PuC triangle - a very special triangle!

After far too long a gap I can at least start to cover the astonishing geometry found incorporating this Market Cross, at Preston, East Lothian.



This photo is of the Market Cross, with the unicorn on top. Bill Buehler is on the right, with his black cowboy hat. Preston Tower can be seen in the background! This was taken in March 2003, after the Sacred Geometry Conference at the Templar Lodge Hotel in Gullane. The weather was lovely spring sunshine for about six weeks. It was in fact the week the Iraq War was started!

In the geometry I will show, I have used the best grid references I can get, using printed off sections of Ordnance Survey 1:25,000, and measuring the fractions as best I could, and calculating to the metre, which in theory is the best achievable! I think that an allowance to ten metres would be reasonable.
Especially in the case of the triangle of which this point is the apex, which covers distances of over two hundred miles, and extends a further 50 miles or so, with astonishing accuracy, the hypotenuse including Schiehallion, summit tip, to a mound at Calanais/Callanish, and the adjacent includes the summit of Ben Cruachan, and extends to Flodagh a small island south of Barra. The short right-angle return side from Flodagh to Calanais runs through the Outer Hebridean land mass.

What is astonishing is that the hypotenuse and adjacent are in the precise ratio of Scots Mile to Imperial/English Mile: 1:1.1212121212, or 37:33, as described by John Reid, in his book of Gardening in 1683, the first book to deal specifically with Scottish conditions. I shall describe this later.

The two lines extend to the south and east, the hypotenuse to Kelso Abbey, and the adjacent to the highest point on Lindisfarne/Holy Island, namely Bible Law.

It is mainly because of this unicorn that I use the name, it signifies so much to me about this whole subject. Its now a kind of side hobby, collecting photos of unicorns, and I have some I will also show later.

This is just a wee intro to this topic. The geometry and calculations will be next.



Preston (Unicorn) Cross triangle sketch





It has taken ages for me to get this stuff together, from notes and re-calcs just to check. (A strange lethargy/inertia came over me over the 'festive' season, part laziness, and part something else I've learned to accept, and patience is needed till 'the time is right' kind of thing!)

These calculations were originally done in 2003/4, as a series over many months, using as detailed Ordnance Survey grid references I could muster.

Explanation of method:

Originally my search area was confined to the Lothians area of Scotland, centered on Midlothian, the area of my youth, which includes Roslin, Arthurs Seat and the Pentland and Moorfoot Hill Ranges, with the Esk River Valley system, and the City of Edinburgh.

This area was available on one map at a scale of 1:50,000, which is ok for Hillwalking and such. The best though is 1:25,000, and for my chosen area three needed to be joined on top of each other. To extend to East and West Lothian, to include Cairnpapple in the west and Traprain Law in the east two wing sections had to be attached. This took up a lot of wall space, and the Firth of Forth with its islands, and Fife would have been on the ceiling, so limits were defined by wall height. As the geometry developed the map became a mess, and distortions in jointing and stretching became inevitable. A number of maps were destroyed in this way, but it was a necessary process. I always knew verification would be necessary and it took many years to get to the stage were I felt confident enough and had access to the on-line Ordnance Survey, 'Get-A-Map', and after practice with the accurate references available for the island of Bornholm, as described by Erling Haagensen and Henry Lincoln in their book, 'The Templar's Secret Island'. That is another topic, which shall be covered eventually.

The geometry in Scotland had expanded to the extremities of Scotland, and further south, and a system had to be found to check it all out.

A concern was how much curvature would there be on this scale, and so what deviation would result. Rough calculations indicated that or Scotland as a whole, if a stretched table cloth was brought down on Scotland, to a height of some 2000feet, there would be sufficient mountain/hill tops over the whole area to allow flat geometry to be done.

What remained to be settled was the accuracy of the Ordnance Survey Grid over such a large area.

Speaking to a Cartographer friend, I was advised that the OS grid was accurate to a meter, but not to expect any better. A meter was/is good enough for me. Subsequently Dr. Nicolas Crane, of the Open University, best known from the series 'Coast', did a TV programme on this very topic. He checked the OS grid in Caithness, as far from the origin of the grid, (Reading, I believe) as practical on the mainland as possible. The discrepancy he found was minuscule, thus verifying the accuracy further.

Being in a phase were I had no computer I used the local library to print out sections of the OS map, 1:25,000 for all the selected points I wanted to check. I then carefully calculated as best I could, the fractions of kilometer squares of the said points, and using the absolute number prefix* in place of the 'usual' letter components obtained twelve figure references, to the meter, and did comparative calculations to verify angle and measures, converted to Miles(Imperial, or English, and also Scottish). The angle was of course vital, to determine orientation, and to confirm a straight line between three or more points. The distance measures were also examined for any whole number correlations with the postulated grid, and the common square root diagonals, namely, sq.rt. two, three and five, and the transcendent numbers, 'phi', 'pi' and natural log. base number 'e'. The latter being so close to the Megalithic Yard determined by Professor Alexander Thom, as to be remarkable that no-one I know, other than Bruce Cathie, makes the connection. These correlations prove to be so numerous, and involved that this present post shall be restricted to confirmation of the triangle with Preston Cross as apex, shown previously.

*Normal Grid references consist of a two letter prefix, and two sets of three numbers, for example: NT 348 765.

For work which crosses map boundaries, a universal grid number can be prefixed to the two number sets, giving eight figure references: NT 348 765, then becomes 3348 6765.

These two sets are called 'eastings' and 'northings', that is, the first set denotes the horizontal component, the second, the vertical component.

These sets are approximately to the hundred meter scale. The last number in each set is a fraction of the kilometer square which is denoted by the first two.

What I did was to measure as best I could and calculate to the meter, giving another two numbers in each section, making twelve figure references to work with: the above example becoming, say; 3348.63 6765.21! Ranging thus from hundreds of kilometers to meters. The decimal point retained as reminder to myself that to the right is my calculations!

The origin of the grid system is in the Atlantic south-west of the Lands End. It is a 'false' origin in that the Meridian 2degrees west is used for orientation, which happens to be the longitude of Berwick upon Tweed. Due to the curvature of the earth and the flat grid a discrepancy develops between the spheroid and the map, 2degrees west is roughly central of the area covered by the grid and the discrepancy kept symmetrical either side of 2dwest. (As explained by my Cartographer friend, from memory, and no fault of his if inaccurate!) Numbers thus increase from west to east, and south to north.

So, having two such attained numbers, calculations can be performed. A purely random pair can be used, for example:

The one already given, 3348.63 6765.21
and another, say,.........1234.56 9876.54
.......................................--------- ---------
subtracting both sets: 2114.07 -3111.33(the negative sign is important for orientation, this minus sign in the northing signifies that the second/bottom site is further north than the first/top, and should be borne in mind. It disappears from the square of the number, and can be ignored when calculating angle to grid, but is essential when drawing)

These numbers give us a horizontal and a vertical, at right angles to each other so the hypotenuse can be determined by using the Theorem of Pythagoras, (at least that's the name we use, although it is fairly certain it was known and used prior to him).

Theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides. X^2 + Y^2 = hypot^2

in our case: (2114.07)^2 + (-3111.33)^2 = 4469291.965 + 9680374.369 = 14149666.33

The square root of this number is 3761.604223.

This is in units of hundred meters. The conversion factor to change meters to feet is 3.28084; thus 3761.604223*100*3.28084 = 1234122.16 feet.

Convert to miles using the factor 5280: 233.7352576 miles(English)

Convert to miles(Scottish) using ratio 33:37, or 0.891891891repeat = 208.4665811 m(S)

These two measures are then analyzed for correlations with whole numbers and the important square roots, irrationals and transcendental numbers mentioned above, but not considered here.

What is now required is the angle this hypotenuse actually makes with the OS grid, for comparison purposes with other alignments, either for straightness or angle between. This achieved using the tangent ratio, from trigonometry, of opposite to adjacent, for sake of consistency the 'eastings' over 'northings', that is, first over second:

2114.07/3111.33 = 0.679474694, (using calculator tan function)=34.195116 degrees.

[I apologize for giving so many decimals after the point, just copying what the calculator gives. In terms of distance measures, in the Imperial/English system it should be 233 miles 1294 yards; and in Scottish measure 208 miles 896 ells. The yard(36 inches) and the ell(37 inches) being the equivalent to the meter, and thus the level of accuracy achievable. The same with degrees, 34d 11m 42.42s is the transposed number above into minutes and seconds. I could limit the decimals to 3 or 4 places but my calculator gives 10 figures so I have by habit just used that format, but aware of the 'nonsense' in practical terms, the final few figures actually are.]

So, with angle and measures comparisons can be made between any two points and any others.


PUC triangle calculations part 1


The first line to be considered is the line from Sandragh to Lindisfarne, through Ben Cruachan, Inchcolm and the apex of the triangle at Preston Cross.



The grid references to be used are, blue circles from left to right;
Sandragh/Carn Ghaltair--: 0640.79 7915.49
Ben Cruachan/summit----: 2069.62 7304.60
Inchcolm/Abbey------------: 3189.70 6826.69
Preston(unicorn)Cross---: 3391.27 6740.57
Lindisfarne/Bible Law-----: 4136.54 6422.64


The hypotenuse of the triangle, from Calanais/Callanish, Schiehallion, Preston Cross apex, and Kelso Abbey:



Grid references, again from left to right:

Calanais/stone circle and cairn--: 1221.50 9325.33
Schiehallion/summit---------------: 2713.83 7547.74
Preston Cross----------------------: 3391.27 6740.57
Kelso Abbey------------------------: 3727.85 6338.90

PUC triangle calculations part 2





First I shall do the Sandragh/Carn Ghaltair point and Preston Cross. Sandragh is not the end of this line, a point very near the island of Flodagh a mile or so further west is the corresponding point at right angles to the Callanish point. This is less important than establishing the two lines, adjacent and hypotenuse, and their angle and relationship to the English/Imperial mile system, and the hypotenuse the Scottish mile system, more on this later:

0640.79 7915.49 Sandragh/Carn Ghaltair
3391.27 6740.57 Preston Cross
------- -------
2750.48 1174.92

Using these two numbers, and the Theorem of Pythagoras, that is adding the squares of these numbers and finding the square root of the result:

(2750.48^2) + (1174.92^2) of which the square root is 2990.916. Now, this is in units of hundreds of metres, so 2990.916 *100 = 299091.6 metres.

Metric is no use to us, so converting to feet using 3.28084 feet/metre, we get

299091.6 * 3.28084 = 981271.685 feet.

There are 5280 feet per English/Imperial miles s;

981271.685 / 5280 = 185.8469 miles(E)-(my way of differentiating between English and Scottish miles.

To convert to Scottish miles the factor is 33:37, so;

185.8469 * (33/37) = 165.75534 miles(S)

So we have the distance between Sandragh and Preston Cross, and I found no neat correspondences with any of the common roots, so next and for now more important is the angle to the O.S. grid.

Using the two obtained numbers again, the tangent of the angle is 'opposite' over 'adjacent', in this case:

2750.48 / 1174.92 = 66.869 degrees

So, that is the method, in words and format, from now on I shall omit most of the explanations.

Next, Preston Cross and Bible Law, the highest point on Lindisfarne:

3391.27 6740.57 PUC
4136.54 6422.64 B.L./Lindisfarne
------- -------
0745.27 0317.93

By Pythag, times 100:

81025.11 metres = 50.34667 miles(E), 44.904 miles(S)

Angle to O.S. grid:

0745.27 / 0317.93 = 66.897 degrees

The difference between these two angles is thus; 66.897 - 66.869 = 0.028 degrees.

As a visual aid, a clock-face has sixty minutes marked, hence each clock-face minute is 6(six) degrees. 6 / 0.028 = 214. Try dividing a clock-face minute visually into 214 parts. This gives an indication of the accuracy here. Over the whole length, 185.85 + 50.35 = 236.2 miles, but using the midpoint as fulcrum the discrepancy either side would be: (236.2/2)* tangent 0.028 = 100 yards at both Lindisfarne and Sandragh. In other words a circle of 100yards radius at either end would contain this discrepancy, with ease.

To finish this first line, for now, the points Ben Cruachan(summit) and Inchcolm Abbey shall be compared to Preston Cross:

2069.62 7304.60 Cruachan summit
3391.27 6740.57 PUC
------- -------
1321.65 0564.03
which Pythagoras times 100 is 143697.20 metres or; 89.2893 miles(E), 79.6364 miles(S)

angle to grid:

tangent (1321.65 / 564.03) = 66.889 degrees.

3189.70 6826.69 Inchcolm Abbey
3391,27 6740.57 PUC
------- -------
0201.57 0086.12

By Pythag( times 100):

21919.653 metres, 13.62 miles(E), 12.148 miles(S).

angle to O.S. grid:

tangent(201.57/86.12) = 66.866 degrees.


Just to compare angles obtained:

066.869
066.897
066.889
066.866
------
267.521 / 4; mean = 66.88025 d.

The greatest discrepancy from mean is that between PUC and Inchcolm, 0.01425 degrees, representing some 18 feet at a distance of some 14 miles(E), or a 420th division of a clock-face minute.

Carn Ghaltair is the high point of the island of Sandragh south of Barra, Cruachan summit, and Bible Law, the highest point on Lindisfarne, and Inchcolm Abbey and the apex of the triangle, Preston Cross, with its unicorn on top, form a straight-line!

PUC triangle calculations part 3



There are four points on the hypotenuse line now to be considered, from top left to bottom right, Calanais/Callanish, stone circle and cairn, Schiehallion summit, Preston Cross, apex, and Kelso Abbey on the line extended through PC.

The line from Callanish through Schiehallion to Preston Cross proves to be straight to an astonishing degree, to 1/177th of one degree, or less than 1/1000th of one clock-face minute.

1221.50 9325.33 Calanais stone circle and cairn
3391.27 6740.57 Preston (Unicorn) Cross
------- -------
2169.77 2584.76

By Pythag.

337475metres, or 209.7miles(E), or 187.03miles(S)

angle to O.S. grid:

40.0117degrees.


2713.83 7547.74 Schiehallion summit
3391.27 6740.57 Preston Cross
------- -------
0677.44 0807.17

By Pythag.

105377.66metres, or 65.48miles(E), or 58.4miles(S)

angle to O.S. grid:

40.0061degrees

The two angles obtained; 40.0117 and 40.0061degrees are as mentioned above, in such close correlation as to be dead straight, 0.0056degrees, taking the reciprocal, 1/178th of one degree, or 1/1070th of one clock-face minute.

At Callanish, from Preston Cross, this is a discrepancy of 108 feet:

tan 0.0056 * 209.7 * 5280 = 108.22feet, at a distance of 209.7 miles(E). An average step or pace being of the order 2.5 feet, this is 43 paces from the centre of the stone circle used in this exercise. The Callanish site is large enough to accomodate this with ease. When I first did this I used a 45 meter spot height on the O.S. map, as being the highest point in the immediate vicinity which gave a distance o just over 188miles(S), and which gave the extended point of Flodagh on the previous line, the adjacent, forming a right angled triangle, and a distance of just over 188miles(E). More on this 188 measure later, the main interest here for now being the fact that the two sides are in 37:33 ratio, the ratio of miles(E) to miles(S).

The proof being that the difference in angle to the O.S. grid of Hypotenuse to Adjacent is 26.888degrees within limits of 1:2000, as given by Professor Lind as being that achievable by 'medieval' surveyors, and then he was talking of a very localized area, I do not think he intended to imply they could achieve the same accuracy at distances in excess of 200miles(E):

Preston Cross to the summit of Ben Cruachan: 66.866d
Preston Cross to the summit if Schiehallion: 40.001d

difference: 26.865d, cosine = 0.892073739; the reciprocal being 1.120983564.

37/33 = 1.121212121repeating:

a correspondence of 99.98%, or better than 1:4000.

I have spent a long time on this relationship and the use of the 37 inch ell, in Scotland, and the six ell fall, as opposed to the 36 inch yard and 5.5 yard pole in the English or Imperial system. Notably, the inch and foot are common, as is the number of falls and poles in the respective miles, namely 320.

It is the (37*6:36*5.5)ratio, or (222:198) which reduces to 37:33!

37 is an interesting prime number. All the triple numbers are multiples of 37; 111, 222, 333, 444, 555, 666, 777, 888, 999. (111 being 37*3)

A friend, Richard Heath, whose work is fascinating, see link to the right)has pointed out that there are 37 lunations in three years, or 36 months! So it may be related to the lunar cycles, this triangle giving the number of lunations on the hypotenuse, in the corresponding number of months on the adjacent. But, why at such a scale?

It was John Reid, in 1683 who insisted that in Scotland, land was measured in falls and ells, in the first book of gardening with focus on the climatic conditions of Scotland.(Chapter 8, see list of books at bottom of page, intro. by Annette Hope). I shall discuss Reid more, later.

The 37 system also shows up in my work on the geometry of Bornholm, whih is a big topic, to covered later, also!

All I intend to do here is establish this triangle, and highlight the positioning of Preston (unicorn) Cross, in relationship to two special mountains, Cruachan and Schiehallion, with the Iona of the east, Inchcolm Abbey, within the triangle, and with the Holy island of Lindisfarne on the adjacent extended, and with Callanish on the hypotenuse, with great accuracy and in this special relationship.

To finish, the hypotenuse extended south-east finds Kelso Abbey, to a fair degree of accuracy, slightly less, but well within the boundary of the Abbey.

3391.27 6740.57 Preston Cross
3727.85 6338.90 Kelso Abbey
------- -------
0336.58 0401.67

By Pythag:

52404.3metres, or 32.5625 miles(E), or 29.04 miles(S)

angle to O.S. grid:

39.9613 degrees.

The difference in angle is 0.0497, at a distance of some 32.5miles(E):

tan0.0497 * 32.5625 * 5280 = 149feet, or say, 50 yards.

The relative distances both show the 188 harmonic to within limits of 1:2000.

32.5625m(E) divided by sq.root 3 = 18.799968 (99.9998% of 18.8)

29.04226m(S)* phi(1.618034) = 46.99136412(99.9816% 47) * 4 = 187.9654565(same 99.98% of 188)(188 being 47 * 4).

There is a lot here still to ponder. I am content though for now, to have laid this down.

I still have to consider the relationships of most of these points to one another, and there are many possible correspondences already noted, but a fuller report later.

PUC triangle calculations part 4



I am not going to spend much time on the third side of the triangle, at present. Should the need arise I can always return to it.

I have in fact spent a lot of time trying to get a best fit, and looked at various measures around the 188 mile, English and Scots measure on the adjacent and hypotenuse respectively. There are just too many options, and most fit in some way or other. The line in fact runs up the spine of the Outer Hebrides, and various landmarks could be used from Mingulay through Vatersay, Barra(personal wish, Caisteal Chiosmuil, or Kishmul's Castle) in Castlebay, is in very good alignment with Roineabhal at the southern end of Harris, at 23.167 degrees to O.S. grid, which is within 0.1 degrees of required angle.

At Callanish/Calanais I used the stone circle with cairn, and mentioned the local high spot to the north west of the Callanish site. Using the gap between the two points gives a band some half-mile wide. At the Barra end this is from a calculated point just off the small island of Flodagh, includes Castlebay, and some of the other islands a bit further south, such as Mingulay/Muighlaigh.

The observant will have noticed other systems and lines drawn on the graphic. There is still a lot to cover and it is to some of these issues I shall turn to next.


Final comments for now!

It is enough for now to have shown this triangle and that Preston Cross is the very accurate apex, and that PC is also a pointer for Eagles Nest, through Borthwick Church, though I did not show the calculations for that system. I shall get back to that some time. The calculations though show the same level of accuracy!

A Google Map of the system:


View Larger Map

I have added temporarily another line. From Lindisfarne to Duart Castle, on Mull, including points such as Cramond Island, Tulliallan Castle, Stirling, Port Of Menteith, and Ben Vorlich.

I intend to try and cut down the number of Google Maps, as they are taking a while to load.