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Megalithic Mathematics 1

The Origins of Mathematics.

4.1.1  Mathematical development in most early societies did not develop much beyond basic arithmetic. In Egypt, Sumeria and the Mayan Empire numeric investigation remained as simple computations specifically designed to support trade, building and ritual or political chronologies. In these societies geometric enquiry went little further than the utilitarian figures used in the construction industries for the measurement of lengths, angles, volumes, and weights. None of these civilisation made great use of arches or wheels nor developed accurate cartography or open ocean navigation.

4.1.2  Modern science and technology is the direct flowering of Pythagorean /Platonic/Euclidean mathematical enquiry unique, it is said, to the Ionian Greek peoples in the Eastern Mediterranean of the 6th century BC. The early Greek philosophers are considered to be the first to hold numbers and shapes as abstract concepts divorced from utilitarian needs.

The Integers. 4.1.3  The original departure taken by the classical Greeks was sparked by the realisation that fundamental processes of nature appeared to be governed by series of whole numbers, ie. integers. It is believed that the great mathematical philosopher Pythagoras of Samos first formulated this concept when he discovered that a harp string will produce notes which harmonise, to the Western ear, with it's ground note if the string is shortened by lengths which are simple fractions of the overall length. 4.1.4   A vibrating string plays the ground note - the fundamental. When the node is fixed half way along, the string plays a note an octave higher. If the node is moved one third of the way along, the string plays one fifth above that. If the still point on the string, the node, does not come at one of these exact points, the sound will be discordant. 4.1.5  Extrapolating from these observations the Greek philosophers surmised that all of nature obeys similar simple numeric imperatives. They applied the same investigations to astronomy, 2D and 3D geometry believing that all the universe was in musical harmony as it was governed by whole numbers, ie. the integers.

Dr. J. Bronowski, The Ascent of Man p.156

4.1.6

'Perhaps Pythagoras was a kind of magician to his followers, because he taught them that nature is commanded by numbers. There is a harmony in nature, he said, a unity in her variety, and it has a language: numbers are the language of nature.'

The Music of the Crystal Spheres. 4.1.7  The ancients thought the Sun and all the planets followed circular orbits revolving around the Earth and that the orbit of the Sun came between Venus and Mars. To explain what held the planets in place they theorised that each one was embedded in it's own sphere, or shell, of pure transparent crystal and that it was the crystal spheres, carrying the planets, which revolved around the Earth.

Integral orbits. 4.1.8  The diameters of these crystal spheres obeyed the simple numerics of musical intervals. Hence the 'Music of the Spheres' was thought to describe the size and scale of the solar system.

Manly Palmer Hall, Hermetic Philosophy 'The Intervals and Harmonies of the Spheres.' 'In the Pythagorean concept of the music of the spheres, the interval between the earth and the sphere of the fixed stars was considered to be a diapason- the most perfect harmonic interval. The following arrangement is most generally accepted for the musical intervals of the planets between the earth and the sphere of the fixed stars:-' 'From the sphere of the earth to the sphere of the moon, one tone; from the sphere of the moon to that of Mercury, one- half tone; from Mercury to Venus, one- half tone; from Venus to the sun, one and one- half tone; from the sun to Mars, one tone; from Mars to Jupiter, one- half tone; from Jupiter to Saturn, one- half tone; from Saturn to the fixed stars, one- half tone, The sum of these intervals equals the six whole tones of the octave.'

These ideas persisted well into the 16th century AD and were contradicted finally by the Polish astronomer Nicholas Copernicus who published in 1543 AD his assertion that all the planets, including the Earth, moved around the Sun.

Non-integers - the incommensurables. 4.1.9  pi. However, the Greek philosophers must soon have noticed anomalies in their world view. One such in mathematics is the constant pi, the ratio of the perimeter of a perfect circle to it's diameter, which equals about 3 1/7 or more accurately 3.1415927... etc. to an infinite number of decimal places. No matter how far this calculation is taken there is no final solution. With digital calculators the value of pi  has been taken to 6 million decimal places without finding an end. pi  is an 'incommensurable' number.

Retrograde planetary movement. 4.1.10  In astronomy the retrograde motions of Mercury, Venus, Mars, Jupiter and Saturn also must have given the ancient Greek philosophers pause for thought as these movements are difficult to explain with the Earth-centred 'Music of the Spheres' theory. Retrograde planetary motion caught by time-lapse photography. This motion is only apparent. The planets do not reverse direction in their orbits but only appear to do so for a brief spell when the Earth catches them up and swings around the ends of its orbit.

4.1.11  Heretical numbers. Pythagoras had little patience with incommensurables nor with anyone who wasted time on them. It is alleged he had a young disciple put to death for daring to investigate the square root of 1. Which number, when multiplied by itself makes 1? This was the question examined by Pythagoras's student. There is no possible number in classical mathematics which satisfies this question. It is a detested incommensurable, abhorred and to be disregarded by any committed Pythagorean. 4.1.12  The Platonic Ideal. Plato postulated that imperfections in the ordering of nature were the result of materialism as this world was 'but a flawed reflection of the Ideal plane of spirit'. Plato also decreed that nothing can be proven absolutely other than by Euclidean geometric design and then only if the drawing instruments used to create this graphic proof are limited to compasses and a straight edge. According to this view the Ideal can only be examined or illuminated in the language of the properties of straight lines, perfect circles and triangles. 4.1.13  Britain circa 2500 -1600 BC. In Britain some 1500 or more years before the times of Pythagoras and Plato there appears to have flourished a body of mathematical enquiry dedicated to creating numerous Euclidean designs by simple peg-and-rope methods marked by standing stone rings. According to the analysis of these designs Alexander Thom believes that a set of strict numeric and geometric constraints may be discerned, which would have been recognisable to early Greek philosopher/mathematicians if they had lived then. These mathematical constraints or rules clearly indicate that the megalithic ring builders had an evolved grasp of numeric and geometric abstraction which gave them access to a large variety of stone ring construction designs. Many of these designs, when deciphered, exhibit very fine standards of surveying, engineering and mathematical acuity with no apparent utilitarean functions. This remarkable flowering of fundamental mathematics, however, appears to have ceased about 1600 B.C.

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