Welcome To JMD COMPUTER TECH hippocrates of chios (ca. 450 BC) http//wwwgroups.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html.Hippocrates of Cos (460-ca. http://www.jmdcomp.com/wldhweb.htm
Lune -- From MathWorld Lune, A figure bounded by two circular arcs of unequal radii. hippocrates of chiossquared the above left lune, as well as two others, in the fifth century BC. http://mathworld.wolfram.com/Lune.html
Extractions: A figure bounded by two circular arcs of unequal radii . Hippocrates of Chios squared the above left lune, as well as two others, in the fifth century BC Two more squarable lunes were found by T. Clausen in the 19th century (Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew and A. W. Dorodnow proved that these are the only five squarable lunes (Shenitzer and Steprans 1994). The left lune above is squared as follows, References Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 1-20, 1990. Heath, T. L. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, p. 185, 1981. Pappas, T. "Lunes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 72-73, 1989. Shenitzer, A. and Steprans, J. "The Evolution of Integration."
Dupcubfin.html discrepancy. In addition to Eudoxus' solution, hippocrates of chios alsodeveloped a working solution of the Delian problem. Hippocrates http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html
Extractions: Duplication of the Cube : Darrell Mattingly, Cateryn Kiernan The ancient Greeks originated numerous mathematical questions, most of which they learned to solve using simple mathematical tools, such as the straight edge and the collapsable compass. Three of these problems persist today, challenging students in contemporary classrooms. This triology of problems, the trisection of a given angle, the squaring of a circle, and the duplication of the cube, have since been proved impossible using exclusively the straight edge and the compass. In the quest to solve these problems using those specific tools, however, mathematicians developed numerous alternate solutions using other mathematical tools. The last problem of the trilogy is the focus of this discussion, and it challenged mathematicians for centuries, due to the restriction of using only the aforementioned tools. Origin of the Problem Proof that NO Platoic Solution Exists for the "Delian" Problem After centuries of mathematicians had worked on this problem, a proof developed that it could not be done using exclusively the straight edge and compass. This proof is based on theorems about the powers of degrees of subfields generated by the x and y coordinates of the side of the cube to be duplicated. Although the desired point can be approximated, it cannot in fact be found based on these theorems.
Selected Older Individuals From Graeco-Roman Antiquity 400 BC. the historian Hellanicus of Lesbos (85). the epic poet Choerilus of Samos(about 80). the mathematician hippocrates of chios (about 70). Socrates (70). http://www.clas.canterbury.ac.nz/oldancientss.html
Extractions: Please send any comments, or suggestions for changes or additions, to Tim Parkin at tim.parkin@canterbury.ac.nz A roughly chronological order, by date of death, is followed ( 500 BC 400 BC 300 BC 200 BC ... AD 400 Where appropriate, reference is made to my Old Age in the Roman World book (referred to here as Old Age ), where further examples are also discussed. Ages at death - exact, approximate, or merely alleged - are given in brackets; no guarantee as to the authenticity or accuracy of any figure, especially when derived solely from ancient sources, can usually be given. The list thus serves also on occasion to highlight the wide variety of figures extant. Homer and Hesiod (?): The
Conic Sections In Ancient Greece A breakthrough of a kind occurred when hippocrates of chios reduced the problem tothe equivalent problem of two mean proportionals , though this formulation http://www.math.rutgers.edu/courses/436/436-s99/Papers1999/schmarge.html
Extractions: The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of "doubling the cube". Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the "Great Geometer" on the basis of his text Conic Sections , an eight-"book" (or in modern terms, "chapter") series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely. In the years following Apollonius the Greek geometric tradition started to decline, though there were developments in astronomy, trigonometry, and algebra (Eves, 1990, p. 182). Pappus, who lived about 300 A.D., furthered the study of conic sections somewhat in minor ways. After Pappus, however, conic sections were nearly forgotten for 12 centuries. It was not until the sixteenth century, in part as a consequence of the invention of printing and the resulting dissemination of Apollonius' work, that any significant progress in the theory or applications of conic sections occurred; but when it did occur, in the work of Kepler, it was as part of one of the major advances in the history of science.
Euclid's Elements, Book I, Proposition 3 later books. Notes. According to Proclus in his Commentary on BookI, hippocrates of chios was the first to write an Elements. Leon http://www.educa.fmf.uni-lj.si/java/pck/ELEMENTS/propI3.html
Extractions: To cut off from the greater of two given unequal straight lines a straight line equal to the less. Let AB and C be the two given unequal straight lines, and let AB be the greater of them. It is required to cut off from AB the greater a straight line equal to C the less. Place AD at the point A equal to the straight line C , and describe the circle DEF with center A and radius AD I.2 Post. 3 Now, since the point A is the center of the circle DEF AE equals AD Def.15 But C also equals AD . Therefore each of the straight lines AE and C equals AD , so that AE also equals C C.N.1 Therefore, given the two straight lines AB and C AE has been cut off from AB the greater equal to C the less. Q. E. F. Now it is clear that the purpose of Proposition 2 is to effect the construction in this proposition. This proposition begins the geometric arithmetic of lines. Explicitly, it allows lines to be subtracted, but it can also be used to compare lines for equality and to add lines. The construction in this proposition is used in Propositions I.5
GalaxyGoo Math Links: Mathematicians, Historical Fuller, Richard Buckminster; Galileo Galilei; Galois, Evariste; Gauss,Johann Carl Friedrich; hippocrates of chios; Hooke, Robert; Kepler http://www.galaxygoo.org/blogs/archives_math/000153.html
Hippias [Internet Encyclopedia Of Philosophy] Hippias was a sophist, a contemporary of Socrates, and an enthusiast for universality.Category Society Philosophy Internet Encyclopedia of Philosophy construction. The lunules of hippocrates of chios belong to it, andHippias, the universal genius, could not be left behind here. He http://www.utm.edu/research/iep/h/hippias.htm
Extractions: A Greek sophist of Elis and a contemporary of Socrates. He taught in the towns of Greece, especially at Athens. He had the advantage of a prodigious memory, and was deeply versed in all the learning of his day. He attempted literature in every form which was then extant. He also made the first attempt in the composition of dialogues. In the two Platonic dialogues named after him ( Hippias Major and Hippias Minor ), he is represented as excessively vain and arrogant.
Earliest Uses Of Symbols From Geometry of points, lines, and planes by a letter or letters was in vogue among the ancientGreeks and has been traced back to hippocrates of chios (about 440 BC http://members.aol.com/jeff570/geometry.html
Extractions: Earliest Uses of Symbols from Geometry Last revision: August 26, 2001 Lettering of geometric figures. The designation of points, lines, and planes by a letter or letters was in vogue among the ancient Greeks and has been traced back to Hippocrates of Chios (about 440 B. C.) (Cajori vol. 1, page 420, attributed to Moritz Cantor). Lettering of triangles. Richard Rawlinson in a pamphlet prepared at Oxford sometime between 1655 and 1668 used A, B, C for the sides of a triangle and a, b, c for the opposite angles. In his notation, A was the largest side and C the smallest (Cajori vol. 2, page 162). Leonhard Euler and Thomas Simpson reintroduced this scheme many years later, Euler using it in 1753 in (Cajori vol 2., page 162). Euler used capital letters for the angles. In 1866, Karl Theodor Reye (1838-1919) proposed the plan of using capital letters for points, lower case letters for lines, and lower case Greek letters for planes in a remarkable two-volume work on geometry, Die Geometrie der Lage (Cajori vol. 1, page 423).
Pedro Pablo Fuentes González with authors as diverse as the Stoics Cornutus and Epictetus, the polygraph andscholar Eratosthenes of Cyrene or the mathematician hippocrates of chios. http://www.ugr.es/~odiseo/Fuentes.html
Historical Ways 450, Zeno of Elea, philosopher; hippocrates of chios, matematician; Callimachusdevelops the Corinthian order; Philolaus of Thebes, astronomer. http://www.mysticsrealm.com/mmgvnts.html
Catálogo Sites Indicados Pelos Participantes Translate this page 12-2002 1900. 3, hippocrates of chios. http//www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html.foto datas dados http://www.prof2000.pt/users/miguel/histmat/af22/Catalogo1.htm
Extractions: site em Ingles, sugestão de: - adam@mail.prof2000.pt em 09-12-2002 01:03 Página de José Miguel Sousa http://www.prof2000.pt/users/miguel É uma das melhores páginas em Português sobre a trissecção do ângulo e a quadratura do círculo. Pode descarregar em pdf a dissertação de Mestrado do José Miguel Sousa.
BSHM: Abstracts -- S of its significance for early Greek geometry, Historia mathematica 22 (1995),119137 Nothing is known of how hippocrates of chios justified his reduction http://www.dcs.warwick.ac.uk/bshm/abstracts/S.html
Extractions: The British Society for the History of Mathematics HOME About BSHM BSHM Council Join BSHM ... Search A B C D ... Z These listings contain all abstracts that have appeared in BSHM Newsletters up to Newsletter 46. BSHM Abstracts - S Sabidussi, Gert, Correspondence between Sylvester, Petersen, Hilbert and Klein on invariants and the factorisation of graphs 1889-1891, Discrete mathematics A collection of 47 letters shedding light on the background and origin of Petersens famous 1891 paper on graph factorisation, and on his abortive collaboration with J. J. Sylvester. Saito, Ken, Doubling the cube: a new interpretation of its significance for early Greek geometry, Historia mathematica Nothing is known of how Hippocrates of Chios justified his reduction of the problem of doubling the cube to that of finding two mean proportionals between two given lines. A reconstruction is proposed, modelled after an argument of Archimedes, leading to a new interpretation of the development of proportion theory in early Greek mathematics. Salmon, Vivian, Thomas Harriot (1560-1621) and the English origins of Algonkian linguistics
Extractions: CHRONOLOGY OF MATHEMATICIANS -1100 CHOU-PEI -585 THALES OF MILETUS: DEDUCTIVE GEOMETRY PYTHAGORAS : ARITHMETIC AND GEOMETRY -450 PARMENIDES: SPHERICAL EARTH -430 DEMOCRITUS -430 PHILOLAUS: ASTRONOMY -430 HIPPOCRATES OF CHIOS: ELEMENTS -428 ARCHYTAS -420 HIPPIAS: TRISECTRIX -360 EUDOXUS: PROPORTION AND EXHAUSTION -350 MENAECHMUS: CONIC SECTIONS -350 DINOSTRATUS: QUADRATRIX -335 EUDEMUS: HISTORY OF GEOMETRY -330 AUTOLYCUS: ON THE MOVING SPHERE -320 ARISTAEUS: CONICS EUCLID : THE ELEMENTS -260 ARISTARCHUS: HELIOCENTRIC ASTRONOMY -230 ERATOSTHENES: SIEVE -225 APOLLONIUS: CONICS -212 DEATH OF ARCHIMEDES -180 DIOCLES: CISSOID -180 NICOMEDES: CONCHOID -180 HYPSICLES: 360 DEGREE CIRCLE -150 PERSEUS: SPIRES -140 HIPPARCHUS: TRIGONOMETRY -60 GEMINUS: ON THE PARALLEL POSTULATE +75 HERON OF ALEXANDRIA 100 NICOMACHUS: ARITHMETICA 100 MENELAUS: SPHERICS 125 THEON OF SMYRNA: PLATONIC MATHEMATICS PTOLEMY : THE ALMAGEST 250 DIOPHANTUS: ARITHMETICA 320 PAPPUS: MATHEMATICAL COLLECTIONS 390 THEON OF ALEXANDRIA 415 DEATH OF HYPATIA 470 TSU CH'UNG-CHI: VALUE OF PI 476 ARYABHATA 485 DEATH OF PROCLUS 520 ANTHEMIUS OF TRALLES AND ISIDORE OF MILETUS 524 DEATH OF BOETHIUS 560 EUTOCIUS: COMMENTARIES ON ARCHIMEDES 628 BRAHMA-SPHUTA-SIDDHANTA 662 BISHOP SEBOKHT: HINDU NUMERALS 735 DEATH OF BEDE 775 HINDU WORKS TRANSLATED INTO ARABIC 830 AL-KHWARIZMI: ALGEBRA 901 DEATH OF THABIT IBN - QURRA 998 DEATH OF ABU'L - WEFA 1037 DEATH OF AVICENNA 1039 DEATH OF ALHAZEN
Pyramid Introduces an ancient method of approximating pi, using Pythagorean triplet triangles. Includes some Category Science Math Recreations Specific Numbers Pi In his lost work he must have mentioned Antiphon and his many other precursors,among them Deinostratus, Hippias of Elis, hippocrates of chios, Bryson of http://www.access.ch/circle/text1.html
Extractions: PRIMARY HILL AND RISING SUN Squaring the Circle Using a Special Kind of Polygon Based on the "Holy Triangle" 3-4-5 and a Sequence of Further So-called Pythagorean Triples - A Method Probably Discovered by Imhotep and Refined by the Unknown Builder of the Great Pyramid at Giza by Franz Gnaedinger, Zuerich 1996 A new way to square the circle that may have been a very old one Please imagine a circle of radius 5 x 5 x 5 x 5 ... = 5, 25, 125, 625 ... units or subunits. Combine the circle with the cross of the axes and a grid of 10 x 10, 50 x 50, 250 x 250, 1250 x 1250 ... squares. The axes define the center of the circle and 4 points of the periphery. 8, 12, 20, 28 ... further points are defined by the following triples beginning with the Holy Triangle: The subsequent triples 1, 2, 3, 4, 5, 6 ... can be derived using two rather simple formulas: triples 1, 3, 5 ... be termed a-b-c (a smaller than b) triples 2, 4, 6 ... be termed A-B-C (A smaller than B) A-B-C follows a-b-c: A = 4b - 3a B = 3b + 4a C = 5c a-b-c follows A-B-C: a = 3B - 4A b = 4B + 3A c = 5C The axes and subsequent triples define 12, 20, 28, 36 ... points of the periphery. If we join them up in turn with lines we obtain a sequence of polygons with 12, 20, 28, 36 ... unequal sides. As the number of sides increases, the polygon slowly converges with the circumscribed circle.
Untitled Document the views of his Greek predecessors from the sixth to the fourth century BC includingthose of Pythagoras (c.560480B-C), hippocrates of chios (fl.440B.C.) and http://www.vigyanprasar.com/dream/mar2001/comets.htm
Extractions: Development of Cometary Thought PART - I Subodh Mahanti Lucius Annaeus Seneca (4B.C.-A.D.65) in Natural Questions ... In thick smoke of human sins, rising every day, every hour, every moment full of stench and horror, before the face of God and becoming gradually so thick as to form a comet, with curled and plaited tresses, which at last is kindled by hot and fiery anger of the supreme Heavenly Judge. Andreas Celichius in The Theologial Reminder of the New Comet (1578) Donald K.Yeomans in Comets : A Chronological History of observations, Science, Myth and Folklore (1991). The development of the scientific understanding about comets has a long and intriguing history. For centuries people (common people and scientists alike) have pondered the appearance of these mysterious apparitions. People's fascination for them, as seneca pointed out, was because they were unusual strange phenomena. They appear rarely. Before the seventeenth century comets were not considered as celestial bodies but as signals at a sinful Earth from God. celichius as quoted above was no doubt expressing the majority view of the comet prevalent in the 16th century. of course, there were opponents, though their number were few. for example Andreas Dudith (1533-89), the Hungarian scholar, countered celichius views by stating that if comets were caused by the sins of the mortals then they would never be absent from the sky.
Mathematicians Oenopides of Chios (c. 450?) *SB. Leucippus (c. 450) *SB *mt. hippocrates of chios(fl. c. 440) *SB. Meton (c. 430) *SB. Hippias of Elis (fl. c. 425) *SB *mt. http://www.chill.org/csss/mathcsss/mathematicians.html
Extractions: List of Mathematicians printed from: http://aleph0.clarku.edu:80/~djoyce/mathhist/mathhist.html 1700 B.C.E. Ahmes (c. 1650 B.C.E.) *mt 700 B.C.E. Baudhayana (c. 700) 600 B.C.E. Thales of Miletus (c. 630-c 550) *MT Apastamba (c. 600) Anaximander of Miletus (c. 610-c. 547) *SB Pythagoras of Samos (c. 570-c. 490) *SB *MT Anaximenes of Miletus (fl. 546) *SB Cleostratus of Tenedos (c. 520) 500 B.C.E. Katyayana (c. 500) Nabu-rimanni (c. 490) Kidinu (c. 480) Anaxagoras of Clazomenae (c. 500-c. 428) *SB *mt Zeno of Elea (c. 490-c. 430) *mt Antiphon of Rhamnos (the Sophist) (c. 480-411) *SB *mt Oenopides of Chios (c. 450?) *SB Leucippus (c. 450) *SB *mt Hippocrates of Chios (fl. c. 440) *SB Meton (c. 430) *SB Hippias of Elis (fl. c. 425) *SB *mt Theodorus of Cyrene (c. 425) Socrates (469-399) Philolaus of Croton (d. c. 390) *SB Democritus of Abdera (c. 460-370) *SB *mt 400 B.C.E. Hippasus of Metapontum (or of Sybaris or Croton) (c. 400?) Archytas of Tarentum (of Taras) (c. 428-c. 347) *SB *mt Plato (427-347) *SB *MT Theaetetus of Athens (c. 415-c. 369) *mt Leodamas of Thasos (fl. c. 380) *SB
500-1000 hippocrates of chios (reduction of the duplication problem, Tunes, arrangement ofthe propositions of geometry in a scientific fashion); Anaxagoras (geometry). http://euphrates.wpunj.edu/courses/math21180/chrono05.htm
Extractions: Possible date of the Sulvasutras (religious writings showing acquaintance with Pythagorean numbers and with geometric constructions); appearance of Chinese rod numerals Battle of Thermopylae. Beginning of Age of Pericles Parmenides (sphericity of the earth). Zeno paradoxes of motion Hippocrates of Chios (reduction of the duplication problem, Tunes, arrangement of the propositions of geometry in a scientific fashion); Anaxagoras (geometry). Antiphon (method of exhaustion). Plague at Athens Hippias of Elis trisection of angles with quadratrix); Theodorus of Cyrene (irrational numbers); Socrates Democritus (atomistic theory). Athens finally defeated by Sparta. Archytas (leader of Pythagorean school at Tarentum, applications of mathematics to mechanics). Death of Socrates Plato (mathematics in the training of the mind, Plato's Academy Theaetetus (incommensurables, regular solids).
