Apollonius of Perga
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Extending Euclidean Constructions with Dynamic Geometry Software
Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) Extending Euclidean constructions with dynamic geometry software Alasdair McAndrew [email protected] College of Engineering and Science Victoria University PO Box 18821, Melbourne 8001 Australia Abstract In order to solve cubic equations by Euclidean means, the standard ruler and compass construction tools are insufficient, as was demonstrated by Pierre Wantzel in the 19th century. However, the ancient Greek mathematicians also used another construction method, the neusis, which was a straightedge with two marked points. We show in this article how a neusis construction can be implemented using dynamic geometry software, and give some examples of its use. 1 Introduction Standard Euclidean geometry, as codified by Euclid, permits of two constructions: drawing a straight line between two given points, and constructing a circle with center at one given point, and passing through another. It can be shown that the set of points constructible by these methods form the quadratic closure of the rationals: that is, the set of all points obtainable by any finite sequence of arithmetic operations and the taking of square roots. With the rise of Galois theory, and of field theory generally in the 19th century, it is now known that irreducible cubic equations cannot be solved by these Euclidean methods: so that the \doubling of the cube", and the \trisection of the angle" problems would need further constructions. Doubling the cube requires us to be able to solve the equation x3 − 2 = 0 and trisecting the angle, if it were possible, would enable us to trisect 60◦ (which is con- structible), to obtain 20◦. -
Es, and (3) Toprovide -Specific Suggestions for Teaching Such Topics
DOCUMENT RESUME ED 026 236 SE 004 576 Guidelines for Mathematics in the Secondary School South Carolina State Dept. of Education, Columbia. Pub Date 65 Note- I36p. EDRS Price MF-$0.7511C-$6.90 Deseriptors- Advanced Programs, Algebra, Analytic Geometry, Coucse Content, Curriculum,*Curriculum Guides, GeoMetry,Instruction,InstructionalMaterials," *Mathematics, *Number ConCepts,NumberSystems,- *Secondar.. School" Mathematies Identifiers-ISouth Carcilina- This guide containsan outline of topics to be included in individual subject areas in secondary school mathematics andsome specific. suggestions for teachin§ them.. Areas covered inclUde--(1) fundamentals of mathematicsincluded in seventh and eighth grades and general mathematicsin the high school, (2) algebra concepts for COurset one and two, (3) geometry, and (4) advancedmathematics. The guide was written With the following purposes jn mind--(1) to assist local .grOupsto have a basis on which to plan a rykathematics 'course of study,. (2) to give individual teachers an overview of a. particular course Or several cOur:-:es, and (3) toprovide -specific sUggestions for teaching such topics. (RP) Ilia alb 1 fa...4...w. M".7 ,noo d.1.1,64 III.1ai.s3X,i Ala k JS& # Aso sA1.6. It tilatt,41.,,,k a.. -----.-----:--.-:-:-:-:-:-:-:-:-.-. faidel1ae,4 icii MATHEMATICSIN THE SECONDARYSCHOOL Published by STATE DEPARTMENT OF EDUCATION JESSE T. ANDERSON,State Superintendent Columbia, S. C. 1965 Permission to Reprint Permission to reprint A Guide, Mathematics in Florida Second- ary Schools has been granted by the State Department of Edu- cation, Tallahassee, Flmida, Thomas D. Bailey, Superintendent. The South Carolina State Department of Education is in- debted to the Florida State DepartMent of Education and the aahors of A Guide, Mathematics in Florida Secondary Schools. -
Unaccountable Numbers
Unaccountable Numbers Fabio Acerbi In memoriam Alessandro Lami, a tempi migliori HE AIM of this article is to discuss and amend one of the most intriguing loci corrupti of the Greek mathematical T corpus: the definition of the “unknown” in Diophantus’ Arithmetica. To do so, I first expound in detail the peculiar ter- minology that Diophantus employs in his treatise, as well as the notation associated with it (section 1). Sections 2 and 3 present the textual problem and discuss past attempts to deal with it; special attention will be paid to a paraphrase contained in a let- ter of Michael Psellus. The emendation I propose (section 4) is shown to be supported by a crucial, and hitherto unnoticed, piece of manuscript evidence and by the meaning and usage in non-mathematical writings of an adjective that in Greek math- ematical treatises other than the Arithmetica is a sharply-defined technical term: ἄλογος. Section 5 offers some complements on the Diophantine sign for the “unknown.” 1. Denominations, signs, and abbreviations of mathematical objects in the Arithmetica Diophantus’ Arithmetica is a collection of arithmetical prob- lems:1 to find numbers which satisfy the specific constraints that 1 “Arithmetic” is the ancient denomination of our “number theory.” The discipline explaining how to calculate with particular, possibly non-integer, numbers was called in Late Antiquity “logistic”; the first explicit statement of this separation is found in the sixth-century Neoplatonic philosopher and mathematical commentator Eutocius (In sph. cyl. 2.4, in Archimedis opera III 120.28–30 Heiberg): according to him, dividing the unit does not pertain to arithmetic but to logistic. -
15 Famous Greek Mathematicians and Their Contributions 1. Euclid
15 Famous Greek Mathematicians and Their Contributions 1. Euclid He was also known as Euclid of Alexandria and referred as the father of geometry deduced the Euclidean geometry. The name has it all, which in Greek means “renowned, glorious”. He worked his entire life in the field of mathematics and made revolutionary contributions to geometry. 2. Pythagoras The famous ‘Pythagoras theorem’, yes the same one we have struggled through in our childhood during our challenging math classes. This genius achieved in his contributions in mathematics and become the father of the theorem of Pythagoras. Born is Samos, Greece and fled off to Egypt and maybe India. This great mathematician is most prominently known for, what else but, for his Pythagoras theorem. 3. Archimedes Archimedes is yet another great talent from the land of the Greek. He thrived for gaining knowledge in mathematical education and made various contributions. He is best known for antiquity and the invention of compound pulleys and screw pump. 4. Thales of Miletus He was the first individual to whom a mathematical discovery was attributed. He’s best known for his work in calculating the heights of pyramids and the distance of the ships from the shore using geometry. 5. Aristotle Aristotle had a diverse knowledge over various areas including mathematics, geology, physics, metaphysics, biology, medicine and psychology. He was a pupil of Plato therefore it’s not a surprise that he had a vast knowledge and made contributions towards Platonism. Tutored Alexander the Great and established a library which aided in the production of hundreds of books. -
Fiboquadratic Sequences and Extensions of the Cassini Identity Raised from the Study of Rithmomachia
Fiboquadratic sequences and extensions of the Cassini identity raised from the study of rithmomachia Tom´asGuardia∗ Douglas Jim´enezy October 17, 2018 To David Eugene Smith, in memoriam. Mathematics Subject Classification: 01A20, 01A35, 11B39 and 97A20. Keywords: pythagoreanism, golden ratio, Boethius, Nicomachus, De Arithmetica, fiboquadratic sequences, Cassini's identity and rithmomachia. Abstract In this paper, we introduce fiboquadratic sequences as a consequence of an extension to infinity of the board of rithmomachia. Fiboquadratic sequences approach the golden ratio and provide extensions of Cassini's Identity. 1 Introduction Pythagoreanism was a philosophical tradition, that left a deep influence over the Greek mathematical thought. Its path can be traced until the Middle Ages, and even to present. Among medieval scholars, which expanded the practice of the pythagoreanism, we find Anicius Manlius Severinus Boethius (480-524 A.D.) whom by a free translation of De Institutione Arithmetica by Nicomachus of Gerasa, preserved the pythagorean teaching inside the first universities. In fact, Boethius' book became the guide of study for excellence during quadriv- ium teaching, almost for 1000 years. The learning of arithmetic during the arXiv:1509.03177v3 [math.HO] 22 Nov 2016 quadrivium, made necessary the practice of calculation and handling of basic mathematical operations. Surely, with the mixing of leisure with this exercise, Boethius' followers thought up a strategy game in which, besides the training of mind calculation, it was used to preserve pythagorean traditions coming from the Greeks and medieval philosophers. Maybe this was the origin of the philoso- phers' game or rithmomachia. Rithmomachia (RM, henceforward) became the ∗Department of Mathematics. -
ANCIENT PROBLEMS VS. MODERN TECHNOLOGY 1. Geometric Constructions Some Problems, Such As the Search for a Construction That Woul
ANCIENT PROBLEMS VS. MODERN TECHNOLOGY SˇARKA´ GERGELITSOVA´ AND TOMA´ Sˇ HOLAN Abstract. Geometric constructions using a ruler and a compass have been known for more than two thousand years. It has also been known for a long time that some problems cannot be solved using the ruler-and-compass method (squaring the circle, angle trisection); on the other hand, there are other prob- lems that are yet to be solved. Nowadays, the focus of researchers’ interest is different: the search for new geometric constructions has shifted to the field of recreational mathematics. In this article, we present the solutions of several construction problems which were discovered with the help of a computer. The aim of this article is to point out that computer availability and perfor- mance have increased to such an extent that, today, anyone can solve problems that have remained unsolved for centuries. 1. Geometric constructions Some problems, such as the search for a construction that would divide a given angle into three equal parts, the construction of a square having an area equal to the area of the given circle or doubling the cube, troubled mathematicians already hundreds and thousands of years ago. Today, we not only know that these problems have never been solved, but we are even able to prove that such constructions cannot exist at all [8], [10]. On the other hand, there is, for example, the problem of finding the center of a given circle with a compass alone. This is a problem that was admired by Napoleon Bonaparte [11] and one of the problems that we are able to solve today (Mascheroni found the answer long ago [9]). -
Theon of Alexandria and Hypatia
CREATIVE MATH. 12 (2003), 111 - 115 Theon of Alexandria and Hypatia Michael Lambrou Abstract. In this paper we present the story of the most famous ancient female math- ematician, Hypatia, and her father Theon of Alexandria. The mathematician and philosopher Hypatia flourished in Alexandria from the second part of the 4th century until her violent death incurred by a mob in 415. She was the daughter of Theon of Alexandria, a math- ematician and astronomer, who flourished in Alexandria during the second part of the fourth century. Information on Theon’s life is only brief, coming mainly from a note in the Suda (Suida’s Lexicon, written about 1000 AD) stating that he lived in Alexandria in the times of Theodosius I (who reigned AD 379-395) and taught at the Museum. He is, in fact, the Museum’s last attested member. Descriptions of two eclipses he observed in Alexandria included in his commentary to Ptolemy’s Mathematical Syntaxis (Almagest) and elsewhere have been dated as the eclipses that occurred in AD 364, which is consistent with Suda. Although originality in Theon’s works cannot be claimed, he was certainly immensely influential in the preservation, dissemination and editing of clas- sic texts of previous generations. Indeed, with the exception of Vaticanus Graecus 190 all surviving Greek manuscripts of Euclid’s Elements stem from Theon’s edition. A comparison to Vaticanus Graecus 190 reveals that Theon did not actually change the mathematical content of the Elements except in minor points, but rather re-wrote it in Koini and in a form more suitable for the students he taught (some manuscripts refer to Theon’s sinousiai). -
Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/306051060 Golden Ratio: A subtle regulator in our body and cardiovascular system? Article in International journal of cardiology · August 2016 DOI: 10.1016/j.ijcard.2016.08.147 CITATIONS READS 8 266 3 authors, including: Selcuk Ozturk Ertan Yetkin Ankara University Istinye University, LIV Hospital 56 PUBLICATIONS 121 CITATIONS 227 PUBLICATIONS 3,259 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: microbiology View project golden ratio View project All content following this page was uploaded by Ertan Yetkin on 23 August 2019. The user has requested enhancement of the downloaded file. International Journal of Cardiology 223 (2016) 143–145 Contents lists available at ScienceDirect International Journal of Cardiology journal homepage: www.elsevier.com/locate/ijcard Review Golden ratio: A subtle regulator in our body and cardiovascular system? Selcuk Ozturk a, Kenan Yalta b, Ertan Yetkin c,⁎ a Abant Izzet Baysal University, Faculty of Medicine, Department of Cardiology, Bolu, Turkey b Trakya University, Faculty of Medicine, Department of Cardiology, Edirne, Turkey c Yenisehir Hospital, Division of Cardiology, Mersin, Turkey article info abstract Article history: Golden ratio, which is an irrational number and also named as the Greek letter Phi (φ), is defined as the ratio be- Received 13 July 2016 tween two lines of unequal length, where the ratio of the lengths of the shorter to the longer is the same as the Accepted 7 August 2016 ratio between the lengths of the longer and the sum of the lengths. -
Phantom Paths from Problems to Equations
Archimedes and the Angel: Phantom Paths from Problems to Equations Fabio Acerbi CNRS, UMR 8163 ‘Savoirs, textes, langage’, Lille [email protected] Introduction This paper is a critical review of Reviel Netz’ The Transformation of Mathematics in the Early Mediterranean World.1 Its aim is to show that Netz’ methods of inquiry are too often unsatisfactory and to argue briefly for a substantially different interpretation of the evi- dence which he adduces. These two aims are pursued in parallel throughout the review and can hardly be disjoined. The review is uncommonly long and uncommonly direct, at times perhaps even a trifle vehement, in criticizing the methods and conclusions displayed in the book. There are two reasons for this. First, the transforma- tion of Academic scholarship into a branch of the editorial business and, very recently, into an expanding division of the media-driven star-system, has dramatically reduced the time left to study primary sources, to get properly informed of works by other scholars,2 and even just to read more than once what was written as a first draft. Second, at the same time, it is increasingly difficult to find serious reviews. Though the number of reviews and book-notices is explod- ing, it is clear that most reviews are written just to get a copy of an otherwise too expensive book. At any rate, the current policy is to keep sharp criticisms, if any, hidden under a seemingly gentle stylistic surface. The present paper is organized as follows. In the first section, I present Archimedes’ problem; in the next, I report the contents 1 R. -
Babylonian Astral Science in the Hellenistic World: Reception and Transmission
CAS® e SERIES Nummer 4 / 2010 Francesca Rochberg (Berkeley) Babylonian Astral Science in the Hellenistic World: Reception and Transmission Herausgegeben von Ludwig-Maximilians-Universität München Center for Advanced Studies®, Seestr. 13, 80802 München www.cas.lmu.de/publikationen/eseries Nummer 4 / 2010 Babylonian Astral Science in the Hellenistic World: Reception and Transmission Francesca Rochberg (Berkeley) In his astrological work the Tetrabiblos, the astronomer such as in Strabo’s Geography, as well as in an astrono- Ptolemy describes the effects of geography on ethnic mical text from Oxyrhynchus in the second century of character, claiming, for example, that due to their specific our era roughly contemporary with Ptolemy [P.Oxy. geographical location „The ...Chaldeans and Orchinians 4139:8; see Jones 1999, I 97-99 and II 22-23]. This have familiarity with Leo and the sun, so that they are astronomical papyrus fragment refers to the Orchenoi, simpler, kindly, addicted to astrology.” [Tetr. 2.3] or Urukeans, in direct connection with a lunar parameter Ptolemy was correct in putting the Chaldeans and identifiable as a Babylonian period for lunar anomaly Orchinians together geographically, as the Chaldeans, or preserved on cuneiform tablets from Uruk. The Kaldayu, were once West Semitic tribal groups located Babylonian, or Chaldean, literati, including those from in the parts of southern and western Babylonia known Uruk were rightly famed for astronomy and astrology, as Kaldu, and the Orchinians, or Urukayu, were the „addicted,” as Ptolemy put it, and eventually, in Greco- inhabitants of the southern Babylonian city of Uruk. He Roman works, the term Chaldean came to be interchan- was also correct in that he was transmitting a tradition geable with „astrologer.” from the Babylonians themselves, which, according to a Hellenistic Greek writers seeking to claim an authorita- Hellenistic tablet from Uruk [VAT 7847 obv. -
Construction of Regular Polygons a Constructible Regular Polygon Is One That Can Be Constructed with Compass and (Unmarked) Straightedge
DynamicsOfPolygons.org Construction of regular polygons A constructible regular polygon is one that can be constructed with compass and (unmarked) straightedge. For example the construction on the right below consists of two circles of equal radii. The center of the second circle at B is chosen to lie anywhere on the first circle, so the triangle ABC is equilateral – and hence equiangular. Compass and straightedge constructions date back to Euclid of Alexandria who was born in about 300 B.C. The Greeks developed methods for constructing the regular triangle, square and pentagon, but these were the only „prime‟ regular polygons that they could construct. They also knew how to double the sides of a given polygon or combine two polygons together – as long as the sides were relatively prime, so a regular pentagon could be drawn together with a regular triangle to get a regular 15-gon. Therefore the polygons they could construct were of the form N = 2m3k5j where m is a nonnegative integer and j and k are either 0 or 1. The constructible regular polygons were 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ... but the only odd polygons in this list are 3,5 and 15. The triangle, pentagon and 15-gon are the only regular polygons with odd sides which the Greeks could construct. If n = p1p2 …pk where the pi are odd primes then n is constructible iff each pi is constructible, so a regular 21-gon can be constructed iff both the triangle and regular 7-gon can be constructed. -
Alexandrea Ad Aegyptvm the Legacy of Multiculturalism in Antiquity
Alexandrea ad aegyptvm the legacy of multiculturalism in antiquity editors rogério sousa maria do céu fialho mona haggag nuno simões rodrigues Título: Alexandrea ad Aegyptum – The Legacy of Multiculturalism in Antiquity Coord.: Rogério Sousa, Maria do Céu Fialho, Mona Haggag e Nuno Simões Rodrigues Design gráfico: Helena Lobo Design | www.hldesign.pt Revisão: Paula Montes Leal Inês Nemésio Obra sujeita a revisão científica Comissão científica: Alberto Bernabé, Universidade Complutense de Madrid; André Chevitarese, Universidade Federal, Rio de Janeiro; Aurélio Pérez Jiménez, Universidade de Málaga; Carmen Leal Soares, Universidade de Coimbra; Fábio Souza Lessa, Universidade Federal, Rio de Janeiro; José Augusto Ramos, Universidade de Lisboa; José Luís Brandão, Universidade de Coimbra; Natália Bebiano Providência e Costa, Universidade de Coimbra; Richard McKirahan, Pomona College, Claremont Co-edição: CITCEM – Centro de Investigação Transdisciplinar «Cultura, Espaço e Memória» Via Panorâmica, s/n | 4150-564 Porto | www.citcem.org | [email protected] CECH – Centro de Estudos Clássicos e Humanísticos | Largo da Porta Férrea, Universidade de Coimbra Alexandria University | Cornice Avenue, Shabty, Alexandria Edições Afrontamento , Lda. | Rua Costa Cabral, 859 | 4200-225 Porto www.edicoesafrontamento.pt | [email protected] N.º edição: 1152 ISBN: 978-972-36-1336-0 (Edições Afrontamento) ISBN: 978-989-8351-25-8 (CITCEM) ISBN: 978-989-721-53-2 (CECH) Depósito legal: 366115/13 Impressão e acabamento: Rainho & Neves Lda. | Santa Maria da Feira [email protected] Distribuição: Companhia das Artes – Livros e Distribuição, Lda. [email protected] Este trabalho é financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e Tecnologia no âmbito do projecto PEst-OE/HIS/UI4059/2011 HYPATIA AND THE IDIOSCYNCRASIES OF CHRISTIANITY IN EGYPT – A STUDY OF THE EVENTS OCCURRED AT EASTER 415 A.D.