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King Lfred's Version Off the Consolations of Boethius
King _lfred's Version off the Consolations of Boethius HENRY FROWDE, M A. PUBLISHER TO THE UNIVERSITY OF OF_0RD LONDON, EDINBURGH_ AND NEW YORK Kring e__lfred's Version o_/"the Consolations of Boethius _ _ Z)one into c_gfodern English, with an Introduction _ _ _ _ u_aa Litt.D._ Editor _o_.,I_ing .... i .dlfred_ OM Englis.h..ffgerAon2.' !ilo of the ' De Con.d.¢_onz,o,e 2 Oxford : _4t the Claro_don:,.....: PrestO0000 M D CCCC _eee_ Ioee_ J_el eeoee le e_ZNeFED AT THE_.e_EN_N PI_.._S _ee • • oeoo eee • oeee eo6_o eoee • ooeo e_ooo ..:.. ..'.: oe°_ ° leeeo eeoe ee •QQ . :.:.. oOeeo QOO_e 6eeQ aee...._ e • eee TO THE REV. PROFESSOR W. W. SKEAT LITT.D._ D.C.L._ LL.D.:_ PH.D. THIS _800K IS GRATEFULLY DEDICATED PREFACE THE preparationsfor adequately commemoratingthe forthcoming millenary of King Alfred's death have set going a fresh wave of popularinterest in that hero. Lectares have been given, committees formed, sub- scriptions paid and promised, and an excellent book of essays by eminent specialists has been written about Alfred considered under quite a number of aspects. That great King has himself told us that he was not indifferent to the opinion of those that should come after him, and he earnestly desired that that opinion should be a high one. We have by no means for- gotten him, it is true, but yet to verymany intelligent people he is, to use a paradox, a distinctly nebulous character of history. His most undying attributes in the memory of the people are not unconnected with singed cakes and romantic visits in disguise to the Danish viii Preface Danish camp. -
Euclid's Elements - Wikipedia, the Free Encyclopedia
Euclid's Elements - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Euclid's_Elements Euclid's Elements From Wikipedia, the free encyclopedia Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise Elements consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems,[1] including the problem of finding the square root of a number.[2] With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises,[3] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. The name 'Elements' comes from the plural of 'element'. The frontispiece of Sir Henry Billingsley's first According to Proclus the term was used to describe a English version of Euclid's Elements, 1570 theorem that is all-pervading and helps furnishing proofs of many other theorems. The word 'element' is Author Euclid, and translators in the Greek language the same as 'letter'. This Language Ancient Greek, translations suggests that theorems in the Elements should be seen Subject Euclidean geometry, elementary as standing in the same relation to geometry as letters number theory to language. -
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◦. -
166 BOOK REVIEWS Gretchen Reydams-Schils, the Roman Stoics
166 BOOK REVIEWS Gretchen Reydams-Schils, The Roman Stoics: Self, Responsibility and Affection. Chicago and London: University of Chicago Press, 2005. xii + 210 pp. ISBN 0-226-30837-5. In the third book of the Histones Tacitus presents us with the ludicrous picture of a Stoic philosopher mingling with the soldiers of the Flavian army as it prepares to invade the city, and preaching unheeded the blessings of peace and the dangers of war (3.81). The purveyor of this intempestiva sapientia was Musonius Rufus, an eques Romanus, a friend of many Roman senators, a political exile, and the hero of this book. The author sets out to prove that the Roman Stoics adapted originally Greek Stoic doctrine to emphasize the importance of social responsibility and social engagement to the life of virtue. Stoicism, for all its emphasis on self-sufficiency, independence of outer circumstances, and spiritual self-improvement, saw the instinct for sociability as implanted by divine providence in human beings, who were imbedded in a society of rational beings comprising men and gods. What the author thinks was the distinctive contribution of the Roman Stoics was to retrieve for the life of virtue the traditional relationships of parenthood and marriage that Plato and the Cynics had tried to denigrate. It is therefore not the Musonius Rufus of Roman politics that interests her, but the advocate of marriage as a union of body and soul between equals and the opponent of infanticide and child exposure. The structure of the book reflects this emphasis, for it ‘reverses the Stoic progression from self to god and universe’ (5). -
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. -
Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus -
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. -
Plato As "Architectof Science"
Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries. -
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. -
Pentagons in Medieval Architecture
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library Építés – Építészettudomány 46 (3–4) 291–318 DOI: 10.1556/096.2018.008 PENTAGONS IN MEDIEVAL ARCHITECTURE KRISZTINA FEHÉR* – BALÁZS HALMOS** – BRIGITTA SZILÁGYI*** *PhD student. Department of History of Architecture and Monument Preservation, BUTE K II. 82, Műegyetem rkp. 3, H-1111 Budapest, Hungary. E-mail: [email protected] **PhD, assistant professor. Department of History of Architecture and Monument Preservation, BUTE K II. 82, Műegyetem rkp. 3, H-1111 Budapest, Hungary. E-mail: [email protected] ***PhD, associate professor. Department of Geometry, BUTE H. II. 22, Egry József u. 1, H-1111 Budapest, Hungary. E-mail: [email protected] Among regular polygons, the pentagon is considered to be barely used in medieval architectural compositions, due to its odd spatial appearance and difficult method of construction. The pentagon, representing the number five has a rich semantic role in Christian symbolism. Even though the proper way of construction was already invented in the Antiquity, there is no evidence of medieval architects having been aware of this knowledge. Contemporary sources only show approximative construction methods. In the Middle Ages the form has been used in architectural elements such as window traceries, towers and apses. As opposed to the general opinion supposing that this polygon has rarely been used, numerous examples bear record that its application can be considered as rather common. Our paper at- tempts to give an overview of the different methods architects could have used for regular pentagon construction during the Middle Ages, and the ways of applying the form. -
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]).