NICOMACHUS: Introduction to Arithmetic
- Pythagoras defined philosophy as the love of wisdom. Before Pythagoras, all men who had knowledge of some art were called wise. Pythagoras restricted the title of wise to those who possessed knowledge and comprehension of reality; this being the only true wisdom according to Pythagoras.
- Wisdom is the knowledge, or science, of the truth in real things. Science is a steadfast and firm understanding of the underlying substance. Real things continue uniformly in the universe and never depart even briefly from their existence; these real things are immaterial, being the underlying substance of everything else that exists and is called “this particular thing.”
- Material things are involved in continuous flow and change. The immaterial things of which we conceive in connection with or together with matter, such as qualities, quantities, relations, etc., are immutable and eternal.
- Things involved in birth and destruction, growth and diminution, are seen to vary continually, and while they are called real things, they are not real by their own nature as the immaterial and eternal things that abide by their own essential being. Bodily things partake in immaterial and eternal things, but they are not entirely real. They are always in a state of becoming; they never are.
- There are things which always are, and there are things which are always becoming but never are. A thing which becomes and passes away never really is.
- A happy life is accomplished only by philosophy – the wisdom or knowledge of the truth of Real Things.
- Some things are unified and continuous; for example, an animal, the universe, and a tree are called magnitudes. Some things are discontinuous and in a side-by-side arrangement or heaps; for example, a flock, a people, and a chorus are multitudes.
- Multitude and magnitude are infinite. Multitude starts at a definite root and never ceases increasing. Magnitude is illustrated when division beginning with a limited whole is carried on to infinity because the dividing process never ceases. Multitude is limitless in the direction of the more, and magnitude is limitless in the direction of the less.
- Sciences are always concerned with limited things. A science concerning magnitude and multitude themselves could never be formulated, but sciences concerning something separated from each of them can be formulated; quantity set off from multitude and size set off from magnitude.
- There are two kinds of quantity. One kind has no relation to anything else, as even, odd, perfect, etc. The other kind is relative to something else and is conceived of together with its relation to another thing, as double, greater, smaller, half, etc. Two scientific methods will deal with the inquiry into quantity: arithmetic for absolute quantity, and music for relative quantity.
- There are two kinds of sizes. One kind is in a state of rest and stability. The other kind is in motion. Geometry deals with the kind that abides and is at rest. Astronomy deals with the kind that moves.
- Without Arithmetic, Music, Geometry, and Astronomy, it is impossible to discover the truth in things – i.e. to attain wisdom – and philosophize correctly. These sciences are like ladders or bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls.
- The truth of the universe is beheld in Mathematics.
- Arithmetic is used for reckoning, exchanges, and partnerships. Geometry is used for sieges, the founding of cities and sanctuaries, and the partition of land. Music is used fro festivals, entertainment, and the worship of the gods. Astronomy, or the doctrine of the spheres, is used for farming and navigation. The eye of the soul is buried and blinded by other pursuits. Mathematics rekindles and arouses the eye of the soul.
- We ought to study arithmetic first because it is the origin of the other sciences. If Arithmetic were abolished, all the other sciences would be abolished with it. Animal is antecedent to man. If we abolished ‘animal’, then ‘man’ would be abolished with it. If man is abolished, it does not follow that animal is abolished.
- That which is younger or posterior implies the other thing with itself. Man implies animal, but animal does not imply man. Thus, if geometry exists, arithmetic exists. It is with the aid of arithmetic that we speak of geometric figures such as triangles, quadrilaterals, etc. A three-sided figure cannot exist unless the number three exists beforehand. On the contrary, three can exists without figures.
- Musical harmonies are arithmetic ratios.
- Astronomic risings, settings, progressions, and all sorts of phases are governed by numerical cycles and quantities.
- The universe seems both in part and as a whole to have been determined and ordered in accordance with number.
- Everything that is harmoniously constituted is knit together by things that are real, different, and have some relation to one another. The most fundamental species of number are even and odd. Even and odd are different from one another, but not of an entirely different genus. They are reciprocally interwoven into harmony with each other.
- Number is limited multitude. The first division of number is even and odd.
- The even is that which can be divided into two equal parts without a unit intervening in the middle. The odd is that which cannot be divided into two equal parts because of an intervening unit.
- The number one is unique because it cannot be divided at all. The number two is unique because it can only be divided into two equal parts; other even numbers can be divided into two equal parts and two unequal parts: 6 can be divided into 3 and 3, and 4 and 2.
- In relation to each other, the even is that number which is one less or one more of the odd, and the odd is that number which is one more or one less than the even.
- Every number is half the sum of the two on either side of itself: 5 is half the sum of 4 and 6, and 3 and 7, and 2 and 8, etc.
- The number 1, or unity, is half merely of the adjoining number 2, because there is only one number adjoining it. Thus, the number one is the natural starting point of all number.
- By subdivision of the even, there are the even-times even, the odd-times even, and the even-times odd. The even-times even and the even-times odd are opposite to one another. The odd-times even is common to them both like a mean term.
- The even-times even is a number which is capable of being divided into two equal parts, which are also capable of being divided into two equal parts until the division reaches Unity. Half of 16 is 8, half of 8 is 4, half of 4 is 2, and half of 2 is Unity, which is naturally indivisible and does not admit of a half.
- The even-times even is a number that, whatever part of it be taken, it is always even-times even in designation. 64 is even-times even. All of its parts are even-times even: 32, 16, 8, 4, 2.
- The method of producing all the even-times even is to begin at unity and proceed by the double ratio to infinity: 1, 2, 4, 8, 16, 32, 64, 128… It is impossible to find others beside these.
- The sum of an even-times even number’s units is one number less than itself. For example: 1, 2, 4, 8. The sum of 1, 2, and 4 is 7, which is one number less than 8. The sum will always be odd. The product of the extreme even-times even numbers in a series is always equal to the product of the means. For example, 1 times 128 is equal to 2 times 64, 4 times 32, and 8 times 16.
- The even-times odd number is even, but specifically opposed to the even-times even number. The two equal parts of an even-times odd number are not divisible into two equal parts. For example, the half of 26 is 13, and 13 cannot be divided into two equal parts.
- Of even-times odd numbers, whatever factor it may be discovered to have is opposite in name to its value, and the quantity of every part is opposite in value to its name. For example, half of 18, which is an even name, is 9, which is odd in value. The third part of 18, an odd designation, is 6, an even value. The sixth part, even in designation, is 3, odd in value. The ninth part 2.
- The method of producing even-times odd numbers is to set forth in proper order to infinity the odd numbers and multiply them by 2. The numbers are: 6, 10, 14, 18, 22, 26, 30, 34… The greater terms always differ by 4 from the next smaller ones. This is because the successive odd numbers exceed one another by 2, and are multiplied by 2; thus, 2 times 2 is 4.
- The even-times odd is opposite in properties to the even-times even because the greatest extreme term alone is divisible, whereas only the smallest extreme of even-times even is indivisible. The product of the extreme parts of even-times even numbers is equal to the product of their mean parts or part [1 times 128 equal 64 times 2, 32 times 4, etc.] The sum of the extreme parts of even-times odd numbers is equal to the sum of the mean parts or part [in 6, 10, 14, 18, 22 the sum of 22 and 6 equals the sum of 14 and 14, and 10 and 18].
- The odd-times even number is an even number whose parts can also be divided, but it cannot carry the division of its parts as far as unity. Such numbers are 24, 28, 40. It has properties of even-times even and even-times odd, and properties which belong to neither of them.
- The method of producing odd-times even numbers is to set forth all odd numbers from 3 to infinity, and set forth all even-times even numbers from 4 to infinity. Begin by selecting 3 from the odd numbers and multiplying by each even-times even number. Then proceed to multiply 5 by every even-times even number, then 7, etc.
- There are three species of odd number: the prime and incomposite, the secondary and composite, and that which is secondary and composite but relatively prime and incomposite.
- The prime and incomposite is found whenever an odd number admits of no other factor save the number itself as a denominator, which always equals unity. For example: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. 3 only has a third part, 5 only a fifth, 7 a seventh, etc. These numbers are called prime because they exist beforehand as the beginning of others, they are origins of other numbers. For every origin is incomposite, into which everything is resolved and out of which everything is made, but the origin itself cannot be resolved into anything or constituted out of anything. [Aristotle includes the number 2 as prime, but Nicomachus only includes odd numbers in his definition]
- The secondary composite number has another part or parts with different denominators. For example: 9, 15, 21, 25, 27, 33, 35, 39. The number 9 has a third part besides a ninth. The number 15 has a third and fifth part besides a fifteenth, etc. These numbers are secondary because they are not elementary like prime numbers, but produced by other root numbers such as 3 in the case of 9.
- The third species of odd number is the one that is in itself secondary and composite, but prime and incomposite relative to another number. Let 9 be compared to 25. Each number is secondary and composite, but relative to one another they are prime and incomposite because they only have unity as a common measure. A third part does not exist in 25 and a fifth part does not exist in 9. The method to determine whether two numbers are primary and incomposite relative to one another is to subtract the smaller number from the larger as many times as possible; then subtract that term from the original smaller number as many times as possible; this method will result in either unity or in a number which will necessarily be odd and shared by the two original numbers as parts. For example, given the numbers 23 and 45, subtract 23 from 45, and 22 is the remainder; then subtract 22 from 23, and 1 is the remainder, subtracting 1 from 23 as many times as possible ends in unity. Thus, 23 and 45 are primary and composite relative to one another. But given 9 and 15, subtract 9 from 15, and the remainder is 6; then subtract 6 from 9, and the remainder is 3, subtracting 3 from 9 as many times as possible ends in 3; thus, 3 is a common part to both 9 and 15, and thus 9 and 15 are secondary and composite relative to one another.
- Evil, disease, disproportion, unseemliness, or nay other such thing could be conceived of without the concepts of excess and deficiency. That which lies between the extremes of excess and deficiency, namely the equal, are virtues, wealth, moderation, propriety, beauty, and the like.
- Of the simple even numbers, some are superabundant, some deficient, and some are perfect.
- A superabundant number is a number whose factors, when added together, exceed the number itself. This is a superabundant number because it exceeds the symmetry which exists between the perfect and its own parts. It’s like a man who has ten limbs, or 5 eyes, or some other kind of grotesque superabundance.12 is a superabundant number; for when its parts are added together – 6, 4, 3, 2, 1 – the sum is 16, which exceeds 12.
- The deficient number is the number whose factors, when added together, are less than the number itself. It’s like a man who has one eye, or 3 fingers, or some other gross deficiency. 8 is a deficient number; for when its factors are added together – 4, 2, 1 – the sum is 7, which is less than 8.
- The perfect number is always equal to the sum of its parts. 6 and 28 are perfect numbers. The sum of the factors of 6 – 3, 2, 1 – equal 6. The sum of the factors of 28 – 14, 7, 4, 2, 1 – equal 28.
- As fair and excellent things are few, while ugly and evil ones are widespread, so also the superabundant and deficient numbers are found in great multitude and irregularly placed – for the method of their discovery is irregular – but the perfect numbers are easily enumerated and arranged with suitable order. One perfect number is found among the units, 6, one among the tens, 28, one among the hundreds, 496, and one among the unit thousands, 8,128.
- The method of producing them is to set forth the even-times even numbers from unity to infinity; then add them together one at a time. If the sum is a prime incomposite number then multiply it by the last number added, and the result will be a perfect number. If the sum is not a perfect incomposite number, then keep adding. For example, the even-times even numbers are: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512… Beginning at 1, 1 plus 2 is 3, a prime incomposite number; 3 times 2 is 6, a perfect number. Continuing, 3 plus 4 is 7, a prime incomposite number, 7 times 4 is 28, a perfect number. Continuing, 7 plus 8 is 15, not prime; 15 plus 16 is 31, a prime incomposite number; 31 times 16 is 496, a perfect number.
- Unity is potentially a perfect number, but not actually; for it is so in very truth, not by participation like the rest. It is perfect potentiality, because it is potentially equal to its own parts, the others actually.
- Of relative quantity, the highest generic division is equality and inequality; for everything viewed in comparison with another thing is either equal or unequal.
- Inequality is divided into greater and lesser. The greater and lesser are divided into 5 species: multiple, superparticular, superpartient, multiple superparticular, and multiple superpartient. The 5 species of lesser merely add the prefix sub- to each of the 5 classifications above.
- A multiple is a number which, when it is observed in comparison with another, contains the whole of that number more than once. For example, 4, 8, 16, 32… are multiples of 2.
- The superparticular is a number that contains within itself the whole of the number compared with it, and some one factor of it besides. The roots are 4:3, 8:6, 12:9, 16:12 and so on to infinity.
- The superpartient is a number that contains within itself the whole of the number compared with it, and more than one part of it. The root-forms are 5:3, 7:4, 9:5, 11:6, 13:7, and so on to infinity.
- The multiple superparticular is a relation in which the greater of the compared terms contains within itself the lesser term more than once and in addition some one part of it. The root-forms are 5:2, 7:3, 9:4, 11:5, and so on to infinity. The second kind is 7:2, 14:4, 21:6, and so on to infinity. The third is 10:3, 20:6, 30:9, and so on to infinity.
- The multiple superpartient a number that contains within itself the whole of a number compared with it more than once, and more than one part of it. 8:3, 16:6, 24: 9, and so on to infinity.
- All the complex species of inequality are produced by equality, as from a mother or root.
“Without the aid of Arithmetic, Music, Geometry, and Astronomy, it is impossible to deal accurately with the forms of being and discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly. These sciences are like ladders or bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those physical things to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls.”
“The truth of the universe is beheld in Mathematics.”
“Relying upon Mathematics, the creator of the universe sets in order his material creations and makes them attain their proper ends.”
“Arithmetic is the origin of the other sciences. Without arithmetic, the other sciences would not exist. Without the other sciences, arithmetic would still exist.”
“The universe both in part and as a whole seems to be determined and ordered in accordance with number.”
Nicomachus divides the sciences of mathematics into four categories: Arithmetic, Music, Geometry, and Astronomy. Arithmetic deals with Number per se. Music deals with numbers in relation to each other. Geometry deals with sizes at rest and stable. Astronomy deals with sizes in motion.
In Book 1, Nicomachus asserts that an understanding of mathematics is essential to philosophize, and because happiness is attained by philosophizing, mathematics is necessary to happiness. As a former teacher of high school math, I am certain that there are thousands of students in the United States who would strongly disagree with Nicomachus’ claim. Nevertheless, I do appreciate the aesthetic qualities of math that arise from its simplicity and harmony.
Nicomachus spends much of the first book describing various qualities of numbers. For example, he explains that prime incomposite numbers are those numbers that are divisible by themselves and unity only, while secondary composite numbers are divisible by other factors besides themselves and unity. Many of Nicomachus’ descriptions of the different properties numbers possess are illustrated by examples – 11 is a prime number because it is divisible by itself and unity only. In the case of mathematics, just as in cases of any abstract reasoning, providing the student with a number of examples that demonstrate the principle being discussed is very helpful in the learning process.
- An element is the smallest thing which enters into the composition of an object, and the least thing into which it can be analyzed. Letters are the elements of literate speech. Sounds are the elements of melody. The four elements of the universe are earth, water, air, and fire.
- Equality is the element, or elementary principle, or relative number. The number 1 and the number 2 are the most primitive elements by which the other numbers are constructed to infinity, and by which the analysis of the other numbers into smaller terms comes to an end.
- Nicomachus explains how different ratios can be produced.
- Unity is non-dimensional and elementary. A point is non-dimensional and elementary. Dimension is first found in 2, then 3, then 4… Dimension is that which is conceived as between two limits.
- A point is non-dimensional and elementary. The first dimension is a line. The second dimension is a surface. The third dimension is a solid.
- Thus, the point is the beginning of dimension, but not itself a dimension; the line is the beginning of surface, but not a surface itself; the surface is the beginning of a body, but not a body itself.
- A triangular number is one which, when it is analyzed into units, shapes into triangular form from the equilateral placement of its parts in a plane. The first triangular number is 3 – one unit above two units. The second is 6 – one unit above two units, above three units. The next triangular number is 10, etc.
- The square is the next number; it shows us four angles. The square numbers are 1, 4, 9, 16, 25, 36, 49, etc.
- The pentagonal numbers: 1, 5, 12, 22, 35, 51, 70, etc. Root numbers are those with a difference of 3 [1, 4, 7, 10, 13, 16, 19, etc.]
- The hexagonal numbers: 1, 6, 15, 28, 45, 66, etc. Root numbers are those with a difference of 4 [1, 5, 9, 13, 17, etc.]; then add them to preceding number.
- The heptagonal numbers: 1, 7, 18, 34, 55, 81, 112, 148, etc. Root numbers are those with a difference of 5 [1, 6, 11, 16, 21, 26, etc.]
- The octagonal numbers have root numbers with a difference of 6.
- The root numbers will be those numbers with a difference of two less than the total number of angles of the figure; i.e. n – 2.
- A square is composed of two triangles. Thus, if we add two consecutive triangular numbers, we will get a square number. If we add a triangular number to a subsequent square, then we get a pentagon, etc.
- He discusses pyramids and cubes.
- The ‘same’ and the ‘other’ are the principles of the universe. The ‘same’ inheres in unity and odd numbers. The ‘other’ inheres in 2 and the even numbers.
- Harmony always arises from opposites; for it is the unification of the diverse and the reconciliation of the contrary-minded.
- A proportion is a combination of two or more ratios, or relations. A ratio is the relation of two terms to one another. Three is the smallest number of terms of a proportion. 1:2 is one ratio, and 2:4 is another ration; 1, 2, 4 is a proportion.
- The first three proportions are the arithmetic, geometric, and harmonic.
- An arithmetic proportion is one in which three or more terms are set forth in succession, and the same quantitative difference is found to exist between them. 1, 2, 3, 4, 5, 6, 7, etc.
- A geometric proportion is one in which three or more terms are in the same ratio with the preceding term. 1, 2, 4, 8, 16, 32, 64, etc.
- A harmonic proportion is one in which among three terms the mean is not the same ratio to the extremes. 2, 3, 6 and 3, 4, 6
- The ten proportions:
- 1, 2, 3
- 1, 2, 4
- 3, 4, 6
- 3, 5, 6
- 2, 4, 5
- 1, 4, 6
- 6, 8, 9
- 6, 7, 9
- 4, 6, 7
- 3, 5, 8
- The most perfect proportion is 6, 8, 9, 12.
“Dimension is that which is conceived as between two limits.”
“All number is composed of unity and dyad, even and odd, and these in truth display equality and inequality, sameness and otherness, the bounded and the boundless, the defined and the undefined.”
Book 2 of the Introduction to Arithmetic was tedious. Nicomachus demonstrates the various patterns that can be found in Mathematics. There was nothing very insightful in this book other than his assertion that the principles of the universe are equality and inequality, manifested in the monad and dyad. Parmenides disagreed with this notion. He believed that the universe was unchanging and uniform; thus, inequality did not exist. On the other hand, Nicomachus is very similar to Heraclitus, who believed in a unity of contrary properties, which is nearly the exact same language used by Nicomachus to describe harmony. Heraclitus also said that no man steps into the same river twice. Some scholars have argued that Heraclitus implies that nothing is ever the same, or even that equality does not exist. Assuming this latter interpretation of Heraclitus’ philosophy, then Nicomachus is a kind of mean between the two extremes of Parmenides and Heraclitus.