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Bibliography The following list of cited works is divided into ancient, medieval and modern writings, where, for convenience, I take the 7th century to separate the ancient and medieval periods. Among the ancient and medieval titles I cite manuscripts when a published version is not available. When multiple entries occur for a given modern author, I have listed the titles in the order of publication. Ancient Sources Anonymus. Prolegomena tes tou Ptolemaiou megales syntaxeos: Vat. ms. gr. 1594 (Vati­ can, 10th cent.), f. lr-9r; Marc. ms. gr. 313 (Venice, 10th cent.), f. lr-28r; B.N. ms. gr. 2390 (Paris, 13th cent.), f. lr-13v -. Introduction to the Syntaxis = Prolegomena ... Syntaxeos -. [Introduction, excerpt on isoperimetric figures]: see F. Hultsch, Pappi Collectio, III, pp. 1138-65 -. see also J. Mogenet (1956) Anthemius of Tralles. Peri paradoxon mechanematon (On Paradoxical Mechanisms, a fragment), ed. J.L. Heiberg, Mathematici Graeci Minores, Copenhagen: A.F. H~st & Son, 1927, pp. 78-87 -. see also G.L. Huxley; and R. Rashed Apollonius. Quae graece exstant, ed. J.L. Heiberg, 2 vol., Leipzig: Teubner, 1891-93 (repr. Stuttgart, 1974) -. Conicorum libri octo, ed. E. Halley, Oxford, 1710 -. see also H. Balsam; T.L. Heath (1896); P. Ver Eecke (1924) Archimedes. Opera Omnia, cum Commentariis Eutocii, ed. lL. Heiberg, 2nd ed., 3 vol., Leipzig: Teubner, 1910-15 (repr. Stuttgart, 1972) -. see also A. Czwalina; T.L. Heath (1897); EJ. Dijksterhuis; C. Mugler; P. Ver Eecke (1921) Aristotle. Opera (I-II, Aristoteles Graece, ed. I. Bekker, 1831; III, Aristoteles Latine, interpretibus variis, 1831; IV, Scholia in Aristotelem, ed. C.A. Brandis, 1836; V, Index 818 Textual Studies in Ancient and Medieval Geometry Aristotelicus, ed. H. Bonitz, 1870), Berlin (Academia litterarum regia borussica): G. Reimer -. Aristoteles Latinus, G. Verbeke (series ed.), Brussels: Desclee de Brouwer and Leiden: Brill Boethius. De institutione arithmetica libri ii, De institutione musica libri v, ed. G. Fried­ lein, Leipzig: Teubner, 1867 -, trans. De sophisticis elenchis, ed. B.G. Dod, in Aristoteles Latinus, VI: 1-3, 1975 CAG = Commentaria in Aristotelem Graeca (Academia litterarum regia borussica), Berlin: G. Reimer Damianus. Kephalaia ton optikon hypotheseon (Chapters of the optical hypotheses), ed. R. Schone, Damianos: Schrift aber Optik, Berlin: Reichsdruckerei, 1897 Damascius. Vitae Isidori Reliquiae, ed. C. Zintzen, Hildesheim: G. Olms, 1967 Diocles. On Burning Mirrors, ed. GJ. Toomer, Berlin/Heidelberg/New York: 1976 -. see also R. Rashed Diophantus of Alexandria. Opera Omnia, ed. P. Tannery, 2 vol., Leipzig: Teubner, 1893-95 -. see also R. Rashed (1984); J. Sesiano Domninus of Larissa. Encheiridion arithmetikes eisagoges (Manual of the Arithmetic Introduction): in J.F. Boissonade, Anecdota Graeca, Paris, 1832 (repr. Hildesheim: G. Olms, 1962), IV, pp. 413-429 -. see also P. Tannery (1906) -. Pos esti logon ek logou aphelein (fragment, "How to subtract a ratio from a ratid'): see C.E. Ruelle Euclid of Alexandria. Elementa, ed. J.L. Heiberg, 4 vol., 1883-85, Leipzig: Teubner (2nd ed., ed. E.S. Stamatis, 4 vol., 1969-73) -. see also T.L. Heath (1926) -. Elementa XlV-XV, Scholia in Elementa, and "Prolegomena critica," ed. J.L. Heiberg (Euclidis Opera, vol. 5), Leipzig: Teubner, 1888 (2nd ed., in two parts, ed. E.S. Stama­ tis, 1977) -. Data (Opera Omnia, vol. 6), ed. H. Menge, Leipzig: Teubner, 1896 -. see also S. Ito -. Optica, Opticorum recensio Theonis, Catoptrica (Opera Omnia, vol. 7), ed. J.L. Heiberg, Leipzig: Teubner, 1895 -. Phaenomena, Scripta musica (Opera Omnia, vol. 8), ed. H. Menge, Leipzig: Teubner, 1916 Eutocius of Ascalon. Commentaria in Conica: in Apollonii quae graece exstant, ed. J.L. Heiberg, vol. II, Leipzig: Teubner, 1893 (repr. Stuttgart, 1974) -. Commentarii in Libros de Sphaera et Cylindro, Dimensionem Circuli, de Planorum Aequilibriis: in Archimedis Opera Omnia, ed. J.L. Heiberg, vol. III, Leipzig: Teubner, 1915 (repr. Stuttgart, 1972) Hero of Alexandria. Opera, 5 vol. (I, Pneumatica, ed. W. Schmidt, 1899; II, Mechanica, ed. L. Nix and [Ptolomeil De speculis, ed. W. Schmidt, 1900; III, Mechanica and Diop­ tra, ed. H. Schone, 1903; IV, Definitiones and Geometrica, ed. J.L. Heiberg, 1912; V, Stereometrica, ed. J.L. Heiberg, 1914), Leipzig: Teubner. -. Metrica: see also E.M. Bruins -. Belopoeica: see E.W. Marsden Hypsicles. Book XIV of the Elements: see Euclidis Opera Omnia, ed. J.L. Heiberg, vol. 5 Iamblichus of Chalcis. In Nicomachi Introductionem Arithmeticam Liber, ed. H. Pistelli, Leipzig: Teubner, 1894 Bibliography 819 -. In Platonis Dialogos Commentariorum Fragmenta, ed. 1.M. Dillon, Leiden: Brill, 1973 Joannes Philoponus. In Aristotelis Analytica Posteriora Commentaria, ed. M. Wallies (CAG, vol. 13, pt. 2), Berlin: Reimer, 1909 Marinus of Neapolis. Vita Procli, ed. 1.F. Boissonade (1814), repro with the English trans­ lation by K.S. Guthrie (1925), in Marinos of Neapolis. The Extant Ubrks, ed. A.N. Oikonomides, Chicago: Ares, 1977 Menelaus of Alexandria. Sphaerica, ed. E. Halley, Oxford, 1758 Nicomachus of Gerasa. Harmonikon Encheiridion, ed. K. von Jan, Musici Scriptores Graeci, no. 5, Leipzig: Teubner, 1895 (repr. Hildesheim: G. Oims, 1962) -. Introductionis arithmeticae libri ii, ed. R. Hoche, Leipzig: Teubner, 1866 Pappus of Alexandria. Collectionis quae supersunt, ed. F. Hultsch, 3 vol., Berlin: Weid- mann, 1876-78 (repr. Amsterdam: A.M. Hakkert, 1965) -. see also A. Jones; P. Ver Eecke (1933) -. Commentaries on Ptolemy: see A. Rome (1931) -. Commentary on Euclid's Book X: see W Thomson and G. Junge Philo of Byzantium. Belopoeica: see E.w. Marsden Philoponus: see Joannes Philoponus Plato. Opera, ed. 1. Burnet, (Oxford Classical Texts), 5 vol., Oxford: Clarendon Press, 1900-07 (repr. 1977) Plutarch of Chaeronea. Scripta Moralia, ed. WR. Paton, I. Wegehaupt, et aI., 7 vol., Leipzig: Teubner, 1925-67 Porphyry. In Platonis Timaeum Commentariorum Fragmenta, ed. A.R. Sodano, Naples: [no publ.], 1964 Proclus Diadochus. In Primum Euclidis Elementorum Librum Commentarii, ed. G. Fried­ lein, Leipzig: Teubner, 1873 (repr. 1967) -. see also G.R. Morrow -. In Platonis Timaeum Commentaria, ed. E. Diehl, 3 vol., Leipzig: Teubner, 1903-06 (repr. Amsterdam: A.M. Hakkert, 1965) Ptolemy of Alexandria. Opera quae exstant omnia, ed. 1.L. Heiberg, 2 vol. (I, 2 pts., Syn­ taxis mathematica, 1898); II, Opera astronomica minora: Phaseis, Hypotheseis, Inscrip­ tio Canobi, Procheiron kanonon diataxis, Analemma, Planisphaerium, etc., 1907), Leipzig: Teubner -. Almagest = Syntaxis mathematica -. see also K. Manitius; GJ. Toomer (1984) Quintilian. Institutionis oratoriae libri xii, ed. L. Radermacher, corr. V. Buchheit, Leip­ zig: Teubner, 1959 Serenus of Antinoeia. Opuscula (De sectione cylindri, De sectione coni), ed. J.L. Heiberg, Leipzig: Teubner, 1896 Simplicius. In Aristotelis de Caelo Commentaria, ed. J.L. Heiberg (CAG 7), Berlin: Reimer, 1894 -. In Aristotelis Physica Commentaria, ed. H. Diels, 2 vol. (CAG 9-10), Berlin: Reimer, 1882-95 Synesius of Cyrene. Opuscula, ed. N. Terzaghi, Rome: Regia officina polygraphica, 1944 -. Hymnes, ed. C. Lacombrade, Paris: Les Belles Lettres, 1978 -. see also A. Fitzgerald Syrianus. In Metaphysica Commentaria, ed. W Kroll (CAG 6, pt. 1), Berlin: G. Reimer, 1902 Themistius. In Aristotelis Physica Paraphrasis, ed. H. Schenkl (CAG 5, pt. 2), Berlin: G. Reimer, 1900 820 Textual Studies in Ancient and Medieval Geometry -. In Libros Aristotelis De Caelo Paraphrasis (Hebrew and Latin), ed. S. Landauer (CAG 5, pt. 4), Berlin: G. Reimer, 1902 Theon of Alexandria. In Claudii Ptolemaei Magnam Constructionem Commentariorum Lib. XI, ed. 1. Camerarius, Basel: 1. Walder, 1538 -. see also A. Rome (1936; 1943) -. Commentaries on Ptolemy's Handy Tables: see 1. Mogenet (1985); A. Tihon (1978) Theon of Smyrna. Expositio rerum mathematicarum ad legendum Platonem utilium, ed. E. Hiller, Leipzig: Teubner, 1878 (repr. New York and London: Garland, 1987) Vitruvius. De Architectura, ed. V. Rose, 2nd ed., Leipzig: Teubner, 1899 -. see also M.H. Morgan Medieval Sources Abu Bakr al-Harawl. [On the trisection of an angle], B.N. ms. arab. 2457, no. 45alt (Paris), f. 194r-195r: see Part II, text A Abu Jacfar al-Khazin. [On the trisection of an angle], Bod!. ms. Hunt. 237 (Oxford), f. 103v-104v: see Part II, text A3 -. Ff stikhraj kha~~ain bain kha~~ain (On the construction of two lines between two lines), B.N. ms. arab. 2457, no. 47 (Paris), f. 198v-199r: see Part II, text F -. see also R. Lorch (1986) Abu 'I-Rashid cAbd al-Hadi. Qaul mansub ila Arshimfdis fl misa~at al-da'ira (Writing attributed to Archimedes on the measurement ofthe circle) [Recension of Dimension 0/ the Circle, with preface, "Theorems applicable to the book of Archimedes"]' Columbia University ms. Smith 45, no. 4 (New York): see Part III, chap. 7, Appendix I Adelard of Bath. [Latin translation of Euclid's Elements]: see H.L.L. Busard (1983) Al).mad ibn Musa. Qaul fl tathlfth al-zawiya (Writing on the trisection ofthe angle), Bod!. ms. Thurston 3 (Oxford), f. 131 v-132r; Bod!. ms. Marsh 720, f. 260v: see Part II, texts B,B* -. see also 1. P. Hogendijk (1981) Anonymus. [Arabic translation of Archimedes' Dimension o/the Circle], ms. Fatih 3414, no. 1 (Istanbul), f. 2v-6v (cf. Sezgin, GAS, V, p. 131): see Part III, Appendices to chaps. 3 and 4 -. [Arabic translation of Archimedes' Sphere and Cylinder], ms. Fatih 3414, no. 2 (Istan­ bul), f. 7r-8r, 9r-59r (cf. Sezgin, GAS, V, p. 129) Anonymus. [Hebrew translation of Archimedes' Dimension o/the Circle], Vat. ms. Hebr. 384, f. 412r-412v: see Part III, Appendices to chaps. 3 and 4 Banu Musa, Kitab macrifa misa~a al-ashkiil al-basf~a wa- 'I-kurfya (Book of the Know­ ledge of Measurement of Plane and Spherical Figures), in the recension of al-Tusi: see al-Tusi, Rasa'il, II, no. 1 -. Verbafiliorum: in M. Clagett, Archimedes in the Middle Ages, I, chap. 4 -. see also H. Suter (1902); and Al).mad ibn Musa al-Bironi, Abu Rail).an Mul). b. Al).mad. Rasa'il.
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    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

    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 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]).