ Descartes Gottfried Wilhem Leibniz Sir Isaac Newton ... Guillaume Franois Antoine l'Hospital, Marquis de Sainte-Mesme TOP Euclid Brown University Library possess a copy of each sixteenth-century translation of Euclid's Elements of Geometry into a modern language. These vernacular editions, grouped around the first Latin edition of 1482, are displayed in chronological sequence, from 1533 (Greek) to 1594 (Arabic). All copies are opened at Book I, proposition 47, "Pythagoras' Theorem," which asserts: "In right-angled triangles the square of the side opposite the right angle is equal to the sum of the squares of the sides containing the right angle." Most of the translations provide proof of this equation (a

64. An Introduction To Pappus' Theorem Pappus Theorem was discovered by pappus of alexandria in the 4th century AD,and has extraordinarily beautiful properties that makeit one of the nicest http://www.math.umd.edu/~wphooper/pappus/intro/

An Introduction to Pappus' Theorem Pappus' Theorem was discovered by Pappus of Alexandria in the 4th century AD, and has extraordinarily beautiful properties that makeit one of the nicest constructions to study in projective geometry. Begin with two lines, l and m , in the projectiveplane. Then choose three points on each line, label the three points on l X X , and X andthe three points on m Y Y , and Y . Then construct the point Z by intersectingthe line with the line .Similarly we can construct the point Z as the point that intersects both the line and the line . Finally we define the point Z to be the intersection of the line and the line . Pappus' Theorem states that the three points we just constructed, Z Z , and Z are collinear. Above is an interactive applet that demonstrates Pappus' Theorem. The blue dots represent the six points X X X Y Y , and Y and the red dots represent the points Z Z , and Z .You can drag the blue dots around and the red dots are forced to change wheneverthe blue dots move. Notice that the red dots remain collinear. Next Step: Permutations andPappus' Theorem Return to Pat's mathematics homepage

65. Untitled Document pappus of alexandria was a Greek Mathematician. In 320 AD he composed a work withthe title Collection (Synagoge). This work was very important because on http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Burrell/Essay3/Essay3.html

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.

66. Essay1 This result was proven by pappus of alexandria in the fourth century. It issometimes referred as the ancient theorem. The proof of Pappus theorem is http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Westmoreland/Essay1/Essay1.html

Arbelos: The shoemaker's knife By Doug Westmoreland, Brad Simmons, and Kimberly Burrell The arbelos is a famous figure believed to have been first studied by Archimedes. It is called the arbelos, from the Greek for "shoemaker's knife," because it resembles the blade of a knife used by ancient cobblers. It is the yellow shaded region in the figure below that is bounded by the semicircles with diameters AB, BC, and AC. B can be any point on AC. For a GSP sketch that you can drag point B along diameter AC and observe the behavior of the arbelos, click here. After checking out the GSP sketch and dragging around point B, one should quickly observe that the length of the arc ADC is equal to the sum of the arcs AEB and BFC. However, no amount of dragging and measuring the arcs will prove the above statement. So, here is a proof of the above conjecture. Given: arbelos in diagram above Prove: arclength AEB + arclength BFC = arclength ADC PROOF Let AO= x, AG= a, thus GO= x-a. And let BH=b, thus OB= x-2b. Since C=2 p r and the radius of AO= x, then the arclength of ADC=

67. Adventures In CyberSound: Euclid In ancient times, Hero and pappus of alexandria and Proclus and Simplicius allwrote commentaries. Theon of Alexandria (4th century AD) brought out a new http://www.acmi.net.au/AIC/EUCLID_BIO.html

A D V E N T U R E S in C Y B E R S O U N D 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. Source: The Vatican Library Euclid , Greek Eucleides (fl. c. 300 BC, Alexandria), the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements Life and work Of Euclid's life it is known only that he taught at and founded a school at Alexandria in the time of Ptolemy I Soter , who reigned from 323 to 285/283 BC. Medieval translators and editors often confused him with the philosopher

68. Maths pappus of alexandria mentions several other works by Apollonius but the onlyother treatise of his that is known to have survived antiquity is the work http://www.kcl.ac.uk/depsta/iss/library/speccoll/exhibitions/gsci/maths.html

Text only ISS: Information Services and Systems Services Information gateway ... Structure Case 5: Mathematics Exhibition curator: Hugh Cahill Diagram of the Archimedes Screw: Archimedes. Londini : excudebat Guil. Godbid, voeneunt apud Rob. Scott, 1675.[Rare Books Collection QA31.Ar1 B2] Archimedes. Londini : excudebat Guil. Godbid, voeneunt apud Rob. Scott, 1675. Rare Books Collection QA31.Ar1 B2 The mathematician and engineer Archimedes (ca.287-212BC) was one of the greatest minds of antiquity, making important discoveries in geometry, statics, hydrostatics and mechanics. For example, in The measurement of the circle he determined the value of Pi, with great accuracy to be between 223/71 and 220/70 and in The sand reckoner On plane equilibriums , the first extant work on statics, Archimedes discovered fundamental theorems concerning the centre of gravity of figures, such as the parallelogram, the triangle, the trapezium and a segment of a parabola. In On floating bodies he laid down the basic principles of hydrostatics. In this treatise, he determines the positions that various solids will assume when floating in a fluid, according to their shape and the variation in their specific gravities, and also establishes a number of general principles, including what has become known as Archimedes' principle. However, Archimedes was also a practical engineer who constructed war machines and other devices for his patron. He is said to have invented a system of compound pulleys capable of lifting entire ships and their contents in order to demonstrate to King Hieron the principle of mechanical advantage. He is also said to have designed the water pump that we now know as the Archimedes Screw in order to raise water from the hold of another King Hieron's ships.

69. Eratosthenes work is known principally from the writings of pappus of alexandria. After studyin Alexandria and Athens, Eratosthenes settled in Alexandria about 255 http://abyss.uoregon.edu/~js/glossary/eratosthenes.html

Eratosthenes After study in Alexandria and Athens, Eratosthenes settled in Alexandria about 255 BC and became director of the great library there. He worked out a calendar that included leap years, and he tried to fix the dates of literary and political events since the siege of Troy. His writings include a poem inspired by astronomy, as well as works on the theatre and on ethics. Eratosthenes was afflicted by blindness in his old age, and he is said to have committed suicide by voluntary starvation. Excerpt from the Encyclopedia Britannica without permission.

70. MATHEMATICS geometer, pappus of alexandria, discovered the final secret which completed thisline of though. In the diagrams 16 and 18 will be seen two lines, http://www.headmap.org/unlearn/alfred/10.htm

@import url(../../ul-css/3-col-nn4-new-main.css); MATHS back to the maths texts index A.N. WHITEHEAD an introduction to mathematics first impression 1911 CONTENTS I THE ABSTRACT NATURE OF MATHEMATICS II VARIABLES III METHODS OF APPLICATION IV DYNAMICS ... XVII QUANTITY BIBLIOGRAPHY NOTE ON THE STUDY OF MATHEMATICS HEADMAP home EUCLID the elements (I-IX) MATHEMATICS AND LOGIC quadrivium SPHAERIC naked eye astronomy SHELTERS shelter shapes WORDS texts CHAPTER X CONIC SECTIONS Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea. There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject : he is anxious at once to get on to the proofs of " important " problems. The history of the science is entirely against him. There are royal roads in science ; but those who first tread them are men of genius and not kings. (1) The plane may cut the cone in a closed oval curve, such as ABA'B' which lies entirely on one of the two half-cones. In this case the plane will not meet the other halfcone at all. Such a curve is called an ellipse ; it is an oval curve. A particular case of such a section of the cone is when the plane is perpendicular to the axis CVC', then the section, such as STU or PQR, is a circle. Hence a circle is a particular case of the ellipse.

71. Egyptology: Alexandria £25.00; CUOMO, S. pappus of alexandria and the Mathematics of Late Antiquity (CUP2000) h/b. £40.00; DECOBERT, C. EMPEREUR, J. Y. Alexandrie Medievale 1 http://www.museumbooks.demon.co.uk/egypt/e_alex.htm

Alexandria Return to book list index Return to Egypt list Home page [A] ... [Z] Entries shown in red indicate forthcoming publications Sciences exactes et sciences appliquees a Alexandrie (Memoires XVI, Centre Jean-Palerne) ASHTON, S. BAKHOUM, S. Dieux Egyptiens a Alexandrie sous les Antonins: Recherches numismatiques et historiques BERNAND, A. Alexandrie: Les Ptolemees BERNAND, E. CHUGG, A. The Lost Tomb of Alexander the Great CUOMO, S. Pappus of Alexandria and the Mathematics of Late Antiquity Alexandrie Medievale 1 EMPEREUR, J. -Y. Alexandria: Jewel of Egypt EMPEREUR, J. -Y. Alexandrina 1 Etudes Alexandrines 1 EMPEREUR, J. -Y. Alexandrina 2 Necropolis 1 FLOWER, D. Shores of Wisdom: The Story of the Ancient Library of Alexandria FRAZER, P. Ptolemaic Alexandria GODDIO, F. Alexandria: the Submerged Royal Quarters, Surveys and Excavations 1992-97 Sunken Egypt: Alexandria Alexandria II - Portus Magnus GRIMM, G. Alexandria: Die erste Konigsstadt der hellenistischen Welt HAAG, M. Alexandria: City of Memory HAAG, M. Alexandria Illustrated: new revised and expanded edition Ancient Alexandria between Egypt and Greece Alexandria, Real and Imagined

72. Krown & Spellman Booksellers: Mathematicae Collectiones... In Latinum Conversae, NOTES pappus of alexandria (b. Alexandria, fl. AD 300350) In the silver ageof Greek mathematics Pappus stands out as an accomplished and versatile http://www.krownspellman.com/cgi-bin/spellman/18179.html

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73. Engineering Database 340, pappus of alexandria writes Synagoge (Collections) which is a guide to Greekgeometry. 390, Theon of Alexandria produces a version of Euclid s Elements http://www.diracdelta.co.uk/science/source/t/i/timeline/source.html

Timeline A brief overview of the major milestones in science and engineering. Palaeolithic peoples in central Europe and France record numbers on bones. Early geometric designs used. Sheep are domesticated in the Middle East. Corn is domesticated in the Oaxaca Valley. Wheat is domesticated in Mesopotamia. Cattle are domesticated in Mesopotamia. Copper artifacts are common in the Middle East. Barley is domesticated in Egypt. Potatoes are domesticated in Peru and Bolivia. Light wooden plows are used in Mesopotamia. Kiln-fired bricks and pots are made in Mesopotamia. Irrigation is developed in Mesopotamia. The first symbols for numbers, simple straight lines, are used in Egypt. Wheeled vehicles are used in Uruk. Square-sailed ships used in Egypt. Draft oxen are used in Mesopotamia. Potter's wheel used in Mesopotamia. Pyramids are built in Egypt. Bronze is developed in Mesopotamia. Horse drawn vehicles are used in Egypt. Horse riding is developed on the Eurasia steppes. Copper is smelted in China. Rice paddies are developed in China. Iron working is developed in the Middle East.

74. Minutes From April 19, 2005 WSS TC A remark of Archimedes quoted by pappus of alexandria. 1. Call to order, rollcall 2. Reading/approving minutes of last meeting (5th April 1 ) 3. http://lists.oasis-open.org/archives/wss/200504/msg00016.html

wss message Date Prev Thread Prev Thread Next Date Next ... List Home Subject Minutes from April 19, 2005 WSS TC From "Weiland, John R. NMIMC GS" <JRWeiland@US.MED.NAVY.MIL> To : wss@lists.oasis-open.org Date : Thu, 21 Apr 2005 13:08:50 -0400 Title: Minutes from April 19, 2005 WSS TC Minutes from last meeting: 1. Call to order, roll call 2. Reading/approving minutes of last meeting (5th April [1] ) 3. Gartner interop demo - status. 5. Kerberos Interop planning status 6. Other business 7. Adjournment http://lists.oasis-open.org/archives/wss/200504/msg00006.html 1. Call to order, roll call Call to Order Role Taken 2. Reading/approving minutes of last meeting (5th April [1] ) Unanimous approval of minutes 3. Gartner interop demo - status. Hal - testing this afternoon event tomorrow. testing going on with usual results, Some work some don't, hopefully everything will be working by the end of the day. Computer Associates withdrew 14 companies now at the interop. Oasis has made up T shirts for event Testing in West Side room by 100 most activities will occur there. Marketing people providing slip sheets and packages. End User speaker from Wachovia care of Data Power.

75. Pappus' Theorem most surprising geometric results probably discovered by pappus of alexandria Pappus theorem and its dual admit slightly different formulations. http://www.cut-the-knot.org/pythagoras/Pappus.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Pappus' Theorem The word Geometry is of the Greek and Latin origin. In Latin, geo- ge- means earth, while metron is measure. Originally, the subject of Geometry was earth measurement. With time, however, both the subject and the method of geometry have changed. From the time of Euclid's Elements rd century B.C.), Geometry was considered as the epitome of the axiomatic method which itself underwent a fundamental revolution in the 19 th century. Revolutionary in many other aspects, the 19th century also witnessed metamorphosis of a single science - Geometry - into several related disciplines The subject of Projective Geometry , for one, is the incidence of geometric objects : points, lines, planes. Incidence (a point on aline, a line through a point) is preserved by projective transformations, but measurements are not. Thus in Projective Geometry, the notion of measurement is completely avoided, which makes the term - Projective Geometry - an oxymoron. In Projective Geometry

76. Reflections Vol20, No3, Aug 95 pappus of alexandria (c. 300 AD) composed the work Collection (Synagogue) ineight books. Book VII, Treasury of Analysis, contained works by Euclid, http://www.mansw.nsw.edu.au/members/reflections/vol23no2grant.htm

MANSW The Mathematical Association of New South Wales, Inc. Promoting Quality Mathematics Education for all. Locus Ken Grant, Loyola College 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

77. Mid Term Papers: Term Papers On Apollonius Of Perga writers, especially pappus of alexandria. As a youth Apollonius studied inAlexandria ( under the pupils of Euclid, according to Pappus ) and subsequently http://www.midtermpapers.com/10514.htm

Home Join Contact Us Sign-In ... Sign-Out Search Keywords: Acceptance Essays American History Anatomy Animal Science ... Zoology Apollonius Of Perga Below is a free term papers summary of the paper "Apollonius Of Perga." If you sign up , you can be reading the rest of this term papers in under two minutes. Registered users should sign-in to view this term paper. Term Paper Title Apollonius Of Perga # of Words # of Pages (250 words per page double spaced) Apollonius of Perga Apollonius of Perga Apollonius was a great mathematician, known by his contempories as " The Great Geometer, " whose treatise Conics is one of the greatest scientific works from the ancient world. Most of his other treatise were lost, although their titles and a general indication of their contents were passed on by later writers, especially Pappus of Alexandria. As a youth Apollonius studied in Alexandria ( under the pupils of Euclid, according to Pappus ) and subsequently taught at the university there. He visited Pergamum, capital of a Hellenistic kingdom in western Anatolia, where a university and library similar to those in Alexandria had recently been built.

78. Euclid In ancient times, commentaries were written by Hero of Alexandria (fl. c. AD 62),pappus of alexandria (fl. c. AD 320), Proclus, and Simplicius of Cilicia http://www.omhros.gr/Kat/History/Greek/Tc/Euclid.htm

Euclid flourished c. 300 BC , Alexandria, Egypt Greek Eukleides the most prominent mathematician of Greco-Roman antiquity, best known for his treatise on geometry, the Elements Life Of Euclid's life nothing is known except what the Greek philosopher Proclus (c. AD BC Elements BC Sources and contents of the Elements Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (fl. c. 460 BC BC BC ). The older elements were at once superseded by Euclid's and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the Platonic solids. A brief survey of the Elements BC ). While Book V can be read independently of the rest of the Elements arithmos antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as a x a x a x a x ); and Book IX proves that there are an infinite number of primes.

79. Pappus Vi 1-11 State U., LA. Return to Vignettes of Ancient Mathematics. pappus of alexandria,Mathematical Collection vi §§111, pp. 474.1-488.25 http://www.calstatela.edu/faculty/hmendel/Ancient Mathematics/Pappus/Bookvi/Papp

Pappus, from the book on spherical geometry trans. by Henry Mendell, Cal. State U., L.A. Return to Vignettes of Ancient Mathematics Pappus of Alexandria, Mathematical Collection Contents: Lemmata: These concern Menelaus trilaterals (spherical triangles whose sides are circular-arcs of great circles less than a semicircle). Prop. 1: Prop. 2: Prop. 3: Prop. 4: Let two arcs be drawn from the vertex of a trilateral to points on the base equidistant (in arcs) from the end points of the base. These form a trilateral inside a larger trilateral. The two sides from the vertex of the larger trilateral are larger than the two sides of the smaller. Theorems: The following theorems use this figure: A great-circle through the poles of two circles, one of which is an equator parallel to some latitudes and the other is at an angle to the equator. The initial great circle is at right angles to the others. We can call this (from its astronomical use), a colure-equator-ecliptic configuration. We will only be concerned with the quadrant between the intersection of the ecliptic/equator and the colure as marked. Prop. 5:

80. Canada Research Chairs - Chairholders pappus of alexandria, for example, drew surprising conclusions concerningarrangements of points in the plane. Matroid theory abstracts the combinatorial http://www.chairs.gc.ca/web/chairholders/viewprofile_e.asp?id=1117&UniversityID=

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