The ratios of ancient metrologyThe application of units of length to problems of measurement, design, comparison or calculation. emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.
The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin HeathEngineer, teacher and author, who discovered the Lunation Triangle (c. 1990), that enabled the lunar year to be rationally related to the solar year. During the 1990s we collaborated to further understand the astronomical and numerical discoveries of the megalithic astronomers.) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.
Such automatism of geometry can also be found in the square which is more popular today, in which the ends of four equal intervals can have their opposite corners forced to be an equal distance apart. These diagonal intervals are the square root of 2 of the side length, long. Unexpectedly, if the astronomical megalithic yardAny unit of length 2.7-2.73 feet long, after Alexander Thom discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. (AMYA megalithic yard which, in inches, expresses the true astronomical ratio of mean solar months to lunar months.) were used to form a square with that side length, the diagonals would exactly equal 96/25 feet – a length we would call 4 root Roman feet of 24/25 feet. The exactitude of this result is directly due to the prime number factors present in the AMY (19.008/7 feet) and Roman foot ratio.
That is, an accurate approximation to the square root of 2 of 140/99 is naturally generated by the AMY when used to express a square of that side length. It seems unlikely megalithic astronomers would not have discovered this fact, whilst looking at ways to generate a right angle, not with another triangle but using a square.
This relates to the previous post, where the “old yard” of 2.97 feet, and its foot of 99/100 feet, gives a diagonal of 7/5 – an inaccurate approximation to root 2. Through reciprocal thinking (which we now use within our equations) 100/99 times 7/5 equal 140/99, arriving at the very same ratio, but then to the English footThe standard prehistoric foot (of 12 inches) representing a unity from which all other foot measures came to be formed, as rational fractions of the foot, a fact hidden within our historical metrology [Neal, 2000]. rather than to the AMY.
As megalithic metrology was rationalized, to become ancient metrology (based upon the English foot and ratios to it,) the original form of this megalithic discovery became a type of foot (100/99) to express it in ancient metrology.
Meanwhile, it seems plausible that when time counting moved from day-inches to lunar months, as megalithic yards, that those megalithic yards were available for work, forming a square at the end of a shorter counting rope. Indeed the lunation triangleThe right-angled triangle within which the lengths of the two longer sides are the relative proportions of the solar and lunar years. is 12 months with a third side of 3. This leads to the use of four squares to form the longest side automatically as the diagonal. Each of those four squares have side length 3 AMY and diagonals 12 Roman feet long (288/25 = 11.25 feet), which is also 10 feet of 9/8 English feet.
That is, once there is a side length equal to 3 AMY, the diagonal is 12 Roman feet of 24/25 feet. It would seem that time, geometry and measures express coincidences which provided simple solutions, enabling the megalithic to easily build their astronomical constructions. Such problems would otherwise have necessitated an arithmetic only developed later, in the historical world of the ancient near east.