# Contributed by John Neal

### Preface by Richard Heath***

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

These modules each had a set of identical variations within, based on one or more applications of just two fractions, Ratio A = 176/175 and Ratio B = 441/440. By this means ail the known historical variations of a given type of foot can be accounted for, in a table of lengths with ratio A acting horizontally and ratio B vertically, between adjacent measures.

In the context of what follows, this means that each of the differently-sized brochs analysed by Neal appear to have used a foot from one or other of these ancient modules, in one of its known variations. That is, the broch builders seem to have chosen a different unit of measure rather than a différent measurement, as we would today, when building a differently sized building. Furthermore, these brochs appear to have been based upon the prototypical yet accurate approximation to pi of 22/7, so that - providing the broch diameter would divide by seven using the chosen module - then the perimeter would automatically divide into 22 whole parts.

Thus, John Neal's discovery that broch diameters divide by seven using a wide range of ancient measures implies that the broch builders had - (a) inherited the original system of ancient measures with its rational interrelations between modules and variations within these, from which they could choose, to suit a required overall size of circular building, often the foundations available: (b) were practicing a design concept found in the construction of stone circles during the Neolithic period.

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

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

*** [note by RDH]I wrote this for Euan MacKie who had resurrected his work on measures found within the brochs of Scotland. Euan was almost a lone voice in support of Alexander Thom's work on metrology in the megalithic, and also the long distance alignments in the Western Isles of Scotland. When he met John Neal at the latter lecture in Glasgow, at which I was present, they appear to have entered into a review of the data and John Neal came back with an interesting theory which would make a full range of historic measures to have been employed in one area of northern Scotand, in the Iron Age. I sent Euan a summary of what ancient metrology appeared to be as a system of ratios and why Neal's finding within MacKie's data would be important. It became the preface for the article called The Roundhouses, Brochs and Wheelhouses of Atlantic Scotland c.700 BC-AD 500: Orkney and Shetland Isles Pt. 1: Architecture and Material Culture (British Archaeological Reports British Series) which I have recovered from a partial proof copy.

Throughout Scotland and the Scottish islands there are in excess of 200 major broch sites. The following analysis is taken from, what I believe to be, the accurately measured inner diameters of 49 of them as supplied by Professor Euan MacKie. The modules are expressed in English feet although the original measurements were taken in metres and converted to feet at the rate of 3.2808427 feet to the metre. The range of diameters extends from the smallest, at Mousa, 18.897654ft, to the greatest at Oxtrow at 44.816311ft.

The evidence would imply that a professional class of masons were employed in their construction throughout the area of their range and the time span of their unique design. The system of measurement employed in the brochs, both in the module lengths and the methods of application, is identical to that of the preceding megalithic -Neolithic/Bronze Age societies, and to the cultures that succeeded them. The most interesting fact that clearly emerges from the cumulative evidence is that the builders applied certain formulaic procedures in their plans. The vast majority of the diameters are multiples of seven in terms of the various feet that are used; and these diameters become exactly seven when known multiples of these feet are employed. For example, diameters that are 21 feet would be seven yards (ancient metrologists sometimes expressed the yard as a double 1½ ft cubit); if they are 28 feet they are seven double two-feet cubits; at 35 feet they are seven five-feet paces (double step) and at 42 feet are seven fathoms.

## INTRODUCTION

As the size of the brochs increase, the numbers of the modules do not; the module itself increases in order to maintain the numerical formulae. The first six brochs of the list illustrate this point:

Mousa; 18.9ft = 21 Assyrian feet of .9ft.

Nybster is 21 English feet.

Ousedale Burn; 21.84756ft = 21 common Greek feet.

Castle Cole; 22.176ft = 21 Persian feet of 1.056ft.

Armadale Burn; 22.94ft = 21 Belgic feet

Dun Carloway; 24ft = 21 royal Egyptian feet of 1.142857ft.

The following 20 brochs, with a couple of notable exceptions, from number 7 in the list, Kiess North, at 28.8934ft to number 27, Clachtoll, at 31.36ft, have diameters that are each of 28 feet which range from Iberian to archaic English.

From number 28, Midhowe, to number 31, Loch of Huxter, are each of 35 feet of the Assyrian variants; the diameters of the next three, from 32 to 34, revert to being 28ft of the greater measures, royal Egyptian and Russian. From numbers 36 to 46 the diameters are all of 35ft in terms of the range of possible measures between the lesser Roman values ascending to the greater values of the royal Egyptian. Finally, when the diameters exceed 40 whole English feet, the division of the final three brochs of the list, are in terms of 42 feet of the common Egyptian and the Persian standards.

The dimensions of the brochs with the measured values and the theoretical absolutes are as follows:

#NameDiameterPerimeter
Notes
1 Mousa 18.897654 59.392627
This diameter has a clear resolution in terms of the Assyrian Root foot of .9ft at 21 or seven yards.
2 Nybster 20.997393 65.991807
Obviously seven three-feet yards diameter, perimeter 22 yards or 66ft. Module - English foot.
3 Ousedale Burn 21.850412 68.672725
The module here is the Common Greek "yard", each three feet of 1.04036ft (Root Geographic) and 22 such yards perimeter or 66 common Greek feet.
4 Castle Cole 22.178497 69.703847
Seven yards of The Persian foot of 1.056ft, (the 5000th part of the Statute mile). Again, 22 such yards or 66ft perimeter. This foot is often encountered in the Gallic 7500 feet leagues.
Obviously this diameter is seven "metres". The metre is composed of three Belgic feet; it is impossible to say if they are Standard Geographic, at 3.284582ft or Root Geographic at 3.277134ft. A plus or minus value on the original measurement would resolve it.
6 Dun Carloway 24.015769 75.47813
One seventh of this diameter (24ft) is three royal Egyptian feet of 1.142857ft.
7 Keiss North 25.918657 81.458637
25.8934ft is 28 feet of the Root Geographic Iberian foot, this is within 1/3 of an inch of the measured length.
8 Dunrobin Wood 26.377975 82.902208
This is eight yards composed of what are termed Sumerian feet of 1.099636ft, yielding a perimeter of 75 Sumerian feet of 1.10592ft, 50 cubits of 1.65888ft or 30 steps of 2.7648ft. This is also the Spanish vara of Burgos, therefore it additionally has a three feet subdivision, so this perimeter could also be viewed as 90 Iberian feet.
9 Brae 26.574826 83.520881
The same solution must be forwarded here as that of Dunrobin Wood, whereas the margin of error at that site is around one part in 2000, the margin here would be one part in 800.
10 West Burra Firth 26.90291 84.552003
11 Borwick 26.90291 84.552003
The most likely solution here is a Roman foot module of .96ft because these feet at 28 to the diameter equal a rational 88 feet perimeter, which is a module number often encountered in the older megalithic monuments. The margin of error is ¼ inch overall.
12 Backies 27.099761 85.170676
This too would have the 28 Roman feet solution, very accurately at 99.983 percent in terms of the Standard Canonical Roman foot of .96768ft.
13 Dunbeath 27.821546 87.439145
With less than a quarter inch error on the diameter, this is 28 Standard Geographic common Egyptian feet of .993071ft (6 sevenths of the royal Egyptian foot). Therefore giving the same numerical solutions as the preceding five circles. Many other modules are compatible with this diameter, it is ten Spanish varas as used in California, 25 Sumerian feet and 24 royal Egyptian feet.
14 Levenwick 27.95278 87.851594
It is difficult to see this as anything other than an intended 28 English feet; it is about ½ inch short.
15 Dun Troddan 28.084014 88.264042
The same applies to this circle, in this case it is an inch too long.
16 Brounaban 28.14963 88.470267
This measure is identifiable as a Root Canonical Greek foot of 1.0057142, as identified by Martin Folkes in 1736; he had noted it as being engraved on a Standards Stone at the Roman Capitol. At 28 to the diameter it offers the same numerical interpretation to the previous circles, as do the following.
17 Dun a' Chaolais 28.8058 90.532511
At exactly 28.8 feet this is 28 common Greek feet at Root Classification.
18 Jarlshof 29.461967 92.594755
29.4668ft is 28 Standard Persian feet of 1.052386ft.
19 Howe of Hoxa 29.724435 93.419652
This length is about a 10th of an inch short of 28 Root Geographic Persian feet of 1.062034ft.
20 Kylesku 29.986902 94.24455
In this case, the diameter is 28 Root Belgic feet of 1.071428571feet, again with about a tenth of an inch in excess in the measured overall distance.
21 Clickhimin 30.24937 95.069448
28 Standard Canonical Belgic of 1.08 feet would be 30.24ft.
22 Burrian 30.77430 96.719243
23 Carrol 30.77430 96.719243
24 Caisteal Grugaig 30.77430 96.719243
These three are within a fifth of an inch overall of 28 Standard Sumerian feet (the basis of the Saxon or Northern foot). Additionally, the perimeters would also be 100 Standard Canonical Roman feet of .96768ft.
This is accurately 28 feet of the Standard Canonical Sumerian foot of 1.10592ft, this perimeter taken as 97.32096ft is the precise inner diameter of the Stonehenge Sarsen circle; 100 Standard Geographic Roman feet.
This distance at a little over half an inch either way would make it either the Standard or Root Canonical classification of the archaic foot of the "yard and full hand" at 1.11111ft.
27 Clachtoll 31.364856 98.575262
This is very clearly 28 of the Standard Canonical classification of the same foot, at 1.12ft.
28 Midhowe 31.627324 99.40016
29 Borrowstone 31.627324 99.40016
Were these circles about half inch longer at 31.68ft, when divided by the next multiple of 7 at 35, it equals the Root Canonical value of the Assyrian foot of .905142ft; thereby making the perimeter rational to the same classification of this foot at 110. This number lends itself ideally to the 2½ ft "step" division at 44 in the perimeter.
30 Yarrows 31.889791 100.22506
This circle offers the same numerical solutions in terms of the Root Geographic classification of the Assyrian foot, 35 of which equal 31.861ft.
31 Loch of Huxter 31.955408 100.43128
The Standard Geographic value used in the same way would yield a diameter of 31.9334ft.
33 Dun Telve 32.283492 101.4624
If these diameters are taken as 32.256ft, about one third of an inch short of the measured distance, there would be 28 Standard Canonical royal Egyptian feet of 1.152ft. Interestingly, the perimeter would be exactly 100 Standard Geographic Greek feet of 1.01367ft.
34 Achvarasdal Lodge 33.070894 103.9371
The solution to this circle is 28 feet of the Root Geographic Russian foot which would be equal to 33.04106ft - less than a half inch difference of the measured length.
35 Dun Boreraig 33.267745 104.55577
At a little over an inch too long on the diameter, it is proposed that this is the least accurate circle yet dealt with at around one part 400. The diameter may be 32 Standard Canonical common Greek feet of 1.0368ft, this would yield a perimeter of 100 Standard Geographic common Greek feet of 1.0427245ft. This is the outer diameter of the Stonehenge lintel ring.
36 Gurness 33.333362 104.76199
This is exactly 440 to 441 35 Root Reciprocal Roman feet at .952381ft.
37 Carn Laith 33.464596 105.17444
35 Root Reciprocal Roman feet, error 1:600 or 1/3 inch.
38 Clumlie 33.661446 105.79312
33.6ft would be about ¾ of an inch short of 35 Root Roman feet. (1:550).
39 Dun Osdale 33.858297 106.41179
This diameter is only about a tenth of an inch short of 35 Roman feet of .96768ft at 33.8688ft. (Standard Canonical).
40 Keiss West 33.98953 106.82424
This length is also at an ideal 33.985 thirty five Roman feet of the Root Geographic .9710ft
41 Forsinain 34.120764 107.23669
This circle is less than ¾ of an inch too long to be exactly 35 Standard Geographic Roman feet of .9732096ft.
42 Dun Beag 35.367484 111.15495
This circle is less than ½ inch short of being 35 Root Geographic Greek feet at 35.4ft.
43 Burray East 36.08927 113.42342
At 36.0818 feet this length is 35 Standard common Greek feet of 1.030909ft, about one tenth of an inch difference on the measured diameter.
44 Keiss South 38.320243 120.43505
There are two possible interpretations of this length; it could be 35 feet diameter of the root Sumerian foot of 1.097142ft, in which case it would be an error of slightly under an inch at 38.4ft. Alternatively at the same length, it could be 28 cubits of the Iberian Root value or 14 varas.
45 Torwoodlee 39.304496 123.52841
At 39.3346ft, about 1/3 of an inch difference, it is equal to 35 feet of the Root Geographic archaic English yard and full hand.
46 Thrumster 40.223132 126.41556
Almost exactly, at 40.22857ft, there would be 35 Root Canonical royal Egyptian feet.
47 Tirefuar 41.732319 131.15872
At 41.709ft this diameter would be 42 common Egyptian feet of .993071ft. It is also 40 common Greek feet of 1.04272ft. The former is the more likely interpretation, as the length would be 28 cubits or 7 fathoms of this foot.
48 Dun Ardtreck 44.488227 139.82014
This measurement is taken from the surviving semi-circle. At 44.4528ft, within less than ½ inch of the measured length, this is exactly 42 Standard Canonical Persian feet of 1.0564ft. This offers the same numerical interpretation as Tirefuar.
49 Oxtrow 44.816311 140.85126
At the value of an extended Persian foot times 42 at 44.808422 - this is less than 1/10th of an inch from the measured value; it offers the same numerical solution.

At the value of an extended Persian foot times 42 at 44.808422 - this is less than 1/10th of an inch from the measured value; it offers the same numerical solution.

Although the above interpretations of the broch dimensions are the simplest, therefore the most likely solutions, within such a tightly related organisation of measure alternative resolutions are possible. Site 1, Mousa for example, although this broch is seven "yards" in terms of the Assyrian foot it may also be viewed as seven steps, the 2½ ft module, whose detection in megalithic monuments gave rise to the belief in the "Megalithic Yard". At Mousa the step would be 2.5 Belgic feet of 1.08ft, therefore 2.7ft. Other values of this Belgic foot as well as variants of the Sumerian feet would yield a range of measures acceptably close to the hypothesised 2.72ft Megalithic Yard. For reasons that have become obvious, it is folly to attempt to define such a module as a habitually employed element of the megalith builders. If the seven division of no. 7, Keiss North, is relinquished for a division by eight, it would be eight Belgic yards whose constituent foot is 1.08ft, the perimeter would then be an integer in terms of modules the 175th part longer. This perimeter would then be 25 such yards composed of feet of 1.08617ft; this could also be expressed as 30 steps of 2.715428ft, the measure recently described by Robin Heath as the "Astronomical Megalithic Yard".

It is also noted that not all of the diameters can be expressed in multiples of seven. Numbers 8,9 and 35 may only be divided by multiples of eight. It is unclear why the seven counting base for diameters is sometimes abandoned; but it is often encountered in ancient metrology. Perhaps there was some compelling reason that a broch or circle had to be exactly a particular size, leaving but small choice as to the module.

There is also a distinct possibility that certain canonical lengths should be expressed in the constructions. Echoes of a far older metrological discipline are perpetuated in certain of the brochs. I had previously noted that certain lengths seem to be equally comfortable when expressed as either a perimeter or a diameter in circular structures. The examples of such occurrences in the brochs are the perimeters of no. 25, Kintradwell and no. 35, Dun Boreraig; they are respectively the inner and outer diameters of the Stonehenge lintel circle. The evidence suggests that such metrological standardizations were common in the Iron Age. One example being the wheel gauge of the chariot recently excavated in Yorkshire; remarkably, at 1.45 metres it is identical to that of the Edinburgh Iron Age chariot burial. This is the lesser value of the five Roman feet "pace", as found at broch 36, Gurness, the diameter of which would be 7 such five feet paces of the chariot gauge. This particular gauge is found in wheel ruts, whether they have been inadvertently or deliberately cut, throughout the ancient world. Notable examples occur in Pompeii, Malta, Corinth and Persia.

Yorkshire Iron Age chariot in situ

Such observations as have been made here concerning the broch dimensions with regard to eliciting the rational sets of numbers, can be equally accurately applied to the older megalithic circles. As indeed they may be applied to interpret later cosmologies, such as the Saxon "King's Girth" or older biblical metrology such as the dimensions of the Mosaic Tabernacle. The Romans used the same criteria in founding their towns, as did those they supplanted. Recent excavations at Silchester have revealed an Iron Age street grid that is in all respects similar to the imposed Roman, but angled at a 45 degrees slant.

The statement, that the system of measures has been accurately maintained from very remote antiquity until the present day is very easily demonstrated. As the best preserved megalithic ring in Britain is deep within the domain of the brochs, on Orkney, the Ring of Brodgar, it is as good a demonstration of this fact as could be wished for. Alexander Thom gave the diameter as 340.7 + .44ft and stated that this was 125 Megalithic Yards. At 340.90909ft, exactly within the measured range, this would yield a Megalithic Yard of 2.727272 feet (Root Reciprocal); this is the vara as preserved in California and is 175 to 176 to the vara of Castile. At precisely 2.742857ft this is the official standard used by the Spanish bureaucracy until very recent times.

However, if one divides this diameter by seven, a more rational module emerges. One seventh of the Brodgar diameter is seen to be 10 five feet paces whose constituent foot is the Root Reciprocal value of the Common Egyptian foot of .97403ft, the perimeter is consequently 1100 common Egyptian feet or 220 paces. Even more obviously this perimeter, at 1071.428ft is exactly 1000 Root Belgic feet giving a closer pace to the human equivalent of 200 at 5.35714ft. It was the detection of this sort of module that led Alexander Thom to call it the Megalithic Fathom, which he tried to pin down to a constant of 5.44 feet. There is no such constant; each ring must be dealt with individually and its metrological solution sought in the rational numbers that emerge. This was the major oversight that prevented Thom from pinning the system down; the fact that he showed no particular preference for his solutions to be in rational numbers of his proposed module.

The fact of the matter is, that the vast majority of the megalithic rings can be metrologically interpreted by the methods that have been demonstrated on the brochs. All that is necessary is knowledge of ancient metrology, the module lengths and multiples, which is nowadays universally lacking. Sadly, this is a development that has come about in the last half-century, it was not always so. Until the demise of Flinders Petrie the majority of archaeologists had a fair working knowledge of the subject. In the older editions of encyclopaedias such as the 1911 and 1915 editions of Britannica, Petrie wrote very extensive articles on the subject, in which he identified and listed in ascending order, examples of all of the modules discussed here. In modern editions scarcely a paragraph is devoted to the subject.

The broch builders therefore preserved methods and modules that had been used by the megalith builders that predated them by millennia and the same modules survived into the present epoch. Although the instruments of measurement may wear out, the standards by which they manufactured them would be accurately maintained in the dimensions of that which was already built. The conjectural purposes of brochs, as well as being the residences of chieftains, council chambers, courts, temples or redoubts could also have been the Weights and Measures bureau in its very dimensions.

The reason that we can now be certain about claims concerning metrology, is that we are dealing with absolute values. No longer may the subject be regarded as arbitrary nor conjecture be utilised to substantiate hypotheses. One very good example of the solidity of the theory is the regularity with which the Assyrian variants occur in all cultures. Oppert positively identified the Root value of .9 English feet from measurements of the ruins of Khorsabad. The value of the 175th part longer at .904514ft is exactly given by the copper bar of Nippur, at four feet long it is reported as 1.1035 metres and four times .904514ft is 1.103549 metres. At the next value in the series, the 175th part longer again, it is exactly the 360th part of the outer perimeter of the Stonehenge lintel ring. This particular value was precisely given by Stecchini taken from the diameter of the Grave Circle at Mycenae. These and other values of the Assyrian foot are also referred to as Oscan, Italic and Mycenaean. It therefore comes as no surprise to find it so prominently in the broch dimensions at Mousa, Midhowe, Borrowstone, Yarrows and Loch of Huxter at exactly these values. Equally strong evidence is extant for each of the other proposed measurements.

Although Livio Stecchini, who has sadly died in recent years, was the most renowned metrologist of his generation he missed the fact that the choice of module must be sought in the sensible ratios and rational numbers. When he identified the Mycenaen foot from the grave circle of Mycenae he measured the diameter as exactly 100 feet of what I have termed the Assyrian foot at its Root Geographic classification of .910315ft. It has been my experience that when such an unsatisfactory number as is this decimal as a diameter, an alternative should be sought. If the distance is divided by seven it is 13.0045ft, this is exactly the 12 feet pertica of the Belgic foot of 1.083708ft. There is little doubt that we are looking at identical construction techniques and formulae over a vast geographic area and span of time.

Few examples of measuring instruments survived in Europe, and no ancient plans or diagrams remain; but it is obvious from the similarity of the broch designs that such detailed plans must have been used. It is to Egypt that we must look for pictorial confirmation of the facts regarding metrology as presented here. An abundance of measuring rods are extant and analyses of the dimensions of ancient buildings in very good condition may be used to confirm many modules. Many working drawings may also be consulted, a good example of which is set out below.

British Museum, wooden board overlaid with gesso. Egypt-5601

The human form is always depicted to canonical proportions. The reason that the drawing above is so interesting is that a whole variety of cubits are portrayed. This is proof that an amalgam of modules was deployed in a single creation. If the grid is terms of the four-digit hand, and the cubit is taken as Root at 1.714285ft then the median of the cubits on the left would be this Royal Egyptian cubit. The one above would be two Roman feet at the precise value found at broch 36, Gurness, and the one below would be the Sumerian cubit of 1.645714ft, exactly 24 to 25 of the Royal Egyptian. Only one cubit may positively measured on the right hand side, and it is a complete curler. For if the seated figure were erect, then he would 3 2/3 royal cubits high which would be four of these cubits. In terms of the English foot his overall height would be twice pi, or 6.285714ft, and the basic foot one third of pi. It is not a measure that I am aware of, but has been encountered; I hesitate to offer an explanation feeling that I may be getting out of my depth trying to decipher Egyptian mysteries.

Many more of their techniques may be extrapolated from this drawing, but the object here is to illustrate that several quite separate modules were in contemporary use in a single culture. Wherever one researches ancient measurement one finds the same modules and all of them are founded on such anthropomorphic bases. There is nothing absurd or even unexpected in finding the identical system used in Scotland, after all, the modules are identical, the Root royal cubit used above would fit exactly 625 times into the perimeter of the Ring of Brodgar.

Virtually every aspect of this amazing and elegant system, particularly with respect to the module identification, would be totally obscured by being expressed in the metric system. As the traditional units such as the English foot are being inexorably phased out, we may confidently say that this knowledge has therefore been rescued in the nick of time. Only somebody who habitually thinks in terms of the English foot could have deciphered it. This is because each number that one is confronted with will have a close solution in terms of the English foot. For example, the length of the mean geographic degree according to ancient metrology is 364953.6 feet, the closest round sensible number to this is 360000, when this degree is divided by this number the result is 1.01376 which is the value of the Standard Geographic Greek foot. Metrological analysis really is this simple.

All cultures used all the measures. Does the system as a whole, which all civilizations used as reference, therefore predate all of them? Because one cannot conjecture that there were any direct cultural contacts between the disparate peoples who used the identical system. Indeed, people who have not previously been regarded as civilized in the literal sense, manifestly utilised this sophisticated measurement system to extraordinary degrees of accuracy. But so thoroughly have assumptions over the centuries become orthodoxies, that the truth when it arises is often regarded as preposterous.