As the whole set of posts covering this whole topic is difficult to follow in sequence due to the nature of the blog format, and as it all involves many calculations, a brief summary post may help!
1. The Reshel system as defined by William S. Buehler, which came to my awareness in the late nineties. This was based on a 20 mile radius centred on St. Mary's Chapel Mount Lothian in Scotland, NT 275 570.
This radius I have since amended to 19.596 miles, being the side of a square with diagonal the same as that of the grid identified on the island of Bornholm which Haagensen and Lincoln show (and prove) to be a Secret Teaching Island of the Knights Templar. This grid in Imperial measure is (16 times the square root of 3) which equals 27.712813 miles (E).
This measure was found to be the exact distance between St. Mary's Chapel and St. Baldred's Chapel on The Bass Rock, NT 602 873.
This as diagonal of a square is in (square root two) relationship with the side, hence the 19.596 miles(E) radius mentioned above, which is the distance approximately to Seafield Tower, NT 279 885, to the north and Dryhope Tower, NT 267 247, to the south. (The double distance between the two Towers giving the diameter of the circle, which is 39.192 miles(E).)
Using the simple geometry of circles and squares, an inner square grid can be constructed naturally, with side half that of the original, 13.8564 miles (E).
2. Using this smaller grid with St. Mary's Chapel as centre, Blackness Castle defines the diagonal approximately, as example, NT 055 802.
It was then found from this grid that a point near Kelso, NT 716 340, and a point near Ericstane, NT 051 116, are both (2 by 1) diagonal points from St. Mary's Chapel, and are naturally a (3 by 1) diagonal distance apart from each other.
Using this as a baseline a line at 90 degrees from the mid-point through St. Mary's Chapel and extended north west is also naturally on a (3 by 1) diagonal, and it is on this line that a system is found corresponding nicely to the basic Reshel format as described by William Buehler.
N.b.: it needs to pointed out here that the orientation of the Bass Rock is some 2.4 degrees clockwise to that of the Rose-line/Tavhara line as commonly understood, and that there is a third in between using Berwick Law NT 556 842, which is used for the following excercise.
This spread of 2.4 degrees could be considered as a 'Selah' spoke, and as the geometry which gives the points below is accurate to the metre, an area of fudge could be considered at all points corresponding to this 2,4 degree spread but centred at St. Mary's Chapel.
3. A) The first point of note on this axis is St. Mary's Chapel itself, as it is naturally the point defining a square with the base-line as diagonal. From Pythagoras' theorem, a triangle with two sides equal to (square root five),which the diagonal of a (2 by 1) rectangle naturally is, with the third side (square root ten), that is the diagonal of a (3 by 1) rectangle, is a right angled triangle, with angles of 45 degrees. The other half of the square is to the south of the base-line, of course! This will be discussed later!
B) The second point on the axis, (and the clincher for me!) is Hillend Fort NT 245 662 which is at the point corresponding to the apex of the Great Pyramid, or Glory Pole in WSB's terminology for the system, 51.86416667 degrees, or 51 degrees, 51 minutes, 51 seconds. I have made many references to the significance of Hillend Fort in the landscape of Lothian, prior even to discovering this fact.
C) The point which defines an equilateral triangle with the base-line is in the Firth of Forth, with Inchmickery and Inchcolm NT 191 826, the nearest islands.
D) There is a pentagon super-imposed with side defined by the tangent from St. Mary's Chapel intersecting the circle with Hillend Fort as radius from the centre of the base-line, and dropping a perpendicular to the base-line, either side of the mid-point of the base-line. The two base line points are NT 116 150, Craigy Middens at Ask Law, and NT 646 318, near Rutherford Lodge, a 67 metre spotheight by a boathouse on the River Tweed at Dalcove.
The apex of this pentagon is to the north of Loch Leven in the village of Milnathort, on Pace Hill NO 123 051. Burleigh Castle is close by at NO 129 047, some half kilometer east.
The 'wings' of the pentagon have points at; i)in the west at NS 793 603, the junction of Biggar and Motherwell Roads, and; ii)in the east at sea NT 651 874, off St. Baldred's Boat in the vicinity of the Bass Rock and Tantallon Castle. St. Baldred's Chapel on the Bass Rock is of course one of the points where this grid originated.
The mid-point of this penta-chord is at Craigleith Avenue, NT 222 738, in Edinburgh near Mary Erskine's school at Ravelston.
The centre of the pentagon is found to be just south of Rosslynlee Hospital at NT 266 599, near a claypit.
Tuesday, 16 December 2008
Friday, 28 November 2008
Friday, 7 November 2008
St. Mary's Chapel grid 3 by 1 diagonal pentagon
In the Reshel basic system shown by William Buehler is a pentagon constructed within the base-line, and this drawing shows my geometric interpretation, based on Bill's drawings.A tangent drawn horizontally from St. Mary's Chapel intersects a circle of radius C - H, centre of base-line to Hillend Fort or G.P apex point. Then a perpendicular is dropped to the base-line defining point P1.
Using the theorem of Pythagoras C-P1 = sq.root[{CG^2) - (CS^2)], where CG is the same as the radius CH, and CS equals the half-base EC;
Using figures calculated in previous post this gives a distance of 17,283343miles(E), or 278.15 O.S. units.
This line extended the same distance towards Kelso through C, gives point P2.
These two points can be determined by calculating the x and y coordinates as before, and subtracting from C for P1 and adding to C for P2:
x: cos17.4742 * 278.15 = 265.32
y: sin17.4742 * 278.15 = 83.52
P1:
x: C-265.32 = 3380.88 - 265.32 = 3115.56
y: C- 83.52 = 6233.87 - 83.52 = 6150.35
P2:
x: C+265.32 = 3380.88 + 265.32 = 3646.20
y: C+ 83.52 = 6233.87 + 83.52 = 6317.39
P1 proves to be Craigy Middens on Ask Law (NT 115 150).
P2 proves to be near a 67 metre spot height at Rutherford Lodge on a bend in the river Tweed at Dalcove (NT 646 317).
Now that the base points are established, the altitude of the pentagon can be calculated.
This is most easily done by using the tangent of 72 degrees multiplied by the half base measure, and oriented to the 17.4742 degree angle. Reminder; The 3 by 1 angle adjusted for the North Berwick Law orientation!
So, MC = tan72 * 278.15 = 856.07 O.S. units
x: C - sin17.4742 * 856.07 = 3380.88 - 257.06 = 3123.82
y: C + cos17.4742 * 856.07 = 6233.87 + 816.56 = 7050.43
This point is Pace Hill(NO124 050), in Milnathort, on the north shore of Loch Leven. Of interest perhaps is Burleigh Castle(NO129 047) some 0.5 KM east.
The line from C to M crosses Loch Leven between Kinross House and Loch Leven Castle Island. Loch Leven also contains St Serf's Island. The line also crosses Benarty Hill to the south, near to the fort there. This latter high point is visible from Hillend Fort, south of the Firth of Forth, and of course the G.P. angle point and much more!
Next, the 'wings' of the pentagon can be calculated using the same methods as before, but the workings are omitted here. The two points are given here labelled W(west) and E(east):
W: 2792.71 6603.40, which is at a crossroad of the Biggar Road, Motherwell Road near Pickerston, (NS793 603).
E: 3651.31 6873.68, in the North Sea, near to St.Baldred's Boat(NT611 849), marked rocks near Seacliff, with St. Baldred's Cave at (NT604 845), with Tantallon Castle close by. St. Baldred's Chapel on The Bass Rock is of course the origin of the grid with St. Mary's Chapel.
The chord of the pentagon between W and E cuts the altitude line at Ravelston in Edinburgh near to Mary Erskine's School(NY218 740) at 3222.01 6738.54.
One other point of the pentagon to be considered is the centre O which is at 3265.92 6599.05, near clay pits just south of Rosslynlee Hospital(NT265 608).
Coincidentally, or otherwise, the distance from O to St. Mary's Chapel calculates to 1.88 miles(E). The distance of 188 miles(E) and (S) was found to be the measure of the sides of the triangle described in previous post from Preston Cross to Flodagh and Callanish respectively.
There are other points or poles that can be determined but for now the main geometrics have been established for one half of the basic Reshel system as defined by William Buehler, and other points on the altitude which may be of interest. And, this is only one half of the system, a mirror system on the other side of the base line should be considered, and that can be done later. For me this system based on the grid found on Bornholm and then in Scotland using the 3 by 1 diagonals and the simple way of determining especially the Great Pyramid point, Hillend Fort, is the clincher. As previously mentioned Hillend Fort is a most important geometric point in the landscape. Inchcolm island being close to the equilateral triangle point is also good. And, that the altitude passes through the gap between Kinross House and Loch Leven castle island, with Benarty Hill giving a good line of sight point is also very impressive. Bearing in mind that the orientation used is that to Berwick Law, with a spread of a degree or so either side, the gap between Kinross House and Loch Leven Castle could be considered as a 'gate' through which the three options pass is interesting. Kinross House is designed looking out to the Castle island with the path leading from the House to the shore points directly towards the island.
The Kelso area is also interesting, with the Abbey, Roxburgh Castle and Floors Castle forming a triangle containing the three options. The Ericstane/Devils Beef Tub area at the other end of the base line is also intriguing, not least because the Rivers Tweed and Annan have their sources there. Kelso also has the Tweed passing through it and where the Teviot joins the Tweed. The Tweed also runs along the Phi-latitude roughly, the title of this blog.
With this exercise I have reached a stage which has taken me so long, due to my personal need to prove to myself that the method I employ is mathematically OK, and just getting the photos and graphics together has also taken time.
I have a few things I need to do yet, and as mentioned above there is still more to do on this system, but for now I can draw relax a bit, go over what I have already done here and improve things a bit. There are also some things to consider as a result of this exercise which I also need to look at. There is one issue I left unfinished way back, that I can now return to also. Another grid, quite specific in the Esk Valley, which contains a feature I have known about for some nine years now, but have not shown here yet, as I needed to establish the full set of geometric systems I have found over the 15 years now I have been researching this subject before speculating further on the who, when and why questions, although a few people have been aware of some of this work. Some have grown impatient with my lack of progress, I guess, but I needed to do this stuff in a way that I felt was necessary.
For me this goes beyond 'mere' ley-lines as commonly perceived. The geometry, IMHO is too accurate, too specific, beyond that considered by some other researchers to be necessary. The work of Henry Lincoln at Rennes Le Chateau is questioned by some, but the work he and Erling Haagensen did on Bornholm shows that there would appear to have been some kind of 'school' system unbeknown to most, working to the same accuracy as I have adopted in Scotland. Then there is the work of William Buehler and the explanations he gives, which I have been aware of for nine years now also, which is still beyond my understanding. Also, as complex and accurate as these systems are, the use of line of sight points, high points in the landscape show how simply it could all be surveyed. Simple, perhaps, but the overall design so complex, covering all of Scotland and incorporating Scottish and English/Imperial measure, and even the metric to some extent, allowing for the slight inaccuracy of the metre as originally calculated. The fact that the foot is such an accurate measure relating to the equatorial circumference of the earth is still an issue to be resolved: (360 * 365.242 * 1000)/ 5280 = 24,902.864 miles(E) which is accurate to within a mile of most authoritative estimates. Who calculated this, when and how???
Using the theorem of Pythagoras C-P1 = sq.root[{CG^2) - (CS^2)], where CG is the same as the radius CH, and CS equals the half-base EC;
Using figures calculated in previous post this gives a distance of 17,283343miles(E), or 278.15 O.S. units.
This line extended the same distance towards Kelso through C, gives point P2.
These two points can be determined by calculating the x and y coordinates as before, and subtracting from C for P1 and adding to C for P2:
x: cos17.4742 * 278.15 = 265.32
y: sin17.4742 * 278.15 = 83.52
P1:
x: C-265.32 = 3380.88 - 265.32 = 3115.56
y: C- 83.52 = 6233.87 - 83.52 = 6150.35
P2:
x: C+265.32 = 3380.88 + 265.32 = 3646.20
y: C+ 83.52 = 6233.87 + 83.52 = 6317.39
P1 proves to be Craigy Middens on Ask Law (NT 115 150).
P2 proves to be near a 67 metre spot height at Rutherford Lodge on a bend in the river Tweed at Dalcove (NT 646 317).
Now that the base points are established, the altitude of the pentagon can be calculated.
This is most easily done by using the tangent of 72 degrees multiplied by the half base measure, and oriented to the 17.4742 degree angle. Reminder; The 3 by 1 angle adjusted for the North Berwick Law orientation!
So, MC = tan72 * 278.15 = 856.07 O.S. units
x: C - sin17.4742 * 856.07 = 3380.88 - 257.06 = 3123.82
y: C + cos17.4742 * 856.07 = 6233.87 + 816.56 = 7050.43
This point is Pace Hill(NO124 050), in Milnathort, on the north shore of Loch Leven. Of interest perhaps is Burleigh Castle(NO129 047) some 0.5 KM east.
The line from C to M crosses Loch Leven between Kinross House and Loch Leven Castle Island. Loch Leven also contains St Serf's Island. The line also crosses Benarty Hill to the south, near to the fort there. This latter high point is visible from Hillend Fort, south of the Firth of Forth, and of course the G.P. angle point and much more!
Next, the 'wings' of the pentagon can be calculated using the same methods as before, but the workings are omitted here. The two points are given here labelled W(west) and E(east):
W: 2792.71 6603.40, which is at a crossroad of the Biggar Road, Motherwell Road near Pickerston, (NS793 603).
E: 3651.31 6873.68, in the North Sea, near to St.Baldred's Boat(NT611 849), marked rocks near Seacliff, with St. Baldred's Cave at (NT604 845), with Tantallon Castle close by. St. Baldred's Chapel on The Bass Rock is of course the origin of the grid with St. Mary's Chapel.
The chord of the pentagon between W and E cuts the altitude line at Ravelston in Edinburgh near to Mary Erskine's School(NY218 740) at 3222.01 6738.54.
One other point of the pentagon to be considered is the centre O which is at 3265.92 6599.05, near clay pits just south of Rosslynlee Hospital(NT265 608).
Coincidentally, or otherwise, the distance from O to St. Mary's Chapel calculates to 1.88 miles(E). The distance of 188 miles(E) and (S) was found to be the measure of the sides of the triangle described in previous post from Preston Cross to Flodagh and Callanish respectively.
There are other points or poles that can be determined but for now the main geometrics have been established for one half of the basic Reshel system as defined by William Buehler, and other points on the altitude which may be of interest. And, this is only one half of the system, a mirror system on the other side of the base line should be considered, and that can be done later. For me this system based on the grid found on Bornholm and then in Scotland using the 3 by 1 diagonals and the simple way of determining especially the Great Pyramid point, Hillend Fort, is the clincher. As previously mentioned Hillend Fort is a most important geometric point in the landscape. Inchcolm island being close to the equilateral triangle point is also good. And, that the altitude passes through the gap between Kinross House and Loch Leven castle island, with Benarty Hill giving a good line of sight point is also very impressive. Bearing in mind that the orientation used is that to Berwick Law, with a spread of a degree or so either side, the gap between Kinross House and Loch Leven Castle could be considered as a 'gate' through which the three options pass is interesting. Kinross House is designed looking out to the Castle island with the path leading from the House to the shore points directly towards the island.
The Kelso area is also interesting, with the Abbey, Roxburgh Castle and Floors Castle forming a triangle containing the three options. The Ericstane/Devils Beef Tub area at the other end of the base line is also intriguing, not least because the Rivers Tweed and Annan have their sources there. Kelso also has the Tweed passing through it and where the Teviot joins the Tweed. The Tweed also runs along the Phi-latitude roughly, the title of this blog.
With this exercise I have reached a stage which has taken me so long, due to my personal need to prove to myself that the method I employ is mathematically OK, and just getting the photos and graphics together has also taken time.
I have a few things I need to do yet, and as mentioned above there is still more to do on this system, but for now I can draw relax a bit, go over what I have already done here and improve things a bit. There are also some things to consider as a result of this exercise which I also need to look at. There is one issue I left unfinished way back, that I can now return to also. Another grid, quite specific in the Esk Valley, which contains a feature I have known about for some nine years now, but have not shown here yet, as I needed to establish the full set of geometric systems I have found over the 15 years now I have been researching this subject before speculating further on the who, when and why questions, although a few people have been aware of some of this work. Some have grown impatient with my lack of progress, I guess, but I needed to do this stuff in a way that I felt was necessary.
For me this goes beyond 'mere' ley-lines as commonly perceived. The geometry, IMHO is too accurate, too specific, beyond that considered by some other researchers to be necessary. The work of Henry Lincoln at Rennes Le Chateau is questioned by some, but the work he and Erling Haagensen did on Bornholm shows that there would appear to have been some kind of 'school' system unbeknown to most, working to the same accuracy as I have adopted in Scotland. Then there is the work of William Buehler and the explanations he gives, which I have been aware of for nine years now also, which is still beyond my understanding. Also, as complex and accurate as these systems are, the use of line of sight points, high points in the landscape show how simply it could all be surveyed. Simple, perhaps, but the overall design so complex, covering all of Scotland and incorporating Scottish and English/Imperial measure, and even the metric to some extent, allowing for the slight inaccuracy of the metre as originally calculated. The fact that the foot is such an accurate measure relating to the equatorial circumference of the earth is still an issue to be resolved: (360 * 365.242 * 1000)/ 5280 = 24,902.864 miles(E) which is accurate to within a mile of most authoritative estimates. Who calculated this, when and how???
Friday, 31 October 2008
St. Mary's Chapel grid - 3 by 1 diagonal
In this post I shall describe the first two elements of the 'Reshel' basic grid, which is the work of William S. Buehler, previously mentioned, constructed by me on the 3 by 1 diagonal system shown in the previous post.
The first element is the Great Pyramid triangle from the two base points, Kelso and Ericstane:
This point proves to be Hillend Fort(HF), a very significant point in the landscape mentioned previously.
The simplest way to calculate this point is to use trig. function tangent of the Great Pyramid angle, 51d 51m 51s, or 51.86416667 degrees, multiplied by the distance Ericstane(E) to base centre(C), to obtain the point on the axis through St. Mary's Chapel(SM):
For this exercise I have used the St. Mary's Chapel(SM) - North Berwick Law(NBL) orientation which gives a 3 by 1 diagonal to O.S grid of 17.4742 degrees, for now, as it is between the other two options, the Bass Rock, and the Roseline through Arthur's Seat.
The distance from E - C is [{16*sq.rt.3}/2]*(sq.rt.10)/2, which is 21.909miles(E), or 352.59 O.S units, (hundred metres)
tan 51.86416667 * 352.59 = 449.10 O.S.units.
The x and y components can be calculated using the 17.4742 deg. offset angle.
x: sin 17.4742 * 449.10 = 134.85 O.S.grid units
y: cos 17.4742 * 449.10 = 428.38 O.S.grid units.
The calculated base mid - point(C) at this orientation was calculated as 3380.88 6233.87; approximately NT 381 234.
Due to the north west north slant the x component will be subtracted, the y component added:
x: 3380.88 - 134.85 = 3246.03
y: 6233.87 + 428.38 = 6662.25
I have Hillend Fort(NT 245 662) as 3245.50 6662.25 universal O.S. grid coordinates, a mere 50 metres west on the x-axis, and exact on the y-axis. The Roseline orientation would take this towards the west a tad. I will get back to this later, to check!
This second graphic shows two grid connections for this point to within fractions of a degree, which could be used for any practical purpose, for this awkward point.
The line from KA(Kelso approx.)through Hillend coincides with the 2 by 3 diagonal to Blackness Castle, and as can be seen the 2 by 1 diagonal connects at the same point.
I have just noticed the 2 by 1 connection, which only reinforces my opinion of the importance of Hillend Fort, and even more convinced of the practicality of using the 3 by 1 grid diagonal, and indeed this particular grid. Found on Bornholm, applied to Scotland, and defining a specific, and very special point in the construct provided by William Buehler. Wow!
This third drawing is simple in comparison. It is the point which forms an equilateral triangle with the two base points.
Again using the half base length, 352.59 O.S units and the tangent of 60 degrees, defines the point on the axis:
Tan 60 * 352.59 = 610.70 O.S. units
x and y components found using the offset angle of 17.4742 degrees:
x: sin 17.4742 * 610.70 = 183.38 O.S. units
y: cos 17.4742 * 610.70 = 582.52 O.S. units
and again using base centre:
x: 3380.88 - 183.38 = 3197.50
y: 6233.87 + 582.52 = 6816.39
Oxcars (NT 202 817) a tiny island in the Forth is the nearest land, about 0.5 km, but the line clips Inchcolm at the eastern end, a distance of approx 0.75 km.
The first element is the Great Pyramid triangle from the two base points, Kelso and Ericstane:
This point proves to be Hillend Fort(HF), a very significant point in the landscape mentioned previously.
The simplest way to calculate this point is to use trig. function tangent of the Great Pyramid angle, 51d 51m 51s, or 51.86416667 degrees, multiplied by the distance Ericstane(E) to base centre(C), to obtain the point on the axis through St. Mary's Chapel(SM):
For this exercise I have used the St. Mary's Chapel(SM) - North Berwick Law(NBL) orientation which gives a 3 by 1 diagonal to O.S grid of 17.4742 degrees, for now, as it is between the other two options, the Bass Rock, and the Roseline through Arthur's Seat.
The distance from E - C is [{16*sq.rt.3}/2]*(sq.rt.10)/2, which is 21.909miles(E), or 352.59 O.S units, (hundred metres)
tan 51.86416667 * 352.59 = 449.10 O.S.units.
The x and y components can be calculated using the 17.4742 deg. offset angle.
x: sin 17.4742 * 449.10 = 134.85 O.S.grid units
y: cos 17.4742 * 449.10 = 428.38 O.S.grid units.
The calculated base mid - point(C) at this orientation was calculated as 3380.88 6233.87; approximately NT 381 234.
Due to the north west north slant the x component will be subtracted, the y component added:
x: 3380.88 - 134.85 = 3246.03
y: 6233.87 + 428.38 = 6662.25
I have Hillend Fort(NT 245 662) as 3245.50 6662.25 universal O.S. grid coordinates, a mere 50 metres west on the x-axis, and exact on the y-axis. The Roseline orientation would take this towards the west a tad. I will get back to this later, to check!
This second graphic shows two grid connections for this point to within fractions of a degree, which could be used for any practical purpose, for this awkward point.
The line from KA(Kelso approx.)through Hillend coincides with the 2 by 3 diagonal to Blackness Castle, and as can be seen the 2 by 1 diagonal connects at the same point.
I have just noticed the 2 by 1 connection, which only reinforces my opinion of the importance of Hillend Fort, and even more convinced of the practicality of using the 3 by 1 grid diagonal, and indeed this particular grid. Found on Bornholm, applied to Scotland, and defining a specific, and very special point in the construct provided by William Buehler. Wow!
This third drawing is simple in comparison. It is the point which forms an equilateral triangle with the two base points.
Again using the half base length, 352.59 O.S units and the tangent of 60 degrees, defines the point on the axis:
Tan 60 * 352.59 = 610.70 O.S. units
x and y components found using the offset angle of 17.4742 degrees:
x: sin 17.4742 * 610.70 = 183.38 O.S. units
y: cos 17.4742 * 610.70 = 582.52 O.S. units
and again using base centre:
x: 3380.88 - 183.38 = 3197.50
y: 6233.87 + 582.52 = 6816.39
Oxcars (NT 202 817) a tiny island in the Forth is the nearest land, about 0.5 km, but the line clips Inchcolm at the eastern end, a distance of approx 0.75 km.
Friday, 17 October 2008
St.Mary's Chapel grid, continued!
The set of graphics below show first what has been established previously, the first two, and then the extension of the inner smaller derived grid, in (1/sq.root two) relationship, using Blackness Castle, and the simple use of grid diagonals, (2 by 1) and (3 by 1) to find a remarkable system linked to the work of William S. Buehler, whose original work, the ''20 mile radius'' system centred on St. Mary's Chapel. The square on this circle, with The Bass Rock as north-east corner, and the subsidiary system oriented on North Berwick Law, and indeed another with Arther's' Seat summit as main North axis at the half-radius point, (which incidentally has Rosslyn Chapel on it, discussed previously) with a spread of approximately 2.5 degrees is shown in the first sketch.
Sketch One
This shows the original circle and square grid derived from it, with a few points marked, St. Marys Chapel(NT275 570) in the centre, The Bass Rock(NT602 873) at the north-east corner, with North Berwick Law(NT556 842) indicated, Seafield(NT279 885) and Dryhope Towers(NT267 247) marking the north and south points respectively. The Arther's' Seat area is marked, and is at the half-radius point, but not indicated as such, but may be considered further, but later! The derived inner grid is found naturally in the geometry!
Sketch Two
This shows the smaller, inner grid, in (1/sq.root two) relationship, with Blackness Castle indicated at the north-west corner.
The length of side of this grid is where the Scot's measure system reveals the phi connection, discussed in previous post. Namely, (16* sq.root 3* 33) / (2* 37) equals 12.35842 miles(S), (half of which is 6.17921).
Incidentally, the discrepancy from phi exact is 6.5 feet, which is lost in the practical margins of error in my method. I work at all times with the limit in practice suggested as the optimum that medieval surveyors could achieve by known methods, by Professor Lind, in connection with the Bornholm work of Erling Haagensen and Henry Lincoln, in The Templar's Secret Island. This limit is 1 in 2000, or 99.95%, as discussed previously. 6.17921 mile(S) is 99.982% of 6.18034 miles(S).
The Scot's measure system I use is that defined by John Reid, 1683, in The Scot's Gard'ner, with the Scot's mile being in the ratio of 37 : 33 with the English/Imperial system. Inches and feet are common. Which raises many questions, not considered here!
Also, extreme accuracy that can be expected from O.S. maps is one metre, as explained to me by a professional cartographer friend, so any coordinates used here will be less than this. I would allow myself a discrepancy of up to ten metres. 6.5 feet is less than two metres (1.98 metres, to be precise)!
The issue of what accuracy the designers/surveyors worked to is not known, but the number of sites/points that are 'spot-on' indicates they got lucky very often.
I have been encouraged to allow for 'telluric'* off-set, and/or geographic/landscape practicalities in the past. I have allowed myself some leeway as at Blackness Castle(NT055 792), or Seafield Tower(NT280 885), where the exact geometric point is off the coast. I have also allowed the consideration of the 2.5 degree spread, and that these two points sit within this spread, as marker points. And, as at Arther's' Seat, the natural area of Holyrood Park, to be a 'unit-point-area', when considering such distances as the 200 mile plus spanning most of Scotland, and previously discussed.
* related to earth forces in some way! No further comment, for now!
These diagrams are simple representations of the geometry found so far.
Sketch Three
In late 1999 I moved to Selkirk to write up my findings till then. As previously mentioned I got connected to the Internet, made contact with William S. Buehler, and the geometry expanded, commensurate with my growing awareness of the landscape of the Tweed Valley and the countryside between Selkirk and Edinburgh. I was a member of the Sauniere Society at the time, and was privileged to hear Alistair Moffat discussing his book, Arthur And The Lost Kingdom, where my attention was drawn to Kelso, and the Roxburghe Castle area, east of Selkirk, where the Tweed and Teviot rivers meet.
On investigation I found that Kelso Abbey(NT727 339) was at the corner of the (2 by 1) grid point, south-east of St. Mary's Chapel:
A quick calculation here, to establish the accuracy of this. I am doing this from memory, and only using a list of O.S. coordinates, and re-doing the calculations as I go. And, I am aware that Kelso Abbey is not the exact point, as it is in the Schiehallion system described previously, but that a point closer to Roxburghe Castle(NT713 337) is the exact point. I shall first consider Kelso Abbey, to establish that we are in the right area:
3275.00 6570.19 St. Mary's Chapel
3728.87 6337.92 Kelso Abbey
--------- ----------
-453.87 232.27
Us usual using Pythagoras' Theorem, a grid unit distance of 509.85 hundreds of metres; which converts to 31.681 miles(E). Converting to Scots measure; using (33/37) gives 28.2557 miles(S).
What needs to be established is the approximation to the ( 2 by 1) diagonal which this distance represents, and the angle to O.S. grid:
We know that the side of the grid square is 12.38542 miles(S), (6.17921 * 2), and that the ( 2 by 1 ) diagonal is side times square root five, and the angle has a tangent of (1/2), or its complementary angle with a tangent of (2/1), which are 26.565 degrees, and 63.435 degrees respectively!
(6.17921 * 2) * square root five = 27.6343 miles(S). This is 0.6214 miles(S) short of Kelso Abbey!
The angle to O.S. grid has as tangent: (453.87/232.27) = 62.8987 degrees. or its complement to 90 degrees of 27.1013 degrees. This latter figure can be added to 90 to allow comparison with the north axis of orientation of the O.S. grid; 90 + 27.1013 = 117.1013. The exact angle for this vector should be (90 + 26.56505) 116.56505 degrees. The difference is 0.53625 degrees, well within the 2.5 degree spread discussed above!
Kelso Abbey could well be considered a marker point, for this grid!
I am content to leave this for now. Should anyone care to do some calculations for themselves I give a few more points around Roxburgh Castle, and also Floors Castle just to the north of Roxburgh Castle:
3716.82 6339.96 'hillock' east of Roxburgh Castle
3710.00 6335.22 'hillock' south of Roxburgh Castle
3713.04 6337.38 mean of two previous
3711.12 6346.62 Floors Castle
Sketch Four
In this part Ericstane Hill(NT059 122) shall be considered in relation to Kelso Abbey as marking the (3 by 1) diagonal south- west, and St. Mary's Chapel, as the (2 by 1) point south and west.
Ericstane Hill is north of Moffat between the A701, and the A74(M), with the 'Devil's Beef Tub' where stolen cattle were apparently penned in seclusion, in the valley to the north-east. It is the area where the Annan river rises, and close to the source of the Tweed. There is a Roman fortlet on its flank, with also the Eric Stane and monument.
3728.87 6337.92 Kelso Abbey
3059.81 6121.89 EricStane Hill(summit)
---------- ----------
669.06 216.03
By Pythagoras' theorem; 703.072 O.S.units(hundred metres); 43.68687 miles(E); 38.964 miles(S).
Now, the diagonal of a (3 by 1) rectangle is square root ten, so dividing we get 12.32132 miles(S), against the square unit side of 12.3584 miles(S), a difference of 0.03708 m(S), a discrepancy of 71.2 ells, or 73 yards, too short.
In real terms it is a discrepancy of 230 yards over the full distance, not great, not bad, good enough for immediate purposes.
The angle to O.S. grid is tangent (669.06/216.03), which gives an angle of 72.1055, and complement to 90 degrees of 17.8945.
The angle of the line from St. Mary's Chapel to Kelso Abbey, from above was found to be 27.1013 degrees.
Now, interestingly, the adjacent angles of a (2 by 1) and (3 by 1) is 45 degrees exactly; so 27.1013 plus 17.8945 equals 44.9958, 45 - 0.0042 degrees
As a check, Ericstane Hill can be compared to St. Mary's Chapel:
3275.00 6570.19 St. Mary's Chapel
3059.81 6121.89 EricStane Hill(summit)
---------- ----------
215.19 448.30
By Pythagoras' theorem 497.2722 O.S. grid units(hundred metres); or 30.899miles(E); or 27.5586 miles(S).
Dividing by square root five(the length of a (2 by 1) diagonal) gives 12.3246 miles(S), a discrepancy of 0.03381 miles(S) from grid square length, 12.3584 miles(S), some 65 ells, or 67 yards, short, similar in scale to the Kelso Abbey measure.
The angle to O.S grid is tangent (215.19/448.3) = 25.64163 degrees, which added to the St. Mary's Chapel - Kelso Abbey angle of 62.8987 degrees is 88.54 degrees, against the 90 degrees it should be. a shortfall of 1.46 degrees, well within the spread of 2.5 degrees, disussed above.
I could 'tweak' things a bit, and get the exact spots at both Kelso and Ericstane, and have done so previously, it's all in my notes somewhere, but for now I shall leave this, content in having established these two areas as containing the corners of the grid centred on St. Mary's Chapel discussed above.
Sketch Five
This sketch shows the next task, establishing the (3 by 1) axis through St. Mary's Chapel, from the mid-point of the Kelso - Ericstane line shown here:
Sketch One
This shows the original circle and square grid derived from it, with a few points marked, St. Marys Chapel(NT275 570) in the centre, The Bass Rock(NT602 873) at the north-east corner, with North Berwick Law(NT556 842) indicated, Seafield(NT279 885) and Dryhope Towers(NT267 247) marking the north and south points respectively. The Arther's' Seat area is marked, and is at the half-radius point, but not indicated as such, but may be considered further, but later! The derived inner grid is found naturally in the geometry!
Sketch Two
This shows the smaller, inner grid, in (1/sq.root two) relationship, with Blackness Castle indicated at the north-west corner.
The length of side of this grid is where the Scot's measure system reveals the phi connection, discussed in previous post. Namely, (16* sq.root 3* 33) / (2* 37) equals 12.35842 miles(S), (half of which is 6.17921).
Incidentally, the discrepancy from phi exact is 6.5 feet, which is lost in the practical margins of error in my method. I work at all times with the limit in practice suggested as the optimum that medieval surveyors could achieve by known methods, by Professor Lind, in connection with the Bornholm work of Erling Haagensen and Henry Lincoln, in The Templar's Secret Island. This limit is 1 in 2000, or 99.95%, as discussed previously. 6.17921 mile(S) is 99.982% of 6.18034 miles(S).
The Scot's measure system I use is that defined by John Reid, 1683, in The Scot's Gard'ner, with the Scot's mile being in the ratio of 37 : 33 with the English/Imperial system. Inches and feet are common. Which raises many questions, not considered here!
Also, extreme accuracy that can be expected from O.S. maps is one metre, as explained to me by a professional cartographer friend, so any coordinates used here will be less than this. I would allow myself a discrepancy of up to ten metres. 6.5 feet is less than two metres (1.98 metres, to be precise)!
The issue of what accuracy the designers/surveyors worked to is not known, but the number of sites/points that are 'spot-on' indicates they got lucky very often.
I have been encouraged to allow for 'telluric'* off-set, and/or geographic/landscape practicalities in the past. I have allowed myself some leeway as at Blackness Castle(NT055 792), or Seafield Tower(NT280 885), where the exact geometric point is off the coast. I have also allowed the consideration of the 2.5 degree spread, and that these two points sit within this spread, as marker points. And, as at Arther's' Seat, the natural area of Holyrood Park, to be a 'unit-point-area', when considering such distances as the 200 mile plus spanning most of Scotland, and previously discussed.
* related to earth forces in some way! No further comment, for now!
These diagrams are simple representations of the geometry found so far.
Sketch Three
In late 1999 I moved to Selkirk to write up my findings till then. As previously mentioned I got connected to the Internet, made contact with William S. Buehler, and the geometry expanded, commensurate with my growing awareness of the landscape of the Tweed Valley and the countryside between Selkirk and Edinburgh. I was a member of the Sauniere Society at the time, and was privileged to hear Alistair Moffat discussing his book, Arthur And The Lost Kingdom, where my attention was drawn to Kelso, and the Roxburghe Castle area, east of Selkirk, where the Tweed and Teviot rivers meet.
On investigation I found that Kelso Abbey(NT727 339) was at the corner of the (2 by 1) grid point, south-east of St. Mary's Chapel:
A quick calculation here, to establish the accuracy of this. I am doing this from memory, and only using a list of O.S. coordinates, and re-doing the calculations as I go. And, I am aware that Kelso Abbey is not the exact point, as it is in the Schiehallion system described previously, but that a point closer to Roxburghe Castle(NT713 337) is the exact point. I shall first consider Kelso Abbey, to establish that we are in the right area:
3275.00 6570.19 St. Mary's Chapel
3728.87 6337.92 Kelso Abbey
--------- ----------
-453.87 232.27
Us usual using Pythagoras' Theorem, a grid unit distance of 509.85 hundreds of metres; which converts to 31.681 miles(E). Converting to Scots measure; using (33/37) gives 28.2557 miles(S).
What needs to be established is the approximation to the ( 2 by 1) diagonal which this distance represents, and the angle to O.S. grid:
We know that the side of the grid square is 12.38542 miles(S), (6.17921 * 2), and that the ( 2 by 1 ) diagonal is side times square root five, and the angle has a tangent of (1/2), or its complementary angle with a tangent of (2/1), which are 26.565 degrees, and 63.435 degrees respectively!
(6.17921 * 2) * square root five = 27.6343 miles(S). This is 0.6214 miles(S) short of Kelso Abbey!
The angle to O.S. grid has as tangent: (453.87/232.27) = 62.8987 degrees. or its complement to 90 degrees of 27.1013 degrees. This latter figure can be added to 90 to allow comparison with the north axis of orientation of the O.S. grid; 90 + 27.1013 = 117.1013. The exact angle for this vector should be (90 + 26.56505) 116.56505 degrees. The difference is 0.53625 degrees, well within the 2.5 degree spread discussed above!
Kelso Abbey could well be considered a marker point, for this grid!
I am content to leave this for now. Should anyone care to do some calculations for themselves I give a few more points around Roxburgh Castle, and also Floors Castle just to the north of Roxburgh Castle:
3716.82 6339.96 'hillock' east of Roxburgh Castle
3710.00 6335.22 'hillock' south of Roxburgh Castle
3713.04 6337.38 mean of two previous
3711.12 6346.62 Floors Castle
Sketch Four
In this part Ericstane Hill(NT059 122) shall be considered in relation to Kelso Abbey as marking the (3 by 1) diagonal south- west, and St. Mary's Chapel, as the (2 by 1) point south and west.
Ericstane Hill is north of Moffat between the A701, and the A74(M), with the 'Devil's Beef Tub' where stolen cattle were apparently penned in seclusion, in the valley to the north-east. It is the area where the Annan river rises, and close to the source of the Tweed. There is a Roman fortlet on its flank, with also the Eric Stane and monument.
3728.87 6337.92 Kelso Abbey
3059.81 6121.89 EricStane Hill(summit)
---------- ----------
669.06 216.03
By Pythagoras' theorem; 703.072 O.S.units(hundred metres); 43.68687 miles(E); 38.964 miles(S).
Now, the diagonal of a (3 by 1) rectangle is square root ten, so dividing we get 12.32132 miles(S), against the square unit side of 12.3584 miles(S), a difference of 0.03708 m(S), a discrepancy of 71.2 ells, or 73 yards, too short.
In real terms it is a discrepancy of 230 yards over the full distance, not great, not bad, good enough for immediate purposes.
The angle to O.S. grid is tangent (669.06/216.03), which gives an angle of 72.1055, and complement to 90 degrees of 17.8945.
The angle of the line from St. Mary's Chapel to Kelso Abbey, from above was found to be 27.1013 degrees.
Now, interestingly, the adjacent angles of a (2 by 1) and (3 by 1) is 45 degrees exactly; so 27.1013 plus 17.8945 equals 44.9958, 45 - 0.0042 degrees
As a check, Ericstane Hill can be compared to St. Mary's Chapel:
3275.00 6570.19 St. Mary's Chapel
3059.81 6121.89 EricStane Hill(summit)
---------- ----------
215.19 448.30
By Pythagoras' theorem 497.2722 O.S. grid units(hundred metres); or 30.899miles(E); or 27.5586 miles(S).
Dividing by square root five(the length of a (2 by 1) diagonal) gives 12.3246 miles(S), a discrepancy of 0.03381 miles(S) from grid square length, 12.3584 miles(S), some 65 ells, or 67 yards, short, similar in scale to the Kelso Abbey measure.
The angle to O.S grid is tangent (215.19/448.3) = 25.64163 degrees, which added to the St. Mary's Chapel - Kelso Abbey angle of 62.8987 degrees is 88.54 degrees, against the 90 degrees it should be. a shortfall of 1.46 degrees, well within the spread of 2.5 degrees, disussed above.
I could 'tweak' things a bit, and get the exact spots at both Kelso and Ericstane, and have done so previously, it's all in my notes somewhere, but for now I shall leave this, content in having established these two areas as containing the corners of the grid centred on St. Mary's Chapel discussed above.
Sketch Five
This sketch shows the next task, establishing the (3 by 1) axis through St. Mary's Chapel, from the mid-point of the Kelso - Ericstane line shown here:
Friday, 10 October 2008
St. Mary's Chapel, Mount Lothian grid contd.
So, having established That there is a grid in Lothian, based on the the exact same dimensions as that found at Bornholm, with St. Mary's Chapel and St. Baldred's Chapel on The Bass Rock forming the diagonal, which links with the side of a square through Dunsappie hill-fort and extends to Seafield Tower between Kinghorn and Kirkcaldy, I shall now show the other diagonal, north-west of St. Mary's Chapel. This diagonal can be fixed by Blackness Castle, and a natural sub-division is found, namely the side of the square fixed by Seafield Tower, becomes the diagonal of a smaller nested square, which shall then be used in the next quite astonishing development. For this exercise though, I shall use the North Berwick Law orientation, confirming that in the landscape, both are relevant. See below!
3275.00 6570.19 St. Mary's Chapel
3055.66 6792.48 Blackness Castle
---------- ----------
219.34 -222.29
Using Pythagoras' Theorem: 312.29 O.S grid units(100 metre), which converts to:
19.4046 miles(E), which is some 337 yards short of the exact figure of 19.596 miles(E)! Now as Blackness juts out into the Firth of Forth, the exact point is in fact off-shore, just like at Seafield Tower.
Through calculation the exact grid reference is found to be :
3275.00 6570.19 St. Mary's Chapel
3055.71 6796.88 Blackness(calculated point)
---------- ----------
319.29 -226.69
again, using Pythagoras' theorem: 315.40 grid units of 100 metres, which converts to: 19.598 miles(E), and allowing for the small rounding off in the calculations is good to 0.002 miles! (I have restricted the figures to two decimal places for convenience here, and to 3 decimal places in the final miles calculation! A discrepancy of some 10 feet. I trust this is acceptable! I normally work to 10 figures on the calculator!)
And the angle to grid north is 44.05 degrees west of grid north, which corresponds to ninety degrees difference to the North Berwick line, 45.95 degrees east of grid north.
The grid squares having diagonal St.Mary's Chapel to Blackness are shown below, with sides equal to 19.598 miles(E)/square root 2 = 13.858 miles(E):
The next section uses this smaller square as the grid unit for the next stage, which extends this grid in all directions.
Now, a final point for now, as I have just realized:
This grid square side length of 13.858 miles(E) is equal to 12.36 miles(S)(33/37 is the onversion factor, see explanation in previous posts, and why I here, always distinguish between the two systems by the (E) and (S).
Now, half of 12.36 miles(S) is 6.18, a harmonic of phi, or little phi, or 1/Phi!!!
This is a new finding, although I may have it in my notes, but I don't recall having found this previously in relation to this system, and has to be of significance, to my mind! Quite astonishing, but then again, that's nothing new in this whole research!
I shall work on this, and see what else is to be found!
Good grief, the time on my computer at this exact moment is 6.19am, BST!!!
I had just done some calculations, one of the dogs barked, and I checked the clock!!!!
This is what I found:
The Bornholm grid axis is 16*square root three miles(E). This equates to 24.71683315miles(S). This divided by 40 gives 0.617920828. The reciprocal is 1.6183303, which squared is 2.61899296. Now that is an approximation of Phisquared, or Phi^2. This multiplied by 6/5; or 1.2 is 3.142791552, which multiplied by 7 = 21.99954086, which is 99.998% of 22. 22/7 is a rough, and often used form of Pi.
So, this grid, and hence the Bornholm grid, in Scottish measure is based on a common form of Pi, and Phi! I had found some correlation with Scottish measure when I was working on the Bornholm grid, but nothing so convincing!
So the full factors involved must all resolve in some way:
[(16*sq.rt3*33/37*40)^2]*5/6 = 7/22; so; (16^2,*3,*33^2,*5,*22)/(37^2,*40^2,*7) = 1;
which resolves to (2^2,*3^2,*11^3)/(5,*37^2) = 47916/47915; which equals 1.00002087, the reciprocal of which is 0.99997913, equivalent to 99.9979% of 1.
Interesting exercise! Or perhaps I should get a life!?
Then again, astonishing find!
Thursday, 2 October 2008
St.Mary's Chapel Bass Rock system further consideration
Having established the Bass Rock line, and considering this to be the diagonal of a square, the vertical will be 45 degrees anti-clockwise, a line which proves to run through the Arthurs' Seat area, and in fact a special rock on Dunsappie Fort, which I call pulpit Rock, whih intrigued me when first found back in the mid-nineties. The line extended north finishes just east of Seafield Tower, just offshore.
A lot of calculations were done using both the Bass Rock and North Berwick Law alignments, but the easiest way here is to just show the Dunsappie calculations first then the Seafield Tower point, just to keep things simple:
3275.00 6570.19 St. Mary's Chapel
3281.32 6731.72 Dunsappie 'notch' or Pulpit Rock
---------- ----------
-6.32 -161.53
Using Pythagoras again the distance is 161.6536, in One hundred metre units which converts to 10.0447 miles(E). This may be of interest but for now it is the angle this line makes to the O.S grid for comparison to the Bass Rock line:
6.32/161.53 = 0.039126, which is the tangent of 2.2406 degrees.
Comparing to the Bass Rock angle of 47.1803 - 2.2406 = 44.94 degrees, which is 0.06 degrees, or 1/100th of one clock-face-minute!
As the diagonal of a square is in square root two relationship with the side, this line extended north to a distance of 19.59592 miles(E), (16*sq.rt.3/sq.rt.2), the point indicated on the map just off-shore at Seafield Tower, which lies between Kinghorn and Kirkcaldy, is found.
Now that the St.Mary's -Bass Rock-Seafield Tower 45 degree right-angled triangle has been established, the full square can be projected, to complete the square on the circle of Bill Buehler's original circular system. And also, some of the sub-divisions also show points of interest.
click on image to see larger version!
This schematic was drawn prior to the calculations, and was what I needed to verify. It is good enough for now, as it shows the extended square and circle and some natural sub-divisions.
AS stand for Arthurs' Seat, and can be seen to be halfway between St. Mary's Chapel and Seafield Tower. It should be pointed out that the exact halfway point is some 440 yards south of Dunsappie, or one quarter mile, which is Duddingston loch, a beautiful spot famed in Scottish art for the painting by Raeburn of the Reverend Walker skating on Duddingston loch. And on the north-east shore lies Duddingston Kirk, the minister at the time was the Reverend John Thomson, who was also an amateur painter and had a studio down by the shore, still extant, an octagonal building which was also the home of the first curling club in the world.
Intriguingly, on the north wall of the kirk is a carved symbol, the same as is shown in 'The Templars' Secret Island', the book of the geometry of Bornholm, by Erling Haaagensen and Henry Lincoln, page 13, where there are examples of stones from Bodilsker, Nylars, Osterlars and Vestermarie, which 'echo the Cross of the Knights Templar'.
It should also be pointed out that there is also the Line from St. Mary's Chapel through Rosslyn Chapel and Arthurs' Seat summit which is the Roseline commonly or as Bill Buehler calls it the Tavhara Line, which passes just to the west of Seafield Tower, previously mentioned. This may constitute a more generous Selah Spoke, with the North Berwick Law line running between Arthurs' Seat and Dunsappie, and we can in passing check this:
3275.00 6570.19 St. Mary's Chapel
3275.28 6729.43 Arthurs' Seat summit
---------- ----------
-0.28 -159.24
and using Pythagoras' theorem: 159.24 O.S.units of 100 meters, which equates to 9.895 miles(E), and the angle to O.S. grid north being: 0.28/159.24 = 0.00176, which is the tangent of 0.1 degrees.
So the gap between this line and the Dunsappie line being 2.24 - 0.1 = 2.14 degrees, which may be considered as a 'Selah spoke'!
I consider the Arthurs' Seat area to be a 'unit point area' at large landscape scale.
So this gap being contained within the Arthurs' Seat area at a landscape scale this may be considered valid!
Also, and more pertinent to the next section is the Blackness Castle/Drem line. This can be seen to be between the diagonals at a distance the same as Seafield Tower, taken as radius. The squares on these sections of diagonals prove to be the unit squares of the next sub-system to be described! See next section!
A lot of calculations were done using both the Bass Rock and North Berwick Law alignments, but the easiest way here is to just show the Dunsappie calculations first then the Seafield Tower point, just to keep things simple:
3275.00 6570.19 St. Mary's Chapel
3281.32 6731.72 Dunsappie 'notch' or Pulpit Rock
---------- ----------
-6.32 -161.53
Using Pythagoras again the distance is 161.6536, in One hundred metre units which converts to 10.0447 miles(E). This may be of interest but for now it is the angle this line makes to the O.S grid for comparison to the Bass Rock line:
6.32/161.53 = 0.039126, which is the tangent of 2.2406 degrees.
Comparing to the Bass Rock angle of 47.1803 - 2.2406 = 44.94 degrees, which is 0.06 degrees, or 1/100th of one clock-face-minute!
As the diagonal of a square is in square root two relationship with the side, this line extended north to a distance of 19.59592 miles(E), (16*sq.rt.3/sq.rt.2), the point indicated on the map just off-shore at Seafield Tower, which lies between Kinghorn and Kirkcaldy, is found.
Now that the St.Mary's -Bass Rock-Seafield Tower 45 degree right-angled triangle has been established, the full square can be projected, to complete the square on the circle of Bill Buehler's original circular system. And also, some of the sub-divisions also show points of interest.
click on image to see larger version!
This schematic was drawn prior to the calculations, and was what I needed to verify. It is good enough for now, as it shows the extended square and circle and some natural sub-divisions.
AS stand for Arthurs' Seat, and can be seen to be halfway between St. Mary's Chapel and Seafield Tower. It should be pointed out that the exact halfway point is some 440 yards south of Dunsappie, or one quarter mile, which is Duddingston loch, a beautiful spot famed in Scottish art for the painting by Raeburn of the Reverend Walker skating on Duddingston loch. And on the north-east shore lies Duddingston Kirk, the minister at the time was the Reverend John Thomson, who was also an amateur painter and had a studio down by the shore, still extant, an octagonal building which was also the home of the first curling club in the world.
Intriguingly, on the north wall of the kirk is a carved symbol, the same as is shown in 'The Templars' Secret Island', the book of the geometry of Bornholm, by Erling Haaagensen and Henry Lincoln, page 13, where there are examples of stones from Bodilsker, Nylars, Osterlars and Vestermarie, which 'echo the Cross of the Knights Templar'.
It should also be pointed out that there is also the Line from St. Mary's Chapel through Rosslyn Chapel and Arthurs' Seat summit which is the Roseline commonly or as Bill Buehler calls it the Tavhara Line, which passes just to the west of Seafield Tower, previously mentioned. This may constitute a more generous Selah Spoke, with the North Berwick Law line running between Arthurs' Seat and Dunsappie, and we can in passing check this:
3275.00 6570.19 St. Mary's Chapel
3275.28 6729.43 Arthurs' Seat summit
---------- ----------
-0.28 -159.24
and using Pythagoras' theorem: 159.24 O.S.units of 100 meters, which equates to 9.895 miles(E), and the angle to O.S. grid north being: 0.28/159.24 = 0.00176, which is the tangent of 0.1 degrees.
So the gap between this line and the Dunsappie line being 2.24 - 0.1 = 2.14 degrees, which may be considered as a 'Selah spoke'!
I consider the Arthurs' Seat area to be a 'unit point area' at large landscape scale.
So this gap being contained within the Arthurs' Seat area at a landscape scale this may be considered valid!
Also, and more pertinent to the next section is the Blackness Castle/Drem line. This can be seen to be between the diagonals at a distance the same as Seafield Tower, taken as radius. The squares on these sections of diagonals prove to be the unit squares of the next sub-system to be described! See next section!
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