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Part Two.


The Mapping of Great Britain
by Chris Knights.

“TRIANGULATION & TRIG POINTS” 

Imagine that we are trying to fix certain locations with precise relationship to other places. A, B, C, D, E and  F are all high points with lines-of-sight to each other (figure 1). These are the points chosen by the surveyors for the triangulation exercise and on these points they will eventually build their “triangulation pillars”  - better known to navigators as ‘trig points’. These points are where the surveyors set up their theodolites, which are really just telescopes fixed to very accurate brass plates on which were engraved degrees, minutes and seconds.  The cross hairs on the telescope had of course to be extremely fine and were originally made from the threads from a spider’s web.

The target high point was usually made easy to see even at a distance of several miles by lighting a phosphorous flame in a copper bowl. This equipment could measure the angle between two target locations with an accuracy of three or four seconds of arc. In fact the great theodolite made by Jesse Ramsden in 1787 had an accuracy of better than two seconds of arc. Compare this with the accuracy with which you can measure the same angles on a modern OS map with a simple protractor – probably no more accurate than half a degree, ie about 2000 seconds of arc!
The whole point of the triangulation method was that distances between fixed points on the ground could be calculated with very great accuracy simply by measuring the angles between points and then applying relatively simple trigonometry. Naturally you needed to start off with just one distance that had been accurately measured on the ground – the ‘base line’. Once that had been done then by chosing a high point visible from the two ends of the known base line and carefully measuring the angle from each end of the line to the high point (to form a triangle) then the distance of the other two legs of the triangle could be calculated, rather than measured. The base line was chosen so that it was flat and therefore easy to measure on the ground.

Once the first ‘triangle’ was thus calculated, each of the three sides could be used as a new base line for further triangles extending ever outwards across the country. Each ‘leg’ had to end at high points which were visible from each other and typically each leg was 7 or 8 miles apart though this varied according to the terrain.

As the ‘triangulation’ gradually proceeded from the south of England to cover the whole country, some accuracy was inevitably lost but when the triangulation eventually reached the northeast of Scotland, near John O’ Groats, which is generally flat, the final leg of the mainland triangles was remeasured on the ground. This leg was about 20 miles long and the difference between the actual measurement and the distance calculated by triangulation was found to be less than 20 metres in 15 miles and this was 800 miles north of the original measured base line!

This is a real tribute to the surveyors of 200 years ago. One of the reasons for inaccuracy was that of ‘spheroidal  excess’. This phenomemon is caused by the curvature of the earth and means that if you measure the three included-angles at each corner of a triangle whose legs are some miles long, the sum of the angles is slightly more than 180°. The surveyors had access to tables which allowed this error to be taken into account.

But to return to our local illustration. Imagine that the leg from Ackleton to Upper Farmcote (line AB on figure 1) has already been calculated as a result of earlier triangulation and is 6.5 kilometres long. We now need to calculate the distances from the high point at Apley Terrace to four other high points:  Upper Farmcote (CB), The Down (CD), Upton Cressett (CE), Shirlett (CF).

 TrigFig1.JPG(Fig one)

The surveyors now trudge up to Apley Terrace (C) and set up their theodolite equipment whilst other surveyors set up equipment at the five points (A, B, D, E and F) where they can all be seen from Apley Terrace. The surveyor at Apley measures the angles between A and B (55½°), between B and D (67½°), between D and E (22½°) and between E and F (30°). Then the surveyor at Upper Farmcote measures the angle between A and C (37½°) and between C and D (61°). Finally the surveyor at Upton Cresset measures the angle between C and D (84½°) and between C and F (61°).  

Now, back in the office, they can begin their trigonometrical calculations. First they calculate any missing angle in triangles where they have only measured two of the angles and make allowances for spheroidal excess.
Then by using the sine rule they can calculate:
 

 

sin 55½  = sin 87           therefore CB = 7.85k

   6.5k          CB   

 sin 51½  =  sin 61          therefore CD = 8.76k

  7.85            CD 

sin 84½  =  sin 73          therefore CE = 8.49k

  8.76            CE 

sin 89   =  sin 61           therefore CF = 7.46k

  8.49          CF

These figures would have to adjusted of course to allow for the fact that all the observation points are at different levels above sea level, but the theodolites used in the original survey allowed for vertical as well as horizontal angles to be measured, so a little Pythogoras would soon take care of that.
Later ‘levelling’ as it was called, would be done rather more accurately with levelling staffs but the principal was similar. See ‘What about the Verticals?….later.

Trig Points Redundant? 

Where these measurements on the local high points were made, the surveyors in the resurvey which started in 1935 usually built a permanent ‘triangulation pillar’ generally made of concrete and containing brass inserts to which their equipment  was fixed and if you visit these points on Figure 1 you will still see them in place. A few years ago the Ordnance Survey accepted that they no longer needed all these trig points and did not want to incur the costs of maintaining them now that most surveying is done by satellite GPS systems. They only wanted to keep a few key points as checks on their GPS measurements so they were faced with a difficult decision. Shoud they remove the unwanted trig points or leave them to deteriorate gradually? Several groups such as The Ramblers Association lobbied them to keep trig points in good order as they were such useful features to their members, particularly in conditions of fog  in otherwise featureless countryside where a trig point looming up in the mist was the only way in which the walkers could pinpoint their position with complete accuracy. The Ordnance Survey decided on a typically British compromise: they put up most of their trig points for “adoption” whereby members of the public could agree to adopt a particular trig point which would then be recorded as ‘belonging’ to that individual. The individual would agree to visit ‘their’ trig point each year and carry out any basic maintenance such as repainting in white and to report to the OS any major damage or deterioration in condition. The OS issue an impressive certificate of adoption, too. 

 warthilltrig1.JPG 

I have adopted the trig point on Wart Hill (ref: 137/401847) pictured here after being repainted, which I have named ‘Trevor Trig Point’. Blimey, that’s sad isn’t it? Visit it sometime. There’s a super view over the whole of the south Shropshire Hills from the top.

What about the verticals... part three!



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