around a person protects him against hostile spirits and perils of the soul; it marks off a place as an inviolable sanctuary; it was commonly used in tracing out the sulcus primigenius , the first furrow, when founding a new city. Apart from being a symbol of stability and protection, the circle, or wheel, had a technological plausibility, as it were, as a suitable element for any machine. But on the other hand, the planetary orbits were evidently not circles; they were eccentric, bulging, oval – of egg-shaped. They could be made to appear as the product of a combination of circles by geometrical artifices, but only at the price of renouncing any semblance of physical reality. There exist some fragmentary remains, dating from the first century A.D., of a small-sized Greek planetarium – a mechanical model designed to reproduce the motions of sun, moon, and perhaps also of the planets. But its wheels, or at least some of them, are not circular – they are egg-shaped. 21 A glance at the orbit of Mercury in the Ptolemaic system on p. 68 shows a similar eggshaped curve staring into one's face. All these pointers were ignored, relegated into limbo as a sacrifice to circle-worship.
And yet there was nothing a priori frightening about oval or elliptic curves. They too were "closed" curves, returning into themselves, and displayed a reassuring symmetry and mathematical harmony. By an ironical coincidence, we owe the first exhaustive study of the geometrical properties of the ellipse to the same man, Apollonius of Perga, who, never realizing that he had the solution in his hands, started the development of the epicyclic monster-universe. We shall see that, two thousand years later, Johannes Kepler, who cured astronomy of the circular obsession, still hesitated to adopt elliptical orbits, because, he wrote, if the answer were as simple as that, "then the problem would already have been solved by Archimedes and Apollonius". 22
6.
The Cubist Universe
Before bidding farewell to the Greek world, an imaginary parallel may help to bring matters into focus.
In 1907, simultaneously with the Cézanne memorial exhibition in Paris, a collection of the master's letters was published. A passage in one of the letters ran:
"Everything in nature is modelled on the sphere, the cone and the cylinder. One must teach oneself to base one's painting on these simple figures – then one can accomplish anything one likes."
And
further:
"One must treat nature by reducing its forms to cylinder, sphere, and cone, all put into perspective, meaning that each side of an object, each plane, is directed towards a central plane." 23
This pronouncement became the gospel of a school of painting known under the misnomer "Cubism". Picasso's first "Cubist" picture was in fact constructed entirely of cylinders, cones and circles; while other members of the movement saw nature in terms of angular bodies – pyramids, and bricks, and octaeders. *
____________________
*
The name of the movement derives from a slighting remark by Matisse, who said of a landscape by Braque that it was "entirely constructed in little cubes". 24
But whether they painted in terms of cubes, cylinders, or cones, the declared aim of the Cubists was to resolve every object to a configuration of regular geometrical solids. Now the human face is not constructed out of regular solids any more than the orbits of the planets are made of regular circles; but in both cases it is possible to "save the phenomena": in Picasso Femme au Miroir , the reduction of the model's eyes and upper lip to an interplay of spheres, pyramids and parallelepipedes, displays the same ingenuity and inspired madness as Eudoxus' spheres pivoting within spheres.
It is rather depressing to imagine what would have happened to painting if Cezanne's Cubist pronouncement had been turned into a dogma, as Plato's spherist pronouncement was. Picasso would have been condemned to go on painting more elaborate cylindrical bowls to
And yet there was nothing a priori frightening about oval or elliptic curves. They too were "closed" curves, returning into themselves, and displayed a reassuring symmetry and mathematical harmony. By an ironical coincidence, we owe the first exhaustive study of the geometrical properties of the ellipse to the same man, Apollonius of Perga, who, never realizing that he had the solution in his hands, started the development of the epicyclic monster-universe. We shall see that, two thousand years later, Johannes Kepler, who cured astronomy of the circular obsession, still hesitated to adopt elliptical orbits, because, he wrote, if the answer were as simple as that, "then the problem would already have been solved by Archimedes and Apollonius". 22
6.
The Cubist Universe
Before bidding farewell to the Greek world, an imaginary parallel may help to bring matters into focus.
In 1907, simultaneously with the Cézanne memorial exhibition in Paris, a collection of the master's letters was published. A passage in one of the letters ran:
"Everything in nature is modelled on the sphere, the cone and the cylinder. One must teach oneself to base one's painting on these simple figures – then one can accomplish anything one likes."
And
further:
"One must treat nature by reducing its forms to cylinder, sphere, and cone, all put into perspective, meaning that each side of an object, each plane, is directed towards a central plane." 23
This pronouncement became the gospel of a school of painting known under the misnomer "Cubism". Picasso's first "Cubist" picture was in fact constructed entirely of cylinders, cones and circles; while other members of the movement saw nature in terms of angular bodies – pyramids, and bricks, and octaeders. *
____________________
*
The name of the movement derives from a slighting remark by Matisse, who said of a landscape by Braque that it was "entirely constructed in little cubes". 24
But whether they painted in terms of cubes, cylinders, or cones, the declared aim of the Cubists was to resolve every object to a configuration of regular geometrical solids. Now the human face is not constructed out of regular solids any more than the orbits of the planets are made of regular circles; but in both cases it is possible to "save the phenomena": in Picasso Femme au Miroir , the reduction of the model's eyes and upper lip to an interplay of spheres, pyramids and parallelepipedes, displays the same ingenuity and inspired madness as Eudoxus' spheres pivoting within spheres.
It is rather depressing to imagine what would have happened to painting if Cezanne's Cubist pronouncement had been turned into a dogma, as Plato's spherist pronouncement was. Picasso would have been condemned to go on painting more elaborate cylindrical bowls to
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