81. Ketch Wellness - Miscellaneous nicomachus of gerasa recorded the process and it was picked up by Iamblichusbefore 350 AD We inherit the work of Leonardo of Pisa, also known as Fibonacci, http://www.ketchwellness.com/(xgefnc45sgdjefatjwd4yxnz)/Products.aspx?m=3&CID=Mi

82. A Mithraic Ritual The statement of nicomachus of gerasa the musician and mystic (second centuryAD), is still clearer; for he not only tells us about the vowels and http://www.hermetic.com/pgm/mithraic.html

A Mithraic Ritual G.R.S. Mead Preamble The last little volume gave the reader a brief outline of what is known of the cult of Mithra and the spread of the Mithriac Mysteries in the Western world. We have now to deal with a Mithriac Ritual of the most instructive and intensely interesting character, which introduces us to the innermost rite of the carefully guarded secrets of the Mithriaca. The credit of unearthing it from the obscurity in which it was buried, and of conclusively demonstrating its parent-age, is due to Dieterich; for though Cumont in his great work quotes several passages from the unrevised text, he does so only to reject it as a genuine Mithriac document. It is dug out of the chaos of the great Paris Magic Papyrus 574 ( Supplement grec de la Bibliotheque nationale ), the date of which is fixed with every probability as the earliest years of the fourth century A.D.. The original text of the Ritual has, however, been plainly worked over by a school of Egyptian magicians, who inserted most of the now unintelligible words and names ( ashma ovomata, nomina barbara, nomina arcana

83. NES/MAA Regional Dinner Meetings - Spring 2004 explorer, and student of D Alembert; and end with recent extensions of thework of the second century (AD) mathematician, nicomachus of gerasa. http://www.southernct.edu/organizations/nesmaa/regdinnermts2005.html

NES/MAA REGIONAL DINNER MEETINGS - Spring 2005 The Northeastern Section Regional Dinner Meeting series has proven to be both popular and stimulating. Therefore, once again, the NES/MAA offers you the opportunity to spend an interesting and enjoyable evening with others involved in mathematics and mathematics education. There will be additional dinner meetings scheduled. Information about dinner meetings will be posted as it becomes available Boston Area Simmons College, April 12, 2005 Central Massachusetts College of the Holy Cross, April 7, 2005 Eastern Massachusetts Framingham State College, April 28, 2005 Rhode Island Providence College, April 21, 2005 Feel free to attend as many of the dinner meetings as you wish. Please send a separate reservation form for each person attending and for each dinner meeting that you wish to attend. Reservations must be made in writing by the respective deadlines since exact numbers must be given in advance. The registration forms are available on-line. However, the form must be mailed to the appropriate coordinator. On-line registration and e-mail registration are not available. Please contact Lucy Kimball, Mathematical Sciences Department, Bentley College, Waltham, MA 02154, (781) 891-2467

84. Jay Kappraff Kappraff, J. “The Arithmetic of nicomachus of gerasa and its. Applications toSystems of Proportion”Nexus Network Journal (an http://www-ec.njit.edu/~kappraff/personal.html

Home Academic Personal Research Site Map Search Web Email ... Links Personal Resume Jay Kappraff Home Address: Professional Address: 102 Rynda Road Assoc. Prof. of Math. South Orange, NJ 07079 New Jersey Institute of Technology Newark, NJ 07102 (973) 596-3490 Education: Ph.D. (Applied Mathematics) New York University M.S. (Chemical Engineering) Iowa State Univ. B.Ch.E. (Chem. Eng.) New York Polytechnic Professional Experience: Academic Appointments - Assoc. Prof (Math.) NJIT 1974 - present - Instructor (Math.) Cooper Union College Non-Academic Employment - Aerospace Engineer NASA (Lewis Res. Lab.) 1962 - 1965 - Teacher of Math. N.Y. City Board of Ed. 1962 - Chem. Eng. Dupont DeNemours Newport Lab. Wilmington, Del. Consultant Dept. of Energy Publications relevent to the work proposed: Kappraff, J. "Beyond Measure: A Guided Tour through Nature Myth, and Number." 480 pages. Singapore: World Scientific (2001) Kappraff, J. "A Secret of Ancient Geometry." In Geometry at Work edited by C.Gorini. Mathematics Association of America Geometry MASA Notes No. 53 (2000) Kappraff, J. Ancient Harmonic Law. unpublished

85. The Research Notebook Of A Beleaguered Hack.: Struik: A Concise History Of Mathe nicomachus of gerasa (c. 100 CE) was one of the earliest Alexandrian mathematiciansof the Roman era. His Arithmetic Introduction is the most complete http://arsenal.media.mit.edu/notebook/archives/000104.html

The research notebook of a beleaguered hack. Struik: A Concise History of Mathematics: Greece As the Mediterranean basin transitioned from the Bronze Age to the Iron Age there were enormous economic and political changes. Iron's introduction changed warfare but also cheapened production, causing a surplus, and allowing more people to be involved in public life. Additionally the introduction of coined money and the alphabet stimulated trade. Culture began to flourish, no longer exclusively the domain of the elite. Sea-raiders upset cultural, scientific, and mathematical progress as they destroyed Minoan civilization, and disrupted Egyptian and Babylonian development. The rise of the Greek city-state in 7th-6th centuries BCE allowed new freedoms to citizens. More leisure (begotten from slavery and wealth) stimulated the growth of rationalism, philosophy, and science. In contrast to oriental approaches, the Greeks did not just "how?" but sought to know "why?" Thales of Milete (who had traveled to Babylon and Egypt) is considered the father of Greek math. In seeking a rational scheme to the universe his tradition added an element of rationalism to mathematics. Only small fragments remain, but through careful analysis a consistent picture of Greek math emerges. The rise of Persia led to conflict and an eventual Greek victory, which expanded the power of Athens. Democratic ideals flourished from 450-400 BCE, paving the path for the Golden Age of Greece. Sophists had greater freedom to examine ideas more abstract than useful. The only complete fragment of this period is written by Hippocrates of Chios. The work shows perfected mathematical reasoning and deals with the impractical: the

86. A Second Summary Of Great Books Collected works Apollonius of Perga Conic Sections nicomachus of gerasa Introduction to Arithmetic Marcus Aurelius The Meditations Apuleius The http://www.radix.net/~bobg/books/summary2.html

... and here's another summary of lists of great books. See also the first (to me, that is). Robert Grumbine bobg@radix.net From andrewd@cs.tamu.edu Wed Apr 20 09:36:44 EDT 1994 Article: 58584 of rec.arts.books Path: news1.digex.net!uunet!cs.utexas.edu!news.tamu.edu!sematech.tamu.edu!andrewd From: andrewd@sematech.tamu.edu (Andrew Duchowski) Newsgroups: rec.arts.books Subject: Greatest books of all time (recompiled) Date: 19 Apr 1994 18:21:59 GMT Organization: Sematech Research, Texas A+M University Lines: 553 Distribution: usa Message-ID: Return to bobg books page Return to bobg main page bobg@radix.net

87. Harmony / Harmonic - Pertaining To Simulaneous Musical Sounds nicomachus of gerasa. c. 100 AD. Manual of Harmonics . English translation andcommentary by Flora Rose Levin (ed.), Phanes Press, Grand Rapids MI, 1994, http://tonalsoft.com/enc/h/harmonic.aspx

Encyclopedia of Microtonal Music Theory A B C D ... Encyclopedia Index harmony / harmonic [John Chalmers, Divisions of the Tetrachord] [In ancient Greek theory,] pertaining to harmonia and harmonike . When applied to chords or scales, it indicates that they are a section of the harmonic series. The mathematical mean of two tones which produces intervals in the reverse order of the arithmetic mean is called the harmonic mean by Archytas because it was felt that this sequence was the more musical. In Western European theory, harmonic refers to harmony. [Joe Monzo, Yahoo tuning group message 46454 (Wed Aug 20, 2003 8:14 am)] ... it really bugs me that so many music-theorists and historians are so quick to believe that "harmony" in music, meaning simultaneous combinations of different pitches, was "invented" by the Franks in the 800s. certainly, there was a long time where the paradigm was monophonic "Gregorian" chant. but what about the period before that? Gregorian chant developed after a period of extreme social upheaval in Europe, namely, the Germanic migrations of the 400s-500s. actually, i believe that the ancient Romans had some form of polyphony, even if it was basically this note-for-note style. Boethius in c. 505 specifically says that a "consonance" is the blending of two or more *simultaneous* sounds, and Nicomachus said the same thing c. 100 AD, when Greece was part of the Roman Empire. see the evidence i present below. note the emphasis on sounds which are "struck at the same time".

88. FAQ - Great Books Academy philosophy (Socrates, Plato), cartography and history (Herodotus, Thucydides),social studies, and even math (Euclid, nicomachus or gerasa). http://www.greatbooksacademy.org/faq.html

[Every Item in this Bookstore is a Part of our Curriculum.] Last Updated 8-12-05 Frequently Asked Questions Great Books Academy For technical questions, click here May I buy materials without enrolling? Yes. Anyone may purchase our materials. If enrolled, may I substitute courses from one grade for another? Yes. Parents are the primary educators of their children. You may choose as many or as few courses as you wish, from whatever grade levels you wish, for any student. Do we have to use Academy materials exclusively for our homeschooling? No. Again, parents are the primary educators. We are here to aid them. While we believe our materials are the finest available we understand that in some circumstances parents may wish to continue using some materials they already have.

89. Footnotes music4 by eg Aristoxenus of Tarentum (fourth century BC), nicomachus ofGerasa (second century), Ptolemy (second century), Aristides Quintilianus (?late http://www-jcsu.jesus.cam.ac.uk/~mma29/essays/dissertation/footnode.html

... musical.' Against the Musicians Source Readings in Music History p. 96 ... Gospels, e.g. Mark 14.26 ... singing; 1 Corinthians 14.15; Colossians 3.16 ... music by e.g. Aristoxenus of Tarentum (fourth century B.C.), Nicomachus of Gerasa (second century), Ptolemy (second century), Aristides Quintilianus (?late third/early fourth century) ... saeculis. Hilary, fragmentary acrostic Hymn 2, CSEL vol. LXV pp. 212 ff. - closing lines: `Christ, returning to heaven victorious, remember my flesh, in which you were born. Unending songs with the angel choirs shall I gladly sing when I rise in this body. Once Satan desired me in death: may he see me reign with you for ever.' [my translation] ... esteemed.' Letter 107, 3, Music in Early Christian Literature p. 68 ... service.' De utilitate hymnorum ... desire.' Jerome, Letter 117, 6 ... totality'; Noise , p. 3 ... musicians. ibid. p. 12 ... masses', Athenaeus, Sophists at Dinner SRMH p. 92 ... men.' pseudo-Plutarch, On Music 1140D-E(27) ... heart.' De utilitate hymnorum ... time; De Spiritu Sancto ... hand.' De utilitate hymnorum ... repeated.' ibid.

90. Physics And Music thought flourished throughout antiquity in such theorists as nicomachus ofGerasa and Ptolemy and was transmitted into the Middle Ages by Boethius. http://www.angelfire.com/wv/jeanwilson/music.html

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "angelfire.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Angelfire 40 Yr Old Virgin Share This Page Report Abuse Edit your Site ... Next Essays from Our Physical World. The Man Who Knew Too Much Music Although many names of musicians are recorded in ancient sources, none played a more important role in the development of Greek musical thought than the mathematician and philosopher Pythagoras of Samos. According to legend, Pythagoras, by divine guidance, discovered the mathematical rationale of musical consonance from the weights of hammers used by smiths. The connections between the two seemingly disparate subjects of mathematics and music become obvious when studying Pythagoras' concept of the harmony of the spheres. Pythagoras believed that all relations could be reduced to number relations, making observations in music, mathematics and astronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. Information on Pythagoras taken from Pre-Socratic Philosophers @ http://ancienthistory.miningco.com/education/

91. Abstracts For MAA INDIANA SECTION MEETING, APRIL 1-2, 2005 At IPFW Two first century (AD) manuscripts, the Introduction to Arithmetic, by Nicomachusof gerasa and Mathematics Useful for Understanding Plato by Theon of http://www.ipfw.edu/math/maa2005/jtabs.html

Abstracts for talks by J. Tattersall MAA INDIANA SECTION MEETING, APRIL 1-2, 2005 Friday Abstract: THREE MATHEMATICAL VIGNETTES; MILLENNIAL, PONTIFICAL, AND NYCTAGINACEOUS Two first century (A.D.) manuscripts, the Introduction to Arithmetic , by Nicomachus of Gerasa and Mathematics Useful for Understanding Plato by Theon of Smyrna were the main sources of knowledge of formal Greek arithmetic in the Middle Ages. The books are philosophical in nature, contain few original results and no formal proofs. They abound, however, in intriguing number theoretic observations. We discuss and extend some of the results found in these ancient volumes. Secondly, we discuss the mathematics of Gerbert the Great, a tenth century educator. We end with the achievements and adventures of Louis Antoine de Bougainville, mathematician, explorer, and student of D'Alembert. Saturday Abstract: EPISODES IN THE EARLY HISTORY OF THE LUCASIAN CHAIR In 1663, Henry Lucas, the long-time secretary to the Chancellor of the University of Cambridge, made a bequest, subsequently granted by Charles II, to endow a chair in mathematics. A number of conditions were attached to the Chair. Among the more prominent Lucasian professors were Newton, Babbage, Stokes, Dirac, and Hawking. We focus attention on the early Lucasians. Many of whom were very diligent in carrying out their Lucasian responsibilities but as history has shown such was not always the case. In the process, we uncover several untold stories and some interesting mathematics.

92. Spring 2003 Sectional Meeting Of The Allegheny Mountain Section Of Abstract In the first century AD, the Introduction to Arithmetic, by Nicomachusof gerasa and Mathematics Useful for Understanding Plato by Theon of Smyrna http://mallard.ds.psu.edu/~ecb5/MAA-Spring-03/tattersall.html

93. Relics From The Dawn Of Time I saw old editions of the works of Archimedes, Apollonius, Clavius, Nicomachusof gerasa, Ptolemy (Regiomantanus translation), Diophantus of Alexandria, http://www.maa.org/features/relics.html

Relics from the Dawn of Time by John Burns I think that the most exciting day of my professional careerexcept, of course, receiving my degrees and employmentwas Monday, August 2, 1999, at Brown University where I was attending the Annual Mathematical Association of America Mathfest I have always been intrigued by the history of the development of science and mathematics in general. However, two of my chief interests were the history of the Calculus before Newton and Leibwitz, and the history of the development of our understanding of the solar system. Each of these interests was partially satisfied, but also stimulated during the many times at California State University, Los Angeles that I taught the "History of Mathematics" course, which naturally contained Astronomy and Physics as well. As Einstein said, "The more that the circle of knowledge increases, the more does the circle of darkness around its edge." Both of these circles expanded greatly for me in the six weeks that I spent in Washington D. C. at the "Institute for the Study of Mathematics and its Use in Teaching" in the summers of 1995 and 1996. It was a wonderful experience, and I loved every minute of it. I also saw the works of the five giants that formed the basis of modern (heliocentric) Astronomy, Copernicus, Ticho Brahe, Kepler, Galileo, and Newton, as well as those who developed the mathmatical foundation of our understanding of the solar system, Clairaut, Euler, d'Alembert, Lagrange, Gauss, Laplace, Hill, Poincaré, and Einstein.

94. ALC III,3: Mathematics Pure And Applied both Mohammedan and Christian, the Introduction to Arithmetic of Nicomachusof gerasa, who lived about one hundred years after the birth of Our Lord. http://www.domcentral.org/study/ashley/arts/arts303.htm

BENEDICT M. ASHLEY, O.P.: THE ARTS OF LEARNING AND COMMUNICATION CHAPTER III Mathematics Pure and Applied PURE MATHEMATICS ALGEBRA AS A SCIENCE We have studied Euclid's Elements as an example of a pure science, the science of geometry, but we saw also that Books VII, VIII, and IX deal with the science of arithmetic, or (as we would now call it) with the science of algebra. If we wish to see such a pure science of number developed even at great length, we should examine the work which became standard among the Greeks, Romans, and medievals, both Mohammedan and Christian, the Introduction to Arithmetic of Nicomachus of Gerasa, who lived about one hundred years after the birth of Our Lord. It deals with that basic study of numbers which we today call Number Theory. The term "algebra" was introduced to indicate a growing interest in the art of calculation, rather than a demonstrative study of the properties of numbers. just as there is a science of logic which seeks to prove the rules of logic (see page 568 ff.) and an art of logic which is concerned with using these rules, so the science of number is concerned with proving the properties of numbers, while the art of calculation is concerned with applying a knowledge of these properties to the solution of particular problems. Today, however, algebra includes both the art of numerical problem-solving and the science of numbers, but the elementary study of algebra usually lays emphasis on its problem-solving aspect. Now that we have analyzed geometry as a science, we need to review algebra from the same point of view and see how it might be set up as a pure science.

95. Progressioni Geometriche by Martin Luther D Ooge, with notes by Frank Egleston Robbins and Louis http://utenti.quipo.it/base5/numeri/progrgeom.htm

HOME - BASE Cinque Appunti di Matematica ricreativa Progressioni geometriche a n = a n-1 q Il cavallo stanco Un cavallo ha percorso 700 miglia in 7 giorni, dimezzando la sua velocità ogni giorno. Quanto ha percorso ogni giorno? Zhang Qiujian Suan Jing Dilapidare la ricchezza Un uomo possiede inizialmente 100 denari e spende ogni giorno 1/10 di ciò che ha. Con quanto rimane dopo 12 giorni? Fibonacci. 1202 Il viaggiatore Un uomo percorre 1, 3, 9, ... leghe in giorni successivi. Continuando a questo ritmo, quante leghe percorrerà in 5 giorni e mezzo? Chuquet, 1484 La botte che si svuota Una botte contiene una quantità di vino pari a 9,5 barili. Il suo contenuto viene trasferito nei barili in modo tale che: il primo barile si riempie in 1 ora; il secondo barile si riempie in 2 ore; il terzo barile si riempie in 4 ore; e così via, raddoppiando ogni volta il tempo. Quanto tempo è necessario per svuotare la botte? Chuquet, 1484 Progressioni bibliche Tratto da Jacques Ozanam, Recreations..., Paris, 1778, Tome 1. Gli ebrei in Egitto Si parte con 210 persone.

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