Geometry and the Cosmos (2) : From the Pre-Socratic Universe to Aristotle’s Two Worlds

 Sequel of the previous post Geometry and the Cosmos (1): Kepler, from polyedra to ellipses 

The Pre-Socratic Universe

Since He [Zeus] himself hath fixed in heaven these signs,
The Stars dividing; and throughout the year
Stars he provides to indicate to men
The seasons’ course, that all things may duly grow.
Aratus, Phaenomena, I, 18.

Although Kepler was the first to determine the motion of the planets by mathematical laws, his search for a rational explanation to the universe was anticipated by numerous earlier thinkers. Even before the time of Socrates a number of philosophers had broken away from accepted mythology and postulated the idea of universal harmony. From the sixth century BC increasingly rational and mathematical ideologies based on the laws of physics began to compete with the traditional belief that the world was controlled by gods with supernatural powers. Most of these thinkers attempted to describe natural phenomena in mechanical terms, with reference to the elements of water, earth and fire. The Ionian philosophers in particular developed new ideas about the heavens, whose signs were used by many of their compatriots to navigate between the islands. Their fundamental notion was that the universe was governed by mechanical laws, by natural principles which could be studied, understood and predicted.

It was Thales of Miletus who propounded one of the first rational explanations of the world, according to which the earth was separate from the sky. Anaximander and Anaximenes, both also natives of Miletus on the coast of Asia Minor, put forward different ideas, which nevertheless derived from the same rationale: they proposed the existence of cosmological systems, explained natural phenomena in terms of a small number of “elements”, and invented new concepts – Anaximander’s “equilibrium” and Anaximenes’ “compression” – which can be regarded as the first recognition of the force of gravity.

The Expanding Universe. According to Empedocles of Acragas (now Agrigento, in Sicily), the universe was held in balance by forces of harmony and conflict, the attractive force of love and the repulsive force of hate alternatively prevailing. This idea of balance can be seen as a mythical precursor of modern astronomical theories whereby the tendency for structures to become compressed by their own gravitational forces is offset by the expansion of the universe, which constantly dilutes all matter.
In Lemaître’s so-called “hesitating universe”, a cosmological model he devised in 1931 from Einstein’s field equations, the evolution of the cosmos is divided into three disctinct phases : two periods of rapid expansion are separated by a period of “stagnation”, representing a sort of equilibrium between the forces of gravitational contraction and expansion.

According to Heraclitus of Ephesus, the day was caused by exhalations from the sun, while the night was the result of dark emissions from the earth. The stars and the planets were bowls of fire which, when turned over, gave rise to eclipses and the phases of the moon. The moon itself, pale and cold, moved in the rarefied air above the earth, whereas the sun, our nearest star, was bright and hot.

Meanwhile, the Greeks were amassing measurements which would enable them to plot the stars more accurately. This required specialised instruments – gnomons to measure the sun’s shadow, compasses to fix the positions of the stars in the sky, etc. – as well as a system of notation which anyone could understand (previously the study of astronomy had been restricted to priests): how many fingers’ width above the horizon was such and such a star; where was due north, and so on. As well as mining the extensive archive of observations made by the Egyptians and Babylonians, the Greeks developed their own system of records. The pre-Socratic thinkers refined and analysed the basic ideas of their predecessors from Miletus with the result that the mechanistic view of the world gradually lost currency and a belief in underlying harmony became de rigueur. As early as 450 BC Anaxagoras of Clazomenae was accused of impiety for referring to the sun as a mass of hot metal, to the moon as a second earth and to the stars as burning stones – views no longer considered seemly. Continue reading

Geometry and the Cosmos (1) : Kepler, from polyhedra to ellipses


The regularity of so much celestial activity has led many cultures to base their models of the universe on concepts of order and harmony. Around the Mediterranean it was the Pythagoreans who first expressed the idea that the universe is characterised by proportion, rhythm and numerical patterns. Plato’s hypothesis was of an organised cosmos whose laws could be deciphered, explained in geometric terms.

The history of physics is nothing other than the story of man’s desire to uncover the hidden order and harmony of things. The most ambitious physicists have attempted to unify apparently discrete phenomena: Galileo with terrestrial and celestial laws; Newton with gravity and the movement of celestial bodies; Maxwell with magnetism and electricity; Einstein with space and time; today’s physists with gravitation and microphysics.

But, as Heraclitus said as long ago as 500 BC, “Nature loves to hide.” Indeed advances in geometry and mathematics have led to new theories of the cosmos which we are unable to comprehend. They provide only abstract images, which do not allow us to visualise the structure of atoms or the dynamics of space-time or the topology of the universe in any direct sense.

It is this fundamental belief in celestial harmony – for which successive generations have found various elaborate expressions: just proportion, equation of the part and the whole, symmetry, constancy, resonance, group theory, strings -that has underlain the development of physics for the past 2,500 years.

Melancholy, or the Spirit of Man in Search of the Secret of the Universe. This etching, dating from 1514 according to the numbers in the square in the top right corner, depicts man contemplating the nature of the world in a state of melancholy, which in medieval times was associated with black bile and with the planet Saturn. The winged man prefigures Johannes Kepler’s interrogations as he calculates how to express the underlying harmony of the cosmos using spheres and polyhedra. The bright light in the sky is the great comet that was observed in the winter of 1513-14. As it shines on the scales (depicting the astronomical sign Libra) it symbolises the end of an earthly cycle, if not the end of time itself. The ladder with seven rungs represents the belief held by the Byzantines that the world would not exist for more than seven thousand years. It is the end of the Middle Ages; Diirer (1471-1528) is to be one of the prime movers of the Renaissance.


Geometry and the Cosmos

“Geometry, which before the origin of things was coeternal with the divine mind and is God himself […], supplied God with patterns for the creation of the world.”
Johannes Kepler, The Harmony of the World, 1619.

The 17th century German astronomer Johannes Kepler was undoubtedly the first to integrate man’s fascination with harmony into an overall vision of the world which can properly be called scientific. For Kepler, as for the natural philosophers of ancient Greece, the cosmos was an organised system comprising the earth and the visible stars. His avowed intention was to investigate the reasons for the number and sizes of the planets and why they moved as they did. He believed that those reasons, and consequently the secret of universal order, could be found in geometry. Kepler wanted to do more than create a simple model or describe the results of his experiments and observations; he wanted to explain the causes of what he saw. This makes him one of the greatest innovators in the history of science and it led him in particular to formulate laws of planetary motion which are still valid today.

Despite his innovative methods, Kepler wrote two studies of the cosmos in the style of the ancient Greeks: Mysterium Cosmographicum (The Secret of the Cosmos) in 1596 and Harmonices Mundi (The Harmony of the World) in 1619. At this turning point between ancient and modern thinking Kepler was steeped in a tradition which connected cosmology explicitly with the notion of divine harmony. But what Kepler sought to express was not the numerical mysticism of the Pythagoreans; his starting point was geometric patterns, which he saw as “logical elements”. His profound desire to devise a rational explanation for the cosmos led him to establish procedures which resembled those of modern science. Continue reading

The Rate of Expansion

There, where worlds seem, with slow steps,
Like an immense and well-behaved herd,
To calmly graze on the ether’s flower.
Giovanni Pascoli, Il Ciocco

A question often asked by the general public interested in cosmology about the expansion of the Universe is the distance scales on which it effectively acts. Before commenting on this, let me recall first some historical facts.

Georges Lemaître in 1927

In 1927, Georges Lemaître published a revolutionary article in the Annales de la Société scientifique de Bruxelles entitled “Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extragalactiques” (“A homogeneous universe of constant mass and increasing radius, accounting for the radial velocity of extragalactic nebulae.” As the title suggests, Lemaître showed that a relativistic cosmological model of finite volume, in which the Universe is in perpetual expansion, naturally explains the redshifts of galaxies, which at that point were not understood. In particular, the article contained a paragraph establishing that forty-two nearby galaxies, whose spectral shifts had been measured, were moving away at speeds proportional to their distances.

Lemaître gave the numerical value of this proportionality factor: 625 km/s per megaparsec, which means that two galaxies separated by 1 megaparsec (or 3,26 million light-years) moved away from each other at an apparent speed of 625 km/s, and that two galaxies separated by 10 megaparsecs moved apart at a speed ten times greater.

The paragraph of Lemaître’s paper in which he derives the law of proportionality between recession velocity and distance, later called the Hubble law.

This unit of measurement, the kilometer per second per megaparsec, shows clearly that the speed of recession depends on the scale. In 1377, in his Book of the Heavens and the World, the scholar Nicole Oresme had noted that, at dawn, one would not notice anything if the world and all living creatures had grown by the same proportion during the night. In Lemaître’s theory, on the contrary, the recession velocity between two points in space grows faster with greater separation, which renders it perceptible.

Eddington and Lemaître

Lemaître’s article, published in French, passed unnoticed until 1931, when it was finally read by Arthur Eddington, who published an English translation. Unfortunately, this version omits the paragraph in which Lemaître established his law of proportionality, see this article for all the details. Meanwhile, in 1929 the great American astronomer Edwin Hubble had published the experimental results he obtained with his collaborators and described a general law, according to which the speed of recession of a galaxy is proportional to its distance. This law, identical to Lemaître’s, with the same proportionality factor, would from now on carry the name of “Hubble’s law.” It forms the experimental basis for the theory of the expansion of the Universe, of which the big bang models are the fruit. Continue reading