Squaring off the Greek math section is Nicomachus of Gerasa's *Introduction to Arithmetic*. Sadly, this work
is long out of print. You can scrounge Amazon or Abe Books for a copy. There
are some paperback versions floating around that are just the translation
without any of the introductory material. They tend to be a lot cheaper.
However, they were intended only for internal use at St. John's College and the
publisher did not give permission for the resale of those books. I went for
original hardcover version. The notes in this edition are truly exceptional.
Most of Nicomachus' explanations and proofs are paired with alternate versions,
often from other Greek authors, in the footnotes. So if Nicomachus loses you,
the footnotes can often put you back on track. The notes also provided
references to a number of other very interesting Greek mathematical works that
I think I would like to check out the next time around.

Nicomachus starts off from a very defensive position. In essence, he tries to explain why anyone should care about arithmetic at all. His very philosophical about it. And the basis of his argument is that the study of "real things" is essential to understanding reality and the universe. From there, he argues that arithmetic is among the things that are truly real. While we moderns would say that a deer in front of us is more real than an abstract concept like "animal", Nicomachus disagrees. A deer can die. However, the idea of "animal" could potentially outlive the existence of animals. In other words, qualities, quantities, and forms exist uniformly throughout all of time, while mere objects and creatures exist for only a slice of time. Therefore, in the grand scheme of the universe, these more abstract things spend more time being real than the seemingly concrete things. And should not the thing that exists for the entire life of the universe be considered more real than the thing that exists for a relative blink of an eye? It is certainly an interesting way of looking at things.

Nicomachus, shortly thereafter, addresses an issue I raised in a previous
post: numbers can fill completely different, e.g. counting versus measuring,
without us giving much thought to that fact. Nicomachus expands on that. He
explains that arithmetic is the study of numbers, particularly numbers of the
countable kind, i.e. integers. While arithmetic enables other types of math, it
is conceptually prior to them all. He argues that the children of arithmetic
are those types of math which study objects, motion, and ratios. These three
types of math Nicomachus says we call geometry, astronomy, and music. While
this may seem like mere word play, I think there is something to it. We easily
become constrained by notions of what math is. But the whole system sometimes
needs radical modification to deal with news problems. The classic example is
Newton's physics requiring the development of Calculus. On the other side of
the coin, Descartes unified geometry and algebra and effectively destroyed the
classical/medieval conception of geometry. Meanwhile, music and its study of
ratios has managed to soldier along largely without numbers and without being
regarded as properly math. And I think there are seriously conceptual
revolutions to be made. I once again point to D'arcy Thompson's *On Growth
and Form*. There you can see how poorly our math copes with measuring a
three-dimensional object's change over time. The problem demands some sort of
calculus of geometry.

Though Nicomachus probably did not see such a division, after his
philosophical expositions comes his purely mathematical ones. He dedicates
himself primarily to figuring out the properties of numbers. He starts with
evens and odds. He then goes into concepts like "even-times even", which we
would call powers of two. Likewise, he goes into all the variants of the even
and oddness of a number's divisors. Prime numbers and the sieve of Eratosthenes
are explained. The list of classifications of numbers is long and fascinating.
I can not really capture its breadth here. Though I am tempted to write a
program that can detect all of Nicomachus' proposed classifications. Going
beyond the properties of single numbers, Nicomachus also covers the 10 classic
proportions in detail, including a bit about their history. Apparently in
Pythagoras' day only first three, arithmetic, geometric, and harmonic, had yet
been discovered. I once again recommend Matila Ghyka's *The Geometry of Art
and Life* for exploration of those proportions in, well, art and life.

The road forward gets trickier from here. Per a friend's advice, I will be
tackling the Latin authors in reverse order from Thomas Aquinas. The reason is
that his Latin is quite easy to read and it provides a good follow-up to the
Vulgate, both conceptually and linguistically. Unfortunately, Aquina's
*Summa* in Latin is eight volumes. It will take quite a while for me to
finish. However, there is some good news. I have recently put effort into
expanding my ability to read academic French to also read literary French. I
have started with Voltaire's *Candide* and, at the current pace, should
be able to write a review of that in a month or so. I will likely continue with
other French authors in the Great Books set. Hopefully this will keep the posts
flowing while I tackle the *Summa*.