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|Name:||Nicomachus of Gerasa - Math Series|
|Birth Year:||c. 100 CE|
|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 , 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 non-mathematicians 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 :
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 two-octave 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
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|Records from Related Period and Category:||Ancient Literature|
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