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.