21. Nicomachus Of Gerasa (fl.c. 100 AD) Library Of Congress Citations N6 A3 Notes At head of title nicomachus of gerasa. Nicomachus Gerasenus;usage nicomachus of gerasa) LC data base, 1026-84 (hdg. 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

22. Personal Names nicomachus of gerasa search TmiWeb for this person nicomachus of gerasa.Lived 3rd C. BC?. Neopythagorean music theorist. His work was lost but it was http://euromusicology.cs.uu.nl:6334/dynaweb/info/persinfo/persons/@Generic__Book

Expand Search Personal names N Nicomachus of Gerasa Nicomachus of Gerasa Lived: 3rd C. BC?. Neopythagorean music theorist. His work was lost but it was quoted by Boethius in the De institutione musica. More information Biography by Baldi (from Cronica

23. Personal Names Nicolini, Domenico Table of Contents nicomachus of gerasa search TmiWebfor this person Nicomachus. Born ; died . Son of Aristotle http://euromusicology.cs.uu.nl:6334/dynaweb/info/persinfo/persons/@Generic__Book

Expand Search Personal names N Nicomachus Nicomachus Born: ; died: . Son of Aristotle

24. Index Of /disciplinas/PAED/temas/nicomach/Nicomachus Of Gerasa - Britannica_com_ Index of /disciplinas/PAED/temas/nicomach/nicomachus of gerasa Britannica_com_files.Name Last modified Size http://piano.dsi.uminho.pt/disciplinas/PAED/temas/nicomach/Nicomachus Of Gerasa

Index of /disciplinas/PAED/temas/nicomach/Nicomachus Of Gerasa - Britannica_com_files Name Last modified Size Description ... Parent Directory 29-Oct-2001 18:12 0k 22-Nov-2001 01:37 12k article;pv=article;sz.. 22-Nov-2001 01:37 1k bl_corner.gif 22-Nov-2001 01:37 1k br_corner.gif 22-Nov-2001 01:37 1k britannica.css 22-Nov-2001 01:37 4k dot.gif 22-Nov-2001 01:37 1k gline.gif 22-Nov-2001 01:37 1k hdr_logo_chnl.gif 22-Nov-2001 01:37 2k onlineopinion.js 22-Nov-2001 01:37 9k red.gif 22-Nov-2001 01:37 1k standard.js 22-Nov-2001 01:37 8k tr_corner.gif 22-Nov-2001 01:37 1k ub2asdark.gif 22-Nov-2001 01:37 1k ub2homedark.gif 22-Nov-2001 01:37 1k ub2mwdark.gif 22-Nov-2001 01:37 1k ub2storedark.gif 22-Nov-2001 01:37 1k

25. Ancient Greek Instruments According to nicomachus of gerasa (Ist cent. AD), the tortoiseshell Lyra wasinvented by god Hermes, who gave it to Orpheus. Orpheus taught Thamyris and http://homoecumenicus.com/ancient_instruments.htm

Ancient Greek Instruments Stringed instruments (Plucked): Cithara, Phorminx, Lyra, Epigonion, Barbiton, Pandouris The graphics and editorial of this page were made by N. Ioannidis, a multi-instrumentalist performer University of Sussex Audio samples of other instrumental performances with modern instruments are available online at 12-string Guitar Works (12 string classical guitar) and Mass insanity (electric and fretless electric guitar). Click to see a replica of an ancient Cithara , and a replica of an ancient Lyra , which were used in the recording of two albums by IOANNIDIS Nikolaos: " The Music of Ancient Greeks - Early Epic and Lyrical Poetry" and "Music and Religion" v Kithara Lyra: originally called Chelys, because of the tortoise shell used as its sound box. According to Nicomachus of Gerasa (Ist cent. AD), the tortoise-shell Lyra was invented by god Hermes, who gave it to Orpheus. "Orpheus taught Thamyris and Linos, and Linos taught Hercules. When Orpheus was killed by the Thracian women, his lyra was thrown into the sea, and washed ashore at Antissa, a city of Lesbos, where it was found by fishermen, who brought it to Terpander, who in turn carried it to Egypt and presented it to the Egyptian priests as his own creation." We don't know how many strings the original Lyras had. By the time of Terpander (8th-7th cent. BC) Lyra was a seven stringed instrument and from many ancient sources we know that this type remained in use for a long time during the classical period. The addition of an eighth string in the 6th century BC is credited by Nicomachus of Gerasa to Pythagoras. By the fifth century there were Lyras with anything from 9 to 12 strings. The strings (neura) were made of animal gut of sinew, but there are also references of strings made of linen or hemp.

26. A History Of Muslim Philosophy nicomachus of gerasa Who lived about 140 AD was one with Plato in holding thatideas were temporally prior to the formation of the world and also in holding http://www.muslimphilosophy.com/hmp/8.htm

Subject Index Search Announcements Feedback ... Support this Site Chapter 5 Alexandrio Syriac Thought Alexandrio Syriac Thought by C.A Qadir A Reason thus assailed could find refuge only in faith. In the period that follows we find philosophy renouncing its independence and becoming merely an instrument of theology. Ritter says, "The feeling of alienation and the yearning after a higher revelation are characteristics of the last centuries of the ancient world; this yearning was, in the first place, but an expression of consciousness of the decline of the classical nations and their cultures, the presentiment of the approach of a new era, and it called into life not only Christianity but also before it pagan and Jewish Alexandrianism and other related developments." It would be incorrect to identify the present geographical boundaries of Syria with its old ones. In Roman days, at the beginning of the Christian era Syria denoted the country west of the Euphrates and north of the Arabian Desert, including Palestine and Palmyra and extending north to the Taurus. The usual language of Syria was Aramaic, a language akin to Hebrew. The Hebrew word "Aram" is rendered as "Syria" and originally the words Aramaean and Syrian were synonymous. After the Hellenization of the country, the Greek language was used by the ruling class and the officials with very little influence on the masses who continued using their dialect. This state of affairs con­tinued till the first/seventh century when after the Muslim conquest Syriac gradually gave way vernacularly and to some extent liturgically to Arabic, though it had great influence on the vocabulary, pronunciation, and even the grammatical forms of Arabic which supplanted it.

27. Great Books Index - Nicomachus nicomachus of gerasa (about 60about 120). An Index to Online Great Books inEnglish Translation. AUTHORS/HOME . TITLES . ABOUT GB INDEX . BOOK LINKS 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

28. CATHOLIC ENCYCLOPEDIA: Neo-Pythagorean Philosophy KAYSER, 187071); the works of nicomachus of gerasa are included in IAMBLICHUS,Theologumena Arithmeticœ (ed. AST, Leipzig, 1817); ZELLER, Philosophie der http://www.newadvent.org/cathen/10745a.htm

Home Encyclopedia Summa Fathers ... N > Neo-Pythagorean Philosophy A B C D ... Z Neo-Pythagorean Philosophy B. C. , disappears completely from history during the fourth century, when philosophy reached the zenith of its perfection at Athens. Here and there, however, there appears a philosopher who reverts to the Pythagorean doctrine of numbers, and in a general way manifests the tendency of the school towards religious ethics and the practices of asceticism. Beginning with the middle of the first century B. C. , a more systematic attempt was made to restore the speculative philosophy of the Pythagoreans and combine it with the practice of astrology and sorcery. The first of these systematic neo-Pythagoreans was Figulus, a Roman philosopher who lived at Alexandria about the middle of the first century B. C. , and was a friend of Cicero. Other Romans also contributed to the movement, the chief of whom were Vatinius and the Sextians. It was, however, at Alexandria that the most influential of the neo-Pythagoreans taught. In the second and third centuries of the Christian era, the philosophers of the school became, so to speak, apostles of the cult, and travelled throughout the Roman Empire. The names most prominently associated with this active philosophical campaign are those of Moderatus of Gades, Apollonius of Tyana, Nicomachus of Gerasa, Numenius, and Philostratus. Like the neo-Platonists (see NEO-PLATONISM), the neo-Pythagoreans definitely placed their philosophy at the disposal of the pagan opponents of

29. 1 nicomachus of gerasa not only mentions the nature of the mathe­matical sciences nicomachus of gerasa. 1926. Introductionis Arithmeticae Liber II. Ed. http://www.brown.edu/Departments/Classics/bcj/12-05.html

Proclus on Mathematics and the Ascent of the Soul Clemency Williams At the end of his Protrepticus , Iamblichus gives us a list of thirty-nine Pythagorean symbols (sumbola) with a brief interpretation of each one. The thirty-sixth Pythagorean symbol is: Honor a figure and a step rather than a figure and three obols. Commentary on the First Book of EuclidÕs Elements , says, Òby this they meant that it is necessary to pursue the science of geometry, which, with each theorem, makes a step upwards and draws the soul to the heightsÓ (Proclus 1977: 84, 16-24). Thus, while Iamblichus asserts that one must not cursorily pursue Òthe mathematical sciences,Ó Proclus specifically prescribes geometry. Further reading of the works of Iamblichus, however, shows that in terms of the upwards progression of the soul, he believes the study of arithmetic best assists such an ascent. Although it is mentioned, mathematics does not play a central role in PlotinusÕ account of the ascent of the soul. In his exposition of this ascent in Ennead I,3, he distinguishes three types of travelers on the roadÑthe philosopher, the lover and the musicianÑand explains how each of the three proceed on the upward journey.

30. Adler And Van Doren. How To Read A Book Germania; nicomachus of gerasa (fl.c. 100 AD) Introduction to Arithmetic;Epictetus (c.60120) Discourses Encheiridion (Handbook); Ptolemy (c.100-170; fl. http://www.interleaves.org/~rteeter/grtadler.html

How to Read a Book by Mortimer J. Adler and Charles Van Doren For more on Adler's ideas about the Great Books, see his essay, "Why Read the Great Books" Frequently Asked Questions Who wrote this list? See the heading above and the credit below to find out who wrote this list. If you don't like the selections in this list or the arrangement, take it up with the author(s). Why isn't my favorite author listed here? This list may not include your favorite author, but he or she may be on other Great Books lists. Check the author index to see. See the Great Books FAQ for more about the Great Books and these lists of them. Homer (9th Century B.C.?) Iliad Odyssey The Old Testament Aeschylus (c.525-456 B.C.) Tragedies Sophocles (c.495-406 B.C.) Tragedies Herodotus (c.484-425 B.C.) History Euripides (c.485-406 B.C.) Tragedies (esp. Medea Hippolytus The Bacchae Thucydides (c.460-400 B.C.) History of the Peloponnesian War Hippocrates (c.460-377? B.C.) Medical Writings Aristophanes (c.448-380 B.C.) Comedies (esp.

31. Great Books: Author-Title Index: Authors N To R nicomachus of gerasa, Greek, fl. ca. 100 CE. Introduction to Arithmetic.Recommended by Adler GBWW SJC (Selections). Niebuhr, Reinhold, American, 18921971 http://www.interleaves.org/~rteeter/grtalphan.html

Great Books: Author-Title Index: Authors N to R This page tells you which authors and titles are included on which great books lists. For more information, see my Great Books page A-B C-D E-G ... U-Z Nabokov, Vladimir, Russian-American writing in English, 1899-1977. Lolita . Recommended by: Bloom BOMC Boston PL Fadiman 3 ... Radcliffe Pale Fire . Recommended by: Bloom Burgess Fadiman 3 Fadiman 4 ... ML Novels Speak, Memory . Recommended by: Fadiman 3 Fadiman 4 ML Nonfiction Ada or Ardor . Recommended by: Ward The Defence . Recommended by: Burgess Nagai Kafu, Japanese, 1879-1959. Kafu the Scribbler . Recommended by: Ward Nagarjuna, Indian writing in Sanskrit, 2nd-3rd C. Vigrahavyvartani . Recommended by: StJE Naipaul, V. S., Trinidadian, 1932- . Nobel Laureate A Bend in the River . Recommended by: Bloom Burgess ML Novels A House for Mr. Biswas . Recommended by: Bloom ML Novels Guerrillas . Recommended by: NYPL Nakanoin Masatada no musume, Japanese, b. 1258. Confessions of Lady Nijo . Recommended by: MW Asian Namier, Lewis, English, 1888-1960. The Structure of Politics at the Accession of George III . Recommended by: TLS Narayan, R. K., Indian writing in English, 1906-2001.

32. Speusippus various authors Anatolius, nicomachus of gerasa, possibly Iamblichus himself, Louis Charles Karpinski, 1926, nicomachus of gerasa Introduction to http://plato.stanford.edu/entries/speusippus/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A ... Z This document uses XHTML/Unicode to format the display. If you think special symbols are not displaying correctly, see our guide Displaying Special Characters last substantive content change MAY The Encyclopedia Now Needs Your Support Please Read How You Can Help Keep the Encyclopedia Free Speusippus ibid . iv 4; Diogenes lists about 30 titles, and his bibliography is on his own admission incomplete). We have very little of what he wrote, if any (we have something from a work later attributed to him, On Pythagorean Numbers , discussed below, but this is not one of Diogenes' titles; and we may have something preserved in Iamblichus, De communi mathematica scientia iv, also discussed below). What we can infer, mostly from Aristotle but with a few supplements from other authors, will be covered under three headings: 1. Metaphysics 2. Epistemology 3. Ethics Bibliography ... Related Entries 1. Metaphysics Aristotle gives us an account of Plato's metaphysical views that goes considerably beyond what we find in Plato's dialogues. In fact, Aristotle gives us what he says are Plato's views and also those of others. From this we learn what little we can about Speusippus and his contemporary, Xenocrates. According to Aristotle, all three of Plato, Speusippus, and Xenocrates constructed their worlds operating with two principles (

33. Access The Great Books [226-300, Page 4 Of 4] nicomachus of gerasa (fl. c. 100 CE) Online Works Introduction to Arithmetic Gifof Greek Multiplication Table Epictetus (c. 60120) More info http://www.anova.org/

Zaine Ridling, PhD Access Foundation Sections Hammurabi (18th c. BCE) to John Milton La Rochefoucauld to Emily Brontë Karl Marx to Arthur Miller Saul Bellow to Annie Dillard Access the Great Books is a compilation of over 240 great authors and their works. The Access Foundation List is liberal in scope, robust in its cataloging, and voluminously linked to other sites that help readers access and study great literature. This list devotes notable attention to works of philosophy and science, and is unrivaled in its organizational structure, background, and biographical content on the authors listed. ACCESS FOUNDATION holds that time spent reading the Great Books is time well spent. Great books lists are not meant to be exclusive of any tradition or culture, but rather form a foundation of knowledge on which to stand. Reading the great books allays the "busyness" of modern life; encourages self-examination, increases reflection, and provokes intellectual curiosity. What follows is a list compiled by ACCESS FOUNDATION from a variety of sources, and based most notably from the one developed for the Great Books collection of the Encyclopædia Britannica by Robert M. Hutchins, Mortimer J. Adler, and Mark Van Doren. It is strongly recommended that the reader visit

34. Ancient Greeks: Prime Numbers And Number Theory nicomachus of gerasa (c. 60 c. 120 AD) provides also a list of properties ofperfect numbers without a proof. The nth perfect number has n digits. http://www.mlahanas.de/Greeks/Primes.htm

Ancient Greeks: Prime Numbers and Number Theory Michael Lahanas Griechische Mathematik: Zahlentheorie und Primzahlen Pythagoras of Samos ( Πυθαγόρας ο Σάμιος) discovered the relation between harmony and numbers. The Pythagoreans saw the number one as the primordial unity from which all else is created. Two was the symbol for the female, three for the male and therefore five (two + three) symbolized marriage. The number four was symbolic of harmony, because two is even, so four (two times two) is "evenly even". Four symbolized the four elements out of which everything in the universe was made (earth, air, fire, and water). Ten that was the sum from one to four was a very special number. The ancient Greeks believed that all numbers had to be rational numbers. 2500 years ago Greeks discovered that if all the common prime numbers were removed from the top and bottom of the ratio then one of the two numbers had to be odd. This we can term reduced form . Obviously, if top and bottom were both even, then both could be divide by the number two and this could be eliminated from both.

35. Raphael's School Of Athens (1/2) or the mathematician and musician nicomachus of gerasa (µa? ? Ge?as) (c.60 c. 120 AD). Nicomachus was the author of Arithmetic Introduction. http://www.mlahanas.de/Greeks/SchoolAthens.htm

Michael Lahanas Part 1 Raffaels Schule von Athen. Wer ist wer? After he had been welcomed very affectionately by Pope Julius, Raphael started to paint in the Stanza della Segnatura a fresco showing the theologians reconciling Philosophy and Astrology with Theology, in which are portraits of all the sages of the world shown disputing among themselves in various ways. The original name of the fresco actually is Causarum Cognitio (Knowledge of Causes) but it is called School of Athens from a 17th century guidebook. The School of Athens (1510-1511), Raphael Virtual Reality Tour (Stanza della Segnatura), Musei Vaticani (with other works of Raphael also completed in the period 1508-1511) Plato and Aristotle as Central Figures walking in a peripatetic manner through the Lyceum. A one-point (linear) perspective is used , one method to show 3-D objects on a 2-D surface. Lines which appear to go away from the viewer meet at a single point on the horizon, the so called vanishing point. The perspective is such that this point is between Plato and Aristotle stressing the importance of these two persons. To an almost equal distance between Plato and Aristotle we have Euclid (23) and Pythagoras (30). I see a transition from Philosophy to pure Science from left to right that also is represented by Apollo and Athena sculptures on the left and right side. Many figures (all except Socrates?) were drawn according to persons of Raphael's epoch such as Plato whose face is that of Leonardo. There is also, although not perfect or proportional, some temporal ordering of the persons from the center.

36. History Of Mathematics: Greece 100 CE); nicomachus of gerasa (c. 100); Theon of Smyrna (c. 125); Ptolemy (ClaudiusPtolemaeus) (100178); Marinus of Tyre (c. http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

37. History Of Mathematics: Chronology Of Mathematicians 100 CE) *MT *SB; nicomachus of gerasa (c. 100) *SB; Zhang Heng (78139); Theon ofSmyrna (c. 125); Ptolemy (Claudius Ptolemaeus) (c. 100-c. http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

Chronological List of Mathematicians Note: there are also a chronological lists of mathematical works and mathematics for China , and chronological lists of mathematicians for the Arabic sphere Europe Greece India , and Japan Table of Contents 1700 B.C.E. 100 B.C.E. 1 C.E. To return to this table of contents from below, just click on the years that appear in the headers. Footnotes (*MT, *MT, *RB, *W, *SB) are explained below List of Mathematicians 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

38. Conari Press This book is the first ever complete translation of The Manual of Harmonics bythe Pythagorean philosopher nicomachus of gerasa (second century AD) http://www.conari.com/estore/product_detail.jsp?product_cat_id=54&product_group_

39. The Manual Of Harmonics The Manual of Harmonics of Nicomachus the Pythagorean Pythagorean philosophernicomachus of gerasa (second century AD) published with a comprehensive, http://www.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.

40. Proofs Without Words In his Introduction to Arithmetic, nicomachus of gerasa (c 100 AD) writes, Everysquare figure diagonally divided is resolved into two triangles and every http://www.cut-the-knot.com/ctk/pww.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Cut The Knot! An interactive column using Java applets by Alex Bogomolny Proofs Without Words July 1998 In the beginning, when there was no language to express general mathematical ideas, proofs without words were the proofs. Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." A classical example concerns triangular numbers: 1 + 2 + 3 + ... + N = N(N+1)/2 which is ascribed to the ancient Greeks. I would argue that, for the Ancients, if they ever drew them, the diagrams were more than an effective aid. In the absence of algebraic symbolism, they might have served as a combination of a statement and its proof in the most concise form available at the time. In his Introduction to Arithmetic , Nicomachus of Gerasa (c 100 A.D.) writes, "Every square figure diagonally divided is resolved into two triangles and every square number is resolved into two consecutive triangular numbers, and hence is made up of two successive triangular numbers." Obviously, he refers to the diagram on the right (although there is no indication that he had ever drawn one), and this is the only

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