DOCSLIB.ORG

Modern Insights Into Ancient Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modern Insights Into Ancient Mathematics Modern Insights into Ancient Mathematics Debora Mroczek September 2019 When we discuss ancient mathematics, the revolutionary ideas of Thales, Euclid, and Pythagoras, it all seems set in stone, both figuratively and literally. Many of these ideas are so old, that they were actually recorded in stone tablets. From the perspective of modern mathematics, one might wonder, with so many great minds that have devoted entire lifetimes to thinking about the fundamental rules of mathematics, what is even left to think about? What contributions could we possibly make after thousands of years? One could also guess that current research in mathematics happens at much higher level than it did in Ancient Greece, so there would be no point in looking back to Ancient mathematics, except for historical purposes. That is true only to a certain extent. While the tools mathematicians have access to nowadays, such as modern number theory and supercomputers, certainly give them an advantage, they are also in part still trying to understand the same aspects of mathematics as the famous Ancient thinkers. Perhaps the most representative example of this idea is the Pythagorean Theorem. For a right angled triangle, a2 + b2 = c2 | a result commonly used by ninth graders and expert mathematicians alike. The oldest mention to the Pythagorean Theorem dates back to Babylonian tablets from 1600 B.C. In the tablets, there were pictures and examples Figure 1: Babylonian tablet that served effectively as proof of special cases of the theo- from around 1600 B.C. with illustration of special cases of rem [1]. For instance, one of the illustrations proved that the the Pythagorean Theorem. 1 p p p diagonal of a square with sides of length one is 2, which effectively says 12 + 12 = 2. It cannot be determined with much credibility if Pythagoras (570 B.C. - 495 B.C.) or a member of his school was the first person to prove the theorem for the general case. Later on, Euclid (365 B.C. - c.300 B.C), in Elements, provided a proof for the theorem in Proposition I.47. Also, in Proposition I.48., Euclid proved that the theorem is reversible, i.e. the converse is also true. A triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled [2]. Although Euclid was the first to provide a robust approach to both the Pythagorean Theorem and its converse, in the two thousand years that followed, Pythagoras's famous theorem collected over 300 proofs, not only for the theorem itself but also for analogues and generalizations. More precisely, in 1968, there were 367 dif- ferent known proofs for the Pythagorean Theorem and its variations [3]. The fact that hundreds of proofs for the same theorem already exist does not mean that people have stopped trying. In 2015, yet another proof of the Pythagorean Theorem was published [4]. Naturally, one might wonder, what is the point of prov- ing the same thing over and over again? It certainly does not make the results more true. So why are we still trying to shed light on something that was proven to be true two thousand years ago? At this point, even a former U.S. pres- Figure 2: The cracked domino, a rather sim- ident, namely James Garfield, has managed to prove that ple visual proof of the 2 2 2 Pythagorean Theorem by for a right angled triangle, a + b = c . The question of Pakoslaw Gwizdalski. intellectual resources might also come into play | why are people who could be otherwise making new discoveries of possibly much greater impact wasting time on something that has been proven true hundreds of times? Regardless of whether these new proofs are driven by a layman's curiosity or by serious and committed mathematical research, there is value in finding new ways to prove old results. Sometimes the value is purely pedagogical. In the case of James Garfield and Kaushik Basu, author of the 2015 proof, both proofs use different methods and concepts 2 to arrive at the final result. Hence, we learned something new about how these particular methods and concepts can be applied. Perhaps that will inspire another reader to use a similar approach to a different problem. Furthermore, by looking at a problem from many different perspectives we can sometimes find new results. In the case of the Pythagorean Theorem, many analogies and generalizations were found. For instance, the law of cosines and the Euclidean distance in R3, both of which have vast applications across mathematics and physics, were derived directly form the Pythagorean Theorem as generalizations of the original result. No matter how many times something has been proved, we always seem to gain from learning new ways to prove it. Although an interesting case, the Pythagorean Theorem is mostly an outlier. It is not the norm for ancient mathematical ideas to get proved over and over again, hundreds of times. Nor is it the case that a relationship discovered thousands of years ago continues to have an impact in modern mathematics. There is, however, serious research taking place all across the world that aims to refine and deepen our understanding of concepts that have been around since ancient times. Let's take number theory, for instance. The study of integers and their properties. That is perhaps the most intuitive field of mathematics, because it concerns our day-to-day reality. After all, the most basic thing one can do with numbers is to count them. Since ancient times people have separated integers, in particular natural numbers, into categories and tried to establish relationships between groups and categories. Yet, number theory remains an active field of mathematics in the academic environment, with many brilliant mathematicians dedicating their lives to the study of the relationships within and between different types of numbers (odd, even, prime, square, Fibonacci, etc). A fascinating example of how modern number theory continued to develop an ancient concept is the case of perfect numbers. Around 100 A.C., Nicomachus of Gerasa (c.60 - c.120 A.C.), published his first book \Arithmetike eisagoge" (Introduction to Arithmetic), in which he makes the distinction between even and odd numbers, then proceeds to refine this categorization and develops the concept of prime and composite numbers. 3 Nicomachus also introduces the concept relatively prime pairs (numbers that only have 1 as a common divider) [5]. Clearly, Nicomachus's work was a game changer for arithmetic (number theory being its modern equivalent), and we might never understand just how much of an impact his work had at the time. One of the concepts Nicomachus discusses in his book is the idea of perfect numbers. Perfect numbers are numbers which parts add up to the number itself. Nicomachus thought of a number's divisors as its parts. For instance, 6 is a perfect number because 6 is divisible by 1, 2, and 3, and 6 = 1 + 2 + 3. Notice that in Nicomachus's definition, 6 itself is left out, since 6 is the number itself and not one of its parts. Ancient Greeks were able to find four of such numbers: 6, 28 496, and 8128. At the time this seemed to hint at a conjecture. We have one perfect number in the units (6), one in the tens (28), one in the hundreds (496), and one in the thousands (8128). One could find it be reasonable to expect a single perfect number Nk for each group of k digits. In fact, in the 16th century, Girolamo Cardano, one of the greatest mathematicians of that time, carelessly made that conjecture, which now we know is false. In fact, the number of digits increases pretty quickly as we move along the list of perfect numbers. The fifth known perfect number, 33; 550; 336, has 8 digits (a four digit jump), and the ninth on the list accumulates an astounding 37 digits. In his book, Nicomachus provides a procedure for find- ing perfect numbers, which is reproduced in Euclid's Ele- ments as Preposition IX.36, \If the sum of the numbers 1; 2; 4; :::; 2(n−1) is prime, then this sum multiplied by the last term will be perfect." This recipe is not necessarily the only way to obtain perfect numbers, but it is the only known pro- cedure. It might seem like having an expression would have made it easy for mathematicians to produce an extensive list of perfect numbers back then, but as we have seen the num- Figure 3: Nicomachus of Gerasa (c. 100 A.C) [6]. ber of digits starts to increase rapidly as we go along. Before 4 high-performance computers became the norm, we only knew a dozen of these perfect numbers. In 1951, the biggest known perfect number had 77 digits. In 1952, when the next perfect number was found, it became clear why the help of technology was needed | the 13th perfect number has 314 digits. Between 1952 and 2018, 38 more perfect numbers were discovered, the largest of them accumulating a mind-blowing 49,724,095 digits. Once again, one might wonder, why does it matter? While there is no practical application for perfect numbers, the fact that modern mathematicians kept digging at this ancient concept continued to raise questions which answers define the underlying structure of natural numbers. These big questions have opened the door to scientific collaboration and discovery. For instance, the recipe given by Nicomachus in his book also happens to generate prime numbers of the form 2n−1 (commonly refered to as Mersenne primes). The search for Mersenne primes resulted in one of the largest collaborative efforts known to mathematics research, the Great Internet Mersenne Prime Search, which ranked in the top 500 most powerful virtual computer systems in the world in 2012 [7].
Load more

Recommended publications
  • 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.
    [Show full text]
  • OUT of TOUCH: PHILOPONUS AS SOURCE for DEMOCRITUS Jaap
    OUT OF TOUCH: PHILOPONUS AS SOURCE FOR DEMOCRITUS Jaap Mansfeld 1. Introduction The view that according to Democritus atoms cannot touch each other, or come into contact, has been argued by scholars, most recently and carefully by C.C.W. Taylor, especially in his very useful edition of the fragments of the early Atomists.1 I am not concerned here with the theoretical aspect of this issue, that is to say with the view that the early Atomists should have argued (or posthumously admitted) that atoms cannot touch each other because only the void that separates them prevents fusion. I wish to focus on the ancient evidence for this interpretation. Our only ancient source for this view happens to be Philoponus; to be more precise, one brief passage in his Commentary on Aristotle’s Physics (fr. 54c Taylor) and two brief passages in that on Aristotle’s On Generation and Corruption (54dand54eTaylor~67A7 DK). These commentaries, as is well known, are notes of Ammonius’ lectures, with additions by Philoponus himself. Bodnár has argued that this evidence is not good enough, because all other ancient sources, in the first place Aristotle, are eloquently silent on a ban on contact; what we have here therefore are ‘guesses of Philoponus, which are solely based on the text of Aristotle’.2 Taylor admits the force of this objection, but sticks to his guns: ‘perhaps’, 1 Taylor (1997) 222,(1999) 186–188, 192–193,(1999a) 184, cf. the discussion in Kline and Matheson (1987) and Godfrey (1990). On the problems related to the assumption that attraction plays an important part see further the cautious remarks of Morel (1996) 422–424, who however does not take into account the passages in Philoponus which suggest that atoms cannot touch each other.
    [Show full text]
  • An Introduction to Pythagorean Arithmetic
    AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail.
    [Show full text]
  • The Origin of Life:Ahistory of Ancient Greek Theories
    Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 1980 Volume V: Man and the Environment The Origin of Life:Ahistory of Ancient Greek Theories Curriculum Unit 80.05.11 by Joyce Puglia This unit presents a history of scientific thought relating to the origin of life as explained mainly by early Greek scientific philosophers. The unit begins with Greek science during the eighth century B.C. and proceeds quickly into the seventh century B.C., concluding with the fourth century B.C. Since the scope is limited to this time period the unit will end with information that is presently, for the most part, outdated. The teacher must constantly remind the students of this fact. The purpose of this unit is not to impart scientific knowledge for its own sake. Rather, it is to show how scientific thinkers came to their conclusions based upon how science was viewed in the scheme of history. There are various high school courses taught, yet no specific course has been designed to relate the development of the academic disciplines to each other. Many science textbooks include the names of scientists who contributed valuable information upon which specific ideas were developed. Yet, most textbooks provide a minimum amount of information relating to the scientists themselves. It is my feeling that students will better understand the development of scientific thought if an opportunity can be provided in which a connection can be made between science and history. There are four general objectives for this unit. Upon completing the unit students will: 1. be familiar with the ideas of early scientific minds, 2.
    [Show full text]
  • Pythagoras and the Pythagoreans1
    Pythagoras and the Pythagoreans1 Historically, the name Pythagoras meansmuchmorethanthe familiar namesake of the famous theorem about right triangles. The philosophy of Pythagoras and his school has become a part of the very fiber of mathematics, physics, and even the western tradition of liberal education, no matter what the discipline. The stamp above depicts a coin issued by Greece on August 20, 1955, to commemorate the 2500th anniversary of the founding of the first school of philosophy by Pythagoras. Pythagorean philosophy was the prime source of inspiration for Plato and Aristotle whose influence on western thought is without question and is immeasurable. 1 c G. Donald Allen, 1999 ° Pythagoras and the Pythagoreans 2 1 Pythagoras and the Pythagoreans Of his life, little is known. Pythagoras (fl 580-500, BC) was born in Samos on the western coast of what is now Turkey. He was reportedly the son of a substantial citizen, Mnesarchos. He met Thales, likely as a young man, who recommended he travel to Egypt. It seems certain that he gained much of his knowledge from the Egyptians, as had Thales before him. He had a reputation of having a wide range of knowledge over many subjects, though to one author as having little wisdom (Her- aclitus) and to another as profoundly wise (Empedocles). Like Thales, there are no extant written works by Pythagoras or the Pythagoreans. Our knowledge about the Pythagoreans comes from others, including Aristotle, Theon of Smyrna, Plato, Herodotus, Philolaus of Tarentum, and others. Samos Miletus Cnidus Pythagoras lived on Samos for many years under the rule of the tyrant Polycrates, who had a tendency to switch alliances in times of conflict — which were frequent.
    [Show full text]
  • Women in Early Pythagoreanism
    Women in Early Pythagoreanism Caterina Pellò Faculty of Classics University of Cambridge Clare Hall February 2018 This dissertation is submitted for the degree of Doctor of Philosophy Alla nonna Ninni, che mi ha insegnato a leggere e scrivere Abstract Women in Early Pythagoreanism Caterina Pellò The sixth-century-BCE Pythagorean communities included both male and female members. This thesis focuses on the Pythagorean women and aims to explore what reasons lie behind the prominence of women in Pythagoreanism and what roles women played in early Pythagorean societies and thought. In the first chapter, I analyse the social conditions of women in Southern Italy, where the first Pythagorean communities were founded. In the second chapter, I compare Pythagorean societies with ancient Greek political clubs and religious sects. Compared to mainland Greece, South Italian women enjoyed higher legal and socio-political status. Similarly, religious groups included female initiates, assigning them authoritative roles. Consequently, the fact that the Pythagoreans founded their communities in Croton and further afield, and that in some respects these communities resembled ancient sects helps to explain why they opened their doors to the female gender to begin with. The third chapter discusses Pythagoras’ teachings to and about women. Pythagorean doctrines did not exclusively affect the followers’ way of thinking and public activities, but also their private way of living. Thus, they also regulated key aspects of the female everyday life, such as marriage and motherhood. I argue that the Pythagorean women entered the communities as wives, mothers and daughters. Nonetheless, some of them were able to gain authority over their fellow Pythagoreans and engage in intellectual activities, thus overcoming the female traditional domestic roles.
    [Show full text]
  • The Project Gutenberg Ebook #31061: a History of Mathematics
    The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith.
    [Show full text]
  • Sources for the Philosophy of Archytas
    Ancient Philosophy 28 (2008) ©Mathesis Publications 1 Sources for the Philosophy of Archytas Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King. By Carl Huffman. Cambridge: Cambridge University Press, 2005. Pp. xv + 665. $175.00 (cloth). ISBN 0521837464. Monte Ransome Johnson Although Archytas of Tarentum was one of the most important philosophers of the classical period, he has, like a second-rate scholar, become consigned to the oblivion of footnotes. Awkwardly, these are often footnotes to Pythagoreanism and fifth century philosophy. Yet he wrote and influenced fourth (not fifth) cen- tury philosophy; and it is an open question to what extent, or in what sense, and even whether, Archytas was really a Pythagorean. The source of the difficulties is to be found in the cruel, irrational, and occa- sionally fraudulent history of textual transmission. According to Huffman, only four genuine fragments of Archytas survive. Paucity of genuine material is a common problem in early Greek philosophy. But there is a further, special prob- lem with Archytas: the number of dubious fragments (or spurious fragments, as they are considered by Huffman) attributed to him in antiquity. While the gen- uine fragments in Huffman’s edition amount to just 73 lines, the standard edition of later Pythagorean material accumulates over 1300 lines under the name ‘Archytas’ (The Pythagorean Texts of the Hellenistic Period Collected and Edited by H. Thesleff [Åbo, 1965] 2-48). As I will argue at the end of this essay, there is hope, although it is not inspired by Huffman, that more of the material collected by Thesleff can be counted as evidence of Archytas’ thought, or at least cannot be ruled out as paraphrasing his words; they should at any rate be included in a complete edition of his philosophy.
    [Show full text]
  • The Beginnings of Greek Mathematics Greeks Were Not Confined to A
    Chapter 3: The Beginnings of Greek Mathematics Greeks were not confined to a neighborhood or the Aegean and the Peloponnesis. They ranged far and wide including present day Italy and Sicily and many lived in North Africa including Egypt. This is of course before the expansion of the Roman Empire (Punic Wars, 3rd Century B.C. and the Second Punic War was 218-201 B.C.) The Greeks introduced abstraction and gave logical arguments. For example Greeks were interested in triangles and not simply triangular fields. Thales of Miletus (ca. 625-547) (Miletus is in Asia Minor). Thales is sometimes known as the father of geometry probably because he was the first to use the deductive method. He proved among other things (see page 83 of the text) that a triangle inscribed inside a semi-circle (with one side the diameter) must be a right triangle. He also observed that the sides similar triangles are proportional. We do not have any mathematical works directly attributable to Thales or Pythagoras. Euclid’s Elements (Euclid 323-285) and Nicomachus of Gerasa’s Introductio Arithmeticae written ca. 100 B.C. are two math texts which we depend on but neither gives credit to the originators and discoverers of the results. Introductio Arithmeticae unlike Elements was a popular textbook for teaching geometry. Thales is credited with using similar triangles to discover the height of a the Great Pyramid of Gizeh (2600 B.C.) He similarly estimated distance to a ship offshore from a tower above the shore. See page 84 of the text. Pythagoras of Samos (580-500 B.C.; or 569-475 B.C.) (Samos is an Island in the northern Aegean near Asia Minor) His father was given honorary citizenship for bringing corn (!) in a time of famine.
    [Show full text]
  • Research and Experiment in Early Greek Thought by Tyler Mayo A
    Research and Experiment in Early Greek Thought by Tyler Mayo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Classical Studies) in the University of Michigan 2019 Doctoral Committee: Professor Francesca Schironi, Chair Professor Sara Ahbel-Rappe Professor Richard Janko Associate Professor Ian Moyer Tyler Edward Mayo [email protected] ORCID iD: 0000-0002-2442-7127 To my mother and father ii ACKNOWLEDGMENTS First, ad consuetudinem maiorum, I would like to begin by acknowledging and expressing my gratitude to the members of my committee: Professors Richard Janko, Ian Moyer, and Sara Abhel-Rappe. Their suggestions and criticisms helped me to rethink some, and alter other, points within this dissertation to its improvement, and saved me from numerous errors within the text, many of which surely still lurk within these pages and are my responsibility alone. However, I owe a special debt to my chair, Francesca Schironi, whose support not only made this dissertation possible, but who acted as a mentor throughout my entire time at Michigan. Her honesty and counsel were always welcome, and her careful philology can be seen within these pages, although I fear only as an imperfect reflection. Outside Angell Hall, others made my studies in Ann Arbor possible. The Posch family— John, Dyanna, Isabel, and most recently, Jude—acted as a family away from home, not least for their generous hospitality in boarding me. I shall always remember and cherish their ξενία. My wife Liz is certainly owed a place of honor in the beginning of this dissertation. Her support was essential in its creation as in all other things.
    [Show full text]
  • The Hellenistic Period1
    The Hellenistic Period1 In this brief section, we show what type of problems mathemati- cians solved, and the extent of Greek mathematics before the collapse of the mathematical world in about 400 CE. 1 Aristarchus of Samos (CA. 310-230 BCE) Aristarchus of Samos was considered by the ancients, particularly Vit- ruvius2, who considered him to be one of the few great men knowl- edgeable in all sciences, especially astronomy and mathematics. Note that his dates are only surmised. For example, he lived before the Sand Reckoner of Archimedes, a work published before 216 BCE. He is also known to have observed the summer solstice in about 280 BCE. Perhaps, foremost among them is that he formulated the Coperni- can hypothesis He was the first to formulate the Copernican hypotheses and is sometimes called the Ancient Copernican. His heliocentric the- ory was never accepted in his day and remained dormant until it was proposed again by Copernicus. His contributions to science and math- ematics are many. He discovered an improved sundial, with a concave hemispherical circle. In supporting his heliocentric theory, he countered the non parallax objection by asserting that the stars to be so far dis- tant that parallax was not measurable. For this he needed the following measure of distance to the stars. distance(earth, stars) distance(radius of the sun) » He also determined a more accurate estimate of the solar year. He wrote as well on light and colors. He wrote On the Sizes and Distances of the Sun and Moon,in which by the way the heliocentric ideas are not mentioned or in any event are not present in the copied version extant.
    [Show full text]
  • Scepticism, Number and Appearances the Ἀριθμητικὴ Τέχνη and Sextus’ Targets in M I-VI
    Philosophie antique Problèmes, Renaissances, Usages 15 | 2015 Questions sur le scepticisme pyrrhonien Scepticism, number and appearances The ἀριθμητικὴ τέχνη and Sextus’ targets in M I-VI Lorenzo Corti Electronic version URL: http://journals.openedition.org/philosant/376 DOI: 10.4000/philosant.376 ISSN: 2648-2789 Publisher Presses universitaires du Septentrion Printed version Date of publication: 24 November 2015 Number of pages: 121-145 ISBN: 978-2-7574-1141-4 ISSN: 1634-4561 Electronic reference Lorenzo Corti, « Scepticism, number and appearances », Philosophie antique [Online], 15 | 2015, Online since 01 November 2018, connection on 19 April 2019. URL : http://journals.openedition.org/ philosant/376 ; DOI : 10.4000/philosant.376 La revue Philosophie antique est mise à disposition selon les termes de la Licence Creative Commons Attribution - Pas d'Utilisation Commerciale - Pas de Modification 4.0 International. SCEPTICISM, NUMBER AND APPEARANCES : The ἀριθμητικὴ τέχνη and Sextus’ targets in M I-VI Lorenzo CORTI Archives Henri Poincaré (Nancy) RÉSUMÉ. Cet article s’interroge sur ce qu’est l’ἀριθμητική τέχνη visée par Sextus dans le Contre les arithméticiens. Après avoir rappelé brièvement le contenu de M IV, on examine la nature de cette discipline. Une fois clarifiée la question de savoir en quoi consistait l’ἀριθμητική τέχνη dans l’Antiquité – et donc ce que visait Sextus dans M IV –, on examine son rapport avec les autres disciplines critiquées par Sextus dans le Contre les Professeurs. Cette enquête mène à mettre en lumière une importante présupposition implicite dans l’attitude de Sextus à l’égard des sciences. SUMMARY. This paper is devoted to Sextus’ target in Against the Arithme- ticians: the ἀριθμητικὴ τέχνη.
    [Show full text]