Pythagoras – the natural scientist
Arithmetician, geometer, astronomer and music theorist: thinking in analogies, the Pythagorean theorem, the forbidden dodecahedron, the central fire and the legend of the smithy.
In every subject of the quadrivium, the mathematical part of the seven liberal arts, essential discoveries are credited to Pythagoras: in arithmetic, geometry, astronomy and music. To these comes the art of healing, in a sense the practical application of philosophy and natural science (see Part 7).
Pythagoras seems almost like a scientist in today’s sense when he valued foreseeing through numbers above reading the future from sacrificial animals. His reasoning, however, also shows the thinker in analogies, a way of thinking that the West has largely lost since Descartes: calculating with numbers corresponds to the heavenly numerical ratios of the gods.
The idea of analogy leads to the Greek word logos, which can also mean “ratio, proportion”. Already in antiquity the comparison of several logoi was called analogia. So two and a half thousand years ago Pythagoras gave the impulse for harmonic research. At an institute of the Vienna music academy it has traced proportions in the structures of nature that can mostly be expressed as consonant intervals.
The arithmetician
For Pythagoras numbers were not abstract quantities. They were concrete steps in an ordered series, such as a musical scale, that one can hear and experience through music not only as quantity but as a quality of soul and spirit. This shows how hard it is to separate the sciences in Pythagoras. The analogical thinker looks over the fence with every insight and searches for correspondences, a method that would do today’s universities good as well.
The principle of analogy also shows in the tetractys, the fourness of the first four numbers, whose sum is the sacred ten. It has its origin in music. On the monochord, a sound box with a single string, Pythagoras discovered that for the consonant intervals the string lengths relate as octave 1 : 2, fifth 2 : 3 and fourth 3 : 4. Three times an odd number faces an even one. The law of the tetractys was then carried over to the opposites of the world (see Part 4), and four, the first square number, became sacred.
The monochord: the bridge divides the string. At 1 : 2, 2 : 3 and 3 : 4 the octave, fifth and fourth sound.
The geometer
Geometry first brings to mind the Pythagorean theorem: in a right triangle the square on the hypotenuse equals the two squares on the other sides together, a² + b² = c². On it rests, for example, the calculation of the distance between two points in analytic geometry.
Whether the theorem goes back to Pythagoras himself cannot be proved. Plato knew it a good hundred years later, as his dialogue Meno shows. The Babylonians already knew it, though without proof. Pythagorean triples already appear on Babylonian clay tablets such as the tablet Plimpton 322, made more than a thousand years before Pythagoras.
The dodecahedron breaks the order of four
What can surely be credited to the Pythagoreans is the discovery of the pentagonal dodecahedron, the solid made of twelve equal pentagons. It caused the greatest stir, because it broke the order of the “world number” four. All the other regular solids that can be inscribed in and circumscribed about a sphere consist of triangles or squares. According to an old marginal note to Euclid, the Pythagoreans knew the tetrahedron, cube and dodecahedron, while the octahedron and icosahedron were first described by Theaetetus, a friend of Plato. The four solids of triangles and squares are:
- the tetrahedron of four equilateral triangles, a pyramid on a triangular base
- the cube (hexahedron) of six squares
- the octahedron of eight equilateral triangles, two pyramids on a square base set together
- the icosahedron of twenty equilateral triangles
That space around a point could be evenly bounded only with equilateral triangles or squares shaped the world picture of the time. In the Timaeus Plato has the Pythagorean philosopher Timaeus explain that the world is built from infinitely small triangles. The smallest surface can only be a triangle, since it has the fewest sides and division always yields triangles again. The smallest body must therefore consist of triangles, equilateral ones, as these are the most beautiful. Why this is so would take too long, says Plato, but whoever proves the opposite wins the prize. Since the sphere is the most perfect form, each of these solids must fit into a sphere.
Even for the cube Plato stresses that its squares are made of four isosceles triangles each, whose right angles meet in the middle. So the divine, creative three reigns in the square too. The four solids were assigned to the four elements, the cube to earth, because it is the most immobile element and needs the firmest bases. This refounded an ancient tradition: three is the number of the gods, four that of the earth.
Into this settled order the dodecahedron fell like a bomb. A solid that could not be built from triangles was to have the same property and fit into the sphere? That shook the foundations of the understanding of the world. Probably out of caution the Pythagoreans kept the discovery a strict secret. Of the Pythagorean who betrayed it (later named Hippasus), Iamblichus reports: because he was the first to make public in writing how to circumscribe a sphere about a dodecahedron, he perished at sea as a betrayer of the mysteries. Plato made a virtue of necessity and assigned the fifth solid to the universe.
The five Platonic solids. The dodecahedron is the only one whose faces are neither triangles nor squares.
More on the solids on the page The Platonic solids.
The astronomer
Pythagoras is said to have been the first to assume that the earth is a sphere. So at least Diogenes Laertius reports, though he also cites Theophrastus, according to whom it was Parmenides. According to this teaching the seven heavenly bodies, also spherical, circle this globe at the centre of the cosmos: the five planets then known, the sun and the moon. The whole sphere of fixed stars also turns around the earth.
This was the beginning of great progress. Soon after, Pythagoreans (Philolaus) taught that the earth, together with the seven spheres and a merely inferred counter-earth, runs in circles around a central fire, the “hearth of the universe”. The earth was thus no longer the centre of the world. What a step in the right direction, considering that from Plato on the geocentric view prevailed again. As the Ptolemaic system it became a matter of faith until well after Copernicus (1473 to 1543), as Galileo Galilei (1564 to 1642) had to learn. Around 400 BC the Pythagoreans Hicetas and Ecphantus even taught that the earth turns on its axis from west to east.
The music theorist
Pythagoras was a pioneer in exploring the laws of music. Iamblichus describes his acoustic experiments in detail, but they do not stand up to testing, perhaps because Iamblichus himself understood too little of the matter.
Charming is the legend of how Pythagoras got the idea. He passed a smithy and heard the hammers on the anvil produce sounds in harmonious intervals in turn, all but one pair. He recognised octave, fifth and fourth, hurried in and found by experiment that the pitch depended on the weight of the hammer, not on the strength of the smith, the shape of the hammer or the position of the iron. He weighed the hammers, hurried home and at once began experimenting with strings. That, Iamblichus concludes, is how he invented music and passed it on to his pupils as a helper to all that is noble. (Physically the story is wrong, since the pitch of a hammer does not simply depend on its weight. On strings, however, the ratios hold exactly.)
Pythagoras certainly did not invent music. But the discovery that musical intervals can be expressed as numerical ratios surely goes back to the Pythagoreans. Its reach becomes clear when one continues the experiment on the monochord: a series of intervals sounds that corresponds exactly to the harmonic series. These overtones, which sound along with every natural tone, were only discovered in the seventeenth century, more than two thousand years later. The musicologist Friedrich Zipp writes in essence that the Pythagoreans, with a rational-mathematical method and without knowing it, found an acoustic law of nature and made visible an ordering principle anchored in nature and in human beings alike.
Harmonic research has shown with many examples that these proportions predominate in nature and the cosmos. That we feel them as consonant is rooted in our psyche. So we are “harmoniously” embedded in the cosmic principle.
Delphi, the tetractys and the Sirens
Pythagoras probably took this insight from the oldest mystery knowledge and joined it with the secret of the tetractys. An old Pythagorean saying runs: What is the oracle of Delphi? The tetractys. It is also the harmony of the Sirens.
The dark words point to knowledge of eternal laws. Delphi literally means “womb”, here the womb of the earth from which all life comes. So the essence of number contains all the laws of heaven and earth. Music makes them sound and lets the soul resonate in divine harmony, for it is itself of divine origin. In the Sirens music is embodied in all its daemonic power, which no one can resist. From this point of view the belief in a music of the spheres is almost a necessity.
Heraclitus: the hidden harmony
His younger contemporary Heraclitus of Ephesus (c. 536 to 470 BC) extended the idea of a primal harmony philosophically. He saw in it the hidden agreement in the interplay of opposing forces and called it harmonia aphanes, the hidden harmony. Today we speak of the law of polarity: of the primal tension from which alone something new arises, and of opposites that are in truth parts of one whole. There would be no harmony without high and low, and no living beings without female and male. A whole philosophy of peace could be built on this thought.