## Numeracy

Three Folkton Chalk Drums found in a young girl's grave {PHOTO: TRUSTEES OF THE BRITISH MUSEUM]

Perhaps as early as 4000 BC, there was a tradition of making chalk drums. Three highly decorated examples were found in a grave dated between 2600 and 2000 BC in Folkton, northern England and one undecorated chalk drum in southern England at Lavant in an upland downs known for a henge and many other neolithic features discovered in a recent community LIDAR project. The Lavant LIDAR project and the chalk drum found there are the first two articles in PAST, the Newsletter of The Prehistoric Society. (number 83. Summer 2016.) It gives the height and radius of both the Folkton drums 15, 16 and 17 and the Lavant drum, presenting these as a graph as below.

Adapted graphic showing inches as well as mm, and possible PI relationships for the chalk drum diameters,
key to the fact that such drums can be rolled.In line with megalithic numeracy,
the simple yet accurate value of 22/7 for PI is shown.

Local chalk is a relatively easily carved yet substantial material and a cylindrical drum can be rolled and, being given a definite diameter, causes the circumference to travel a known distance on the earth. Folkton 17 has a 4 inch diameter which gives an 88/7 inch circumference (using PI = 22/7) which equals 22/21 feet of twelve inches. In Sacred Number and the Lords of Time, 167-171, I point out that the microvariations to be found between measures of the same module (in historical metrology), [identified first by John Michell in Ancient Metrology and then John Neal in All Done With Mirrors] include 176/175. This ratio is the product of two early versions of PI since 176/175 = 8/25 times 22/7. leading to the fact that 1/3 foot diameter (= 4") will give a circumference of 22/21 feet. (see panel). The module is 25/24 feet, varied by 176/175 to give 176/175. Since the accurate PI of 22/7 is present in the ratio 176/175, then a circumference of 22/21 feet times 7/22 gives a diameter of 22/21 feet times 7/22 equalling a 1/3 foot diameter or the 4 inch diameter of Folkton 17.

There are three levels of interpretation:

• Firstly, the four inch radius appears to imply that inches were the native units of measure and the five inch radius of Folkton 16 appears to support this. It is uncontested (though not peer reviewed) that Le Manio Quadrilateral used inches to count days to resolve the megalithic yard, as the day-inch difference when counting three lunar and three solar years.
• Secondly, if the later megalith builders had evolved a network of fractional measures based upon the English foot as unit, 1/1, then new measures were made from that foot by laying out right triangles in feet, with a different number of whole units in the two longest sides, numbers we would call the numerator and the denominator of a fraction. In the case of 22 feet for hypotenuse and 21 for base, the 21 divisions of the base can be made to rise at right angles to define 21 divisions on the hypotenuse, each 22/21 feet long. The chalk drum would perform 21 revolutions in travelling the 22 feet of such a hypotenuse. The diameter has to be 1/3 of a foot so that 1/3 times 22/7 equals 22/21 feet.
• Thirdly, if one wishes to make a circumference of 22 feet or 21 rotations of this chalk drum, then the diameter must be 21 times 1/3 feet or seven feet. 22/21 is in fact the Thoth ratio found between the 1/6 arc on the circumference relative to the straight distance between the ends of the arc. Egyptian Thoth presents this in his iconography, because PI was a sacred invariant to geometers and this brings us to decorated drums being found in a high status burial, if the family involved were the geometers who laid out megalithic monuments and pathways. The undecorated drum found at Lavant shows signs of usage as if it had been rolled many times.

The largest of the three drums, Folkton 15, appears to have a circumference of 18 inches and, if so, the drum transfers the idea of rationality to the circumference so that the diameter is an irrational number of inches, 7 and 9/11th inches. Such a drum would be able to lay out cubits of 3/2 = 1.5 feet in a line, enabling yards to work extensively. One also notices that the designs on the tops of drums has a possible role in dividing the rotation of the drum like a rotary ruler or in the angular sense.

Picture of Lavant Chalk Drum, showing a plain drum
with a single boss and double rim on its side separated by a depression.

grabbed with thanks from 3D SketchFab by Marta Diaz-Guardamino

The Lavant drum, shown above, appears to have been a practical rolling device equivalent to a short and fat axle with no wheels.

1. The radius is measured to be 115mm or 4.528 inches (in our notation).
2. Multiplied by 22/7 this is 14.23 inches or 1.18578 feet.

To understand this length requires that the largest type of foot using in ancient metrology, called the Russian foot but essentially 7/6 (1.16feet which, after two yards, becomes seven English feet. But here the geographical variant is used of 7/6 times 1.01376, the Geographical Constant.

Geographical constant is a term within the modern subject called Ancient Metrology [see Michell. 1981. and Neal. 2000.], so-called because it defines an important relationship between the equator of the earth and its mean circumference. In brief,

1. the sun moves one day in angle every day and that angular distance can be measured on the equator (from other parallels of latitude) as being 360,000 feet.
2. the mean radius of the earth can be known from the polar radius and equatorial radius and each degree of latitude on the mean earth would then be 69.12 miles long.

Despite the north-south deformation of the earth (due to its daily rotation) one degree of latitude, between 51 and 52 degrees, corresponds to the mean earth in measuring 69.12 miles. Within this latitude lies Stonehenge and Avebury, separated by one quarter of this degree length. Lavant lies just below 51 degrees. So why be using the geographical constant at Lavant?

John Michell, in The Dimensions of Paradise and elsewhere, deconstructed the geographical constant as being 3168/3125 and observed that temples in prehistory were metrologically organised to express 3168 units around their perimeter, making such structures models of the mean earth which, by inference, gives signifies the mean earth as being the spiritual earth. Not only were monuments being built within or near the degree of latitude corresponding to the mean degree within southern Britain, but also those monuments needed to be laid out so that their circumferences could correspond to the 3168 units required, and using the geographical units of measure corresponding to that mean degree. In turn, to understand the Lavant chalk drum we might need to deconstruct 3168 as being 22 times 144.

The number 3168, in being 144 x 22, is to be the perimeter of sacred spaces which in circular monuments means the circumference needs to be 22 times 144 units long. The diameter of the monuments, calculated as 3168 divided by 22 and multiplied by 7 must give a required radius of 72 (144/2) times 7 which equals 1008. (Using 2 PI = 44/7)

Since six revolutions of the chalk drum gave seven geographical feet, 72 x 6 rotations would have defined a rope from the central peg which could achieve a perimeter of 3168 geographical feet when used to define a circle. The foot of 7/6 is revealed to be ideal in solving a particular problem of generating on the ground a radius which a pegged rope can turn into a perimeter iof 3168, representing a part of the actual earth as the mean earth, seen to be sacred as part of the mean earth.

### Conclusions

The chalk drums are probably evidence of their use  in southern Britain as metrological contraptions whose power lay in their natural expression of PI as 22/7.

1. Folkton 17 (4" diameter) allowed the creation of a unique variation of 22/21 feet on its circumference when rolled. This ratio is one third of PI and
1. this allowed one third of a foot to transform 22/7 into 22/21
2. a foot of 22/21 enabled a hexagon to be constructed with sides 21 feet within a circle whose circumference would then be
2. Folkton 16 (5" diameter) is 5/4 = 1.2 of these feet in circumference and hence could be useful within the same module
3. Folkton 15 appears designed to provide an explicit cubit of 1.5 feet (18") with every rotation of travel by being (7 x 18)/22 (= 63/11) inches in diameter.
4. The undecorated and worn Lavant drum seems contrived to generate 7/6 geographical feet in circumference so as to allow the formation of circular perimeters for sacred spaces measuring 3168 geographical feet or multiple thereof, an activity connected to its latitude and perimeters of other contemporary structures.

The drums appear to be within a range of sizes large enough to be accurate and small enough to be simply used in the task of laying out distances on the earth, perhaps by a single trained individual who belonged to a craft tradition and this may have led to a premature death being marked in the burial of three decorated chauk drums at Folkton, up to a thousand years after the Lavant drum was disposed of. If so then the drums may represent the iconography of the times perhaps representing PI or a god connected to geodetic constructions. Further examples could strengthen this interpretation.

An interactive demonstration of how accurate rational values of PI can be formed using denominators one to ten.
The last three all appear in ancient metrology; 22/7 (the best), 25/8 and 2 PI of 63/10.
[using oCanvas with thanks to Johannes Koggdal]

There are many good rational versions of PI but 22/7 is remarkably good and arrives early in the number field, making it amenable to ancient forms of numeracy.

### 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".

In 2009/10, my brother (Robin) and I identified surveyed and verified the existence of a day-inch count for three lunar years (1063.1 inches) within Le Manio's Quadrilateral, which then extends to the end of stone 37 by a megalithic yard of 32.625 inches, a megalithic yard, to the end of a three solar year count (1095.75 inches). The ratio of stones of the southern kerb 36:37 appears to symbolise and approximate the lunar months involved in those times periods.

Figure 2 Plan of the Quadrilateral from Thom's Megalithic Sites in Britain and Brittany with geometry of day-inch counting overlaid. The solar years appear as counted aligned to the midsummer solstice sunrise to point G. The monument appears to memorialise the origin of the megalithic yard as the difference between three years of the solar and lunar kind.

This points to the origins of the megalithic yard as having been most naturally generated as a unit of length when megalithic astronomers were comparing the three year near-anniversary of sun and moon, using day-inches to count circa. 4000 BC. As a unit representing the ratio between lunar and solar years (1.035), the more accurate anniversary of 19 years can give a later refinement, almost exactly seven lunar months different in nineteen years. This identifies the astronomical megalithic yard (AMY of 19/7 feet long) as being the number of years in which the solar year count is ahead of the lunar year count by a single lunar month of difference (2.7154 lunar years or 32.585 lunar months). Adding one lunar month gives 33.585 lunar months which is 2.7154 solar years of counting.

Figure 3 The geometry of solar versus lunar years as a "lunation triangle" (HEATH, Robin. 1998) almost perfectly matches the diagonal angle of a four-square triangle. The three lunar year period in day-inches is the near square of the difference (32.625 inches) times the superparticular ratio (N:N+1) of this triangle, where N = 32.585 (see figure 4), the number of inches in the astronomical megalithic yard found later in the megalithic (3500 BC onwards) [HEATH, Richard. 2014]

Since the AMY is widely found in later monuments then one is drawn to look for an explicit monument which, through counting over 19 lunar and 19 solar years (instead of three) might have shown a difference of seven lunar months between 228 and 235 lunar months respectively. It seemed possible that the distance to Le Geant from the Quadrilateral could have allowed this - but there is insufficient evidence for that and so one has to consider the alternative ways the AMY unit might have been resolved. For instance there is a property of right angled triangles which could have been exploited within the three year counting at Le Manio, to isolate the AMY of 32.585 inches, as well as the megalithic yard of 32.625 inches, twin factors that sum to give three lunar years as a day count in inches.

Figure 4 The near square of day-inches in three solar years is the megalithic yard times the astronomical megalithic yard for purely mathematical reasons, yet the nearness of this to a square number is a coincidence belonging to the sun-moon-earth system.

If the two counts could somehow be divided up by the differential unit between them (the MY of 32.625 inches), then the astronomers would have found 32.585 megalithic yards in the three lunar year count and 33.585 megalithic yards in the three solar year count. The amount left over, in either case, is then 19 (.0669) inches. That is, three lunar years is 32 x MY = 1044 plus nineteen inches making 1063 inches. The solar year is 33 x MY = 1076.625 plus nineteen inches making 1095.625.

The idea of dividing up the day-inch counts (into how many megalithic yards would fit them) arrives at a factorization demonstrating another feature of the counted lengths as forming a nearly square rectangular number when seen as being divided by the differential unit of length held between the base of the triangle and the hypotenuse. The difference can normalize the triangle: the difference becomes equal to one, revealing the unique numerical signature of the ratio between the two lengths.

Figure 5 Triangles such as the three solar and lunar year triangles have a differential length which, divided into their two longer sides, leaves the difference as unity whilst the two sides are nececarily then super-particular, that is of the form N:N+1 and in this case N is the AMY and three lunar years divided by the difference becomes the AMY as a pure number, as per figure 4's near square property.

Having established  that N = 32.585 rather than the almost identical 3 year difference of 32.625 days, one can resort to a little-known property of super-particular right triangles; that their difference multiplied by N gives the original measurement of the baseline. In this case, the original measurement is the lunar year count over three years, of 1063.1 days, whilst N must therefore be that count divided by 32.625 (the MY difference), a value already very close to N. This therefore makes it possible to divide up the count 32.585 times using the MY, but then what meaning or utility would a new count of 32.585 megalithic yards have?

It is here that stone age numeracy can display its ability to factorise and re-proportion. The process of dividing 1063.1 by 32.625 to obtain 32.585 (the true value of N for the solar-lunar ratio) must be achieved using geometrical metrology. The MY is 32.625 or 32 and five eighths, so that this length can be shown as 261/8 seen then as a number of eighths of an inch; a limit of likely resolution when using inches.

The numerator 261 then has the factors 3 x 3 x 29 whilst the denominator is 2 x 2 x 2 = 8.

That is 261 = 29 x 9/8

Since dividing 261 into 1063 was not directly possible in the megalithic, as a first step, and knowing the above prime factors of 261 (as the stone age was able to, from simple experimentation with numbers as objects) the lunar year count of 36 months could be divided into nine parts leaving each divided part as 118.12 day-inches, or four lunar months (36 lunar months are 9 x 4 lunar months). This division by nine could then be followed by a multiplication by 8, the denominator of the divisor and so a multiplier for the required calculation. This results in 4 x 8 = 32 lunar months (in day-inches).

It then appears that, in order to finally divide by 29 (and complete this "calculation") the scaling effect between base and hypotenuse within any right angled triangle could be used to form a hypotenuse of 32 lunar months above a base of 29 lunar months, so as to reveal that the required value of N = 32.625 must be 32/29 of the lunar month in day-inches, and indeed the lunar month of 29.53 x 32/29 equals 32.5854786, effectively exact. (see Appendix for algebraic version of the solution)

Figure 6 The method of manipulating factors to derive the AMY (instead of creating a nineteen year triangle) could possibly have been shown within the Quadrilateral. If the kerb from the solar gate to stone 29 was taken to be 29 lunar months and to stone 32 that number of lunar months, then 32 lunar months is 29 astronomical megalithic yards. The method of converstion then follows convenience, such as using a four lunar month count of 118.125 inches and obtaining four AMY on the hypotenuse.

Each of the twenty nine lunar months on the base of this triangle can be seen in horizontal width to select a length 32.585 day inches on the hypotenuse, each the length of the true Astronomical Megalithic Yard (AMY) which is also the value for N for the lunar-solar ratio between respective years. In feet, this length is 2.7154 , which is the number of these years which then differ by a single lunar month and are 32.585:33.585 lunar months long. The units of this 2.7154 figure are solar years per lunar month of difference. In principle the figure below shows how the day-inch count for a single lunar month translates into the astronomical megalithic yard when a 29:32 right triangle is used.

Figure 7 The methodology of using a 29:32 triangle to generate astronomical megalithic yards can employ near integer numbers of days found in multiple lunar months. The three year period of 36 months is one such, being 1063.1 inches long, so that there must be 36 AMY at a 25 degree slope angle above three lunar years. In practice, once the slope angle of 29:32 is constructed, any convenient number of lunar months on the base will correspond with the same number on the length of the hypotenuse above so that there is no need to divide by 29 at all!

### Conclusions

The 32 to 29 relationship of AMY to the lunar month in day-inches appears to be defining; alongside the near-square nature of three lunar years as the difference (32.625) times the AMY (32.585). Perhaps it has a causal explanation? It has also been found that Le Manio is an implex of many more interrelating themes than first thought, indicating it was probably a memorial to determined facts rather than an original instrument for elucidating them.

Figure 8 Le Manio Quadrilateral, viewed from the north east. Extreme left is grooved stone G and extending right is the northern kerb and behind that the southern kerb, terminated by the taller "Sun Gate", from where one can view the midsummer solstice sunrise above the grooved stone.

### Appendix: Seeing this Megalithic Method in Algebraic Terms

We can say 3 x LY = D x N = MY x AMY (1) where

• LY = lunar year
• D = difference between three solar and three lunar years = megalithic yard
• N = the base number for normalised triangle of solar and lunar years
• MY = 32.625 inches = 261/8 inches
• AMY = N =  the superparticular ratio governing the relationship of solar and lunar year, in inches

Therefore N = 3 x LY / D (2)

where N is the desired unit length expressing the solar lunar relationship.

If we factorise 261 in MY it equals 29 x 9

so that MY = 29 x 8/9 (3)

So (2) becomes

N = 3 x LY x 8 / 29 x 9

then N = 36 x 8 / 29 x 9 [lunar months]and then

N = 4 x 8 / 29 [lunar months] (4)

1. The astronomical megalithic yard = N = 32/29 of the lunar month (in day-inches).
2. But 32/29 is a unitless ratio, albeit for conversion between the lunar month and the AMY, allowing the true value of N to be derived from the lunar month in day-inches.
3. We note that the lunar month in day-inches is 3/4 metres and 4 of them are equal to three metres, a metric clearly visible on the ground at the Le Manio Quadrilateral.