Update to Case Histories

Michael Behrend, 2014

The following tables show the expected number of alignments according to the Furness formula, and the average number of alignments actually found in computer simulations.  It can be seen that the Furness formula, which was used in the original Linear Dream, tends to underestimate the number of alignments.  Thus the computer simulations confirm, or indeed somewhat strengthen, the negative results found by Bob Forrest. 

Case 1.Relevant to Paul Screeton’s Quicksilver Heritage, p. 43, on the South Durham leys. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 19262378.852379.5089.95
4-pointers 4059.714060.00238.58
5-pointers 222271.35271.0725.69
5-pointers 442.91443.1465.99
6-pointers 1923.0823.076.16
6-pointers 36.7036.8717.74
7-pointers 11.571.581.46
7-pointers 2.462.514.82

Case 1: Map 40 × 45 km; ley width 0.02 mile; 400 sites; 20,000 simulations

Case 2.Salisbury Plain. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 25426  45794.845868.7356.5
4-pointers 183119.2183154.14913.1
5-pointers 7132  14869.114834.2331.3
5-pointers 53311.153294.92612.9
6-pointers 1499  3487.63487.4153.2
6-pointers 11793.411760.31037.8
7-pointers 252  649.5648.350.4
7-pointers 2111.92089.3374.5
8-pointers 35.2100.6100.615.5
8-pointers 318.7307.2130.8
9-pointers 4.213.413.35.1
9-pointers 41.737.440.9

Case 2: Map 40 × 40 km; ley width 0.02 mile; 1006 sites; 500 simulations

Case 3.Relevant to Janet and Colin Bord Mysterious Britain, p. 200, leys in the Bedford and Luton area. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 6622  9305.139315.02197.67
4-pointers 20655.7920665.77835.24
5-pointers 1144  1630.231629.5680.60
5-pointers 3391.643396.58293.63
6-pointers 148  211.15211.1622.83
6-pointers 423.36425.7694.41
7-pointers 15  21.8521.986.27
7-pointers 42.7943.8234.16
8-pointers 1  1.891.901.71
8-pointers 3.644.0214.27
9-pointers 0.10.140.150.45
9-pointers 0.270.395.45

Case 3: Map 40 × 45 km; ley width 0.02 mile; 600 sites; 2,000 simulations

Case 4.Relevant to John Michell’s The View Over Atlantis, p. 40, concerning Dorchester and Weymouth 1:50,000 OS sheet 194. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 402   541.08541.1636.67
4-pointers 778.52778.4973.41
5-pointers 29   41.2141.178.35
5-pointers 57.4557.4016.85
6-pointers 1.6 2.352.341.75
6-pointers 3.213.203.81
7-pointers 0.070.110.110.36
7-pointers 0.140.140.87

Case 4: Map 40 × 40 km; ley width 0.02 mile; 250 sites; 50,000 simulations

Case 5.Again relevant to John Michell’s The View Over Atlantis, from the notes and illustrations p. xx, concerning alignments of moats in East Anglia. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
3-pointers 92   *** TO DO ***109.5710.70
3-pointers 182.88182.9027.33
4-pointers 12.5 *** TO DO ***15.625.19
4-pointers 24.1524.1611.87
5-pointers 1.1 *** TO DO ***1.441.50
5-pointers 2.142.143.41
6-pointers 0.08*** TO DO ***0.100.36
6-pointers 0.140.140.83

Case 5: Region 9 × 14 km; ley width 0.075 km; 38 sites; 200,000 simulations

Case 6.Also from notes and illustrations p. xx of John Michell’s book, being the small map there presented showing an alignment of 5 moats between Athelington and Laxfield. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 191   217.322217.29221.728
4-pointers 306.077306.10242.063
5-pointers 14   15.54915.5334.938
5-pointers 21.24921.2589.733
6-pointers 0.8 0.8320.8311.031
6-pointers 1.1171.1232.136
7-pointers 0.030.0360.0370.207
7-pointers 0.0470.0480.398

Case 6: Map 40 × 45 km; ley width 0.03 mile; 172 sites; 100,000 simulations

Case 7.Relevant to John Michell The View Over Atlantis, Notes and illustrations p. xvi: Alignments of Norfolk Castles and Mounts. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 5    4.26024.26062.2981
4-pointers 4.69614.69882.8609
5-pointers 0.1  0.08410.08460.3046
5-pointers 0.09190.09220.3864
6-pointers 0.0020.00120.00120.0364
6-pointers 0.00130.00130.0480

Case 7(a): Map 40 × 45 km; ley width 0.02 mile; 75 sites; 200,000 simulations

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
6-pointers 1923.07723.0656.123
6-pointers 36.70235.89918.017

Case 7(b): Map 40 × 45 km; ley width 0.02 mile; 400 sites; 15,000 simulations

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
5-pointers 65   72.18972.13311.529
5-pointers 104.224104.13724.802
6-pointers 4.2 4.5584.5432.508
6-pointers 6.4506.4495.943
7-pointers 0.220.2310.2320.531
7-pointers 0.3230.3251.387
8-pointers 0.010.0100.0100.112
8-pointers 0.0140.0140.291

Case 7(c): Map 40 × 45 km; ley width 0.02 mile; 300 sites; 50,000 simulations

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
5-pointers 181   203.373203.26722.523
5-pointers 351.755351.31863.586
6-pointers 16   19.44019.3775.746
6-pointers 32.65432.57018.162
7-pointers 1.2 1.4871.4731.429
7-pointers 2.4502.4595.008
8-pointers 0.080.0950.0960.353
8-pointers 0.1550.1601.177

Case 7(c): Map 40 × 45 km; ley width 0.03 mile; 300 sites; 15,000 simulations

Case 8.Relevant to John Michell, The View Over Atlantis, pl. xxix, facing p. 45: leys through Gare Hill, Wiltshire. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 38675503.05508.0148.9
4-pointers 11106.911115.0524.2
5-pointers 556840.5841.053.5
5-pointers 1600.61602.4166.8
6-pointers 6095.195.114.2
6-pointers 175.1175.247.7
7-pointers 58.68.73.7
7-pointers 15.515.413.4

Case 8: Map 40 × 40 km; ley width 0.02 mile; 500 sites; 5,000 simulations

Case 9.Relevant to Alfred Watkins, The Old Straight Track, Fig. 6, of South Radnor Leys. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 219    *** TO DO ***288.06  26.35 
4-pointers 436.34  436.34  58.15 
5-pointers 16.5  *** TO DO ***25.09  6.70 
5-pointers 36.98  36.98  15.02 
6-pointers 0.925 *** TO DO ***1.67  1.52 
6-pointers 2.42  2.42  3.74 
7-pointers 0.041 *** TO DO ***0.091 0.339
7-pointers 0.129 0.129 0.897
8-pointers 0.0015*** TO DO ***0.00410.070
8-pointers 0.00580.00570.200

Case 9: Region 20 × 40 km; ley width 0.02 mile; 178 sites; 100,000 simulations

Case 10.Relevant to Major F. C. Tyler’s The Geometrical Arrangement of Ancient Sites, Fig. 2, of leys in the region of Tiverton, Devon. 

  Expected by
Furness formula
Expected by
strip formula
Mean of
simulations
St. dev. of
simulations
4-pointers 227    *** TO DO ***264.337924.2871
4-pointers 369.2372369.092246.5233
5-pointers 15    *** TO DO ***18.36565.3925
5-pointers 25.019724.993010.5007
6-pointers 0.75 *** TO DO ***0.97151.1134
6-pointers 1.29691.28702.2308
7-pointers 0.03 *** TO DO ***0.04100.2236
7-pointers 0.05460.05050.3942
8-pointers 0.001*** TO DO ***0.00110.0349
8-pointers 0.00190.00130.0533

Case 10: Map 27 × 18 miles; ley width 0.02 mile; 200 sites; 50,000 simulations