Introduction to Arithmetic

Summary

Nicomachus of Gerasa wrote the Introduction to Arithmetic (Arithmetike eisagoge) around AD 100, in Greek, as a two-book primer in the Pythagorean study of number. By arithmetic he does not mean computation, which the Greeks called logistic, but the theory of the kinds, properties and relations of whole numbers. The book contains few proofs; it proceeds by definition, classification and example, often with tables, and it treats number as something worth contemplating for its own order and beauty. Book I opens with philosophy. Following Plato, Nicomachus says wisdom is knowledge of the things that truly exist, the immaterial and unchanging, and that the mathematical sciences are the ladder to it. He divides quantity into multitude and magnitude and assigns four sciences: arithmetic studies multitude in itself, music multitude in relation, geometry magnitude at rest, and astronomy magnitude in motion. Arithmetic comes first because it is presupposed by the others and because, he says, number existed beforehand in the mind of the creating God as the pattern of the world. This fourfold division became the medieval quadrivium. He then defines number as limited multitude and works through its kinds: even and odd, with the subdivisions even-times-even, even-times-odd and odd-times-even; prime and composite, with the 'sieve' of Eratosthenes for finding primes; and the superabundant, deficient and perfect numbers, those equal to the sum of their factors, such as 6, 28, 496 and 8128. He draws a moral parallel: perfect numbers are rare and orderly as virtues are, while excess and deficiency abound like vices. The last chapters of Book I classify ratios: multiple, superparticular, superpartient and their combinations. Book II shows how all kinds of inequality can be generated from equality by three simple rules, and then turns to figured numbers: triangular, square, pentagonal and other polygonal numbers that can be drawn as patterns of dots, solid numbers such as pyramids and cubes, and oblong numbers of the form n(n + 1). It ends with proportions, especially the arithmetic, geometric and harmonic means, and the 'most perfect' proportion 6, 8, 9, 12 on which musical intervals rest. Through Boethius's Latin adaptation the book became the standard arithmetic of medieval schools for nearly a thousand years. It suits grades 8-12 as a hands-on introduction to number theory and as a primary source for what the quadrivium was. The library attaches no text and gives no quotations; Martin Luther D'Ooge's English translation (1926) is the usual one.

Plot or Argument

• Book I.1-6: Wisdom, the four sciences and the priority of number

  Nicomachus defines philosophy as love of wisdom and wisdom as knowledge of true being. Quantity is divided into multitude and magnitude, yielding arithmetic, music, geometry and astronomy. Arithmetic is prior because it is implied by the others and because number was the pattern in the mind of the creator.

• Book I.7-13: Even, odd, prime and composite

  Number is defined as limited multitude, a combination of units. Even numbers are divided into even-times-even (powers of two), even-times-odd and odd-times-even; odd numbers into prime, composite and relatively prime. Chapter 13 describes the sieve of Eratosthenes.

• Book I.14-16: Superabundant, deficient and perfect numbers

  Comparing a number with the sum of its factors gives three classes. Perfect numbers are found 'in order' as Euclid's method shows, one in each order of ten: 6, 28, 496, 8128. Nicomachus compares their rarity to the rarity of virtue.

• Book I.17-23: Kinds of ratio

  Relative quantity is equal or unequal; the unequal is classified into ten relations (multiple, superparticular, superpartient, multiple superparticular, multiple superpartient, and their reciprocals), each illustrated with tables.

• Book II.1-5: Inequality born from equality

  Starting from three equal terms, three rules of addition generate every kind of ratio in turn, showing that equality is the element and source of the unequal.

• Book II.6-20: Figured numbers

  Linear, plane and solid numbers are built from points: triangular, square and higher polygonal numbers, pyramids and cubes, and the oblong (heteromecic) numbers. Nicomachus shows how square and oblong numbers relate as 'the same' and 'the other'.

• Book II.21-29: Proportions and the musical mean

  The arithmetic, geometric and harmonic proportions are defined and illustrated, followed by further proportions to make ten in all. The work closes with the 'most perfect' proportion 6, 8, 9, 12, which contains all three means and the ratios of the fourth, fifth and octave.

Lessons

• Arithmetic as the first rung of the quadrivium

  Nicomachus's division of mathematics (multitude in itself, multitude in relation, magnitude at rest, magnitude in motion) is the origin of the four arts that medieval schools called the quadrivium. It rests on a claim about order: you cannot study musical intervals without ratios, or geometric figures without number. Students who know the trivium can see here, in a primary source, why the arts were arranged as they were.

  • Restate Nicomachus's fourfold division in your own words. Where would modern subjects such as statistics or physics fit?
  • Why does he think arithmetic must come before music and geometry? Do you agree?
  • Nicomachus says number existed first in the mind of God as a pattern. How does that claim relate to Wisdom 11:20, 'thou hast ordered all things in measure and number and weight'?

• Perfect numbers and the rarity of virtue

  A perfect number equals the sum of its factors: 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14. Nicomachus notes that such numbers are rare and appear in regular order, while superabundant and deficient numbers are common and disorderly, and he compares this with virtue as a mean between the many forms of excess and defect. The analogy is ancient moral arithmetic, not proof, but it shows how the Pythagoreans saw ethics and mathematics as reflecting one order.

  • Check by hand that 496 is perfect. Is 12 superabundant or deficient?
  • Is Nicomachus's comparison between perfect numbers and virtue an argument, an illustration, or a poetic image?
  • Augustine later said that 6 is not perfect because God made the world in six days, but God made the world in six days because 6 is perfect. What do you make of that reasoning?

• Seeing number in figures

  Figured numbers can be built with pebbles or dots: 1, 3, 6, 10 form triangles; 1, 4, 9, 16 form squares; two successive triangular numbers make a square. Nicomachus's Book II turns these patterns into a systematic theory, and his tables let students discover relationships for themselves before any algebra is introduced. This is a good place to let students do mathematics the way ancient students did, by arranging and counting.

  • Draw the first five triangular and square numbers. What do you notice when you add consecutive triangular numbers?
  • Why might ancient mathematicians have preferred figures and tables to symbolic formulas?
  • Nicomachus gives patterns but rarely proves them. What is gained and lost by that method compared with Euclid's?

• Means, music and the most perfect proportion

  The book's last chapters define the arithmetic mean (equal differences), the geometric mean (equal ratios) and the harmonic mean. In 6, 8, 9, 12, the numbers 9 and 8 are the arithmetic and harmonic means of 6 and 12, and the ratios 12:6, 12:8 and 12:9 give the octave, the fifth and the fourth. For Nicomachus this showed that musical harmony and numerical order are one, an idea that runs through Boethius, the cathedral schools and Kepler.

  • Show that 8 is the harmonic mean of 6 and 12 using Nicomachus's definition: the greatest term is to the least as the difference of the greater pair is to the difference of the lesser.
  • Why would a musician care about ratios?
  • What would it mean for the world to be built on harmony, and how might a Christian reader receive that idea?

Context

Nicomachus came from Gerasa, a prosperous Hellenistic city of the Decapolis in Roman Syria (modern Jerash, Jordan). He belongs to the Neopythagorean revival of the first and second centuries AD, which joined Platonic metaphysics to the Pythagorean conviction that number is the key to the cosmos. His other surviving work is the Manual of Harmonics; a Theology of Arithmetic on the symbolism of the numbers one to ten is attributed to him and survives only in excerpts.

Composition

The work is usually dated to about AD 100 on the basis of Nicomachus's other writings and citations; no exact date is known. It was written in Greek in two books. It draws on Plato (especially the Republic and Timaeus), on earlier Pythagorean tradition and on Euclid's arithmetical books (Elements VII-IX), though it presents results by example rather than proof.

Reception

Iamblichus and later Neoplatonists wrote commentaries on it, and Apuleius is said to have made a Latin version, now lost. Boethius's De institutione arithmetica (c. AD 500), a close Latin adaptation, made Nicomachus's arithmetic the textbook of the medieval quadrivium. Modern historians of mathematics value it as a witness to ancient number theory while noting its lack of rigour. It appeared in volume 11 of the first edition of the Great Books of the Western World (1952) and volume 10 of the second (1990), in D'Ooge's translation.

Comparisons

• Elements, Book I

  Euclid's Elements, written about four centuries earlier, proves its results from definitions, postulates and common notions; Nicomachus states and illustrates his. Elements Book I is geometric, but Euclid's arithmetical books (VII-IX) cover even, odd, prime and perfect numbers, and Nicomachus's method of finding perfect numbers is Euclid's IX.36. Reading them together shows two ancient styles of mathematics, demonstrative and contemplative.

• An Introduction to Mathematics

  Whitehead's 1911 introduction, like Nicomachus's, aims to show a general reader why mathematics matters rather than to train a technician. But Whitehead praises abstraction and symbolism, where Nicomachus relies on pictures of number and on its moral and theological meaning. The pair shows how the idea of mathematical education changed between antiquity and the modern era.

Quotes

Creative Prompts

• Classroom

  Using dried beans or counters, build the first six triangular, square and pentagonal numbers, record them in a table like Nicomachus's, and write down three patterns you find.

• Dialogue

  Write a dialogue between Nicomachus and a modern mathematics teacher on whether 6 is 'perfect' in any sense beyond the sum of its factors.

• Essay

  Write an essay explaining why medieval schools placed arithmetic first among the four mathematical arts, using Nicomachus's own reasons.

Further Reading

• Introduction to Arithmetic, Nicomachus of Gerasa, trans. Martin Luther D'Ooge, with studies by F. E. Robbins and L. C. Karpinski

  The standard English translation (1926), with long studies of Greek arithmetic.

• De institutione arithmetica, Boethius

  The Latin adaptation through which the medieval West learned Nicomachus's arithmetic.

• Elements, Books VII-IX, Euclid

  Euclid's arithmetical books, which give proofs for many results Nicomachus states.

• A History of Greek Mathematics, Thomas L. Heath

  Heath's classic survey, with a chapter on Nicomachus and the Neopythagoreans.