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	1926	2378.85	2379.50	89.95
4-pointers	—	4059.71	4060.00	238.58
5-pointers	222	271.35	271.07	25.69
5-pointers	—	442.91	443.14	65.99
6-pointers	19	23.08	23.07	6.16
6-pointers	—	36.70	36.87	17.74
7-pointers	1	1.57	1.58	1.46
7-pointers	—	2.46	2.51	4.82

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	45794.8	45868.7	356.5
4-pointers	—	183119.2	183154.1	4913.1
5-pointers	7132	14869.1	14834.2	331.3
5-pointers	—	53311.1	53294.9	2612.9
6-pointers	1499	3487.6	3487.4	153.2
6-pointers	—	11793.4	11760.3	1037.8
7-pointers	252	649.5	648.3	50.4
7-pointers	—	2111.9	2089.3	374.5
8-pointers	35.2	100.6	100.6	15.5
8-pointers	—	318.7	307.2	130.8
9-pointers	4.2	13.4	13.3	5.1
9-pointers	—	41.7	37.4	40.9

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	9305.13	9315.02	197.67
4-pointers	—	20655.79	20665.77	835.24
5-pointers	1144	1630.23	1629.56	80.60
5-pointers	—	3391.64	3396.58	293.63
6-pointers	148	211.15	211.16	22.83
6-pointers	—	423.36	425.76	94.41
7-pointers	15	21.85	21.98	6.27
7-pointers	—	42.79	43.82	34.16
8-pointers	1	1.89	1.90	1.71
8-pointers	—	3.64	4.02	14.27
9-pointers	0.1	0.14	0.15	0.45
9-pointers	—	0.27	0.39	5.45

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	541.08	541.16	36.67
4-pointers	—	778.52	778.49	73.41
5-pointers	29	41.21	41.17	8.35
5-pointers	—	57.45	57.40	16.85
6-pointers	1.6	2.35	2.34	1.75
6-pointers	—	3.21	3.20	3.81
7-pointers	0.07	0.11	0.11	0.36
7-pointers	—	0.14	0.14	0.87

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 ***	109.57	10.70
3-pointers	—	182.88	182.90	27.33
4-pointers	12.5	*** TO DO ***	15.62	5.19
4-pointers	—	24.15	24.16	11.87
5-pointers	1.1	*** TO DO ***	1.44	1.50
5-pointers	—	2.14	2.14	3.41
6-pointers	0.08	*** TO DO ***	0.10	0.36
6-pointers	—	0.14	0.14	0.83

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	217.322	217.292	21.728
4-pointers	—	306.077	306.102	42.063
5-pointers	14	15.549	15.533	4.938
5-pointers	—	21.249	21.258	9.733
6-pointers	0.8	0.832	0.831	1.031
6-pointers	—	1.117	1.123	2.136
7-pointers	0.03	0.036	0.037	0.207
7-pointers	—	0.047	0.048	0.398

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	4.2602	4.2606	2.2981
4-pointers	—	4.6961	4.6988	2.8609
5-pointers	0.1	0.0841	0.0846	0.3046
5-pointers	—	0.0919	0.0922	0.3864
6-pointers	0.002	0.0012	0.0012	0.0364
6-pointers	—	0.0013	0.0013	0.0480

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	19	23.077	23.065	6.123
6-pointers	—	36.702	35.899	18.017

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	72.189	72.133	11.529
5-pointers	—	104.224	104.137	24.802
6-pointers	4.2	4.558	4.543	2.508
6-pointers	—	6.450	6.449	5.943
7-pointers	0.22	0.231	0.232	0.531
7-pointers	—	0.323	0.325	1.387
8-pointers	0.01	0.010	0.010	0.112
8-pointers	—	0.014	0.014	0.291

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	203.373	203.267	22.523
5-pointers	—	351.755	351.318	63.586
6-pointers	16	19.440	19.377	5.746
6-pointers	—	32.654	32.570	18.162
7-pointers	1.2	1.487	1.473	1.429
7-pointers	—	2.450	2.459	5.008
8-pointers	0.08	0.095	0.096	0.353
8-pointers	—	0.155	0.160	1.177

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	3867	5503.0	5508.0	148.9
4-pointers	—	11106.9	11115.0	524.2
5-pointers	556	840.5	841.0	53.5
5-pointers	—	1600.6	1602.4	166.8
6-pointers	60	95.1	95.1	14.2
6-pointers	—	175.1	175.2	47.7
7-pointers	5	8.6	8.7	3.7
7-pointers	—	15.5	15.4	13.4

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 ***	0.0041	0.070
8-pointers	—	0.0058	0.0057	0.200

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 ***	264.3379	24.2871
4-pointers	—	369.2372	369.0922	46.5233
5-pointers	15	*** TO DO ***	18.3656	5.3925
5-pointers	—	25.0197	24.9930	10.5007
6-pointers	0.75	*** TO DO ***	0.9715	1.1134
6-pointers	—	1.2969	1.2870	2.2308
7-pointers	0.03	*** TO DO ***	0.0410	0.2236
7-pointers	—	0.0546	0.0505	0.3942
8-pointers	0.001	*** TO DO ***	0.0011	0.0349
8-pointers	—	0.0019	0.0013	0.0533

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