Measures
The measurement of length involves a count of standard units of length and, in prehistory, such counts could represent time periods such as days in a longer period. It was soon found that integer lengths interacted according to the prime numbers they contained. Many interrelated units were generated using right angled triangles and the historical metrology noted in ancient world constructions was born; a sophisticated prehistoric number science.
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These are some sections I prepared for the (contested) Megalithic Yard page on Wikipedia. This is part of an effort to defend the historical development of this measure (first proposed by Alexander Thom) from scientific attempts to "airbrush it out of history", along with Thom's role in surveying Britain's monuments for posterity.
Firstly, the arguments for a geometrical origin need to be presented and I have placed this on the Wikipedia page today.
Secondly, I cannot add my own contribution, with Robin Heath, in which the megalithic yard is seen as likely to be derived from a differential day count between three solar and three lunar years, conducted in one inch per day and evident in the Quadrilateral at Le Manio.
Thirdly, Colin Renfrew's 2010 compendium on measure within archaeology has an article in which Japanese researcher Saburo Sugiyama, after decades of working with new survey data, has found a unit identical to the megalithic yard [83cm], at the city of Teotihuacan in the Mexico Basin, called a TMU or Teotihuachan measurement unit. It is quite typical for such work to not research old world historical metrology since there is a scientific ban on theories that Amerindian traditions might have derived partly through ideas from both the Pacific Asian and Atlantic European systems of ocean currents and prevailing winds.
Arguments for a Geometric Derivation
Some commentators upon Thom's megalithic yard (John Ivimy and then Euan Mackie^{[1]}) have noted how such a measure could relate to geometrical ideas found historically in two Egyptian metrological units; the remen of about 1.2 feet and royal cubit of about 1.72 feet. The remen and royal cubit were used to define land areas in Egypt: "On documentary and other evidence Griffith came to the conclusion that the square on the royal cubit was intended to be twice that the square on the remen; and Petri identified the remen as a length of 20 digits" ^{[2]}.
Explains how some have derived Thom's
Megalithic Yard unit of measure
from metrological land measure relationships
established historically in Egypt's Dynastic periods
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as appended to Sacred Number and the Origin of Civilisation.
* This Web page originally appeared in MatrixofCreation.co.uk, hacked then superceded.
This article will be expanded to include other proposed metrological systems, more modules of the ancient system, and so on.
There used to be an interest in metrology  the Ancient Science of Measures  especially when studying ancient monuments. However the information revealed from sites often became mixed with the religious ideas of the researcher leading to coding systems such as those of Pyramidology and Gematria. The general effect has been that metrology, outside of modern engineering uses, has been left unconsidered by modern scientific archaeology.
CONTENTS
 The Twelve Main Measures
 The Role of Accuracy
 The System of Variations
 The Grid for Variations within a Module
 Other Metrological Systems

REFERENCE TABLE OF VARIED ANCIENT MEASURES
Metrology seemed a very complex subject before John Neal and John Michell redefined it in a very compelling and much more simple fashion. All the ancient measures were first found in different regions of the world and so became known by the name of a civilisation or country. This implied and later led to the assumption that these measures had (a) been uniquely developed there in (b) an arbitrary fashion.
But ancient measures are not arbitrary and indeed are all related to a single and unified system. This simplicity would have been obvious had measures not been slightly “varied”, for precise reasons. Aside from these variations, John Neal has identified that the English foot is the basis of the whole system – used as the number one within it – and all the other types of foot are, at root, rationally related using integer fractions of an English foot. What might appear to be a rather partisan approach should be understood in the knowledge that the English foot did not come from England.
It is also important to base such a discussion on the length naturally called feet since, whilst it is only one of many longer and shorter units of length, each such greater length is simply made up of feet according to a formula. Subdividing a foot can yield 10, 12, 16 or other divisions, such as inches or fingers, in different measures. A yard is generally three feet and a pace two and a half. As with a new language the exceptions such as that a cubit sometimes has one and a half feet and other times two can be learnt later. Feet all appear to lie within a given range, plus or minus, of the English foot.
Because the ancient feet largely use low numbers in their fractions of the English and most often are superparticular (where the denominator and numerator vary by one as in 8/7, the royal foot) then many of them represent musical tones and the measures are interrelated in the same ratios found in musical harmony (chapter 2). This is shown in figure one but has not been an important consideration so far in applying this metrological system.
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Contributed by John Neal
Preface by Richard Heath***
John Neal has demonstrated elsewhere [All Done With Mirrors, John Neal, 2000] that ancient metrology was based upon a "backbone" of just a few modules that each related 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 = 176/175 and Ratio B = 441/440. By this means ail 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 differentlysized brochs analysed 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 différent measurement, as we would today, when building a differently sized building. Furthermore, these brochs appear to have been based upon the prototypical yet accurate approximation to pi of 22/7, 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 ancient measures implies that the broch builders had  (a) inherited the 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 foundations available: (b) were practicing a 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 Britain, but are historically associated with locations hundreds if not thousands of miles distant. 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.
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The Prologue
Metrology can seem quite arbitrary in its choice of aggregate measures, that is as to how a given foot measure is divided up into subunits or multiplied into a common range of greater lengths, defined for a given foot or MODULE*.
The idea of a foot module comes from the NEED to create a set of measures NUMERICALLY interrelated to each other, around a foot length equal to one english foot.
It was John Neal who discovered that ALL the modules of Ancient Metrology (discovered in many lands so as to form Historical Metrology) were linked together in small number ratios (i.e. significant but rational differences in length), these rational differences ONLY employing just prime numbers 2, 3, 5, 7, 11, though often being microvaried within each module by larger number ratios such as 441/440 and 176/175 (smaller rational differences) that appear to have had special uses such as providing versions of PI (so as to retain whole numbers between any radius/diameter and the circumference of the circle it defines.)
Therefore, although there were many modules or types of foot in ancient near eastern metrology, each module had exactly the same set of larger aggregates, subdivisions and smaller microvariations. Examples are
 AGGREGATES: Cubits of 3/2 ft, Steps of 5/2 (2.5 ft), Yards of 3 ft, Fathoms of 5 ft, Chains of 22 ft, Furlongs of 600 or 660 ft, Miles of 5000 ft
 SUBDIVISIONS: Digit, Inch (thumb), Palm,
 MICROVARIATIONS: 441/440 =(), 176/175 = (), their sum of 126/125 (= 1.008 ft), their complex product 3168/3125 (= 1.01376 ft), 225/224
It appears that these toolkits of modular lengths were generated using rescaling of certain standard aggregates and microvariations, probably using right triangles to reproportion between modules and microvariations within modules. Please see Appendix Two of Sacred Number and the Origins of Civilization for an idea of how this pattern formed our historical measures.
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The relationship of the radius or diameter of a circle to the size of its circumference is governed by the irrational constant Pi = 3.1415... where the fractional part is endless. This means that, in theory, using a given number of length units to form the radius R will mean that the circumference (2 times Pi times R) cannot be made up of a whole number of the same units. Ancient metrology solved this and other problems by developing a whole range of interrelated units of length  a concept alien to our selves where our unit of length is usually just the metre or the English foot.
In other words, by having a range of units related to one another, a type of calculation was possible that today we would achieve using trigonometry and other techniques based upon the notational mathematics we now use. This makes ancient metrology a candidate for the prehistoric mathematics that is implied by megalithic monuments.
A number of rational approximations to Pi existed in the ancient world but the simplest accurate one is 22/7. which means that a diameter of a circle seven units in length will produce a circumference 22 units long. However, to be really useful, a metrology needs to be able to divide up a circle into any number as required, just as we do when we divide a circle into 360 parts and call the angle from the centre of each, one degree. If we can place 360 around a circle then a degree scale can be produced and how are degree scales made anyway?
Ancient metrology decided on a single unit and called this unit one  a fact revealed conclusively by John Neal in his All Done With Mirrors. Having worked for many years with this fact it continues to reveal the hidden nature of metrological buildings from the Megalithic to the Gothic period.
To build the system, the unit one is then extended or contracted into new units of measure that are a rational fraction of a foot, the English foot (as we call it today). Thus a Sumerian foot in its simplest form is 12/11 feet and this unit can be found in the vertical dimension of the Great Pyramid.
Read more: How Ancient Metrology placed Numbers on a Circumference
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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 interrelate.
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 coincide 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 premetric 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 EtruscanMycenaean 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 subdivisions when sourcing a measure.
Read more: John Neal's integration of the Megalithic Yard (WIP)