July 2003 AOM: The Science of Metrology

I welcome this opportunity to broach a subject that has become a virtual obsession; those who have seen the point of what I have to offer have likened the content to the Rosetta Stone. This may be an exaggeration, but time will tell. It intimately concerns the pre-historic world and those who access these pages have been drawn here through an interest in this subject, just the sort of people whom I would wish to interest. It ain’t easy but it is very rewarding, and that’s no exaggeration.

Although I may have certain specific criticisms of the work of Graham, our host, it is impossible to deny the fact that the main thrust of his work is to propagate an interest in the ancient world, which is identical to my own. This he has done in no uncertain terms and my work will be entirely supportive of this point of view.

Many of the hypotheses forwarded for an advanced civilization in remote history concern measurements. Very few of them agree, one with another, as to the values used, causing fundamental flaws in their arguments. Once it is grasped, what I have to offer is a level place where all can stand, that never existed before. As Stecchini said:

It is a basic principle of epistemology that our ability to reject erroneous theories increases in proportion with the precision and accuracy of the measurements.

The Science of Metrology

The most important study of the ancient world is that of its metrology, for this is the medium through which its architects and artisans expressed much of their intention. All ancient civilizations have their unique and distinctive styles and personas, one has only to see objects of everyday use, examples of artistic expression or buildings to enable us to identify them and the culture to which they belong. Literally, from all over the world, there is a distinctive similarity of technique that underlies all of these seemingly disparate cultures; this is because all that was created had to conform to a canon of order in order to acceptable. The canon could be likened to music where a limited number of governing factors allow a limitless number of harmonious arrangements. These cultural similarities extend to their creation myths, functions of the gods, astronomy, counting methods and cosmologies. Although any of these linkages may be open to criticism or made to seem arbitrary, the one unifying factor, which cannot, is their metrology. Measurement is at the root of all civilizations and at the root of metrology is pure number.

It has become customary for us to label certain metrological values with a cultural appellation, such as Sumerian, Egyptian, Greek, Roman, etc. Although many bureaucracies adopted certain values for the regulation of their societies, they were not the sole or dominant modules used by that nation. All of the modules from all of the societies form an elegant structure and were concurrently used in their entirety by each ethnic group. As this measurement system was in use in its sophisticated wholeness at the very dawn of the historical period, the reasonable assumption is that it was developed in very remote antiquity. The earliest monuments from the prehistoric period overtly display the use of the system — with the most astonishing degrees of accuracy. So exactly are the modules and their rational multiples seen to be in use at widely dispersed locations by civilizations that were no longer aware of each others existence — one may also assume that at one time, all the nations of the world had access to a common canon of reference. The metrologist, Berriman, posed the question “Was the earth measured in remote antiquity?” The short answer is yes — but not by any culture from the historical record.

For around two centuries, theories regarding the geodetic origins of ancient metrology have been regularly mooted. Before this, it was generally accepted that this was the case. Our ancestors were traditionally regarded as being more advanced in all of the arts and sciences. Newton sought the dimensions of the earth in ancient measures; Kepler made guarded references to his knowledge being of an Egyptian provenance. Galileo stated astronomical facts that he could not have known from observation and renaissance scholars are known to have reckoned the geographic degree of latitude to be 75 Roman miles. Classical texts are strewn with facts that the authors could not have calculated but must have been known from even more ancient records, now destroyed; mere fragments of higher learning had percolated down to the Greeks who are our premier reference. Before we elaborate on these points it is necessary to establish the general structure of the subject.

The constituent feet and their fractional integration.

Throughout history, scholars have sought to unravel the mysteries of metrology. Regarded as a whole it is a very confusing picture and when viewed in millimetres, becomes utterly impenetrable, it is fair to say that the enigmas of metrology would never have been resolved through the medium of the decimal system. One noted scholar (De Sanctis) remarked that “(Ancient) metrology is not a science, it is a nightmare.” As is any branch of science — if one is unversed in its peculiarities. Nevertheless, once the principal threads that bind the subject together are grasped, its connective links can fall into place with remarkable rapidity. It is a quote from another author (De Morgan) that hints at how the subject should be approached: “The

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.

This means that if you are going to study comparative metrology, the module of comparison should be an ancient foot measure. The English foot is just such a touchstone and when decimally expressed is the most suitable medium to represent the complexities of the subject. Secondly, all of the measures being compared should be reduced to their constituent foot value. By that, it is meant that there are many modules other than the sixteen digit or twelve-inch feet that are often called feet. Certain of them are really half cubits, some are 18 or 20 digit modules, and once identified they may be discarded for comparative purposes. Also, in all ancient cultures there was a range of variations of the modules in what are essentially the same foot measure, and when taken at their extremes, are often recorded as different feet. Thereby one may discard some of the variants, until a single example remains. Because the observed variations in the modules are the same in all cultures, one must learn to classify them and identify what is termed the Root in each one, then one can compare like with like. Once this process of reduction to the relevant modules is complete, it is apparent that there are probably only 12 basic feet used throughout the world. These are listed below at their parity with the English foot:

Table 1

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. Variants at one and one half of these feet as a cubit would be the Arabic black cubit, also the Egyptian cubit of the Nilometer.

The number of decimal points used to express these values is sometimes criticised, but what we are actually looking at is an elaborate series of rational fractions. Exactly how these, which have been previously regarded as separate measures, fractionally integrate into a single organisation is shown in the table below.

Table 2

The above tables are two methods of expressing the same data.
The values given in the second column of the top table is all of the foot measures at Root classification.

The variational spread of the above values

Variants and variables in the above standards are in no wise arbitrary or 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. However, 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 feet) are as follows:

Table 3

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

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 of table 1 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 from the Root of 1.05ft:

Table 4

Least 1.038102	Reciprocal 1.044034	Root 1.05 Standard	Canonical 1.056	Geographic 1.062034
1.040461	1.046406	1.052386	1.0584	1.064448

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 recorded length of the Mycenaen foot at .910315ft (Stecchini) 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.

Reasons for the variants and the importance of pi

Pi, or the ratio between the diameter and perimeter of a circle, held a position of enormous importance in the cosmologies, philosophies, practicalities and the metrologies through which they were numerically expressed. Pi would be important to those who realised the spherical nature of the heavenly bodies, lived upon a sphere, had a circular horizon and observed circular orbits. On a day-to-day level of practicalities, storage and measuring vessels, and certain buildings were of circular construction. So important was this ratio that the greatest building ever constructed is a monument to the equation. True pi is a transcendental number that can never be fully accurately expressed. Nature does not use it, because nothing is perfectly spherical or circular, slight distortions from perfection mar everything that is manifest. It is an ideal. Therefore close approximations that preserve integers in calculations were, and still are, the norm.

22/7 was the most commonly used value, which is accurate to about one part in 2400 (about 2 feet in a mile), and all other values that were used correct calculations back to be this ratio. The next most used was the value 25/8 or 3.125, this differs by the fraction as 25/8 being 175 to 22/7 being 176, which is the identical fraction that separates the measures across the tables. This means that diameters that are not multiples of seven may be measured in the lesser value and the units in the perimeters will be a whole number by virtue of being the 175th part longer. I.e. 4ft x 22/7 = 12.57142ft, but 4ft x 25/8 = 12.5. Accuracy to the 22/7 value is maintained by virtue of the fact that if the circle is 4 English feet diameter, then its perimeter is 12.5 Greek feet of 1.0057142ft, or Root Canonical, these are variations in the same module.

Similarly, the fraction 441/440 serves to maintain integers in diameters and perimeters, but of differing modules. When attempting to identify a possible module in an ancient structure, one must seek not only integers in the perimeter, but rational numbers, those that are associated with perimeters. The best example of this is to take the simple and familiar number 360 as a circumference; then the diameter is 114.545454. If these numbers are regarded as English feet, the diameter is one hundred royal Egyptian feet of the Standard classification, but the royal Egyptian foot is exactly one and one seventh English feet, or Root at its correct relationship. The number has converted by the addition of the 440th part to Standard. This is but one of the proofs that metrology is altogether composed directly from the behaviour of number itself, and the English foot is the datum, or number one, which all else metrologically develops.

In fact, other values of pi and values directly related to pi may be arranged in a tabular format identical to the variational tables. The value from the Rhind papyrus, which is the area of the circle obtained by squaring eight 9ths of the diameter, is 3.160493, this is virtually 22/7 plus the 175th part, or 3.160816 (99.99% accuracy). Other values are 3.15, 3.168 and the value given by Fibonacci 3.141818. The difference between the Fibonacci value and 22/7 is a very small, but enormously significant fraction, that of 3025 to 3024, it will be elaborated upon as the text develops, but at this juncture the significance of the other values take priority. 3.168 is a very notable number, the best example of its use is taken from the description of the ideal city as described by Plato. It is a circular construction with a radius of 5040 and 316800 units perimeter. Other temple-city-world descriptions incorporate these numbers, as given by Eziekiel and St. John the Divine, it is detectable in both megalithic structures and mediaeval sacred architecture — as well as being the numerical value, by the art of gematria, of the title Lord Jesus Christ = 3168. It is a thinly disguised reference to the pi formula, at 100.8ft (100 Standard Canonical Greek feet) and 3168 ft perimeter = 22/7. As in the metrological tables these values may be arranged thus:

Table 5

3.125 x 176/175	3.142857 x 176/175	3.160846
	x 441/440	x 441/440
	3.15 x 176/175	3.168

As it is the value 3.15 that is of importance to the following, its significance will be explained in the text.

The geodetic relationships of metrology

Three of the metrological variants — Standard Canonical, Root Geographic and Standard Geographic — have a distinct and unmistakable affinity with the length of the earth’s meridian. As one progresses northward from the equator to the pole, each degree is measurably longer. Near the equator the degrees are only about one meter longer than the preceding degree, this gradually increases, until around 45° where each degree is 25 metres longer, toward the pole the differences have once again decreased to about a metre. There are many inherent difficulties in using this elliptical quadrant as a datum or basis from which develop a standard of measure. The French did it by measuring many of the degrees at differing latitudes and dividing their estimate of the total length of the 90° quadrants into ten million parts and calling them metres. It was soon pointed out that their estimate of the total length was in error by what amounts to 25 metres in every degree. Judging by the evidence, in remote antiquity this problem was tackled entirely differently.

The differences in the lengthening meridian degree are recorded in the values given by ancient metrology. The geographic foot is reckoned to be 360000 to the degree, the Standard Canonical Greek foot of 1.008ft fits this criteria at 10° latitude, then the Root Geographic, 1.011461ft, at around 38° and the Standard Geographic, 1.01376ft at around 50°. These latter values reveal a most startling relationship between the variable meridian and the stable polar axis. It has often been proposed that the polar axis would be a far more sensible datum on which to base an earth commensurate international standard of measure (originally by Sir John Herschel). It would seem that in remote antiquity that there was a perfect reconciliation of both methods.

If one uses the simple number 176/175 (or 1.0057142), and squares it, then in terms of metrology this number, at 1.01146122, is the length of the geographic foot at three sevenths of the distance equator to pole. If the total perimeter of the earth were taken to be degrees of this length then by using 22/7 for pi, the polar radius would be 20854491ft. Increase the polar radius by the 440th part and it would be equal to 20901888ft (6370890 metres) and this is the accepted mean radius of the earth. Calculate the length of the geographic foot from this length, also by using 22/7, then it is a length of 1.01376ft, which is the given value of the geographic or Olympian Greek foot (309mm), which I term Standard Geographic. It must be stressed that although this length is very close, at 364953.6ft, to the average degree — it is calculated from the mean radius. If one wished to calculate the polar diameter from this mean degree one would use the value for pi of 3.15, this is 22/7 plus the 440th part. We have seen this identical fraction added to the Egyptian foot module of the diameter to be 360ft perimeter as a result of the integer distortion in pure number. Remarkably, the earth itself distorts by the identical fraction, the polar radius being 440 to 441 of the mean, thereby giving compatible numbers (Standard Geographic), in both the mean degree and polar diameter.

Flinders Petrie, the most credible metrologist ever, expressed strong doubts about the geodetic nature of ancient metrology, reasoning that the Greeks were noticeably lame at long distance angular measurement. However, he did point out that had the British adopted the Belgic foot instead of the imperial at the time of the 1305 statutes of Edward I, then three of these feet would be very closely equal to the metre. This comparison may be demonstrated to be exact by the following method. The metre calculated from the mean degree would be a length of 3.284582ft, one third of which is 1.09486ft. We know from the records of Nero Claudius Drusus that the exchange for the Belgic foot was taken to be nine eighths of the Roman. Therefore, 8/9 of the Belgic foot is .9732096ft which is precisely one of the values of the Roman foot identified by Petrie himself, which he termed a Pelasgo foot, hereby classified as a Standard Geographic Roman foot. It is this foot, which is the basis of the Roman mile of which 75 equal a mean geographic degree.

Thus we have seen not only very elegant solutions to the difficulties inherent in the oblateness of the earth upon which to rationalise a standard, but many modules have been identified as absolutes — which perfectly fit the empirical evidence. A ten millionth part of the quadrant is a Belgic yard of 3.284582ft and a ten millionth of the polar radius is a sacred cubit of 2.0854491ft. This is the cubit upon which Newton pondered and was referred to, (although not accurately identified), as “Cubit A” by Berriman, it is a module of two common Greek feet and is six hands breadth to five of the royal Egyptian cubit.

By these simple methods, the dimensions of the earth can be very accurately portrayed. The two important radii, that of the polar and mean are very decently pinpointed, the equatorial radius may also be calculated by the same fractional method. It is a peculiarity of the spheroid that the longer equatorial radius is exactly half the difference that obtains between the polar and mean. If the polar is 440 to 441 of the mean, then the equatorial is 883 to 882 of the mean. Thus, the simple progression of the three radii is 880, 882 and 883. How this compares to modern estimates is as follows.

Earth dimensions from WGS84 satellite data
Polar: 20855442ft	mean: 20902215ft	equatorial: 20925602ft
Earth dimensions from ancient metrology
Polar: 20854491ft	mean: 20901888ft	equatorial: 20925586ft
Levels of correspondence as percentages
Polar: 99.9954%	mean: 99.9984%	equatorial: 99.99992%

The monuments as repositories of measure

The Great Pyramid. The clear simplicity of the Great Pyramid design has been very much occluded by the mountainous volume of hypotheses forwarded by the doubtlessly sincere and intelligent, among others. The first person to observe that the Pyramid was a solid rendition of the pi ratio was John Taylor, in 1863, and since then, very few have pinpointed the correct pi ratio value in conjunction with its correct dimensions. Berriman states that the pyramid, in clear terms reveals it as 22/7. This is the conclusion reached by Flinders Petrie stating that its base was intended to be 440 royal Egyptian cubits and its height 280. Petrie also identified the length of the cubit that had been used, and this translates perfectly as 1.71818ft, the Standard value, which is exactly 441 to 440 of the Root of 12/7 English feet. Therefore, the south side must be the datum at exactly 756 English feet, and the height exactly 481.090909ft which is the shorter Roman 500ft stadium of Standard Roman feet of .9621818ft (or 441 to 440 of .96ft).

In spite of his working with the erroneous dimensions of the pyramid as supplied by Howard Vyse, John Taylor also proposed that the Great Pyramid was a scale model of the earth. Again, although many have picked up on this statement, none have been able to convincingly convey how this may be the case. As we have just discussed the matter of the earth dimensions, perhaps it would be best to explain exactly how this may be demonstrated before we elaborate upon the metrology.

As we have seen, the dimensions of the Pyramid have a clear resolution that reveals an uncompromising value of 22/7 as the pi value. Yet, many of those who speculate on the Pyramid as a model of the earth claim it to be a model of the northern hemisphere, with the height representing the polar radius and the circumference the equator. How absurd, this would distort the pi value to 3.1536, thereby reducing the height by over 1½ foot. Because the Pyramid is clearly designed to embody the pi ratio of 22/7, it must be regarded as a model of the mean radius and mean circumference of the earth. Knowing the mean radius to be 20901888ft one seeks to find a numerically rational scale with the height of the Pyramid of 481.0909ft. It is virtually impossible for one who is not familiar with the principles of ancient metrology to see the significance of the resulting ratio of 1:43446.857. The procedure to follow is to seek for a familiar canonical number close to this ratio, it is obvious to select 432000 and 432000 is 175 to 176 of 43446.857. Thus, the ratio conforms to the general principals of metrology, that is, Canonical number transforming to Geographic number by the addition of its 175th part. As is so often the case with these geographic ratios it may summed to a neat equation, in this case it is the height of the pyramid x 24³ x pi = 20901880ft. As another example, John Michell states that the earth’s mean circumference is 10 x 12⁵ English miles, and by using 22/7 for pi, this equation it also yields the radius as 20901888ft.

Much has been speculated as to the relationship of the sides or perimeter of the Pyramid to the geographic degree. In comparatively recent times, the very first western scholar to analyse the Pyramid mathematically was Edmé Jomard, in 1805. He came breathtakingly close to the truth of the matter by stating that 480 times the side of the pyramid equals the average length of the degree in Egypt. Although his equation was correct by using his figures, he was wrong because his estimate of the Pyramid base was too long. Modern researchers believe that 120 times the measured perimeter of the base equals a degree of longitude at the equator. They may be right in a literal fashion but quite wrong to believe it as a design intention for this fact. The reason for the error in their thinking is that of putting the very deliberately varying sides of the Pyramid together into a single unit. In short, they are taking an average and in the field of metrology, averages have ever been the enemy of exactitude. By taking an average of the variations in the measures one ends up with a dreadful chimera, for the simple reason that the variations are quite deliberate and have a mathematical function. Each value one comes across has been deliberately produced or expressed as a standard and should be investigated comparatively. Therefore, the variations in the sides of the Pyramid, which are exquisitely determinable within the parameters of Cole’s Report, have a deliberate function. This is obviously the case, because the inner chambers and passages over their considerable distances are constructed with the most infinitesimal of errors. Consequently, had they wished to construct the base as perfectly square, it was well within their abilities to have done so.

When he stated that the side of the pyramid multiplied by 480 equals the degree of longitude, how incredibly close Jomard had come the truth of the matter may now be appreciated. The following solutions are the more extraordinary when it is realised that this was no calculation of Jomard’s — he had consulted ancient Greek texts which stated that this was the case. Indeed, 480 times the base side of 756 feet equals 362880ft or 360000 Greek feet of 1.008ft, which is the length of the first development of the basic feet to reach geographic proportions at 10° latitude.

However, the Great Pyramid is designed in Standard values of the modules, greater by the 440th part to the Root, and 441 to 440 of 362880 is 363704ft, which is the length of the northernmost degree of Egypt at 31°. Therefore, the correct way to manipulate the Pyramid ratios is not to multiply by 480, but by 481.090909. Put very simply, the height of the pyramid in English feet multiplied by the base in either English feet or any other module, gives the length of the geographic degrees in Egypt in the module used for the base. This then, is the reason for the differing lengths in the Pyramid’s sides, at values taken from within the narrow parameters of Cole’s Report. The height times the south side gives the degree at 31°, then times the shorter east side gives 30°, which is the degree upon which the pyramid stands, and the west side gives the next consecutive degree at 29°. The north side is the decisive factor because at 230.254 metres or 755.428ft, it is a great deal shorter than the south side (about 7 ins) and gives the solution to a degree that is considerably to the south of the other consecutive northernmost degrees. It translates to the degree that encompasses the point that is two sevenths north of the equator at 25.7°. This is highly significant because it is the location of the temple of Amun at Thebes, where Reisner found the ornate stone omphalos from which it is reputed that all distances were calculated. The length of this degree, at 363428.57ft releases a symphony of number with distinct geographic associations that will not be explored here. If it is not already grasped, then it is recommended that the reader reread this paragraph, as many times as is necessary, as it should produce some reaction when it is understood.

The metrological lessons of the Pyramid are extraordinarily informative; it was by exploring its shape in conjunction with the correct dimensions that the integration of metrology became clear. Although the pyramid is designed with modules of the Standard classification, modules of other classifications are also integral with the dimensions. For reasons already explained, the south side should be taken as the datum and four times this number, 756ft x 4 or 3024ft, is regarded as the conceptual model. As this is 1760 Standard royal cubits of 1.71818ft, it is (obviously) also 1750 cubits of the 1.728ft Standard Canonical value. Additionally, the base side is 440 Standard cubits and 441 Root cubits. It perfectly exhibits the principals that govern metrology, in clear numbers, through the medium of the exactly identified cubit values.

As the Pyramid is a multiple of seven, at 280, in terms of the cubit of design then the base is an integer of the same classification (Standard). The decisive factor of the correctness of this line of reasoning is the fact that the height of the Pyramid is also 500 Roman feet. Therefore, the base is consequently 1000 times pi, as a number in terms of this foot of .9621818ft; but is it is a fractured number at 3142.857. In order to be rendered into an integer, the base may be divided, by the175th part longer, Standard Canonical Roman foot of .96768ft; it is then 3125, also by the Root foot of .96 at 3150 and the Root Reciprocal .9545454ft at 3168. These observations illustrate the functions of the pi value in the maintenance of integers, which, the evidence suggests, is what the system is primarily designed to do, (or rather, does of its own volition).

This is merely the beginning of a number of astonishing metrological functions of the Great Pyramid, but, to those unversed in the niceties of metrology, in other words, just about everybody, they must be approaching an information overload. Constrained by time and a reasonable article length, we reluctantly move on, with much left unsaid.

Stonehenge. The vast majority of archaeologists or professional antiquarians will not be best pleased at even a hint that there could be some link between these monuments. However, this link is so incontrovertible that it basically requires no explanatory justification; metrological analysis of the dimensions should be sufficient. Nevertheless, some background as to the nature and purpose of monuments and temples is essential to the purpose here.

Speaking of temples in general, John Michell, in his “Ancient Metrology”, gives the most succinct precis as to their role that I have come across:

Among the functions of the Temple, as known from those at Jerusalem and Heliopolis and of traditionalist societies throughout the world, are to preserve standards and measures and to provide a microcosmic representation of the whole world. The temple had many other scientific attributes, some of which were inherited by the mediaeval cathedrals. It served as an observatory, as a monumental record of astronomic lore, as an instrument for measuring time, seasons and planetary cycles and as a centre for the rituals throughout the year by which the order of society was regulated.

As well as the preservation of standards, as established by Petrie, it was the custodians of the temple who manufactured and issued instruments of weights and measures. If their standard models were lost of destroyed, they could be accurately reconstructed from the dimensions of the temple itself. At Stonehenge, this function is permanently preserved in the carefully dressed ring of lintel stones that are laboriously placed atop the massive sarsens that form the outer circle, far above the wear and tear they would have received at ground level. It is a source of amazement how much data may be encapsulated in two simple circles of the inner and outer diameter of the lintels. Although the Great Pyramid is built to ratios, Stonehenge is built to scales, however, in all monuments, simple ratios of ten determine the principle metrological units of design. It is obvious that the Pyramid was constructed with units of the Standard classification i.e. the 440th part greater than Root, (all units multiplied by 1.002272), Stonehenge is designed in units of the Standard Geographic classification, that is, 1.01376 times greater than Root.

As the Great Pyramid has a clear resolution as to its intended dimensions, so does Stonehenge. The inner diameter of the sarsens as determined by Petrie in 1880 is 97.32096ft, or 100 Standard Geographic Roman feet, Pelasgo feet as he termed them. The width of the lintels by direct measurement is around 3.5ft. By calculation this may be rendered as 3.4757485ft which is 3 Royal Egyptian feet or two royal Egyptian cubits at the Standard Geographic classification. Therefore the outer diameter is 104.272457ft, which is a 15 to 14 ratio with 97.32096ft. All of the feet modules which fit these dimensions are as follows:

The inner diameter	100 Roman feet of	.9732096ft
	98 Common Egyptian feet of	.993071ft
	96 Greek feet of	1.01376ft
	84 Royal Egyptian feet of	1.158583ft
The outer diameter	105 Common Egyptian feet of	.993071ft
	100 Common Greek feet of	1.0427245ft
	96 Belgic feet of	1.086171ft
	90 Royal Egyptian feet of	1.158583ft

The ratio expressed here is 14 to 15 or 1:1.0714285. It will be noticed that this simple ratio number is the Root value of the Belgic foot expressed in English feet, such ways of determining modules from ratios is universal, reiterating that metrology stems from the behaviour of number itself interpreted through the medium of the English foot. All the values used in the Henge are 1.01376 times greater than the Root values of the original comparative table.

Naturally, the Belgic foot of 1.086171ft is perhaps the most interesting, this is because the module is writ large in the original station stone rectangle, which is over a millennium older than the subsequent sarsen and bluestone circles. The station stone rectangle is a Pythagorean rectangle of the proportions 5, 12 and 13. In terms of a two-foot cubit of the Belgic 2.17234ft it is 50, 120 and 130. 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. As the Sarsens are 48 Belgic cubits in diameter and the Rectangle 50 cubits along its shorter side, then a perfect squint of one cubit in width between the Station stones and the sarsens would focus upon the point on the horizon where these east-west maximum declinations occur. 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.

Also, at this latitude the length of the Geographic foot is that taken from the mean radius of the earth at 1.01376ft. The Henge contains a host of other data encapsulated within its dimensions, perhaps the most interesting of which is the interpretation of the outer diameter in terms of the sacred Jewish cubit, (2 common Greek feet). There are 50 sacred cubits of 2.0854491ft, Stonehenge is consequently built to a scale with the Polar radius as 1:400,000, this is the exact value deduced elsewhere. Therefore, the perimeter of the outer diameter would give values compatible with the degree at 38 degrees north, and in rational numbers if the perimeter is measured into 360 divisions it is exactly the Mycenaen foot as defined by Stecchini, of .910315ft, which is the 400,000th part of the degree at 38 degrees.

Stonehenge contains a wealth of metrological data in other modules, again too much to analyse here. But a parting reference will be made to another stone circle, far removed from Stonehenge that perfectly underwrites the evidence already gathered and gives credence to the universality of the system. It also broaches the subject of the Megalithic Yard, which is now a clearly understood and well-defined range of values. It is a circle in the Orkney Islands, called the Ring of Brodgar.

Because of the well-preserved condition of this enormous circle it may be surveyed with great accuracy and Alexander Thom stated it to be 125 Megalithic Yards in diameter. If that is the case then he has clearly identified a unit of 2.727272ft. This is a very well known module of the ancient world, called a step, a half 5ft pace. Therefore, at two and a half feet, the constituent foot is 1.090909ft and this is clearly identifiable as a Root Reciprocal Sumerian foot (x 176/175 = Root at 1.097142ft). It may be appreciated from this observation why the hunt for the “Megalithic Yard” has been such a wild goose chase, perpetrated by well qualified scholars who are totally ignorant of the principles governing ancient metrology. Any of the feet of table one multiplied by 2½ is a step, Megalithic Yard is a complete misnomer. The range of Sumerian and Belgic feet x 2½ converts all of the approximations postulated by Thom into accurate absolutes.

Although Thom believed that the megalithic surveyors strove to render their circles into integral perimeters, he completely missed the point that these integers had to be significant rational multiples. Brodgar beautifully displays this characteristic by being exactly 1000 Root Belgic feet of 1.07142ft in circumference. Each quadrant is therefore exactly 100 steps of two and a half of these feet. I find this to be highly significant, as the four stations of the circle could be accurately paced in the course of rituals, which were probably designed around such mnemonic devices.

A direct link with Brodgar and the whole ritualised landscape of the district encompassing Stonehenge and Avebury, is the fact that the twin inner circles within the vast Avebury enclosure have the same dimensions. At certain times, there would be gatherings, such as the Olympiads, certainly from all over Britain and perhaps the majority of Europe, at these monuments. This would serve the scientists of that age to meet and not the least of their purposes would be to have their surveying and instruments of measurement calibrated. Little else can explain the uniformity of culture that was maintained over enormous distances and ages of time.

This exercise has merely introduced the subject of metrology, an enormous amount of corroboratory evidence supports all of the claims made here, and because the structure of the subject is so newly discovered, there is a great deal yet to be explored.

John Neal, June 2003.