This paper proposes that an unfamiliar type of circumpolar astronomy was practiced by the time Le Menec was built, around 4000 BCE. This observatory enabled the rotation of the earth and ecliptic location of eastern and western horizons to be known in real time, by observing stellar motion by night and solar motion by day. This method avoided stellar extinction angles by measuring the circular motion of a circumpolar marker star as a range in azimuth, which could then be equated with the diameter of a suitably calibrated observatory circle. The advent of day-inch counting and simple geometrical calculators, already found at Le Manio’s Quadrilateral, enabled the articulation of large time periods within Carnac’s megalithic monuments, the Western Alignments being revealed to be a study of moonrises during half of the moon’s nodal period. Le Menec’s Type 1 egg is found to be a time-factored model of the moon’s orbit relative to the earth’s rotation. This interpretation of Le Menec finds that key stones have survived and that the gaps seen in the cromlech’s walls were an essential part of its symbolic language, guiding contemporary visitors as to how its purpose was to be interpreted within the pre-literate megalithic culture.

Two key lengths are found at Le Manio and Le Menec: The first, of 4 eclipse years is a day-inch count of the Octon eclipse cycle; the second is a four solar year count that, with the first, forms a triangle, marked clearly by stones at Le Menec. The principles worked out at Le Manio appear fully developed in Le Menec’s western cromlech, including the use of an 8 eclipse year day-inch count, consequently forming a diameter of 3400 megalithic inches which equals in number the days in half a nodal period. The scaling of the Western Alignments is found to be 17 days per metre, a scaling naturally produced by the diagonal of a triple square geometrical construction. A single sloping length on the top of the central stone initiating row 9, indicates a single lunar orbit at 17 days per metre, a length of 1.607 metres. This control of time counting within geometrical structures reveals that almost all of Le Menec’s western cromlech and alignments express a necessary form, so as to represent a megalithic study of (a) circumpolar time as having 365 time units, (b) the moon’s orbit as having 82 times 122 of those units and (c) the variations of successive moonrises over most of a lunar nodal period of 18.6 solar years.

There used to be an interest in metrology – the Ancient Science of Measures – especially when studying ancient monuments. However the information revealed from sites often became mixed with the religious ideas of the researcher leading to coding systems such as those of Pyramidology and Gematria. The general effect has been that metrology, outside of modern engineering uses, has been left unconsidered by modern scientific archaeology.

Metrology seemed a very complex subject before John Neal and John Michell re-defined it in a very compelling and much more simple fashion. All the ancient measures were first found in different regions of the world and so became known by the name of a civilization or country. This implied and later led to the assumption that these measures had (a) been uniquely developed there in (b) an arbitrary fashion.

But ancient measures are not arbitrary and indeed are all related to a single and unified system. This simplicity would have been obvious had measures not been slightly “varied”, for precise reasons. Aside from these variations, John Neal has identified that the English foot is the basis of the whole system – used as the number one within it – and all the other types of foot are, at root, rationally related using integer fractions of an English foot. What might appear to be a rather partisan approach should be understood in the knowledge that the English foot did not come from England.

It is also important to base such a discussion on the length naturally called feet since, whilst it is only one of many longer and shorter units of length, each such greater length is simply made up of feet according to a formula. Subdividing a foot can yield 10, 12, 16 or other divisions, such as inches or fingers, in different measures. A yard is generally three feet and a pace two and a half. As with a new language the exceptions such as that a cubit sometimes has one and a half feet and other times two can be learnt later. Feet all appear to lie within a given range, plus or minus, of the English foot.

Because the ancient feet largely use low numbers in their fractions of the English and most often are superparticular (where the denominator and numerator vary by one as in 8/7, the royal foot) then many of them represent musical tones and the measures are interrelated in the same ratios found in musical harmony (chapter 2). This is shown in figure one but has not been an important consideration so far in applying this metrological system.

There are twelve main measures and usually any measure found is a variation (see later) of one of these rather than the root value. Each root value forms what is called a module. Neal and Michell have identified TWELVE MODULES:

Assyrian Foot 9/10 = 0.9ft 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 Foot 32/35 = .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 Foot 24/25 = .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 Foot 48/49 = 0.979592ft One of the better-known measures, being six sevenths of the royal Egyptian foot. English/Greek Foot 1ft The English foot is one of the variations of what are accepted as Greek measure, variously called Olympian or Geographic.

The English and Greek Foot 1 = 1.000000 ft Defined from the Equator of the Earth as one 360,000th of a DAY (angular movement of the Sun in a day) whilst also being one seventh of 1/12^{6}th of the mean Earth radius – see chapters 3 and 4

Common Greek Foot 36/35 = 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 Foot 21/20 = 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 Foot 15/14 = 1.071428ft Develops into the Drusian foot or foot of the Tungri. Detectable in many Megalithic monuments.

Sumerian Foot 12/11 = 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 Foot 11/10 = 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 Foot 8/7 = 1.142857ft The most discussed and scrutinised historical measurement. Examples of the above length are plentiful.

Russian Foot 7/6 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.

To this list can be added a thirteenth,

Inverse Iberian Foot 35/32 = 1.09375 This module that is 1 part is 8000 different to the present Metre is found at Carnac.

One immediate problem with these measures appears to be their accuracy, as in the number of decimal places shown in the modules above. However, this is an artifact of the decimal system that only employs the two primes 2 and 5, which constitute 10. The above modules are often rational fractions of an English foot involving other primes, notably 3 leading to 0.333333… written as 0.3 using the underline to represent a repeating fraction. Seven in the denominator, as in 8/7, leads to the repeating fraction 1.142857 for the Royal foot. The problem lies not in the number base used but rather in the accuracy of say the Roman foot at 0.96; it should be written 0.960000 or similar showing that in practice the full fraction is shown but the practice of metrology will always be measurable according to our instruments. Correcting a measurement to show the intent within the system is a movement towards the meaning of a monument such as that it represents the polar radius of the Earth by employing a module with 11/7 in its mix of primes, in feet.

Each of the above can be and are usually found systematically varied from the above ROOT values of a module.

The first variation is by one or more applications of the ratio A = 176/175, which contains 11 above and 7 below (also 16 above and 25 below), which 11/7 allows the ratio to function in various ways,

It can resolve the difference between the diameter and circumference of a circle so that whole numbers of two different variations of the same module are present despite the fact that pi is irrational – because the accurate 22/7 approximation has 11/ 7 within it.

It can convert a module with no prime numbers apart from 2, 3 and 5 in it into a Polar Measure – a measure that divides into the polar radius of the Earth because it contains 11/ 7 within it.

Since the meridian of the Earth between 10^{o} and 51^{o} generates a degree of latitude length that varies by this amount, then two measures differing by this ratio will divide the two latitudes to give the same count in feet.

This latter use was deduced by Michell leading to his early categorisation of measures as Tropical and Northern, the latter longer by 176/175. Analysis of the different historical measures led Neal to the simpler idea that a table of variations was involved and that each module was based upon a simple fraction of the English foot, a ROOT value that leads to a module of variations using just these two geodetic ratios, 176/175 and 441/440.

This second ratio used for variation, B = 441/440, is the ratio of the Earth’s mean radius to its polar radius, again discovered by Michell. It has 7^{2} above and 11 below and can be used in similar ways to the first.

As with ratio A, ratio B when suitably employed can resolve the difference between the diameter and circumference of a circle.

It can convert a module with 7 in its denominator into a Polar Measure.

The meridian of the Earth again generates degrees of latitude length that vary by this amount, so that two measures differing by this ratio will divide the two latitudes to give the same count in feet.

Thus, the two ratios used to vary measures were those that already related to the ellipse of the Meridian, the size of the Earth and its deformation due to daily rotation.

Since the English foot is one for the system, then its own variations demonstrate the ratios used for varying a root value within any module. Neal generates such a grid using 176/175 for horizontal differences and 441/440 for vertical ones, as in this table,

Least

Reciprocal

English

Canonical

Geographic

ROOT

0.988669

0.994318

1.000000

1.005714

1.011461

STANDARD

0.990916

0.996578

1.002273

1.008000

1.013760

A = 176/175 = 1.0057143 and B = 441/440 = 1.002272. 1.01376 is therefore one times A times A times B since two columns and one row are involved. 0.98867 is 1/A^{2}.

Other modules are then varied similarly, as in the case of the Persian Foot that is 21/20 feet long in its Root value.

Least

Reciprocal

Persian

Canonical

Geographic

ROOT

1.038102

1.044034

1.050000

1.056000

1.062034

STANDARD

1.040461

1.046407

1.052386

1.058400

1.064448

1.0644486 is therefore 1.05 times A times A times B since two columns and one row are involved. 1.038102 is 1.05/A^{2}.

The names have come to be adopted out of experience but having become part of a coherent system, they are then employed to communicate the variety of a given module. Thus, the foot behind the Astronomical Megalithic Yard (AMY) of 19.008/7 feet (2.715428) is one third of this yard, which rationally is 792/875 feet (0.905142857). However, this foot is 176/175 times the Assyrian Foot of 9/10 or 0.9 feet. Thus it should be referred to as being a “root canonical Assyrian foot”.

Least

Reciprocal

Assyrian

Canonical

Geographic

ROOT

0.889802

0.894886

0.900000

0.905143

0.910315

STANDARD

0.891824

0.896920

0.902045

0.907200

0.912384

The AMY is also a pace of two and a half feet (of 1.0861714285 feet) if the Drusian foot of 27/25 is similarly varied, i.e. a “pace of root canonical Drusian feet” could also describe it. A yard of Assyrian feet is 27/10 feet showing how the Drusian and Assyrian modules are naturally related in this way.

Least

Reciprocal

Drusian

Canonical

Geographic

ROOT

1.067762

1.073864

1.080000

1.086171

1.092378

STANDARD

1.070189

1.076304

1.082455

1.088640

1.094861

In practice tables can be prepared for reference or, as is more common, ad hoc calculations are done on a calculator to measure the variation of a measurement within a possible module. For instance, in chapter 8:

John James associated the Roman foot with Chartres in The Master Masons of Chartres. He used a figure that was 6.8 feet to seven Roman feet, but this would not belong to ancient metrology for its formula would then contain the prime number 17. The nearest of Neal’s values would be the root geographical Roman foot where the root value of 24/25 feet is increased twice by 176/175 to give 0.971003 feet, a difference of less than 0.04 %. This illustrates the necessary system of correction to arrive at an ancient measure from a modern measurement whose accuracy limited by monumental damage, instrumental errors, ambiguous reference points and so on. Such correction returns to the original number symbolism of measures and the model of the Earth itself and differs from the modern idea of measurement as an operational and controlling activity.

A number of non-academic books incorporate some metrology whilst academic books are largely catalogues of historic values and approximate relationships between the above modules. Virtually nobody has adopted the system of Neal and Michell and what is generally found is at best sufficient for the work being done but more often is (a) not ancient and (b) interprets monuments in an eccentric fashion relative to the ancient and prehistoric norms.

The Bronze Age Computer Disk by Alan Butler is typical of how the co-incidences within the numerical environment start to re-discover facets of the ancient system. Whilst it is possible that there were other, different metrological systems, the evidence points to a single system of measure wherever metrology has been employed. This is crucial; that what must have been a thousand years in the making was the greatest cultural artifact ever created and was part of a world-view that was transmitted between antediluvian civilizations down to historical times. Most of the results developed in this book [Sacred Number] can step beyond surmise only because of this metrological system. Just as a new telescope can lead automatically to new celestial objects, the application of metrology to ancient sites appears to reveal interesting facts that cannot be seen otherwise. Almost all other lines of approach are hampered by the ravages of both time and later civilizations.

Each type of foot generates MODULES of varied feet according to just two ratios, as explained above:

Assyrian Foot 9/10 = 0.9ft 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.

Least

Reciprocal

Assyrian

Canonical

Geographic

ROOT

0.889802

0.894886

0.900000

0.905143

0.910315

STANDARD

0.891824

0.896920

0.902045

0.907200

0.912384

Iberian Foot 32/35 = .9142857ft This is the foot of 1/3rd of the Spanish vara, which survived as the standard of Spain from prehistory to the present.

Least

Reciprocal

Iberian

Canonical

Geographic

ROOT

0.903926

0.909091

0.914286

0.919510

0.924765

STANDARD

0.905980

0.911157

0.916364

0.921600

0.926866

Roman Foot 24/25 = .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.

Least

Reciprocal

Roman

Canonical

Geographic

ROOT

0.949122

0.954545

0.960000

0.965486

0.971003

STANDARD

0.951279

0.956715

0.962182

0.967680

0.973210

Common Egyptian Foot 48/49 = 0.979592ft One of the better-known measures, being six sevenths of the royal Egyptian foot. English/Greek Foot 1ft The English foot is one of the variations of what are accepted as Greek measure, variously called Olympian or Geographic.

Least

Reciprocal

Common Egyptian

Canonical

Geographic

ROOT

0.968492

0.974026

0.979592

0.985190

0.990819

STANDARD

0.970693

0.976240

0.981818

0.987429

0.993071

The English and Greek Foot 1 = 1.000000 ft Defined from the Equator of the Earth as one 360,000th of a DAY (angular movement of the Sun in a day) whilst also being one seventh of 1/12^{6}th of the mean Earth radius – see chapters 3 and 4

Least

Reciprocal

English

Canonical

Geographic

ROOT

0.988669

0.994318

1.000000

1.005714

1.011461

STANDARD

0.990916

0.996578

1.002273

1.008000

1.013760

Common Greek Foot 36/35 = 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.

Least

Reciprocal

Common Greek

Canonical

Geographic

ROOT

1.016916

1.022727

1.028571

1.034449

1.040360

STANDARD

1.019227

1.025052

1.030909

1.036800

1.042725

Persian Foot 21/20 = 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.

Least

Reciprocal

Persian

Canonical

Geographic

ROOT

1.038102

1.044034

1.050000

1.056000

1.062034

STANDARD

1.040461

1.046407

1.052386

1.058400

1.064448

Belgic Foot 15/14 = 1.071428ft Develops into the Drusian foot or foot of the Tungri. Detectable in many Megalithic monuments.

Least

Reciprocal

Belgic

Canonical

Geographic

ROOT

1.059288

1.065341

1.071429

1.077551

1.083708

STANDARD

1.061695

1.067762

1.073864

1.080000

1.086171

Inverse Iberian Foot 35/32 = 1.09375 This module that is 1 part is 8000 different to the present Metre is found at Carnac.

Least

Reciprocal

Carnac

Canonical

Geographic

ROOT

1.081356

1.087536

1.093750

1.100000

1.106286

STANDARD

1.083814

1.090007

1.096236

1.102500

1.108800

Sumerian Foot 12/11 = 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.

Least

Reciprocal

Sumerian

Canonical

Geographic

ROOT

1.078548

1.084711

1.090909

1.097143

1.103412

STANDARD

1.080999

1.087176

1.093388

1.099636

1.105920

Yard and full hand Foot 11/10 = 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.

Least

Reciprocal

Yard & Hand

Canonical

Geographic

ROOT

1.087536

1.093750

1.100000

1.106286

1.112607

STANDARD

1.090007

1.096236

1.102500

1.108800

1.115136

Royal Egyptian Foot 8/7 = 1.142857ft The most discussed and scrutinised historical measurement. Examples of the above length are plentiful.

Least

Reciprocal

Royal

Canonical

Geographic

ROOT

1.129907

1.136364

1.142857

1.149388

1.155956

STANDARD

1.132475

1.138946

1.145455

1.152000

1.158583

Russian Foot 7/6 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.

Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).

In reviewing some ancient notes of mine, I came across an interesting comparison between the Golden Mean (Phi) and PI. They are more interesting in reverse:

A phi square (area: 2.618, side: 1.618) has grown in area relative to a unit square by the amount (area: 0.618) plus the rectangle (area:1 ). This reveals the role of phi’s reciprocal square (area: 0.384) in being the reciprocal of the reciprocal so that in product they return the unity (area: 1).

The ratios of ancient metrology emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.

The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin Heath) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.

The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules.

So why was this foot and its yard important, in the Scottish megalithic and in later, historical monuments?

If one forms a square with side equal to the old yard, that square can be seen as containing 9 square feet, and each of those has side length 99/100 feet. This can be multiplied by the rough approximation to 1/√ 2 of 5/7 = 0.714285, to obtain a more accurate 1/√ 2 of 99/140 = 0.70714285.