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???

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