Introductio Arithmetica

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Introductio Arithmetica

Nicomachus was said to be born in Palestine, particularly at Gerasa which is now known as Jordan. He came from a well-off family, his family was said to be a merchant who traded with some of the well known aristocrats. He learned a lot from outside of his city particularly in Alexandria where the most famous days of Pythagoreanism is known (“Nicomachus of Gerasa”). His work Introductio arithmetica was credited because he was the first ever person to discuss arithmetic separately from geometry.

Although his work was based on the word arithmos, which in English means numbers, he really don’t calculate it mathematically but rather the answers are based through speculative thinking or are just a reasonable guesses. Nicomachus considers his views through the combination of mathematics which discusses about numbers and philosophy which explains it as Plato stated that the ideas are numbers and the philosophy but number-ideas do not exist in a transcendental world of their own: rather they were ideas on the divine mind, which is a philosophy of the Jew, the middle Platonism and the Neo-Platonism.

Nicomachus is not so much known or popular though he achieved so much success; he wrote a lot of other works but then unfortunately did not survive such as introduction to geometry and a life of Pythagoras. His works complement one another based in the neo-Pythagorean understanding of cosmos, the numbers as well as the ratios that exist between them define the geometry’s discipline, also music, and even in the aspect of architecture (“Nicomachus of Gerasa”).

His work with the others soon became more understanding school of philosophy known as the neo-platonic; their school stated that there was a higher reality rather than the material world and that the only way to understand this level of reality is through using the intellectual reasoning (“Nicomachus of Gerasa”). Nicomachus on the other hand has a lot of other works and some of it is inclined with music such as his Manual of Harmonics. In this work of Nicomachus, he wrote musical notes as well as its intervals with the influence of Pythagorean tradition.

He explains notes and intervals basing on numerical ratios hence he tried to accept some of Aristoxenus’ developed idea about music. Some of the ideas that Nicomachus adapted is; to define whether a certain interval is said to be consonant and that the related hearing perception is an important consideration (“Nicomachus”). He was the first to discuss the weights of the 4 different hammers known as 12, 9, 8, and 6 that determined the differences between the pitches Pythagoreans heard. Nicomachus made a methodical measuring understanding how the sound was made and then trying his experiments with the string.

He observed how the tension and lengths were related to each other and repeatedly doing it with mathematical calculation then he formulated a law (“Nicomachus”). this was quoted by JOC/EFR in the year 1999’s article ; “assignments of numbers and numerical ratios to notes and intervals, his recognition in the indivisibility of the octave and the whole tone… but, unlike Euclid, who attempts to prove musical proposition through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings” (JOC/EFR “Nicomachus of Gerasa “).

Another work of Nicomachus was the theology of arithmetic in which he stated that such numbers possesses an arcane power that sooner can lead to a closer connection or understanding to supernatural powers and or divine powers. Numbers according to Nicomachus’ statement is both matter and form they could be changeless, eternal, and they could also be never ending changing(“Nicomachus of Gerasa”). His work about theology of arithmetic discusses about trying to understand the said natural and supernatural worlds and he derived both of it using the properties of numbers (“Nicomachus of Gerasa”).

He also discussed that different gods in the Greek mythology were made to be identified by the different properties of number. Monad, which defines the primary element of the number one, was said to be identified with the primordial chaos which existed before the gods did at the same time it is also identified with the goddess of the sun whom is Apollo. The dyad which is primarily elemented by the number two was connected with deities ranging from the goddesses Zeus and Demeter and Artemis and Aphrodite.

The triad was associated with the goddesses Hecate and Athena and the tetrad was associated with the goddesses Hermes and Heracles (“Nicomachus of Gerasa”). Each of these numbers are also associated in mythical aspect; monad is cannotated as the power of mind and chaos, dyad is cannotated with matter as well as equality, triad inclined with marriage, tetrad cannotated as harmony and so on with the other numbers. For Nicomachus, arithmetic is an essential passage for people to understand both the material physics as well as the spiritual metaphysics (“Nicomachus of Gerasa”).

Nicomachus’ great contribution to Pythagoreans is his discussion about perfect numbers in his work introduction arithmetica. In his time, he divided numbers using the odd as the masculine numbers and the even numbers as the feminine numbers. They also considered numbers in the concept of abstract representation such as 1 stands for reason, 2 first true female stands for opinion, 3 first true male stands for harmony, 4 stands for justice and etc (“Perfect Numbers “) In Nicomachus’ discussion, he divided the numbers into certain three types of classification.

The first one in the subdivisions of number is known as the superabundant numbers in which he explained that the sum of the number’s aliquot parts is expected to be greater than the number itself, the second one is the deficient numbers which is explained as the very contradict property of superabundant numbers because the sum of the number’s aliquot parts is expected to be lesser than that of the number itself, and lastly among the three subdivisions is the perfect numbers which explained that the property of the number has the balance or equal sum of the aliquot parts to the number itself (“Perfect Numbers”).

The first two are said to be extreme opposite to each other as defined by much little and too much and the one that is said to be equal defines perfect (E F Robertson J J O’Connor). Nicomachus also said that there is a religious significance about the number of 6 which they considered and believed as perfect because people in his period believed that the number was chosen by God, and that god chose the number of 6 because it was perfect as it was said in the bible that God took 6 days of work to create the whole universe and all the other creatures found in it (“Perfect Numbers”).

Nicomachus’ words were quoted; “six is a number perfect in itself, and not because God created all things in six days” (“Perfect Numbers”) but because the statement is a fact. God created all things in six days because the number is perfect (E F Robertson J J O’Connor; E F Robertson J J O’Connor; Mathematics; JOC/EFR “Perfect Numbers”; “Perfect Numbers”). Nicomachus considered numbers in moral term such as superabundant; exaggeration and or abuse, while deficient; defaults and insufficiencies and perfect; virtue and beauty.

He also considered it in biological means such as superabundant; twenty eyes and ten legs and deficient; one eyed and or one hand (E F Robertson J J O’Connor). Nicomachus said in his work that a perfect number of nth has its n digits, all numbers end in an alternate pattern of 6 and 8 and also included Euclid’s algorithm to generate perfect numbers will give all perfect numbers for example: all perfect numbers underlie to the form 2k-1(2k – 1), for some k > 1, where in 2k – 1 is prime and that perfect numbers are infinitely many (E F Robertson J J O’Connor).

The discussion of perfect numbers includes the method of generating these certain numbers. It is explained by Nicomachus that if you set an even number powered by two, then the next number should be totaled and the result would be perfect e. g. 42, 82,642, and infinite and if you are to generate perfect number using composite numbers or odd numbers, you should add them to the next term and the result will also be perfect always e. g. 1+1 would equal into 2, and 3+3 would equal into 6 which are considered perfect.

These are some quotes that follow the description of Nicomachus’ alogarithm; Nicomachus stated that “only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, which is below ten thousand, 8128. And it is their accompanying characteristics to end in alternately 6 and 8 or, and always to be even” (E F Robertson J J O’Connor; Mathematics; JOC/EFR “Perfect Numbers”; “Perfect Numbers”).

Another quote from Nicomachus says that “when these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next…the result is 496, in the hundreds; and then comes 8128 in the thousands, and so on, as far as it is convenient for one to follow” (JOC/EFR “Perfect Numbers”; Mathematics; “Perfect Numbers”) He subdivided the even numbers through his discussion such as: even-even — 2n, even-odd — 2 (2m+1), and odd-even — 2n+1(2m+1) (Allen).

On the other hand, he subdivided the odd numbers into prime and composite — these are ordinary primes that exclude 2, secondary and composite – these are ordinary composites which includes prime factors only, and relatively prime – includes two composite numbers but prime and composite to another number e. g. 9 and 25 (Allen). In Nicomachus’ book of Introductio arithmetica, he included Philolaus’ definition of prime numbers or incomposite numbers as well as secondary numbers or those composite numbers.

The work explains that such a prime kind of number is rectilinear which means that it only can be set in one dimension; the composite is explained as a number which can be measured by another number, and explained that two numbers can be prime if and only if they have more than that of only one greatest divisor (Allen). Despite the truth that this studies and works of Nicomachus are not justified and ungiven proofs by him, these studies where accepted and was said to be believed through years as a fact (E F Robertson J J O’Connor).

There were also Arabian mathematicians who were inspired by the perfect numbers; Thabit ibn Qurra was one of the inspired Arabians and his work was entitled treatise on amicable numbers the discussion he wrote about was searching how p becomes the prime number which is said to be perfect only in the form of 2np (JOC/EFR “Perfect Numbers”; Mathematics; “Perfect Numbers”). Another one is Ibn al-Haytham who gave a proof about Eclucid’s unpublished work. He showed it in his work treatise on analysis and synthesis.

He stated it in his discussion that the certain satisfying condition of a number is in the form of 2k-1 (2k – 1) where 2k – 1 is prime (E F Robertson J J O’Connor). The greatest among the Arab’s who made a study about the Greek’s work was Ismail Ibn Ibrahim Ibn Fallus. He wrote his work version of treatise basing on Nicomachus’ Introductio arithmetica (E F Robertson J J O’Connor). In his work he discussed about Nicomachus’ classification of numbers but his version was purely based on mathematical facts which excluded Nicomachus’ moral comments (E F Robertson J J O’Connor).

Another topic on the books discussion is about the amicable numbers these numbers have a property such as the sum of all the said proper divisors of the first given number without including itself exactly equals the next second number while the sum of all the proper divisors of the said second number without also including itself is exactly equal to the first number (Smithhisler). The most common example is the numbers 220 and 284.

The number set of 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 are the proper divisors for 220 if we will tend to add up all of this numbers we will see that the result will be 284. Now if we will try to find the divisors for the number 284, well come up to numbers 1, 2, 4, 71, and 142 which when you’ll add up, the sum would be 220 (Smithhisler). The numbers 220 and 284 are the first known amicable numbers. This numbers explain the relationship between two different numbers.

The Pythagoreans called it as the amicable numbers because Pythagoras considered it as a symbol of friendship (Smithhisler). Nicomachus made his works with the links of the original works of Pythagoras and soon developed it in many different ways. He then began to include mystical elements into the philosophy (“Nicomachus of Gerasa”). In his greatest work Introductio arithmetica of introduction to arithmetic he discussed all important aspects in the world of mathematics such as odd, even, prime, composite, figurate and perfect (“Nicomachus of Gerasa”).

Works Cited Allen, Don. “Pythagoras and the Pythagoreans. ” 1997. J J O’Connor, E F Robertson. “Perfect Numbers. ” 2001. J J O’Connor, E F Robertson “Perfect Numbers. ” 1998. JOC/EFR. “Nicomachus of Gerasa “, 1999. —. “Perfect Numbers. ” 2001. Mathematics, MacTutor History of. “History Topic: Perfect Numbers. ” 2001. “Nicomachus. ” 2006. “Nicomachus of Gerasa. ” 2006. “Perfect Numbers “, 2007. “Perfect Numbers. ” 1998. “Perfect Numbers. ” 2001. Smithhisler, Damian. 2007.


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  • University/College: University of California

  • Type of paper: Thesis/Dissertation Chapter

  • Date: 9 August 2016

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