A special line in the landscape!

Hillend Fort on skyline, with the T-wood just below, and Dalmeny Kirk mid photo, and Hawes Pier area, taken from the midpoint of the Forth Road Bridge, May, 2009.

When I started to investigate the landscape of Lothian for geometry similar to what Henry Lincoln had described around Rennes Le Chateau some 15 years back, I came across a bare system in Andrew Sinclair's: Sword and the Grail, but which I no longer have to hand. One line I did find intriguing, as I had found the same line, but with somewhat different marker points!

Where-as Sinclair had noted the points: Hawes Pier, at South Queensferry, the T-Wood, or Cross Wood, above Swanston Village, Rosslyn Chapel, and Temple Kirk; I had: Dalmeny Kirk, T-Wood, Hillend Fort, Rosslyn Chapel and Temple Kirk.

Those who have followed this blog will be aware, Hillend Fort is one of the main points in the landscape, not least as the G.P apex point in the (3 by 1) diagonal Reshel system, and one of the best vantage points for the whole of the Lothian landscape, and can be seen from Roslin village, looking along Main Street.

I eventually noted that this line extended to the exact mid-point of the Forth Road Bridge, surely coincidence, but intriguing none-the-less!(?)

While living in Selkirk(1999-2001) I would quite often drive up to Edinburgh via Innerleithen, quite deliberately to the point where the road came out of the Moorfoot Hills to a superb vantage point on the other side of the Esk Valley from Hillend, before turning sharply right, (north-east), and try to see the points Temple and Rosslyn in the direct line that the map showed them to be. The exact point is about 200 yards down this road. Rosslyn Chapel is identifiable, now with the white coloured canopy, but Temple is not visible as it sits by the river in the steep sided valley.

I also extended this line on the map not so surprisingly came to the Eildon Hills, by way of Carcant Hill, the slope of which this vantage point is on.

Three years ago when I moved to Inverkeithing came the opportunity to check this line often from the midpoint of the Road Bridge. I managed to identify Dalmeny Kirk, and tried whilst a passenger in a friends car to take a decent photo. This proved awkward in that the main cable at the centre obscures the view, and being on the move was a bit troublesome also. I knew I would have to walk to the middle of the Bridge sometime, but it took me till May this year to get round to it!



This is a recent shot taken from South Queensferry at Hawes Pier to the mid-point of the Forth Road Bridge. The hills below allow a visual guide for the extended line to the north-west. A rough check in Google Earth leads across Loch Tay, just west of Ben Lawers, and directly in line is Maggernie Castle in Glen Lyon. This shall be checked later.



The Eildon Hills, from the north-west. The two used in this report are left and centre.

The Great and second Great Pyramid Geometry

This figure as found in Tompkins, pages 260-261, basically a four square with (1 by 1) and (2 by 1) diagonals, contains the necessary information to construct both of the great pyramids at Giza.

I've been a bit pre-occupied for the past few weeks, re-reading 'Secrets Of The Great Pyramid, by Peter Tompkins, and the appendix on Ancient Measure by Livio Stecchini. Slowly it dawned on me that the basic geometry the Great Pyramid is based upon is so simple, yet I have not seen it explained previously. I also came to realize that the Second 'Great' Pyramid is also derived as simply from a (2 by 1) rectangle, or double square.

Tompkins in a discussion about the above figure describes the idea of Tons Brunes who in The Secrets of Ancient Geometry
who 'shows that the Great Pyramid, like most of the great temples of antiquity, was designed on the basis of an advanced but hermetic geometry known only to initiates. only fragments o which percolated to the classic and Alexandrine Greeks.'

Tompkins states: Brunes shows how the ancient Egyptians used the basic design of a circle inscribed in a square to divide both circle and square geometrically into equal parts from 2 to 10, and all their possible multiples, without recourse to measuring or arithmetical calculations, with the aid of nothing but a straightedge and a compass - common emblems, along with the Pyramid, of the Masonic orders of yesterday and today.

He continues: In Brunes' reconstruction o the secret geometry, the cross emerges as the first geometric addition to the circle and square, and is the key not only to the solution of geometric problems but to the development of numerals and the alphabet.
By including the diagonals, every number both Latin and Arabic and all the letters of several alphabets may be obtained.
According to Brunes, both mathematics and the alphabet sprang from geometry, not the reverse. He says that nowadays we use numbers as the primary factor in our calculations, and geometry only as a subsidiary, whereas he believes the Egyptians reversed the order. He uses a detailed analysis of the Rhind Mathematical Papyrus to demonstrate that the ancient Egyptian system of counting was directly governed by geometric factors and that their ideas and theories were bound in geometric rules.
Brunes found that the circle was indeed considered sacred by the Egyptians, as were the square and the cross and the triangle, all of which are intimately incorporated into the Great Pyramid with its square base and triangular faces designed to represent the ''sacred'' circle.
Brunes demonstrates how the circle inscribed in a square and quartered by a cross enabled the ancient Egyptian geometer to inscribe in a circle the basic figures of pentagon, hexagon, octagon and decagon.

end quote.

Tompkins points out that the golden section is formed automatically between the sides and chords of the pentagon. This is so!

What is also true though is that very simply the golden section can be deirved directly from the (2 by 1) rectangle, as the diagonal is square root five, and if the unit side is added and the whole length halved, this is Phi in relation to the side. (1.618034 : 1)

It was musing on this and that the apothem of the Great Pyramid is Phi in relation to half of the base that the geometric construct of the Great Pyramid became apparent, as described below.


Right click on images, and 'Open in New Window' to see at full size:




figure 1: The basic (2 by 1) rectangle a circle is drawn centred on the common side with radius half the base. The points on the diagonals cut by this circle are 1.618034 units, with square side being 1 unit. Aa = Bb = Cc = Dd = 1.618034, Phi.

Diagonal = Square root five, 2.236068.

2.236068 / 2 = 1.118034

1.118034 + 0.5 = 1.618034.



figure 2.



figure 3.

Figure 2 is the construct of the Great Pyramid geometry. Figure 3 is labelled.

From points A and B with length Aa and Bb, or Phi, arcs are scribed to cut the axis, at P. The figure so formed is a cross section of the Great Pyramid, with base 2, apothem or slant height, Phi, and height square root Phi.

The angles PAQ and PBQ equals 51.8273 degrees.



figure 4.



figure 5.

Figures 4 and 5 show how the Second Pyramid geometry is found.

Stecchini, pages 378/379, gives Petrie's figures for the Second Pyramid at Giza, which he reckons best. He shows that the cross sectional triangle, half base, height and apothem are in the relationship, 3,4,5, which gives a base angle of 53.13 degrees.

In figure 4 the angle formed by the intersection of the two diagonals at O, is 53.13 degrees, so a construct of the Second Pyramid is possible from this fact alone. Angle BOM is 53.13. An arc is scribed from point B with length BO, to the intersection with the diagonal extended to M.

Figuire 5 shows a variation using the length Phi, Bb, to construct a triangle with the same apothem as the Great Pyramid, namely Phi.

Triangle BKL is a 3,4,5 triangle. KL is 3 units, BK is 4 units, BL is 5 units.

As the two pyramids are constructed I can't find any convincing correlation in dimensions, the second Pyramid being some 97% roughly smaller in base and height. Apparently, the Second Pyramid is on a higher level than the Great Pyramid so appears to be somewhat higher.

I find it fascinating that both can be found so simply from the same basic figure, namely figure 1 above.

Finally, the (2 by 1) rectangle diagonals form angles of 26.5650512 degrees, and 63.43495 degrees. The angle 26.5650512 degrees is very close to the angles given for the Ascending and descending passages of the Great Pyramid, adding further, perhaps, to the links shown here.

March 10, 2009, full moon

Tuesday night shift, I went a bit Lunar, it was a stunning full moon rise in the evening. I took photos all through the night and even got the sunrise in the morning.

I'll get round to adding some text some time soon! I even took some of the Grassmarket as it is now after renovation.

The last one is in Inverkeithing walking home and I have incorporated phi into the composition.

Web album

full moon, March10, 2009

Grid Reshel Basic summary

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.

Reshel basic google map

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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???

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.