PROPOSITION

. *THEOREM*

If the square drawn on one side of a triangle is equal to the squares drawn on the other two sides, then the angle contained by those two sides is a right angle. |

It is this proposition that informs us that if the sides of a triangle are 3-4-5 — so that the squares on them are 9-16-25 — then the triangle is right-angled. Whole-number sides such as those are called

.

Pythagoras and the Pythagoreans

Pythagoras is remembered as the first to take mathematics seriously in relation to the world order. He taught that geometry and numbers should be studied with reverence, because we are entering into knowledge of the Divine. Mathematics is therefore more than just intellectual stimulation, and its relationship to the Universe is more than just coincidence.

Pythagoras was born on the Greek island of Samos. He left to found what we might call a religious sect in the city of Crotona in southern Italy, which was then part of greater Greece. His devotees were called Pythagoreans ("Pi-thag-o-REE-ans"), and many of them, both men and women, lived communally. They had their rituals and their dietary laws, and they made important contributions to the medicine and astronomy of their time. They were among the first to teach that the earth is round and that it revolves about the sun.

They also taught the continuity and reincarnation of life; and that through philosophy (literally, love of wisdom) there is purification and thus escape from the cycle of births. Hence they practiced equality between one another, and they showed compassion to all creatures, because we are all forms of One.

The Pythagoreans were the first to systematically investigate both arithmetic and geometry. Not only did they discover many theorems, but they gave an ethical and spiritual significance to each of the figures they drew. As a secret society, it must be said that they were extremely successful — because not a word has survived! In this regard, Pythagoras is credited with discovering incommensurable magnitudes. (See Topic 10 of The Evolution of the Real Numbers.) They considered their doctrine of incommensurables their most esoteric teaching — one of their members was treated as dead for having even spoken of it to an outsider. The significance they gave to it has never been revealed.

As a scientific researcher, Pythagoras discovered how musical harmonies depend on ratios of whole numbers. He found that we

hear the interval called an octave when the length of a vibrating string is halved, that is to say, when the length of the plucked string is to the whole string in the ratio 1:2 (Fig 1). We hear the interval called a perfect fifth (G above C) when two thirds of a string is plucked (Fig. 2), that is, when the stopped string is to the whole string in the ratio 2:3. And we hear a perfect fourth (F above C) when three fourths of the string is sounded (Fig. 3); and so on, for each interval in the musical scale.

In fact, Pythagoras taught that all things are known through number (not symbols for numbers). Or rather, "All things are assimilated to number." This means that just as one population becomes assimilated by another — they become just like those others — so all things are like numbers. In that likeness, all things meet and are intelligible.