Vectors in Prehistory 2

In early education of applied mathematics, there was a simple introduction to vector addition: It was observed that a distance and direction travelled followed by another (different) distance and direction, shown as a diagram as if on a map, as directly connected, revealed a different distance “as the crow would fly” and the direction from the start.

The question could then be posed as “How far would the plane (or ship) be, from the start, at the end”. This practical addition applies to any continuous medium, yet the reason why took centuries to fully understand using algebraic math, but the presence of vectors within megalithic counted structures did not require knowledge of why vectors within geometries like the right triangle, were able to apply vectors to their astronomical counts.

Firstly, the creation of geometrical frameworks or grids, at right angles, were evidently helpful in establishing square, rectangular and triangular geometries, calibrated by exact measures. At Le Manio for example, Dan Palmateer and used an 82 inch grid oriented to the cardinal directions, over the 2010 survey.

Almost all major points of the site were placed using this grid, except some belonging to cosmic measurements using time counts. Similarly work has shown rectangular grids and structures, once established, can aid the layout of the flattenned and egg-shaped circle designs later used in Britain, while the use of grids in wall-art and temple plans of Egypt is well attested.

Secondly, the addition of vectors of travel as a composite sum was intuitively available using right angled triangles millennia before the analytical algebraic methods of recent centuries. One can see the difference between the analytical and the geometrical if a famous equation is presented graphically. A five square rectangle will have a diagonal at angle arctan(1/5) to its base, an angle of 11.3099 degrees, and multiplied four times gives 45.2397 degrees. One eighth of a 360 degrees is 45 degrees, and the excess of 0.2397 degrees is arctan(1/239) so that “4 arctan (1/5) – arctan (1/239) = 45 degrees” (or pi/4 radians).

The difference between 45 degrees (the diagonal of a square) and four 5-squares when stacked from a common origin.

Thus it is that practical geometrical working can discover the proximity of four 5-squares to 45 degrees, and see the exact excess using a square grid framework, something only strayed across by early theorists of analytical geometry studying the “imaginary” square root of minus one or i. This issue can be seen as inherent in the fact that two 5-squares, when stacked equal the internal vertex angle of a {5 12 13} and the Quadrilateral at Le Manio and Station Stone Rectangle at Stonehenge incorporates this fully rational triangle as below.

The {3 4 5} geometry to the solstitial sun can itself be arrived at by stacking two 3-squares and chapter two of Sacred Number and the Lords of Time indicates that early rational rectangles (and their pairs of inner triangles) are often rational also in angle, allowing many stacking possibilities, suggesting that grids in rotation, as with the later Egyptian Canevas, was able to find and exploit interesting relationships that, at Carnac in Brittany and in other places, corresponded with horizon events at different latitudes upon the Earth.

Therefore, when one compares the visual geometrical approach to the equivalent algebraic equations required to abstract how geometry works, one can see that the West had turned away from using geometry to do things, in order to understand the mathematical reasons why such geometries worked as they do. This developed our modern incapacity to comprehend megalithic geometry, as a valid route to studying the patterns of time latent in the planetary motions seen in the sky on Earth.

The only antidote against the modern chauvinism against these stone monuments is to recognise the counting of time, alongside the powers of geometrical forms, provided a valid path to a different type of astronomy to our own, this based upon the average motions of celestial bodies and the numerical coincidences between these. It was this form of astronomy that led to their being sacred numbers at all!

The next post will look at how the sun’s path created time for the megalithic astronomers. One often hears that time began at a significant megalithic site when in fact these sites were the beginning of our understanding of time, using the megalithic methods of time counting, factorization of numbers, and the properties of geometrical forms.