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Articles by John Neal

• The Definition of the Measures of Gaul

• Ancient Measurement Systems

• Megalithic Measurement System?

• About the Author

The Definition of the Measures of Gaul

All contents Copyright © 2003 John Neal

In a work entitled Arithmetical Books, written by Augustus De Morgan in 1847
(Augustus De Morgan, Arithmetical Books, sub. From The Invention Of Printing Till The Present Time, Taylor and Walton, London 1847), he neatly identified the primary cause of modern confusion regarding ancient metrology. He stated:
"There runs through all these national systems a certain resemblance in the measures of length; and, if a bundle of rods were made of foot rules, one from every nation, ancient and modern, there would not be a very unreasonable difference in the lengths of the sticks."

The difficulties inherent in the clear definition, indeed, the very identification of ancient modules of linear measure is obvious to any who have researched the subject. The reason for this is that all of the measurement systems of antiquity display a variation in the specific modules, and, as De Morgan observed, many of the modules attributed to separate cultures have very closely related values. In the course of comparisons this often results in the lesser variation of a distinct measure - that is essentially longer than the measure of comparison - to be shorter in length than the greater variations of the lesser measure.

Through an extensive and protracted study of ancient metrology I have been able to propose that, in a general sense, the extent of the variations of a particular module is as much as one part in forty. Yet the Roman scale compared to the common Egyptian measures is forty-nine to fifty, and the common Egyptian is forty-eight to forty-nine of the Greek. They therefore overlap in magnitude at certain of their variations, but by using the methods that I have discovered we are now able to state which of the modules is the target of the writer, architect or surveyor. This is because the variations of the individual modules are intentional, exact and mathematical; not, as has been previously believed, due to an off-hand or arbitrary approach to metrology on the part of historical peoples.

Although it may be stated in a general sense that the range of variations of an individual module are in the region of one in forty, certain of the evidence would imply that this fractional range might be greater; in some cases it can definitely be stated to be the case. This is certainly true of some of the modules termed by historians as Gallic, Belgic, Germanic, Saxon or Northern. As these are mostly taken to be the same basic measurement, it underlines the lack common agreement even as to terminology, let alone concurrence on what are the precise values.

When comparing the linear measures of ancient nations, one encounters many modules that are multiples of the basic digits. Here again, De Morgan's statement contains a broad hint at the methods of analysis to be employed in comparative evaluation of these modules. They should all be compared at the basic foot length; far too often a unit is compared with a foreign multiple-digit (or compound) module to little avail. For example the foot of one nation is often compared with a cubit, pygon, palimpes or some other unit containing a number of the basic digits other than the sixteen to the foot, of another nation. This has been so since recorded history, Pliny the Elder noted that the Roman foot was five ninths of a royal Egyptian cubit. This proves to be the case, but only at certain of its variations and not by the tested method of the addition or subtraction of single fractions, which means that by Pliny's method of comparison, the wrong classifications are linked. (The correct way to view Pliny's observation would be to compare a two-feet cubit of the Roman feet with the royal Egyptian cubit - the linking fraction is then nine Egyptian to ten Roman. This was the practise of ancient metrologists who strove to link number by single fractions, multiple fractions such as five-ninths could not be expressed, but would have to be articulated as one half plus one eighteenth). The importance of this will later be appreciated during the explanation of the difficulties posed in the classification of the Germanic and Belgic modules. The modern search for the definition of the "Megalithic Yard" has been dogged by such misguided comparisons. This is largely caused by the contemporary custom of comparing ancient measures in terms of the yard, inch and even more hopelessly with the metre. Comparative ancient metrology would forever have remained impenetrable when expressed in the metric system. This is because there is no sensible sub-division of the metre that compares to any module in the ancient world, and the extremely close correspondences of certain of the ancient feet cannot be appreciated in terms of the millimetre. This is also true of the habit of the researchers who use the English language - of expressing the modules in decimal inches. They have carried the subdivision one step too far; the comparisons become much clearer when expressed in decimal feet.

The variations of the same module.

Very early in my studies of metrology it was realised that in the course of comparisons that it was an advantage to express the ancient modules in terms of the decimal foot; largely because the English foot is itself an extremely ancient measurement, it therefore lent itself to comparative valuations perfectly. It soon became apparent that the preserved length of the English foot was pivotal to the whole structure of metrology.
It has often been remarked in recent historical texts that the English foot more closely resembles the Grecian than the Roman. This is quite correct, for after extensive comparisons of the varying recorded values of the Greek foot, it became obvious that the English foot was one in the series of the variations to which the Greek foot is subject.
It would appear from most of the empirical evidence that the full range of the variations in a single module, here given in terms of the variations of the Greek-English foot, are as follows: 

Table 1

For reasons dealt with elsewhere, the above terminology is used as descriptive in the classification of the variations. It was realised from the beginning that all of these variations were impossible to express in an ascending order. They must be tabulated in two rows, the fraction linking each of the variations across the rows is 175:176, and each of the values in the top row is linked to the value directly below as 440:441. "Root" prefixes the descriptive terminology from Least to Geographic in the top row and "Standard" in the bottom row. For example, 1.008 is Standard Canonical and 1.0114612 is Root Geographic etc. As well as these values being measurements, they are also regarded as the formulae by which any other module is classified.

It may be argued that the closeness of the disparities would allow for any proposed value to be categorised by using such an inclusive set of variations; the foot length difference between the Root and Standard classification is only .69 of a millimetre and the difference between Root and Root Canonical is .74 of a millimetre and the same close relationships occur between all of the neighbouring values. 

This argument is countered in several ways:

Firstly:
Very close and reliably reported estimates of lengths between the lesser and greater values agree to such a level of equivalence with the proposed values from the above tables, often to a single part in many thousands. Similarly, with the lengthening of the modules, to the level of yards, paces or fathoms etc, the differences become distinctly measurable. It becomes apparent that we are dealing with absolute theoretical values, therefore, without margins of error.

Secondly:
The fractional differences, which have been identified as separating these variations of the same module, have practical mathematical purposes. This purpose would seem to be the maintenance of integers in all aspects of a design. Circular designs in particular, because these fractional variations are related to the values of pi (the ratio between the diameter and perimeter of a circle) that were used in the ancient world. The fractional difference of 175:176 serves the purpose of maintaining integers of the same module in both the diameters and perimeters of circles if the diameters are multiples of four. For example, in discussing the odometer, Vitruvius stated that a wheel of four feet in diameter travels 12 ½ feet in one revolution. This is a very inaccurate pi ratio of 25/8 or 3.125, but is a value that is frequently encountered throughout the ancient world. Far more commonly used is the quite accurate 22/7 or 3.1428571, and the difference between 25/8 and 22/7 is exactly 175 to 176. There is thus an eminently practical reason for this fractional separation in the modules: if Vitruvius' carriage wheel were four Roman feet of .96768ft (175) then the perimeter is indeed exactly 12 ½ feet, but of the related foot of .9732096ft (176).
The fraction 440/441 also serves the purpose of maintaining integers in diameters and perimeters, but of different modules. This will be explained in the text as the examples arise.

Thirdly:
Due to the fractional integration of the ancient national systems, once a module has been identified as an absolute value, all others become so expressible. A widely acknowledged example of this integration was demonstrated by the noted metrologist, Livio Stecchini, as so:

Many more, if not all of the national systems are similarly linked, by a single fraction, but have escaped recognition because of the variations of the individual feet. In order to see the integration of the modules they must be compared at the correct classification, i.e. Root Geographic to Root Geographic etc. Otherwise one is looking at a compound fraction - the fraction that separates the individual modules of comparison, plus the variational fraction(s), they then exhibit no sensible mathematical relationship. As an example of this extended single fraction connection, the Persian foot of 1.05ft (half the "cubit of Darius the Great" of 2.1ft) is added to the above list:

The potential measures

The above expansion of Stecchini's original three comparative measures can be continued through virtually all of the foot measures that are recorded from antiquity. Seldom are all of the values of any module found as examples, but where certain of the values are absent, its acknowledged relative may be present. For example, it is one of the better-established relationships that illustrate this point, that of the Roman to Greek association of 24 to 25. There is no accepted value of the Greek foot recorded (that I thus far know of) at 1.002272ft (1 1/440th), yet 24 to 25 of this value at .9621818ft is a very well attested length of a Roman foot. And so it is through all of the national modules, enough examples of all of them are given to establish the general theory. Where one variant is missing in the record, it is inferentially there because of the presence of its relative neighbour.

The Persian foot was selected to illustrate the expansion of the measures from Stecchini's foundation of three related national modules because it is the most important of the Gallic measures, in spite of the fact that we term the Root value of 1.05ft "Persian": 

Table 2

A fact that I have noted on the nature of cubits is that when they achieve or exceed a length of 1.8ft they become divisible by two, as opposed to the 1½ ft multiplication that is normally associated with the cubit definition. This is the case with the Persian cubit of Darius; it has a two feet division each of 1.05 feet. It, and its variations which are found throughout Asia, Europe and North Africa, I will refer to it as Persian because at this value it is one and one twentieth of an English foot, therefore a Root value. All of its variations, up to 1.064448ft, which was termed a Hashimi foot throughout the Arabian Empire, and pied de roi throughout the Frankish, will be appended Persian. For example, this value of 1.064448ft would be termed a Standard Geographic Persian foot (i.e. 1.05 x 1.01376); similarly, a variant of this foot survives in the English system as being 5000 to the English mile of 5280ft. At this value, 1.056ft, it would be termed a Root Canonical Persian foot (i.e. 1.05 x 1.005714). (That this is indeed the derivation of the English mile will be firmly established by analysing the Gallic leagues).

This has been my method of classifying the variable modules; in the majority of cases, whichever of the variants comes to correspondence with the English foot, or Root, by a single substantial fraction the other variants are appended by that nomenclature. For example, a well attested value of the royal Egyptian foot is one and one seventh of an English foot, therefore wherever its variants be found, be they quite out of context with anything "Egyptian", it will still be termed royal Egyptian because of this association at Root level with the English foot. Classification of measurement has always been the difficulty with metrology, for example what are universally known as "Roman" feet are often referred to as "Attic", implying a Greek, specifically Athenian origin. Almost invariably, measures are called after the location in which they are proven to have been used, implying an exclusive origin from that nation or state. This is misleading, as the evidence points to the fact that all ancient measures form a single organisation and were used universally and concurrently.

Prehistoric Measures of Europe

Persian foot: Ancient Persian measure was based upon the royal Persian cubit of Darius the Great; its length was 2.1ft. The measurement standard later adopted by the maturing Arabic Empire was the Hashimi cubit, given as 649mm. This is two Persian feet at the Standard Geographic value, as described above. (2.1 x 1.01376) This was the measure adopted by Charlemagne as one of the cementing links between his Frankish Empire and that of the great Arabian league of states represented by the Caliph Harun-al-Rashid. Many other values as predicted from the above tables have been very accurately noted, principally by Flinders Petrie, but also recorded by John Greaves as early as 1640. Its variants are known to have been in continuous use from the most ancient records throughout the Middle East and Egypt.
It is widely believed that on the adoption, in 785 AD, of the Hashimi cubit by the Franks as two pied de roi, that this is when it first made its appearance in Europe. But Jacques Dassié has produced some very convincing proofs that the lineage of the pied de roi in France extends into prehistory.

The Roman Foot: M. Dassié has written extensively upon the Great Gallic Leagues. Miles and leagues were the principal itinerary measures used throughout Roman and, apparently, pre Roman Gaul. The term "mile" stems from the Latin mille - thousand, the Roman mile, or mille passum, is therefore one thousand paces.
A pace is composed of two steps and is five feet, the Roman mile being 5000 feet; the league is one and a half miles, or 7500 feet (5000 cubits). He notes the earliest researches into these distances in Gaul, were conducted by Bourguignon d' Anville in 1760, who calculated from the distances between the cities of Gaul a Roman league that equates to 2211 metres. The Standard Canonical value of the Roman foot is .96768ft and 7500 of them equal 2212 metres. But the methods employed by him for his estimates of the distance are considered unreliable. However, he knew of this Roman value.

Using more methodical techniques, in 1770, De La Sauvagére arrived at a value of 2225 metres for the Roman league and 7500 feet of the Standard Geographic classification is 2224.75 metres. There is much to be said in favour of using this Standard Geographic classification as it is often found in the itinerary modules in Britain. Below is the table of the Roman foot variations, the majority of which have been positively confirmed from analysis of buildings, etc. and surviving measuring devices: (table 3)

However, M. Dassié is also critical of De La Sauvagére's interpretations, although I suspect he may be vindicated.

Gallic Leagues other than "Roman": Now things become very interesting. Pistollet de Saint-Ferjeux, in 1858 becomes the first to propose a longer league of pre Roman origin. He is stated to have calculated the league as 2415 metres, and one and a half English miles is 2414 metres. Therefore the basic foot may be stated to have been 1.056ft - which is the Root Canonical value of the Persian foot and directly related to the original pied de roi by a ratio of 1:1.008.

Aurés, the author of 14 memoirs concerning the Gallic league, in 1865 proposed a value of 2436 metres. Liévre later confirmed this distance in 1893; he described the methods he used in this thorough determination following the route from Tours to Poitiers, a distance of 102.3 kilometres. He consulted an ancient document known as the Table of Peutinger, a 13th century copy of a map of the Roman Empire, which stated that this distance was 42 leagues giving the same length for the league as Aurés' 2436 metres. This is within 2½ metres of the original reckoning of the pied de roi adopted by Charlemagne.

This may be regarded as total accuracy; the margin of error over the entire distance of nearly 64 miles is within 100 metres, the original positions of the milliary posts being debatable. Due to the fact that this same level of accuracy is displayed by distances between all of the major cities of Gaul, as measured by Dassié from information given on the Roman milliary columns, it proves two major claims. Firstly the extreme longevity of measures - defined in remote antiquity and surviving into modern times, secondly the accuracy of the ancient surveyors being equal to those of the modern. It also demonstrates how the extremely close values of the original feet become distinctly measurable when reaching these itinerary multiplications. The Persian foot of the 1½ English mile league is 2414m and the league of the Persian foot 1.008 times greater as the pied de roi is about 20 metres longer. (2414 and 2436)

Dassié is something of a pioneer regarding the methods of accurate identification of these itinerary distances. He extensively uses aerial photography in conjunction with global positioning satellites, then verifies these distances on large-scale maps, making all due allowances for curvature etc. He has proven that the Roman inscriptions relating to these itinerary distances are not all recorded in terms of the Roman league. As we have seen, he has accurately identified a variety of distances that were termed "leagues", because they have been repetitively found in the course of many hundreds of comparisons. There are more.

The Belgic Foot: In 15 BC, Nero Claudius Drusus, brother of Tiberius, became governor of the states north of Italy. Gaul had three distinct divisions; they encompassed all of France and overlapped what are now parts of Spain, Germany and the Low Countries. He adopted the foot of the Germanic Tungri as an exchange linear standard with these northern territories and Rome. Their administrative centre was at Tongeren, in what is now Belgium, where their standards were kept. The reason for this adoption is that the Belgic foot was 18 digits to the 16 digits Roman foot. This was a unit already familiar to the Romans through the similarity of Greek system in which the 18 digits module was termed a pygme.

Once again, the acceptance of the 9 to 8 ratio with the Roman is not, strictly speaking, quite correct. This is because, the evidence suggests, they achieve this correspondence at the wrong classifications, as a perusal of the potential values of the Belgic (also called Drusian) foot, depicted below, illustrate: 

Table 4

The Root value of the Roman foot is .96ft, 9 to 8 of this length is 1.08ft, which is a Standard Canonical value. This classification shift is identical to that of Pliny's comparison, previously mentioned - of the Roman module with the royal Egyptian, the 5 of the Roman foot is Root to 9 of the Standard Canonical Royal Egyptian cubit. This comparative integral classification shift is quite common in the science of metrology and hints at depths to the study, which have yet to be fully understood.

A perusal of the above "Germanic" table in comparison to the previous "Persian or Gallic" table perfectly illustrates the points raised in the opening paragraph of this article, concerning the lesser values, as of the above table, overlapping the greater values of the previous. In the past, this made the accurate identification of the intended module a virtual impossibility, leading people to believe the subject is more complicated than it actually is. First, one must identify what is the Root value of a module by comparing it to the English foot; as stated the Persian is 1 1/20th to the English and the Germanic - Belgic is 1 1/14th. The use of these units may be fully substantiated by analyses of the modules used in the construction of ancient monuments, one of the functions of which is the embodiment of standards; they are constructed with such extreme accuracy that it leaves little doubt as to this purpose.

This overlapping of values in the variations of the distinctly separate feet may explain the confusion regarding metrology in pre revolutionary France - because the accepted value of the Hashimi cubit, two original pied de roi, is a maximum, or Standard Geographic classification, yet the values used in France at that time often exceeded this length, where they strayed into lesser values of the Belgic standards. This unsatisfactory definition of standards was the primary reason that led to the development of the metre, a length designed for universal scientific agreement.

Had the French, or the scientific community at large, realised that three feet of the Standard Geographic Belgic foot was more accurately a metre than that which they eventually devised after a hundred years of experiment, much expense would have been spared. But, the Belgic foot would be taken from the table below:

Table 5

If the exchange rate of Drusus is taken literally, and the tables are arranged so that they begin at Root 1.08ft in order for them to be a ratio of 9 to 8 with all of the values of the Roman feet, then the below is an extension of the original table (4). In certain cases therefore, there are more than 10 potential variations of the same measure:

The emboldened numbers are those that are duplicated in tables 4 & 5, 
illustrating how in certain cases the directly related measures extend 
beyond the classifications of the ten values.

That these extended values are valid is confirmed by the fact that an additional value of the Gallic league proposed by Dassié is one of 2490m, and 7500 Belgic feet of (1.08864ft a/a) is 2488.6m. This is a doubly interesting value because, as stated, three of these Belgic feet from the above at 1.094861ft would be the metre that is more precise than that devised by the pre revolutionary French from their measurement of the meridian degree. Many other exact examples of both these modules exist.

The Northern or Saxon foot: This is yet another quite separate module that is often confused with the Gallic-Persian and Belgic-Germanic measures. Close variants of this module are found throughout Eurasia and North Africa. Its rating in England was exactly 1.1 feet and was used almost exclusively for land measurement. The English acre was a strip of land one furlong by one chain; 660 x 66 English feet and the rod or perch was 16.5 feet. In terms of the Saxon foot of 1.1ft these distances make better sense; the furlong is a 600ft stadium, as in all other cultures, the chain is the sexagesimal 60ft and the perch is 10 cubits. Although this measure seems quite straightforward at 1 1/10th English feet, its derivation is rather more complex.
It is the measure that is broadly known as Sumerian, although it must be stressed that all of these nationalistic terminologies as applied to the modules are a modern convention, about which, there is little universal agreement. These Sumerian values are quite accurately represented by examples of every classification. German archaeologists, principally directed by Robert Koldeway, worked intensively in Mesopotamia for many years in the early 20th century. Between them they closely identified the foot lengths represented in the table below:

Table 6

As can be seen, none of these values is exactly 1.1ft. Extraordinarily close is the Standard value at 1.099636 and this was rendered into rational correspondence with the English foot by the addition of another repetitively found fraction, that of 3025 to 3024. There can be little doubt that this convention was observed by the ancient metrologists, so often is it found. Possibly the best example is from that of the most studied and clearly defined ancient measurement - the royal Egyptian cubit. For simplicity, the side of the Great Pyramid is 756ft or 440 royal cubits of 1.71818ft, (Standard) an extremely close value to this is the often-quoted 20.625 inches or1.71875ft, and the difference between them is 3024 to 3025. Although the difference at the cubit length is minute, it would add exactly .25 of a foot to the side of the pyramid were the greater length used; reiterating the statement that the greater the distance measured, the clearer these distinctions become.

The operative perception in these observations is that these "Standard" values of the modules are rendered into rational numbers in terms of the English foot, and from a protracted study of metrology my conclusion is that the English foot is the base one, or datum, from which all metrological calculations stem. This minute adjustment is observable from geographic distances to individual modules. Once again, it is often encountered as an adjustment to the pi value - it is the difference between the universal 22/7 or 3.1428571 and 864/275 or 3.1418181, (the pi value deduced by Fibonacci). It was not ignorance of the nature of pi in the ancient world, but conventions of convenience that governed the choice of the values; they were used in the maintenance of integers or rational numbers.

Just one other example will be given because it is pertinent to the next module to be considered. The British exchange of the variable Spanish vara was that of Madrid, taken as 2.74285ft. As an absolute value, this may calculated as (Root) 2.7428571 and 441 to 440 of this is (Standard) 2.74909ft - (the vara of Mexico is given as 2.749ft). 3025 to 3034 of this Standard value is exactly 2.75ft. Thus, all of the Standard values of the modules, such as the Saxon foot, could be made numerically rational in terms of the English foot.

The Spanish Vara: Gallia Belgica, northern Gaul, was of mixed Celtic and Germanic nations and Gallia Lugdunensis, central Gaul, principally Celtic, is the general area associated with the above modules. Southern Gaul - Gallia Aquitania, was a mix of Celtic and Iberian nations and the measurement most associated with Iberia is the vara. The equivalent of the vara would be 2 ½ feet of the above Sumerian table from which the Saxon measures derived, and this step or half pace was a module that was used widely in northern Gaul and Britain. That it is a step may be established by the fact that it has been demonstrated to have a 40 digit sub division. Its range of values, many of which survived into modern times as variations of the vara, are as follows:

Table 7

However, in Iberia these modules had a three feet sub division, so may more properly be called a yard. Additionally, each foot was divided into 12 pulgadas, or inches rather than digits. Was this division of the vara adopted at the time of the Roman occupation in order to bring the module into line with Roman subdivisions? Probably not, as the foot of the vara integrates very well with many other modules - it is exactly 20 to 21 of the Roman. An alternative division of the vara was one of four palmos, which far better suits the northern interpretation of 2 ½ feet divisions because it is forty digits. This is quite common in ancient metrology; subdivisions of the modules being dictated by convenience; for example, if a module needed to be one and one third of the constituent foot, then it must be sub divided into 12 inches because one cannot have an integral third of 16 digits. I term such modules nodes, where a multiple of one unit exactly corresponds with a different multiple of another.

This correspondence is common at itinerary lengths; for example, a widely used Arabic measurement was a Black cubit, first recorded in Egypt, with a value of 1.77408ft. There were 4000 cubits to an Arabic mile that was also exactly 7000 Greek feet of 1.01376ft. The league in Iberia was 5000 varas, roughly 2.6 miles or 4.2 kilometres. This length is exactly 2/3 of the Egyptian schoenus of 12000 royal Egyptian cubits. The Spanish also used a league that exactly corresponds to the ancient Egyptian schoenus. It was known legua géographique of 7500 varas, taken to be 17 ½ to the degree of the meridian it was made obligatory to use on the scales of maps. This is truly remarkable because Phillip V passed the statute as early as 1718 which was a full 70 years before the French measured the degree to similar accuracy; and this because at certain of its known values it is indeed 17 ½ to the meridian degree. Did the Spanish pre-empt the French in their laborious determination of this length? Or did the Spanish refer to traditions that stretched back to ancient Egypt?
Five palmos of four to the vara is three royal Egyptian feet. There is little doubt that we are viewing a single association of international measures whose origins or dispersal have nothing to do with the Roman Empire.

The Exemplar of the Temple: The first millennium BC is remarkable for mass migrations and wars on what must be a previously unprecedented scale, culminating in the subjugation of Gaul and Britain by the Romans. At the time of the Roman invasion of Gaul the Gallic nations were under extreme pressure from the insurgent Helvetians. In 58 BC, Caesar slaughtered 30000 of them at the battle by the Arar River, later that same year some 150000 were massacred before the remnants of their nation retreated back to Switzerland. In the course of his 8 years Gallic campaigns and his expeditions against the Germanic tribes across the Rhine, upwards of a further million died at the hands of the Romans. Prior to this the Greeks had migrated and slaughtered throughout the world known to them, such unsustainable expansions with casualties of modern war proportions seem to have typified the staggered closing of what we term the Bronze Age.

Even before this it would seem that things were very different and far better organised. All of the disparate nations of Gaul, Germany, Scandinavia, Britain and Iberia would seem to have been under the sway of a unified culture. Even this view is limited as to area, but let us stay within the bounds of the sceptic. Roughly, between 5000 and 1500 BC, throughout this area the megalithic monuments were erected, the most impressive of these megaliths are clustered in Britain and Brittany. There is such uniformity in design and orientation of these great stones they had to be products of a single culture that governed this enormous territory.

At the time of the Roman invasion of Britain, Gallic and Belgic tribes inhabited the southern part of the country; overall, the culture was Celtic whose traditions were maintained by the Druids. Although the Druids conducted ceremonies at megalithic sites, they had had little to do with their erection or design. They had been abandoned as inhabited college sites at least a millennium before the arrival of the Romans, the destruction of, then the deterioration of Stonehenge into a ruin was well under way, and the arts associated with monumental masonry long disused.

Among its other functions, Stonehenge is the monument that best exhibits the role of the temple as a repository of measure. It evolved into its final form over a period of around 1500 years, begun in the fourth millennium BC and completed in the second. Through metrology it may be established that this development adhered to a single theme. Those from the scientific community who have evaluated the geometrical-astronomical orientation of the Henge have established that its precise latitude north governed the foundation of the station stone rectangle. Because by sighting along the lesser sides they are oriented to the most northerly sunrise and the most southerly sunset, and by sighting along the greater sides of the rectangle they are aligned upon the most northerly moonset; and in the reverse direction, the most southerly moonrise. Furthermore, by sighting along the diagonal in a westerly direction it is aligned to the most southerly moonset. If Stonehenge were sited just a few miles north or south of its location, the rectangle would lose its integrity as to this purpose and become a parallelogram.

Most often in ancient structures an inbuilt proportion determines its module of construction. Significantly, the Stonehenge rectangle is Pythagorean in as much that two adjacent sides and the diagonal are in the proportion of 5, 12 and 13; one unit of these integers is a length of ten Standard Geographic Belgic two feet cubits - 21.72342ft (table 4), or 8 steps of 2 ½ of these feet. Overall, the shorter side is 100 Belgic feet, the longer side 240 and the diagonal 260. This rectangle was one of the earliest features of the whole complex; its diagonal length is identical to the diameter of the Aubrey hole circle upon which it is founded. Originally there were four station stones of which only two remain, it is estimated that that they were placed as early as 3000 BC and it was a full 1200 years from this date that the final phase, the erection of the sarsen circle, was complete. Yet, by means of the metrological evidence, all of the phases may be demonstrated to form an elegant and coherent whole.

For simplicity, a no more ingenious device could be conceived than the outline of the sarsen circle and the station stone rectangle for the preservation and exemplification of metrological standards. As we have seen, the station stone rectangle is laid out with values of the Standard Geographic variation of the modules; therefore, one would seek to find other modules throughout the monument at this classification, which proves to be the case.

The most important component of the massive stone circle is the surmounting ring of the lintel stones. Unlike the rough hewed sarsen uprights, the lintel circle is carefully dressed; it may therefore be measured with accuracy. There were thirty uprights and thirty lintels and the lintel width was one thirtieth of the circle's diameter; as measures in ancient monuments are invariably linked to ratios, one should seek the modules of the design in these in-built numbers. Taking as the inner diameter of the sarsens the length from Flinders Petrie's careful examination, it is 100 Roman feet of .9732096ft, which is the Standard Geographic classification (table 3). The outer diameter may therefore be computed be 100 common Greek feet of 1.0427245 feet, also of the Standard Geographic. The width of the lintel, at 3.4757485ft is a double royal Egyptian cubit of the same classification; this could also be legitimately termed a three feet "royal" yard.

Immediately one can recognise the integrated nature of ancient metrology and also gain an insight into its purpose - which appears to be the maintenance of integers in all aspects of a design. This integration continues through all the features of this simple design - the outer edge of the lintel stones, or one thirtieth of the perimeter, is exactly 10 Belgic feet of 1.092378ft the "Drusian foot" at 9/8ths of the Roman (it is one pertica or perch taken from extended table 4). However, this is a Root Geographic value; which is 440 to 441 of the Standard Geographic. These niceties can be appreciated when it is realised that one is dealing with absolute values. Another way to interpret the module of the outer perimeter is to divide the overall length by a number that is traditionally associated with circular design. In this case the canonical 360 - which yields the value of the Mycenaen foot of .910315ft, also a Root Geographic classification. (This value of the Mycenaean foot is that tendered by Stecchini from the 100 feet diameter of the Grave Circle adjacent the Lion Gate of Mycenae).

Only one module fits all of the lintel dimensions at the same Standard Geographic Classification, and this is the common Egyptian foot of .993071ft (6/7ths of the royal Egyptian), at 98 in the inner diameter, 105 in the outer, 308 in the inner perimeter and 330 in the outer. Feet that fit the dimensions in integers:

Many other modules fit these dimensions but they may all be demonstrated to be compound measures, by which it is meant they are multiples of the 6 types of feet as above, or digit multiples based upon these feet. Of particular interest is the Belgic foot regarding the expansion of the monument's proportions beyond the sarsen circle. 
It is illustrated in the - not strictly to scale - diagram below. The module is taken from the numerical proportion is a cubit of two of these Belgic feet - there are 48 in the outer sarsen diameter and 50 in the station stone rectangle, therefore one cubit separates the two above and below the circle:

Many other modules fit this rectangle in integers but the above is the most satisfactory and is the probable design intention, being x 10 the rectangle and diagonal proportion of 5,12,13.

Thus we have seen that the measures of Germania, Gaul and Iberia are identical to the units used in Southern England during pre history, and had been anciently used here long before the intrusions of those who used them at the time of the Roman invasion.

- - - - -

Copyright © 2003 John Neal

Ancient Measurement Systems
- Their fractional integration

All contents Copyright © 2003 John Neal

From the beginning of the 20th century the study of ancient metrology has faded into the background of academic research. Before this, it was a topic of lively debate among the scientific and archaeological communities. It was considered important to clearly define the ancient modules in order to interpret architectural intentions in the ancient monuments, understand itinerary distances, the statements of the classical writers and even biblical descriptions that abound with reference to measures.

Interest in the subject was briefly revived during the 1960s by the claim of Alexander Thom who asserted that the builders of the Megalithic structures had consistently used a common unit of measure throughout the British Isles and Brittany. The claim, to this day, has been neither confirmed nor disproved. This is entirely due to the fact that there is a prevailing ignorance of the subject of ancient metrology. The statisticians who number-crunched the Megalithic data, Thom, Broadbent, Kendall, Freeman and dozens of highly qualified authors, simply failed to recognise the modules that their analyses produced. None of them had made a detailed study of ancient systems and methods of mensuration.

At first glance, the subject seems formidable, causing one learned academic to exclaim that "Ancient metrology is not a science, it is a nightmare". It may seem so, there were a great many modules that were commonly used, various spans, feet, digit multiples such as pygme, remen, cubit and feet multiples - step, yard, pace, fathom, pertica and various bracchia; intermediate measures of the various furlongs and stadia and the itineraries of miles, leagues, schoenus etc. However, the approach to the subject may be simplified by simply considering the basic measure of each system, which is invariably the foot. Augustus De Morgan gave a broad hint at this method of assessment in 1847, he stated:

"There runs through all these national systems a certain resemblance in the measures of length; and, if a bundle of rods were made of foot rules, one from every nation, ancient and modern, there would not be a very unreasonable difference in the lengths of the sticks."

It was simply by comparing the foot lengths of the different national systems that a very elegant order was perceived, leading to the conclusion that all of these "national systems" formed a single organisation. It has become customary for us to name a unit from the society in which it is found to have been used, but most often, the bureaucracies have adopted particular units during the historical period. Obviously a universal system had been fragmented into these various cultures. The difficulties of research are compounded by the lack of agreement as to nomenclature, for example, what are universally known as Roman feet are often called Attic, and at one of its variations, Pelasgo.

Which brings us to the most confusing of all the aspects of metrology - the variations. In all ancient societies, there is a quite broad variation throughout the modules, which has been wrongly regarded as slackness in the maintenance of standards. It would seem that the range of variations and the fractions by which they vary, are not merely similar from nation to nation, but identical. Once these fractions - and the simple mathematical reasons for them - had been established, it became possible to then classify these dissimilarities of the same module. The feet of the various national standards could then be compared at their correct relationships. By seeing the fractional integration through the basic foot structure, many modules could be discarded for comparative purposes, until very few Root feet remained. In fact, there are probably only twelve distinct feet from which all other "feet" are extrapolated. For example the Pythic foot is a half Saxon cubit, and many modules attributed to different cultures are in fact variations of the same basic foot, such as Saxon and Sumerian, or pied de roi and Persian. 

These feet in ascending order, in terms of the English foot are as follows:

• Assyrian .9ft 
  When cubits achieve a length of 1.8ft such as the Assyrian cubit they are divisible by two, instead of the 1 ½ ft division normally associated with the cubit length. Variations of this measure are distinctively known as Oscan, Italic and Mycenaean measure.
• Iberian .9142857ft 
  This is the foot of 1/3rd of the Spanish vara, which survived as the standard of Spain from prehistory to the present.
• Roman .96ft 
  Most who are interested in metrology would consider this value to be too short as a definition of the Roman foot, but examples survive as rulers very accurately at this length.
• Common Egyptian .979592ft 
  One of the better-known measures, being six sevenths of the royal Egyptian foot.
• English/Greek 1ft 
  The English foot is one of the variations of what are accepted as Greek measure, variously called Olympian or Geographic.
• Common Greek 1.028571ft 
  This was a very widely used module recorded throughout Europe, it survived in England at least until the reforms of Edward I in 1305. It is also the half sacred Jewish cubit upon which Newton pondered and Berriman referred to as cubit A.
• Persian 1.05ft 
  Half the Persian cubit of Darius the Great. Reported in its variations throughout the Middle East, North Africa and Europe, survived as the Hashimi foot of the Arabian league and the pied de roi of the Franks.
• Belgic 1.071428ft 
  Develops into the Drusian foot or foot of the Tungri. Detectable in many Megalithic monuments.
• Sumerian 1.097142ft 
  Perhaps the most widely dispersed module of all, recorded throughout Europe, Asia and North Africa, commonly known as the Saxon or Northern foot.
• Yard and full hand 1.111111ft 
• This is the foot of the 40 inch yard widely used in mediaeval England until suppressed by statute in 1439. It is the basis of Punic measure and variables are recorded in Greek statuary from Asia Minor.
• Royal Egyptian 1.142857ft 
  The most discussed and scrutinised historical measurement. Examples of the above length are plentiful.
• Russian 1.166666ft 
  One half of the Russian arshin, one sixth of the sadzhen. One and one half of these feet as a cubit would be the Arabic black cubit, also the Egyptian cubit of the Nilometer.

Variants and variables in the above descriptions are in no wise arbitrary regional fluctuations but follow a distinct discipline. The extent of the variations covers a range of values that amounts to about one fortieth part. Immediately one can see one of the prime difficulties in the identification of ancient modules, because some of the distinct foot values are related by lesser fractions; the Roman is 48 to 49 of the common Egyptian and the common Egyptian is 49 to 50 of the Greek/English. They therefore overlap at certain of their variations, in the course of comparisons this often results in the lesser variation of a distinct measure - that is essentially longer than the measure of comparison - to be shorter in length than the greater variations of the lesser measure. Metrologists continually confuse the Belgic, Frankish and Saxon/Sumerian, the latter has also been appended Ptolemaic. But, the differences become distinctively identifiable at the lengths of the pertica, chain, furlong, stadium, mile etc.

It would appear from most of the empirical evidence that the full range of the variations in a single module, here given in terms of the variations of the Greek-English foot, (the English foot being one of the series of the Greek foot) are as follows:

The above terminology is used as descriptive in the classification of the values. It was realised from the beginning that all of these variations were impossible to express in an ascending order. They must be tabulated in two rows, the fraction linking each of the variations across the rows is 175:176, and each of the values in the top row is linked to the value directly below as 440:441. "Root" prefixes the descriptive terminology from Least to Geographic in the top row and "Standard" in the bottom row. For example, 1.008 is Standard Canonical and 1.0114612 is Root Geographic etc.

As well as these values being measurements, they are also regarded as the formulae by which any other module is classified. That is, any of the listed feet could occupy the Root position in the above table, and all of its variants would be subject to the multiplications of the tabulated values. As an example, the Persian foot when subjected to this process:

Thus, whichever of the measures shows a direct fractional link to the English foot, such as the one and one twentieth, as above, is Root, then the maximum value of 1.064448ft is both the Hashimi foot and the original pied de roi, both could be classified as a Standard Geographic Persian foot (1.05 x 1.01376). Or the given length of the Mycenaen foot at .910315ft could be classified as a Root Geographic Assyrian foot (.9 x 1.0114612ft) and so forth. Then, whenever one is making cultural comparisons of modules, the correct classification must be selected, Root Reciprocal to Root Reciprocal etc. otherwise one is looking at a compound fraction, i.e. the fraction separating the distinctive foot plus the fraction of the variation(s), which may then show no apparent rational relationship.

These fractional separations of the rows and columns have a practical purpose; they are designed to maintain integers in circular structures and artefacts such as storage and measuring vessels. If a diameter is multiple of four or a decimal, by using 22/7 as pi this results in a fractured number perimeter. Therefore 3.125 or 25/8 would be used to give an integer or rational fraction for the perimeter. Accuracy is maintained by using the longer version - by the 176th part - of the measure in the perimeter; this is because 25/8 differs from 22/7 as 175 to 176. Similarly, the fraction 441 to 440 maintains integrity of number in diameter and perimeter, but of different modules. If one has a canonical perimeter number such as 360 English feet, then the diameter will be exactly 100 royal Egyptian feet, but, the royal Egyptian foot that is directly related by a ratio 8 to 7 of the English at 1.142857ft (Root), is supplanted in the diameter by the foot that is the 440th part longer at 1.145454ft (Standard). Another example is to use as a diameter 100 Standard common Greek feet, then the perimeter is 360 Assyrian feet but of the Root classification - the 440th part less. This is clearly indicative of the integrated nature of the original system, the purpose of which was the maintenance of integers in what would mostly be fractured numbers were a single standard measure used, which is what we have today. Ancient metrology is very simply based upon how number itself behaves.

Copyright © 2003 John Neal

Megalithic Measurement System?

All contents Copyright © 2003 John Neal

The existence of a rigidly maintained prehistoric system of measurements, based upon a unit he called the Megalithic yard, was first proposed by Alexander Thom in 1968. He claimed that the Megalithic yard and associated modules governed the construction of Megalithic monuments throughout a wide geographic area. For the first time in 100 years an ancient unit of measurement became internationally newsworthy, attracting both approving and condemnatory responses from the surprisingly wide spectrum of those who were interested. This situation persists until the present day. For far too long, the existence or otherwise of this Megalithic yard has been an unresolved topic of debate among archaeologists. Despite the apparent dryness of the subject, the debate is often acrimonious. The orthodox view is that the idea of precision measurement among Neolithic and Bronze Age societies is preposterous; the progressive view is that a high level of sophistication governed both their science and social structure.

Because metrology has been more or less dropped from the study of both archaeology and the history of science, there are no longer any experts to consult on the matter. This has led to the argument on the validity of the Megalithic yard being conducted by virtually unqualified people in this field. Representatives of either faction act out an absurd "Oh yes it is", "Oh no it isn't", circular impasse of academic bickering; the "evidence" called upon to support either view is scientifically flimsy.

The resolution of the validity of the Megalithic yard can only be reached via a proper understanding of the rules governing ancient metrology. Megalithic culture extends far beyond the regions explored by Thom and therefore overlaps with ancient territories whose systems of metrology are understood. This article is therefore more than an explanation of the Megalithic yard; it requires a brief introductory account of the principles that the evidence suggests, govern historical measurement systems generally.

The Metrological Background

Gaetano de Sanctis, a lecturer at the University of Rome during the 1930s, remarked: "Ancient metrology is not a science, it is a nightmare".¹ The only leap of faith required of the reader is the acceptance that far from being a nightmare, the basic subject might be both simple and sublime. A plethora of modules were used concurrently in the ancient world and there are many differing values for each of them. At first sight the subject appears arbitrary, random and confused. The variable values have for long been regarded as evidence for slackness in the maintenance of standards, but this is not the case. Luca Petto, the renaissance antiquarian, remarked that if any of these variables were found to exactly agree, then this would imply an intended standard.² Seen from this angle, so many of the ancient modules are of precisely the same length that one must conclude that the variations are indeed deliberate. But what exactly are these variations - and what is their purpose?

W.M.F. Petrie was the most prolific, accurate and engaged researcher into ancient metrology. From his Inductive Metrology, (1877), to the publication of his Measures and Weights, (1934), he noted many of the variations evidenced by singular modules, certain of these variations were regularly of the 450th and the 170th part.³ (As evidence now stands, these figures should be exactly refined as variations of the 440th and the 175th part). Petrie had reached his very accurate results by measurement alone, but the solution to ancient metrology is partially numerical in terms of certain absolute measurements, which leaves no tolerance at all in the definition and identification of related modules. That is, Petrie, and the majority of metrologists, have had to express their given values between certain narrow parameters, but once certain measures are unequivocally identified, in an integrated system, all others become expressible as absolutes in the same way.

It is ultimately thanks to an insight of John Michell that it has been possible to make significant advances. In his Ancient Metrology, (1981), he unwittingly established some of the governing principles that reveal the subject as a true branch of science. Much of his work concerns the geodetic appearance of certain of the modules, particularly the Greek, in that certain values of the Greek foot were sexagesimal fractions of the meridian degree. He was not of course the first to point this out, for the hypothesis has been regularly raised over the past 200 years.

Most importantly, Michell recognised that that the ancient systems were most easily interpretable through the medium of the decimally expressed English foot. That is, fractional relationships in comparisons of differing measures are virtually undetectable when expressed in either millimetres or feet and inches, for this reason the systems of metrology have remained impenetrable. (Expressed in decimal feet, one regularly sees the repetitive eleven fraction such as .1818 recurring or the septenary .142857, or 12 multiples 1.728 etc.). Most significantly he was also able to identify the fractional difference between singular measures as 175 to 176 - for example the Roman foot of .96768ft relates as 175 to 176 of the Roman foot of .9732096ft. Furthermore, he noted that this regular separation also occurred in the Greek, the royal Egyptian, the common Egyptian and sacred Jewish systems and modules. Thus, there were absolute values able to be expressed without margins of error, whereby one is able to detect direct linkages between cross-cultural systems.⁴

Livio Stecchini, probably the best-informed metrologist of the latter 20th century, noted many of these fractions linking ancient national standards. He graphically expressed certain of these links in the following manner:⁵

• Mycenaean or Italic foot 15 9
• Roman foot 16 24
• Greek foot 10 25

Had he continued to regard other systems from this point of view he might have recognised that they are all similarly related. Furthermore, of the measures just listed, he never understood that it is the Greek foot that is pivotal to these interrelationships. He was thus unable to recognise the purely numerical structure of the metrological values that he could identify. From his wide experience and ability to make comparisons from an enormously broad array of values, he was able to very closely identify certain definitive values of the Roman and Greek feet. The Roman foot, he claimed, was exactly .9709501ft, and the Greek at the universally accepted 25 to 24 of this value he gave as 1.0114064ft. These values are almost exactly 440 to 441 of the values claimed by Michell, and for reasons adequately dealt with elsewhere it is the values proposed by Michell that proved to be the absolutes.⁶ The higher value Michell stated of the Greek foot is 1.01376ft and 440 to 441 of this value is 1.0114612ft, there is little doubt, therefore, that the value of the Greek foot almost given by Stecchini is one of a series. Because in numerical terms of the English foot this Greek foot is (176/175)2, which is twice the fraction identified by Michell for differing values of the same module; obviously there are considerably more than the two values of each module he has identified. And, most importantly, what we call the English foot is shown to be one of the variables of the acknowledged series of the Greek foot: 1ft x 1.0057143 x 1.0057143 = 1.0114612ft.

It was the numerical structure underlying ancient metrology, had he but recognised it, that first attracted Algernon Berriman to the subject. (His Historical Metrology, (1953), for all its faults, errors and conjecture, remains the most frequently quoted authority on the subject of mensuration - a fact which simply underscores the prevailing ignorance of this highly important branch of our historical inheritance). He noted that that the sexagesimal 129.6 English inches as the perimeter of a circle, expressed a closely accepted value of the royal Egyptian cubit as the radius.⁷ Unfortunately he used true pi to divide the sexagesimal "canonical" number, 129.6, his resultant cubit thereby lost its precise definition. Had he used 22/7 as pi, a figure he clearly knew was an approximation used in ancient Egypt, then, the cubit would have been the numerical absolute. The value arrived at by Berriman is 1.71887ft, but the precise value is 1.7181818ft. This is exactly the length of the lesser value identified by Michell (related to a cubit as 175 to one of 176 equalling 1.728 ft). This is also the value arrived at by Petrie, principally from his examination of the king's Chamber of the Great Pyramid, but he was obliged to give the solution as 20.62 + .005 inches; yet 1.7181818ft is exactly 20.61818 inches.8 (This point of canonical numbers as perimeters in terms of the English values will be laboured here because of its importance in the solutions of the stone circle designs).

If we apply the rule proposed in the explanation of the variations of the Greek foot and reduce the 1.71818ft cubit by its 441st part, then it is exactly 12/7 or 1.714285 English feet. This value has also been recognised as one of the standard expressions of the royal cubit. This royal Egyptian cubit is therefore exactly one English cubit of eighteen inches plus its seventh part. Consequently, since the English foot is one of a series of the Greek feet, then the seventh part added to any of the many values of the Greek cubit will equal a known value of the royal Egyptian cubit.
For many reasons, none of them patriotic, it is the foot called English that is the basis or "Root" from which all calculations involving ancient metrology should begin. It would appear from most of the empirical evidence that the full range of the variations in a single module, here given in terms of the variations of the Greek-English foot, are as follows:

Least  .98867 /.990916
Reciprocal  .994318 /.996578
Root 1 Standard /1.002272
Canonical 1.0057143 / 1.008
Geographic  1.0114612 /1.01376
   

For reasons dealt with elsewhere, the above terminology is used as descriptive in the classification of the variations. It was realised from the beginning that all of these variations were impossible to express in an ascending order. They must be tabulated in two rows, the fraction linking each of the variations across the rows is 175:176, and each of the values in the top row is linked to the value directly below as 440:441. "Root" prefixes the descriptive terminology from Least to Geographic in the top row and "Standard" in the bottom row. For example, 1.008 is Standard Canonical and 1.0114612 is Root Geographic etc.9 As well as these values being measurements, they are also regarded as the formulae by which any other module is classified. For example, an established value of the Persian foot of Darius is 1.05ft. This is one and one twentieth of an English foot, therefore a Root value (at Root, the modules relate to the English foot by a single fraction, 1 1/10th, 1 1/14th, or 9/10 etc.). This Persian foot multiplied by 1.01376 = 1.064448ft which is the recorded value of the Arabian Hashimi foot and the basis of the French pied de roi, this would therefore be classified as a Standard Geographic Persian foot (the Persian being the "Root" value). Few ancient modules are encountered that cannot be classified by the above methods. The principal reason that the regular interrelationships of ancient systems has not previously been recognised is that researchers have been trying to assess the element at the wrong distinction. They are therefore viewing a compound fraction that appears to possess no mathematical relationship, (i.e. the fraction that separates the modules plus the fraction(s) of the module variation). The modules must be compared at the correct variation, that is, in the same column. For example, Freidrich Hultsch spent a lifetime attempting to link the common Egyptian foot of approx. 300mm with the Roman foot of approx 296mm, but since the former is a Root Canonical (300.28mm) value and the latter a Root Geographic (295.96mm), it was an exercise in futility.10 The ratio connecting the two at the same classification is exactly 49 Roman to 50 common Egyptian.

All of the ancient systems to which we erroneously give nationalistic names are similarly connected - by a single fraction. This is in keeping with the ancient systems of mathematics that are a matter of record (multiple fractions such as 4/7ths would have to be expressed as ½ + 1/14th; and depending on the complexity of the original fraction, series of diminishing single fractions approach the solution). This is how ancient metrology fits together, the variables of the singular measures relate to their neighbours by a single fraction, and the measures relate to differing measures at the correct classification also by a single fraction. As the research progressed, such comparisons provoked the obvious conclusion - that the disparate "national" metrological structures were branches of a single system that was originally designed to be used concurrently.

The fractional difference of 175:176 serves the purpose of maintaining integers of the same module in both the diameters and perimeters of circles if the diameters are multiples of four. For example, in discussing the odometer, Vitruvius stated that a wheel of four feet in diameter travels 12 ½ feet in one revolution.11 This is a very inaccurate pi ratio of 25/8 or 3.125, but is a value that is known to have been used throughout the ancient world. Far more commonly used is the quite accurate 22/7 or 3.1428571, and the difference between 25/8 and 22/7 is exactly 175 to 176. There is thus an eminently practical reason for this fractional separation in the modules: if Vitruvius' carriage wheel were four Roman feet of .96768ft (175) then the perimeter is indeed exactly 12 ½ feet, but of the related foot of .9732096ft (176).

The fraction 440:441 also serves the purpose of maintaining integers in diameters and perimeters but of differing modules, particularly if the perimeters are canonical numbers, i.e. sexagesimal and duodecimal solutions - numbers that are traditionally associated with circle perimeters. We saw this with Berriman's resolution of the royal Egyptian cubit, where the canonical number 129.6ins (correctly 10.8ft, also canonical) as a perimeter rightly yields a radius of 1.71818ft. Thus, there is a perimeter expressed in English feet and a radius expressed in royal Egyptian cubits, but the cubit related to the English is 12/7ft or 1.714285ft - 1½ ft + 1/7th, and this is the 440th part less than 1.7181818ft. The best example of this phenomenon is the division of the perimeter by 360; if this is regarded as feet, then the diameter is 114.5454ft, or 100 royal Egyptian feet each of (1ft + 1/7th) +1 /440th. Therefore the module of the perimeter must have the 440th part added to its related module of the diameter, so there is Root classification perimeter and Standard classification diameter, again, the separation of the 440th part is shown to be a corrective fraction related to the pi ratio and designed to maintain integers. This may be difficult to grasp but will be further explained as we analyse Megalithic circles. This rule is consistently observable with all of the canonical perimeter numbers in feet and the differing modules of the diameters.

The Vigesimal Counting Base

The last aspect of metrology to take into account before considering the Megalithic measurements is the nature of these modules other than feet and cubits. The basic unit of any system of measurement is the foot. It may be sub-divided into either inches, or with more versatility, into digits of which there would be sixteen, various multiples of these digits form other modules. Many of these modules persisted in the British imperial system into the 20th Century, but their antiquity was not remarked upon, being largely obscured by the sub-division of the foot exclusively into inches. On modern steel tapes the sixteen-inch division is still marked, but no longer has a terminology such as pygon, remen or pygme. The five feet pace was also commonly used until recent times. The yard, of course, is a double cubit, and a cubit is simply a device to alter the counting base to twos instead of threes, as in terms of the yard and fathom.

In Egyptian metrology there was a 20-digit measurement called a remen. The measurement systems of the Greeks and the Romans are directly related to the Egyptian and both had a 20-digit module, (this is not to say that the digit was the same length - they all differed by the same proportion as did their respective feet). In Greek, this unit was called a pygon and in Rome a palimpes. The double remen was therefore a step (gradus) of 40 digits and the double step was a passus or pace. As there are 80 digits in a pace, it is therefore five feet, each of sixteen digits. There are many digit multiples that were recognised as modules of measure. Knuckle, palm, hand, lick, handlength, pygme, cubit and so forth, and of these it is the handlength that interests us. From the heel of the hand to the fingertips this length, at ten digits, is a half remen, this, to this day, is the basis of the counting system that is still used by the Welsh.

A twenty, or vigesimal, counting base is not peculiar to the Welsh, but was relatively common in the historical world; it is basically made up from two tens. In Welsh, twenty is di-deg - two tens; forty is then two two-tens, or di di-deg; sixty is tri di-deg, or three two-tens and eighty is pedwar di-deg, four two-tens, and so forth. That this method of counting - by the score - has survived from at least the Neolithic and Bronze Age to the present day should come as no surprise; our societies are permeated with ancient cultural distinctions that we unconsciously preserve. As well as cultural differences in the ancient world, there were also very distinct cultural similarities and these are nowhere as obvious as in the counting systems and measurements that have survived from pre-history. It is the measurement system used by the megalith builders that reveals them as "fully paid up members" of their contemporary world. This is because their method of measurement is identical in all respects to that of the majority of the systems of the ancient world; it is not unique to the Megalithic arena.

Alexander Thom, although he identified a module that had been consistently used in the construction of Megalithic monuments failed to accurately identify it. The modern thought processes require us to conform to a single standard; heretofore it has been the yard, rapidly being supplanted by the metre. One must not assume that the inhabitants of the ancient world viewed the subject of mensuration in this fashion. To them the subject was a science in its own right, rather than a simple method of quantifying. It was Thom's modern thought processes that forced him to try to identify a single module and an invariable value of that module. He mistakenly believed the basic measure to be the Megalithic yard, but this is essentially a compound measure. From his examinations of the monuments he identified rather more than the basic Megalithic yard, and taken altogether, these multiples and sub-divisions which he noted distinguish the branch of ancient metrology from whence they originate.

He noted that the stone circles, egg shapes and elliptical structures were often an odd number in terms of his Megalithic yard in their diameters; therefore there must be half-yard module in the radius. A length of two Megalithic yards was detectable in the longer distances, which he termed a Megalithic fathom, and he claimed that the perimeters of the circles were designed to be in multiples of 2½ yards, which he termed a Megalithic rod. If this were all of the information available, then the Megalithic series would have remained an irresolvable enigma, believable only by statistical mathematicians who would be unable to clearly demonstrate its existence to the layman, which has been the case. But Thom recognised another unit that seems to have escaped scrutiny and clearly identifies the system to which the measures belong.

The great stones are often patterned with "cup and ring" markings which are thought to be contemporary with their host megaliths, and it was through analysis of their spacing and geometrical layout that Thom identified the basic sub-division of the longer modules. This was the Megalithic inch - forty to the Megalithic yard. With crystal clarity this enables the whole series to be categorised. All of Thom's appellations of his modules are misleading misnomers. The "Megalithic inch" is clearly a digit, the half-yard is a remen, pygon or palimpes, the yard is a step or gradus, the fathom is the 5 feet pace and the rod is 10 palms, which suggests that the 10 digit palm was the basic module of the Megalithic engineers. 

Note how the versatile digit structure allows for many counting bases, particularly duodecimal, but the Megalithic is clearly the remen or groups of twenty - two-tens.

The Sumerian Cubit and the Megalithic Inch

When the Megalithic "inch" is compared with known ancient metrological systems at the commonly used multiples of 16 to the foot and 24 to the cubit, Thom's measurements are clearly what we call Sumerian measures.

Roman and Greek measures relate to each other by the ratio 24 to 25, Sumerian and royal Egyptian measures relate to each other by the same ratio, the Greek and royal Egyptian measures relate to each other by the ratio of 7 to 8. The Roman measures are therefore 7 to 8 of the Sumerian. In addition to the Megalithic yard, all these "systems" (among others) are significantly represented in British stone circles with the most minuscule margins of error; indeed, to ratios that far exceed the accuracy of 1:400 of which Thom claimed the builders were often capable.

Although no measuring devices survive from Sumeria, the varying values of the Sumerian cubit have been accurately established from the dimensions of buildings, the measurement of bricks, and from cuneiform tablets that record these dimensions. The most widely acknowledged value of the cubit is taken from that of the half-cubit represented as a rule on the statue of Gudea from Lagash, and given as 248mm. Thoreau-Dangin had previously given a related value of 495mm from cuneiform texts and plans of temple dimensions, which he subsequently excavated. As an absolute value it may be stated as 495.93mm, which is the classification Root Least (1.627066ft).12 The other values of the Sumerian cubit drop into their proposed positions of the tabular arrangement - (at the ratios forwarded as above for the Greek feet) - with high degrees of accuracy, all the way to the maximum, Standard Geographic value, of 1.66836ft. This final measure is accurately given by the eastern side of the Ziggurat of Etemenanki at Babylon, as excavated under Robert Koldeway, and given as 91.52 metres, or 180 cubits, (calculated here as 91.532 metres).13 These are the proposed values of the Sumerian cubit that are convincingly in agreement with the empirical evidence:

Least1  .627066 /1.630764 
Reciprocal  1.636363 /1.640082
Root  1.645714 Standard /1.649454
Canonical  1.655118 /1.65888 
Geographic  1.664576 /1.668359
  

Although the Sumerian cubit was most commonly sub-divided into 30 shu-si, Stecchini remarked:

"The texts do not draw any distinction between different types of cubits, except to state that the cubit usually divided sexagesimally into 30 fingers is at times divided into 24 fingers as in the rest of the ancient world".
It is this 24th division of the Sumerian cubit that is the Megalithic inch.

Although the royal Egyptian cubit had 28 divisions, they are not regular, the reasons for which have not yet been satisfactorily explained. However, Petrie, largely by his analysis of artist's grids that were accurately chalk-marked onto the walls of tombs, was able to state that they used a unit in multiples of 1/25th of a royal Egyptian cubit:

"Of these engraved lists the first two have a unit of a decimal division of the cubit; in No. I the spaces are 16/100 of a cubit wide, and 20/100 high, or 4/25 and 5/25; and in No.2 the spaces are 14/100 wide, and 16/100 high, or 7/50 and 8/50. The cubit of No. I would be 20.45 ± .05, and of No.2, 20.58 ± .08. This is of course inferior as a cubit standard to the determinations from large buildings but it is very valuable as showing the decimal division of this cubit, which is also found in other countries".
(Additionally, Petrie has here accurately pinpointed two of the deduced values of the royal cubit, Root Reciprocal is 20.4545 inches and Root is 20.57142 inches).

Obviously this digit is the Sumerian, 24 to that cubit, 25 to the royal Egyptian and 40 to the Megalithic yard. More examples of the application of this Megalithic "inch" are cited below in the course of the Stonehenge description.

But rather than labour through all the given values of the Sumerian cubit to substantiate the general theory, let us move directly on to the module which is the survival of the Megalithic yard into modern times.

The Spanish Vara

In his comparisons of the Megalithic yard with surviving measures, Thom gave much attention to the varied values of the Spanish vara.14 After millennia of non-recognition, the most widely accepted value of the vara, that of Madrid and Castile proves to be the principal, or Root value, of the Megalithic yard. The exchange value of this vara is given as 2.7425ft and the value of a 40-digit module from the Root 24 digit Sumerian cubit is 2.7428571ft, this is a ratio of accuracy in the region of 1:7700. Even this negligible discrepancy is probably to be accounted for by the rounding down of a decimal. The vara is divided into three feet of 12 pulgadas or inches, and in all probability these divisions were adopted at the time of the Roman occupation so as to come into line, so far as counting bases go, with the Roman unica. The alternative division of the vara is a length of 4 palmos, although these would each be of 9 pulgadas, the length would also be that of 10 Sumerian digits or Megalithic inches; ten digits being the palm length of the Greco-Roman measurement systems, and the basis of an essentially decimal digit count. Having identified the Castilian vara as the Root value of the Megalithic yard, the related values may be expressed as follows:

Least  2.711777 /2.717939 
Reciprocal  2.727272 /2.733470 
Root  2.742857Standard /2.749090 
Canonical  2.758531 /2.768400 
Geographic  2.774293 /2.780598

As well as regional variations of the vara within Spain, the Spaniards took them to all of the countries where they settled. Many of them subsequently fell from general use in the home country, but were preserved in the colonies. Thom noted some of these variations, and with additional variations from elsewhere they are tabulated below with their ratios of accuracy to the absolute values (derived from the Sumerian 40 digit module) as listed above:

There are many more variations of the vara that accurately represent all of the proposed values and it is the values of the Root Least, Standard Least and Root Reciprocal that were very closely identified by Thom from his surveys. He admitted that his definitive value of 2.72 feet was obtained by averaging, not realising that the variations that he witnessed were quite deliberate. The widespread practice of averaging in the study of ancient metrology has totally obscured its structure.

Land areas and itinerary distances also yield close values to those proposed in the above table. The length of the legua, a distance of 5000 varas, as used in the American Southwest is given as 2.6305 miles,15 which differs from the Root Canonical by only 17.5 feet. The slight discrepancy may be accounted for by the acceptance of this value as 33 1/3 inches for convenience of conversion, instead of the 33.291 inches, which is the absolute. For a measure that is certainly of prehistoric derivation, it is a great tribute to the artisans, civil servants and scientists, who, over countless years, have maintained these standards.

The Survival of Standards

Petrie speculated that constant copying had caused many of the variations of singular modules. Reasoning that standards, over the years, must lengthen, the copyists would err on the side of generosity because an error could be corrected were the rod cut too long, but would have to be discarded were it too short. This is obviously not the case. It was the custodians of the temples that manufactured and issued weights and measuring devices, as Petrie himself established. Ritual stone rods were kept in the temples as standards from which others were copied or compared. Were these rods ever lost or destroyed, they could be reconstituted from the accurately known dimensions of the temple itself, which among its other functions, was regarded as the permanent repository of measure. Additionally, civic buildings would be of known dimensions and were often engraved with standards of measure upon officially approved stones as checks for market traders and artisans. By such methods, but principally by constant usage, standards remained unchanged for millennia. A new rod could be calculated from known constants.

These facts immediately answer the primary criticisms of Megalithic measures theory:

"How could such a unit be kept standard over more than a millennium and a geographic area of thousands of square miles? What would the standard be made of (wood? stone?) and where would it be kept? How could a population of dispersed migratory tribes maintain standardized measures, or even want to?" ¹⁶

The answer to the first question is that the units would be kept standard in the dimensions of permanent monuments. The second answer is stone, the stones of the monuments. The final criticism is specious, because to hypothesise that the Neolithic people were "dispersed and migratory tribes", remains just that, a hypothesis.

The massiveness and uniformity of the structures that have survived indicate that these people were highly organised from centralised points throughout their territorial range. E.W. Mackie has acutely observed that the situation could not have come about unless there was a centralised training centre whereby instruction would be imparted in the necessary skills to achieve such uniformity.17 He saw the whole scheme as analogous to the Roman Church, with its training colleges and rigidly hierarchical organisation. These centralised training points would be attended by the most able to receive instruction in the necessary skills of astronomy, geometry, geography and mensuration; of necessity, this would be accomplished under the umbrella of religion.

Although statisticians may be convinced of the regular unit construction of the Megalithic monuments, their analytical data is not comprehensible to the non-mathematician. This is because the analysts seem to show no particular necessity for the units that they identify within the monuments - and with which they were supposed to be designed - in any sensible integers within the constructions. This situation is entirely unsatisfactory; far better - John Michell:

"A tradition which has been credited by many learned men over the centuries is that the ancients encoded their knowledge of the world in the dimensions of their sacred monuments. If that is so, any attempt to elicit that knowledge must be preceded by a study of ancient metrology, for to interpret any set of dimensions it is of course necessary to establish the units of measure in which they were originally framed".

Naturally, we should first establish the precise magnitude of that which is under investigation, then seek the sensible integers that fit those measurements. Should those round numbers be discovered in modules that have been previously identified, these may then be considered to be a basis for a sensible theorem. Considered from this perspective, the Megalithic yard, fathom and rod at the values forwarded by Thom should, quite rightly, be dismissed. But if a consideration of the variables of the basic modules that he deliberated shows that they are present in the arrangements of the megaliths to great degrees of accuracy, this should help to convince the sceptical.

Stonehenge

Stonehenge is the monument most amenable to such analysis. It is an intricate and unique structure, but to establish the regular modules used in its design it is not necessary to analyse every feature. The most important element of this complex structure is the sarsen lintel circle, which proves to be extremely informative. In its function as the repository of constant modules, a no more ingenious a device could be imagined. The majority of the stones of Stonehenge are roughly shaped, but the imposts of the trilithons and the lintels of the sarsen circle are carefully dressed. The lintels are far above the ground and by accident or design have been preserved from damage and wear. They are carefully mortised and tenonned to the uprights and tongued and grooved together, and this circle may therefore be calculated in both its intended inner and outer dimensions with the utmost accuracy. Although the lintel circle itself is reduced to six stones of the original thirty, its inner diameter is identical to the inner diameter of the sarsen circle, the most accurate estimate of whose intended length is that of Petrie's examination of 1877.18 He gives this dimension as 1167.9 + .7 inches, which he identified as 100 Roman feet and since 100 Roman feet of the Standard Geographic classification (97.32096ft) is 1167.851 inches, it must be recognised as such. (That such information may be deduced from a monument four millennia after its construction, should answer the question "Where would such a standard be kept"?).

Although the widths of the lintels have been roughly measured, no fully accurate estimate of the outer diameter of the lintels has been made until the publication of Michell's "Ancient Metrology". He gave it as 104.2724571ft, an absolute that can be expressed within the parameters of the estimates of other surveys.19 It is a correct solution for a number of reasons, not the least of which is the fact that temples were constructed to certain proportions whereby the modules of their design were deduced from ratios. It is significant that there are 30 lintels in the perimeter and that the width of the lintel is also one 30th of the overall diameter; the ratio of the inner to outer diameter is therefore 14 to 15. One fourteenth of 97.32096ft and one fifteenth of 104.2724571ft is 6.951497ft, and as a single module this is one Megalithic rod. The constituent Megalithic yard of this "rod" is the surviving length of the Standard Geographic Spanish vara whose absolute value is 2.780598ft. Because there are 14 rods in the inner diameter, there are exactly 44 in the inner perimeter, but the outer diameter being of 15 rods yields a perimeter that is not integral in terms of this module - there are 47.25 rods of the Root Geographic classification of 6.93573ft. Another solution must therefore be sought, in terms of a different module, for the outer perimeter.

The Megalithic yard also fits these diameters very informatively. Because there are exactly 35 in the inner diameter, it betrays the fact that there is a half yard module, (remen), in the radius. But most interestingly, there are exactly 37 ½ Megalithic yards in the outer diameter; therefore, as predicted from the appearance of the Megalithic yard module as a double remen, with a vigesimal counting base there would be a ten-digit palm in this radius.

The dimensions of Stonehenge incorporating Thom's Megalithic rod are entirely unsatisfactory. Although he had identified a value of the Spanish vara in excess of 2.78ft, it never seemed to occur to him to experiment with these increased lengths, as opposed to those that he thought definitive of the Megalithic yard module. He stated that the intention of the design was to contain the sarsens (or lintels) between concentric circles with circumferences of 45 and 48 Megalithic rods. Since the difference between the two is .48 of a rod of 6.806ft, this implies a lintel width of 3.26688ft, which as every survey of the stones reveals, is far too small.20 He thus had to interpolate a whole rod into the inner circumference by using too small a value. The inner diameter he took as 97.41ft and employed a Megalithic yard of 2.7224ft, which fits this length in no sensible integer at all. Small wonder that the rationalists have dismissed his results.

It is beyond dispute that Stonehenge in addition to whatever its other functions may have been, served as a repository of measures. The principal dimensions are designed in values of the Standard Geographic classification, (which means that in order to obtain the parity of the modules with the English foot, they must be divided by 1.01376, which reduces them to Root). Exactly how all the measurement systems of remote antiquity once formed an integrated system is beautifully exemplified in Stonehenge with some of the more interesting modules:

The inner diameter 

100 Roman feet of .9732096ft
98 Common Egyptian feet of .993071ft
96 Greek feet of 1.01376ft
56 Royal Egyptian cubits of 1.737874ft
35 Megalithic yards (varas) of 2.780598ft
14 Megalithic rods of 6.951497ft

The outer diameter 

105 Common Egyptian feet of .993071ft
100 Common Greek feet of 1.0427245ft
90 Royal Egyptian feet of 1.158583ft
60 Royal Egyptian cubits 1.737874ft
50 Jewish sacred cubits of 2.0854491ft
37.5 Megalithic yards (varas) of 2.780598ft
15 Megalithic rods of 6.951497ft

Interesting connections become clearer by such methods of tabulation. Frank Skinner, who had charge of the Weights and Measures Department of the Science Museum during the 1950s, noted that the common Greek foot related as 3/5ths of the royal Egyptian cubit, and since the sacred Jewish cubit relates to the royal Egyptian as 6 to 5 it is clear that the common Greek foot is a half sacred cubit.21 More such loops can be easily spotted in the above table. The sacred Jewish cubit, that of Moses and Ezekiel, expressed as "a cubit and a hands breadth" refers to the royal Egyptian as the basic cubit. Since a hands breadth is 5 digits, the implication is that 5 such hands comprise the royal Egyptian and 6 the sacred Jewish, this digit is therefore the Megalithic "inch": 15 to the "common" Greek foot, 24 to the Sumerian cubit, 25 to the royal Egyptian cubit, 30 to the sacred Jewish cubit and 40 to the Megalithic yard. There is little doubt that we are regarding a single organization of measurement in what has previously been viewed as quite separate systems.

One of the most intriguing solutions to these numerical harmonies is that of the outer circumference of Stonehenge, 327.713ft, being exactly 360 Italic or Mycenaean feet of .910315ft as an absolute. This canonical perimeter number is of the Root Geographic classification, the 440th part less than the Standard Geographic measures of the diameter, this is the corrective fraction, previously mentioned, that maintains integers and happens arithmetically. There are 100 common Greek feet in the diameter and the common Greek foot is a Mycenaean foot plus the seventh part. The modules of diameters are therefore composite measures if the perimeters are basic measures in canonical multiples. A pertinent example is to view the compound module of 100 Standard Megalithic yards as a diameter of 274.90909ft, the perimeter is then the canonical number 864 English feet. It is to be noted that the composite measures of the diameters are invariably expressed decimally and the perimeters duodecimally. These dual counting bases in harness have other metrological properties, but space precludes an explanation here. It would thus appear that the system had been observed or discovered through simple arithmetic, rather than contrived.

The Rollright Stones

Stonehenge is often thought as a great temple with a surrounding necropolis of the burial mounds of Neolithic Bronze Age royalty or heroes. The monument is supposedly unique, with no ancestors or descendents. But a metrological examination of other Megalithic monuments reveal similar, though lesser solutions, in precisely defined values to such degrees of accuracy that they may only be recognised as intentional, and thereby directly related to the design of Stonehenge.

Without considering the complexities of the repetitive shapes of the rings identified by Thom, the existence of a regular measurement system is more simply demonstrable if confined to the straightforward analysis of true circles. The Rollright Stones of Oxfordshire22 have a very obvious and direct metrological relationship with Stonehenge. Although many of the stones had been removed early in the 19th century they were replaced in 1866 when such "restorations" were fashionable, and their intended dimensions may thus be accurately assessed. Given by Thom as 103.6ft, the diameter is within .96 of inch of its presumed value of 103.68ft; because 103.68ft is exactly 175 to 176 of the outer diameter of Stonehenge at 104.27245ft. The modules in the Rollrights are therefore all calculated at the Standard Canonical classification of measures, 1.008 greater than Root. All the elements which fit the Stonehenge dimension therefore fit the Rollrights at this classification. This indicates a margin of error in the region of one part in 1300, which of course is no error at all; any surveyor would allow such latitude. The Megalithic yard of this classification is that of the vara of Burgos, 2.7648ft, at 37.5 in the diameter; the principal "Megalithic" measurement here is likely to be in counts of the 10 digit palmo at 75 in the radius, or, as at Stonehenge, 15 Megalithic rods diameter. The most likely intention at the Rollrights is 100 common Greek feet diameter and 360 Mycenaean feet perimeter. Thom claims that the diameter is 38.1 Megalithic yards, this yields 2.71916ft, which is numerically unsatisfactory and unlikely as part of any design.

The Merry Maidens

The Merry Maidens at St. Buryan is another true circle, also connected to the Stonehenge measurements in as much it is constructed with the maximum, Standard Geographic modules. Thom remarks that it was re-erected, but it comprises stones of such sturdy simplicity that it is certain that it reflects the original design. He gives the diameter as 77.8ft or 28.6 Megalithic yards and perimeter as 35.9 Megalithic rods, which is an arbitrary Megalithic yard of 2.72027ft, and it is little wonder that his reasoning of a constant unit is disputed.23 The diameter of this circle is 80 Roman feet of .9732096ft to within less than ¾ of an inch, but because this is a multiple of four it is unlikely that this is the module with which to interpret the metrological design. (The four multiple cannot give an integral perimeter by the established method of adding the 175th part, because at the Standard Geographic this classification is already the apparent maximum). 80 Roman feet is, however, also 70 Sumerian feet, which yields an eminently suitable 220 Sumerian feet of Standard Geographic of 1.11224ft as a perimeter. This is also 88 Megalithic yards of 2.78059ft (the vara of California) and 35.2 Megalithic rods. Both 88 and 352 are significant numbers in metrology, and it would seem from this observation that Thom was correct in much of his reasoning; in this case, in his claims that the designers were obsessed with maintaining perimeter integrity. Thom believed that they "sacrificed" diameter integrity to sustain this; but with a variable base measure both may be obtained with exactitude. Again, it would seem from this interpretation of the Merry Maidens that the addition of the seventh part to a module has a function related to pi in the maintenance of integers. (Mycenaean plus the seventh is common Greek, Roman plus the seventh is Sumerian and "geographic" Greek - of which the English is a component - plus the seventh is royal Egyptian).

Perhaps the most widely acknowledged connection between what are often wrongly believed to be separate systems of measure, is that of the Roman to Greek at 24 to 25. A Roman furlong of 625 feet is the equivalent of a Greek stade of 600 feet. This too, is related to the pi ratio. One Roman foot radius is six Greek feet perimeter. But once again, there is a classification change in the modules; the foot of the perimeter is the 175th part greater than its relative of the radius. (Berriman noted this exact relationship). Therefore, one Sumerian foot as a radius is six Egyptian feet as a perimeter. Six royal Egyptian feet (or four royal cubits) is therefore 6.25 Sumerian feet, which is exactly the Megalithic rod (2 ½ Megalithic yards which are each 2 ½ Sumerian feet).

The Ring of Brodgar

The Ring of Brodgar on mainland Orkney provides a perfect example of the kind of ambiguities thrown up in the interpretation of measurements. Thom found the diameter to be 340.66 + .44ft and rightly gave the distance as 125 Megalithic yards. However, he took the lesser estimate and took the Megalithic rod to be 6.813ft, stating:

"The reason for using a circle diameter 125 Megalithic yards is that it gives a perimeter of 392.7 Megalithic yards which is very nearly an integral number of Megalithic rods". ²⁴

Of course, it is nothing of the sort; at 157.08 Megalithic rods it is quite meaningless. The correct interpretation of the geometry of Brodgar, however, yields perfect integers in previously identified modules. The diameter is intended 125 Megalithic yards of the Root Reciprocal 2.7272ft and 340.909ft is perfectly within the measured length. As well as being exactly 50 m rods it is also 200 royal Egyptian cubits. Because both are a decimal count the resultant perimeter will be in modules of the Root classification, the 175th part longer than those of the diameter. The ensuing perimeter is 1071.4285ft and the temptation is to leave it there, in that it is 1000 feet of 1 1/14th English feet, and although this module is difficult to categorise, similar and related lengths are reported from many ancient sources. It is in perfect agreement with other aspects of metrology by being at Root an increase of the English foot by a single fraction. At certain of its values it overlaps, exactly, the so-called Germanic or Drusian foot, which is 18 digits to the Roman 16.25 In Greek terminology this Germanic foot would be termed a pygme at its Roman reduction. The perimeter of Brodgar is also exactly 625 Royal Egyptian cubits of 12/7 English feet, but this is a numerically unsatisfactory number for a circumference. Certainly, this length cannot be interpreted in terms of either the Megalithic yard or the Megalithic rod, but the most interesting aspect of Brodgar is perhaps that it has exactly the same diameter as the two inner circles of Avebury.

This gives credence to the claims of cultural regularity and wide dispersal of monuments that had an obvious scientific purpose, if only as an exercise in geometry. Obviously it was much more. This is apparent by what we know of the enormous difficulties that were experienced in the development of the metric system, which can only be used to quantify. In comparison to the system of antiquity, it is childishly simplistic.

From these few examples we can see how modules that have been precisely defined elsewhere, fit British Megalithic geometry without margins of error. The plausibility of the argument is supported by the fact that the modules that have been identified also fit the monuments in rational sets of numbers. Such is the integration of metrology. Are we then dealing with a "Megalithic" system of measures? Probably not - nor any other that could be labelled. It would seem that all of the systems which we brand with a nationalistic nomenclature are simply contrived from singular standards which have been drawn from an older complete cosmology by the bureaucrats who sought to quantify, tax and regulate the societies into which it was fragmented. Were it not for the survival of the vara into modern times then Thom's claims of a consistent unit would have remained forever enigmatic. For all of the evidence that Thom brings to bear on the subject, from his own statistical methods to those of Broadbent, Kendall and Freeman, whom he cites, they remain virtually useless in the definite establishment of the Megalithic yard.26 Because for all the fine terminologies in which statistical methodologies are cloaked, they simply remain other methods of averaging. Averages are the enemy of precise definition, and in most cases are misleading; they have certainly obscured the clarity of ancient metrology.

Although the few examples that have been included here are persuasively indicative of a regular and systematic usage of a single system across a wide geographic area, only around 30% or so of Megalithic monuments (apparently) conform to the numerical scheme as outlined. So the argument for the Megalithic yard remains open, but a greater beast has been unleashed.




__________________________________________________
REFERENCES

1 Stecchini, L.C. http://www.metrum.org/measures/whystud.htm 
2 http://www.metrum.org/measures/romegfoot.htm 
(Few works are published by Stecchini, but his memorial web site is the equivalent of several books)
3 Petrie, W.M.F. Encyclopaedia Britannica, Eleventh Edition, 1910, (Ancient Historical) Weights and Measures, p 481
4 Michell, J. 1981 Ancient Metrology, p 17
5 Stecchini, appendix to Tompkins, P. Secrets of the Great Pyramid, 1971, p 352
6 Neal, J. All Done With Mirrors, 2000, pp 69-75
7 Berriman, A.E. 1953 Historical Metrology, pp18-19
8 Petrie, W.M.F. 1883 The Pyramids and Temples of Gizeh, ch 20, para137
9 Neal, J. http://www.secretacademy.com/pages/greeksystem.htm 
10 Stecchini, appendix to Tompkins, P. Secrets of the Great Pyramid, 1971, p 309
11 Vitruvius, On Architecture, Book 10 ch. 9 (Trans Frank Granger, 1934, from the Harleian manuscript).
12 Skinner, F. Weights and Measures, 1967, HMSO Science Museum, p 41
13 Stecchini, L. http://www.metrum.org/measures/length_u.htm  para 7
14 Thom, A. and Thom, A.S. 1978, Megalithic Remains in Britain and Brittany, p 43
15 Rowlett, R. How Many? A Dictionary of Units of Measurement, (revised 2001) University of N. Carolina at Chapel Hill
16 Knorr, W. R. The geometer and the archaeoastronomers: on the prehistoric origins of mathematics. Review of: Geometry and algebra in ancient civilizations [Springer, Berlin, 1983; MR: 85b:01001] by B. L. van der Waerden. British J. Hist. Sci. 18 (1985), no. 59, part 2, 197--212. SC: 01A10, MR: 87k:01003.
17 MacKie, E. 1977, The Megalith Builders, Phaidon Press
18 Petrie, W.M.F. 1877 Stonehenge: Plans, Description, and Theories. p 23
19 Michell, J. 1981 Ancient Metrology, p 20
20 Stone, E. Herbert. 1924 The Stones of Stonehenge, p 6
21 Skinner, F. Weights and Measures, 1967, HMSO Science Museum, p 35
22, 23, 24 Thom, A. and Burl, A. 1980, Megalithic Rings
25 Skinner, F. Weights and Measures, 1967, HMSO Science Museum, p 40
26 Thom, A. and Thom, A.S. 1978, Megalithic Remains in Britain and Brittany, p 4

About the Author

The author, John Neal,  has no qualifications whatever, either as a mathematician or as a writer. This may make it difficult for academicians either to read this book, or to accept the findings herein. But were I a scholar, then I may have had a vested interest in not writing it at all. Because the content so contradicts the prevailing orthodoxy regarding our historical origins, that it may have been career-threatening to reveal it from within the ranks. Having no career to threaten, I communicate this good news with enthusiasm
This is written in my sixtieth year, and I have had nothing more than a pure interest in history and archaeology since childhood. Ancient buildings, monuments, standing stones and earthworks have always exerted a solemn fascination upon me and I was ever aware that they embodied something communicable within their mystery. This communication was finally established through an understanding of the units of measurement that were enshrined in their proportions, they speak in numbers. The architects speak to us across the ages through this forgotten language, as clearly as though their voices were heard in their antique tongue. Although the final solution proved to be utterly simple, the convoluted paths which had to be followed towards the resolution of the enigmas of metrology, defy description. Countless hours of calculation and comparisons over a period of some thirty years, finally yielded a priceless pearl that becomes brighter the more it is contemplated and handled. If the reader does not appreciate this, it is entirely due to the inexperience and the presentation of the author. I recommend that those who have difficulties with numbers resolve them and read on until they do understand.

Related Links

• ALL DONE WITH MIRRORS 
• Ancient Metrology
• The Great Pyramid
• World Mysteries - The Great Pyramid