Introduction

The numbers from one to ten

For the Pythagoreans every number up to ten had a nature of its own: the monad as origin, the dyad as opposition, five as marriage and seven as the motherless virgin. What the old books of numbers say about each one, what of it is mathematics and what interpretation.

For us numbers are tools. We count with them, we calculate with them, and none is more important than another. For the Pythagoreans it was different. Every number up to ten had a nature of its own, a shape of its own and names of its own. They laid numbers out in pebbles and looked at the figure that arose, and from the figure and the arithmetical relations they read the number’s character. This teaching is called arithmology, and it is one of the roots of number symbolism to this day.

The most important sources are Nicomachus of Gerasa, who around AD 100 wrote an Introduction to Arithmetic and a lost Theology of Numbers, and a book from the fourth century, the Theologumena arithmeticae, which collects excerpts from older writings. Then there is Philo of Alexandria, who interpreted the numbers of the creation story in the Pythagorean way, and scattered remarks in Aristotle and Plutarch.

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The numbers from one to ten as figures of points, with their Greek names. A tap on a number shows its figure and what the old books of numbers saw in it.

One and two

The monad, the unit, was for the Greeks not yet a number at all. Euclid defines number as “a multitude composed of units”, and a multitude needs at least two. One is the origin from which all numbers proceed, just as the point has no magnitude and yet begins every line. It was called God, form, centre and seed.

The dyad, twoness, is the first multitude. With it begin separation, the other and opposition. The Pythagoreans set the unlimited against the one, Plato in his oral teaching turned it into the “indefinite dyad”, from which together with the One everything arises. It is the first even number and therefore stood for the feminine (see The masculine and the feminine). Two points determine a line.

Three and four

For Aristotle the triad is the first number that denotes the whole, for it has beginning, middle and end. “Of two things we say both, not all,” he remarks, only of three do we say “all”. With three points the first surface arises, the triangle, and so three became the number of completeness, in religion as in Freemasonry (see Polygons and star polygons in Freemasonry).

The tetrad is the first square number, and it has a property no other number has: 2 + 2 = 2 × 2. What it gives it receives back, and so for the Pythagoreans it was the image of justice, which repays like with like. Four points enclose the first solid, the tetrahedron. And with four the tetractys is complete, 1 + 2 + 3 + 4 = 10 (see From point to solid).

Five and six

The pentad unites the first even and the first odd number, 2 + 3, and was therefore called marriage. Plutarch explains it thus: the even is female, the odd male, and five arises from their union. Its figure is the pentagon with the pentagram inside it, according to late reports the sign of recognition of the Pythagoreans, who called it “health” (see The pentagram and the golden ratio).

The hexad is the first perfect number: it equals the sum of its divisors, 1 + 2 + 3 = 6. At the same time it is their product, 1 × 2 × 3 = 6. Philo writes that God created the world in six days, not because he needed time but because order needed a number and six was the most perfect. At the end of the ninth book of the Elements Euclid showed how even perfect numbers are found, and two thousand years later Euler proved that there are no other even ones. Whether odd ones exist is still open today. The next ones are 28, 496 and 8128.

Seven

No number occupied the ancients like the heptad. Seven planets moved across the sky, the seven days of the week were named after them, and the moon changes its shape roughly every seven days. The Pythagoreans named it after the goddess Athena, who was born without a mother from the head of Zeus and had no children herself. The reason is arithmetical: among the numbers up to ten seven is the only one that neither arises as a product of others nor brings forth another by doubling or multiplying. Two produces four, three nine and five ten. Seven produces nothing. It is motherless and virgin.

But there is a visible seven too: around a circle fit exactly six circles of the same size, and together there are seven (see Circle, triangle and square). For Philo seven is the day of rest on which creation is completed.

Eight and nine

The ogdoad is the first cube, 2 × 2 × 2. The cube has 12 edges, 8 corners and 6 faces, and 12 : 8 : 6 is a harmonic proportion, the same numbers that give octave, fifth and fourth on the string. That is why followers of Philolaus called the cube “geometric harmony” (see Number and sound). For the Gnostics the ogdoad is the first stage of the pleroma (see The Gnostics and the thirty aeons).

The ennead is the square of three and the last number before ten. It was called “horizon” because it stands at the boundary beyond which counting begins again, and “Oceanus”, after the stream that in the old picture flows around the earth.

Ten

The decad closes the circle. With it counting returns to unity, and in most languages a new round begins after it, because we count on ten fingers. For the Pythagoreans it was the perfect number par excellence, for it is the sum 1 + 2 + 3 + 4 and so contains all the steps from point to solid. It was called cosmos, heaven and the all, and in the Pythagorean oath it appears as the tetractys: they swore by him who “handed down the tetractys, which contains the source and root of ever-flowing nature” (see The ancient sources).

What of this is mathematics

Whoever reads the old books of numbers finds two things in them, and it pays to separate them (see Chance and pattern).

The left column is certain, anyone can check it. The right one is interpretation. It is not arbitrary, for it ties on to the left, but it does not follow from it. That 2 + 2 = 2 × 2 does not make four just. It makes it a beautiful image of justice, and the best of the old authors claimed no more than that.

How it continued

The Pythagorean books of numbers were continued in the Middle Ages, by Augustine, by Boethius and by the scholars of the school of Chartres (see Measure, number and weight). In the Kabbalah the ten sefirot can be read in the same pattern, though that is an interpretation (see The Kabbalah). And in 1591 Giordano Bruno published a whole book, De monade, numero et figura, in which he presented the numbers from one to ten as figures in a circle (see Giordano Bruno).

Exercise

Take ten coins or pebbles and lay out each number from one to ten in as many ways as possible, as a row, as a rectangle, as a triangle or as a circle. Note which numbers can be laid as a rectangle and which cannot, which as a triangle and which as a square. You will find the primes, the triangular numbers and the square numbers without calculating once. That is how the Pythagoreans worked.

Sources and further reading

Sources

• Nicomachus of Gerasa, Introduction to Arithmetic, around AD 100
• Theologumena arithmeticae (The theology of arithmetic), attributed to Iamblichus, 4th century AD, with excerpts from Nicomachus, Anatolius and Speusippus
• Philo of Alexandria, On the Creation of the World 13 to 14 and 89 to 128
• Aristotle, Metaphysics A 5, 985b to 986a, and On the Heavens I 1, 268a
• Plutarch, On the E at Delphi 8 and On Isis and Osiris 56
• Euclid, Elements VII, definitions 1 and 2, and IX 36
• Giordano Bruno, De monade, numero et figura, Frankfurt 1591

Further reading

• Robin Waterfield (trans.), The Theology of Arithmetic, Grand Rapids 1988
• Walter Burkert, Lore and Science in Ancient Pythagoreanism, Cambridge, Mass. 1972