DOCSLIB.ORG

C:\Users\Steve Williams\Documents\Classes\300\Fall 2011\Midterm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
C:\Users\Steve Williams\Documents\Classes\300\Fall 2011\Midterm Name: KEY MthEd/Math 300– Williams Fall 2011 Midterm Exam 1 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. There is a handy list of Greek mathematicians on the last page. Section 1: Fill in the name of the Ancient Greek mathematician who best fits the description. One point each. 1. Proved the Law of the Lever from axioms. Archimedes 2. Worked on the problem of squaring the circle, and succeeded in squaring a lune. Hippocrates 3. Developed the method of exhaustion, a theory of proportions, Eudoxus and an astronomical model involving nested spheres. 4. Was the first Greek mathematician and philosopher, supposedly Thales studied mathematics in Egypt. 5. Wrote a definitive treatment of conic sections. Apollonius 6. Used the method of exhaustion to approximate π as being Archimedes 1 10 between3 7 and 3 71 . 7. Wrote the Mathematical Collection, which came to be known as Ptolemy the Greatest Collection, and which encompassed astronomy and trigonometry. 8. Compiled and organized the mathematical knowledge of his Euclid time into what became the most published mathematics book ever. 9. Leader of a group that studied the geometric properties of whole Pythagoras numbers (.e.g. square and triangular numbers). 10. Was the first woman mathematician of which we have record. Hypatia 11. In his Arithmetica, he studied indeterminate equations and Dophantus developed a symbol system for equations. 12. Alexandrian mathematician who wrote commentaries on earlier Pappus, but also Theon works. or Hypatia. Section 2: Match the following descriptions with the correct ancient culture by circling B for Babylon, E for Egypt, or G for Greece. There may be more than one correct answer for some questions. One point each. 13. No original sources for studying their mathematical history are G available. 14. Flourished mathematically between 600 BC and 400 AD G 15. Seemed to be concerned with the problem of trading bread and E beer of various strengths. 16. In at least one source, provided a very accurate approximation to B 2 . 17. Were aware of and used the Pythagorean Theorem. BG 18. Provided demonstrative proofs for mathematical propositions. G 19. In most sources, used an approximation of 3 to determine the B area of a circle. 2 E 1 20. In a number of sources, used the formula dd to determine 9 the area of a circle. 21. Used a number system based on groups of 10 EG 22. Multiplied by doubling and adding. E 23. Divided by multiplying by reciprocals. B 24. Seemed to rely on reciprocal tables do to do some of their B arithmetic. 25. Dealt extensively with the notion of incommensurabililty. G 26. Used the notion of seked as a measure of steepness. E 27. Had trouble dealing with the fraction 1/7 in their number system. B 28. Distinguished between magnitudes and numbers G 29. Wrote all fractions (except 2/3) in terms of unit fractions. E 30. Could solve quadratic relations using geometry. BG Section 3: Problems and Short Answers. Five points each. 31. Complete the following Egyptian division of 392 by 12, by A) marking the correct rows with an “X”, and B) adding up the appropriate entries in the left hand column. 112 224 448 896 16 192 32 384 X 1/3 4 2/3 8 X 32 2/3 32. Find 356 ÷ 3 in the Babylonian style by completing this multiplication: 5 , 56 ; ×0 ;20 18 40 140 1 58; 40 33. Describe the method of exhaustion. Give the names of two mathematicians who used or developed this method. Eudoxus and Archimedes worth one point each; I also gave a point for Euclid. Finding an unknown area or volume of a figure or solid by successively approximating it with figures (or solids) of known area/volume which were inscribed within it, and which could be modified to better approximate the area / volume step by step, eventually “exhausting” the area or volume of the unknown figure/solid. It was sometimes combined with circumscribed figure or solids, and sometimes combined with a reductio ad absurdum argument to prove the result was correct. Note: I took a point off for being too specific, e.g. just talking about finding the area of a circle. I also took a point off for not capturing the idea of successively changing the inscribed figure to better and better approximations. 34. Describe what is meant by two magnitudes being incommensurable. There is no third magnitude which measures, or divides evenly into, each of the two magnitudes. Usually the only problem with this was some confusion over whether numbers were incommensurable (which of course, from a Greek viewpoint, they are not, because every pair of rational numbers has a common measure which is the unit fraction with their common denominator). 35. What is meant by geometrical algebra? Where do we read about it in Greek mathematical works? Give an example of a theorem of this kind. I gave two point for expressing the idea that geometrical algebra was the use of geometry theorems to accomplish what we today would generally accomplish using algebra. I gave one point for noting that it was found in Elements, Book II I gave two points for giving an example. Valid examples are Propositions II-4, II-5, II-6, II-11, II-14, all given on pp. 63-64 of Katz. Others were also possible, such as the construction of the square root of x given x. 36. Describe briefly what the “method” was in Archimedes’ The Method. There were several important parts to this, including: 1. It is a method of comparing two areas or volumes 2. It involved slicing the areas or volumes into segments or disks 3. It involved using the properties of the figures to... 4. ... compare distances and lengths along a line used as a lever. 5. It uses the Law of the Lever along with 3 and 4 above to get relative “weights” of the segments and disks. 6. It assumes the same relationships found in 5 apply for the whole area or volume, taking into account centers of gravity. Of these, 2, 3-4 and 5 are most critical. Points were awarded based on how many of these points you made. You didn’t have to hit all 6 to get full credit. 37. Give three reasons Diophantus’ work in his Arithmetica might qualify him to be the Greek “Father of Algebra.” 1. He used symbolism. 2. He stated general rules for dealing with subtraction (i.e. “negatives,” but not really negative numbers). 3. He stated general strategies for reducing equations to forms with known solutions. 4. He gave solutions to equations, without recourse to geometry. His solutions were numeric, not geometric. Any three of these four would work. Typically I deducted a point if you said something that just didn’t make sense. 38. Describe the Babylonian tablet called Plimpton 322 and briefly say why it is significant. Description (4 points): It was a tablet with several rows and four columns. One column was just the row number. A second column gave the hypotenuse of a right triangle (or the largest of a Pythagorean triple). A third column gave the smaller leg of the triangles (or the smallest of the Pythagorean triple). The fourth column gave the square of the ratio of the hypotenuse and the longer side (the side not given); or alternatively, the square of the ratio of the largest to the middle of the Pythagorean triple. This corresponds to the square of the secant of the angle opposite the smaller leg. (Optional but interesting and the source of my confusion – if you take the arccsc of the square root of the numbers in that fourth column, they correspond to angles growing steadily from about 45 to 58 degrees.) The significance (1 point) is that it showed that 1) the Babylonians knew about Pythagorean triples and probably knew how to generate a large number of them, and/or 2) they new about right-triangle trigonometry in some form. Section 5: Slightly longer questions; 10 points each. 39. Regarding Euclid’s Elements: a. Briefly describe the mathematical content of the Elements (include at least 5 major topics). Plane geometry; circles inscribed and circumscribed; geometric algebra; ratios and proportions of magnitudes and of numbers; similarity; polygons and their construction; number theory, including primes, composites, etc.; solid geometry; polyhedra; method of exhaustion. (5 points for five of those.) b. Why was this work so important? What “footprint” did it leave in today’s mathematics? Some combination of: 1. It brought together most of the mathematics known at the time of its publication, supplanting all other sources. 2. It introduced an axiomatic approach or axiomatic system. 3. Many mathematicians used it as a basic text; they learned mathematics from it and learned what mathematics was like from it. 4. It became “the” pattern for how to do mathematics; we still use the basic structure of postulates/definitions/ theorem/proof today. 40. Briefly describe the sources we have for our knowledge of the mathematics done in each of the following early cultures. Be sure to address whether the sources are original, and if not, describe how close to the originals they are. A. Egypt The Ahmes or Rhind Papyrus and the Moscow Papyrus, along with a few others (2 points). These were mostly original, although some were copied from earlier manuscripts, they still come directly from the time of Ancient Egypt (1 point). B. Babylon A few hundred clay tablets, mainly problem tablets and table tablets (2 points). They were mostly original.
Load more

Recommended publications
  • Lesson 3: Rectangles Inscribed in Circles
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection .
    [Show full text]
  • 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.
    [Show full text]
  • Plato As "Architectof Science"
    Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries.
    [Show full text]
  • Three Points on the Plane
    THREE POINTS ON THE PLANE ALEXANDER GIVENTAL UC BERKELEY In this lecture, we examine some properties of a geometric figure formed by three points on the plane. Any three points not lying on the same line determine a triangle. Namely, the points are the vertices of the triangle, and the segments connecting them are the sides of it. We expect the reader to be familiar with a few basic notions and facts of geometry of triangles. References like “[K], §140” will point to specific sections in: Kiselev’s Geometry. Book I: Planimetry, Sumizdat, 2006, 248 pages. Everything we assume known, as well as the initial part of the present exposition, is contained in the first half of this textbook. Circumcenter. One says a circle is circumscribed about a triangle (or shorter, is the circumcircle of it), if the circle passes through all vertices of the triangle. Theorem 1. About every triangle, a circle can be circum- scribed, and such a circle is unique. Let △ABC be a triangle with vertices A, B, C (Figure 1). A point O equidistant from1 all the three vertices is the center of a circumcircle of radius OA = OB = OC. Thus we need to show that a point equidistant from the vertices exists and is unique. Indeed, the geometric locus 2 of points equidistant from A and B is the perpendicular bisector to the segment AB, i.e. it is the line (denoted by MN in Figure 1) perpendicular to the segment AB at its midpoint (see [K], §56). Likewise, the geometric locus of points equidistant from B and C is the perpendicular bisector P Q to the segment BC.
    [Show full text]
  • A Concise History of the Philosophy of Mathematics
    A Thumbnail History of the Philosophy of Mathematics "It is beyond a doubt that all our knowledge begins with experience." - Imannuel Kant ( 1724 – 1804 ) ( However naïve realism is no substitute for truth [1] ) [1] " ... concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to comprehend the problem of the á priori as posed by Kant", from "The Problem of Space, Ether, and the Field in Physics" ( "Das Raum-, Äether- und Feld-Problem der Physik." ), by Albert Einstein, 1934 - source: "Beyond Geometry: Classic Papers from Riemann to Einstein", Dover Publications, by Peter Pesic, St. John's College, Sante Fe, New Mexico Mathematics does not have a universally accepted definition during any period of its development throughout the history of human thought. However for the last 2,500 years, beginning first with the pre - Hellenic Egyptians and Babylonians, mathematics encompasses possible deductive relationships concerned solely with logical truths derived by accepted philosophic methods of logic which the classical Greek thinkers of antiquity pioneered. Although it is normally associated with formulaic algorithms ( i.e., mechanical methods ), mathematics somehow arises in the human mind by the correspondence of observation and inductive experiential thinking together with its practical predictive powers in interpreting future as well as "seemingly" ephemeral phenomena. Why all of this is true in human progress, no one can answer faithfully. In other words, human experiences and intuitive thinking first suggest to the human mind the abstract symbols for which the economy of human thinking mathematics is well known; but it is those parts of mathematics most disconnected from observation and experience and therefore relying almost wholly upon its internal, self - consistent, deductive logics giving mathematics an independent reified, almost ontological, reality, that mathematics most powerfully interprets the ultimate hidden mysteries of nature.
    [Show full text]
  • Math2 Unit6.Pdf
    Student Resource Book Unit 6 1 2 3 4 5 6 7 8 9 10 ISBN 978-0-8251-7340-0 Copyright © 2013 J. Weston Walch, Publisher Portland, ME 04103 www.walch.com Printed in the United States of America WALCH EDUCATION Table of Contents Introduction. v Unit 6: Circles With and Without Coordinates Lesson 1: Introducing Circles . U6-1 Lesson 2: Inscribed Polygons and Circumscribed Triangles. U6-43 Lesson 3: Constructing Tangent Lines . U6-85 Lesson 4: Finding Arc Lengths and Areas of Sectors . U6-106 Lesson 5: Explaining and Applying Area and Volume Formulas. U6-121 Lesson 6: Deriving Equations . U6-154 Lesson 7: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas . U6-196 Answer Key. AK-1 iii Table of Contents Introduction Welcome to the CCSS Integrated Pathway: Mathematics II Student Resource Book. This book will help you learn how to use algebra, geometry, data analysis, and probability to solve problems. Each lesson builds on what you have already learned. As you participate in classroom activities and use this book, you will master important concepts that will help to prepare you for mathematics assessments and other mathematics courses. This book is your resource as you work your way through the Math II course. It includes explanations of the concepts you will learn in class; math vocabulary and definitions; formulas and rules; and exercises so you can practice the math you are learning. Most of your assignments will come from your teacher, but this book will allow you to review what was covered in class, including terms, formulas, and procedures.
    [Show full text]
  • INFINITE PROCESSES in GREEK MATHEMATICS by George
    INFINITE PROCESSES IN GREEK MATHEMATICS by George Clarence Vedova Thesis submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1943 UMI Number: DP70209 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI DP70209 Published by ProQuest LLC (2015). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 PREFACE The thesis is a historico-philosophic survey of infinite processes in Greek mathematics* Beginning with the first emergence, in Ionian speculative cosmogony, of the tinder lying concepts of infinity, continuity, infinitesimal and limit, the survey follows the development of infinite processes through the ages as these concepts gain in clarity, -it notes the positive contributions of various schools of thought, the negative and corrective influences of others and, in a broad way, establishes the causal chain that finally led to the more abstract processes of the mathematical schools of Cnidus (Eudoxus) and Syracuse
    [Show full text]
  • Plato Journal
    What did Plato read? Autor(es): Kutash, Emilie F. Publicado por: Imprensa da Universidade de Coimbra URL persistente: URI:http://hdl.handle.net/10316.2/42217 DOI: DOI:https://doi.org/10.14195/2183-4105_7_3 Accessed : 26-Sep-2021 21:07:36 A navegação consulta e descarregamento dos títulos inseridos nas Bibliotecas Digitais UC Digitalis, UC Pombalina e UC Impactum, pressupõem a aceitação plena e sem reservas dos Termos e Condições de Uso destas Bibliotecas Digitais, disponíveis em https://digitalis.uc.pt/pt-pt/termos. Conforme exposto nos referidos Termos e Condições de Uso, o descarregamento de títulos de acesso restrito requer uma licença válida de autorização devendo o utilizador aceder ao(s) documento(s) a partir de um endereço de IP da instituição detentora da supramencionada licença. Ao utilizador é apenas permitido o descarregamento para uso pessoal, pelo que o emprego do(s) título(s) descarregado(s) para outro fim, designadamente comercial, carece de autorização do respetivo autor ou editor da obra. Na medida em que todas as obras da UC Digitalis se encontram protegidas pelo Código do Direito de Autor e Direitos Conexos e demais legislação aplicável, toda a cópia, parcial ou total, deste documento, nos casos em que é legalmente admitida, deverá conter ou fazer-se acompanhar por este aviso. impactum.uc.pt digitalis.uc.pt JOURNAL DEZ 2007 ISSN 2079-7567 eISSN 2183-4105 PLATO 7 Established 1989 http://platosociety.org/ INTERNATIONAL PLATO SOCIETY PLATO INTERNATIONAL PL ATO Société Platonicienne JOURNALInternationale Associazione Internazionale dei Platonisti Sociedad Internacional de Platonistas Internationale Platon-Gesellschaft JOURNAL DEZ 2007 ISSN 2079-7567 eISSN 2183-4105 PLATO 7 Established 1989 http://platosociety.org/ INTERNATIONAL PLATO SOCIETY PLATO INTERNATIONAL PL ATO Société Platonicienne JOURNALInternationale Associazione Internazionale dei Platonisti Sociedad Internacional de Platonistas Internationale Platon-Gesellschaft E.
    [Show full text]
  • A Short History of Greek Mathematics
    Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers.
    [Show full text]
  • Greek Mathematics
    Greek Mathematics We’ve Read a Lot. Now, Can We Make a Good Review? What’s New and Important? • Compared to Mesopotamian and Egyptian Mathematics, Greek Mathematics: – Used proof, logic, demonstration. – Was much less “practical” in its orientation. – Was more geometrical and less arithmetical. – Was considered good for the mind, soul. What About our Sources? • No primary sources remain. • Secondary sources (or worse). • Mainly mentions in books written much later. The First Greek Mathematician We Know about Was: • Thales of Miletus (600 BC) – Father of Demonstrative logic – One of the Seven Sages – Proved five basic geometry theorems • Vertical angles, triangle inscribed in a semicircle, isosceles triangle theorem, ASA theorem, circle bisected by diameter – Not very nice; predicted eclipse; predicted olive crop; walked into holes. Next in Line is: • Pythagoras (572 BC) – Founded Pythagorean Brotherhood – Pythagorean Theorem – Incommensurability of segments – Connections to music – Mystical; believed in transmigration of souls; cult‐ like following. Next in Line is: • Hippocrates of Chios – Original Elements (lost) – Quadrature of 3 lunes – Lost his money before becoming a scholar. Next in Line is: • Eudoxus – Theory of Proportions that didn’t depend on commensurability – Method of exhaustion Next in Line is: • Euclid (300 BC) – Compiled and Organized – Clever and timeless proofs – Still held in honor. About the Elements: • Why has it been so influential? – Axiomatic approach provided foundation. – Seeing how it was done has inspired countless students. About the Elements: • What’s in the Elements? – Plane Geometry – Ratios and proportions – Number theory – Geometric Algebra – Solid Geometry About the Elements: • What are it’s weaknesses from our modern viewpoint? – Definitions are not “careful.” – Unstated assumptions are used After Euclid, we have the great.
    [Show full text]
  • 92 Some Early Greek Attempts to Square the Circle A. WASSERSTEIN
    Some Greek to the circle early attempts square A. WASSERSTEIN HE quadrature of the circle, one of the most famous mathematical problems of antiquity no less than of more modern times, appears comparatively early in the history of Greek mathematics. In the Sth century it had an established position as one of the three traditional problems, viz. quadrature or rectification of the circle, duplication of the cube and trisection of a given angle. Thinkers as different from each I 2 other as Hippocrates of Chios Anaxagoras the philosopher and 3 Antiphon the Sophist worked on it; and by 414 B.C. the endeavour "to square the circle" had become the recognized sign of the crank.4 In what follows I shall (a) examine the evidence that we have for some of the reported attempts to square the circle, and (b) attempt to reconstruct the historical relation that subsists between these attempts. Aristotle mentions an attempt by Antiphon the Sophist to square the circle.s - "No man of science is bound to solve every kind of difficulty that may be raised, but only as many as are drawn falsely from the principles of the science: it is not our business to refute those that do not arise in this way: just as it is the duty of the geometer to refute the squaring of the circle by means of segments, but it is not his duty to refute Antiphon's proof." (Translation by R. P. Hardie in the Oxford translation of Aristotle.) We owe our knowledge, such as it is, of Antiphon's method to the commentators on the Physics.6 92 Themistius, in Phys.
    [Show full text]
  • Appendix I: on the Early History of Pi
    Appendix I: On the Early History of Pi Our earliest source on 7r is a problem from the Rhind mathematical papyrus, one of the few major sources we have for the mathematics of ancient Egypt. Although the material on the papyrus in its present form comes from about 1550 B.C. during the Middle Kingdom, scholars believe that the mathematics of the document originated in the Old Kingdom, which would date it perhaps to 1900 B.C. The Egyptian source does not state an explicit value for 7r, but tells, instead, how to compute the area of a circle as the square of eight-ninths of the diameter. (One is reminded of the 'classical' problem of squaring the circle.) Quite different is the passage in I Kings 7, 23 (repeated in II Chronicles 4, 21), which clearly implies that one should multiply the diameter of a circle by 3 to find its circumference. It is worth noting, however, that the Egyptian and the Biblical texts talk about what are, a priori, different concepts: the diameter being regarded as known in both cases, the first text gives a factor used to produce the area of the circle and the other gives (by implication) a factor used to produce the circumference.1 Also from the early second millenium are Babylonian texts from Susa. In the mathematical cuneiform texts the standard factor for multiplying the diameter to produce the circumference is 3, but, according to some interpre­ tations, the Susa texts give a value of 31.2 Another source of ancient values of 7r is the class of Indian works known as the Sulba Sfitras, of which the four most important, and oldest are Baudhayana, .Apastamba, Katyayana and Manava (resp.
    [Show full text]