Introduction to Arithmetic

Overview

The Introduction to Arithmetic is two books, about a hundred and twenty pages, written in Greek around 100 CE by Nicomachus of Gerasa, a town in Roman Syria. Almost nothing is known about him except that he was a Neopythagorean and that he also wrote a Manual of Harmonics and a lost theological work on numbers.

It is not a work of original mathematics. Almost everything in it was known, and most of it is in Euclid’s arithmetic books, which are better. What Nicomachus did was take the material out of the demonstrative form and put it into a shape that could be taught, with definitions, classifications, worked examples and tables, and no proofs at all.

That decision determined its career. Boethius produced a Latin version, De Institutione Arithmetica, in the early sixth century, and Boethius was the channel through which the Roman world’s learning reached the medieval West. For the next thousand years, when a student in a cathedral school learned arithmetic, this was the arithmetic he learned, and the vocabulary of perfect and abundant numbers, of superparticular ratios and figurate numbers, comes into English from here.

Read it as a document of what was taught rather than as mathematics, and it becomes genuinely interesting.

The argument

Book I, chapters 1 to 6: why arithmetic first. The philosophical frame, and the part worth reading even if you skip the rest.

Nicomachus begins by distinguishing what truly exists, which is unchanging and immaterial, from what merely becomes. Wisdom concerns the first. Since magnitude and multitude are the two forms of unchanging quantity, there are four sciences: arithmetic, which treats multitude in itself; music, which treats multitude in relation; geometry, which treats magnitude at rest; and astronomy, which treats magnitude in motion. That is the quadrivium, and this is where the four-part scheme that governed education for a millennium is first set out.

Arithmetic comes first, and not by convention. Remove number and geometry disappears, since figures are counted; remove geometry and number survives. Arithmetic is therefore prior, and Nicomachus takes the further step that makes him a Pythagorean rather than a mathematician: number existed in the mind of the creator as the pattern for everything else, so studying it is studying the plan.

Book I, chapters 7 to 23: the classification of numbers. Numbers divide into even and odd. Even numbers divide into evenly even, evenly odd and oddly even, each with its own properties.

Then the classification that mattered most. Take a number and add up its proper divisors. If they exceed it, the number is abundant, like twelve. If they fall short, it is deficient, like eight. If they exactly equal it, it is perfect, like six, whose divisors are one, two and three.

Nicomachus lists the first four perfect numbers, six, twenty-eight, four hundred and ninety-six, and eight thousand one hundred and twenty-eight, and then makes two general claims which he does not prove and which are both wrong: that there is exactly one perfect number in each order of magnitude, one among the units, one among the tens, one among the hundreds, one among the thousands, and that they end alternately in six and eight. The fifth perfect number is 33,550,336, which breaks the first claim badly, and it was not found for more than a thousand years. Both errors were transmitted as fact through Boethius and repeated by generations of readers.

The moralising around the classification also travelled. Perfect numbers are rare and beautiful, abundance and deficiency are excesses to be avoided, and the whole thing is presented in the vocabulary of virtue.

He also treats prime and composite numbers, and gives Eratosthenes’ sieve for finding primes, which is one of the most useful things in the book.

Book II, chapters 1 to 20: figurate numbers. Numbers arranged as shapes. Triangular numbers are those that make a triangle of dots, one, three, six, ten; then squares, pentagonals, hexagonals, and the solid figures, pyramidal, cubic. Nicomachus gives the relations between them, including that consecutive triangular numbers sum to a square, and shows them in tables.

This is genuinely attractive material and it is where the Pythagorean conviction is most visible: number is not counted, it is shaped, and the shapes are the reason the classification means anything.

Book II, chapters 21 to 29: proportion and the means. The last section, and the one that connects arithmetic to music. Nicomachus defines ten kinds of mean and dwells on three: the arithmetic, where the middle term exceeds the first by the same amount the last exceeds it; the geometric, where the ratios are equal; and the harmonic, where the middle term exceeds the first and is exceeded by the last by the same fraction of each.

The harmonic mean is what gives the musical intervals, and its treatment here, and in Boethius’ companion work on music, is the reason the medieval quadrivium treated music as a branch of mathematics rather than as an art. The book ends with the perfect proportion, a four-term relation Nicomachus says the Babylonians discovered and Pythagoras brought to Greece.

How to read it

Read Book I chapters 1 to 6 for the philosophy and the quadrivium, and then chapters 13 to 16 for the perfect numbers. That is perhaps twenty pages and it is the part with consequences.

Then browse Book II for the figurate numbers, which are pleasant, and the three means, which explain why music was a mathematical subject.

Use D’Ooge’s translation, which is free and comes with a long introduction by Frank Egleston Robbins and Louis Charles Karpinski on the history and on Nicomachus’ errors. The apparatus is more valuable than the text.

Read Euclid Books VII to IX beside it, even briefly. The contrast is the whole lesson: Euclid proves that there are infinitely many primes and gives a formula for perfect numbers; Nicomachus lists four and generalises wrongly. And Euclid was not the book that got taught.

Take the false generalisations seriously as a case study. Two unproved claims entered the European curriculum and were repeated for a thousand years because the book that carried them was the one in the schools. That is a fact about how knowledge is transmitted and it is more instructive than the arithmetic.

And notice what the Pythagorean frame is doing. If number is the creator’s pattern, then classifying numbers is theology, and the moral vocabulary around perfect and deficient is not decoration. Kepler was still working inside this assumption fifteen hundred years later.

Questions it raises

• Nicomachus asserts and Euclid proves. Why did the book without proofs become the one that was taught?
• Two of his generalisations are false and were transmitted as fact for a millennium. What does that say about the authority of textbooks?
• Is there anything to the idea that number is the pattern of the world, or is it a metaphor that happened to be productive?
• The quadrivium put music with arithmetic, geometry and astronomy. What is gained and lost by classifying it that way?
• Perfect, abundant and deficient are moral words applied to numbers. Does the vocabulary shape what questions get asked?
• Kepler was looking for harmonies in the heavens on the strength of assumptions like these and found the third law. Can a false framework be a good research programme?