Sacred Number

When I first researched Earth-based astronomy it was based upon the properties of right angled triangles, first of all related just to the periodicity of the Sun and Moon seen from Earth. These properties make right angled triangles, proportional calculators of a remarkable and simple kind. Let me explain since this naturally extends the capability of ancient peoples to perform complex mathematical tasks once they had mastered counting within astronomical cycles. It also led on naturally to the use of triangulation and the measurement of angles in such a way that advanced results could be achieved within a non-scientific (in our sense) culture. 

 Let me show a simple but very sophisticated example. First of all, suppose we have an English foot and want to make the Persian Foot. This can be done using a right angled triangle as per:

Our first requirement is to make a right angle using the simplest pythagorean triangle of 3 by 4 by 5 feet long, the perimeter being 12 feet or, conceivably, 13 knots in a rope with three stakes. More difficult then is to measure a base of 20 feet and an hypotenuse of 21 feet that touches a vertical formed by the extension of the 4 side of the 3:4:5 triangle.

Each of the 20 feet on the base now projects similarly onto the hypotenuse to define a length equal to one root Persian foot of 21/20 feet.  In the same way, assuming you can do it, all the other ancient measures in the ancient metrologists toolbag can be reconstituted from a single standard.

 Given that the ancient builders were known to work to one part in 1000 or better, and that measures were consistent, then as Neal and Michell propose their technology for maintaining measures must have been as accurate. However, Neal and Michell do not suggest how these root measures might then have been varied according to two ratios, 441/440 and 176/175. Instead of explaining these again here, the main point for us is that these are very small differences, 441/440 being just 1.002273 feet in the English module. How could this be achieved?

It so happens (see Sacred Number) that 441/440 can be decomposed, and one way is into 21/20 times 21/22. Since we have the Persian foot of 21/20 we can now repeat the process only making the triangle in Persian not English feet and forming the base out of 21 of these and making the hypotenuse 22 Persian feet long.

Once you decide you need a lot of one kind of foot, you have the option of generating over a much larger distance the exact length you want using triangulation over the landscape.  I will provide the complementary triangle that is one way to geometrically generate 176/175. Lying behind it all is a practical skill and facility with numbers that is only now being revealed by trying to explain the extraordinary capabilities of prehistoric cultures.

 Here the root Roman foot can be employed to fulfill the decomposition of the ratio 176/175 as 24/25 times 22/21. The Roman foot is  going to be enlarged by 22/21 by reading from the base to the hypotenuse.