Introduction to Arithmetic

by Nicomachus of Gerasa

Introduction to Arithmetic (The Classics of the St. John's Program) by Nicomachus of Gerasa. Translated by Martin Luther D'Ooge. 1955 paperback published by The St. John's College Press, Annapolis, Maryland.

Genres: Mathematics, Nonfiction, Science, Classics, Philosophy, Education

200 pages, Textbook Binding

First published January 1, 100

Reviews

[Roy Lotz · Rating 3 out of 5]

One would naturally think, judging from the title of this work, that this is a manual of addition, subtraction, multiplication, and so on. But it is nothing of the sort. Nicomachus was a Neopythagorean philosopher, influenced by both Plato and Aristotle, who considered numbers to be the basis of reality. Thus, there is a quasi-mystical aspect to this work, as Nicomachus’s categorization of different numbers and ratios are supposed to reveal something about the order of the cosmos:

All that has by nature with systematic method been arranged in the universe seems both in part and as a whole to have been determined in accordance with number, by the forethought and the mind of him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preexistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things, time, motion, the heavens, the stars, all sorts of revolutions.

The result is rather strange for modern readers. Nicomachus goes through what, to us, appear to be basic and trivial aspects of number—even, odd, square, cubed, etc.—in such a way that neither explains the cosmos nor sheds any light into the mathematics. The result is uninspiring, since nowadays we are not likely to think that different sorts of ratios or number sequences have any intrinsic interest. Further, Nicomachus includes no proofs, and instead illustrates every one of his mathematical idea with a simple example, which renders the work of little value to mathematics.

It is as a window into history, then, that I think the work has the most interest. For example, the very fact that it was copied and preserved through the Middle Ages demonstrates the appeal this mathematical mysticism had for Christians. We can also see how abstract mathematical concepts could be turned into moral ones. Aristotle’s idea of virtue being a mean between two extremes is given a mathematical explanation here, and some of Plato’s proto-scientific ideas (such as that the universe was composed of mathematical elements) are echoed by Nicomachus. Even so, I think even the most generous reading of this work will find much of it to consist of uninteresting classifications of different sorts of numbers. Luckily, it is quite short.

[Kristofer Dubbels]

Nota bene: A homeschooler looking for practical advice on Nicomachus’ place in teaching the quadrivium might want to skip to the final section for my (non-expert) opinion on how to approach Nicomachus.

Any modern reader hoping for an introduction to arithmetic will be disappointed by what Nicomachus has to offer. But surely anyone capable of reading this sentence thus far already has a reasonably sound grasp of “arithmetic” and requires no introduction to the subtle arts of adding, subtracting, multiplying, and perhaps even dividing. Indeed, arithmetic as a discipline is uniquely suited to the nature of modern compulsory public education: if anything can be turned into a “worksheet” - the paradigmatic mode of learning in late modernity, or at least in the United States - it is arithmetic.

Nicomachus’ Introduction to Arithmetic, however, is not primarily a book about arithmetic at all, or at least not as we understand “arithmetic” today - the basic discipline of the aforementioned numerical operations of addition, subtraction, multiplication, and division. Predating the modern disciplinary divisions within mathematics, Nicomachus’ work encompasses metaphysics, philosophy of mathematics, as well as some recognizably modern “arithmetic”, particularly in part 1, where he includes a 10x10 multiplication table, apparently a novelty for the time. The Introduction, however, is primarily a work of what we would now recognize as “number theory.”

Unlike arithmetic, number theory is generally not taught as a standard part of most primary math education curriculums. Although some ideas from number theory may be taught in high school depending on the choice of textbook, systematic and dedicated study of it is generally reserved for college mathematics students. Number theory is primarily dedicated to the study of integers and arithmetic functions - study of the properties of prime numbers, for example, is characteristic of modern number theory.

According to the traditional understanding within the canon of the quadrivium, arithmetic encompasses the order of discrete things, taken in themselves. One way of conceptualizing the discipline, then, is to place it alongside the other three disciplines of the quadrivium, geometry, music (or harmonics), and astronomy. Arithmetic consists of the order of discrete things, whereas harmonics considers the order of discrete things in motion. Geometry considers extended things, while astronomy considers extended things in motion.

Arithmetic for Nicomachus, then, is not the simple business of adding and subtracting; it is something much more elevated. It is the study of numbers in themselves (that is, the aforementioned “discrete things”), with the ultimate aim of acquiring wisdom itself (pt 1, ch. II). Quoting Androcydes, Nicomachus explains, “just as painting contributes to the menial arts towards correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom” (pt 1, ch. III). That is, knowledge of numbers is valued not for its practical applications; rather, arithmetic is valuable as an aid to acquiring wisdom. Importantly, Introduction to Arithmetic is intended as a complement to his other works on the quadrivium - his Manual of Harmonics is extant, but his Introduction to Geometry has not survived (if Nicomachus composed a similar introductory work on astronomy, no evidence for its existence survives).

As Nicomachus explains, there is a definite order in which the subjects constituting the quadrivium must be learned, and arithmetic comes first:

Which then of these four methods must we first learn? Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others. And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan … but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them. … Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry. (pt. 1, ch. IV)

Thus, for Nicomachus, arithmetic is the “mother and nurse” of the rest of the quadrivium. This opening discussion on the nature of the quadrivium is perhaps the most inherently interesting content of the entire tract; it is this section of the book that will be unfamiliar to a modern audience, while the mathematical content will likely be a rather excruciating slog.

Peter Ulrickson’s A Brief Quadrivium makes the case that mathematics is fundamentally about proofs, not computation; it is also fundamentally about what is real, about tangible, familiar objects. By this standard, Nicomachus has surely failed to produce a work of mathematics - there is not a single proof contained in the Introduction. In the absence of proofs and without any meaningful conception of algebra, Nicomachus relies on truly excruciating prose explanations for concepts that a modern reader could grasp quite easily with modern notation.

So - having read Nicomachus’ Introduction to Arithmetic, have I learned anything new about “arithmetic”? The answer is still a highly qualified yes. Some concepts in the Introduction I was previously unfamiliar with: so-called “perfect numbers,” for example (That is, an integer such that it is the sum of its own divisors. The classic example is 1+2+3=6.) However, anything treated in Nicomachus can almost certainly be more readily understood by simply consulting the topic’s Wikipedia entry, and reading until the point where the article becomes too difficult for the lay reader to comprehend.

The strength of Euclid’s Elements is that a student is immediately presented with two things they have possibly never encountered before during a lifetime of “worksheets” - they are introduced to propositions that can be proven and moreover, they are to prove them with implements (the straightedge and compass) that show these propositions to be true. There is nothing like this in Nicomachus.

Now, one might also reasonably argue that quadrivial “arithmetic” (that is, something like elementary number theory) is itself perhaps a bit of an outlier in the quadrivium. With geometry, a student can immediately begin drawing equilateral triangles and so on. With music, a student can strike a monochord and thus both see and hear an octave. With astronomy, of course, one can observe the movements of celestial bodies. Typically, when a new learner comes to the quadrivium, it is music that has the immediate appearance of an outlier. I would argue, however, that the relationship between geometry and music becomes quite apparent when armed with the tools of the respective trades: straightedge and compass and monochord. Geometry and music (or, more properly, harmonics) are about ratio. Once this is understood and appreciated, the groundwork has been laid to appreciate their relationship to Ptolemaic astronomy. But quadrivial arithmetic - at least as presented by Nicomachus - has a less than obvious place in this schema.

Now, having read Nicomachus, do I feel any wiser? After all, the end goal here is wisdom, with mathematics a mere way station along the journey - perhaps Nicomachus’ presentation of the (mostly) familiar has shed new light on the material? Sadly, Nicomachus did not inspire any particular insight, at least on my initial reading. (Perhaps he will reward re-reads in the future?) The Introduction is perhaps most interesting from the lens of intellectual history - this was how people in the West learned number theory for centuries. In the form of Boethius’ version, along with Euclid, this just was the extent of the West’s mathematical knowledge for a very long time. In that respect, it is well worth reading, or, more likely, worth skimming after one has read the opening philosophical chapters of book one.

That said, one is then naturally led to wonder just what could account for the book’s apparent popularity in the history of the West - it does not compare particularly favorably with Euclid or Archimedes or Apollonius or Ptolemy. Presumably it would hold more interest to philosophers than mathematicians, but it doesn’t have anything even remotely approaching the philosophical depth of Plato or Aristotle. Indeed, it is perhaps a paradigm case of making the simple appear complex.

What makes Nicomachus feel like such particularly thin gruel is that while citing Plato and Aristotle in defense of the quadrivial arts, it is never quite clear just what it is that binds the quadrivium together, aside from being “quantitative” in an imprecisely defined way. Nicomachus provides a clear categorial schema for how to understand the areas of concern of the quadrivial arts, but then the exact relation of the components to the whole is less clear. And their collective relationship to the trivium is entirely unclear.

Moreover, how all of this leads one to wisdom is not so clear, at least as it is presented here. Of course one might retort that Nicomachus’ book is merely the first step in the sequence of the quadrivium. To expect a deeper treatment of the quadrivium by telling - rather than showing - is to put the cart before the horse. The expectation that the “meaning” of the quadrivium be outlined clearly from the start of the journey is perhaps already to miss the point of the whole endeavor.

*****

As a practical matter, I assume anyone who has read this far is likely interested in a traditional quadrivial curriculum, and potentially a homeschooler as well (I am the former, not the latter). As a text for instruction, I am inclined to advocate skipping Nicomachus altogether if one is actually trying to teach the mathematics involved. Peter Ulrickson’s A Brief Quadrivium covers most of the actual number theory in Nicomachus with a vastly superior presentation for a modern audience. I would cover the section on arithmetic in Peter Ulrickson’s book, and then consider reading Nicomachus - if at all - afterwards, as a glimpse into (one of) the original quadrivial sources for the material Ulrickson covers.

It is also worth noting that nearly everything covered in the Introduction is contained already in Euclid’s Elements, books VII-IX. Euclid’s presentation will also look odd to modern audiences - Euclid defines a number as “a multitude composed of units,” which is then explicated by describing ratios between lines. This is often hard to follow relative to an algebraic presentation, but it is certainly far more rigorous and straightforward than the ponderous mode of explanation in Nicomachus. Anyone dedicated to using traditional sources may want to read these three books of the Elements in tandem with Nicomachus, in order to get a sense of how Nicomachus’ assertions might be proven, as opposed to merely illustrated.

Another note to prospective homeschoolers - there is only one modern translation of Nicomachus in English, that of Martin Luther D’Ooge. This is the version that is available on Amazon as a public domain on-demand reprint, and it is also the version in the Britannica Great Books series. The standalone edition has a scholarly introduction that covers practically every aspect of Nicomachus’ thought, his relationship to classical authors such as Euclid and Boethius, and also contains excerpted phrases and passages in untranslated Greek. That is to say, this introductory section is for a specialist audience. This might be worth skimming as a homeschooler, but there is precious little in it that will help one actually teach quadrivial arithmetic. It is, however, one of the few sources of in-depth information on Nicomachus for anyone who is truly interested.

[Evan Leach]

This book is a textbook on mathematics from around the year 100. It was not renowned for its originality - rather, the book was famous in antiquity and in the middle ages for being a useful primer covering the mathematical knowledge of its time.

I am no mathlete, and I'll admit some of the esoteric concepts and convoluted proofs that Nicomachus throws out there made my eyes glaze over. For me, the more interesting aspects of this book concerned Nicomachus' philosophy. Nicomachus was a Neo-Pythagorean who embraced a mystical, symbolism-heavy view of numbers and mathematics. Most of his thoughts about philosophy are outside the scope of this book, but the Introduction will occasionally allude to them. Nicomachus subscribed to the Platonic idea of eternal forms, with the wrinkle that numbers are a superior kind of form out of which the other forms are made and under which they are classified. Numbers are the highest forms and the properties seen primarily in them are also the essential properties of things in the world, conferred upon them by number. The “scientific” numbers (the numbers we deal with everyday on earth) are related to, but not identical to, the ideal “platonic” numbers.

This philosophy led its practitioners to search for all kinds of meaning in numbers. They believed that through mathematical proofs, they could uncover universal "truths" - the number seven was associated with the leader of the universe, odd numbers were considered male and even numbers female, etc. This attitude towards mathematics persisted for centuries, at least among certain factions, but frankly it is hard for me to really grasp the rationale behind it. To modern minds, all of the time and energy spent on "proving" these mystical numerological proofs seems pointless.

This book gave me a little bit of insight into how Nicomachus and the thinkers who followed in his footsteps thought about numbers, which was interesting at times. But I have no idea who I would recommend this to, other than specialists interested in the history of mathematics (and even then, this probably takes a distant backseat to Euclid). I don't know how to even begin to rate this one, so I'm just going to give it a pass. A strange little book that opens a window onto an even stranger way of viewing the universe.

[David Shane · Rating 3 out of 5]

This short book was OK, we're going to read it with 10th grade math students in the Fall. I actually found the Greek philosophy at the beginning the most interesting part... the idea that the bodiless abstractions (like math!) are the real things, not the material ever-changing stuff we interact with. (Interestingly, some modern physicists would kind of agree with that claim.) A nice explanation of what the "original guys" meant by the four pieces of the quadrivium too.

But most of the book Nicomachus spends being a "botanical mathematician", putting stuff into categories... eh. Historically interesting to see how math developed over time, but that's about it. I sometimes tell students they should be grateful for modern notation, because it helps our thinking and it makes the writing so much shorter. Just look at the longwinded sideways ways people had to go about saying the same stuff before modern definitions and notation!

[Arkar Kyaw · Rating 3 out of 5]

"You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld."
-Plato

[Sarah · Rating 4 out of 5]

4 stars: really liked it.

According to Nicomachus in around AD 100, arithmetic is the most important mathematical science. He distinguishes between divine number, by which the universe is created and ordered by God, and scientific number, which is what we see expressed in the world on a lower plane. Nicomachus' Introduction to Arithmetic is a work on scientific number.

This is as much a work of philosophy as mathematics. The emphasis was on studying arithmetic not for the activities of human life (commerce, war, navigation), but for the cultivation of wisdom. Ancient mathematicians were seeking truth about number, not about using number.

I was surprised to see that this is a compilation and organization of centuries of thought, not new theories. I admire and respect Nicomachus for undertaking the important task of summarizing and presenting the prevailing theories in a cohesive and understandable way.

I am impressed and amazed by the minds that thought through these lengthy and precise definitions and gave words to the framework of arithmetic. We all know what is meant by terms such as "odd" and "even," but we know that only because of the early mathematicians who (1) determined that two "species of number" was sufficient and (2) identified their properties.

This is an important foundational work for Western civilization.

[Tyler · Rating 5 out of 5]

Fascinating. This book was not particularly difficult, but it was just... well, fascinating. All of the different properties of numbers and even some of how they apply to music were expounded in this fine piece of Greek literature. If you are interested even remotely in the mystical, Pythagorean properties of numbers, or even a little bit in just arithmetic in general, then this book is definitely for you. As opposed to its name 'Introduction to Arithmetic', you don't really read this if you're JUST interested in Arithmetic. You would read this after you've read Timaeus by Plato. It's THAT kind of book. I honestly suggest you do, too, considering this book references Plato constantly. It's also nice to be familiar with a few Euclidean definitions.

Overall, beautifully done work of art. The whole book ends with four numbers perfectly exemplifying the Arithmetic, Geometric, and Harmonic ratios: 6, 8, 9, 12. Very well done, nice work of genius right here.

[Esther]

Most of it I had to struggle to understand, but it was worth it. The way he approached numbers and their relationships to each other made me wish I had learned math this way instead of almost pure calculation and formulas which they teach in public school.

[Carter Reads Classics · Rating 2 out of 5]

Bleh. I think some of the concepts are brilliant and I was surprised his writing was so well.

[Becky Carlan · Rating 4 out of 5]

I only read book 1 but it is really cool how the ancients thought so much more deeply about math!

[Wellington J Pindar · Rating 4 out of 5]

A great book for math majors who dropped out.

[JP · Rating 4 out of 5]

He flourished around 100 a.d, writing the introduction as a summary of what was known about numbers at that time. Much of the focus is on the various categories -- odd, even, perfect, superlative, deficient, prime, etc. There are quite a few orders found within a variety of series. All of this tied back to philosophy as the early mathematicians made so many connections. Perhaps not since that time has anyone thought in such away about multiple-dimensions and dualism.

[Michael · Rating 3 out of 5]

Not at all what one might expect, but reading it one can easily see how the Ancients attributed all the mystical properties to numbers. The correspondences are amazing. How numbers are the building blocks of everything. Sort of the creation's alphabet. A difficult read, because of my own unwillingness to put in the requisite attention.

[Claire · Rating 3 out of 5]

I kept up for about 2/3 of Book 1, but got lost in all the "superpartients" and "sesquialters." (Yes, iPhone, those are real words.) in Book 2 I was able to catch snippets around the geometric principles. Surprisingly, I sort of enjoyed this book - at least some of it. Mathematics is a kind of elegant unifying idea, despite my inability to grasp it. Glad I read it!

[M. Filozofs - Lauvassirds · Rating 5 out of 5]

Great stuff. Reading for the first time, I grasped the overview of the ideas, and I very liked the philosophy. Definitely will need to read for the second time to get into details, and maybe even third and the fourth time... separate parts of it. Maths never was so exciting as for the ancient Greeks...

[Jill · Rating 3 out of 5]

So amazing to see the pattern and simplicity in numbers, to recognize that we discovered what was already there; meaning the organization of numbers and quantities, equality and its opposite in all its species, existed prior to our understanding it. Mathematics proves natural law.

[Jennifer · Rating 4 out of 5]

I don't think I agree with Nicomachus about the Nature of God but wow, he discovered some amazing patterns and relationships with numbers. I'll need to study this one again.

[Robert Kaufman · Rating 4 out of 5]

As a math teacher I love the simplicity and patterns in his numbers. There is beauty in math!

[Jim · Rating 3 out of 5]

When you're talking about fundamentals, you're talking about Metaphysics. This "Arithmetic" is harder than I thought it would be, and I was a Math major.