The Symbolic and Mathematical Influence of Diophantus's Arithmetica
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The Diophantine Equation X 2 + C = Y N : a Brief Overview
Revista Colombiana de Matem¶aticas Volumen 40 (2006), p¶aginas31{37 The Diophantine equation x 2 + c = y n : a brief overview Fadwa S. Abu Muriefah Girls College Of Education, Saudi Arabia Yann Bugeaud Universit¶eLouis Pasteur, France Abstract. We give a survey on recent results on the Diophantine equation x2 + c = yn. Key words and phrases. Diophantine equations, Baker's method. 2000 Mathematics Subject Classi¯cation. Primary: 11D61. Resumen. Nosotros hacemos una revisi¶onacerca de resultados recientes sobre la ecuaci¶onDiof¶antica x2 + c = yn. 1. Who was Diophantus? The expression `Diophantine equation' comes from Diophantus of Alexandria (about A.D. 250), one of the greatest mathematicians of the Greek civilization. He was the ¯rst writer who initiated a systematic study of the solutions of equations in integers. He wrote three works, the most important of them being `Arithmetic', which is related to the theory of numbers as distinct from computation, and covers much that is now included in Algebra. Diophantus introduced a better algebraic symbolism than had been known before his time. Also in this book we ¯nd the ¯rst systematic use of mathematical notation, although the signs employed are of the nature of abbreviations for words rather than algebraic symbols in contemporary mathematics. Special symbols are introduced to present frequently occurring concepts such as the unknown up 31 32 F. S. ABU M. & Y. BUGEAUD to its sixth power. He stands out in the history of science as one of the great unexplained geniuses. A Diophantine equation or indeterminate equation is one which is to be solved in integral values of the unknowns. -
The Natures of Numbers in and Around Bombelli's L
THE NATURES OF NUMBERS IN AND AROUND BOMBELLI’S L’ALGEBRA ROY WAGNER Abstract. The purpose of this paper is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Re- naissance mathematicians of the 14th–16th centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for under- determined meanings combine with shifting principles of organisation to change the character of algebra. 1. Introduction 1.1. Scope and methodology. In the year 1572 Rafael Bombelli’s L’algebra came out in print. This book, which covers arithmetic, algebra and geom- etry, is best known for one major feat: the first recorded use of roots of negative numbers to find a real solution of a real problem. The purpose of this paper is to understand the semiotic processes that enabled this and other, less ‘heroic’ achievements laid out in Bombelli’s work. The context for my analysis of Bombelli’s work is the vernacular abba- cus1 tradition spanning across two centuries, from a time where almost all problems that are done in the abbacus way reduce to the rule of three2 to the Renaissance solution of the cubic and quartic equations, and from the abbacus masters, who taught elementary mathematics to merchant children, through to humanist scholars. My purpose is to track down sign Part of the research for this paper was conducted while visiting Boston University’s Center for the Philosophy of Science, the Max Planck Institute for the History of Science and the Edelstein Center for the History and Philosophy of Science, Technology and Medicine. -
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. -
Hypatia of Alexandria A. W. Richeson National Mathematics Magazine
Hypatia of Alexandria A. W. Richeson National Mathematics Magazine, Vol. 15, No. 2. (Nov., 1940), pp. 74-82. Stable URL: http://links.jstor.org/sici?sici=1539-5588%28194011%2915%3A2%3C74%3AHOA%3E2.0.CO%3B2-I National Mathematics Magazine is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Sun Nov 18 09:31:52 2007 Hgmdnism &,d History of Mdtbenzdtics Edited by G. -
The Theorem of Pythagoras
The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. -
Arithmetic Sequences, Diophantine Equations and the Number of the Beast Bryan Dawson Union University
Arithmetic Sequences, Diophantine Equations and the Number of the Beast Bryan Dawson Union University Here is wisdom. Let him who has understanding calculate the number of the beast, for it is the number of a man: His number is 666. -Revelation 13:18 (NKJV) "Let him who has understanding calculate ... ;" can anything be more enticing to a mathe matician? The immediate question, though, is how do we calculate-and there are no instructions. But more to the point, just who might 666 be? We can find anything on the internet, so certainly someone tells us. A quick search reveals the answer: according to the "Gates of Hell" website (Natalie, 1998), 666 refers to ... Bill Gates ill! In the calculation offered by the website, each letter in the name is replaced by its ASCII character code: B I L L G A T E S I I I 66 73 76 76 71 65 84 69 83 1 1 1 = 666 (Notice, however, that an exception is made for the suffix III, where the value of the suffix is given as 3.) The big question is this: how legitimate is the calculation? To answer this question, we need to know several things: • Can this same type of calculation be performed on other names? • What mathematics is behind such calculations? • Have other types of calculations been used to propose a candidate for 666? • What type of calculation did John have in mind? Calculations To answer the first two questions, we need to understand the ASCII character code. The ASCII character code replaces characters with numbers in order to have a numeric way of storing all characters. -
94 Erkka Maula
ORGANON 15 PROBLÊMES GENERAUX Erkka Maula (Finland) FROM TIME TO PLACE: THE PARADIGM CASE The world-order in philosophical cosmology can be founded upon time as well as .space. Perhaps the most fundamental question pertaining to any articulated world- view concerns, accordingly, their ontological and epistemological priority. Is the basic layer of notions characterized by temporal or by spatial concepts? Does a world-view in its development show tendencies toward the predominance of one set of concepts rather than the other? At the stage of its relative maturity, when the qualitative and comparative phases have paved the way for the formation of quantitative concepts: Which are considered more fundamental, measurements of time or measurements of space? In the comparative phase: Is the geometry of the world a geometry of motion or a geometry of timeless order? In the history of our own scientific world-view, there seems to be discernible an oscillation between time-oriented and space-oriented concept formation.1 In the dawn, when the first mathematical systems of astronomy and geography appear, shortly before Euclid's synthesis of the axiomatic thought, there were attempts at a geometry of motion. They are due to Archytas of Tarentum and Eudoxus of Cnidus, foreshadowed by Hippias of Elis and the Pythagoreans, who tend to intro- duce temporal concepts into geometry. Their most eloquent adversary is Plato, and after him the two alternative streams are often called the Heraclitean and the Parmenidean world-views. But also such later and far more articulated distinctions as those between the statical and dynamic cosmologies, or between the formalist and intuitionist philosophies of mathematics, can be traced down to the original Greek dichotomy, although additional concepts entangle the picture. -
Survey of Modern Mathematical Topics Inspired by History of Mathematics
Survey of Modern Mathematical Topics inspired by History of Mathematics Paul L. Bailey Department of Mathematics, Southern Arkansas University E-mail address: [email protected] Date: January 21, 2009 i Contents Preface vii Chapter I. Bases 1 1. Introduction 1 2. Integer Expansion Algorithm 2 3. Radix Expansion Algorithm 3 4. Rational Expansion Property 4 5. Regular Numbers 5 6. Problems 6 Chapter II. Constructibility 7 1. Construction with Straight-Edge and Compass 7 2. Construction of Points in a Plane 7 3. Standard Constructions 8 4. Transference of Distance 9 5. The Three Greek Problems 9 6. Construction of Squares 9 7. Construction of Angles 10 8. Construction of Points in Space 10 9. Construction of Real Numbers 11 10. Hippocrates Quadrature of the Lune 14 11. Construction of Regular Polygons 16 12. Problems 18 Chapter III. The Golden Section 19 1. The Golden Section 19 2. Recreational Appearances of the Golden Ratio 20 3. Construction of the Golden Section 21 4. The Golden Rectangle 21 5. The Golden Triangle 22 6. Construction of a Regular Pentagon 23 7. The Golden Pentagram 24 8. Incommensurability 25 9. Regular Solids 26 10. Construction of the Regular Solids 27 11. Problems 29 Chapter IV. The Euclidean Algorithm 31 1. Induction and the Well-Ordering Principle 31 2. Division Algorithm 32 iii iv CONTENTS 3. Euclidean Algorithm 33 4. Fundamental Theorem of Arithmetic 35 5. Infinitude of Primes 36 6. Problems 36 Chapter V. Archimedes on Circles and Spheres 37 1. Precursors of Archimedes 37 2. Results from Euclid 38 3. Measurement of a Circle 39 4. -
History of Algebra and Its Implications for Teaching
Maggio: History of Algebra and its Implications for Teaching History of Algebra and its Implications for Teaching Jaime Maggio Fall 2020 MA 398 Senior Seminar Mentor: Dr.Loth Published by DigitalCommons@SHU, 2021 1 Academic Festival, Event 31 [2021] Abstract Algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities (letters and other general sym- bols) defined by the equations that they satisfy. Algebraic problems have survived in mathematical writings of the Egyptians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts. In this paper, we will discuss historically famous mathematicians from all over the world along with their key mathematical contributions. Mathe- matical proofs of ancient and modern discoveries will be presented. We will then consider the impacts of incorporating history into the teaching of mathematics courses as an educational technique. 1 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 2 Maggio: History of Algebra and its Implications for Teaching 1 Introduction In order to understand the way algebra is the way it is today, it is important to understand how it came about starting with its ancient origins. In a mod- ern sense, algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they sat- isfy. Algebraic problems have survived in mathematical writings of the Egyp- tians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts, but these concepts had a heavier focus on geometry [1]. The combination of all of the discoveries of these great mathematicians shaped the way algebra is taught today. -
Arithmetical Proofs in Arabic Algebra Jeffery A
This article is published in: Ezzaim Laabid, ed., Actes du 12è Colloque Maghrébin sur l'Histoire des Mathématiques Arabes: Marrakech, 26-27-28 mai 2016. Marrakech: École Normale Supérieure 2018, pp 215-238. Arithmetical proofs in Arabic algebra Jeffery A. Oaks1 1. Introduction Much attention has been paid by historians of Arabic mathematics to the proofs by geometry of the rules for solving quadratic equations. The earliest Arabic books on algebra give geometric proofs, and many later algebraists introduced innovations and variations on them. The most cited authors in this story are al-Khwārizmī, Ibn Turk, Abū Kāmil, Thābit ibn Qurra, al-Karajī, al- Samawʾal, al-Khayyām, and Sharaf al-Dīn al-Ṭūsī.2 What we lack in the literature are discussions, or even an acknowledgement, of the shift in some authors beginning in the eleventh century to give these rules some kind of foundation in arithmetic. Al-Karajī is the earliest known algebraist to move away from geometric proof, and later we see arithmetical arguments justifying the rules for solving equations in Ibn al-Yāsamīn, Ibn al-Bannāʾ, Ibn al-Hāʾim, and al-Fārisī. In this article I review the arithmetical proofs of these five authors. There were certainly other algebraists who took a numerical approach to proving the rules of algebra, and hopefully this article will motivate others to add to the discussion. To remind readers, the powers of the unknown in Arabic algebra were given individual names. The first degree unknown, akin to our �, was called a shayʾ (thing) or jidhr (root), the second degree unknown (like our �") was called a māl (sum of money),3 and the third degree unknown (like our �#) was named a kaʿb (cube).