1. Nicomachus is distinguishing
between changeable material things (bodies) and unchangeable immaterial things
(qualities). It is the unchangeable that philosophy is interested in, according
to Nicomachus, while things that are changeable are, by their lack of persistence,
less "real." This idea is developed in the next chapter.

2. The distinction
between what Nicomachus calls "magnitudes" and "multitudes"
is an important one in mathematics (and especially in the foundations of calculus),
and what we read here is a very early consideration of these ideas. Today we
use the corresponding terms "continuous" and "discrete"
to describe these concepts. A "magnitude" is **continuous**,
since it extends across an unbroken interval of values, any portion of which
is arbitrarily finely divisible into smaller pieces. However, a "multitude"
is **discrete**, since its elements are separately distinguishable,
and is not arbitrarily finely divisible, as its smallest building blocks can
be counted one by one.

3. Discrete multitudes are characterized by the property that they are countable, while continuous magnitudes are arbitrarily finely divisible. (See the previous note.)

4. The real world is
only perceived in finite bits, so "multitude" and "magnitude"
are measured by **finite** continuous and **finite**
discrete numbers.

5. As we alluded to
in the introduction to this text, a number (like 6, or 28) is **perfect**
if it is the sum of all its divisors (6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 +
14).

6. The four *mathemata*
of the *quadrivium*: arithmetic, music, geometry, and astronomy.

7. These musical terms
are defined in the upcoming text. One of the most important discoveries of Pythagorean
mathematics was the
fundamental theory of harmony, that vibrating strings whose lengths were
in simple ratios of small whole numbers produced the most pleasant intervals.
The octave (produced, for instance, by playing both middle C and the next highest
C on a piano) is described by strings whose lengths are in the ratio 2:1; Nicomachus
calls this interval the diapason. The **perfect fifth** (the interval
from C to G) is what Nicomachus terms the diapente, and the **perfect
fourth** (the interval from C to F) the diatessaron.

8. This is a perfect
example of what Pythagorean arithmetic was like: an exploration of the simplest
properties of whole numbers.

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last modified 8/24/04

Copyright (c) 2000. Daniel E. Otero