## Pythagoras of Samos

#### Commentary on the text

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.