Pythagoras and number

The teaching

As the Pythagorean school that Aristotle, Nicomachus and Plato record teaches it.

All things are number

The Pythagoreans taught that number is the principle of all things: that things are made of number, or are like numbers, and that to know the numbers of a thing is to know the thing. The elements of number are the odd and the even. The odd is limited and the even unlimited, and the one, which is both odd and even, comes from both and is the source of all the rest.

The ten pairs of opposites

Some of them set out the principles as ten pairs of opposites, in two columns: limit and the unlimited, odd and even, one and many, right and left, male and female, at rest and moving, straight and bent, light and darkness, good and bad, square and oblong. The first of each pair belongs to limit, order and the good; the second to the unlimited.

The tetractys

Their emblem is the tetractys, ten points in four rows of one, two, three and four, making a triangle. The rows climb through the kinds of magnitude: one point; two points, which make a line; three, which make a surface; four, which make a solid. The rows add to ten, the perfect number that holds all the others, and the same four numbers hold the ratios of harmony, two to one, three to two and four to three.

They swore by it: by him who gave to our soul the tetractys, the fount and root of ever-flowing nature.

Figured numbers

The Pythagoreans laid out numbers as pebbles and read their shapes. The triangular numbers, 1, 3, 6, 10, grow by rows, and the tetractys is the fourth of them. The square numbers, 1, 4, 9, 16, grow by adding the odd numbers in turn as an L-shaped border, the gnomon: one and three make four, five more make nine, seven more sixteen, and the shape stays a square.

Adding the even numbers to two instead makes the oblong numbers, 2, 6, 12, 20, whose sides always differ by one and whose shape changes as it grows. That is why square stands with limit, and oblong with the unlimited, in their table of opposites.

The numbers from one to ten

Each number of the decad has its nature. One, the monad, is unity and the source, and is not itself counted as a number. Two, the dyad, is otherness and division, the first even, female. Three, the triad, is the first true number, with a beginning, a middle and an end, the first odd, male. Four, the tetrad, is the first square, equal times equal, and so it is justice. Five, the pentad, joins the first female and the first male, two and three, and is called marriage.

Six is the first perfect number, the sum of its own parts, one, two and three. Seven is the virgin, since within the ten it neither divides any number nor is made from any by multiplying, and it is given to Athena, born of no mother. Eight is the first cube, two times two times two. Nine, three times three, is the last before ten, the boundary. Ten, the decad, holds them all and returns to one.

The ratios of the string

On the monochord the ear hears what the numbers say. Stop the string at half its length and it sounds the octave, the ratio of two to one; at two thirds, the fifth, three to two; at three quarters, the fourth, four to three. A fifth and a fourth together make the octave, since three halves times four thirds is two. The difference between them, three halves divided by four thirds, is nine to eight, the whole tone. The octave and a fifth is three to one, the double octave four to one, and all of these come from one, two, three and four.

The soul of the world in the Timaeus

Plato's Timaeus builds the soul of the cosmos from these ratios. The maker takes portions in the numbers 1, 2, 3, 4, 9, 8 and 27: the doubles 1, 2, 4 and 8, and the triples 1, 3, 9 and 27. Into each gap he sets two means, one exceeding the smaller term and falling short of the larger by the same fraction of each, the other by the same amount, and these give the ratios three to two, four to three and nine to eight. Then he fills every four to three with tones of nine to eight, and a small remainder is left over, 256 to 243.

The four arts of number

From this school the later teachers drew the four arts of number, which Boethius called the quadrivium, the fourfold way. Arithmetic is number in itself; music is number in relation to number; geometry is magnitude at rest; astronomy is magnitude in motion. Together they were the road a student walked from the things that are seen toward the order behind them.