The Sleepwalkers Read Online Free PDF Page A

fifth) + 6 = 12 (4 : 3, quart)

    In a similar manner, "cubic" numbers and "pyramidal" numbers were obtained. Mnesarchos had been a gem engraver, so Pythagoras in his youth must have been familiar with crystals whose form imitated those of pure number-shapes: quartz the pyramid and double-pyramid, beryl the hexagon, garnet the dodocaeder. It all went to show that Reality could be reduced to number-series and number-ratios, if only the rules of the game were known. To discover these was the chief task of the Philosophos , the Lover of Wisdom.
    An example of the magic of numbers is the famous theorem, by which alone Pythagoras is consciously remembered today – the visible peak of the submerged iceberg. * There is no obvious relationship between the lengths of the sides of a right-angled triangle; but if we build a square over each side, the areas of the two smaller squares will exactly equal the area of the larger. If such wonderfully ordered laws, hitherto hidden from the human eye, could be discovered by the contemplation of number-shapes, was it not legitimate to hope that soon all secrets of the universe would be revealed through them? Numbers were not thrown into the world at random; they arranged themselves into balanced patterns, like the shapes of crystals and the concordant intervals of the scale, according to the universal laws of harmony.
    ____________________
*
Ironically, Pythagoras seems to have had no complete proof of the Pythagorean theorem.

    3.
"Soft Stillness and the Night"
    Extended to the stars, the doctrine took the form of the "Harmony of the Spheres". The Ionian philosophers had begun to prise open the cosmic oyster, and to set the earth adrift; in Anaximander's universe the earth-disc no longer floats in water, but stands in the centre, supported by nothing and surrounded by air. In the Pythagorean universe the disc changes into a spherical ball. 3 Around it, the sun, moon and planets revolve in concentric circles, each fastened to a sphere or wheel. The swift revolution of each of these bodies causes a swish, or musical hum, in the air. Evidently each planet will hum on a different pitch, depending on the ratios of their respective orbits – just as the tone of a string depends on its length. Thus the orbits in which the planets move form a kind of huge lyre whose strings are curved into circles. It seemed equally evident that the intervals between the orbital cords must be governed by the laws of harmony. According to Pliny, 4 Pythagoras thought that the musical interval formed by earth and moon was that of a tone; moon to Mercury, a semi-tone; Mercury to Venus, a semi-tone; Venus to Sun, a minor third; Sun to Mars, a tone; Mars to Jupiter, a semi-tone; Jupiter to Saturn, a semi-tone; Saturn to the sphere of the fixed stars, a minor third. The resulting "Pythagorean Scale" is si, do, re bemol, fa, sol, la bemol, do – though the accounts of the scale given by different writers vary slightly.
    According to tradition, the Master alone had the gift of actually hearing the music of the spheres. Ordinary mortals lack this gift, either because they are from the moment of birth, unknowingly but constantly bathed in the celestial humming; or because – as Lorenzo explains to Jessica – they are too grossly constituted:
    ... soft stillness and the night
Become the touches of sweet harmony ...
Look how the floor of heaven
Is thick inlaid with patines of bright gold;
There's not the smallest orb which thou behold'st
But in his motion like an angel sings ...
Such harmony is in immortal souls;
But, whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it. 5
    The Pythagorean dream of musical harmony governing the motion of the stars never lost its mysterious impact, its power to call forth responses from the depth of the unconscious mind. It reverberates through the centuries, from Kroton to Elizabethan England; I shall quote two more versions of it – with a purpose that will become