Post-Euclid Greek Mathematics
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Greeks Doing Algebra
Greeks Doing Algebra There is a long-standing consensus in the history of mathematics that geometry came from ancient Greece, algebra came from medieval Persia, and the two sciences did not meet until seventeenth-century France (e.g. Bell 1945). Scholars agree that the Greek mathematicians had no methods comparable to algebra before Diophantus (3rd c. CE) or, many hold, even after him (e.g. Szabó 1969, Unguru and Rowe 1981, Grattan-Guinness 1996, Vitrac 2005. For a survey of arguments see Blåsjö 2016). The problems that we would solve with algebra, the Greeks, especially the authors of the canonical and most often studied works (such as Euclid and Apollonius of Perga), approached with spatial geometry or not at all. This paper argues, however, that the methods which uniquely characterize algebra, such as information compression, quantitative abstraction, and the use of unknowns, do in fact feature in Greek mathematical works prior to Diophantus. We simply have to look beyond the looming figures of Hellenistic geometry. In this paper, we shall examine three instructive cases of algebraic problem-solving methods in Greek mathematical works before Diophantus: The Sand-reckoner of Archimedes, the Metrica of Hero of Alexandria, and the Almagest of Ptolemy. In the Sand-reckoner, Archimedes develops a system for expressing extremely large numbers, in which the base unit is a myriad myriad. His process is indefinitely repeatable, and theoretically scalable to express a number of any size. Simple though it sounds to us, this bit of information compression, by which a cumbersome quantity is set to one in order to simplify notation and computation, is a common feature of modern mathematics but was almost alien to the Greeks. -
Great Inventors of the Ancient World Preliminary Syllabus & Course Outline
CLA 46 Dr. Patrick Hunt Spring Quarter 2014 Stanford Continuing Studies http://www.patrickhunt.net Great Inventors Of the Ancient World Preliminary Syllabus & Course Outline A Note from the Instructor: Homo faber is a Latin description of humans as makers. Human technology has been a long process of adapting to circumstances with ingenuity, and while there has been gradual progress, sometimes technology takes a downturn when literacy and numeracy are lost over time or when humans forget how to maintain or make things work due to cataclysmic change. Reconstructing ancient technology is at times a reminder that progress is not always guaranteed, as when Classical civilization crumbled in the West, but the history of technology is a fascinating one. Global revolutions in technology occur in cycles, often when necessity pushes great minds to innovate or adapt existing tools, as happened when humans first started using stone tools and gradually improved them, often incrementally, over tens of thousands of years. In this third course examining the greats of the ancient world, we take a close look at inventions and their inventors (some of whom might be more legendary than actually known), such as vizier Imhotep of early dynastic Egypt, who is said to have built the first pyramid, and King Gudea of Lagash, who is credited with developing the Mesopotamian irrigation canals. Other somewhat better-known figures are Glaucus of Chios, a metallurgist sculptor who possibly invented welding; pioneering astronomer Aristarchus of Samos; engineering genius Archimedes of Siracusa; Hipparchus of Rhodes, who made celestial globes depicting the stars; Ctesibius of Alexandria, who invented hydraulic water organs; and Hero of Alexandria, who made steam engines. -
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. -
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. -
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. -
He One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’S Elements?
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT VOL. 59/2014 • ISSN 0066–6874 Zbigniew Król he One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’s Elements? ABSTRACT: his paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite diferent intuitive model (or frame), without ininite space, ininite lines and surfaces. his ancient intuitive model of Euclidean geometry is reconstructed in connection with Plato’s unwritten doctrine. he model cre- ates a kind of “hermeneutical horizon” determining the explicit content and mathematical methods used. In the irst section of the paper, it is argued that there are no actually ininite concepts in Euclid’s Elements. In the second section, it is argued that ancient mathematics is based on Plato’s highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled. KEYWORDS: Euclidean geometry • Euclid’s Elements • ancient mathematics • Plato’s unwrit- ten doctrine • philosophical hermeneutics • history of science and mathematics • philosophy of mathematics • philosophy of science he possibility to imagine all of the theorems from Euclid’s Elements in a Tquite diferent intuitive framework is interesting from the philosophical point of view. I can give one example showing how it was possible to lose the genuine image of Euclidean geometry. In many translations into Latin (Heiberg) and English (Heath) of the theorems from Book X, the term “area” (spatium) is used. For instance, the translations of X. 26 are as follows: ,,Spatium medium non excedit medium spatio rationali” and “A medial area does not exceed a medial area by a rational area”2. -
The Fascinating Story Behind Our Mathematics
The Fascinating Story Behind Our Mathematics Jimmie Lawson Louisiana State University Story of Mathematics – p. 1 Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Story of Mathematics – p. 2 Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Story of Mathematics – p. 2 Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Story of Mathematics – p. 2 Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. -
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. -
Greece: Archimedes and Apollonius
Greece: Archimedes and Apollonius Chapter 4 Archimedes • “What we are told about Archimedes is a mix of a few hard facts and many legends. Hard facts – the primary sources –are the axioms of history. Unfortunately, a scarcity of fact creates a vacuum that legends happily fill, and eventually fact and legend blur into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible, to eradicate.” – Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, p. 1. Archimedes • Facts: – Lived in Syracuse – Applied mathematics to practical problems as well as more theoretical problems – Died in 212 BCE at the hands of a Roman soldier during the attack on Syracuse by the forces of general Marcellus. Plutarch, in the first century A.D., gave three different stories told about the details of his death. Archimedes • From sources written much later: – Died at the age of 75, which would put his birth at about 287 BCE (from The Book of Histories by Tzetzes, 12th century CE). – The “Eureka” story came from the Roman architect Vitruvius, about a century after Archimedes’ death. – Plutarch claimed Archimedes requested that a cylinder enclosing a sphere be put on his gravestone. Cicero claims to have found that gravestrone in about 75 CE. Archimedes • From sources written much later: – From about a century after his death come tales of his prowess as a military engineer, creating catapults and grappling hooks connected to levers that lifted boats from the sea. – Another legend has it that he invented parabolic mirrors that set ships on fire. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
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.