The Music of Pythagoras (58 page)

*
You get the same result by playing scales using only the white keys on a piano but starting on different notes. The Ionian mode (start on C) is the same as the major scale, the Dorian mode (start on D), the Phrygian mode (E), the Lydian mode (F), and the Mixolydian mode (G). The Aolian mode (start on A) is the same as the minor scale.

*
The three-man cell was used again in Vietnamese communism, in Algeria in the 1950s, and in the USSR in the late 1960s.

*
Picture a collection of coins. Call it Set A. A collection of coins is an example of a “set” that cannot be a member of its own set. In other words, a collection of coins is not a coin. Picture, then, another set (call it Set B) that contains all things that are not coins. This Set B itself is not a coin, so it must be a member of itself. In other words, Set B is a member of Set B. Now picture a third set—Set C. This one contains all the sets that are not members of themselves. Is Set C a member of itself or not? You will find that it is if, and only if, it is not.

*
However, the simplest form of a problem is not always the form in which it is first encountered. If it were, the history of mathematical and scientific discovery would have gone much more smoothly!