Geometry 5: Easy application of numerical ratios

above: Le Manio Quadrilateral

This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.

The last lesson showed how right triangles are at home within circles, having a diameter equal to their longest side whereupon their right angle sits upon the circumference. The two shorter sides sit upon either end of the diameter (Fig. 1a). Another approach (Fig. 1b) is to make the next longest side a radius, so creating a smaller circle in which some of the longest side is outside the circle. This arrangement forces the third side to be tangent to the radius of the new circle because of the right angle between the shorter sides. The scale of the circle is obviously larger in the second case.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

A triangle within a radial circle gives us the opportunity to easily make ratios out of the different lengths of their  longer sides. To illustrate, in Fig. 2 the hypotenuse has a length of 32 units and the radius has a shorter length of 29 units.

The second side of the triangle becomes a tangent when the third side becomes a radius (Fig. 1).  by this different usage but a new opportunity arises: ratios can easily be constructed as different lengths. To illustrate, the hypotenuse can be seen as a length of 32 squares and the radius as a shorter grid of 29 units (Fig. 2).

Figure 2 Calibrating the two longest sides to an integer ratio, leaving the third side unmeasured.

Each unit on the new radius, when projected at a tangent to it, divides the hypotenuse into units expanded by 32/29, illustrated (Fig. 2) by the full radius of 29 having a tangent exactly striking the 32nd unit, lying 3 units beyond the circle. Perpendiculars from the radius always follow the ratio of the hypotenuse over the radius (Fig. 3).

From the highlighted blocks of figure 3 we see these units on the hypotenuse larger than those on the radius, by 32/29, a ratio very closely equal to 11/10 (1.1) per unit .

Figure 3 Units on the radius will be expanded on the hypotenuse by the difference of lengths, here by 32/29.

A fundamental characteristic of ancient metrology is that different modular measures form integer ratios to other modules, and the reason for this is now more clear: the megalithic had to measure using integers and perform transforms upon irrational fractions such as π, using ratios like this one to maintain integers. By viewing a right triangle with two integer sides as forming a tangent between a radius and the other small side, all integer ratios could be generated between such lengths.

A Case Study at Le Manio Quadrilateral

Our example of 32/29 was of particular interest to megalithic astronomy at Le Manio

since, it was seen that the 32nd stone was 945 inches from the sun gate, which began a long count of three lunar years (36 lunar months).

32:  On dividing 945 day-inches by 32, a very accurate approximation to the lunar month appears, of 945/32 or 29.53125 days. When compared to the modern figure of 29.53059 days, 29.53125 is only one minute longer.

29: The number 29 emerges when one compares the lunar month to the astronomical megalithic yard (AMY), which is 32/29 of the lunar month.

Since the lunar month is 945/32 day-inches long, then the AMY must be 945/29 inches long and this means that three lunar years, of 36 lunar months – when placed as the radius of the right triangle of Fig. 2 & 3 – must form an hypotenuse 36 AMYs in length.

Figure 4 Re-use of a three lunar year day-inch count at Le Manio Quadrilateral, to generate 36 astronomical megalithic yards.

  1. At Le Manio, the hypotenuse of the desired triangle (32 lunar months) was aligned to the east-west axis of the equinoctial sun. 
  2. A radius rope was formed (P to Q) so as to move the lunar count to the angle of the new bearing (for a 29,32 right triangle) above the east-west axis, rotating the rope about P.
  3. On this east-west bearing from the sun gate, 29 lunar months must be made to stand directly south of 32 lunar months along the new radius count, by setting the radius rope to the correct angle of 25 degrees (Fig. 4, purple 29:32 triangle).
  4. This new radial bearing then rode a few degrees above the southern kerb, both three lunar years long, and both starting at the sun gate P.
  5. The tangent to the circle’s radius will then cross the east-west axis to provide the 36 AMY† length, This was only quantified by the tangent to the circle.

†The astronomical megalithic yard (AMY = 32.585 inches) is a term introduced by Robin Heath to differentiate it from other megalithic yards (such as 32.625 inches or Thom’s 2.72 feet). Unlike these, it is a ratio, in feet or inches, by which the solar year is longer than the lunar year.

The example given at Le Manio is a very early indication that the AMY had been evolved using this new technique: in which an integer radius to a circle could place the end of the hypotenuse and the whole of the third side, beyond the circle to achieve a proportional increase in units. This technique was more flexible than later metrology, in allowing any integer ratio to be manipulated using numbers and lengths.

It is also worth mentioning that, on a philosophical level, the radius represents the intensive world of the circle (division by a denominator) whilst the hypotenuse represents the extensive world of linear measure (multiplication by a numerator). In this example, the lunar count is evidently phenomenal whilst the 32 side is noumenal or ideal; since the AMY expresses the ratio between the solar and lunar years.