Earth, Pi, Miles and the Barleycorn
With a touch of Stone Circles and the Megalithic Yard
I took a sneaky look at one of my Christmas presents, and through the wrapping paper I could see that it was a small square book with a dust jacket in black. Centrally on the front emblazoned in white, was the Greek letter Pi (π). I had heard of the book, it is called 'The Joy of Pi' by David Blatner. I was so pleased I wanted to tear the paper off and immerse myself in it right away. Idle thoughts came into my head. π is connected to circles, why was the book not round, if one can have a round or spherical book! It wasn't the usual rectangular book shape either, but square. A black square with p in the middle. I measured it. It was just about 6½ inches square. If it had been 6.283... inches it would have been 2p inches square, but it wasn't. Did it's squareness mean something? What if it were to represent p square units of area? Then I could escribe a circle around it which would have a circumference of 7.87.... units.
So we have a circle around a square of p sq. units. It has a diameter of 2.5066....units, radius 1.253...units and an area of 4.9348022...sq.units.
A strange circle, for R3 (radius cubed) equals a quarter of the circumference, which means that p = 2(R2), (2 times radius squared).
Well that is it - almost. What would happen if I were to multiply the square of p sq. units by 10,000,000? The circle's area would also be 10 million times as big, giving a circumference of 24902.31984... units, diameter 7926.65... units, radius of 3963.327... units, and the side of the inscribed square would be 5604.99... units.
Look again at the circumference and call the units 'MILES', good old fashioned English statute miles. Then look up in your reference books to find the Equatorial dimensions of the Earth. Satellites now give the Equator as being 24901.46... miles. ( Enc.Britannica. radius 6378137 m.)
Taking the satellite figure means that our current mile is too long by 2.155 inches - the thickness of two thumbs. Which works out at less than the radius of the earth over a distance from here to the Sun.
This minute discrepancy of a couple of inches can hardly be measured on a one to one map ( if we had one!), and means that our mile has been an accurate geodesic measure since goodness knows when.
If this remarkable relationship which the Mile has with π had been recognised before the French started playing around with 'Meters', the Mile would now be ruling the Millennium.
(This little mathematical coincidence of Miles and the Earth is a perfect Joy which I have wrangled from p . I am unaware that it has been previously noted, so I can give no reference.)
I wonder what the odds must be of inventing an arbitrary unit of length (as the mile is deemed to be), then using it to measure the Earth and finding that the unit you have chosen has a unique relationship with p and the Earth itself? It is regrettable that measurements and drawings of archaeological material now use the metric system which hides so many interesting facts, and it is a nuisance to keep converting to Imperial measure. Who would dream that 81366935.57... sq. kilometers was equivalent to 10,000,000 π sq. miles? Or that 31.676.. m. was 60 (√ 3 )ft.?
(N.B. Conversion factors are those used in "Changing to the Metric System", National Physical Laboratory, HMSO. 1969).
There is (or was), a British 'Telegraph Nautical Mile', equivalent in the SI system to 1.85532 km. This works out at 6087.0062 ft. per minute of arc at the Equator, 69.170525 miles per degree, against my mathematical one of 69.1731 miles per degree. So by whom or when was this accurate Telegraph Mile determined? And why make such an accurate measure obsolete along with fathoms, rods poles and perches? And eventually the Mile?
Having got this far, a dormant interest in miles and measure was rekindled, and I wondered what else could be produced from doodling with a calculator. Might there not be more to the myth that the Mile and it's companion Acre were of vast antiquity? First, a delving into the informed origins of the mile gave cause for concern when I discovered my first fact. The Roman general Nero Claudius Drusus campaigning in lower Germany in 12 BC, was forced to adopt the long established and deeply ingrained Northern foot of the Germanic tribes in preference to the smaller Roman foot. The longer measure of these North European people, was 4800 Northern feet (N.ft.) to the 'mile'. What their name was for this measure I do not know, perhaps the Romans insisted it should be 'Mille Passus', a thousand paces. What we do know is that 4800 N.ft. are equivalent to 5280 ft. - our current mile.
A second clue is that this Northern foot was deemed to be 36 Barley corns laid end to end, or 13.2 present inches. It can be traced back to 3000 BC to the pre-Aryan races of the Indus valley civilisation. The length is marked on Egyptian wooden cubit rods of 1900 BC, and on Royal cubits 1567- 1320 BC
The Sumerians were aware of it also, for their cubit was equivalent to ¾ of the Northern 'cubit' of 2 N.ft., and half the Sumerian cubit was known as the Natural or Pythic foot which was preferred by the Celtic races, yet still with a Barleycorn base. (Skinner F.G., 'Weights and Measures', Science Museum, H.M.S.O. 1967).
So there is a little history for you. For me it leads to the conclusion that there was a race of people in Northern Europe well acquainted with units of measure which are allied to our present mile way before the time of the Romans. The Celts with the Natural foot were here prior to the Romans, the German/Saxon tribes with the Northern foot came along after they had left. It is not beyond the bounds of possibility that even earlier visitors to these shores, be they Stone, Bronze or Iron age peoples, came over with a measurement system based on barleycorns, the foot and who knows, maybe the mile.
I am going to spend a little time with this little item, not to discuss it's brewing properties, but to show how my calculator fits it into the grand scheme in various ways.
Firstly there are three Barleycorns (Bc) to the inch, not our present inch, but the old Northern inch, (1.1 present inches), i.e. 36 Bc. to 13.2 inches. Thus making each one 0.366666 ins. and giving 172800 Bc. to the Mile.
The formulae which follow I shall call ''Franklin's Formulae".
Now, take a square with sides of 360 Bc., the area will be 13.444... sq.yds, which I do believe is 1/360th of an Acre. (4840 sq.yds.). Multiply the sides of this square by 6, and the area will be 1/10 th of an Acre or 1sq.chain (22yds. by 22yds.). An unexpected result.
Going one step further with this duo-decimal/sexagesimal system, it can be shown that :-
The rotational speed in Barleycorns per second, of any perfect sphere rotating once in 24 hours, at any latitude, will be twice the number of miles in the circumference of that latitude.
For example, at the Equator :-
24902.31984 * 63360 / (86400 * 0.3666666..) = 49804.6396... Bc's per sec.
Moving on to show another example of the powers of this little Barleycorn, take a square with sides of 20 Bcs. and escribe a circle around it. The circumference of this circle will be 2.715095129...ft., and ...........
When the rotational speed in ft.per.sec. of our perfect sphere, rotating once in 24 hours, (at any latitude) is divided by the above 2.715095129..ft., we arrive at a number which is 1/10th of the number of miles in the side of the inscribed square at that latitude.
For example, using the Equator again :-
Rotational speed = 24902.31984.. * 5280 / 86400 = 1521.808.. .ft.per.sec
1521.808.. / 2.715095129.. = 560.499.... ,
Equatorial inscribed square is 5604.99.. Miles
Here is another gem which my calculator turned up. A square with sides of 2.715095129.. ft., has a diagonal of 3.8397..ft. ((√ 2) * 2.715...), and I can say that :-
The number of ((√ 2) * 2.715...)ft. per degree of arc at any latitude, is always equal in number to the number of Miles in the diameter at that latitude, multiplied by 12.
It is true for any circle when using a division of 360 °, and means also that the diameter in miles * 4320 is the number of them in any circle circumference. If you want to have 1 unit of 3.8379 ft per degree, you are stuck with a circle of 2 π miles, and 8640° in the circle. So Miles in Diameter * Factor (x) = No. of 3.8397. .ft in 1 degree., and as you can choose any number of degrees you would like to have in a circle, Factor (x) = 4320 / degrees required .).
e.g. (1). Taking for a change, latitude 60°, circumference of our perfect sphere 12451.159. miles (cos 60° * Equator circ)
Diameter is 3963.327.. miles, times 12 = 47559.927...
Circumference / 360 = 34.586.. miles = 182617.01 ft per degree
182617.01.. ft. / 3.83973.. = 47559.927...
e.g. (2). Circle diameter = 550 ft. = 0.104166.. miles, times 4320 = 450.
450 * 3.83973.. = 1727.87.., the circumference. / π = 550
The mile is now connected, via the Barleycorn, to the division of a unit radius circle into degrees. One is led to choosing a convenient factor of 8640, and 24 * 360 looks pretty good. An odd looking number 3.83973.. ft., but it is 40 π Barleycorns. Try working these neat exercises with millimetres and Kilometres!
Now the rotational speed of our perfect sphere at 60° is 760.9042.. ft.per sec., this, when converted to Northern Ft. (Nft.), is 691.73... Nft.per.sec. By another quirk in our number system, there are 69.173.. miles to one degree at the Equator.
The question is, 'Why is this so?', and I think one of the answers is that if we have a square with sides of 10 miles inscribed in a circle which rotates once in 86400 seconds, then the rotation speed is :-
(10(√ 2))π * 5280 / 86400 = 2.71509.. ft.per.sec.
I am intrigued with this number, and my calculator invites me to continue the play.
Each degree of arc at latitude 50.036.. ° , (inscribed square 3600 miles ), is 10 (√ 2)pi miles , (44.4288..). Rotational speed of 977.434... ft.per.sec., which is equal to 888.576.. Nft.. The number of Barleycorns in 2.71509.. ft. is 88.8576. (circumference around a 20 Bc. square.).
Oddly enough, a circle of 977.434.. ft. will have an inscribed square of 220 ft., 9 squares of which would cover an area of 10 Acres.
Now if the circumference of a circle rotates in 86400 secs. , an arc equal to the side of the inscribed square, would pass by in
19446.832.. seconds, and 5280 ft. / 2.71509.. ft. is 1944.6832..
A circumference of 1 Mile, say it was centred on the North Pole, would have a rotational speed of 2 Barleycorns per.sec. (.06111.. ft.per.sec.). So it can be said that any circle of latitude in miles, divided by 16.3636...., will give it's speed of rotation in ft.per .second.
Did you note that 2.715095... ft. is very close to Professor Thom's Megalithic Yard of 2.72 ± .003 ft.
The MEGALITHIC YARD
This is the measure which the late Professor Thom found in his studies of Stone circles from Callanish in Scotland to Carnac in France. (There is also a Carnac place name in the Isle of Lewis, as well as the better known sites in Egypt and France!). He claims it was used over a period of a thousand years or more, and by various races of people.
He regarded it as being an extremely accurate measure which must have been in the form of a rod, issued from some central source and carried to all parts of N.W.Europe. The practical implications are enormous, and so the idea has been dismissed by archaeologists who tend to regard it's existence as being the average length of a human pace. I am not saying that my 2.715095.. ft. is a firm candidate, for there is another similar number derived from geometry which equally fits the bill, and I shall come to that shortly.
What if there were two systems in use from Neolithic times, both deriving from the mile? The Northern foot, which I hope I have shown is a distinct possibility, and our current foot of 5280 to the Mile?
Historically our foot is a fairly newcomer to the scene, that is if it is deemed to have started by the Statute for Measuring Land, 33 Edward 1, Stat.6 (AD 1305), although a 'foot' measure is mentioned in 1196 AD. It may well be worth while quoting from Edward's statute :-
"It is ordained that three grains of Barley, dry and round make an inch, twelve inches make a foot, three feet make an "Ulna", five and a half Ulne make a rod, and forty rods in length and four in breadth make an Acre. And it is to be remembered that the Iron Ulna of our Lord the King contains iii feet and no more, and the foot must contain xii inches measured by the correct measure of this kind of Ulna; that is to say the thirty-sixth part of the said Ulna makes one inch neither more nor less; and five and a half Ulne make 1 rod, sixteen feet and a half, by the aforesaid Ulna of our Lord the King."(my italics)(Skinner)
'This kind of Ulna' indicates that there was or were other kinds in use, and not that this was a new invention. One of the other kinds was the Saxon Elne or Ulna, the Northern cubit of antiquity (2 Nft.). Edward's 'Ulna' was later to be called the Yard, and is unlikely to have differed from the current Imperial British Yard by more than .04 inches (1 mm.) He had a choice of units deriving from the Mile, and calculating would have been much easier if he had chosen the ancient Saxon foot, but it is my premise that both feet were running in parallel from time out of mind; he set an official standard but not a new one, both converging together as a unit of the mile, at 22 yards.
It is not easy to imagine that our foot was used in prehistoric times, but as Thom's Megalithic Yard turns up time and time again in the measurement of monuments, it is worth searching for it's origin. Thom was confident that some of the flattened stone circles (stone circle web site under construction) that he examined, were deliberate efforts to come to terms with the irrationality of π, and that their perimeters were attempts to make the relationship of the diameter to perimeter equal to three and not 3.1415... Not only that, but perimeters and diameters of normal circles were found to be in whole numbers of Megalithic Yards (MY) and multiples, mainly 2.5 MY which he called a Megalithic Rod.
I want to show you a Geometrical Megalithic Yard which relates to π, the foot, and Earth square Miles.
I have a circle with a radius of 1 foot, (i.e. a unit radius circle), circumference 2π ft. Inside this circle I draw the six sided figure of a hexagon, and add to it an inscribed circle. This will have a circumference of 5.44139... ft, half of which is 2.720699.... ft. Call this number 'M', as it is to all intents and purposes exceedingly close to Thom's figure for his Megalithic Yard (MY)., and is thus my second candidate for this position.
The area of a (√ 3) rectangle, constructed on Earth's diameter is 27206990.45 square miles.
'M' can also be found from a √ 3 triangle with an hypotenuse of π , the third side being π(√ 3)/2, which is 'M', or 2.720699...
I shall deal with M and Stone circles in detail in the near future, just remember that Thom's measure for the outer Sarsen circumference at Stonehenge, was 120 MY, as was his missing ring of posts at Woodhenge.
I have mentioned that archaeologists can not accept that a measuring rod could have remained a constant length over a few thousand years, even less that Stone circle builders could have used our 'foot'. A unit radius circle, be it in cubits, Greek feet or what have you, produces the number M: when one finds this number appearing in units in a particular measuring system, then it would appear to be deliberate intent. When it is found in Imperial feet, then I am confident that 'Feet' were the units used. Our Mile has existed for millennia, and our current foot, historically, for 700 known years, so acceptance of longevity should be no barrier for them (the archaeologists).
Just in case you think that 'M' with its π make up is rather unusual, let me introduce you to Arabian measures which I have extracted from ProKon, a computer conversion programme.
(Harold Schwartz, ShowMe Software, P.O.Box 104482, Jefferson City, MO 65110)
|Arabian feet||English feet||Equivalent to|
|Barid||72000||75398.548556||24000 π ft.|
|Mille||6000||6283.2122703||2000 π ft.|
|Qasab||12||12.5664247||4 π ft.|
|Foot||1||1.04720206||π / 3 ft.|
|Cabda||1/4||0.26246719||π / 12 ft.|
|One Cubit||-||1.7734269||(√ π)ft.|
|One Assba||-||0.06561679||(√ π) * (√ 2) * π / 120|
(N.B. These units do not appear in the Dent Dictionary of Measurement. 1994.)
Take a careful look at the Cubit, 1.7734.. ft. It is to all intents the side of the inscribed square in the circle discussed at the top of the page, which led to Earth and area dimensions, and the Assba is 1/120th of the surrounding circle circumference.
There is another Arabian measure which doesn't fit in with the forgoing scheme. It is the 'Galva', of 756 English Feet., and for Pyramid buffs this is a pretty good measure for the base length of the Gt. Pyramid. However, take
10,000M / 36 = 755.749735 ft., an even closer measure to the pyramid base!
STONEHENGE the Sarsen Circle
I did mention that Thom said the outer circumference of the Sarsen Stone circle was 120 Meg.Yards. Trying it with 120 M ( difference on circumference = 1.0066 inches) produces some interesting figures, and the numbers are recognisable as being the 'Precessional' numbers so beloved by Graham Hancock in 'Finger Prints of the Gods'.
|Circle Area||2700 π sq.ft.|
|Inscribed square area||5400 sq.ft.|
|(√ 5) rectangle area||4320 sq.ft.|
|Circle in this rectangle||540 π sq.ft.|
|1.813799 radian sector area||900 M sq.ft.|
|Arc of sector||30 π ft.|
|1.666 radian sector area||2250 sq.ft.|
|Arc of sector||50 (√ 3) ft.|
|Diameter||60 (√ 3) ft.|
|Sides of (√ 5) rectangle||√ 8640 and √ 2160|
Coming outside of the circle, and making a shoe-box of the double square, the diagonal from far top left to near bottom right is 113.841 ft., or √ 12960. Dividing this by (√ 3) gives √ 4320.
√ 12960 / √ 8640 = 1.22474, which is 1/60 th of the side of the inscribed square. (73.484.. /60).
I am quite impressed with these numbers appearing from the outer circumference of the Sarsen Circle at Stonehenge. At the bottom of page 264, Hancock, referring to his source authors (discussing precessional numbers) says .....
'If they are wrong, we need to find some other explanation for how such specific and interrelated numbers (the only obvious function of which is to calculate precession) could by accident have got themselves so widely imprinted on human culture.'
To me they fall out naturally by playing with π and squares. I expect they did too for the Maya, Chinese and in India.
You may wonder why I have included the odd 1.813799 radians in the above table. For fun, it would be nice to have a segment of a circle to contain the same number of degrees as there are units in the diameter. If the Sarsen diameter is 60 √ 3 ft. (103.923.. ft, radius 51.9615.. ft.), then the number of radians has to be 103.923..° / (180/π), or 1.813799 rads. Odd number or not so odd? It is 4.9348022 / M, or π/√ 3, or 2M/3. If one tries this number of radians with any other circumference it will not work unless the number of degrees in the circle is changed, and the number of degrees in the new circle will need to be Diameter * 2 * √ 3. So, 2M/3 radians and 360 degrees are unique to Stonehenge.
Which leads me to ponder as to why we always expect to find buildings ,stone circles, monuments etc., staked out in whole round units of measure. I can imagine a different mind-set, far more subtle, which worked with relationships of numbers, numbers related in one way or another to π , to the squares of numbers, to 360 and to unity. (As was done in Arabia?).
There is another way in which Stonehenge can be related to the number 360.
Imagine a π type pyramid erected on the inscribed square of a 120 M circle. Base 73.484.. ft..
Height = (base*√ 3) / M, = 46.7818.. ft. Transfer this height to that of a regular Tetrahedron, and it's Base will be 57.295.. ft. with a surrounding circle of 120 (√ 3) ft. Isn't 57.295.. the radius of a 360 unit circle, and the number of degrees in a Radian? (Base of tetrahedron = height * √ 3 / √ 2)
(A π - type pyramid is one in which the ratio of the height to the perimeter of the base, is the same as that of the radius of a circle to the circumference, 1 : 2 π). Five times the height of this pyramid on the inscribed square at Stonehenge, is 233.9... ft., the height of the pyramid of the Sun at Teotihuacan.
If the Sarsen Circle was taken to be the meridian of a sphere (or spherical temple), it would have a volume of :- 12,000 π √ 3 cubic yards, or 24,000 M cu.yds., or 216000 M cu.ft., or 108000 π √ 3 cu.ft.
If one can accept that the circumference is 120 M, then many interesting things can be determined from the remaining geometry of the Trilithon and Bluestone circles, together with the Bank, Ditch and the Hele Stone, but as yet I have not completed the required explanatory drawings. I shall return.
The world is not a perfect sphere, and my Mile system is 'nearly but not quite'. (The satellite figure is 99.9966% of mine). But during the Ice Ages, might not the pressure and weight at the poles have made the swelling at the Equator nearer to my figure? (Not that there was anyone around to figure out the 'system' then! ). . How you measure it's length on the ground after inventing it is another matter. Eratosthenes may not have been the first. Might not the length of the mile have changed slightly over the centuries?.
Is there any meat on the mile measures that Henry Lincoln (author 'The Holy Place') and others pursue at Rennes le Chateau in the South of France? (French countryside laid out in English Miles, mon dieu!). The Barleycorn has been used to decorate pottery since Neolithic times, I have no doubt that if it were used as a measure for 'weight', (we still have the 'grain'), it was played with to produce a measure of length. Basic geometry, circles, squares, triangles etc. was and is a common world wide language, you can always produce proportionate √ 3 lines without really knowing anything about sq.roots. And 360 is just about the best ever division that a circle could have. I shall elaborate on that at some future date. My knowledge of Maths and Geometry is elementary and naive, I am just enthralled with the pattern and coincidence of numbers which fall from my pocket calculator. I hope you will be too. Somehow, there are grounds here for retaining our Imperial measures of length, now that it can be seen that they actually relate to the earth we walk upon.
hugh franklin. March 2000