Tetractys

Among the sayings the Pythagorean school transmitted orally, the acusmata, later antiquity preserved an oath: by him who handed down to our soul the tetractys, the source and root of ever-flowing nature. The one sworn by is Pythagoras himself, who is not named; the thing he handed down is a triangle of ten points in rows of one, two, three and four.

Walter Burkert's study of early Pythagoreanism, the standard modern treatment, explains why this small figure could carry so large a claim. Its rows hold the numbers one to four, and in those numbers the school had found something astonishing: the concordant intervals of music. A string stopped at half its length sounds the octave, the ratio 1:2; at two thirds, the fifth, 2:3; at three quarters, the fourth, 3:4. Harmony, the most immaterial of experiences, obeyed the first four integers exactly. If sound was ordered by number, the school held, then so was everything: the tetractys was the seed from which the ordered cosmos unfolds, and its sum, ten, the perfect number of the decad, the complete gathering of number itself. How much of this goes back to the historical Pythagoras, and how much was elaborated by later generations of the school, is a question Burkert weighed with great care and left genuinely open. What is certain is the reverence: a community that swears its oaths by a diagram has told us, more plainly than any treatise could, what it held sacred.

The tetractys is the earliest famous example of a figurate number, a number shown as a shape. Set out one point, then a row of two, then three, then four, and the total, ten, appears as an equilateral triangle; the Greeks called such totals triangular numbers, and the tetractys is the fourth of them. The figure rewards contemplation in more directions than one. Read downward it was taken as a generative sequence: the single point stands for the monad, the pure unit; two points determine a line; three, the first surface; four, the first solid, since four points not in one plane define a tetrahedron. In four steps the figure walks from unity to body, from the One to the world of extended things, which made it a diagram of emanation as much as of arithmetic. Read as ratios between its rows it holds the musical concords, and read as a sum it delivers the decad, the number the school held perfect and complete.

Burkert's reconstruction of early Pythagoreanism shows this arithmology already interwoven with the school's religious life, transmitted in the acusmata alongside rules of diet and burial, which is to say that for the Pythagoreans it was never mathematics on one side and piety on the other. The tetractys stands at the head of a long tradition, running through Neopythagorean and Platonist number lore into the number mysticism of the West, in which the first ten numbers are not quantities but powers, and this atlas's entries on those numbers walk in its shadow.