1. The Arithmetic Of Nichomachus Of Gerasa By Jay Kappraff For The Nexus Network Jo The Arithmetic of nicomachus of gerasa. and its Applications to Systems of Proportion nicomachus of gerasa (Figure 1, at left) has gained a position of importance in the history of http://www.nexusjournal.com/Kappraff.html

Abstract. Mathematician Jay Kappraff discusses a pair of tables of integers found in the Nicomachus's Introduction to Arithmetic and shows how they lead to a general theory of proportion. He shows how the system of musical proportions developed by the neo-Platonic Renaissance architects Leon Battista Alberti and Andrea Palladio, the Roman system of proportions described by Theon of Smyrna, and the Modulor of Le Corbusier are derived naturally from the Nicomachus tables. The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion Jay Kappraff New Jersey Institute of Technology Department of Mathematics University Heights Newark, New Jersey 07102 USA To speed up loading times, this text is hyperlinked to illustrations on a separate page. To go directly to the illustrations to the text, click here INTRODUCTION N icomachus of Gerasa (Figure 1, at left) has gained a position of importance in the history of ancient mathematics due in great measure to his Introduction to Arithmetic [1]. This book is one of the only surviving documentations of Greek number theory. Little is known about the life of Nicomachus, and the period of his life can only be estimated to lie between the middle of the first century and the middle of the second century AD, making him contemporary with Theon of Smyrna and Ptolemy. I will discuss a pair of tables of integers found in the

2. Nicomachus Of Gerasa BABYLONIAN and EGYPTIAN. nicomachus of gerasa (c. A.D http://www.brainfly.net/html/nicomach.htm

Greek and Roman Literature: HOME GREEK and ROMAN BABYLONIAN and EGYPTIAN Nicomachus of Gerasa (c. A.D. 60-120) Introduction to Arithmetic .........none.htm .......Disk 1 Manual of Harmonics .........none.htm .......Disk 1 ... .........none.htm .......Disk 1

3. Great Books Index - Nicomachus nicomachus of gerasa (about 60about 120) Have you written an online publication about Nicomachus? Please send the URL so it may be considered for a link. http://books.mirror.org/gb.nicomachus.html

GREAT BOOKS INDEX Nicomachus of Gerasa (about 60about 120) An Index to Online Great Books in English Translation AUTHORS/HOME TITLES ABOUT GB INDEX BOOK LINKS Nicomachus' Arithmetic Links to Information About Nicomachus Biography of Nicomachus (History of Math Archive) [Back to Top of Page] GREAT BOOKS INDEX MENU Great Books Index Home Page and Author List List of All Works by Author and Title [90KB] About the Great Books Index Links to Other Great Books and Literature Sites ... Literary Cryptograms Support for the Great Books Index web pages is provided by Ken Roberts Computer Consultants Inc URL: http://books.mirror.org/gb.nicomachus.html Last revised January 11, 1999 by Ken Roberts e-mail ken@mirror.org

4. Nicomachus nicomachus of gerasa. nicomachus of gerasa is mentioned in a small number ofsources and we can date him fairly accurately from the information given. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html

5. Nicomachus Biography of Nicomachus (60120) nicomachus of gerasa. Born about 60 in Gerasa, Roman Syria (now Jarash, Jordan) nicomachus of gerasa is mentioned in a small number of sources and we can date http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html

6. References For Nicomachus Books MLD Ooge (trs.), nicomachus of gerasa, Introduction to Arithmetic (New York,1926). TL Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921). http://www-gap.dcs.st-and.ac.uk/~history/References/Nicomachus.html

7. Great Books And Classics - Nicomachus Of Gerasa Great Books and Classics nicomachus of gerasa (c. 60?-120?) http://www.grtbooks.com/nicomachus.asp?idx=0&yr=60&aa=NI&at=AA

8. Nicomachus Of Gerasa --  Encyclopædia Britannica Encyclopædia Britannica, nicomachus of gerasa Encyclopædia Britannica Article. MLAstyle nicomachus of gerasa. Encyclopædia Britannica. 2004. http://www.britannica.com/eb/article?eu=57159

9. A-Z Browse Results - Encyclopædia Britannica Nicolet, Jean Nicolle, CharlesJules-Henri Nicollet, Joseph Nicolas Nicolls, RichardNicolson, Sir Harold Nicomachus nicomachus of gerasa Nicopolis Actia http://www.britannica.com/eb/alpha?seq=46362&alphakey=N

10. Final Chapters Of Greek Mathematics: Nicomachus nicomachus of gerasa ( Late 1st century AD) Biography. Contributions. A Closer Look. Additional Resources. Credits. Home. Biography. Very little is known about the life of Nicomachus. He was from Gerasa, a Greek town in Judea. M L D'Ooge (trs.), nicomachus of gerasa, Introduction to Arithmetic http://www.math.utk.edu/~m400/ch5/Nicomachus.html

Nicomachus of Gerasa (Late 1st century AD) Biography Contributions A Closer Look Additional Resources ... Home Biography Very little is known about the life of Nicomachus. He was from Gerasa, a Greek town in Judea . It is not known exactly when he was born or died but it is estimated that he lived from about 60-120 A.D. based on him being mentioned in a small number of sources. We can assume that he probably studied in Alexandria because much of his work is heavily influenced with Pythagorean ideas. There are two surviving works done by Nicomachus: Introduction to Arithmetic and Introduction to Harmonics . The books are written with great detail but contain no proofs. Nicomachus relied on examples rather than proofs. Contributions Introduction to Arithmetic Introduction to Harmonics A Closer Look Introduction to Arithmetic was split into two books. The 1st book dealt with the classification of integers and their relations. For example, the classification of even and odd numbers into subgroups. Even Even-times-even (powers of 2): 2, 2*2*2=8 Even-times-odd (doubles of odd numbers): 2*3=6, 2*9=18

11. Figure 4 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And Its Applic Click here to go to the NNJ homepage. Figures for Jay Kappraff s The Arithmeticof nicomachus of gerasa and its Applications to Systems of Proportion. http://www.nexusjournal.com/Kappraff-fig04.html

Figures for Jay Kappraff's The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion ... Top of Page

12. Great Books And Classics - Nicomachus Of Gerasa Selected Reading List All Works ? Change Selected Language AllChange. Author Chronological, nicomachus of gerasa (c. 60?-120?), http://www.grtbooks.com/nicomachus.asp?idx=0&yr=60

13. The Manual Of Harmonics The Manual of Harmonics of Nicomachus the Pythagorean. Translation and Commentary. by Flora R. Levin translation of The Manual of Harmonics by the Pythagorean philosopher nicomachus of gerasa (second century A and helps to situate Nicomachus' thought in the history of ideas http://phanes.com/manhar.html

The Manual of Harmonics of Nicomachus the Pythagorean Translation and Commentary by Flora R. Levin In ancient Greek thought, the musical scale discovered by the philosopher Pythagoras was seen as a utopian model of the harmonic order behind the structure of the cosmos and human existence. Through proportion and harmony, the musical scale bridges the gap between two extremes. It encapsulates the most fundamental pattern of harmonic symmetry and demonstrates how the phenomena of nature are inseparably related to one another through the principle of reciprocity. Because of these relationships embodied in its structure, the musical scale was seen as an ideal metaphor of human society by Plato and other Pythagorean thinkers, for it is based on the cosmic principles of harmony, reciprocity, and proportion, whereby each part of the whole receives its just and proper share. This book is the first ever complete translation of The Manual of Harmonics by the Pythagorean philosopher Nicomachus of Gerasa (second century A.D.) published with a comprehensive, chapter-by-chapter commentary. It is a concise and well-organized introduction to the study of harmonics, the universal principles of relation embodied in the musical scale. Also included is a remarkable chapter-by-chapter commentary by the translator, Flora Levin, which makes this work easily accessible to the reader today. Dr. Levin explains the principles of Pythagorean harmony, provides extensive background information, and helps to situate Nicomachus' thought in the history of ideas. This important work constitutes a valuable resource for all students of ancient philosophy, Western cosmology, and the history of music.

14. - Great Books - nicomachus of gerasa (c. 100), nicomachus of gerasa is mentioned in a small numberof sources and we can date him fairly accurately from the information given. http://www.malaspina.com/site/person_887.asp

Nicomachus of Gerasa (c. 100-) Nicomachus of Gerasa is mentioned in a small number of sources and we can date him fairly accurately from the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates. On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates. One of the most interesting references is by Lucian , the rhetorician, pamphleteer and satirist who was born about 120 AD, who makes one of his characters say: You calculate like Nicomachus . Clearly Nicomachus had achieved fame for his arithmetical work! Nicomachus was a Pythagorean . This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagoreans School. Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples. However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus stated a result that is simply false and then illustrated it with an example that happens to have the properties described in the result. We must deduce from this that some of the results are merely guesses based on the evidence of the numerical examples (and in some cases perhaps even based on one example!).

15. FWD: Greek & Latin Texts On The Net By Antreas P. Hatzipolakis is our =the arithmetician nicomachus of gerasa. His text (in Beta English Translation Author Nicomachus, of Gerasa Title Introduction to arithemetic / nicomachus of gerasa http://mathforum.com/epigone/math-history-list/vimpzhimpblimp

reply to this message post a message on a new topic Back to math-history-list Subject: Author: xpolakis@hol.gr Date: our server The Math Forum

16. - Great Books - Nero (3768), Antiquity History 65. New Testament (c. 140-), Antiquity Literature66. nicomachus of gerasa (c. 100-), Antiquity Literature/Music/Science 67. http://www.malaspina.com/site/results_p17_page1.htm

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17. Geometer's Angle No. 5 By Marcus The Marinite For The Nexus Network Journal Vol. The Arithmetic of Nicomachas of Gerasa and its Applications to Systems of Proportion in Nexus Network Journal2. Nicomachus. 1926. nicomachus of gerasa An Introduction to Arithmetic http://www.nexusjournal.com/GA-v3n4.html

Abstract. Geometer Marcus the Marinite explores how the geometric and harmonic means can be employed compositionally within a frame; how means function in the development of organizational field grids; how means work to develop linear perspective and proportional grids. The Geometer's Angle No. 5: Geometric and Harmonic Means and Progressions Marcus the Marinite 667 Miller Avenue Mill Valley, California 94941 USA T he three principal means most commonly held useful to architects and artists are the arithmetic, the geometric, and the harmonic means. The subject of the three means can be found both in ancient Greek sources and in crucial Renaissance architectural treatises such as those of Leon Battista Alberti and Andrea Palladio.[ As a very brief introduction to means, let me give a simplified definition and explanation of what a mean is. Mathematically, a mean is a quantity having a value intermediate between two other values, as determined by a specific formula or set of conditions. A mean is a number that stands in some proportionality (or other relationship) to a number lesser than it in value and another number greater than it in value. The outer two numbers are called extremes.[ ] A mean is a kind of average between extremes, but the various types of means include averages that are more complex than the familiar concept of 'halfway in between' (i.e., the arithmetic mean).

18. Nicomachus Of Gerasa (fl.c. 100 AD) Library Of Congress Rare and Hardto-Find Books from Alibris nicomachus of gerasa (fl.c. 100 AD) Library of Congress Citations The Little Search Engine that Could. http://www.mala.bc.ca/~mcneil/cit/citlcnico.htm

Nicomachus of Gerasa (fl.c. 100 A.D.) : Library of Congress Citations The Little Search Engine that Could Down to Name Citations LC Online Catalog Amazon Search Book Citations [6 Records] Author: Nicomachus, of Gerasa. Uniform Title: Introductio arithmetica. English Title: Introduction to arithmetic. Translated into English by Martin Luther D'Ooge, with studies in Greek arithmetic by Frank Egleston Robbins and Louis Charles Karpinski. New York, Macmillan, 1926. Published: [New York, Johnson Reprint Corp., 1972] Description: ix, 318 p. illus. 23 cm. LC Call No.: QA31 .N553 1972 Dewey No.: 513 Notes: Original ed. issued as v. 16 of University of Michigan studies. Humanistic series. Bibliography: p. 311-312. Subjects: Mathematics, Greek. Arithmetic Early works to 1900. Series Entry: University of Michigan studies. Humanistic series ; v. 16. Control No.: 73039141 //r934 Author: Nicomachus, of Gerasa. Uniform Title: Introductio arithmetica. French Title: Introduction arithmbetique / Nicomaque de Gberase ; introd., traduction, notes et index par Janine Bertier. Published: Paris : J. Vrin, 1978. Description: 254 p. ; 26 cm. Series: Histoire des doctrines de l'antiquitbe classique ; 2 LC Call No.: QA31 .N554 Dewey No.: 513 ISBN: 75.00F Notes: Translation of Introductio arithmetica. Bibliography: p. [227]-232. Includes index. Subjects: Mathematics, Greek. Arithmetic Early works to 1900. Other authors: Bertier, Janine. Control No.: 79338551 //r933

19. Just Intonation: Origins Of Length Ratios 400 years later in the Manual of Harmonics by the mathematician nicomachus of gerasa (b. c which probably explains why neither Nicomachus nor Plutarch contributed a single original http://www.chrysalis-foundation.org/origins_of_length_ratios.htm

Musical Mathematics a practice in the mathematics of tuning instruments and analyzing scales © 2004 Cristiano M.L. Forster www.Chrysalis-Foundation.org FLEXIBLE STRINGS Part V: Musical, Mathematical, and Linguistic Origins of Length Ratios For descriptions of the arithmetic and harmonic divisions of musical intervals that determine the 'minor' and 'major' tonalities in Western music, see Ibn Sina, Stifel, and Zarlino. Chapter 3: Section 17 We may use the preceding example, which describes the tuning process of two unfamiliar frequency ratios, 7/5 and 7/2, to analyze the musical, mathematical, and linguistic origins of length ratios. Although aural experiences of sound depend on frequencies, our anatomy does not enable us to quantify rates of vibration from oscillating strings. As such, the frequencies produced while playing musical intervals have always and will always remain incomprehensible quantities. Therefore, if we spontaneously tune two canons strings to an unfamiliar musical interval, we cannot identify the new interval by counting cycles per second and constructing a frequency ratio. To comprehend such an experience, we must measure the corresponding string lengths, and, through the construction of a length ratio, identify the unfamiliar interval. Conversely, to intentionally tune two strings to a new musical interval, we cannot locate the required position of a canon bridge by simply listening to frequencies. Instead, we must divide the strings according to a length ratio that

20. Phorum - Nicomachus Of Gerasa nicomachus of gerasa DR. ELLIOT S NORTH AMERICAN GREAT BOOKS TOURCOMINGTO A BOOK STORE NEAR YOU WRITERSWORD.COM Open Source http://jollyroger.com/classics/cforum/list.php?f=181

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