Hippocrates' Quadrature of the Lune

Total Page：16

File Type：pdf, Size：1020Kb

Math 305, Section 1 • Mathematics from a Historical Perspective • UNM, Fall 2008 Hippocrates’ Quadrature of the Lune Part 1 Question Set 1 Due: 9.3.2008 1. What is the term “awakening” used to describe in relation to the history of mathematics? 2. What were the two main contributions of the Pythagorean brotherhood to modern mathematics? 3. What does it mean for two quantities to be commensurable? 4. Hippasus discovered that the side of square and its diagonal are not commensurable. How does this amount to the discovery of irrational numbers, speaking in today’s terms? 5. Why was Hippasus discovery regarded as a triumph of geometry over arithmetic? 6. What is it meant by the word “quadrature,” and why were the Greeks concerned with finding out which figures were quadrable? 7. Draw a compass and straight-edge picture that shows a Lune known to be quadrable today. 8. Using a compass and a straight-edge only, construct a hexagon inscribed in a circle. Describe, using a detailed series of steps, the construction in question. 9. Hippocrates of Chios, Lindemann, Tschebatorew, and Dorodnow each contributed to the results known today concerning quadratures of Lunes. Explain what each of these individuals/collaborators did and how it built on and/or formed a foundation for the work of the others. 10. What is a constructible number? Give an example of a number that is constructible, and explain how you know it is. Give an example of a number that is not constructible and explain how you know it is not. 11. State Lindemann’s result on the transcendental nature of π. Explain how this result can be used to prove that the circle is not squarable. 12. Write a summary of Chapter 1 of Dunham’s book that fits in at most two (single-sided) pages. The emphasis should be on describing how the different results presented weave a piece of mathematical history through time..

Copy Link

Recommended publications

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]
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]
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]
Euclid of Alexandria the Writer of the Elements
Euclid of Alexandria The Writer of the Elements OLLI Summer 2014 1 BIO • Little is known of Euclid's life. According to Proclus (410-485 A.D.) in his Commentary on the First Book of Euclid's Elements , he came after the first pupils of Plato and lived during the reign of Ptolemy I (306-283 B.C.). Pappus of Alexandria (fl. c. 320 A.D.) in his Collection states that Apollonius of Perga (262-190 B.C.) studied for a long while in that city under the pupils of Euclid. Thus it is generally accepted that Euclid flourished at Alexandria in around 300 B.C. and established a mathematical school there. 2 Ptolemy I (323 – 285 BC) • Upon the death of Alexander the Great in 323 BC, the throne of Egypt fell to Ptolemy one of Alexander’s trusted commanders • It is said that Ptolemy once asked Euclid if there was in geometry any shorter way than that of the Elements. Euclid replied that there was no royal road to geometry. • From the time of Archimedes onwards, the Greeks referred to Euclid as the writer of the Elements instead of using his name. 3 Euclid’s Elements (300 BC) The greatest mathematical textbook of all time • Book I. The fundamentals of geometry: theories of triangles, parallels, and area. • Book II. Geometric algebra. • Book III. Theory of circles. • Book IV. Constructions for inscribed and circumscribed figures. • Book V. Theory of abstract proportions. • Book VI. Similar figures and proportions in geometry. • Book VII. Fundamentals of number theory. • Book VIII. Continued proportions in number theory.
[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. These Notes In this set of notes I will leisurely trace the history of this problem. By “leisurely” I mean that we will take many detours and examine side issues that are of interest (to me at least), both mathematical and historical, as well as sociological and philosophical. 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.
[Show full text]
Doubling the Cube: a New Interpretation of Its Significance for Early Greek Geometry
HISTORIA MATHEMATICA 22 (1995), 119-137 Doubling the Cube: A New Interpretation of Its Significance for Early Greek Geometry KEN SAITO Faculty of Letters, Chiba University, Yayoi-cho, lnage-ku, Chiba, 263 Japan It is widely known that Hippocrates of Chios reduced the problem of doubling the cube to the problem of finding two mean proportionals between two given lines. Nothing, however, is known about how this reduction was justified. To answer this question, propositions and patterns of arguments in Books VI, XI, and XII of the Elements are examined. A reconstruction modelled after Archimedes' On Sphere and Cylinder, Proposition II-1, is proposed, and its plausibility is discussed. © 1995 AcademicPress, Inc. I1 est admis qu'Hippocrate de Chio a r6duit la duplication du cube au probl~me de la recherche de deux moyennes proportionnelles entre deux segments donn6s, mais rien n'est certain quant ~ la justification de cette r6duction. Pour r6pondre ~ cette question, sont exami- n6es les propositions comme les argumentations caract6ristiques des livres VI, XI, et XII des Elements. Une reconstruction sera propos6e d'apr~s De la sphere et du cylindre, proposition II-1 d'Archimbde, et son caract6re plausible sera discut6. © 1995 AcademicPress, Inc. Es ist wohlbekannt, daft Hippocrates von Chios das Problem der Verdoppelung des Wtirfels auf das Problem der Auffindung von zwei mittleren Proportionalen zwischen zwei gegebenen Geraden zurtickfiihrte. Jedoch ist nichts bekannt tiber die Rechffertigung dieser Reduktion. Um diese Frage zu beantworten, werden S~itze und Strukturen von Argumenten aus den Btichern VI, XI, und XII der Elemente untersucht. Eine Rekonstruktion wird dann vorgeschla- gen, die sich auf Satz II-1 in Archimedes Uber Kugel und Zylinder stiitzt, und ihre Plausibilitat wird diskutiert.
[Show full text]
Mathematical Analysis in Ancient Greece
WDS'08 Proceedings of Contributed Papers, Part I, 27–31, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS Mathematical Analysis in Ancient Greece K. Cernˇ eková Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The article deals with ideas of mathematical analysis which appeared in the ancient Greece. Our goal is to present the method, that gave us the modern perspective on the history of these ideas. We start with the theory of exhaustion used in the twelfth book of Euclid’s Elements (ca. 300 B.C.), which bears the idea of correct integration techniques based on Proposition X,1. However, Euclid did not invented this theory, nor referred to the author. Later on, Archimedes mentioned in his works that earlier geometers used this lemma to prove important theorems of plane geometry, and Eudoxus (4th century B.C.) was the first one, who proved some of them. This indicates, and is nowadays believed, that Eudoxus proved Proposition X,1 and many other theorems from the twelfth book of Euclid’s Elements. On the other hand Simplicius stated that one of those important geometrical theorems was already proved by Hippocrates of Chios in the 5th century B.C. But he did not mention, how Hippocrates proved it. This yields us a dilemma whether to call a founder somebody who discovered the idea and intuitively used it in one case, or someone who firmly formulated and proved the basic idea and explained its usage in many problems. Ideas of Mathematical Analysis The theory of mathematical analysis from a modern point of view was established in the 17th century, but we can trace the ideas throughout history.
[Show full text]
Theoretical Mathematics: Born of Play by the Leisure Class, Flourished As a Social Status Signal by Allen Chai in His Lecture No
Theoretical Mathematics: Born of Play by the Leisure Class, Flourished as a Social Status Signal By Allen Chai In his lecture notes, Dr. Ji states that the history of Ancient Greece, as a civilization, can be divided into four periods, and he gives specific start and end dates: the Greek Dark Ages (1100- 750 B.C.), the Archaic Period (750-480 B.C.), the Classical Period (500-323 B.C.), and the Hellenistic Period (323-146 B.C.). When his first ten lectures discuss Ancient Greek mathematics, it is primarily to introduce great Greek mathematicians of the Classical Period and their achievements1: Zeno of Elia (490-430 B.C.), Hippocrates of Chios (470-410 B.C.), Hippias of Elis (born about 460 B.C.), Plato (427- 347 B.C.), Eudoxus of Cnidus (390-337 B.C., according to Wikipedia2), and Aristotle (384-322 B.C.). When his next ten lecture notes (11-20) (2016) discuss Ancient Greek mathematics, it is primarily to introduce great Greek mathematicians of the Hellenistic Period; Archimedes of Syracuse (287-212 B.C., lecture 12), Diophantus of Alexandria (about 284-200 B.C., lecture 13), Apollonius of Perga (262-190 B.C., lecture 11); review the broad contributions and limitations of 1. Only Thales of Miletus (624-547 B.C.), Pythagoras of Samos (born between 580-572, died 497 B.C.), and Euclid of Alexandria (lived around 300 B.C.) don’t belong to the Classical Period. Thales and Pythagoras belong to the Archaic Period, but it is reasonable to question their inclusion as great mathematicians, or mathematicians at all.
[Show full text]
MA 341 Fall 2011
MA 341 Fall 2011 The Origins of Geometry 1.1: Introduction In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. The Greek historian Herodotus (5th century BC) credits the Egyptians with having originated the subject, but there is much evidence that the Babylonians, the Hindu civilization, and the Chinese knew much of what was passed along to the Egyptians. The Babylonians of 2,000 to 1,600 BC knew much about navigation and astronomy, which required knowledge of geometry. Clay tablets from the Sumerian (2100 BC) and the Babylonian cultures (1600 BC) include tables for computing products, reciprocals, squares, square roots, and other mathematical functions useful in financial calculations. Babylonians were able to compute areas of rectangles, right and isosceles triangles, trapezoids and circles. They computed the area of a circle as the square of the circumference divided by twelve. The Babylonians were also responsible for dividing the circumference of a circle into 360 equal parts. They also used the Pythagorean Theorem (long before Pythagoras), performed calculations involving ratio and proportion, and studied the relationships between the elements of various triangles. See Appendices A and B for more about the mathematics of the Babylonians. 1.2: A History of the Value of π The Babylonians also considered the circumference of the circle to be three times the diameter. Of course, this would make 3 — a small problem.
[Show full text]