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.

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