A triangle made of ten points
The tetractys (Greek for “a set of four”) is a triangular arrangement of ten points in four rows: one point at the top, then two, then three, then four at the base. Adding the rows gives 1 + 2 + 3 + 4 = 10. It is the fourth of the triangular numbers, the series 1, 3, 6, 10 that counts dots arranged in a growing triangle, and it was the most important of them for the Pythagorean school.
Its appeal was partly arithmetic and partly musical. The first four numbers contain the ratios that Pythagorean thinkers identified with the main concordant intervals of music: 2:1 for the octave, 3:2 for the fifth and 4:3 for the fourth. Each ratio can be read off two adjacent rows (the second and first, the third and second, and the fourth and third). So a figure that was simple to draw seemed to hold both the number ten and the proportions of harmony, which is why later authors called it the “source” of the school’s wisdom.
Why ten mattered
Ten had a special place in Greek counting, as the base of the everyday number system, and the Pythagoreans treated it as a complete number. Aristotle, in discussing the school, says that they thought ten to be perfect, and the numbers one to four that make it up were counted as its parts. The idea of ten and completion in Pythagorean thought comes through chiefly from his reports and from those of later commentators.
The oath
The oath is the best-known ritual detail. According to ancient reports, members of the school swore by the tetractys, or rather by the one who had handed it down. The usual form is a pledge in the name of “him who gave our generation the tetractys, which holds the source and root of ever-flowing nature”. This wording is taken from later sources, not from any text of Pythagoras:
- The verse collection known as the Golden Verses, a poem attributed to Pythagoras but of uncertain, later date, contains a line of this kind.
- The sceptic Sextus Empiricus, writing in the second or third century CE, quotes a similar formula.
- Later biographers such as Iamblichus also mention the practice.
Two points are worth noting. First, the person who “gave” the tetractys is not named. Later authors take him to be Pythagoras, and the habit of not naming the founder is often explained as a sign of reverence, though it may also reflect a later neat tradition. Second, nothing in these sources tells us the setting, whether the oath was spoken at initiation, at meetings or on some other occasion. Its details should be treated as plausible but not firmly established.
The satirist Lucian of Samosata, in the second century CE, gives a humorous scene in which Pythagoras sells himself at a slave auction and asks a buyer to count: the buyer says one, two, three, four, and is told that this is ten, a perfect triangle and the oath of the school. As a joke it only works if the tetractys was already a widely known emblem, which suggests that by that time the story of the oath was familiar.
What the rows were taken to mean
Later writers read the figure in many ways. Reports of such readings vary, and none can be firmly traced to the earliest Pythagoreans.
- Point, line, surface, solid. One point is a single position, two points define a line, three points define a plane figure (a triangle), and four points not in one plane define a solid (a tetrahedron). On this reading the rows mark the steps from nothing extended to a full body.
- The four elements. Some authors linked the four rows to fire, air, water and earth, or to the four seasons or ages of life. These are correlations in the style of the four-fold schemes of the period.
- The cosmos and the soul. Neoplatonic writers treated the tetractys as a diagram of stages from the one to the many, and linked it to the soul. These are philosophical interpretations built on the original arithmetic.
Because the figure is so simple, it can be made to support almost any fourfold scheme, and that flexibility explains both its longevity and the difficulty of knowing its first meaning.
The tetractys after antiquity
Medieval and Renaissance writers inherited the figure through Latin and Arabic authors. Modern occult movements, beginning in the nineteenth century, adopted the tetractys and compared it with the Kabbalistic sefirot, the ten emanations of Jewish mystical thought, which can also be drawn as a pattern of ten points. The two systems arose separately, and the resemblance is a later comparison, not a historical link. The Golden Dawn and some modern Pythagorean-inspired groups use the tetractys in their ceremonies, and it also features in the history of numerology as a loosely cited ancestor of number symbolism.
Today it is used mostly as an emblem. Seeing it as a piece of ancient ritual requires care. The surviving evidence supports a figure that was revered, a number that was treated as complete, and an oath that was reported by several later authors. Beyond that lies reconstruction. Among ritual numbers in the ancient world, ten and its triangle stand out less for what we can prove about their use than for the clarity with which a few dots express an idea about how number organises the world.