Friday, 11 April 2008
Schiehallion - Galachlaw - Rubers Law
Back in the year 2000, I was driving south to England from Selkirk, early morning, to be in Lichfield for 9.30am.
From Hawick I took the A6088 towards Bonchester Bridge, and as the road took a turn east just after Kirkton, I was stunned by the hill silhouetted by the early dawn light:
On the way back later that day, I took another photo of the same hill:
I confirmed this hill to be Rubers Law, O.S grid reference: NT 579 155.
I had been working on some geometry, see previous Schiehallion - Eildon line post, and was discovering the landscape of the Borders as I went. Rubers Law was part of the discovery process. I can't quite describe the feeling of shocked awe when I turned east that morning to see the hill in the dawn light, dark and brooding in silhouette. I had to check the maps to see if and where it fitted in. With great amazement I found it to be in a direct line with Schiehallion and Galachlaw, the entre of the original pentagonal system previously described. Especially so since I had already confirmed the Schiehallion -Arthurs Seat - Eildon line.
When I got round to doing the detailed calculations, I was even more amazed to find that Galachlaw was indeed very close to the Phi-point.
A few calculations to show this:
Grid References (full, and normal forms)):
2713.83 7547.74 Schiehallion summit NN 714 547
3253.65 6683.21 Galachlaw cairn NT 253 683
3249.37 6687.44 Water Works mast( exact Phi-point) NT 249 686
3580.35 6155.75 Rubers Law summit NT 579 155
Schiehallion:
first calculation; Schiehallion - Rubers Law:
2713.83 7547.74 Schiehallion
3580.35 6155.75 Rubers Law
--------- ----------
0866.52 1391.99
By Pythagoras' theorem; 1639.66
which converts to 101.88 miles(E), and 90.87 miles(S)
The angle to grid north is 31.9024 degrees, using the tangent of the two calculated values above.
Second Calculation; Schiehallion - Galachlaw cairn:
2713.83 7547.74 Schiehallion
3253.65 6683.21 Galachlaw cairn
--------- ----------
0539.82 0864.53
By Pythag. 1018.22
which converts to 63.33miles(E)and 56.48miles(S)
angle to grid; 31.98075 deg.
using the English miles distances of both we ind the relationship to be 99.424% of Phi:
101.88/63.33 = 1.60872, which is 99.424% of 1.618034.
Galahlaw Cairn:
The exact Phi-point proves to be some 660 yards to the north west of the cairn, at point (3249.37 6687.44), which is in the Water works where a Mast is erected, normal grid reference NT 249 686.
view of gap between Hillend fort on right and Galachlaw with Water Works Mast to the left:
This mast can be seen to the left of this photo, with Hillend Fort to the right.
Comment:
All I can say about this is that it was around this time that I came to strongly accept a 'sense of process' involving time and place, and which I for sure am not in control of. It is not just an excuse for me to be lazy in presenting all this material, it has taken years to get to, and I have learned to accept patience as an essential. All the geometry and correlations have in a sense been presented as and when 'this process' deems necessary. I no longer question this, it just happens when it does.
supplementary
When plotting this line on Google Maps, I checked Fordell Castle which I had noted originally was on this line, and on calculation was found to be at the centre, Fordell Hill precisely, within the estate. Fordell was the home/seat of Nicholas 'Nicky' Fairbairn, a Tory MP, government Minister and somewhat extrovert.
As Galachlaw is at the phi-point of the line, there is another equivalent on the other side of the centre-point. On examination and confirmed by calculation, it was found to be White, or Hood Hill, NO 042 018, near Tullibole Castle, by Crook of Devon.
There is also a radio/electronics mast on Hood Hill!? (Once is chance, twice is coincidence...!)
(There was reported that a sign-post in Crook Of Devon had neatly printed under it: 'Twinned with Thief Of Baghdad'! - btw! lol)
In addition it was noted that Lauriston Castle is on this line, and Cramond Island. Lauriston Castle came to my attention in connection with John Napier of Merchiston, an Alexander Napier having owned it. Lauriston Castle proved subsequently to be a geometric node, and still to be described.
Cramond has been highlighted recently by Bill Buehler, in connection with the Roman lion staue with man in its mouth, which I also linked with John Napier, or the double lion-gate at Merchiston, in an e-mail exchange!
From Hawick I took the A6088 towards Bonchester Bridge, and as the road took a turn east just after Kirkton, I was stunned by the hill silhouetted by the early dawn light:
On the way back later that day, I took another photo of the same hill:
I confirmed this hill to be Rubers Law, O.S grid reference: NT 579 155.
I had been working on some geometry, see previous Schiehallion - Eildon line post, and was discovering the landscape of the Borders as I went. Rubers Law was part of the discovery process. I can't quite describe the feeling of shocked awe when I turned east that morning to see the hill in the dawn light, dark and brooding in silhouette. I had to check the maps to see if and where it fitted in. With great amazement I found it to be in a direct line with Schiehallion and Galachlaw, the entre of the original pentagonal system previously described. Especially so since I had already confirmed the Schiehallion -Arthurs Seat - Eildon line.
When I got round to doing the detailed calculations, I was even more amazed to find that Galachlaw was indeed very close to the Phi-point.
A few calculations to show this:
Grid References (full, and normal forms)):
2713.83 7547.74 Schiehallion summit NN 714 547
3253.65 6683.21 Galachlaw cairn NT 253 683
3249.37 6687.44 Water Works mast( exact Phi-point) NT 249 686
3580.35 6155.75 Rubers Law summit NT 579 155
Schiehallion:
first calculation; Schiehallion - Rubers Law:
2713.83 7547.74 Schiehallion
3580.35 6155.75 Rubers Law
--------- ----------
0866.52 1391.99
By Pythagoras' theorem; 1639.66
which converts to 101.88 miles(E), and 90.87 miles(S)
The angle to grid north is 31.9024 degrees, using the tangent of the two calculated values above.
Second Calculation; Schiehallion - Galachlaw cairn:
2713.83 7547.74 Schiehallion
3253.65 6683.21 Galachlaw cairn
--------- ----------
0539.82 0864.53
By Pythag. 1018.22
which converts to 63.33miles(E)and 56.48miles(S)
angle to grid; 31.98075 deg.
using the English miles distances of both we ind the relationship to be 99.424% of Phi:
101.88/63.33 = 1.60872, which is 99.424% of 1.618034.
Galahlaw Cairn:
The exact Phi-point proves to be some 660 yards to the north west of the cairn, at point (3249.37 6687.44), which is in the Water works where a Mast is erected, normal grid reference NT 249 686.
view of gap between Hillend fort on right and Galachlaw with Water Works Mast to the left:
This mast can be seen to the left of this photo, with Hillend Fort to the right.
Comment:
All I can say about this is that it was around this time that I came to strongly accept a 'sense of process' involving time and place, and which I for sure am not in control of. It is not just an excuse for me to be lazy in presenting all this material, it has taken years to get to, and I have learned to accept patience as an essential. All the geometry and correlations have in a sense been presented as and when 'this process' deems necessary. I no longer question this, it just happens when it does.
supplementary
When plotting this line on Google Maps, I checked Fordell Castle which I had noted originally was on this line, and on calculation was found to be at the centre, Fordell Hill precisely, within the estate. Fordell was the home/seat of Nicholas 'Nicky' Fairbairn, a Tory MP, government Minister and somewhat extrovert.
As Galachlaw is at the phi-point of the line, there is another equivalent on the other side of the centre-point. On examination and confirmed by calculation, it was found to be White, or Hood Hill, NO 042 018, near Tullibole Castle, by Crook of Devon.
There is also a radio/electronics mast on Hood Hill!? (Once is chance, twice is coincidence...!)
(There was reported that a sign-post in Crook Of Devon had neatly printed under it: 'Twinned with Thief Of Baghdad'! - btw! lol)
In addition it was noted that Lauriston Castle is on this line, and Cramond Island. Lauriston Castle came to my attention in connection with John Napier of Merchiston, an Alexander Napier having owned it. Lauriston Castle proved subsequently to be a geometric node, and still to be described.
Cramond has been highlighted recently by Bill Buehler, in connection with the Roman lion staue with man in its mouth, which I also linked with John Napier, or the double lion-gate at Merchiston, in an e-mail exchange!
Monday, 7 April 2008
Google Maps trial
First attempt at drawing lines on Google Maps. This is the Schiehallion to Eildon Hills North line:
View Larger Map
It seems to be slightly off at Arthurs Seat in comparison to calculations and Google Earth. Needs to be considered further!?
I have also marked Newtongrange and Borthwick on this line now.
View Larger Map
It seems to be slightly off at Arthurs Seat in comparison to calculations and Google Earth. Needs to be considered further!?
I have also marked Newtongrange and Borthwick on this line now.
Monday, 24 March 2008
sense of place - photo-album
After all that math of the previous posts, I have spent the day 'playing' with photos, and have managed to upload some as a web-album in Picasa.
Here's some photos of Lothian and the Borders of Scotland, un-captioned as of yet, and pretty much random.
Here's some photos of Lothian and the Borders of Scotland, un-captioned as of yet, and pretty much random.
Friday, 7 March 2008
Schiehallion alignments part one - Arthurs Seat to Eildon
The grid references to be used in this section are:
2713.833 7547.736 Schiehallion summit(S)
3275.280 6729.430 Arthurs Seat(AS)
3554.330 6329.100 Eildon north Ring Fort(E)
3333.750 6642.000 Newtongrange Church(N)
3580.350 6155.750 Rubers Law(RL)
3253.645 6683.210 Galachlaw Cairn(G)
3249.370 6687.440 Water Works, calculated exact golden-section point near Galachlaw
3189.702 6826.692 Inchcolm Abbey(I)
The calculations to follow on the three lines on the drawing. What I shall show is: 1) that Schiehallion, Arthurs Seat and Eildon Ring Fort, and Newtongrange Church are all on a straight line, within very tight margins.
2) That Galachlaw Cairn, the center of the original penta-system, is very close to the EXACT phi-point(golden section/mean) between Schiehallion summit and Rubers Law summit. The exact point is in fact in the local water-works, where a radio mast stands(!?).
3) That Inchcolm Abbey is at one degree, very lose, to the Arthurs Seat line, at Schiehallion, which may be relevant.
On the diagram, it can be seen that the Inchcolm line extends to a point labelled (FC), this is Fatlips Castle near Minto in the Borders. The calculations to Fatlips have not been included here.
The axis of the original penta-system is shown, from Roslin(R) through Galahlaw(G) to Inchcolm(I).
There is a vertical line, indicating what has been called the 'Roseline', or in Bill Buehler's terminology, the Tavhara, with Dryhope Tower(DT) at the bottom, and Seafield Tower(ST) at the top, both on the circle circumference, centre St Mary's chapel(St M), radius 20 miles. This is still to be explained in detail.
Part 2
The line highlighted in the diagram, between Schiehallion(S) and The Eildon Hills(E), runs through the Arthurs Seat(AS) area, with a ratio of 2:1, and also includes Newtongrange Church(N), which is one of the determining points of the Rennes Le Chateau radius as determined by Henry Lincoln, with Mavisbank as centre.
Schiehallion:
The Eildon Hills, known to the Romans as Trimontium, has Melrose Abbey nestled in the low ground near the river Tweed. The example given here uses Eildon North, with a large ring-fort on its summit, and was used as a signal station by the Romans.
Eildon Hills:
I shall later also use Eildon Mid-hill, which has an O.S. trig-point on its summit.
Arthurs' Seat also has a trig-point on its summit, and it is the summit which really defines the 'Roseline', through Rosslyn Chapel(R), and is shown on the diagram, with centre at St. Mary's Chapel(St.M), and a 20mile radius circle, with Seafield Tower(ST) on the 'Roseline' to the north, and Dryhope Tower(DT) to the south. The Arthurs Seat complex covers quite a lage area, from Holyrood Abbey/Palace/Parliament Building, St. Anthony's chapel, Dunsappie Fort, Duddingston Church, Crow Hill and Whinny Hill, and Salisbury Crags. There is a lot to consider and this post has been somewhat delayed due to my need to compare a whole set of calculations between Arthurs' Seat and the Eildon Hills, with some interesting correspondences and ratios of distances and angles.
Arthurs Seat, from Hillend Fort,(very important geometric point, to be discussed), over Galachlaw, (centre of original pentagon, and to be discussed in the Schiehallion to Rubers Law line, Galachlaw at Phi-point):
Schiehallion summit is taken as a constant. It has already been seen to lie on the hypotenuse of the Preston Cross triangle.
All three sites are special, with their associated myths and legends, in a sense they are all considered 'magic' mountains.
Using the Grid references listed in the previous post, and using the same procedure as before:
Calculation (1)
2713.833 7547.736 Schiehallion
3275.280 6729.430 Arthurs Seat
------------ ------------
0561.447 0840.167
By Pythagoras theory;
0992.395
which converts to;
61.664556 miles(E); or 54.998 miles(S)
The angle to the O.S. grid, again using the tangent of the ratio of the two derived grid components is 34.4544 degrees
Calculation(2)
2713.833 7547.736 Schiehallion
3554.330 6329.100 Eildon North(ring-fort)
------------ ------------
0840.497 1218.636
By Pythag; 1480.375
which converts to;
91.9862 miles(E); or 82.042 miles(S)
Angle to O.S.grid; 34.59415 degrees
First of all the angles to grid are:
34.45440deg.
34.59415deg.
------------
0.13975deg. difference
This is 1/43rd of one clock-face-minute(cfm), (reminder that one cfm is six degrees).
This discrepancy at Arthurs' Seat summit is some 265 yards.
The exact line runs to the north east, between the summit and Dunsappie fort. If the Arthurs' Seat line is used and projected to the Eildon, it passes through the gap between Eildon North and Mid-hills.
The Schiehallion/Eildon North line runs through the gap between Arthurs Seat summit and this stone, convenient marker(!), on Whinny Hill:
So, Schiehallion, Arthurs' Seat and the Eildon Hills are in a straight line!
The distance ratio is 61.664556:91.9862 = 2:2.98344(99.45% of 3)
The exact distance of Arthurs Seat to Schiehallion should be 61.324 miles(E) for a 3:2 ratio. A discrepancy of 600 yards, or 0.34 miles(E), north towards St. Anthony's Chapel and Holyrood Abbey.
Holyrood Abbey/Palace, lower foreground and St. Anthony's Chapel, the ruin left centre:
This is relevant perhaps to geometrical grid work, the 3:1 ratio having shown up elsewhere. More needs to be done on this aspect at a later date. Just to note in passing that the diagonal of a 3 by 1 grid is the square root of ten!
William Buehler(Bill) has recently described a temple format using this ratio, and I have found it to be in use in a grid based on the twenty mile radius in the diagram, which was first defined, to my knowledge, by Bill.
Enough on that topic for now!
Next to consider the position of Newtongrange church:
2713.833 7547.736 Schiehallion
3333.750 6642.000 Newtongrange Church
----------- ------------
0619.917 905.736
By Pythag. 1097.568
converts to; 68.1997 miles(E); or 60.827 miles(S)
Angle to O.S. grid: 34.389 degrees.
Comparing this angle to that of Schiehallion/Eildon North:
34.59415 Schiehallion/Eildon(N)
34.38900 Newtongrange churh
-------------
0.20515 deg. (1/29th cfm, or some 430yards at Newtongrange) A quarter of a mile may seem a lot, but at this scale it is in reality not much, and later I shall show that Newtongrange is more accurately in line with Eildon Mid-hill and Arthurs Seat.
Keeping this in bite-sized chunks I shall leave this here for now!
Supplementary one
The exact 3 by 1 ratio mentioned in the previous post, existing between Schiehallion, Arthurs Seat and Eildon North can be shown to be just over 60 yards from St Anthony's Chapel, O.S.grid reference (3275.755 6737.100).
I'm not bothering with full calculation sets here, but shall give O.S.grid references used, so can be verified by anyone with a calculator, and an interest!
St. Anthony's chapel is also 323606 feet, (61.289 miles(E)) from Schiehallion, a 2*Phi relationship.
2*Phi is 2 * 1.618034 = 3.236068, hence correspondence is 99.9997%, ignoring decimal points.
The calculated distance from St. Anthony's Chapel and Eildon North is 30.698 miles(E), or 162084 feet, a discrepancy of 93.5 yards from perfect of 161803.4 feet
As calculated in previous post, the exact 2:3 point at Arthurs Seat is 61.324 miles(E) from Schiehallion, and St.Anthony's Chapel is 61.289 miles(E), the difference being 61.6 yards.
The angle to O.S.grid is found to be 34.7292 degrees, and Duddingston Church (O.S. grid ref. used 3283.113 6726.172), an angle of 34.72 degrees, at a distance of 62.1075 miles(E).
These two angles are nearly identical, (a difference of 0.0092 deg, at a distance of 0.8185 miles(E), or 4321 feet, or 1440 yards (!?) is 0.7 feet, exact given the limit of practicality in my method. Anything within ten yards is too close to differentiate!)
So, Duddingston Church and St. Anthony's Chapel are in direct line with Schiehallion summit!
Also, St. Anthony's Chapel is also remarkably at the exact Phi harmonic distance in feet, from Schiehallion, and also marks the 2:3 point!!
And a final point of note concerning Duddingston Church and St. Anthony's Chapel:
the ratio of their distances from Schiehallion is: 62.1075 : 61.289, which is 1.013355 : 1! This ratio is greater than 99.97% of the Comma of Pythagoras as found in the work of Robert Temple, The Sirius Mystery, for example (page 24); the ratio of 3^12 : 2^19, or 531,441 : 524,288, or 1.0136433.
Duddingston Church and loch:
So, the area between both can be considered as the area of harmonic fluctuation between the ideal and the real in geometric terms, in accordance with the theories of the Pythagoreans. This area contains most of the Arthurs Seat area, including the summit. I consider this to be a perfect example of what I term a 'unit point area', which varies in size with distance, especially at special sites like Arthurs Seat and the Eildon.
Also, St. Anthony's Chapel is situated very close to the projection of the Tavhara line mentioned previously, to the north of Arthurs Seat summit.
Thus St. Anthony's Chapel can be considered a very special geometric marker.
Duddingston, too, is also a special point, including the area of the loch which it sits by. The exact line from Schiehallion to Eildon North passes by the western end.
added 20/03/08. Taking the tangent of the difference in angles between Duddingston and St.Anthony's to Schiehallion:
34.792 - 34.72 = 0.0092, the tangent of which is 0.00016057, times the distance from Schiehallion to St. Anthony's Chapel, 61.289 miles(E), equals 0.0098412 miles(E), which equals 51.9615eet, or 17.32049908 yards, which is an incredible 99.99995% harmonic of the square root of three!
I have difficulty coming to terms with a finding like this. It implies, an intent in the positioning of St.Anthony's and Duddingston, with an accuracy so great to beggar belief. Root three is of course an integral component of an equilateral triangle, namely the altitude in relation to the side.
The correspondences between St. Anthony's chapel and Duddingston and Schiehallion are astonishing, and leave much room for thought.
Supplementary two
As mentioned in main post, Newtongrange Church is more exactly in line with Arthurs Seat summit and Eildon Mid-Hill summit (O.S.grid reference used: 3548.19 6323.03)
Arthurs Seat to Eildon Mid-Hill, the distance is 30.418 miles(E), and the angle to O.S. grid is; 33.8826 degrees.
Newtongrange Church to Eildon Mid-Hill, the distance is 23.88252 miles(E), and the angle to O.S. grid; 33.9125 degrees.
The distance from Arthurs Seat and Newtongrange Church is some 6.5355 miles(E), and the difference in angle, 0.03 degrees; and using the tangent of this angle a discrepancy of 18 feet, near enough to be considered exact! (0.03degrees is 1/200th of one clock-face-minute!)
Finally, worthy of note, perhaps, is that there is found to be a 'near' pi, (99.97%) relationship between the two distances:
A.S. to E.M.-H.; 30.418 miles(E)
N.C. to E.M.-H.; 23.88252 miles(E)
23.88252/30.418 = 0.785144322, multiplied by four gives; 3.1406, (99.97% calculator pi, or 99.93% of 22/7, a commonly used version).
Supplementary three
Consideration of a significant geometric centre, already discussed in the Borthwick series of posts, namely Borthwick Church and Castle.
The references in the form used here are:
Borthwick Church 3368.785 6596.075
Borthwick Castle 3370.00 6597.597
I do not intend giving the full calculations here, just to point out that the exact line from Schiehallion to Eildon North, is at an angle to the O.S. grid of 34.594 degrees.
The angles to Borthwick Church, 34.5365 degrees, and to Borthwick Castle, 34.62 degrees.
From this it can be seen that the line passes between the two Borthwick points. It passes some 58 yards from the castle, and 126 yards from the church.
The distances are 71.7844 miles(E), or 64.024 miles(S) for Borthwick Church, and 71.765 miles(E), or 64.0067 miles(S). for Borthwick Castle.
The distance in Scottish miles, of 64 miles(S), is worthy of note, being 8 squared!
So, the line from Schiehallion to Eildon North passes through Borthwick.
Borthwick Castle, left, Church, right, looking south-east towards The Eildon, way beyond horizon:
This is looking north-west over Arthurs Seat, summit just visible, towards Schiehallion, faintly discernible on horizon, (click on image for larger version). Borthwick is on other side of foreground rise, not visible. This is taken from the point on the A7 south o Middleton, where the line crosses:
Finally, there are several points on this line, from a scan of Google Earth, but not calculated:
Drum estate, and the corner at The Wisp, near Danderhall, and Edmonstone House, just south of Arthurs Seat. North, Starbank Park on the coast, and Aberdour harbour on the Fife coast.
Extending south to the North Sea, the line passes through Lincolnshire and The Wash. A crop circle was found in Lincolnshire in Google Earth, very close the line. This is an area which a fellow researcher is looking at.
Main Finding from investigation, (so far)
The main finding rom this investigation, so far, is that Arthurs Seat/Holyrood Park sits at the 2/3rds distance between Schiehallion and The Eildon Hills. St.Anthony's Chapel in Holyrood Park can be assumed as the marker point. Not only that, but St. Anthony's Chapel is 'exactly' 323,606 feet from Schiehallion, and approximately 161,804 feet from Eildon North. Phi, (golden section) is 1.618034, so both distances are an exact harmonic of Phi!
This is a simple representation of the three by one rectangle, using the diagonal as the Schiehallion - Eildon line. I could have done it as the side of the rectangle, but chose the diagonal, as later I shall be discussing the 3 by 1 rectangle in connection with a grid, which has a 3 by 1 diagonal as a baseline for a very special geometric system. The diagonal of a 3 by 1 rectangle is in fact, square-root ten.(By Pythagoras, (3^2 plus 1 equals 10. The diagonal is thus square-root 10)
Thursday, 24 January 2008
PuC triangle - a very special triangle!
After far too long a gap I can at least start to cover the astonishing geometry found incorporating this Market Cross, at Preston, East Lothian.
This photo is of the Market Cross, with the unicorn on top. Bill Buehler is on the right, with his black cowboy hat. Preston Tower can be seen in the background! This was taken in March 2003, after the Sacred Geometry Conference at the Templar Lodge Hotel in Gullane. The weather was lovely spring sunshine for about six weeks. It was in fact the week the Iraq War was started!
In the geometry I will show, I have used the best grid references I can get, using printed off sections of Ordnance Survey 1:25,000, and measuring the fractions as best I could, and calculating to the metre, which in theory is the best achievable! I think that an allowance to ten metres would be reasonable.
Especially in the case of the triangle of which this point is the apex, which covers distances of over two hundred miles, and extends a further 50 miles or so, with astonishing accuracy, the hypotenuse including Schiehallion, summit tip, to a mound at Calanais/Callanish, and the adjacent includes the summit of Ben Cruachan, and extends to Flodagh a small island south of Barra. The short right-angle return side from Flodagh to Calanais runs through the Outer Hebridean land mass.
What is astonishing is that the hypotenuse and adjacent are in the precise ratio of Scots Mile to Imperial/English Mile: 1:1.1212121212, or 37:33, as described by John Reid, in his book of Gardening in 1683, the first book to deal specifically with Scottish conditions. I shall describe this later.
The two lines extend to the south and east, the hypotenuse to Kelso Abbey, and the adjacent to the highest point on Lindisfarne/Holy Island, namely Bible Law.
It is mainly because of this unicorn that I use the name, it signifies so much to me about this whole subject. Its now a kind of side hobby, collecting photos of unicorns, and I have some I will also show later.
This is just a wee intro to this topic. The geometry and calculations will be next.
Preston (Unicorn) Cross triangle sketch
It has taken ages for me to get this stuff together, from notes and re-calcs just to check. (A strange lethargy/inertia came over me over the 'festive' season, part laziness, and part something else I've learned to accept, and patience is needed till 'the time is right' kind of thing!)
These calculations were originally done in 2003/4, as a series over many months, using as detailed Ordnance Survey grid references I could muster.
Explanation of method:
Originally my search area was confined to the Lothians area of Scotland, centered on Midlothian, the area of my youth, which includes Roslin, Arthurs Seat and the Pentland and Moorfoot Hill Ranges, with the Esk River Valley system, and the City of Edinburgh.
This area was available on one map at a scale of 1:50,000, which is ok for Hillwalking and such. The best though is 1:25,000, and for my chosen area three needed to be joined on top of each other. To extend to East and West Lothian, to include Cairnpapple in the west and Traprain Law in the east two wing sections had to be attached. This took up a lot of wall space, and the Firth of Forth with its islands, and Fife would have been on the ceiling, so limits were defined by wall height. As the geometry developed the map became a mess, and distortions in jointing and stretching became inevitable. A number of maps were destroyed in this way, but it was a necessary process. I always knew verification would be necessary and it took many years to get to the stage were I felt confident enough and had access to the on-line Ordnance Survey, 'Get-A-Map', and after practice with the accurate references available for the island of Bornholm, as described by Erling Haagensen and Henry Lincoln in their book, 'The Templar's Secret Island'. That is another topic, which shall be covered eventually.
The geometry in Scotland had expanded to the extremities of Scotland, and further south, and a system had to be found to check it all out.
A concern was how much curvature would there be on this scale, and so what deviation would result. Rough calculations indicated that or Scotland as a whole, if a stretched table cloth was brought down on Scotland, to a height of some 2000feet, there would be sufficient mountain/hill tops over the whole area to allow flat geometry to be done.
What remained to be settled was the accuracy of the Ordnance Survey Grid over such a large area.
Speaking to a Cartographer friend, I was advised that the OS grid was accurate to a meter, but not to expect any better. A meter was/is good enough for me. Subsequently Dr. Nicolas Crane, of the Open University, best known from the series 'Coast', did a TV programme on this very topic. He checked the OS grid in Caithness, as far from the origin of the grid, (Reading, I believe) as practical on the mainland as possible. The discrepancy he found was minuscule, thus verifying the accuracy further.
Being in a phase were I had no computer I used the local library to print out sections of the OS map, 1:25,000 for all the selected points I wanted to check. I then carefully calculated as best I could, the fractions of kilometer squares of the said points, and using the absolute number prefix* in place of the 'usual' letter components obtained twelve figure references, to the meter, and did comparative calculations to verify angle and measures, converted to Miles(Imperial, or English, and also Scottish). The angle was of course vital, to determine orientation, and to confirm a straight line between three or more points. The distance measures were also examined for any whole number correlations with the postulated grid, and the common square root diagonals, namely, sq.rt. two, three and five, and the transcendent numbers, 'phi', 'pi' and natural log. base number 'e'. The latter being so close to the Megalithic Yard determined by Professor Alexander Thom, as to be remarkable that no-one I know, other than Bruce Cathie, makes the connection. These correlations prove to be so numerous, and involved that this present post shall be restricted to confirmation of the triangle with Preston Cross as apex, shown previously.
*Normal Grid references consist of a two letter prefix, and two sets of three numbers, for example: NT 348 765.
For work which crosses map boundaries, a universal grid number can be prefixed to the two number sets, giving eight figure references: NT 348 765, then becomes 3348 6765.
These two sets are called 'eastings' and 'northings', that is, the first set denotes the horizontal component, the second, the vertical component.
These sets are approximately to the hundred meter scale. The last number in each set is a fraction of the kilometer square which is denoted by the first two.
What I did was to measure as best I could and calculate to the meter, giving another two numbers in each section, making twelve figure references to work with: the above example becoming, say; 3348.63 6765.21! Ranging thus from hundreds of kilometers to meters. The decimal point retained as reminder to myself that to the right is my calculations!
The origin of the grid system is in the Atlantic south-west of the Lands End. It is a 'false' origin in that the Meridian 2degrees west is used for orientation, which happens to be the longitude of Berwick upon Tweed. Due to the curvature of the earth and the flat grid a discrepancy develops between the spheroid and the map, 2degrees west is roughly central of the area covered by the grid and the discrepancy kept symmetrical either side of 2dwest. (As explained by my Cartographer friend, from memory, and no fault of his if inaccurate!) Numbers thus increase from west to east, and south to north.
So, having two such attained numbers, calculations can be performed. A purely random pair can be used, for example:
The one already given, 3348.63 6765.21
and another, say,.........1234.56 9876.54
.......................................--------- ---------
subtracting both sets: 2114.07 -3111.33(the negative sign is important for orientation, this minus sign in the northing signifies that the second/bottom site is further north than the first/top, and should be borne in mind. It disappears from the square of the number, and can be ignored when calculating angle to grid, but is essential when drawing)
These numbers give us a horizontal and a vertical, at right angles to each other so the hypotenuse can be determined by using the Theorem of Pythagoras, (at least that's the name we use, although it is fairly certain it was known and used prior to him).
Theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides. X^2 + Y^2 = hypot^2
in our case: (2114.07)^2 + (-3111.33)^2 = 4469291.965 + 9680374.369 = 14149666.33
The square root of this number is 3761.604223.
This is in units of hundred meters. The conversion factor to change meters to feet is 3.28084; thus 3761.604223*100*3.28084 = 1234122.16 feet.
Convert to miles using the factor 5280: 233.7352576 miles(English)
Convert to miles(Scottish) using ratio 33:37, or 0.891891891repeat = 208.4665811 m(S)
These two measures are then analyzed for correlations with whole numbers and the important square roots, irrationals and transcendental numbers mentioned above, but not considered here.
What is now required is the angle this hypotenuse actually makes with the OS grid, for comparison purposes with other alignments, either for straightness or angle between. This achieved using the tangent ratio, from trigonometry, of opposite to adjacent, for sake of consistency the 'eastings' over 'northings', that is, first over second:
2114.07/3111.33 = 0.679474694, (using calculator tan function)=34.195116 degrees.
[I apologize for giving so many decimals after the point, just copying what the calculator gives. In terms of distance measures, in the Imperial/English system it should be 233 miles 1294 yards; and in Scottish measure 208 miles 896 ells. The yard(36 inches) and the ell(37 inches) being the equivalent to the meter, and thus the level of accuracy achievable. The same with degrees, 34d 11m 42.42s is the transposed number above into minutes and seconds. I could limit the decimals to 3 or 4 places but my calculator gives 10 figures so I have by habit just used that format, but aware of the 'nonsense' in practical terms, the final few figures actually are.]
So, with angle and measures comparisons can be made between any two points and any others.
PUC triangle calculations part 1
The first line to be considered is the line from Sandragh to Lindisfarne, through Ben Cruachan, Inchcolm and the apex of the triangle at Preston Cross.
The grid references to be used are, blue circles from left to right;
Sandragh/Carn Ghaltair--: 0640.79 7915.49
Ben Cruachan/summit----: 2069.62 7304.60
Inchcolm/Abbey------------: 3189.70 6826.69
Preston(unicorn)Cross---: 3391.27 6740.57
Lindisfarne/Bible Law-----: 4136.54 6422.64
The hypotenuse of the triangle, from Calanais/Callanish, Schiehallion, Preston Cross apex, and Kelso Abbey:
Grid references, again from left to right:
Calanais/stone circle and cairn--: 1221.50 9325.33
Schiehallion/summit---------------: 2713.83 7547.74
Preston Cross----------------------: 3391.27 6740.57
Kelso Abbey------------------------: 3727.85 6338.90
PUC triangle calculations part 2
First I shall do the Sandragh/Carn Ghaltair point and Preston Cross. Sandragh is not the end of this line, a point very near the island of Flodagh a mile or so further west is the corresponding point at right angles to the Callanish point. This is less important than establishing the two lines, adjacent and hypotenuse, and their angle and relationship to the English/Imperial mile system, and the hypotenuse the Scottish mile system, more on this later:
0640.79 7915.49 Sandragh/Carn Ghaltair
3391.27 6740.57 Preston Cross
------- -------
2750.48 1174.92
Using these two numbers, and the Theorem of Pythagoras, that is adding the squares of these numbers and finding the square root of the result:
(2750.48^2) + (1174.92^2) of which the square root is 2990.916. Now, this is in units of hundreds of metres, so 2990.916 *100 = 299091.6 metres.
Metric is no use to us, so converting to feet using 3.28084 feet/metre, we get
299091.6 * 3.28084 = 981271.685 feet.
There are 5280 feet per English/Imperial miles s;
981271.685 / 5280 = 185.8469 miles(E)-(my way of differentiating between English and Scottish miles.
To convert to Scottish miles the factor is 33:37, so;
185.8469 * (33/37) = 165.75534 miles(S)
So we have the distance between Sandragh and Preston Cross, and I found no neat correspondences with any of the common roots, so next and for now more important is the angle to the O.S. grid.
Using the two obtained numbers again, the tangent of the angle is 'opposite' over 'adjacent', in this case:
2750.48 / 1174.92 = 66.869 degrees
So, that is the method, in words and format, from now on I shall omit most of the explanations.
Next, Preston Cross and Bible Law, the highest point on Lindisfarne:
3391.27 6740.57 PUC
4136.54 6422.64 B.L./Lindisfarne
------- -------
0745.27 0317.93
By Pythag, times 100:
81025.11 metres = 50.34667 miles(E), 44.904 miles(S)
Angle to O.S. grid:
0745.27 / 0317.93 = 66.897 degrees
The difference between these two angles is thus; 66.897 - 66.869 = 0.028 degrees.
As a visual aid, a clock-face has sixty minutes marked, hence each clock-face minute is 6(six) degrees. 6 / 0.028 = 214. Try dividing a clock-face minute visually into 214 parts. This gives an indication of the accuracy here. Over the whole length, 185.85 + 50.35 = 236.2 miles, but using the midpoint as fulcrum the discrepancy either side would be: (236.2/2)* tangent 0.028 = 100 yards at both Lindisfarne and Sandragh. In other words a circle of 100yards radius at either end would contain this discrepancy, with ease.
To finish this first line, for now, the points Ben Cruachan(summit) and Inchcolm Abbey shall be compared to Preston Cross:
2069.62 7304.60 Cruachan summit
3391.27 6740.57 PUC
------- -------
1321.65 0564.03
which Pythagoras times 100 is 143697.20 metres or; 89.2893 miles(E), 79.6364 miles(S)
angle to grid:
tangent (1321.65 / 564.03) = 66.889 degrees.
3189.70 6826.69 Inchcolm Abbey
3391,27 6740.57 PUC
------- -------
0201.57 0086.12
By Pythag( times 100):
21919.653 metres, 13.62 miles(E), 12.148 miles(S).
angle to O.S. grid:
tangent(201.57/86.12) = 66.866 degrees.
Just to compare angles obtained:
066.869
066.897
066.889
066.866
------
267.521 / 4; mean = 66.88025 d.
The greatest discrepancy from mean is that between PUC and Inchcolm, 0.01425 degrees, representing some 18 feet at a distance of some 14 miles(E), or a 420th division of a clock-face minute.
Carn Ghaltair is the high point of the island of Sandragh south of Barra, Cruachan summit, and Bible Law, the highest point on Lindisfarne, and Inchcolm Abbey and the apex of the triangle, Preston Cross, with its unicorn on top, form a straight-line!
PUC triangle calculations part 3
There are four points on the hypotenuse line now to be considered, from top left to bottom right, Calanais/Callanish, stone circle and cairn, Schiehallion summit, Preston Cross, apex, and Kelso Abbey on the line extended through PC.
The line from Callanish through Schiehallion to Preston Cross proves to be straight to an astonishing degree, to 1/177th of one degree, or less than 1/1000th of one clock-face minute.
1221.50 9325.33 Calanais stone circle and cairn
3391.27 6740.57 Preston (Unicorn) Cross
------- -------
2169.77 2584.76
By Pythag.
337475metres, or 209.7miles(E), or 187.03miles(S)
angle to O.S. grid:
40.0117degrees.
2713.83 7547.74 Schiehallion summit
3391.27 6740.57 Preston Cross
------- -------
0677.44 0807.17
By Pythag.
105377.66metres, or 65.48miles(E), or 58.4miles(S)
angle to O.S. grid:
40.0061degrees
The two angles obtained; 40.0117 and 40.0061degrees are as mentioned above, in such close correlation as to be dead straight, 0.0056degrees, taking the reciprocal, 1/178th of one degree, or 1/1070th of one clock-face minute.
At Callanish, from Preston Cross, this is a discrepancy of 108 feet:
tan 0.0056 * 209.7 * 5280 = 108.22feet, at a distance of 209.7 miles(E). An average step or pace being of the order 2.5 feet, this is 43 paces from the centre of the stone circle used in this exercise. The Callanish site is large enough to accomodate this with ease. When I first did this I used a 45 meter spot height on the O.S. map, as being the highest point in the immediate vicinity which gave a distance o just over 188miles(S), and which gave the extended point of Flodagh on the previous line, the adjacent, forming a right angled triangle, and a distance of just over 188miles(E). More on this 188 measure later, the main interest here for now being the fact that the two sides are in 37:33 ratio, the ratio of miles(E) to miles(S).
The proof being that the difference in angle to the O.S. grid of Hypotenuse to Adjacent is 26.888degrees within limits of 1:2000, as given by Professor Lind as being that achievable by 'medieval' surveyors, and then he was talking of a very localized area, I do not think he intended to imply they could achieve the same accuracy at distances in excess of 200miles(E):
Preston Cross to the summit of Ben Cruachan: 66.866d
Preston Cross to the summit if Schiehallion: 40.001d
difference: 26.865d, cosine = 0.892073739; the reciprocal being 1.120983564.
37/33 = 1.121212121repeating:
a correspondence of 99.98%, or better than 1:4000.
I have spent a long time on this relationship and the use of the 37 inch ell, in Scotland, and the six ell fall, as opposed to the 36 inch yard and 5.5 yard pole in the English or Imperial system. Notably, the inch and foot are common, as is the number of falls and poles in the respective miles, namely 320.
It is the (37*6:36*5.5)ratio, or (222:198) which reduces to 37:33!
37 is an interesting prime number. All the triple numbers are multiples of 37; 111, 222, 333, 444, 555, 666, 777, 888, 999. (111 being 37*3)
A friend, Richard Heath, whose work is fascinating, see link to the right)has pointed out that there are 37 lunations in three years, or 36 months! So it may be related to the lunar cycles, this triangle giving the number of lunations on the hypotenuse, in the corresponding number of months on the adjacent. But, why at such a scale?
It was John Reid, in 1683 who insisted that in Scotland, land was measured in falls and ells, in the first book of gardening with focus on the climatic conditions of Scotland.(Chapter 8, see list of books at bottom of page, intro. by Annette Hope). I shall discuss Reid more, later.
The 37 system also shows up in my work on the geometry of Bornholm, whih is a big topic, to covered later, also!
All I intend to do here is establish this triangle, and highlight the positioning of Preston (unicorn) Cross, in relationship to two special mountains, Cruachan and Schiehallion, with the Iona of the east, Inchcolm Abbey, within the triangle, and with the Holy island of Lindisfarne on the adjacent extended, and with Callanish on the hypotenuse, with great accuracy and in this special relationship.
To finish, the hypotenuse extended south-east finds Kelso Abbey, to a fair degree of accuracy, slightly less, but well within the boundary of the Abbey.
3391.27 6740.57 Preston Cross
3727.85 6338.90 Kelso Abbey
------- -------
0336.58 0401.67
By Pythag:
52404.3metres, or 32.5625 miles(E), or 29.04 miles(S)
angle to O.S. grid:
39.9613 degrees.
The difference in angle is 0.0497, at a distance of some 32.5miles(E):
tan0.0497 * 32.5625 * 5280 = 149feet, or say, 50 yards.
The relative distances both show the 188 harmonic to within limits of 1:2000.
32.5625m(E) divided by sq.root 3 = 18.799968 (99.9998% of 18.8)
29.04226m(S)* phi(1.618034) = 46.99136412(99.9816% 47) * 4 = 187.9654565(same 99.98% of 188)(188 being 47 * 4).
There is a lot here still to ponder. I am content though for now, to have laid this down.
I still have to consider the relationships of most of these points to one another, and there are many possible correspondences already noted, but a fuller report later.
PUC triangle calculations part 4
I am not going to spend much time on the third side of the triangle, at present. Should the need arise I can always return to it.
I have in fact spent a lot of time trying to get a best fit, and looked at various measures around the 188 mile, English and Scots measure on the adjacent and hypotenuse respectively. There are just too many options, and most fit in some way or other. The line in fact runs up the spine of the Outer Hebrides, and various landmarks could be used from Mingulay through Vatersay, Barra(personal wish, Caisteal Chiosmuil, or Kishmul's Castle) in Castlebay, is in very good alignment with Roineabhal at the southern end of Harris, at 23.167 degrees to O.S. grid, which is within 0.1 degrees of required angle.
At Callanish/Calanais I used the stone circle with cairn, and mentioned the local high spot to the north west of the Callanish site. Using the gap between the two points gives a band some half-mile wide. At the Barra end this is from a calculated point just off the small island of Flodagh, includes Castlebay, and some of the other islands a bit further south, such as Mingulay/Muighlaigh.
The observant will have noticed other systems and lines drawn on the graphic. There is still a lot to cover and it is to some of these issues I shall turn to next.
Final comments for now!
It is enough for now to have shown this triangle and that Preston Cross is the very accurate apex, and that PC is also a pointer for Eagles Nest, through Borthwick Church, though I did not show the calculations for that system. I shall get back to that some time. The calculations though show the same level of accuracy!
A Google Map of the system:
View Larger Map
I have added temporarily another line. From Lindisfarne to Duart Castle, on Mull, including points such as Cramond Island, Tulliallan Castle, Stirling, Port Of Menteith, and Ben Vorlich.
I intend to try and cut down the number of Google Maps, as they are taking a while to load.
This photo is of the Market Cross, with the unicorn on top. Bill Buehler is on the right, with his black cowboy hat. Preston Tower can be seen in the background! This was taken in March 2003, after the Sacred Geometry Conference at the Templar Lodge Hotel in Gullane. The weather was lovely spring sunshine for about six weeks. It was in fact the week the Iraq War was started!
In the geometry I will show, I have used the best grid references I can get, using printed off sections of Ordnance Survey 1:25,000, and measuring the fractions as best I could, and calculating to the metre, which in theory is the best achievable! I think that an allowance to ten metres would be reasonable.
Especially in the case of the triangle of which this point is the apex, which covers distances of over two hundred miles, and extends a further 50 miles or so, with astonishing accuracy, the hypotenuse including Schiehallion, summit tip, to a mound at Calanais/Callanish, and the adjacent includes the summit of Ben Cruachan, and extends to Flodagh a small island south of Barra. The short right-angle return side from Flodagh to Calanais runs through the Outer Hebridean land mass.
What is astonishing is that the hypotenuse and adjacent are in the precise ratio of Scots Mile to Imperial/English Mile: 1:1.1212121212, or 37:33, as described by John Reid, in his book of Gardening in 1683, the first book to deal specifically with Scottish conditions. I shall describe this later.
The two lines extend to the south and east, the hypotenuse to Kelso Abbey, and the adjacent to the highest point on Lindisfarne/Holy Island, namely Bible Law.
It is mainly because of this unicorn that I use the name, it signifies so much to me about this whole subject. Its now a kind of side hobby, collecting photos of unicorns, and I have some I will also show later.
This is just a wee intro to this topic. The geometry and calculations will be next.
Preston (Unicorn) Cross triangle sketch
It has taken ages for me to get this stuff together, from notes and re-calcs just to check. (A strange lethargy/inertia came over me over the 'festive' season, part laziness, and part something else I've learned to accept, and patience is needed till 'the time is right' kind of thing!)
These calculations were originally done in 2003/4, as a series over many months, using as detailed Ordnance Survey grid references I could muster.
Explanation of method:
Originally my search area was confined to the Lothians area of Scotland, centered on Midlothian, the area of my youth, which includes Roslin, Arthurs Seat and the Pentland and Moorfoot Hill Ranges, with the Esk River Valley system, and the City of Edinburgh.
This area was available on one map at a scale of 1:50,000, which is ok for Hillwalking and such. The best though is 1:25,000, and for my chosen area three needed to be joined on top of each other. To extend to East and West Lothian, to include Cairnpapple in the west and Traprain Law in the east two wing sections had to be attached. This took up a lot of wall space, and the Firth of Forth with its islands, and Fife would have been on the ceiling, so limits were defined by wall height. As the geometry developed the map became a mess, and distortions in jointing and stretching became inevitable. A number of maps were destroyed in this way, but it was a necessary process. I always knew verification would be necessary and it took many years to get to the stage were I felt confident enough and had access to the on-line Ordnance Survey, 'Get-A-Map', and after practice with the accurate references available for the island of Bornholm, as described by Erling Haagensen and Henry Lincoln in their book, 'The Templar's Secret Island'. That is another topic, which shall be covered eventually.
The geometry in Scotland had expanded to the extremities of Scotland, and further south, and a system had to be found to check it all out.
A concern was how much curvature would there be on this scale, and so what deviation would result. Rough calculations indicated that or Scotland as a whole, if a stretched table cloth was brought down on Scotland, to a height of some 2000feet, there would be sufficient mountain/hill tops over the whole area to allow flat geometry to be done.
What remained to be settled was the accuracy of the Ordnance Survey Grid over such a large area.
Speaking to a Cartographer friend, I was advised that the OS grid was accurate to a meter, but not to expect any better. A meter was/is good enough for me. Subsequently Dr. Nicolas Crane, of the Open University, best known from the series 'Coast', did a TV programme on this very topic. He checked the OS grid in Caithness, as far from the origin of the grid, (Reading, I believe) as practical on the mainland as possible. The discrepancy he found was minuscule, thus verifying the accuracy further.
Being in a phase were I had no computer I used the local library to print out sections of the OS map, 1:25,000 for all the selected points I wanted to check. I then carefully calculated as best I could, the fractions of kilometer squares of the said points, and using the absolute number prefix* in place of the 'usual' letter components obtained twelve figure references, to the meter, and did comparative calculations to verify angle and measures, converted to Miles(Imperial, or English, and also Scottish). The angle was of course vital, to determine orientation, and to confirm a straight line between three or more points. The distance measures were also examined for any whole number correlations with the postulated grid, and the common square root diagonals, namely, sq.rt. two, three and five, and the transcendent numbers, 'phi', 'pi' and natural log. base number 'e'. The latter being so close to the Megalithic Yard determined by Professor Alexander Thom, as to be remarkable that no-one I know, other than Bruce Cathie, makes the connection. These correlations prove to be so numerous, and involved that this present post shall be restricted to confirmation of the triangle with Preston Cross as apex, shown previously.
*Normal Grid references consist of a two letter prefix, and two sets of three numbers, for example: NT 348 765.
For work which crosses map boundaries, a universal grid number can be prefixed to the two number sets, giving eight figure references: NT 348 765, then becomes 3348 6765.
These two sets are called 'eastings' and 'northings', that is, the first set denotes the horizontal component, the second, the vertical component.
These sets are approximately to the hundred meter scale. The last number in each set is a fraction of the kilometer square which is denoted by the first two.
What I did was to measure as best I could and calculate to the meter, giving another two numbers in each section, making twelve figure references to work with: the above example becoming, say; 3348.63 6765.21! Ranging thus from hundreds of kilometers to meters. The decimal point retained as reminder to myself that to the right is my calculations!
The origin of the grid system is in the Atlantic south-west of the Lands End. It is a 'false' origin in that the Meridian 2degrees west is used for orientation, which happens to be the longitude of Berwick upon Tweed. Due to the curvature of the earth and the flat grid a discrepancy develops between the spheroid and the map, 2degrees west is roughly central of the area covered by the grid and the discrepancy kept symmetrical either side of 2dwest. (As explained by my Cartographer friend, from memory, and no fault of his if inaccurate!) Numbers thus increase from west to east, and south to north.
So, having two such attained numbers, calculations can be performed. A purely random pair can be used, for example:
The one already given, 3348.63 6765.21
and another, say,.........1234.56 9876.54
.......................................--------- ---------
subtracting both sets: 2114.07 -3111.33(the negative sign is important for orientation, this minus sign in the northing signifies that the second/bottom site is further north than the first/top, and should be borne in mind. It disappears from the square of the number, and can be ignored when calculating angle to grid, but is essential when drawing)
These numbers give us a horizontal and a vertical, at right angles to each other so the hypotenuse can be determined by using the Theorem of Pythagoras, (at least that's the name we use, although it is fairly certain it was known and used prior to him).
Theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides. X^2 + Y^2 = hypot^2
in our case: (2114.07)^2 + (-3111.33)^2 = 4469291.965 + 9680374.369 = 14149666.33
The square root of this number is 3761.604223.
This is in units of hundred meters. The conversion factor to change meters to feet is 3.28084; thus 3761.604223*100*3.28084 = 1234122.16 feet.
Convert to miles using the factor 5280: 233.7352576 miles(English)
Convert to miles(Scottish) using ratio 33:37, or 0.891891891repeat = 208.4665811 m(S)
These two measures are then analyzed for correlations with whole numbers and the important square roots, irrationals and transcendental numbers mentioned above, but not considered here.
What is now required is the angle this hypotenuse actually makes with the OS grid, for comparison purposes with other alignments, either for straightness or angle between. This achieved using the tangent ratio, from trigonometry, of opposite to adjacent, for sake of consistency the 'eastings' over 'northings', that is, first over second:
2114.07/3111.33 = 0.679474694, (using calculator tan function)=34.195116 degrees.
[I apologize for giving so many decimals after the point, just copying what the calculator gives. In terms of distance measures, in the Imperial/English system it should be 233 miles 1294 yards; and in Scottish measure 208 miles 896 ells. The yard(36 inches) and the ell(37 inches) being the equivalent to the meter, and thus the level of accuracy achievable. The same with degrees, 34d 11m 42.42s is the transposed number above into minutes and seconds. I could limit the decimals to 3 or 4 places but my calculator gives 10 figures so I have by habit just used that format, but aware of the 'nonsense' in practical terms, the final few figures actually are.]
So, with angle and measures comparisons can be made between any two points and any others.
PUC triangle calculations part 1
The first line to be considered is the line from Sandragh to Lindisfarne, through Ben Cruachan, Inchcolm and the apex of the triangle at Preston Cross.
The grid references to be used are, blue circles from left to right;
Sandragh/Carn Ghaltair--: 0640.79 7915.49
Ben Cruachan/summit----: 2069.62 7304.60
Inchcolm/Abbey------------: 3189.70 6826.69
Preston(unicorn)Cross---: 3391.27 6740.57
Lindisfarne/Bible Law-----: 4136.54 6422.64
The hypotenuse of the triangle, from Calanais/Callanish, Schiehallion, Preston Cross apex, and Kelso Abbey:
Grid references, again from left to right:
Calanais/stone circle and cairn--: 1221.50 9325.33
Schiehallion/summit---------------: 2713.83 7547.74
Preston Cross----------------------: 3391.27 6740.57
Kelso Abbey------------------------: 3727.85 6338.90
PUC triangle calculations part 2
First I shall do the Sandragh/Carn Ghaltair point and Preston Cross. Sandragh is not the end of this line, a point very near the island of Flodagh a mile or so further west is the corresponding point at right angles to the Callanish point. This is less important than establishing the two lines, adjacent and hypotenuse, and their angle and relationship to the English/Imperial mile system, and the hypotenuse the Scottish mile system, more on this later:
0640.79 7915.49 Sandragh/Carn Ghaltair
3391.27 6740.57 Preston Cross
------- -------
2750.48 1174.92
Using these two numbers, and the Theorem of Pythagoras, that is adding the squares of these numbers and finding the square root of the result:
(2750.48^2) + (1174.92^2) of which the square root is 2990.916. Now, this is in units of hundreds of metres, so 2990.916 *100 = 299091.6 metres.
Metric is no use to us, so converting to feet using 3.28084 feet/metre, we get
299091.6 * 3.28084 = 981271.685 feet.
There are 5280 feet per English/Imperial miles s;
981271.685 / 5280 = 185.8469 miles(E)-(my way of differentiating between English and Scottish miles.
To convert to Scottish miles the factor is 33:37, so;
185.8469 * (33/37) = 165.75534 miles(S)
So we have the distance between Sandragh and Preston Cross, and I found no neat correspondences with any of the common roots, so next and for now more important is the angle to the O.S. grid.
Using the two obtained numbers again, the tangent of the angle is 'opposite' over 'adjacent', in this case:
2750.48 / 1174.92 = 66.869 degrees
So, that is the method, in words and format, from now on I shall omit most of the explanations.
Next, Preston Cross and Bible Law, the highest point on Lindisfarne:
3391.27 6740.57 PUC
4136.54 6422.64 B.L./Lindisfarne
------- -------
0745.27 0317.93
By Pythag, times 100:
81025.11 metres = 50.34667 miles(E), 44.904 miles(S)
Angle to O.S. grid:
0745.27 / 0317.93 = 66.897 degrees
The difference between these two angles is thus; 66.897 - 66.869 = 0.028 degrees.
As a visual aid, a clock-face has sixty minutes marked, hence each clock-face minute is 6(six) degrees. 6 / 0.028 = 214. Try dividing a clock-face minute visually into 214 parts. This gives an indication of the accuracy here. Over the whole length, 185.85 + 50.35 = 236.2 miles, but using the midpoint as fulcrum the discrepancy either side would be: (236.2/2)* tangent 0.028 = 100 yards at both Lindisfarne and Sandragh. In other words a circle of 100yards radius at either end would contain this discrepancy, with ease.
To finish this first line, for now, the points Ben Cruachan(summit) and Inchcolm Abbey shall be compared to Preston Cross:
2069.62 7304.60 Cruachan summit
3391.27 6740.57 PUC
------- -------
1321.65 0564.03
which Pythagoras times 100 is 143697.20 metres or; 89.2893 miles(E), 79.6364 miles(S)
angle to grid:
tangent (1321.65 / 564.03) = 66.889 degrees.
3189.70 6826.69 Inchcolm Abbey
3391,27 6740.57 PUC
------- -------
0201.57 0086.12
By Pythag( times 100):
21919.653 metres, 13.62 miles(E), 12.148 miles(S).
angle to O.S. grid:
tangent(201.57/86.12) = 66.866 degrees.
Just to compare angles obtained:
066.869
066.897
066.889
066.866
------
267.521 / 4; mean = 66.88025 d.
The greatest discrepancy from mean is that between PUC and Inchcolm, 0.01425 degrees, representing some 18 feet at a distance of some 14 miles(E), or a 420th division of a clock-face minute.
Carn Ghaltair is the high point of the island of Sandragh south of Barra, Cruachan summit, and Bible Law, the highest point on Lindisfarne, and Inchcolm Abbey and the apex of the triangle, Preston Cross, with its unicorn on top, form a straight-line!
PUC triangle calculations part 3
There are four points on the hypotenuse line now to be considered, from top left to bottom right, Calanais/Callanish, stone circle and cairn, Schiehallion summit, Preston Cross, apex, and Kelso Abbey on the line extended through PC.
The line from Callanish through Schiehallion to Preston Cross proves to be straight to an astonishing degree, to 1/177th of one degree, or less than 1/1000th of one clock-face minute.
1221.50 9325.33 Calanais stone circle and cairn
3391.27 6740.57 Preston (Unicorn) Cross
------- -------
2169.77 2584.76
By Pythag.
337475metres, or 209.7miles(E), or 187.03miles(S)
angle to O.S. grid:
40.0117degrees.
2713.83 7547.74 Schiehallion summit
3391.27 6740.57 Preston Cross
------- -------
0677.44 0807.17
By Pythag.
105377.66metres, or 65.48miles(E), or 58.4miles(S)
angle to O.S. grid:
40.0061degrees
The two angles obtained; 40.0117 and 40.0061degrees are as mentioned above, in such close correlation as to be dead straight, 0.0056degrees, taking the reciprocal, 1/178th of one degree, or 1/1070th of one clock-face minute.
At Callanish, from Preston Cross, this is a discrepancy of 108 feet:
tan 0.0056 * 209.7 * 5280 = 108.22feet, at a distance of 209.7 miles(E). An average step or pace being of the order 2.5 feet, this is 43 paces from the centre of the stone circle used in this exercise. The Callanish site is large enough to accomodate this with ease. When I first did this I used a 45 meter spot height on the O.S. map, as being the highest point in the immediate vicinity which gave a distance o just over 188miles(S), and which gave the extended point of Flodagh on the previous line, the adjacent, forming a right angled triangle, and a distance of just over 188miles(E). More on this 188 measure later, the main interest here for now being the fact that the two sides are in 37:33 ratio, the ratio of miles(E) to miles(S).
The proof being that the difference in angle to the O.S. grid of Hypotenuse to Adjacent is 26.888degrees within limits of 1:2000, as given by Professor Lind as being that achievable by 'medieval' surveyors, and then he was talking of a very localized area, I do not think he intended to imply they could achieve the same accuracy at distances in excess of 200miles(E):
Preston Cross to the summit of Ben Cruachan: 66.866d
Preston Cross to the summit if Schiehallion: 40.001d
difference: 26.865d, cosine = 0.892073739; the reciprocal being 1.120983564.
37/33 = 1.121212121repeating:
a correspondence of 99.98%, or better than 1:4000.
I have spent a long time on this relationship and the use of the 37 inch ell, in Scotland, and the six ell fall, as opposed to the 36 inch yard and 5.5 yard pole in the English or Imperial system. Notably, the inch and foot are common, as is the number of falls and poles in the respective miles, namely 320.
It is the (37*6:36*5.5)ratio, or (222:198) which reduces to 37:33!
37 is an interesting prime number. All the triple numbers are multiples of 37; 111, 222, 333, 444, 555, 666, 777, 888, 999. (111 being 37*3)
A friend, Richard Heath, whose work is fascinating, see link to the right)has pointed out that there are 37 lunations in three years, or 36 months! So it may be related to the lunar cycles, this triangle giving the number of lunations on the hypotenuse, in the corresponding number of months on the adjacent. But, why at such a scale?
It was John Reid, in 1683 who insisted that in Scotland, land was measured in falls and ells, in the first book of gardening with focus on the climatic conditions of Scotland.(Chapter 8, see list of books at bottom of page, intro. by Annette Hope). I shall discuss Reid more, later.
The 37 system also shows up in my work on the geometry of Bornholm, whih is a big topic, to covered later, also!
All I intend to do here is establish this triangle, and highlight the positioning of Preston (unicorn) Cross, in relationship to two special mountains, Cruachan and Schiehallion, with the Iona of the east, Inchcolm Abbey, within the triangle, and with the Holy island of Lindisfarne on the adjacent extended, and with Callanish on the hypotenuse, with great accuracy and in this special relationship.
To finish, the hypotenuse extended south-east finds Kelso Abbey, to a fair degree of accuracy, slightly less, but well within the boundary of the Abbey.
3391.27 6740.57 Preston Cross
3727.85 6338.90 Kelso Abbey
------- -------
0336.58 0401.67
By Pythag:
52404.3metres, or 32.5625 miles(E), or 29.04 miles(S)
angle to O.S. grid:
39.9613 degrees.
The difference in angle is 0.0497, at a distance of some 32.5miles(E):
tan0.0497 * 32.5625 * 5280 = 149feet, or say, 50 yards.
The relative distances both show the 188 harmonic to within limits of 1:2000.
32.5625m(E) divided by sq.root 3 = 18.799968 (99.9998% of 18.8)
29.04226m(S)* phi(1.618034) = 46.99136412(99.9816% 47) * 4 = 187.9654565(same 99.98% of 188)(188 being 47 * 4).
There is a lot here still to ponder. I am content though for now, to have laid this down.
I still have to consider the relationships of most of these points to one another, and there are many possible correspondences already noted, but a fuller report later.
PUC triangle calculations part 4
I am not going to spend much time on the third side of the triangle, at present. Should the need arise I can always return to it.
I have in fact spent a lot of time trying to get a best fit, and looked at various measures around the 188 mile, English and Scots measure on the adjacent and hypotenuse respectively. There are just too many options, and most fit in some way or other. The line in fact runs up the spine of the Outer Hebrides, and various landmarks could be used from Mingulay through Vatersay, Barra(personal wish, Caisteal Chiosmuil, or Kishmul's Castle) in Castlebay, is in very good alignment with Roineabhal at the southern end of Harris, at 23.167 degrees to O.S. grid, which is within 0.1 degrees of required angle.
At Callanish/Calanais I used the stone circle with cairn, and mentioned the local high spot to the north west of the Callanish site. Using the gap between the two points gives a band some half-mile wide. At the Barra end this is from a calculated point just off the small island of Flodagh, includes Castlebay, and some of the other islands a bit further south, such as Mingulay/Muighlaigh.
The observant will have noticed other systems and lines drawn on the graphic. There is still a lot to cover and it is to some of these issues I shall turn to next.
Final comments for now!
It is enough for now to have shown this triangle and that Preston Cross is the very accurate apex, and that PC is also a pointer for Eagles Nest, through Borthwick Church, though I did not show the calculations for that system. I shall get back to that some time. The calculations though show the same level of accuracy!
A Google Map of the system:
View Larger Map
I have added temporarily another line. From Lindisfarne to Duart Castle, on Mull, including points such as Cramond Island, Tulliallan Castle, Stirling, Port Of Menteith, and Ben Vorlich.
I intend to try and cut down the number of Google Maps, as they are taking a while to load.
Friday, 4 January 2008
Happy New Year, to all!
Belated, but it's still that season, so:
A guid New Year tae ane an aa!
Lots of ground still to cover this year, the calculations to show, and many more geometric systems to describe, and photos to add!
A guid New Year tae ane an aa!
Lots of ground still to cover this year, the calculations to show, and many more geometric systems to describe, and photos to add!
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