Archimedes and Liu Hui on Circles and Spheres Joseph W
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The Evolution of Mathematics in Ancient China: from the Newly Discovered Shu and Suan Shu Shu Bamboo Texts to the Nine Chapters
The Evolution of Mathematics in Ancient China: From the Newly Discovered Shu and Suan shu shu Bamboo Texts to the Nine Chapters on the Art of Mathematics*,† by Joseph W. Dauben‡ The history of ancient Chinese mathematics and texts currently being conserved and studied at its applications has been greatly stimulated in Tsinghua University and Peking University in Beijing, the past few decades by remarkable archaeological the Yuelu Academy in Changsha, and the Hubei discoveries of texts from the pre-Qin and later Museum in Wuhan, it is possible to shed new light periods that for the first time make it possible to on the history of early mathematical thought and its study in detail mathematical material from the time applications in ancient China. Also discussed here are at which it was written. By examining the recent developments of new techniques and justifications Warring States, Qin, and Han bamboo mathematical given for the problems that were a significant part of the growing mathematical corpus, and which * © 2014 Joseph W. Dauben. Used with permission. eventually culminated in the comprehensive Nine † This article is based on a lecture presented in September of 2012 at the Fairbank Center for Chinese Studies at Har- Chapters on the Art of Mathematics. vard University, which was based on a lecture first given at National Taiwan Tsinghua University (Hsinchu, Taiwan) in the Spring of 2012. I am grateful to Thomas Lee of National Chiaotung University of Taiwan where I spent the academic Contents year 2012 as Visiting Research Professor at Chiaota’s Insti- tute for Humanities and Social Sciences, which provided sup- 1 Recent Archaeological Excavations: The Shu port for much of the research reported here, and to Shuchun and Suan shu shu ................ -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Archimedes of Syracuse
5 MARCH 2020 Engineering: Archimedes of Syracuse Professor Edith Hall Archimedes and Hiero II’s Syracuse Archimedes was and remains the most famous person from Syracuse, Sicily, in history. He belonged to the prosperous and sophisticated culture which the dominantly Greek population had built in the east of the island. The civilisation of the whole of ancient Sicily and South Italy was called by the Romans ‘Magna Graecia’ or ‘Great Greece’. The citis of Magna Graecia began to be annexed by the Roman Republic from 327 BCE, and most of Sicily was conquered by 272. But Syracuse, a large and magnificent kingdom, the size of Athens and a major player in the politics of the Mediterranean world throughout antiquity, succeeded in staying independent until 212. This was because its kings were allies of Rome in the face of the constant threat from Carthage. Archimedes was born into this free and vibrant port city in about 287 BCE, and as far as we know lived there all his life. When he was about twelve, the formidable Hiero II came to the throne, and there followed more than half a century of peace in the city, despite momentous power struggles going on as the Romans clashed with the Carthaginians and Greeks beyond Syracuse’s borders. Hiero encouraged arts and sciences, massively expanding the famous theatre. Archimedes’ background enabled him to fulfil his huge inborn intellectual talents to the full. His father was an astronomer named Pheidias. He was probably sent to study as a young man to Alexandria, home of the famous library, where he seems to have became close friend and correspondent of the great geographer and astonomer Eratosthenes, later to become Chief Librarian. -
A Centennial Celebration of Two Great Scholars: Heiberg's
A Centennial Celebration of Two Great Scholars: Heiberg’s Translation of the Lost Palimpsest of Archimedes—1907 Heath’s Publication on Euclid’s Elements—1908 Shirley B. Gray he 1998 auction of the “lost” palimp- tains four illuminated sest of Archimedes, followed by col- plates, presumably of laborative work centered at the Walters Matthew, Mark, Luke, Art Museum, the palimpsest’s newest and John. caretaker, remind Notices readers of Heiberg was emi- Tthe herculean contributions of two great classical nently qualified for scholars. Working one century ago, Johan Ludvig support from a foun- Heiberg and Sir Thomas Little Heath were busily dation. His stature as a engaged in virtually “running the table” of great scholar in the interna- mathematics bequeathed from antiquity. Only tional community was World War I and a depleted supply of manuscripts such that the University forced them to take a break. In 2008 we as math- of Oxford had awarded ematicians should honor their watershed efforts to him an honorary doc- make the cornerstones of our discipline available Johan Ludvig Heiberg. torate of literature in Photo courtesy of to even mathematically challenged readers. The Danish Royal Society. 1904. His background in languages and his pub- Heiberg lications were impressive. His first language was In 1906 the Carlsberg Foundation awarded 300 Danish but he frequently published in German. kroner to Johan Ludvig Heiberg (1854–1928), a He had publications in Latin as well as Arabic. But classical philologist at the University of Copenha- his true passion was classical Greek. In his first gen, to journey to Constantinople (present day Is- position as a schoolmaster and principal, Heiberg tanbul) to investigate a palimpsest that previously insisted that his students learn Greek and Greek had been in the library of the Metochion, i.e., the mathematics—in Greek. -
Hawai'i Reader In
Traditional Chinese Culture Edited by Victor H. Mair Nancy S. Steinhardt and Paul R. Goldin University of Hawai'i Press Honolulu 34 | Cao Pi, "A Discourse on Literature" CAO PI (187-226) was the second son of the formidable military dictator Cao Cao (155-220), whose story is told in Romance of the Three Kingdoms fSanguoyanyi'), and the elder brother of the poet Cao Zhi (192-232). He ended the puppet rule of the last Eastern Han emperor and founded the Wei dynasty in 220; thus he is also known as Emperor Wen of the Wei. Like his father and brother, Cao Pi was an accomplished writer and poet. His "Discourse on Literature" (Lun wen) survives from a critical work entided A Treatise on the Classics (Dianlun), of which only some frag- ments are extant. One of the most notable things about this essay is Cao Pi's use of the notion of qi, breath or vital force, as a concept of literary criticism that has had an immense influence on classical Chinese literary thought. "A Discourse on Literature" is itself a beautiful literary composition. It is particularly touch- ing because we see in it a writer who was secretly anxious and insecure about the value of his work and the uncertain possibility of immortal fame. A great deal of the first part of the essay is devoted to the reasoning that one's literary talent is an inherent quality that cannot be learned or obtained by hard efforts (li qiang); in the second part of the essay, he claims that a person may achieve literary immortality by exerting himself (qiang li). -
Unit 3, Lesson 1: How Well Can You Measure?
GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3, Lesson 1: How Well Can You Measure? 1. Estimate the side length of a square that has a 9 cm long diagonal. 2. Select all quantities that are proportional to the diagonal length of a square. A. Area of a square B. Perimeter of a square C. Side length of a square 3. Diego made a graph of two quantities that he measured and said, “The points all lie on a line except one, which is a little bit above the line. This means that the quantities can’t be proportional.” Do you agree with Diego? Explain. 4. The graph shows that while it was being filled, the amount of water in gallons in a swimming pool was approximately proportional to the time that has passed in minutes. a. About how much water was in the pool after 25 minutes? b. Approximately when were there 500 gallons of water in the pool? c. Estimate the constant of proportionality for the number of gallons of water per minute going into the pool. Unit 3: Measuring Circles Lesson 1: How Well Can You Measure? 1 GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3: Measuring Circles Lesson 1: How Well Can You Measure? 2 GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3, Lesson 2: Exploring Circles 1. Use a geometric tool to draw a circle. Draw and measure a radius and a diameter of the circle. 2. Here is a circle with center and some line segments and curves joining points on the circle. Identify examples of the following. -
Archimedes and Pi
Archimedes and Pi Burton Rosenberg September 7, 2003 Introduction Proposition 3 of Archimedes’ Measurement of a Circle states that π is less than 22/7 and greater than 223/71. The approximation πa ≈ 22/7 is referred to as Archimedes Approximation and is very good. It has been reported that a 2000 B.C. Babylonian approximation is πb ≈ 25/8. We will compare these two approximations. The author, in the spirit of idiot’s advocate, will venture his own approximation of πc ≈ 19/6. The Babylonian approximation is good to one part in 189, the author’s, one part in 125, and Archimedes an astonishing one part in 2484. Archimedes’ approach is to circumscribe and inscribe regular n-gons around a unit circle. He begins with a hexagon and repeatedly subdivides the side to get 12, 24, 48 and 96-gons. The semi-circumference of these polygons converge on π from above and below. In modern terms, Archimede’s derives and uses the cotangent half-angle formula, cot x/2 = cot x + csc x. In application, the cosecant will be calculated from the cotangent according to the (modern) iden- tity, csc2 x = 1 + cot2 x Greek mathematics dealt with ratio’s more than with numbers. Among the often used ratios are the proportions among the sides of a triangle. Although Greek mathematics is said to not know trigonometric functions, we shall see how conversant it was with these ratios and the formal manipulation of ratios, resulting in a theory essentially equivalent to that of trigonometry. For the circumscribed polygon We use the notation of the Dijksterhuis translation of Archimedes. -
Archimedes Palimpsest a Brief History of the Palimpsest Tracing the Manuscript from Its Creation Until Its Reappearance Foundations...The Life of Archimedes
Archimedes Palimpsest A Brief History of the Palimpsest Tracing the manuscript from its creation until its reappearance Foundations...The Life of Archimedes Birth: About 287 BC in Syracuse, Sicily (At the time it was still an Independent Greek city-state) Death: 212 or 211 BC in Syracuse. His age is estimated to be between 75-76 at the time of death. Cause: Archimedes may have been killed by a Roman soldier who was unaware of who Archimedes was. This theory however, has no proof. However, the dates coincide with the time Syracuse was sacked by the Roman army. The Works of Archimedes Archimedes' Writings: • Balancing Planes • Quadrature of the Parabola • Sphere and Cylinder • Spiral Lines • Conoids and Spheroids • On Floating Bodies • Measurement of a Circle • The Sandreckoner • The Method of Mechanical Problems • The Stomachion The ABCs of Archimedes' work Archimedes' work is separated into three Codeces: Codex A: Codex B: • Balancing Planes • Balancing Planes • Quadrature of the Parabola • Quadrature of the Parabola • Sphere and Cylinder • On Floating Bodies • Spiral Lines Codex C: • Conoids and Spheroids • The Method of Mechanical • Measurement of a Circle Problems • The Sand-reckoner • Spiral Lines • The Stomachion • On Floating Bodies • Measurement of a Circle • Balancing Planes • Sphere and Cylinder The Reappearance of the Palimpsest Date: On Thursday, October 29, 1998 Location: Christie's Acution House, NY Selling price: $2.2 Million Research on Palimpsest was done by Walter's Art Museum in Baltimore, MD The Main Researchers Include: William Noel Mike Toth Reviel Netz Keith Knox Uwe Bergmann Codex A, B no more Codex A and B no longer exist. -
On Archimedes' Measurement of a Circle, Proposition 3
On Archimedes’ Measurement of a circle, Proposition 3 Mark Reeder February 2 1 10 The ratio of the circumference of any circle to its diameter is less than 3 7 but greater than 3 71 . Having related the area of a circle to its perimeter in Prop. 1, Archimedes next approximates the circle perimeter with circumscribed and inscribed regular polygons and then finds good rational estimates for these polygon perimeters, thereby approximating the ratio of circumference to diameter. The main geometric step is to see how the polygon perimeter changes when the number of sides is doubled. We will consider the circumscribed case. Let AC be a side of a regular circumscribing polygon, and let AD be a side of a regular polygon with the number of sides doubled. C D θ O A θ B To make Archimedes’ computation easier to follow, let x = AC, y = AD, r = OA, c = OC, d = OD. We want to express the new ratio y/r in terms of the old ratio x/r. But these numbers will be very small after a few subdivisions, so they will be difficult to estimate. Instead, we will express r/y in terms of r/x. These are big numbers, which can be estimated by integers. 1 From Euclid VI.3, an angle bisector divides the opposite side in the same ratio as the other two sides of a triangle. Hence CD : DA = OC : OA. In our notation, this means x − y c x c + r r r c = , or = , or = + . y r y r y x x From Euclid I.47, we have r c r2 = 1 + , x x2 so that r r r r2 = + 1 + (1) y x x2 Thus, the new ratio r/y is expressed in terms of the old ratio r/x, as desired. -
Post-Euclid Greek Mathematics
Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Some high points of Greek mathematics after Euclid Algebra Through History October 2019 Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Outline 1 Archimedes 2 Apollonius and the Conics 3 How Apollonius described and classified the conic sections Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Who was Archimedes? Lived ca. 287 - 212 BCE, mostly in Greek city of Syracuse in Sicily Studied many topics in what we would call mathematics, physics, engineering (less distinction between them at the time) We don’t know much about his actual life; much of his later reputation was based on somewhat dubious anecdotes, e.g. the “eureka moment,” inventions he was said to have produced to aid in defense of Syracuse during Roman siege in which he was killed, etc. Perhaps most telling: we do know he designed a tombstone for himself illustrating the discovery he wanted most to be remembered for (discussed by Plutarch, Cicero) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Figure: Sphere inscribed in cylinder of equal radius 3Vsphere = 2Vcyl and Asphere = Acyl (lateral area) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works On the Equilibrium of Planes (2 books) On Floating Bodies (2 books) Measurement of a Circle On Conoids and Spheroids On Spirals On the Sphere and Cylinder (2 books) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works, cont. -
Papers Presented All Over World Inc
THE GREAT WALL OF CHINA: The World’s Greatest Boundary Monument! John F. Brock, Australia Keywords: Ancient China, surveyors, Pei Xiu, Liu Hui, The Haidao Suanjing, Great Wall(s) of China, Greatest Boundary Monument. ”… in the endeavors of mathematical surveying, China’s accomplishments exceeded those realized in the West by about one thousand years.” Frank Swetz – last line in The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China. ABSTRACT It is said that the Great Wall of China is the only manmade structure on Earth which is visible from space (not from the Moon)! The only natural feature similarly identifiable from the outer reaches past our atmospheric zone has been named as Australia’s Great Barrier Reef. This Fig. 1 The moon from The Great Wall instead of natural wonder of the sea is vice versa which cannot actually occur !!! continuous while the Great Wall of China is actually made up of a series of castellated walls mainly erected along ridge lines causing major variations in the levels of its trafficable upper surface. Some of the barriers built are not formed from stone but from rammed earth mounds. The purpose for these walls was primarily to facilitate protection from hostile adjoining tribes and marauding hordes of enemy armies intent on looting and pillaging the coffers of its neighbouring wealthier Chinese Dynasty of the time. As the need for larger numbers of military troops became required to defeat the stronger opponents, which may sometimes have formed alliances, the more astute provincial rulers saw a similar advantage in the unification of the disparate Chinese Provinces particularly during the Ming Dynasty (1368-1644). -
Order in the Cosmos
11/12/2015 Order in the Cosmos: how Babylonians and Greeks Shaped our World 1 11/12/2015 2 11/12/2015 Two distinct periods of flowering: • Old Babylonian astronomy: during and after First Babylonian dynasty (Hammurabi) 1830‐1531 BCE • New Babylonian/Chaldean astronomy: Neo‐Babylonian (Nebuchadnezzar) 626‐539 BCE Medo‐Persian 539‐331 BCE Seleucid 335‐141 BCE Parthian 129 BCE‐224 AD timeline Babylonian astronomy Evans 1998 3 11/12/2015 Babylonian Astronomers: ∏ most consistent, systematic and thorough astronomical observers of antiquity ∑ First to recognize periodicity astronomical phenomena (e.g. eclipses !), and apply mathematical techniques for predictions ∑ Systematically observed and recorded the heavens: ‐ Records spanning many centuries (> millennium) ‐ Archives of cuneiform tablets ‐ Famous Examples: Enuma Anu Enlil 68‐70 tablets Kassite period (1650‐1150) tablet 63: Venus tablet of Ammisaduga MUL.APIN 700 BCE oldest copy: 686 BCE • Several types of astronomical texts in Babylonian astronomy. • Four principal types: 1) astronomical diaries 2) goal year texts 3) ephemerides 4) procedure texts • Ephemerides: ‐ listing of positions of planets and their meaning (eg. extreme points retrograde path) ‐ predictive: positions based on calculations (based on scheme) ‐ ephemerides for Moon ‐ ephemerides for planets • Procedure texts: description of procedure(s) to calculate ephemerides 4 11/12/2015 Old text, probably Kassite period (1595‐1157 BCE) • Amajor series of 68 or 70 tablets • dealing with Babylonian astrology. • bulk is a substantial collection of omens, estimated to number between 6500 and 7000, • interpreting a wide variety of celestial and atmospheric phenomena in terms relevant to the king and state 2. If with it a cloudbank lies on the right of the sun: the trade in barley and straw will expand.