Five-minute digest
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.
Top 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.
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.
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.
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.