User:Michael saunders

The Source of Ancient Metrology.

Two things cannot be rightly put together without a third; there must be some bond of union between them. …and the fairest bond is that which makes the most complete fusion of itself and the things which it combines, and proportion (analogia) is best adapted to effect such a union. For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it, and again, when the mean is to the first term as the last term is to the mean - then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one [Timaeus 31b-32a].

Graeco-Egyptian Metrology - The 28 finger system

First, came the 360 day year and the base 60 system. Second, came the finger whose value in the Egyptian hieroglyphic world was 10,000. Third, came the Earth that was cut into 1/60th, that were cut into 1/60th, that were cut into 1/60th, threefold. That is the size of the Graeco-Egyptian Earth you see, 60 * 60 * 60 * 10,000 fingers and 60 * 60 * 60 stade [Herodotus].

In fact Egypt laid out her boundaries according to this system and recorded it to posterity with the division of the two lands. There were two classification of cubit then, called common and royal. The royal cubit was 7/6th of the common cubit and at 31 degrees north, the circle of latitude is 6/7th of the equator. Hence forth the boundaries would be set. Six degrees for upper Egypt, 1 degree for lower Egypt placing the southern boundary at 24 degrees north and the division of the two lands at the latitude of Giza (unification under Menes is synonymous with the carbon dating of the great pyramid). The boundaries of Egypt are not so straightforward however as 23d51', 24d00' and 24d06'north are used for various reasons. One of the reasons is the well of Syene at 24d06' North.

Three stone cubit rods were recovered from the Temple of Amun. Each had identical inscriptions saying that the distance from Behdet to Syene was 106 Atur. Behdet was once the predynastic capital of Egypt and means 'Crown' as in Jewish 'Kether'. Placed at 31.5 degree north it demarcated the boundary line of the Oikhoumene of the ancient world along with the 31st parallel. 106 Atur was equivalent then to 7.4 degrees.

The Greeks however, especially Eratosthenes always made a point of stating that the length of Egypt was 5000 stade of 300 royal cubits. They always use the south boundary or the well of Syene as 1 end of the distance. Herodotus for his part threw a can of worms in the academic world that most intellectuals never quite mastered. He stated the royal cubit was 28 fingers and the common cubit was 3 fingers less. For this clarification in metrology, Herodotus was cast by many into the dustbin. Everyone knew the ratio was 24 fingers to 28 or 6:7. They knew better and in their wisdom became blinded to the metrological reality of history.

Herodotus' stade was composed of 600 feet which we'll call 12 geodetic inch. The common cubit he stated as 1.5 feet. So 1 stade was 400 common cubit which is 10,000 fingers. Therefore 1 common cubit is 25 fingers. The royal cubit of 28 fingers therefore becomes 20.16 geodetic inch. So 5000 stade of 300 cubits of 28 fingers becomes exactly 7 degrees, the length of the division of the two lands. This then was 100 Atur in length with 1 Atur being 50 * 300 royal cubit or 150 Khet of 100 cubit. This is not where the story ends however. The fact the common cubit is 24 or 25 fingers is fundamental (nay crucial) for a proper grasp of Graeco-Egyptian Metrology.

The Greeks and Romans when mapping referred to a cubit of 700 stade of 300 cubit per degree. This equates to exactly 144/7 geodetic inch. The Egyptians for their part mainly used a remen royal cubit of 20.625" for their architecture (the diagonal of a square with sides of 20 fingers). However they do have a special cubit called the black cubit of 540mm found carved upon black stones. All of these cubits thus far mentioned are part of a 5 cubit set that occur due to the 24 or 25 finger definition of the common cubit. A second 3 cubit set is also used as this is more accurate for the remen cubit and degree based geodetic cubit. This then is the deeper meaning behind the words of Plato in the Timaeus:

5 Set Royal Cubits of the 24:25 finger ratio 0.96:1 (4th root 0.96 scale)
- 21.0000 geodetic inch (100) = 21.2688 British inch = 540.2297mm (Black Cubit)
- 20.7867 geodetic inch (099) = 21.0529 British inch = 534.7444mm (See Nile Cubits)
- 20.5757 geodetic inch (098) = 20.8391 British inch = 529.3148mm (700 stade of 300 per degree Royal Cubit)
- 20.3667 geodetic inch (097) = 20.6275 British inch = 523.9404mm (Remen diagonal Royal Cubit)
- 20.1600 geodetic inch (096) = 20.4181 British inch = 518.6205mm (28 finger Royal Cubit)

3 Set Royal Cubits of the First, Last and Mean ratio 0.98:1 (2nd root 0.98 scale)
- 20.5714 geodetic inch (100) = 20.8348 British inch = 529.2046mm (700 stade of 300 per degree Royal Cubit)
- 20.3646 geodetic inch (099) = 20.6254 British inch = 523.8858mm (Remen diagonal Royal Cubit)
- 20.1600 geodetic inch (098) = 20.4181 British inch = 518.6205mm (28 finger Royal Cubit)

All of these measures use their own individual scale and observe the following rules...

1 Schoene = 60 stade of 400 common cubit = 70 stade of 300 royal cubit (of 144/7 inches) [see Note 1]
1 Atur = 50 stade of 300 royal cubit (of 28 fingers) [see note 1]
1 Stade = 300 royal cubit OR 400 common cubit
1 Khet = 100 royal cubit
1 Fathom = 4 cubits = 6 Feet AND 7 Feet.
1 Royal Cubit = 7 Palms = 28 fingers.
1 Common Cubit = 6 Palms = 24 fingers.
1 Remen = 5 Palms = 20 fingers.
1 Foot = 4 Palms = 16 fingers.
1 span = 3 palms = 12 fingers.
1 hand = 4 thumbs. (4/3rd remen royal cubits of 7 hands are held in the museums. Normally 27.5" or 698.5mm)
1 palm = 3 thumbs = 4 fingers.

Anglo-Sumerian Metrology - The 30 finger system

Much is known about the Sumerian fascination with the 20:21:29 Pythagorean triangle that forms near squares. This predisposition is carried through to their metrology and in fact, forms the basis of it. This system should not be thought of as unique from the Graeco-Egyptian 28 finger system. It is a twin that runs parallel with it. They are two alternative views of the same system.

The Tower of Babel (ziggurat Etemenanki in the Temple Esagila of Babylon) was surveyed in Great detail by Robert Koldewey in 1913 and shows two squares on which the Tower was based. The dimensions of this building were found on a tablet copied from an older document and written up by George Smith in 1876 (Louvre AO 6555). It was composed of 7 floors of 9 rooms (3x3) totalling 63 units [suggest research 7 circuit labyrinth, mathematical eye of Horus as 63/64th, Egptian Mathematical Papyrus Problem 48/50 - "squaring the circle"]. Koldeweys' survey dimensions given for the sides of the Tower are thus:

Outer Square.
- 91.66m North = 509.22mm cubit.
- 91.52m East = 508.44mm cubit.
- 91.10m South = 506.11mm cubit.
- 91.44m West = 508.00mm cubit.
- 91.43m Mean = 507.94mm cubit.

Inner Square.
- 61.10m North = 509.16mm cubit.
- 61.15m East = 509.58mm cubit.
- 61.20m South = 510.00mm cubit.
- 61.15m West = 509.58mm cubit.
- 61.15m Mean = 509.58mm cubit.

The size of the inner square provided by the tablet is 1 Iku of 120 cubit of 30 fingers (or 60 double cubits). The side then is 60*60 fingers as befits the Sumerian base 60 system. The outer square is 120 Great cubit or 180 cubit of the 30 finger type. This outer square is exactly 60*60 imperial British inches. The difference then between the 28 finger system and 30 finger systems is that one uses a 21 inch cubit and the other uses a 20 inch cubit. This can be demonstrated in the most simple way. The Sumerian foot is 18 fingers and goes with the 30 finger system. The Graeco-Egyptian foot of 16 fingers goes with the 28 finger system.

20 inches * (18/30 fingers) = 12 thumb inches (1.5 x 18 fingers = 18 inch common cubit)
21 inches * (16/28 fingers) = 12 thumb inches (1.5 feet = 1 common cubit)

This system is however developed further with the area of land measure through the Remen. The Egyptians used a 20 finger remen of their 21 inch cubit or 28 finger. In alternative manner (due to the 20:21 mechanism) the Sumerians use a 21 finger remen of their 20 inch cubit or 30 finger system. It can be demonstrated like this:

21 * square root 2 = 29.6984......29.6984 / 30 fingers = 0.9899 (2nd root 0.98).
20 * square root 2 = 28.2842......28 / 28.2842 fingers = 0.9899 (2nd root 0.98).

The result is a 3 cubit set that works in identical manner to the Egyptian remen 3 cubit set in the opposing direction.

3 Set Royal Cubits of the First, Last and Mean ratio 0.98:1 (2nd root 0.98 scale).
- 20.0000 British inch (100) = 508.0000mm (Babel Cubit).
- 19.7989 British inch (099) = 502.8943mm (Sumerian Remen diagonal Royal Cubit).
- 19.6000 British inch (098) = 497.8400mm (Gudea Cubit).

[In 1880, Ernest de Sarzec raised a group of Statues of the Sumerian regent Gudea (2150-2100bc). Statues B and H (now in the Louvre) have a measuring rule. They both are shown holding a tablet. The tablet on one statue shows the plan of a building. This rule was composed of 15 fingers which was reckoned to be 1/2 a cubit of 30 fingers with a resulting length of 498.4 - 498.6mm. This is the minor cubit of the sumerian remen 3 cubit set.]

All of these measures use their own individual scale and observe the following rules...

120 cubit = 1 Iku (Aslu or "literally measuring Cord") = 60*60 fingers or 60*60 thumb inches (1/16th mille passus).
1 double cubit = 40 thumb inch = 60 fingers.
1 great cubit = 30 thumb inch.
1 cubit = 20 thumb inch = 30 fingers.
1 sumerian remen = 14 thumb inch = 21 fingers.
1 foot = 12 thumb inch = 18 fingers.
3 fingers = 2 thumb inch.

Indus Valley Metrology - The 5 part foot

This is currently a place holder for ongoing research into the 13.2 inch Northern/Saxon/Celtic foot that is the basis of measure in the Indus valley from several millenia BC.

I am currently looking at the 13 inch foot used in Greece and Britain along with the 12 inch foot and how it relates to a 5 part foot ie 13*13 - 12*12 = 5*5. This would be metrology based on 5:12:13 Pythagorean as opposed to a 20:21:29 Pythagorean triangle. It would also explain the predominance of 144 inches and the concept of the dozen and the bakers dozen. This might hold the key to the polar radius as 250,000,000 inches. The Highest base 10 figure of the Egyptian system was 10^7 or 10,000,000 and was signified by the shen ring, the precursor to the cartouche of the Pharaoh. Hence the finger would represent the stade base unit of 10,000 fingers (*60*60*60), the shen ring would represent the radius of a circle (earth) 1/10,000,000th of which would be 25 megalithic inch (25.0265 British inch). The hieroglyph for re incidentally is not a ring but a circle that has a dot in the middle (see pyramid texts). No sun or moon or any other body has such an appendage in reality. Its only function then is to show that Ra/Day/Year equates to a circle or path around a central point.

Graeco-Egyptian Geodetic to Anglo-Sumerian conversion

The clue to the relationship between the Graeco-Egyptian Geodetic inch and the Anglo-Sumerian british inch comes about through realising that the Iku is 1/16th of a mille passus.

180 sumerian cubits = 120 great cubits = 1 minute of march = 60*60 british inch (Smith tablet).

This is the length of one side of the larger square of the tower of Babel. By sumerian reckoning then, the mille passus (land mile) is 16 minutes of march. The nautical mile being 5/4 longer would equate to 20 minutes of march. As the nautical mile is 10 stade then 1 stade (of the 7200 inch variety) = 2 minutes of march. 1 second would equate to 5 feet of 60 inches. Now 60 inches is the conventional definition of the passus (double pace). Hence the sumerian great cubit is a 1/2 passus (1 pace) 25:24th larger then usual. This takes us back to the 5 cubit set of the graeco-egyptian system with a 0.96:1 spread.

The mille passus is defined by Rome as 1000 passus (double pace) of 5 feet of 60 inches. This roman foot originates from the Ionic/Cycladic greek foot that is 24/25th of the Geodetic foot we have discussed previously. The roman 60,000 inch = 57,600 geodetic inch = 8 stade of 7200 geodetic inch (10,000 fingers). The 60 roman inch Passus is therefore 80 fingers or 4 egyptian 20 finger remen. Now This explains the Stade as 300 royal cubit or 400 common cubit:

300*20 inches = 6000 inches = 1/10th of 60,000 inch = 1 mille passus (land mile)
400*18 inches = 7200 inches = 1/10th of 72,000 inch = 1 nautical mile (1 angular minute or sea mile)

[In greece 600 feet = 1 double minute of march = 1 stade of 7200 inch. 1 second of march is therefore 60 inches again or 1 passus.]

The question then is, "how do you measure a mile" and whats the link between the Geodetic and British inch ? The odometer provides the key to this puzzle. For instance;

The central fire altar of the hindu temples follow a systematic design pattern that approximates squaring the circle. This fire alter represents an 8 spoked wheel of the sun. This design remit from the hindu texts is believed to be several thousand years old.
Polychrome bowls from the late Halaf period (6000-5400bc) from Tell Arpachiyah in Iran depict 32 petals of whats called the sunburst. Around the edge of such bowls are several rings of squares (a double set of 50 and triple set of 72) surrounded by 288 teeth on the rim. [It is intersting to note the obvious 2x50=100 3x72=216 216x100=360x60 which is the nautical mile division of earth or 1 angular minute of arc]

The basis of the odometer is that 400 revolutions of a wheel will measure out 1 mille passus of 60,000 minor inch or 57,600 major inch. 57,600 / 400 = 144 inches for the circumference of this wheel. To know how big a circumference your wheel has requires knowledge of Pi. So what do you use as your radius to get the required circumference ??

14.4 british inches / 2 pi = 2.29183 british inches for the radius.
- 2.29183 * (7/22) = 0.72922 british inch finger
- 0.72922 * 28 (196/7) fingers = 20.4181 = 20.16 geodetic inch
- 0.72922 * 20 * square root 2 (198/7) fingers = 20.6254 = 14.4 x sqaure root 2 geodetic inch
- 0.72922 * 200/7 fingers = 20.8348 = 144/7 geodetic inch
- For example (9 * 14.4) / (2 * Pi) = 144/7 * 0.99 geodetic inch

John Legon (among others) has done extensive work especially on the Bent pyramid from the 3rd dynasty and shown the Egyptians used the concept of the sqaure root of 2 using 140,99 and 70 royal cubits. Along with this came the approximation of Pi as 22/7. Some believe the great pyramid squares the circle such that the height is the radius of a circle whose circumfernce equals the pyramid perimeter. This would identify it with the fire altar of the hindu temple. This is why the dimensions of the pyramid are often quoted as 280 cubits high by 440 cubit sides. Herodotus adds some weight to this concept.

têi de puramidi autêi chronon genesthai eikosi etea poieumenêi: tês esti pantachêi [every side] metôpon [face] hekaston [each one] oktô [eight]. plethra [100 feet or 100x100 feet] eousês [to be/exist] tetragônou [square] kai [and] hupsos [height] ison [equal], lithou de xestou te kai harmosmenou ta malista: oudeis tôn lithôn triêkonta podôn elassôn.

"......every side face, each one 8 plethra (to be square) and height equal."

This only has 1 interpretation. The gradient of the pyramid described is 'square root PHI:1'. The height is square root 80,000 units or 282.8427. Now 280 / 282.8427 = 0.9989 which is absolutely identical to the cubit ratios described above. The Egyptian Plethra of 100 unit side being the setjat of 100*100 royal cubit. So it would appear that the face of the Great pyramid is 100x100 royal cubit of 28 fingers and the height is the same. This equates to 280 remen diagonal royal cubits. Remember, Herodotus compared the 25 finger common cubit and 28 finger royal cubit.

The mile and area.

modern statute mile

4800 * 13.2" foot = 5000 * 12.672" (12.672" is 16 fingers. 12 of these fingers made the ploughmans or celtic foot of 9.504")

original mile - british roman mille passus

4800 * 12" foot = 5000 * 11.52"

divisions of the modern mile

63360 / 8 / 40 / 5 / 3 = mile / furlong / rod(pole) / yard / foot (13.2")

equivalent divisions of the original mile

57600 / 8 / 40 / 5 / 3 = mille passus / stade / 10 cubit / 2 cubit(greek pygon/roman cubitus) / foot (12")

Modern area of the statute mile.

The acre is normally defined by the amount of land ploughed in 1 day.
4 strips of land 1 furlong in length by 1 rod in width.
1 furlong was 40 rod in length

1 square mile = 640 acres.
1 acre = 4 rood. (1/640th)
1 rood = 40 sqr pole. (1/2560th)
1 sqr pole = 15 * 15 foot (13.2") (1/102400th)

equivalent area of the mille passus.

1 mille passus = 640 acres.
1 acre = 4 rood
1 rood = 40 sqr pole
1 sqr pole = 15 * 15 foot (12")

Roman mile and Roman area

Area (11.52 greek inch = 12 roman inch)

1 saltus (1 square mille passus) = 57,600 * 57,600 greek inches (1/12th 12.154" greek foot) = 4000 * 4000 Egyptian remen (or Roman Palmipes)
centuria (1/4th saltus : 1/2nd length) = 28,800 * 28,800 greek inches = 2000 * 2000 remen
Heredium (1/400th saltus : 1/20th length) = 2,880 * 2,880 greek inches = 200 * 200 remen
Iugerium = 1 days plough (1/800th saltus) = 2,036.5 * 2036.5 greek inches = 100 * 100 egyptian remen royal cubit (1 setjat)
actus quadrans (1/1600th saltus : 1/40th length) = 1,440 * 1,440 greek inches = 100 * 100 remen
clima (1/6400th saltus : 1/80th length) = 720 * 720 greek inches = 50 * 50 remen
actus minimus (1/192000th saltus) = 131.453 * 131.453 greek inches
scripulum (1/230400th : 1/480th length) = 120 * 120 greek inches = 10 * 10 greek feet (12.154")
pes quadratus (1/23040000th area : 1/4800th length) = 1 * 1 greek foot (12.154")

Length

Leuga = 7500 pedes
mille passus = 5000 pedes
stadium = 625 pedes
actus = 120 pedes = plough length
pertica = 10 pedes
passus = 5 pedes
gradus = 40 fingers
cubitus = 24 fingers
palmipes = 20 fingers
pes = 12 unciae (11.667")
palmus = 4 digitus
unciae = 1.33' digitus
digitus

Relationship between roman version of british mile and roman area

300 actus minimus = 10 clima = 2.5 actus = 1 acre (2276.84 * 2276.84 inches) [see note 2]
75 actus minimus = 2.5 clima = 1 rood (1138.42 * 1138.42 inches)
1 clima = 4 * 4 rods/poles (720 * 720 inches)

Conlcusions on british mile and areas.

The length of the acre ploughed by oxen was 1 stade of 7200" Greek or 7500" roman (1 furrow long or Furlong).
The width of the acre was 720" Greek of 1 actus which was 4 strips of land 180" long called the rod.
The saxons used a foot 1.1x larger then the british 12" foot and all measures were scaled up en masse to suit this 13.2" foot.

Megalithic Metrology (British, Saxon and Celtic) - and the 1.1x larger feet

Running alongside the 12 inch and 11.52 inch British feet is the celtic foot (also called ploughmans foot) and the Saxon foot (also called northern or Indus). The Celtic foot described is composed of 12 fingers of 0.792 british inch (1.1 * 0.72). The conventional foot of 16 fingers would equate to 12.672 british inch which is 24/25th of the 13.2 inch Saxon foot. Understanding that there is a 24:25 ratio to these feet as well, should make it clear that there is a relationship to the 11.52 and 12 inch British feet.

12.672 * 5000 = 13.2 * 4800 = 63360 inches (statute mile)
11.52 * 5000 = 12 * 4800 = 57600 inches (mille passus)
63360 / 57600 = 1.1 : 1

Lots of metrologists have observed and write about occurences of feet 1.1x larger then standard which they explain in many different ways. In my view they occur in the following manner and apparently arise because someone had accurate knowledge of the size of the earth and its ellipticity. Which seems apparent from the Geodetic inch and its relationship to the polar circumference. The size of the earth according to the WSG84 satellite data is thus:

1/298.257223563 flattening
6,378,137,000 mm Equatorial Radius Modern Millimeter
Earth Using Modern Meter
- 1/298.257223563 flattening
- 0.99664718933525251928015447138148 axis ratio
- 6,378,137,000 mm Equatorial Radius Modern Millimeter
- 6,356,752,314 mm Polar Radius Modern Millimeter
- 40,075,016,686 mm Equatorial Circumference Modern Millimeter
- 40,007,862,917 mm Polar Circumference Modern Millimeter

In fractional terms the ratio between the circumference of the earth through the poles and the circumfernce of the earth via the equator is 40,075,016,686 / 40,007,862,917 = 1.001678514 = 1 + 1/595.76. In simple terms then, the ratio is 597 equatorial:596 polar. If we wanted to demonstrate this knowledge we would use it in our metrology. If we want to base it on a land mile previously described or our 0.72" finger, we would use 8*7200 inch stade or 4800 x 12 inch foot. The common demominator between these is 24 (2*24=48 3*24=72 et cetra). In the same token, if we used 1.1x measures we would use 264 (264*2=528 264*3=792). In fact if we were really astute and clever, we would use both.

40,075,016,686 / 597 = 67127331.1323 ...(/ 264 = 254270.1936)
40,007,862,917 / 596 = 67127286.7734 ...(/ 264 = 254270.0256)
6,356,752,314 / 254270.1936 = 24999.9900
6,356,752,314 / 254270.0256 = 25000.0065

Here then using a unit of 254270mm we would get a polar radius of 25,000 units. The scary thing is that a unit based on this has already been proposed by Professsor Charles Piazzi Smyth called the primitive inch of 1.00106 british inch. But the relationship with the statute mile and polar to equatorial ratio has hitherto remained hidden.

The equatorial circumference then = 2,640,000 * 597 = 1,576,080,000 primitive inch.
The polar circumference then = 2,640,000 * 596 = 1,573,440,000 primitive inch.
The polar radius = 250,000,000 primitive inch.
The circle of polar radius = 2,640,000 * 595 = 1,570,800,000 primitive inch.
Resulting Pi value = 1,570,800,000 / 250,000,000 / 2 = 3.1416
Origin of Pi value = 5/6th Pi (3.1416) = Phi squared = Phi + 1 = 2.618
Origin of 24 hours = 2 x 3.1416 / 24 = 0.2618

- 1,576,080,000 / 63360 = 24,875 statute mile.
- 1,573,440,000 / 63360 = 24,833.333' statute mile.
- 1,570,800,000 / 63360 = 24,791.667 statute mile.

- 1,576,080,000 / 60000 = 26,268 Barley mile. (597 x 44)
- 1,573,440,000 / 60000 = 26,224 Barley mile. (596 x 44)
- 1,570,800,000 / 60000 = 26,180 Barley mile. (595 x 44)

- 1,576,080,000 / 57600 = 27,362.5 mille passus.
- 1,573,440,000 / 57600 = 27,316.667 mille passus.
- 1,570,800,000 / 57600 = 27,270.833' mille passus.

The historical Earth Polar circumference is...

1,620,000,000" Roman (900)
1,575,000,000" Eratosthenes (875)
1,555,200,000" English Greek (864)
1,512,000,000" Common Greek (840) (See John Neal ~ The Structure of Metrology).

The Shen Ring that became the Cartouche, represented all that was ruled/measured under the Suns path, with a circle and a diameter bar attached by coiled rope with a value of 100, or 100 solar days march diameter at 1 passus per second.

100 days * 24 hours * 60 minutes * 60 seconds * 60 inch passus * 25/8 (Anthropomorphic Pi used in Sumeria and described by d'Vinci with 'Vitruivian Man') = 1,620,000,000.
1,620,000,000" Roman is 27,000 mille passus or 100 * 270 (The 100 day diameter to form the basis of the Hekatompedon Temple diameter).
250,000,000 * 2 * 63/20 = 1,575,000,000 (with 144/55 * 6/5th = 864/275, 55/21 * 6/5th = 22/7, 21/8 * 6/5th = 63/20, 8/3 * 6/5th = 3.2 & 3/1 x 6/5th = 3.6.. thus 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1)
250,000,000 * 2 * 864/275 * 99/100 = 1,555,200,000

As 1 Greek Remen diaganol Royal Cubit Rolls 9x Eratosthenes Remen Arcs using Fibonacci Pi...

15" Greek x 99/70 (integer square root 2) x 2 x 864/275 (6/5th 144/55 Fibonacci or 864/275) x 875/864 (Greek to Eratosthenes conversion) = 9x Arcs of 15" Eratosthenes.

And the Bematists Cart wheel originates circa 3100bc, at the time of the Unification under Narmer or the Scorpion King, that defines the 24:30:31 degree boundaries...

1,575,000,000 * 864/275/Pi = 1,575,113,065.887" Imperial British inches rolled by a true wheel and...
1,575,000,000 / 1,573,440,000 * 864/275/Pi = 1.0010633 Polar Inches per British Inch.

We can see how and why the Imperial weight relates to Eratosthenes inches, yet the Volume relates to Polar inches.

The mile is however 57600 major inches (60000 minor) and hence to get a 250,000,000 unit polar radius, requires 12 inch foot but to make the 63360 inch mile 57600 inches requires using a foot and all other measures 1.1x larger resulting in the 13.2 inch foot.

The megalithic yard and megalithic rod deduced by Alexander Thom fits perfectly into this system. The megalithic rod was 2.5 times larger then the Yard and used for larger monuments. 2.5x is a very curious multiple to use unless working in units of 5 or similar which is the key to this unit.

11.52 * 2 * square root 2 * 1.00106 = 32.618 british inch (2.718 feet)

This is the exact figure quoted for the British Megalithic Yard. This is the diagonal of a double foot using primitive inch. The megalithic rod at 2.5 x greater is the diagonal of a passus (double pace of 5 feet) square. This megalithic rod then, happens to be a fathom of 4 remen diagonal royal cubit, as shown by the metrological relief held within the Ashmolean museum.

(ie 14.4" remen x 4 x sqr rt 2 x 1.00106 = 57.6" passus x sqr rt 2 x 1.00106 = 11.52" foot x 5 x sqr rt 2 x 1.00106).

This also explains the figure quoted by Flinders Petrie for Stonehenges Sarsen circle rings internal diameter as a Hekatompedon, with the the outer diameter as 15/14th, with a thickness of 2 Greek English cane cubits of 144/7".

100 x 12 x 35/36 x 1.00106 = 1167.9033' Imperial British
100 x 12 x 24/25 x 25/8 = 3600 English Greek

Metrological conclusions

There was two competing metrological systems in existance. One was the megalthic system and one was the mediteranean system. Originally the british inch was 1.00106x larger and geodeticaly based Like the graeco-egyptian geodetic inch. The ratio would have been 596 * 2,640,000 primitve inch = 360 * 60 * 72000 graeco-egyptian inch (1639:1620). At some point two modifications occurred; firstly the british inch became 1/1,575,000,000th of the earth polar circumference recorded by Eratosthenes (possibly using 3.141818' and 22/7 as Pi), secondly this became rationalised as the circumference of a wheel with 22/7 graeco-egyptian fingers resulting in the error between the anglo-sumerian and megalithic primitive inch.

One break away almost definitely occurred at some time before Eratosthenes quote that Egypt was 1/50th of the earth and 5000 stade of 300 cubits of 21 inch in length. This earth is 1,575,000,000 inches vs the nominal greek 1,555,200,000 inches and explains the reason we have 4.536kg per Pound. Orginally it was believed that 20 ounces should always equal 1 pint of water. Our weights and measures are supposed to be based on the cubic yard where the romans was based on the cubic foot. I have yet to see the maths for the british yard based weights. What I do know, is there is a certain relationship with the megalithic yard. The scenario is as follows:

1 cubic megalithic yard = 125 gallons = 1000 pints = 1250 pound weight where
20 ounces = 1 pint
160 ounces = 8 pint = 1 gallon = 10 pound weight.
1 litre of distilled water = 1000 cc = 1kg

If we say 1 british inch = 25.4mm, If we use our 22/7 geodetic finger basis to the modern british inch, If we use our 1,575,000,000 Eratosthenes proto british inch.

We will find that 1 cubic megalithic yard based on the Eratosthenes inch equals 125 gallons @ 4536cc or 4.536kg or 10 modern british pound

14.4 / 2 / pi / (22/7) * 10,000 * 60 * 60 * 60 = 1575113065.887 modern british polar circumference
1,575,113,065.887 / 1,575,000,000 = 1.0000717878647096007586587022098 (divider to convert British to Eratosthenes)
((11.52 x 2 x sqr rt 2) ^ 3) / 125 = 276.74667 cu. in. (4535.0654cc)
4535.0654 x (1.0000717878647096007586587022098 ^3) = 4536.04216 cc (10 pound = 4.536kg)

We will find that 1 cubic megalithic yard based on the Primitive inch equals 125 gallons @ 4549cc or 4.549kg or 1 modern british gallon

14.4 / 2 / pi / (22/7) * 10,000 * 60 * 60 * 60 = 1575113065.887 modern british polar circumference
1,575,113,065.887 / 1,573,440,000 = 1.0010633172456528370957901159243 (divider to convert British to Primitive)
((11.52 x 2 x sqr rt 2) ^ 3) / 125 = 276.74667 cu. in. (4535.0654cc)
4535.0654 x (1.0010633172456528370957901159243 ^3) = 4549.54743 cc (1 gallon volume = 4546 cc)

Now it is very difficult to preserve measuring rules. They break, they erode, they rot, they shrink and grow with environment, they change with number of copies. The only way to preserve a standard of measure over time is to use a weight to store all the metrological information. So it does not surprise me in the least that our 1 pound weight is identical to 1 pint using the Eratosthenes inch. The fact a gallon by volume and weight are not equal is a sure sign of a conflict in interests...differing samples with equal footing. How this coincidence should come about using a cubic megalithic yard as 1000 pints or 125 gallons is anybodies guess. The odds against such a chance coincidence must be absolutely staggering

Notes

[1] The Schoene and Atur.

I have come across various references to skoinous, Schoene, Atur, Iteru among others. The schoene as defined by Herodotus/Strabo is 60 stade and 1/60th of 1/60th of the earth into which it was divided. The Atur has been previously defined by the tablets recovered from the temple of Amun that make 7.4 degrees as 106 Atur. The Atur was thus 50 stade of 300 cubit of 28 fingers (the earth being 60 * 60 * 60 * 10,000 fingers). The length of Upper Egypt (6 degrees or 1/60th Earth) and lower Egypt was 7 degrees with divisions at 31 (cos^-1 6/7), 30 and 24 degrees north.

The Atur I also sometimes see quoted as 210 Khet. The measures of the Schoene and Atur appear to be confounded here. The mainstay of geodetic cubits was that defined by the romans and greeks as 700 stade of 300 cubit to the degree. This royal cubit is not 28 fingers but 200 fingers/7 by the geodetic description above. It equates to 10 egyptian remen / 7. The Schoene being 0.1 degrees will be 70 stade of 300 cubit of this 200/7 finger cubit (and not the 196/7 finger cubit). If Herodotus foot is considered 12 geodetic inches then;

1 Schoene = 210 Khet = 70 stade of 300 cubit of 20.5714 geodetic inch (200/7 fingers)
1 Atur = 150 Khet = 50 stade of 300 cubit of 20.16 geodetic inch (196/7 fiingers)
(60 * 60 * 60 * 10000) / (252000 stadia * 300 cubit) = 28.5714 fingers (200/7)

The simplest explanation for the existance of the Atur and Schoene is the cubit relationships and the remen diagonal.

1 Schoene = 0.1 degree. (60 stade of 400 common cubits of 18 geodetic inch or 25 fingers)
1 Atur = 0.07 degree (50 stade of 300 royal cubits of 20.16 geodetic inch or 28 fingers)
1 Atur = 0.07071 degree (50 stade of 300 royal cubit of 14.4 x sqaure root 2 geodetic inch or diagoanal of a remen square)

The Atur using a remen royal cubit becomes the side of a square whose diagonal is 1 schoene.

“Wonderful as is the Labyrinth, the work called the Lake of Moeris, which is close by the Labyrinth, is yet more astonishing. The measure of its circumference is sixty schoenes, or three thousand six hundred furlongs, which is equal to the entire length of Egypt along the sea-coast. The lake stretches in its longest direction from north to south, and in its deepest part is of the depth of fifty fathoms. It is manifestly an artificial excavation, for nearly in the centre there stand two pyramids, which rise to the height of fifty fathoms above the surface of the water, and extending as far beneath, crowned each of them with a colossal statue sitting upon a throne. Thus these pyramids are one hundred fathoms high, which is exactly a furlong (stadium) of six hundred feet: the fathom being six feet in length, or four cubits, which is the same thing, since a cubit measures six, and a foot four, palms. – Herodotus 484-418 BC”

“In essential accord with all this are the views of Hipparchus. He says that, having taken as hypothesis the measurement of the earth as stated by Eratosthenes, he must then abstract the inhabited world from the earth in his discussion; for it will not make much difference with respect to the celestial phenomena for the several inhabited places whether the measurement followed is that of Eratosthenes or that given by the later geographers. Since, then, according to Eratosthenes, the equator measures two hundred and fifty two thousand stadia, the fourth part of it would be sixty three thousand stadia; and this is the distance from the equator to the pole, namely, fifteen sixtieths of the sixty intervals into which the equator is divided. And the distance from the equator to the summer tropic is four sixtieths; and the summer tropic is the parallel drawn through Syene. Now the several distances are computed from the standard measures that are obvious to our senses. The summer tropic, for instance, must pass through Syene, because there, at the time of the summer solstice, the index of the sun-dial does not cast a shadow at noon. And the meridian through Syene is drawn approximately along the course of the Nile from Meroë to Alexandria, and this distance is about ten thousand stadia; and Syene must lie in the centre of that distance; so that the distance from Syene to Meroë is five thousand stadia. And when you have proceeded about three thousand stadia in a straight line south of Meroë, the country is no longer inhabitable on account of the heat, and therefore the parallel though these regions, being the same as that through the Cinnamon-producing Country, must be put down as the limit and the beginning of our inhabited round on the South. Since, then, the distance from Syene to Meroë is five thousand stadia, at total distance from Syene to the confines of the inhabited world would be eight thousand stadia. But the distance from Syene to the equator is sixteen thousand eight hundred stadia (for that is what the four sixtieths amounts to, since each sixtieth is estimated at four thousand two hundred stadia), and therefore we should have eight thousand nine hundred stadia left as the distance from the confines of the inhabited world to the equator, and from Alexandria twenty-one thousand eight hundred.” Strabo

[NB. Here Strabo quotes the distance from Syene/Elephantine to Alexandria as 31.1428 - 24 degrees as 5000 stadia according to Eratosthenes. This is a half truth. Alexandria sits at 24 + 7.2 degrees north. The Earth that Eratosthenes was referring to in regard to the measurement of the Earth, was in fact 1/50th Earth being 5000 stade of 300 cubit of 21" or 1,575,000,000" and 250,000 stade of 300 cubit of 21". This is why we have the 35/36th ratio between Roman (1,620,000,000") and Eratosthenes, or 24/25th Between Greek English (1,555,200,000") and Eratsothenes. This 100 foot diameter 3600" circumeference, requires using 24/25th scale with 25/8th Pi, leading to the 12.5" common Greek foot and 1,512,000,000" Earth. Vitruvius acknowledges this with his 4 foot diameter wheel marking 12.5 feet. Hence the 14:15 ratio between the Stonehenge Sarsen circles inner and outer diameter, ranging from 100 Roman feet to 100 common greek feet, where 36/35 x 25/24 = 15/14 and 25/24 / 36/25 = 875/864 = 1,575,000,000 / 1,555,200,000]

[2] The link between Phi, square roots, the royal cubit and the acre.

square root ((57600 * 57600) / 640) = 2276.839915 (1 acre side in inches)
14.4 x sqaure root 2 = 20.364675 (1 remen diagional royal cubit)

2276.839915 / 20.364675 = 111.803398 (1 acre side in royal cubit)
square root 12,500 = 111.803398.

The square root of 1.25 has some very interesting values and is directly related to Phi or the Golden Section.

Phi = (square root 1.25) + 1/2 = 1.6180339.

The square root of 1.25 forms the basis of a table to find the square roots of all whole numbers. for example....

base = sqrt 1.25, side = 1 and side = square root of 2

...will give you an apex for a vertical line that is common to all the root numbers. If you now place sides with....

base = sqrt 1.25, side = square root of 2 and side = square root of 3
base = sqrt 1.25, side = square root of 3 and side = square root of 4
base = sqrt 1.25, side = square root of 4 and side = square root of 5

...the apexes of all these triangles will fall on the same vertical line. Hence by knowing the square root of 1 number you can find the square root of all. And all of these sqaure roots have Phi in common.