## Circle of Same Area as a Square and 9/8

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In *Stone Circle Design and Measurement*, G J Bath says

The Ancient Egyptians developed a procedure to determine the area of a circle by subtracting one-ninth from the diameter and squaring the result.

This can be best visualised as:

*Figure 1 The near integer relationship between the half side length of a square and the circle of equal area.*

## Astronomical Rock Art at Stoupe Brow, Fylingdales

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I recently came across Rock Art and Ritual by Brian Smith and Alan Walker, (subtitled Interpreting the Prehistoric landscapes of the North York Moors. Stroud: History Press 2008. 38.). It tells the story: Following a wildfire, thought ecologically devastating, of many square miles of the North Yorkshire Moors, those interested in its few decorated stones headed out to see how these antiquities had fared.

### Background

Fire had revealed many more stones carrying rock art or in organised groups. An urgent archaeological effort would be required before the inevitable regrowth of vegetation.

*Figure 1 Neolithic stone from Fylingdales Moor | Credit: Graham Lee, North York Moors National Park Authority. *From <this site>

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## Le Manio: How ancient maths manipulated factors

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### Abstract

This article is based upon notes made in 22 May 2014 whilst the 32/29 relationship between AMY and day-inch lunar month was discovered by autumn 2009 (after this paper), being driven to use day-inch counting to explain the origins of megalithic monuments evidently created in the pursuit of astronomical knowledge yet measuring time as lengths. The movement from counting days as inches to using megalithic yards to stand for lunar months was partly explained in my *Sacred Number and the Lords of Time* by the fact that the excess of lunar months in the solar year is 7/19ths of a lunar year, a residue which adds up over nineteen years to lead to the Metonic period having 235 lunar months in nineteen years. If the AMY is 19/7 feet then it cancels with the residue if and when the megalithic astronomers counted in lunar months.Until last year, there seemed no way to derive the astronomic megalithic yard short of a Metonic scale of monument but the work on this site, on numeracy, and a growing set of techniques such as scaling, proportioning to cancel factors from denominators and hence "clear" fractions, has revealed the 32/29 relationship as deducible in the megalithic and necessary for the quantification of N = 32.585 inches, the measure Robin Heath refers to as the astronomic megalithic yard.

### Le Manio's Quadrilateral

This unique monument (figure 8), located east of the Carnac Alignments, has been interpreted as being a kerb monument, possibly once filled in as a mound. However, the kerbs follow a very purposeful geometrical design and have a south west to north east diagonal equal to four solar years in day-inch counting. The southern kerb (figure 1) expresses three years from a sun gate (a backsight for both summer and winter solstice sunrises), of two types - three lunar years and three solar years. The relations between these is then projected into the Quadrilateral as a right angled triangle (figure 2). The astronomers at Carnac appear to have understood the right angled triangle as a means to define the ratio (or interval) between time periods as a super-particular ratio of the form N (the base) to N+1 (the hypotenuse), as well as enabling units of measure to be reproportioned in order to "clear" the residues in their measures (that we call fractions.) Fractions can be avoided by choosing units of measure which divide into a measured distance *a whole number of times*. But in order to achieve this, the whole of a given problem had to be matched with different parts of their toolkit: metrological triangles. Instead we would flatten such problems into arithmetical solutions, and can ignore fractions by using the decimal system.

*Figure 1 The sillouette of the southern kern of Le Manio's Quadrilateral made from a photo survey by the author.The stones are numbered from the Sun Gate, see below, and reach three lunar years at Q atop stone 36 and three solar years at Q' at eastern end of stone 37. See plan in figure 2 and photo of "gate".*

## John Neal's integration of the Megalithic Yard (WIP)

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John Neal makes a masterful job of considering the megalithic yard in the context of historical metrology, a metrology that he has managed to forge into a single conceptual scheme in which measures known to history from different lands all inter-relate.

Neal's book, *All Done With Mirrors*, is one of the most fundamental and significant contributions to the ancient understanding of numbers but to read it is no easy matter since he takes no prisoners and fully expects readers to resolve through calculation what he does not explicitly state. This makes his approach different to mine in which I try to present as easily a possible aids to the visualisation and registration of a pattern of facts. However, neither approach can really substitute for what one has to do for oneself in order to understand and John gave his "Secret Academy" idea the catch line "We can't give it away" because of the often deafening silence with which his work is met.

The aim here is to co-incide some workings based on Neal's book, to give others a taste of what lies beneath what is written and also to further my own interests in the Megalithic Yard. My brother's biography, *Alexander Thom: Cracking the Stone Age Code,* reveals that Thom's lack of metrological background led to both an original approach but also a disconnect to what is known about historical metrology. One particular mystery is how measures appear to propagate unchanged across millenia.

Neal says on page 47:

Thom made a comparison of his Megalithic Yard with only one other known unit of measurement. This was the Spanish vara, the pre-metric measurement of Iberia, its value 2.7425 feet. Related measurements to the vara survive all over the Americas wherever the Spanish settled, from Peru to Texas. Although the vara is exactly one of the lengths of the m.y. the fact that it is divided into three feet makes this relationship uncertain. These feet are thought to be Roman but this belief is also unlikely, and they would appear to be related to the earlier Etruscan-Mycenaean units. This is a good example of an intermediate measure being thought to be related because of a similarity in length, and illustrates the importance of considering the sub-divisions when sourcing a measure.

Read more: John Neal's integration of the Megalithic Yard (WIP)

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