Summary of Nicomachus' Introduction to Arithmetic, book 1

My answer to lesson 23 in the classical arithmetic course in the Classical Liberal Arts Academy, where upon finishing the first book of Nicomachus' Introduction to Arithmetic I was tasked to write a short essay summarizing the content of the book. (with some amendments)

In the Classical Arithmetic course the subject of study is discrete and absolute quantity, which is one of Aristotle's ten categories. Quantity is partitioned into the continuous and discrete, producing the species of magnitude and multitude respectively. The continuous species is subsequently divided into resting and moving magnitudes, viz. geometry and astronomy, and multitutde into the relative and the absolute, the first being music and the latter arithemtic. These four arts and sciences of quantity is often referred to as the quadrivium.

Arithmetic is prior to the other mathematical arts in that it being subverted, the others also will be subverted, but not vice versa. For subverting geometry, nothing hinders that the number three or the superpartient still exists, but subverting absolute multitde, neither relative multitdes, triangles nor squares or any of the kind may exist. Arithmetic is therefore the first art in the quadrivium to be engaged.

Studying these arts of the quadrivium elevates the human mind beyond the ever changing world of material things and opens up a view to the eternal and immutable, which acording to Nicomachus and Pythagoras alike, is more real than what is always becoming, but never really is. The quadrivium is therefore understood to be sciences about this higher, more real, reality.

Of the absolute multitudes, then, there are two species, the even and the odd. The even being those numbers which admits of division into two halfs without any rest. While the odd is that which does not admit of such a division without a rest.

The two species are again constituted of lesser species, three each to be precise. They are given their species according to their production, which is studied in Nicomachus' first book, along with their qualities and peculiarities.

We also learn of the perfect and imperfect numbers. The former being that which is equal to the sum of its parts, and the imperfect that wich is not, either on account of being lesser or greater. The perfect numbers are all even numbers and may be orderly produced from the species of the even-times even.

Following this is given an introduction to relative numbers, which belongs primarily to the science and art of music. Relative numbers subsists in two species, the equal and the unequal. The equal being one and the same, having no species as all that is equal is equal synonymously. The unequal, howeverl, is subdivided into lesser species according to how the numbers relate to each other. And these are the multiple, superparticular, superpartient, multiple-superparticular and multiple-superpartient. The production of these species may be procured in an artistic and orderly fashion, starting always with three equal terms, demonstrating to us that inequality has its origin in equality as a root and base. This fact also testifies to the truth of Nicomachus' words when saying:

"... that which is fair and limited, and which subjects itself to knowledge, is naturally prior to the unlimited, incomprehensible, and ugly, and furthermore that the parts and varieties of the infinite and unlimited are given shape and boundaries by the former, and through it attain to the fitting order and sequence ..."

What Nicomachus says here is understood by the ancients to be applicable to the whole of the natural universe, including human beings and our faculties.

Everything in short, in this book Nicomachus teaches the basics of arithmetic, which is the first of the mathematical arts and hence the first science we must undertake in comming to know the truth of things real and immaterial.