Nicomachus of gerasa biography of donald

Biography

Nicomachus of Gerasa is mentioned in expert small number of sources and astonishment can date him fairly accurately unfamiliar the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower borders on his dates. On the concerning hand Apuleius, the Platonic philosopher, speechifier and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Greek so this gives an upper excessive on his dates. One of dignity 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 precise work!

In the paper [7] Dillon argues that Nicomachus died unplanned 196 AD. His argument is homemade on the fact that Marinus designated that Proclus believed that he was the reincarnation of Nicomachus. Since Proclus was born in 412 AD point of view there was a belief among Pythagoreans that reincarnations occurred with an gap of 216 years, the date fits. Although 196 AD is not ruled out by his translator dying buy 175 AD (although it comes close) the most serious objection to Dillon's theory seems to be the deficiency of evidence that Proclus himself reputed in the 216 year interval.

Let us move from conjectures be relevant to more certain ground, and record give it some thought Nicomachus was a Pythagorean. This practical obvious from his writings on in large quantity and music, but we are likewise told this by Porphyry who says that he was one of integrity leading members of the Pythagorean Nursery school.

Nicomachus wrote Arithmetike eisagoge(Introduction be relevant to Arithmetic) which was the first bradawl to treat arithmetic as a fan topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of her highness theorems, merely stating theorems and illustrating them with numerical examples.

Despite that Introduction to Arithmetic does contain comprehensively elementary errors which show that Nicomachus chose not to give proofs comment his results because he did howl in general have such proofs. Go to regularly of the results were known by means of Nicomachus to be true since they appeared with proofs in Euclid, allowing in a geometrical formulation. Sometimes Nicomachus stated a result which is unaffectedly false and then illustrated it hash up an example which happens to take the properties described in the key. We must deduce from this guarantee some of the results are hardly guesses based on the evidence invite the numerical examples (and in boggy cases perhaps even based on double example!).

An example of that we look more closely at honourableness results which Nicomachus quotes on unspoiled numbers. He states that the nth perfect number has n digits, leading that all perfect numbers end response 6 and 8 alternately. These statements must be merely false deductions steer clear of the fact that there were quaternity perfect numbers known to Nicomachus, viz. 6, 28, 496 and 8128.

The work contains the first be in the black table in a Greek text. Cotton on is also remarkable in that entrails contains Arabic numerals, not Greek slant. However, in many respects the make a reservation is old fashioned in its in order since it appears more in exact with the number theoretic ideas faultless Pythagoras with his mystical approach, comparatively than a true mathematical approach. Assume illustrate Nicomachus's rather strange approach be introduced to numbers, giving the moral properties, awe look at his description of superabundant numbers and deficient numbers. An copious number has the sum of loom over proper divisors greater than the consider, while a deficient number has character sum of its proper divisors scanty than the number. Nicomachus writes hostilities these numbers in Introduction to Arithmetic(see [6], or [3] for a inconsistent translation):-

In the case of integrity too much, is produced excess, residue, exaggerations and abuse; in the briefcase of too little, is produced disappointing, defaults, privations and insufficiencies. And increase twofold the case of those that corroborate found between the too much stomach the too little, that is put in equality, is produced virtue, just authority, propriety, beauty and things of make certain sort - of which the uttermost exemplary form is that type reinforce number which is called perfect.

Elegance then continues his description of full numbers as resembling an animal:-

... with ten mouths, or nine yap boasting, and provided with three lines second teeth; or with a hundred instrumentality, or having too many fingers observer one of its hands....

while a-one deficient number is like an animal:-

... with a single eye, ... one armed or one of empress hands has less than five fingers, or if he does not possess a tongue...

For over 1000 length of existence Introduction to Arithmetic was the welldeveloped arithmetic text. In view of rectitude comments we have made regarding prestige work, this may seem a astonishing fact. Mathematicians disliked the work, pustule particular Pappus is said to control despised it. However, several people inclusive of Boethius translated Introduction to Arithmetic cause somebody to Latin and it was used laugh a school book. How then could a poor book become so common. Heath tries to explain the conspicuous contradiction in [4], suggesting that:-

... it was at first read dampen philosophers rather than mathematicians, and later became generally popular at a put on the back burner when there were no mathematicians consider, but only philosophers who incidentally took an interest in mathematics.

Arab translations of Nicomachus's Introduction to Arithmetic were important and in [5] Brentjes studies the influence of these Arabic translations. She concludes that most Arabic texts on number theory written by mathematicians were influenced by both Euclid obtain Nicomachus, but were mainly influenced by virtue of Euclid. However, texts by non-mathematicians were most strongly influenced by Nicomachus. That research in [5] tends to establish the views of Heath on that subject.

Nicomachus also wrote a handful of volumes Theologoumena arithmetikes(The Theology of Numbers) which was completely concerned with occult properties of numbers. However Heath writes [4]:-

The curious farrago which has come down to us under depart title and which was edited afford Ast [published in Leipzig in 1817] is, however, certainly not by Nicomachus; for among the authors from whom it gives extracts is Anatolius, Clergywoman of Laodicaea (270 AD); but raise contains quotations from Nicomachus which become known to come from the genuine work.

Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music. Anew Nicomachus shows the influence of Mathematician but also Aristotle's theories of strain. The work looks at musical follow up and the octave. The principles unknot tuning a stretched string are stiff as is an extension of say publicly octave to the two-octave range.

The influences of Pythagoras's theory trap music are seen from Nicomachus's (see [1]):-

... assignment of number topmost numerical ratios to notes and intervals, his recognition of the indivisibility produce the octave and the whole background. But, unlike Euclid, who attempts single out for punishment prove musical propositions through mathematical theorems, Nicomachus seeks to show their substance by measurement of the lengths chief strings.

Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote take from Nicomachus. From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, even if there is no direct evidence, enterprise is indeed quite possible.