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Category:  Ancient Science Ancient Literature Ancient Music  
Name:  Nicomachus of Gerasa  Math Series  
Birth Year:  c. 100 CE  
Death Year:  na  
Representative Image:  
Biography, Lectures, and Research Links: 
Blog Nicomachus of Gerasa An example of this we look more closely at the results which Nicomachus quotes on perfect numbers. He states that the nth perfect number has n digits, and that all perfect numbers end in 6 and 8 alternately. These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128. The work contains the first multiplication table in a Greek text. It is also remarkable in that it contains Arabic numerals, not Greek ones. However, in many respects the book is old fashioned in its style since it appears more in tune with the number theoretic ideas of Pythagoras with his mystical approach, rather than a true mathematical approach. To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers. An abundant number has the sum of its proper divisors greater than the number, while a deficient number has the sum of its proper divisors less than the number. Nicomachus writes of these numbers in Introduction to Arithmetic: In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort  of which the most exemplary form is that type of number which is called perfect. He then continues his description of abundant numbers as resembling an animal: ... with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands... while a deficient number is like an animal: ... with a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue... For over 1000 years Introduction to Arithmetic was the standard arithmetic text. In view of the comments we have made regarding the work, this may seem surprising. Mathematicians disliked the work, in particular Pappus is said to have despised it. However, several people including Boethius translated Introduction to Arithmetic into Latin and it was used as a school book. How then could a poor book become so popular. Heath tries to explain the apparent contradiction in [4], suggesting that: ... it was at first read by philosophers rather than mathematicians, and afterwards became generally popular at a time when there were no mathematicians left, but only philosophers who incidentally took an interest in mathematics. Most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus, but were mainly influenced by Euclid. However, texts by nonmathematicians were most strongly influenced by Nicomachus. This research tends to support the views of Heath on this subject. Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers. However Heath writes [4]: The curious farrago which has come down to us under that title and which was edited by Ast [published in Leipzig in 1817] is, however, certainly not by Nicomachus; for among the authors from whom it gives extracts is Anatolius, Bishop of Laodicaea (270 AD); but it contains quotations from Nicomachus which appear to come from the genuine work. Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music. Again Nicomachus shows the influence of Pythagoras but also Aristotle's theories of music. The work looks at musical notes and the octave. The principles of tuning a stretched string are studied as is an extension of the octave to the twooctave range. The influences of Pythagoras' theory of music are seen from Nicomachus: ... assignment of number and numerical ratios to notes and intervals, his recognition of the indivisibility of the octave and the whole tone... But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings. Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus. From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible. [Adapted from MacTutor] The Great Books: Nicomachus of Gerasa Please browse our Amazon list of titles about Nicomachus of Gerasa. For rare and hard to find works we recommend our Alibris list of titles about Nicomachus of Gerasa. Post Comments, Questions or Suggestions! This database is maintained by Malaspina Great Books.  
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Best Choice Books, Music, Art:  Introduction to Arithmetic The Manual of Harmonics  
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Online Research:  Music Research Online at Questia  
Records from Related Period and Category:  Ancient Literature Ancient Music Ancient Science 
About this Database: 
This web page is part of a biographical database on Great Ideas. These are living ideas that have shaped, defined and directed world culture for over 2,500 years. By definition the Great Ideas are radical. As such they are sometimes misread, or distorted by popular simplifications. Understanding a Great Idea demands personal engagement. Our selection of Great Ideas is drawn from literature and philosophy, science, art, music, theatre, and cinema. We also include biographies of pivotal historical and religious figures, as well as contributions from women and other historically underrepresented minorities. The result is an integrated multicultural and multidisciplinary database built upon the framework of the always controversial Great Books Core List published in 1940 by the late Great Books Pioneer Mortimer Adler (19022001). Most of the works on that list are available in the 60 volume Great Books of the Western World. 
