1. Nicomachus Of Gerasa (fl.c. 100 A.D.) Library Of Congress Citations Book Citations 6 Records. Author Nicomachus, of Gerasa. Uniform Title Introductio arithmetica. 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

2. 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

Nicomachus of Gerasa Born: about 60 in Gerasa, Roman Syria (now Jarash, Jordan) Died: about 120 Click the picture above to see a larger version Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index 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! In the paper [7] Dillon argues that Nicomachus died in 196 AD. His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus. Since

3. Nicomachus Biography of Nicomachus (60120) nicomachus of gerasa. Born about 60 in Gerasa, Roman Syria (now Jarash, Jordan) http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html

Nicomachus of Gerasa Born: about 60 in Gerasa, Roman Syria (now Jarash, Jordan) Died: about 120 Click the picture above to see a larger version Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index 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! In the paper [7] Dillon argues that Nicomachus died in 196 AD. His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus. Since

4. 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

References for Nicomachus Biography in Dictionary of Scientific Biography (New York 1970-1990). Biography in Encyclopaedia Britannica. Books: M L D'Ooge (trs.), Nicomachus of Gerasa, Introduction to Arithmetic (New York, 1926). T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921). Articles: S Brentjes, Untersuchungen zum Nicomachus Arabus, Centaurus M Crubellier and J Sip, Looking for perfect numbers, History of Mathematics : History of Problems (Paris, 1997), 389-410. J M Dillon, A date for the death of Nicomachus of Gerasa, Classical Review Rev. Histoire Sci. H R Muzafarova, Nicomachos' arithmetic in a work by Qutb ad-Din as-Shirazi (Russian), in Mathematics and the methodology of its teaching (Russian) I (Dushanbe, 1974), 124-131. Trans. Amer. Philos. Soc. Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR April 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Nicomachus.html

5. - Great Books - nicomachus of gerasa is mentioned in a small number of sources and we can date him fairly accurately from the http://www.mala.bc.ca/~mcneil/nico.htm

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6. - 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!).

7. - Great Books - 10. nicomachus of gerasa (c. 100), Antiquity Literature/Music/Science11. Nielsen, Carl (1865-1931), Modern Music 12. Nietzsche http://www.malaspina.com/site/results_iN_page1.htm

Index Listings for: N 13 listings found Browse again by first initial of authors authors last name A B C D ... M N O P Q R ... Nabokov, Vladimir (1899-1977), Modern Literature/Science Nafis (-al), Ibn (1210-1288), Medieval Science Naogeorgus, Thomas (1511-1563), Renaissance Theatre Neel, Alice (1900-1984), Modern Art Nero (37-68), Antiquity History New Testament (c. 140-), Antiquity Literature Newman, John Henry (1801-1890), Romantic Literature Newton, Isaac (1642-1727), Baroque Literature/Science Nibelungenlied and Saga of Burnt Njal (c. 1250-), Medieval Literature Nicomachus of Gerasa (c. 100-), Antiquity Literature/Music/Science Nielsen, Carl (1865-1931), Modern Music Nietzsche, Friedrich Wilhelm (1844-1900), Romantic Literature Nightingale, Florence (1820-1910), Romantic History A B C D ... M N O P Q R ... Z

8. Figure 4 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And Its Applic 4. Depressing the string of a monochord at its midpoint and sounding it with a bow yields a tone one octave above the fundamental tone characteristic of the whole string. http://www.emis.de/journals/NNJ/Kappraff-fig04.html

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

9. The Arithmetic Of Nichomachus Of Gerasa By Jay Kappraff For The Nexus Network Jo INTRODUCTION nicomachus of gerasa (Figure 1, at left) has gained a position ofimportance in the history of ancient mathematics due in great measure to his 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

10. Figure 8 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And 8. A design by Mark Bak which contains S, SR, and RR at three different scales. back to text top of page http://www.hms.gr/EMIS/journals/NNJ/Kappraff-fig08.html

11. Figure 5 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-fig05.html

Figures for Jay Kappraff's The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion ... Fig. 10 2, the sacred cut. Four sacred cuts mark the vertices of a regular octagon and lead to a subdivision of the square into three species of rectangles: a square of proportion 1:1; a square root rectangle of proportion 1: 2; and a rectangle of proportion 1: q, top of page back to text Top of Page

12. Figure 6 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And 6. The algebraic properties of the Roman system can be made palpable by considering the equivalent geometric properties based on the interrelation of the proportions 1, Ö2, and q. http://www.hms.gr/EMIS/journals/NNJ/Kappraff-fig06.html

13. Figure 10 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And Its Appli 10. The exterior of Le Corbusier's Unité d'Habitation with its windows exhibiting the Modulor tiling. top of page back to text http://ftp.univie.ac.at/EMIS/journals/NNJ/Kappraff-fig10.html

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

14. Malaspina.com - Nicomachus Of Gerasa (fl. Ca. 100 AD) Launch Previous Entry in New Window Malaspina Literature Database Launch NextEntry in New Window nicomachus of gerasa (fl.c. 100 AD) MacTutor. http://www.mala.bc.ca/~mcneil/nico1.htm

Nicomachus of Gerasa (fl.c. 100 A.D.) [MacTutor] Great Books Biography [Malaspina] Amazon Search Form] Library of Canada Online Citations [NLC] Library of Congress Online Citations [LC] Library of Congress Offline Citations [MGB] COPAC UK Online Citations [COPAC] Free Online Practice Exams [Grad Links] Canadian Book Orders! Chapters-Indigo Save on Textbooks! [Study Abroad] Used Books Search Form Alibris Dummies Books Amazon Books from Amazon Amazon EBay! Ebay Books from Amazon UK Amazon UK Books from Chapters Canada Chapters Amazon's 100 Hot Books Amazon The Manual of Harmonics of Nichomachus the Pythagorean Amazon Introduction to Arithmetic Amazon Ancient Greek Music (book Review) [Brown University] The Manual of Harmonics Paper, 208 pages, ISBN 0-933999-43-7, $18.00 Published by Phanes Press Top of Page

15. Figure 5 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And Its Applic 5. The geometrical construction referred to by Tons Brunés, a Danish engineer and student of ancient geometry, as the Sacred Cut. http://ftp.univie.ac.at/EMIS/journals/NNJ/Kappraff-fig05.html

Figures for Jay Kappraff's The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion ... Fig. 10 2, the sacred cut. Four sacred cuts mark the vertices of a regular octagon and lead to a subdivision of the square into three species of rectangles: a square of proportion 1:1; a square root rectangle of proportion 1: 2; and a rectangle of proportion 1: q, top of page back to text Top of Page

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17. Figure 9 For Kappraff's "The 'Arithmetic' Of Nicomachus Of Gerasa And Its Applic 9. A square tiled in three ways with the same species of rectangles whose lengths and widths are numbers from the Red and Blue sequence. top of page back to text http://webdoc.sub.gwdg.de/EMIS/journals/NNJ/Kappraff-fig09.html

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

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19. Nicomachus Of Gerasa - Britannica.com ENCYCLOPÆDIA BRITANNICA. nicomachus of gerasa fl. AD 100,, Gerasa,Roman Syria now Jarash, Jordan. NeoPythagorean philosopher http://piano.dsi.uminho.pt/disciplinas/PAED/temas/nicomach/Nicomachus Of Gerasa

ENCYCLOPÆDIA BRITANNICA Nicomachus Of Gerasa fl. AD 100,, Gerasa, Roman Syria [now Jarash, Jordan] Neo-Pythagorean philosopher and mathematician who wrote Arithm e tik e eisag o g e Introduction to Arithmetic ), an influential treatise on number theory. Considered a standard authority for 1,000 years, the book sets out the elementary theory and properties of numbers and contains the earliest-known Greek multiplication table. In his Arithm e tik e Nicomachus, unlike Euclid, gave no abstract proofs of his theorems. Instead he merely enunciated the theorems and illustrated them with specific numerical examples. A Latin translation of the Arithm e tik e by Apuleius of Madaurus ( c. AD 125) is lost, but Boethius' version survived and was used as a schoolbook up to the Renaissance. Nicomachus' Encheiridion Harmonik e s ("Handbook of Harmony") is on the Pythagorean theory of music. He also wrote

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