Text: The natural series of numbers (that is 1,2,3,4,5,6,7?) is part of the family of numerical progressions called "Arithmetic" They are created by the continual addition of the same number-unit, which is determined by the difference between the first and second terms. Arithmetic proportion, which is established from any three terms taken from an arithmetic progression, must have a mean term that is always halfway between the two extremes, what we call an average. (2,3,4) 3 being the middle term and the average. In the Arithmetic progression addition is the means of growth, whereas in the progression called geometric (2,4,8,16) each successive term unfolds by being multiplied by the first term. (imagine a line on paper, has two terminal points, stretch it out into a square, it has 4, stretch it out again into a cube it has 8, if in your mind you could make a 4th dimensional hypercube then it would have 16 - this is geometry) A third, closely related family of progressions is formed by establishing a mean term that results from the multiplication of any two extremes, followed by the division of this product by their average or arithmetic mean. Eg. 2 multiplied by 6 = 12 divided by 4 = 3, which gives us the progression 2,3,6. This type of progression, which combines both the additive and multiplicative growth procedures of the other two progressions is called Harmonic, and its proportion has the characteristic that the mean term always exceeds the smaller extreme, and is less than the larger extreme, by the same fractional proportion. Eg. In the series 2,3,6? three exceeds two by one half of two (i.e. one), and is less than six by one half of six (i.e. 3) Buth the wierdest thing about this form of harmonic, proportional progression is the fact that the inverse of any harmonic progression is an arithmetic progression. So 2,3,4,5?. Is an ascending arithmetic progression, while the inverse (1/2, 1/3, 1/4, 1/5?) is a descending harmonic progression. In music it is the insertion of the harmonic and arithmetic means between the two extremes in double ratios - such as 6 and 12 ? representing the octave double, which gives us the progression known as the musical proportion: that is 6,8,9,12. In other words, the arithmetic and harmonic means between the double geometric ratios are the numerical ratios which correspond to the tonal intervals of the major fourth and major fifth, the basic consonances in nearly all musical scales. These two parallel yet inversing progressions not only provide the foundation of music, but more generally provide a mathematical model with which to investigate the complementary or opposed symmetries of a dualised, yet harmonically integrated, whole. For this reason the musical metaphor was the cornerstone of ancient philosophy, applicable to both physical and meta physical domains. You can get this from Timaeus, although its pretty hidden, I recommend a book I think you will have trouble putting down, called Rediscovering Sacred Science, it has chapters by Keith Critchlow, Robert Lawlor, Anne Macaulay, Kathleen Raine and Arthur G. Zajonc (Floris Books) ISBN 0-86315-197-3. Do a search under Robert Lawlor and sacred geometry - its all there and very interesting.

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