## Summary

In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court. Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15). The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7The simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods., generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.

The two ratios involved, 22/7 and 63/20, each an approximation to π, become 9.9 (99/100) when they are multiplied together, as an approximation to π squared. Figure 1 shows that these two ratios, if 22/7 differently used as its reciprocal 7/22, also generates the ratio between the mean and polar radii of the Earth, since 63/20 x 7/22 = 441/440. The ancient Meridian length could be calculated from 396 when multiplied by using the most accurate rational π noted by Fibonacci as 864/275. The 396 units, of 10 miles per foot, was a practical distance to have realized in the megalithic without arithmetic, to store the 3960 mile mean radius of the earth, since the mile of 5280 feet is 4/3 of 3960; that is, 396 x 4/3 equals 528, implying that this model was conceived of within a decimal framework but without the base-10 positional notation of arithmetic. We show that the methods of calculation used can only have seen numbers-as-lengths as being composed of factors of just the first five prime numbers {2 3 5 7 11} and that this limitation upon numbers created a metrologyThe application of units of length to problems of measurement, design, comparison or calculation. in which fractional units of measure could manipulate lengths to multiply and divide them through addition and subtraction of the powers of these primes.

#### Contents

- Summary
- Introduction.
- Main Thesis.
- Pre-arithmetic Calculation using Powers of {2 3 5 7 11}
- Combining Prime Number Composites.
- Appendix 1 Extract from
*Science and Society in Prehistoric Britain*. - Appendix 2: Preface: The Metrology of the Brochs.
- Metrological Bibliography.

### 2. Introduction

The megalithic had to realise numbers and ratios using lengths-as-numbers and triangular geometries. This required a common standard unit be chosen to represent the root number 1. For ancient metrology, this root was the foot we now call English, the root for the ancient Greek module of measures[3]. At every megalithic building site, understanding the numeracy employed as being metrological reveals more examples of this ancient metrology at work, and the same is true of the ancient world, which inherited the same system. The system arose from the quantification of astronomical day-counting of celestial time periods, then stored through using numbers-as-lengths rather than the ancient arithmetical systems of calculation which treated numbers as symbols arranged in a positional notation, such as the base of 60 of the Babylonians and its decimal successor, of base-10 arithmetic. Functional arithmetic methods of addition, subtraction, multiplication, and division, can applied to transform basal numbers, but the megalithic had no knowledge of arithmetic so that geometrical methods involving triangles, squares and circles and, as we propose here, prime number analysis of counted lengths, could bypass the need for arithmetic, and employ a simple number theory of primes, as an alternative with considerable but little understood abilities in quantifying and transforming numbers-as-lengths.

The Great Pyramid of Giza demonstrates that the length of the earth’s Pole and its mean radius had been quantified by the middle of the third millennium BC, in the same way modern surveys had to, through surveys conducted at different parallels of astronomical latitude which get longer towards the Pole. The height of the Pyramid is 440 Sumerian feet (12/11 feet) and at its pinnacle, now lost, would be one more Sumerian foot. The southern side length of 756 feet [Petrie] is 440 Giza cubits [Petrie] as well as 441 Royal cubits (12/7 feet) or alternatively as 800 “of our feet” by Herodotus of Samos (c.484), whose Samian foot was 0.945 (189/200) feet. Metrology, it seems, is the key to late stone-age numeracy, embodied in their building works.

A reasonably good approximation to pi, of 22/7 (or 3.142857), involves the early prime numbers {2 7 11} and it takes the geometric form of a circle of diameter 11 whose out-square has perimeter 4 x 11, the circumference of a circle diameter 14 (figure 1). A less good approximation of 63/20 (or 3.15) involves {2 3 5 7 11} and, when multiplied by 22/7, the sevens cancel to give 99/10. When one of these approximations (22/7) is instead reciprocated, to 7/22, the product approximates the number 1 as the ratio 441/440 which, as already stated above, relates the earth’s mean radius to the polar radius, as in figure 2.

We then see that, through management of the prime numbers within, in this case approximations of π, the metrology of the megalithic can be explained as necessary *because there was no arithmetic*. Instead, there had to be the *remeasuring actions* upon the lengths of time-counts, such as lengths-of-days, using a set of measuring sticks whose lengths were the first five prime numbers {2 3 5 7 11} in feet.

### Main Thesis

In the elliptical courtyard of Dun Torcuill, in North Uist, a procedure was implied in the size and shape of the ellipse. The major axis of 40 feet and two approximate pi ratios of 63/20 feet (the difference in the axes) and 22/7 were to be directly multiplied. That is 40 x 63/20 x 22/7 (or 40 x π squared) yield the mean earth radius number as 396 feet {4 9 11}, at a scale of 10 miles per foot, rather than the 3958.690 miles (7 x 12^{6}) of the spinning earth.

Though these two approximations to π were familiar to me, in the broch they were not expressing the 441/440 ratio between the two planetary radii. Instead, they were used to create a significant integer value for the mean earth in miles, a value belonging to *an earlier and simpler model of the Earth than that seen in the Great Pyramid*; despite the fact that the broch is a later Iron Age building. If true, the broch may have expressed continuity of an ancient tradition that represents the knowledge of a megalithic model of the earth, prior to the Egyptian model seen in the Pyramid.

The excess of major over minor is more clearly visible in figure 3 (right) as 3.15 (63/20) feet: in metrology, this is a yard of root Persian feet (as our historical metrology would name it) of 21/20 (1.05) feet.

My starting point for proposing *how* *this calculation* of the mean earth radius (as 396) was to be *performed*, was with the major axis of exactly 40 feet [MacKie, 1977, 65] as a multiplier[4] and the 63/20 unit as the multiplicand. If the 3.15-foot length were repeated, end-to-end, as many times as there were feet in the 40-foot major axis, this would constitute the **multiplication** of 63/20 by 40, equalling an integer value of 126 when measured in feet[5]. The number 126 is 18 x 7 so that, if it is multiplied by pi again, but by the “good” pi of 22/7, their sevens will cancel out and one could lay down 22 feet 18 times to obtain another integer result of 396, when measured in feet. Both 126 and 396 are important numbers in what appears to have been an intellectual system for the megalith builders, whose meaning was both astronomical and geodeticUnits of measures and monumental measurements relating to the numerical definition of the shape of the Earth by the late megalithic. model of their visible world.

With a fraction like the 63/20 foot yard, this will resolve to 63 feet after placing the yardstick 20 times. In each successive application of 63/20 feet “on the flat” then by the end one can see the third side of a right triangle of 20 feet and that the hypotenuse “is” the steady accumulation of height (figure 4).

### Pre-arithmetic Calculation using Powers of {2 3 5 7 11}

If my interpretation is correct, the “less good” pi of 3.15 feet (made explicit through an eccentricity of the monument’s ellipse) and its major axis of 40 feet were involved in a megalithic method of multiplication defining, in this case, the generated length 126, whose factors are easily found using prime number measuring sticks, in the multiplier of 40 {8 5} and the multiplicand of 63/20 {9 7} / {4 5}. That such factorization of numbers was vital to the megalithic is seen in the fact that metrological units of measure were wisely restricted to the first five prime numbers {2 3 5 7 11}. That a number was an integer was clear in the lack of a denominator in the result, the stick used had nothing further or less to consume. The ability to divide an integer number by the five prime number lengths of measuring stick, enabled the megalithic to determine whether a number-as-a-length contained a given prime number and so whether it was a factor of that an integer length is commensurate with the prime factors it contains.

This prime number view of numbers-as-lengths explains how what are called Diophantine numbers were called “regular numbers” when restricted to the harmonic primes of {2 3 5}, where 2 x 3 x 5 = 30 which is half of the 60 of sexagesimal arithmetic of the Sumerians by 3,000 BC, then carried forth by cuneiform and arithmetic methods. Before arithmetic we culturally expect to see very little numeracy, but numbers-as-lengths can only be “read” according to prime number sticks and any primes larger than {2 3 5 7 11} would then be which we might call the measurement while the first five primes speak of musical harmony and geometrical harmony between two numerical lengths.

Division, like multiplication, was not yet invented for numbers-as-lengths so that, instead, a physical process involving the repetitive application of measuring stick to either (a) **divide:** use a prime measure to test a measurement for a given metrological length or (b) **multiply:** repetitively use a measure (like 63/20 feet) a defined number of times (such as the 40 foot length of the major axis as above).

In figure 4, one could see that a fractional length such as 63/20 is like a right triangle in which time is horizontal and, as time moves right from the vertex, the hypotenuse is the cumulative growth in height as the measure is successively applied until, on the 20^{th} application the horizontal side equals 63 feet and (by the 40^{th} application) the height is 126 feet, twice 63 feet. That is, the inherent process within time of creating a time count is an explicit presentation of multiplication by time of a unit of measure, such as a foot or inch per day. In practice, it is easier to work with rectangles in which the hypotenuse becomes the diagonal of the rectangle of two, contra-flowing triangles. And again, this is probably why, near CarnacAn extensive megalithic complex in southern Brittany, western France, predating the British megalithic., the latitude of 47.5 degrees north enabled horizon alignmentsA name special to Carnac's three successive groups of parallel rows of stones, starting above Carnac called Le Menec, Kermario, and Kerlescan and another found near Erdevan. to solstitial sun, and lunar maximum and minimum to follow the simple geometries of {3 4 5} triangle and single and double squares, respectively.

### Combining Prime Number Composites

When using the prime numbers in the numerator and denominator, the megalithic method recognised an integer in its *lack of a denominator* so that, in effect they were using the fractional form which may be summarised as being like a capital H in which the top was a space within which the prime factors of the numerator were placed and in the bottom half the prime factors of the denominator. One can see this in many antique forms such as the vertical double square of the Egyptians or indeed the triple H format which might have shown the multiplicand, multiplier and product, in which case calculations within the realm of numbers containing just the first five primes would be especially simple, more simple than decimal arithmetic since one simply cancels the shared reciprocal primes in the multiplicand and multiplier.

Figure 5 illustrates such a technique in which calculations are formed using three Hs. In the top register are the three numbers and ratios that are part of the calculation as *three constants*: the major axis, the simple pi (as out-circle of the ellipse) and the less good pi of the 63/20 foot yard.

The middle part of figure 5 offers the resolution of the two different pi ratios, through cancellation of similar primes, top and bottom. The bottom register then uses this as a multiple for the multiplicand of 40 feet which creates an integer with no primes as denominator.

If the two types of pi used are multiplied, we can see that 63/20 x 22/7 = 99/10 (or “prime squared” of 9.9), then used to multiply 40 and so obtain 396 feet. This result is {4 9 11} or 36 repeated eleven times, where only the 4 has survived from the broch’s major axis of 40 feet.

### Appendix 1 Extract from *Science and Society in Prehistoric Britain*

The most important information for my paper provided by MacKie is the accurate length of the two axes of the elliptical court of Dun Torceill as 40 feet and 36.85 feet. These give an eccentricity of 63/800 so that, times 40, the foci are 63/20 feet apart. In this the confidence is high whilst other speculations about megalithic yards and rods operate at levels of inaccuracy accepted if the measures used were not standardized or that thinking something was accurate was a sufficient criterion – arguments based upon inaccuracy unlike the computer derived ellipticity of the axes, then needing ancient metrology beyond that of Alexander ThomScottish engineer 1894-1985. Discovered, through surveying, that Britain's megalithic circles expressed astronomy using exact measures, geometrical forms and, where possible, whole numbers. to interpret the monument as part of that corpus. Indeed MacKie came to acknowledge that the different modules of ancient metrology were used to access numerical designs within highly variable platforms for their constructions, where space to build was absolutely limited, requiring the scaling mechanisms modules provided.

**p64:** **Elliptical brochs**

Of the 37 brochs surveyed, seven appeared not to be set out as true circles but three of these were buried under too much of their own rubble for their exact shape to be ascertained. Of the remaining four, two, Dun Torcuill in North Uistand Dun Borodale on Raasay Island, were oval and two, Ness in Caithness and Torwood in Stirlingshire, appeared to be either egg-shapes or flattened circles of some kind. The surveyed points on the inside wallface of Dun Torcuill were processed by the computer which had been programmed to test such data for the accuracy of their fit to ellipses.

As a result, the broch was found to have been built very precisely around an ellipse with eccentricity of 0.93. (The eccentricity of an ellipse is defined as the proportion of the long axis of that part of it not between the foci. In the example given the foci are close together and only 0.07 of the long axis is between them.) The fit was particularly impressive because since the broch has never been excavated the survey of the inside wallface had to be done at a height of 5-6 ft above the floor and in places the wall was leaning noticeably inwards. The ellipse exactly fits the wallface where curvature is smooth and diverges from it in those parts where it is obviously leaning in or out. The axes of the best fit ellipse, deduced by the computer through trial and error, were 40.00 ft and 36.85 ft in length and the perimeter 121.33 ft. That the axes are exactly divisible into 15 and 14 by a unit of 2.65 ft can hardly be a coincidence.

The perimeter of this figure is exactly 44 MY long (121.38 ft); it is 0.2 megalithic rods less than the nearest whole number of these ( 18) . It is worth considering whether the slightly smaller size of the Dun Torcuill ‘yards’ [2.65 feet?] might be explained as the result of a desire to make the perimeter close to an integral number of megalithic rods. By using axes of 14 and 15 megalithic yards (2.72 ft), the perimeter would have been 123.99 ft or 45.49 MY, that is 0.24 MY more than 18 rods (122.41 ft). Thus, there seems no reason to suppose that the question of the number of rods which would be in the perimeter played any part in the calculations of the designer of Dun Torcuill. Neither is the elliptical shape likely to have been dictated by the situation of the broch: Dun Torcuill stands on a small islet in a loch which is little larger than the building itself, but there would have been no difficulty, as far as one can judge, in making it circular if this had been thought necessary.

### Appendix 2: Preface: The Metrology of the Brochs

I wrote the preface below for Euan MacKie, who had resurrected his work on measures found within the brochs of Scotland. Euan was almost a lone voice in support of Alexander Thom’s work on metrology

The application of units of length to problems of measurement, design, comparison or calculation, in the megalithic, and also the long distances of alignments to horizon events has been neglected by archaeological academia, since the death of Alexander Thom, due to its contradiction of a deeply held paradigm, that before the ancient near east, the stone age could not have had such a thing.

A name special to Carnac’s three successive groups of parallel rows of stones, starting above Carnac called Le Menec, Kermario, and Kerlescan and another found near Erdevan.

in the Western Isles of Scotland. When he met John Neal at the latter lecture in Glasgow, at which I was present, they appear to have entered into a review of the data and John Neal came back with an interesting theory which would make a full range of historic measures to have been employed in one area of northern Scotand, in the Iron Age. I sent Euan a summary of what ancient metrology appeared to be as a system of ratios and why Neal’s finding within MacKie’s data would be important. It became the preface for the article called The Roundhouses, Brochs and Wheelhouses of Atlantic Scotland c.700 BC-AD 500: Orkney and Shetland Isles Pt. 1: Architecture and Material Culture (British Archaeological Reports British Series) which I have recovered from a partial proof copy.

**PREFACE **by Richard Heath

John Neal has demonstrated elsewhere [John Neal, 2000] that ancient metrology was based upon a “backbone” of just a few modules, each relating as simple rational fractions to the “English” Foot. Thus a Persian foot was, at its root value, 21/20 English feet, the Royal foot 8/7 such feet, the Roman, 24/25 feet and so on. By this means, one foot allows the others to be generated from it.

These modules each had a set of identical variations within, based on one or more applications of just two fractions, Ratio A = 176/175Ratio crucial to maintaining integers (see geometry lesson 2) between radii and circumference of a circle, and crucial to the micro-variation of foot modules in ancient metrology. and Ratio B = 441/440. By this means all the known historical variations of a given type of foot can be accounted for, in a table of lengths with ratio A acting horizontally and ratio B vertically, between adjacent measures.

In the context of what follows this means that, each of the differently-sized brochs analyzed by Neal, appear to have used a foot from one or other of these ancient modules, in one of its known variations. That is, the broch builders seem to have chosen a different unit of measure rather than a different measurement, as we would today when building a differently-sized building of same design. Furthermore, these brochs appear to have been based upon the prototypical yet accurate approximation to pi or π: The constant ratio of a circle’s circumference to its diameter, approximately equal to 3.14159, and in ancient times approximated by rational approximations such as 22/7, 25/8, 63/20 and 864/275 which, as Fibonacci discovered, is nearly exact. Pi as 22/7 was the best, simplest approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods, so that – providing the broch diameter would divide by seven using the chosen module – then the perimeter would automatically divide into 22 whole parts.

Thus, John Neal’s discovery, that broch diameters divide by seven using a wide range of different ancient measures, implies **the Iron Age broch builders** had

**inherited an original system of ancient measures**with its rational interrelations between modules and variations within these, from which they could choose, to suit a required overall size of circular building, often the only foundation available and,**were practicing a circular design concept**found in the construction of stone circles during the Neolithic period.

These measures, used in the brochs, are not often found elsewhere in Iron Age Britain, but they became historically associated with locations hundreds or thousands of miles distant due to empires and ages of exploration. This suggests that the historical identification of such measures is only a record of the late use of certain modules in different regions, after the system as a whole had finally been forgotten, sometime after the brochs were constructed.

Such conclusions, if correct, are of such a fundamental character that they present a compelling case for ancient metrology and its forensic power within the archaeology of ancient building techniques.

### Metrological Bibliography

1877. Petri, W. M. Flinders. *Inductive Metrology*. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

1953. Berriman, A. E. *Historical Metrology*. London: J. M. Dent and Sons.

1967. Thom, Alexander. *Megalithic Sites in Britain*. Oxford: Clarendon.

1977. MacKie, Euan. *Science and Society in Prehistoric Britain*. London Elek.

1978. Thom, A and A. S. *Megalithic Remains in Britain and Brittany*. Oxford: Clarendon.

1981. Michell, John. *Ancient Metrology*. Bristol, England: Pentacle Press, 1981.

2000. Neal, John. *All Done with Mirrors*. London: Secret Academy, 2000.

2007. Heath, Richard. Sacred Number and the Origins of Civilization. Inner Traditions.

2006. Heath, Robin, and John Michell. *The Measure of Albion*. reprinted as *The* *Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients*. Kempton, Ill: Adventures Unlimited Press.

2007. Heath, Richard. Sacred Number and the Origins of Civilization. Inner Traditions. Appendix 2.

2016. Neal, John. *Ancient Metrology*. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016.

2017. Neal, John. *Ancient Metrology*. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.

2021. Heath, Richard. *Sacred Geometry: Language of the Angels*. Inner Traditions.

#### End Notes

[1] by Historical Environment Scotland.

[2] *Science and Society in Prehistoric Britain*, Euan MacKie, London: Paul Elek 1977.

[3] All the root values of the different modules are simple ratios of the English, around which systematic variations were created for different numerical effects. The Greek module is not the common Greek module whose root is 36/35 feet and all modules are linked to the Greek via a simple rational fraction, as with the Royal foot of 8/7 feet and the Saxon of 11/10 = 1.1 feet. [Neal, 2000]

[4] See Skeat for multiplier: to increase many times, make more numerous its roots being multi and plex in the sense of folding.

[5] This length of 126 or the yet bigger 396 feet would not easily fit the broch, whose role was to store the method for generating the size of the mean earth radius.

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