Tag: lessons

Astronomy 1: Knowing North and the Circumpolar Sky

about how the cardinal directions of north, south, east and west were determined, from Sacred Number and the Lords of Time, chapter 4, pages 84-86.

Away from the tropics there is always a circle of the sky whose circumpolar stars never set and that can be used for observational astronomy. As latitude increases the pole gets higher in the north and the disk of the circumpolar region, set at the angular height of the pole, ascends so as to dominate the northern sky at night.

Northern circumpolar stars appearing to revolve around the north celestial pole. Note that Polaris, the bright star near the center, remains almost stationary in the sky. The north pole star is constantly above the horizon throughout the year, viewed from the Northern Hemisphere. (The graphic shows how the apparent positions of the stars move over a 24-hour period, but in practice, they are invisible in daylight, in which sunlight outshines them.)
[courtesy Wikipedia on “circumpolar star”, animation by user:Mjchael CC-ASA2.5]

Therefore, the angular height of the pole at any latitude is the same angle we use to define that latitude, and this equals the half angle between the outer circumpolar stars and the pole itself. For example, Carnac has a latitude of 47.5 degrees north so that the pole will be raised by 47.5 degrees above a flat horizon, while the circumpolar region will then be 95 degrees in angular extent.

It is perhaps no accident that the pole is called a pole since to visualize the polar axis one can imagine a physical pole with a star on top, like a toy angel’s wand. The circumpolar region is “suspended” around the pole like a plate “held up” by the pole. Therefore, a physical pole, set into the ground, can be used to view the north pole from a suitable distance south (i.e., with the pole’s top as a foresight for the observer’s backsight). Such an observing pole would probably have been set at the center of a circle drawn on the ground, representing the circumpolar region around the north pole. This arrangement, a gnomon,* existed throughout history but usually presented as part of a sun dial.

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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.

My thanks to Dan Palmateer of Nova Scotia
for his graphics and dialogue for this series.

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. diagram of Dan Palmateer.

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

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Geometry 4: Right Triangles within Circles

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.

My thanks to Dan Palmateer of Nova Scotia
for his graphics and dialogue for this series.

This lesson is a necessary prequel to the next lesson.

It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally defined relative to the ratios of a right triangle’s sides, conform to the properties of a circle.

A triangle with sides {3 4 5} demonstrates the general fact that, when a right triangle’s hypotenuse is the diameter of a circle, the right angle touches the circumference.
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Geometry 3: Making a circle from a counted length

The number of days in four years is a whole number of 1461 days if one approximates the solar year to 365¼ days. This number is found across the Le Manio Quadrilateral (point N to J) using a small counting unit, the “day-inch”, exactly the same length as the present day inch. It is an important reuse of a four-year count to be able to draw a circle of 1461 days so that this period of four years can become a ouroboros snake that eats its own tale because then, counting can be continuous beyond 1461 days. This number also permits the solar year to be counted in quarter days; modelling the sun’s motion within the Zodiac by shifting a sun marker four inches every day.

Figure 1 How a square of side length 11 will equal the perimeter of a circle of diameter 14
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Geometry 2: Maintaining integers using fractions

understanding the megalithic: circular structures: part 2

The megalithic sought integer lengths because they lacked the arithmetic of later millennia. So how did they deal with numbers? There is plenty of evidence in their early monuments that today’s inch and foot already existed and that these, and other units of measure, were used to count days or months. From this, numbers came to be known by their length in inches and later on as feet, and longer lengths like a fathom of five feet, the cubit of 3/2 feet and, larger still, furlongs and miles – to name only a few.

So megalithic numeracy was primarily associated with lengths, a system we call metrology. Having metrology but not arithmetic, the integer solutions to problems became a necessity. Incidentally, it was because of their metrological numeracy that the megalithic chanced upon a rich seam of astronomical meaning within the geocentric time world that surrounds us, a seam well-nigh invisible to modern science. Their storing of numbers as lengths also led to their application to the properties geometrical structures have, to replicate what arithmetic and trigonometry do, by using right triangles and a system of fractional measures of a foot (see later lesson – to come). In what follows, for both simplicity and veracity, we assume that π was too abstract for the megalithic, since they first used radius ropes to create circles, so that 2π was a more likely entity for them to have resolved.

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Geometry 1: π

understanding the megalithic: circular structures: part 1
What is the relationship of a circles perimeter to its width, called its “diameter”?

It would require 3 and a bit diameters to wrap around the circle – the ratio of 3 and a bit diameters to the perimeter is known as “Pi”, notated by the Greek symbol “π”. Half of the diameter, from the circle’s center to its edge, is named its radius.

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