From Democritus' Atom to Quarks
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Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
Democritus C
Democritus c. 460 BC-c. 370 BC (Also known as Democritus of Abdera) Greek philosopher. home to the philosopher Protagoras. There are several indications, both external and internal to his writings, The following entry provides criticism of Democritus’s that Democritus may have held office in Abdera and life and works. For additional information about Democ- that he was a wealthy and respected citizen. It is also ritus, see CMLC, Volume 47. known that he traveled widely in the ancient world, visit- ing not only Athens but Egypt, Persia, the Red Sea, pos- sibly Ethiopia, and even India. Scholars also agree that he INTRODUCTION lived a very long life of between 90 and 109 years. Democritus of Abdera, a contemporary of Socrates, stands Democritus is said to have been a pupil of Leucippus, an out among early Greek philosophers because he offered important figure in the early history of philosophy about both a comprehensive physical account of the universe and whom little is known. Aristotle and others credit Leucippus anaturalisticaccountofhumanhistoryandculture. with devising the theory of atomism, and it is commonly Although none of his works has survived in its entirety, believed that Democritus expanded the theory under his descriptions of his views and many direct quotations from tutelage. However, some scholars have suggested that his writings were preserved by later sources, beginning Leucippus was not an actual person but merely a character with the works of Aristotle and extending to the fifth- in a dialogue written by Democritus that was subsequently century AD Florigelium (Anthology) of Joannes Stobaeus. lost. A similar strategy was employed by the philosopher While Plato ignored Democritus’s work, largely because he Parmenides, who used the character of a goddess to elu- disagreed with his teachings, Aristotle acknowledged De- cidate his views in his didactic poem, On Nature. -
Archimedes' Principle
General Physics Lab Handbook by D.D.Venable, A.P.Batra, T.Hubsch, D.Walton & M.Kamal Archimedes’ Principle 1. Theory We are aware that some objects oat on some uids, submerged to diering extents: ice cubes oat in water almost completely submerged, while corks oat almost completely on the surface. Even the objects that sink appear to weigh less when they are submerged in the uid than when they are not. These eects are due to the existence of an upward ‘buoyant force’ that will act on the submerged object. This force is caused by the pressure in the uid being increased with depth below the surface, so that the pressure near the bottom of the object is greater than the pressure near the top. The dierence of these pressures results in the eective ‘buoyant force’, which is described by the Archimedes’ principle. According to this principle, the buoyant force FB on an object completely or partially submerged in a uid is equal to the weight of the uid that the (submerged part of the) object displaces: FB = mf g = f Vg . (1) where f is the density of the uid, mf and V are respectively mass and the volume of the displaced uid (which is equal to the volume of the submerged part of the object) and g is the gravitational acceleration constant. 2. Experiment Object: Use Archimedes’ principle to measure the densities of a given solid and a provided liquid. Apparatus: Solid (metal) and liquid (oil) samples, beaker, thread, balance, weights, stand, micrometer, calipers, PASCO Science Workshop Interface, Force Sensors and a Macintosh Computer. -
Great Inventors of the Ancient World Preliminary Syllabus & Course Outline
CLA 46 Dr. Patrick Hunt Spring Quarter 2014 Stanford Continuing Studies http://www.patrickhunt.net Great Inventors Of the Ancient World Preliminary Syllabus & Course Outline A Note from the Instructor: Homo faber is a Latin description of humans as makers. Human technology has been a long process of adapting to circumstances with ingenuity, and while there has been gradual progress, sometimes technology takes a downturn when literacy and numeracy are lost over time or when humans forget how to maintain or make things work due to cataclysmic change. Reconstructing ancient technology is at times a reminder that progress is not always guaranteed, as when Classical civilization crumbled in the West, but the history of technology is a fascinating one. Global revolutions in technology occur in cycles, often when necessity pushes great minds to innovate or adapt existing tools, as happened when humans first started using stone tools and gradually improved them, often incrementally, over tens of thousands of years. In this third course examining the greats of the ancient world, we take a close look at inventions and their inventors (some of whom might be more legendary than actually known), such as vizier Imhotep of early dynastic Egypt, who is said to have built the first pyramid, and King Gudea of Lagash, who is credited with developing the Mesopotamian irrigation canals. Other somewhat better-known figures are Glaucus of Chios, a metallurgist sculptor who possibly invented welding; pioneering astronomer Aristarchus of Samos; engineering genius Archimedes of Siracusa; Hipparchus of Rhodes, who made celestial globes depicting the stars; Ctesibius of Alexandria, who invented hydraulic water organs; and Hero of Alexandria, who made steam engines. -
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. -
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. -
Greek Numbers 05/10/2006 12:02 PM
Greek numbers 05/10/2006 12:02 PM History topic: Greek number systems There were no single Greek national standards in the first millennium BC. since the various island states prided themselves on their independence. This meant that they each had their own currency, weights and measures etc. These in turn led to small differences in the number system between different states since a major function of a number system in ancient times was to handle business transactions. However we will not go into sufficient detail in this article to examine the small differences between the system in separate states but rather we will look at its general structure. We should say immediately that the ancient Greeks had different systems for cardinal numbers and ordinal numbers so we must look carefully at what we mean by Greek number systems. Also we shall look briefly at some systems proposed by various Greek mathematicians but not widely adopted. The first Greek number system we examine is their acrophonic system which was use in the first millennium BC. 'Acrophonic' means that the symbols for the numerals come from the first letter of the number name, so the symbol has come from an abreviation of the word which is used for the number. Here are the symbols for the numbers 5, 10, 100, 1000, 10000. Acrophonic 5, 10, 100, 1000, 10000. We have omitted the symbol for 'one', a simple '|', which was an obvious notation not coming from the initial letter of a number. For 5, 10, 100, 1000, 10000 there will be only one puzzle for the reader and that is the symbol for 5 which should by P if it was the first letter of Pente. -
Velázquez's Democritus
Velázquez’s Democritus: Global Disillusion and the Critical Hermeneutics of a Smile javier berzal de dios Western Washington University Velázquez’s Democritus (ca. 1630) presents a unique encounter: not only are there few depictions in which the Greek philosopher appears with a sphere that shows an actual map, but Velázquez used a court jester as a model for Democritus, thus placing the philosopher within a courtly space. When we study the painting in relation to the literary interests of the Spanish Golden Age and its socio-political circumstances, we can see the figure of Democritus as far from just another instantiation of a conven- tional trope. The philosopher’s smile and his crepuscular globe entrap the viewer in a semiotic game with pedagogical and ethical goals. While the scholarship on the painting has dwelt extensively on the identification of the figure, this essay moves beyond the superficial aspects of subject identity in order to explore how the painting articulates and requests a profoundly philosophical engagement. I thus exa- mine Democritus in relation to contemporary literary and philosophical themes, many of which were present in Velázquez’s own personal library: the period’s understanding of the philosopher, cartographic spheres, and treatises on laughter. Considered in this manner, Velázquez’s figure is not responding to the folly of humanity in general, as is commonly the case in representations of the philosopher, but is rather presented through a courtly prism in which conquest, geography, and politics are inescapably interrelated. Velázquez’s Democritus emphasizes the philosophical and moral qualities of a learned and decorous laughter, which performs a critical and ethical role framed by Spain’s political difficulties. -
Chapter Two Democritus and the Different Limits to Divisibility
CHAPTER TWO DEMOCRITUS AND THE DIFFERENT LIMITS TO DIVISIBILITY § 0. Introduction In the previous chapter I tried to give an extensive analysis of the reasoning in and behind the first arguments in the history of philosophy in which problems of continuity and infinite divisibility emerged. The impact of these arguments must have been enormous. Designed to show that rationally speaking one was better off with an Eleatic universe without plurality and without motion, Zeno’s paradoxes were a challenge to everyone who wanted to salvage at least those two basic features of the world of common sense. On the other hand, sceptics, for whatever reason weary of common sense, could employ Zeno-style arguments to keep up the pressure. The most notable representative of the latter group is Gorgias, who in his book On not-being or On nature referred to ‘Zeno’s argument’, presumably in a demonstration that what is without body and does not have parts, is not. It is possible that this followed an earlier argument of his that whatever is one, must be without body.1 We recognize here what Aristotle calls Zeno’s principle, that what does not have bulk or size, is not. Also in the following we meet familiar Zenonian themes: Further, if it moves and shifts [as] one, what is, is divided, not being continuous, and there [it is] not something. Hence, if it moves everywhere, it is divided everywhere. But if that is the case, then everywhere it is not. For it is there deprived of being, he says, where it is divided, instead of ‘void’ using ‘being divided’.2 Gorgias is talking here about the situation that there is motion within what is. -
Nietzsche's Views on Plato Pre-Basel
Sophia and Philosophia Volume 1 Issue 1 Spring-Summer Article 6 4-1-2016 Nietzsche's Views on Plato Pre-Basel Daniel Blue [email protected] Follow this and additional works at: https://repository.belmont.edu/sph Part of the Ancient History, Greek and Roman through Late Antiquity Commons, German Language and Literature Commons, History of Philosophy Commons, Logic and Foundations of Mathematics Commons, and the Metaphysics Commons Recommended Citation Blue, Daniel (2016) "Nietzsche's Views on Plato Pre-Basel," Sophia and Philosophia: Vol. 1 : Iss. 1 , Article 6. Available at: https://repository.belmont.edu/sph/vol1/iss1/6 This Article is brought to you for free and open access by Belmont Digital Repository. It has been accepted for inclusion in Sophia and Philosophia by an authorized editor of Belmont Digital Repository. For more information, please contact [email protected]. S.Ph. Essays and Explorations 1.1 Copyright 2016, S.Ph. Press Nietzsche’s Views on Plato Pre-Basel Daniel Blue In an essay published in 20041 Thomas Brobjer surveyed Nietzsche’s attitudes toward Plato and argued that, far from entering into a dedicated agon with that philosopher, he had little personal engagement with Plato’s views at all. Certainly, he did not grapple so immediately and fruitfully with him as he did with Emerson, Schopenhauer, Lange, and even Socrates. Instead, he merely “set up a caricature of Plato as a representative of the metaphysical tradition … to which he opposed his own.”2 This hardly reflects the view of Nietzsche scholarship in general, but Brobjer argued his case vigorously by ranging broadly over Nietzsche’s life, collating his assessments of Plato, and then noting certain standard views which he believes to be overstated. -
Anaximander and the Problem of the Earth's Immobility
Binghamton University The Open Repository @ Binghamton (The ORB) The Society for Ancient Greek Philosophy Newsletter 12-28-1953 Anaximander and the Problem of the Earth's Immobility John Robinson Windham College Follow this and additional works at: https://orb.binghamton.edu/sagp Recommended Citation Robinson, John, "Anaximander and the Problem of the Earth's Immobility" (1953). The Society for Ancient Greek Philosophy Newsletter. 263. https://orb.binghamton.edu/sagp/263 This Article is brought to you for free and open access by The Open Repository @ Binghamton (The ORB). It has been accepted for inclusion in The Society for Ancient Greek Philosophy Newsletter by an authorized administrator of The Open Repository @ Binghamton (The ORB). For more information, please contact [email protected]. JOHN ROBINSON Windham College Anaximander and the Problem of the Earth’s Immobility* N the course of his review of the reasons given by his predecessors for the earth’s immobility, Aristotle states that “some” attribute it I neither to the action of the whirl nor to the air beneath’s hindering its falling : These are the causes with which most thinkers busy themselves. But there are some who say, like Anaximander among the ancients, that it stays where it is because of its “indifference” (όμοιότητα). For what is stationed at the center, and is equably related to the extremes, has no reason to go one way rather than another—either up or down or sideways. And since it is impossible for it to move simultaneously in opposite directions, it necessarily stays where it is.1 The ascription of this curious view to Anaximander appears to have occasioned little uneasiness among modern commentators. -
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar Ve Yorumlar
Thales of Miletus Sources and Interpretations Miletli Thales Kaynaklar ve Yorumlar David Pierce October , Matematics Department Mimar Sinan Fine Arts University Istanbul http://mat.msgsu.edu.tr/~dpierce/ Preface Here are notes of what I have been able to find or figure out about Thales of Miletus. They may be useful for anybody interested in Thales. They are not an essay, though they may lead to one. I focus mainly on the ancient sources that we have, and on the mathematics of Thales. I began this work in preparation to give one of several - minute talks at the Thales Meeting (Thales Buluşması) at the ruins of Miletus, now Milet, September , . The talks were in Turkish; the audience were from the general popu- lation. I chose for my title “Thales as the originator of the concept of proof” (Kanıt kavramının öncüsü olarak Thales). An English draft is in an appendix. The Thales Meeting was arranged by the Tourism Research Society (Turizm Araştırmaları Derneği, TURAD) and the office of the mayor of Didim. Part of Aydın province, the district of Didim encompasses the ancient cities of Priene and Miletus, along with the temple of Didyma. The temple was linked to Miletus, and Herodotus refers to it under the name of the family of priests, the Branchidae. I first visited Priene, Didyma, and Miletus in , when teaching at the Nesin Mathematics Village in Şirince, Selçuk, İzmir. The district of Selçuk contains also the ruins of Eph- esus, home town of Heraclitus. In , I drafted my Miletus talk in the Math Village. Since then, I have edited and added to these notes.