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| Dr. J. Bronowski, The Ascent of Man p.158 |
| Not only the natural world as we experience it, but the world as we construct it is built on that relation....(the triad)...It has been so since the time that the Babylonians built the Hanging Gardens, and earlier, since the time that the Egyptians built the pyramids. These cultures already knew in a practical sense that there is a builder's set square in which the numerical relations dictate and make the right angle. The Babylonians knew many, perhaps hundreds of formulae for this by 2000 BC. The Indians and the Egyptians knew some. The Egyptians, it seems, almost always used a set square with the sides of the triangle made of three, four, and five units. It was not until 550 BC or thereabouts that Pythagoras raised this knowledge out of the world of empirical fact into the world of what we should now call proof. |
| 3.5.2 Alexander Thom has identified the use of Pythagorean special triangles as base elements in the construction of two classes of megalithic ring design; the ellipses and egg rings. |
| A.Thom, Megalithic Sites in Britain p.27. |
| The Pythagorean Special Triangles.
The basic figure of their geometry, as of ours, is the triangle. Today every- one knows the Pythagorean theorem which states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. We do not know if Megalithic man knew the theorem. Perhaps not, but he was feeling his way towards it. One can almost say that he was obsessed by the desire to discover and record in stone as many triangles as possible which were right-angled and yet had all three sides integers. The most famous of the so-called Pythagorean triangles is the 3, 4, 5-right-angled because (3x3)+(4x4) = (5x5) He used this triangle so often that he may well have noticed the relation. Limiting the hypotenuse to 40 there are six true Pythagorean triangles. These are: |
(2) 5, 12, 13 (3) 8, 15, 17 (4) 7, 24, 25 (5) 20, 21, 29 (6) 12, 35, 37 |
| Megalithic man knew at least three of these. He may have known all six and we simply have not yet found the sites where they were used, but we shall see later that there were other conditions to be fulfilled and these certainly restricted the use of some of these triangles. The remarkable thing is that the largest, the 12, 35, 37, was known and exploited more than any other with the exception of the 3, 4, 5.
3.5.3  |
| A.Thom, Megalithic Sites in Britain p.29. |
| A.Thom Megalithic Sites in Britain, p31. | |
| The Ellipses
A circle has a constant radius but an ellipse has the average of the two lengths. When Megalithic man set out a circle with a diameter of 8 units he found the circumference very nearly 25 units but in general he could not get nice whole numbers like these for both the diameter and the circumference simultaneously. Probably the attraction of the ellipse, and we know of over 30 set out by these people, was that it had an extra variable (F1 F2) and so it was easier to get the circumference near to some desired value. But the ellipse has, as it were, two diameters, the major and minor axes. How is it possible to get both of these and at the same time the focal distance F1 F2 all integral? Looking at Fig. 4.5 we see that a, b, and c are the sides of a right-angled triangle and if the triangle is Pythagorean we can have the major axis, the minor axis, and the focal distance all integral. Just as for the egg-shaped rings so for the ellipses it was desirable to start with a Pythagorean triangle. For both eggs and ellipses Megalithic man had a further very difficult task, namely, to get the perimeter integral. |
| There are four construction lengths to an ellipse; the major axis, the minor axis, the focal length (distance between centres) and the perimeter. According to Thom when we examine the plan of an elliptical stone ring integrality in these measurements should be sought for.
Further there should be evidence for the employment of one of the six special Pythagorean triangles at the base of the design. Following are digital examinations made of two of Thom's stone ring surveys. Both the major and minor diameters and distance between foci are measured to a precision of better than 1 inch, (25 mm). The nearest best- fit Pythagorean special triangle was then deduced and layered in red onto the survey. The coherence with nominal theory is easily apparent. |
3.5.4 Postbridge and Sands of Forvie.
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Postbridge stone ring. Digital measurements of the four construction lengths in the Postbridge design have been made. They give good indications of the intent to establish nominal integrality in all four dimensions;- minor axis = 9.996 MY, (nominal = 10). focal length = 3.07 MY, (nominal) = 3. and perimeter = 12.89 MR. (nominal) = 13. On the ground the difference of the actual lengths from the nominal would scarcely be noticable. Two mirrored Pythagorean triangles lie at the base of this construction set. The triangle chosen is no. 4 in the series, the 7/24/25/, superimposed in red. The integer used for this special triangle sides is 3/7 MY = 17 MI, (Megalithic Inch), and, as we can see, it is a perfect fit. |
| Sands of Forvie stone ring.
The digital survey here shows that again the four construction lengths are close to nominal integrality as;- maj.= 16.433 MY = 16.5. min.= 15.63 MY = 15.5. focal= 6.01 MY = 6. perimeter.= 50.37 MY = 20.15 MR = 20. Shown in red, the special Pythagorean triangle no. 2 in the series, the 5/12/13, makes a very close match with the minor radius and semi focal length. Here the integer chosen for the triangle side lengths is 3/5 MY. |
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