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.

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:

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!

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!

Bornholm Grid Part Two - Lothian Scotland

In this section, The grid, based on 16*sq.root three miles(E), found on Bornholm is confirmed with the very first calculation applied to Scotland. In 1999, I first made contact with Bill Buehler, an Earth-Grid researcher of some 30 plus years, and promoter of what is known as Reshel grid dynamics, and much more, (of which more later!). I had known of his Rosslyn Chapel geometric analysis, from two A4 photocopies that were lying in the shop at the Chapel, a couple of years previously. I don't know why there were two piles of these on a shelf opposite the counter, and the assistant/volunteer gave the ok to take a copy of each, presumably there for use by a tour party. One was a plan of the Chapel, and the other an elevation, both with complex geometric constructs, and labeled with terms totally foreign to me, with explanations mostly beyond my comprehension, seeming to hint at a life spirit/force based on principles of geometric design, which the Chapel Design seemed to follow perfectly. It was the first hint that the geometry I had been finding, was somehow more precisely purposed than a mere cartographic exercise. I had ound of course precise pentagonal geometry, which did in fact incorporate Rosslyn Chapel, and related in measure to what Henry Lincoln had described in the south of France. More on this later!

When I had joined a discussion forum there was a topic in which this same mysterious terminology was referred to, so I posted my interest, and described something of what I had found. Biil Buehler answered, explaining that what I had obtained in the Chapel Tea-Room was his work. He congratulated me on my work, very pleased with the pentagonal system I had found, and even more so when I described a feature in the landscape, which I still have to describe, which for him was a sure sign of activation of a Reshel Grid system in the Edinburgh/Rosslyn area. I have been receiving posts from Bill for some Nine years now, and there have been many fascinating developments in that time.

One of the first systems he described in the landscape was based on the Roseline, or in Bill's terminology the Tavhara line, which passed through Arthurs' Seat Summit, Rosslyn Chapel, and St. Mary's Chapel to the south at Mount Lothian. This line I knew of, it's the one everyone finds. Bill described though a Reshel system, centred on St. Mary's Chapel, and a 20 mile radius circle, from Seafield Tower in the north, between Kinghorn and Kirkcaldy, in Fife, and Dryhope Tower in the Yarrow valley, near St. Mary's Loch to the south.



copyright William S. Buehler

It is the recti-linear grid derived from this graphic that I had been working on for a few years, finding The Bass Rock to be at the north east corner of the square constructed on the 40 mile diameter outer circle. This was one of the main systems that had developed when I had moved to Selkirk in the Scottish Borders, and the geometry kind of opened up or expanded with my investigations of the landscape, as I commuted back and forth to Edinburgh. I shall expand on this later.

After my Bornholm exercise, it was the diagonal from St. Mary's to The Bass Rock, I wanted to try first. I obtained the two relevant sections of 1:25,000 O.S. maps from the 'Get-A-Map' feature on their website, from the local library. And the obvious place to try on The Bass, was St. Baldred's Chapel. I worked out the grid references to the metre for both points, and did the calculations. To my total amazement, it was to within a few feet 16*Square root three miles (E), the exact same as the grid found on Bornholm.

Deleted google map, for now, see top of page!

The three main lines for now are the St.M's to The Bass Rock, St. M's to N. Berwick Law, and the St.M's to Dunsappie extended line.

Monday 22 Sept, 2998, equinox!
OK, after a few weeks break, I should get back to work!

Calculations

1. St. Mary's Chapel, Mount Lothian, (NT 275 570) to St. Baldred's Chapel, The Bass Rock. (NT 602 873).

3275.00 6570.19 St Mary's Chapel
3602.26 6873.26 St. Baldred's Chapel
---------- ----------
-327.26 -303.07

Using Pythagoras Theorem: 446.04 0.S.grid units of 100metres

which converts to 27.71556 miles (E)

divided by sq.root three = 16.0016, which is a 99.99% correlation with 16!

I am redoing the calculations as I go, with my hand-held calculator, as a check, and this still amazes me, how close it is to the Bornholm grid size. It is 8.4 feet and the references are calculated to the metre, 3.28084 feet, in theory only, in practice my calculations involve small areas of map, and pencil and ruler, so there is an added discrepancy inherent.

The angle of this line to the O.S map can be compared to the others and the angles to each other compared!

Using the two calculated figures above to give the tangent ratio the angle can be obtained.

(-327.06)/(-303.07) = 1.07915663, this gives the angle as 47.1802838.

The North Berwick Law line from St.Mary's Chapel, Mount Lothian

OK, time again to do some more. It is now Obtober first, another month of distraction and laziness.

Having established the Bass Rock line to St. Baldred's Chapel as being the exact same as the Bornholm grid, I need to show another possibility, on a slightly diferent orientation. This time using North Berwick Law:

3275.00 6570.19 St Mary's Chapel
3556.32 6842.23 N. Berwick Law
---------- ----------
-281.32 -272.04

again using Pythagoras' theorem: 391.34 O.S.grid units of 100 metres
which converts to 24.31672 miles(E)

which divided by square root three, gives 14.0393, which is a 99.72% correlation with 14. This discrepancy is some 69 yards, so not to the same 'exactness' as I normally allow, but North Berwick Law and The Bass Rock do seem to interact in the landscape from the area of Midlothian in the landscape.

The angle to O.S. grid again is found rom the tangent ratio of the two calculated components.

281.32/272.04 = 1.034113, which gives the angle 45.96 degrees.

I note this alignment due to the fact that Nylars Church on Bornholm marks the 13/16 point on the grid axis.

Compared to the Bass Rock line of 47.18 - 45.96 = 1.22 degrees, or 'roughly' one fifth of one clock-face-minute!(one c.f.m. is six degrees, there being 60 minutes in one hour, or 360 degrees!)

This could be considered as a 'Selah-spoke' in Bill Buehler's terminology, which he normally expects in a 'spinner' system. In discussion he would actually want it a bit greater, up to 3 degrees. I shall from here on consider only the Bass Rock alignment, but it should be kept in mind that this second alignment is there.

Bornholm Island, Baltic Sea, grid findings. Part One.

Introduction

This section summarizes the main results of many months of calculations on the data supplied by the Danish Government mapping office, Kort & Matrikelstyrelsen System 45 Bornholm, as provided by Erling Haagensen and Henry Lincoln in 'The Templars' Secret Island', page 177, published in the year 2000.

I restrict the findings to what is most relevant to the landscape geometry of the Lothians and Border regions of Scotland.

For this purpose I require only to use three site coordinates, those of Point Christianso*, where it is believed that a compass rose was carved in the bedrock of the small island, some 12 miles north-east of Bornholm, and two of the four round churches on the actual island of Bornholm, Osterlars* and Nylars. A fourth found/calculated point, at sea, is also used, and labeled by the authors Point C.

(* there should be a diagonal stroke through the 'O' of Osterlars, and the small case 'o' at the end of Christianso. Apologies if the omission offends, I don't have the correct characters to hand.)

The coordinates given here are the theoretical coordinates calculated by Haagensen and Lincoln, based on the altars to the east of both churches and the calculated position of the compass rose, which was blown up for building material at the end of the 17th century, and is mentioned in an extant letter by the officer in charge of the defensive construction at the time.

The actual Kort & Matrikelstyrelsen coordinates are for the tips of the conical roofs of the two circular churches. to the west of the altars, and for the Store Tarn on Christianso.

Haagensen's calculations were 'checked' by Distinguished Professor Emeritus Niels C. Lind at the University of Waterloo, Victoria, BC. His letter of reply is on page 144.

His first point of consideration was the accuracy of Haagensen's calculations:

(1) I calculated the coordinates of 12 churches and four auxiliary points according to the layout you specify in your Appendix, using a double precision computer spreadsheet. I have not discovered any errors in your calculations.**

(Neither did I, and took that as in some way verifying my methods, and workings. TG)

**My italics and bold. Lind merely has the final statement in italics!

This is in essence all that is necessary for the main point under discussion here. His fourth point I quote as it also gives a rule of thumb margin I use for accuracy of all the geometry I show in this blog. Namely 1:2000, or 99.95% accuracy:

(4)It is interesting to consider how medieval surveyors could have laid out a design such as ''the map'' in the field and positioned the churches. I have several years experience with similar field work, albeit using mid-20th century technology. I have no knowledge of what instruments and procedures they can have used, but they probably laid out open traverses in the terrain, sighting by eye and chaining distances with metal chains without correcting for temperature, sag and slope. I believe they could not achieve accuracies better than 1:2000 in the measured lengths over 10-20 km distances in fairly wooded and hilly terrain and 0.01 degree in directions. This would give RMS* errors of at least 7 m, roughly. Again, my italics - TG!

* RMS I take to mean Root Mean Square, of which I am not accustomed to using, and take his word on this final statement which he gives in italics which I also em-bold-en:

An RMS error of about 24.8 m, as found in (2) above** is not incompatible with the belief that the churches were located according to a plan such as ''the map''.

** not included here!

This letter is dated March 22, 1999.

Their book was published in 2000, and I was lucky enough to be at the launch at the Sauniere Symposium at Newbattle, in Midlothian, Scotland. I knew then I would have to study the material. It is now 2008 and am only getting to the stage of presenting it all! Time, continuous new findings, reading, computer resources/skills and so on.

I did my work on this in 2003/4, and wrote a report dated 10th November 2004, and distributed to a few friends/associates, along with an additional report covering the follow-up investigation into the landscape geometry of Scotland on 16th December, 2004.

Calculations

Keeping this section as simple as possible, the first and most important point to show is the span of the grid which follows by implication, namely the distance between the two furthest points on the line, from the island of Christianso through Osterlars and Nylars on Bornholm to the Point C, found by Haagensen and Lincoln:

the theoretical coordinatesin metres:

Y-component..... X-component
73,240.92 ............31,071.52 .........Point Christianso
39,444.21 ............60,223.48 .........Point C
------------ ............------------
33796.71 ............-29151.96

distance, by Pythagoras = 44632.44 metres

=146431.88 feet;
=27.7333 miles(E)

This distance is very close to 16*sq.root3, or27.7128 miles(E), a correspondence of 99.926%, or 0.0205 miles, or 108.24 feet, or 36 yards, over a distance of 27.7+miles(E).

This is all that is necessary for the next section when this 16*sq.rt.3 miles(E) is applied to a specific system in Scotland, namely St. Mary's Chapel in Midlothian to St.Baldred's Chapel on the Bass Rock in the Firth of Forth, near North Berwick.

I first came upon the 16*square root three miles(E)unit whilst doing the Christianso - Nylars distance:

Y-component..... X-component
73,240.92 ............31,071.52 .........Point Christianso
45,803.24.............54,738.38...........Nylars
-------------.............-------------
27,437.68.............-23,666.86

By Pythagoras' theorem: 36,234.604 metres, which converts to; 118,888 feet, or; 22.51514 miles(E), which divided by 'square root three' is 12.999122, which is a 99.99325% correspondence to 13.

And this also correlates to the system in Scotland, and complicates things somewhat as it indicates a second grid a mere degree or so off the main one, St. Baldred's Chapel/Bass Rock version mentioned above, but the second version using North Berwick Law at 14/16ths units of grid measure, as I shall cover in part two! Both systems centred on St. Mary's Chapel, Mount Lothian.

[I would like to include a bit on the Osterlars - Nylars measurement, which is on the same axis, and identified as the controlling radius of the system described by Haagensen and Lincoln.

y - coordinate..... x - coordinate

56,658.79............45,374.73..........Osterlars
45,803.24............54,738.38..........Nylars
-------------............-------------
10,855.55............-9,363.65

which is, by Pythagoras' theorem; 14,336 metres precisely(to within 4/100ths of a millimetre)!

For now I just wish to note that this radius gives a circle circumference of 56 miles(E) to 99.9875%, using pi - 22/7, and 99.945% using calculator pi, as indeed pointed out by (H & L).

There are some points of note which are interesting in themselves regarding this measure and the full grid measure, but not necessary for the immediate concern, applying the '16*square root three' miles(E) to the landscape of Scotland, centred on St. Mary's Chapel, Mount Lothian.]

A Google Map of Bornholm and main points. Osterlars and Nylars define the orientation, and Olsker Nyker extended defines Point C and meets axis at 30 degrees, hinting at hexagonal geometry.

The axis from Point C to Christianso is divided into 16 sections, each of 1.732 miles(E), (or the square root three):


View Larger Map

Tinto Hill - Preston Cross - Isle of May

Having had a break of some weeks from the geometry, I was checking a sketch I had done a few years back, and noticed a line I had not checked by calculation. It passes again through the unicorn Cross at Preston(NT 391 740), and links two points not previously mentioned, Tinto Hill(NS 952 343) and the Isle of May, or May Isle(NT 658 990, which is the Grid Reference for St. Adrian's Chapel).

[When I started my investigation, I was working with 1:25,000 scale maps, and the Isle of May was too far north of Lothian, as was Fife, and so no casual links could be made. The same was true for Tinto Hill, but to the south. It was when I was living in Selkirk that Tinto came to my awareness, as the part of a grid I shall be describing soon.]

The full grid references I shall use:

2952.75 6343.79 Tinto Hill

2965.23 6345.47 Scout Hill, a hill a mile to the east of Tinto, which is found to be a more exact point in line with the southern point on The May Isle.

3391.27 6740.57 Preston Cross, {unicorn}

3658.68 6990.19 St. Adrian's Chapel, May Isle.

3662.83 6988.49 South Ness, May Isle.



3658.68 6990.19 May Isle, St. Adrian's Chapel. The May Isle is roughly at 45 degrees to grid, from North-west to south-east and is a bit more than the diagonal of a grid kilometre square, or a mile approximately. The range is from North Ness {NT 651 999} to South Ness (NT 662 988}.
Marked features on the Island include, between St.Adrian's Chapel and South Ness, Pilgrim's Haven, Pilgrim's Well, Maiden Hair, The Pillow and Kettle Ness. To the north-west of the central Lighthouse are features marked as The Bishop, St. Andrews Well, Altarstanes and Standing Head. Near the Lighthouse, on the western shore, is Mill Door, a natural arch. The island is a designated Nature Reserve with sea-bird colonies on the impressive cliffs. It is less than a half-mile wide.

What I found was that the line of Tinto Hill through Preston Cross extended just misses the southern tip of The May Isle. To the east of Tinto is Scout Hill(NS 965 345), which is a mere 1/32nd of a clock-face-minute(6 degrees) off the Preston Cross - St.Adrian's Chapel line(0.188 degrees).

The bunch of angles range from 46.6 to 47.6 degrees to O.S grid north, which could be taken as being approximately 43 degrees north of east. This is possibly a midsummer sunrise line, if looking north-east from Tinto, or from Preston to the May Isle, across the Firth of Forth. At latitude 55/56 north, midsummer sunrise is approximately 45 degrees dependent on altitude. There is also a minor allowance required for the O.S. grid being 'normal' to true-north at 2 degrees west(Berwick upon Tweed). As this line is roughly in the area of 3 degrees west the adjustment would be minimal, and too complex for me to account for. It would seem likely therefore that midsummer sunrise from Preston Cross, or nearby Tower, would occur over the May Isle. The May Isle I am sure could be seen from the Tower. It would be interesting to get photo/video of the midsummer sunrise from the Tower. Perhaps next year, but access would need to be arranged.

Mid-winter sunrise would be seen in the opposite direction.

Calculations:

Only two included here, Scout Hill to Preston Cross, and Preston Cross to St. Adrian's Chapel.

1. Scout Hill/Preston Cross:

2965.23 6345.47 Scout Hill
3391.27 6740.57 Preston Cross
---------- ----------
-426.04 -395.10

By Pyhtagoras' theorem:

58105 metres
or 36.1045 miles(E), or 32.2 Miles(S), (20 times phi !?)

angle to grid north, 47.16 degrees, (426.04/395.1 = 1.0783; which is tan 47.16deg.)

2. Preston Cross/St. Adrians Chapel, May Isle

3391.27 6740.57 Preston Cross
3658.68 6990.19 St. Adrian's Chapel
---------- ----------
-267.41 -249.62

By Pythag. theorem:

365812 metres (120017 feet)

or 22.73 miles(E), or 20.273 miles(S)(remarkably 20 times the Comma of Pythagoras, 1.0136433, to within 99.999%, or 9.5 inches!!!)

[5/08/08 - just noticed that 22.73 miles(E) is close to 13*'square-root three', 22.5167 miles(E), after doing the following post on Bornholm! This is found to be the distance from Christianso to Nylars! I shall come back to this! The discrepancy is 375 yards, so something may show! TG.]


angle to grid north; 46.97 degrees.

angles difference: 47.16 - 46.97 = 0.19 deg., (1/32nd of one c.f.m.)

[Although not included in these calculations, the distance from Tinto Hill to St. Adrian's Chapel is 314034 feet, a 99.96% correlation with calculator pi, or 99.92% of 22/7.

A circle of this radius would have a circumference of 373,7 miles(E), or 333.3 miles(S).]

Tinto shall be mentioned in following posts, as well may The May be.

I shall do a google map for this line, but I will, no doubt, play about with a bit to see what turns up, using Google Maps and G-Earth.