A Reassessment of the High Precision Megalithic Lunar Sightlines,2: Foresights and the Problem of Selection

Bessel's refraction tables, in Pryde J. (ed.), Chambers's seven-figure mathematical tables, Part 2: Miscellaneous tables (Edinburgh, 1958), 388–9.

Morrison L. V. , “On the analysis of megalithic lunar sightlines in Scotland”, Archaeoastronomy, no. 2 (1980), S65–77, p. S73.

Declinations quoted in AA2 and TBAR assume a horizontal parallax of 56′·4 or 57′·4, and various temperatures and pressures, according to the position of the site and lunar event assumed (for two typical calculations shown in full, see JHA, 175). These declinations typically differ from our mean values by between 0′·5 and l'·0, the largest difference being l'·2 at Callanish (Line 14). Declinations quoted in MLO, 76 agree with our values to within 0′·4, except for a 0′·9 difference in the case of Corogle Burn (Line 16).

MLO, 76.

MRBB, 134.

Using the formula for rate of change of declination with azimuth for latitude ø between 52° and 60° and small altitude h, we obtain for minor standstill and for major standstill with a most probable value around 0·3.

JHA, 176, Table 3.

Cooke J. A. Few R. W. Morgan J. G. and Ruggles C. L. N. , “Indicated declinations at the Callanish megalithic sites”, Journal for the history of astronomy, viii (1977), 113–33; see pp. 117–18 for details of surveying technique.

See A map of the standing stones and circles at Callanish, Isle of Lewis, with a detailed plan of each site (Glasgow, 1978), produced by the Department of Geography, University of Glasgow, under the direction of Dr D. A. Tait.

MLO, 29–32.

MLO, 93.

MLO, 75.

As deduced from the term 0·01 D (K_n–K_o) in Thom's formula (3·9) (MLO, 32), taking the maximum variation in K as 8, in line with MLO, 30.

Patrick J. D. , “A reassessment of the lunar observatory hypothesis for the Kilmartin stones”, Archaeoastronomy, no. 1 (1979), S78–85, p. S82.

MRBB, 125.

MRBB, 124, Fig. 10.3.

MLO, 68; see MLO, 123 for the method.

MLO, 56.

MLO, 48.

MLO, 66.

MLO, 59.

MLO, 73.

For a site plan see MLO, 71.

MLO, 71 and 72.

MRBB, 126; for a profile diagram see MRBB, 124, Fig. 10.4.

MRBB, 173.

AA2, S81; TBAR, 32–33.

Compare Thom's diagram, MLO, 46, inset (a).

See MLO, 65, Fig. 6.10.

MLO, 57.

MLO, 60, Fig. 6.2.

For a site plan and nomenclature see MLO, 46.

For a site plan and nomenclature see Ponting M. R. and Ponting G. H. , “Decoding the Callanish complex—some initial results”, in BAR, 63–110, p. 80.

See Thom's site plan, MLO, 94.

Thom A. and Thom A. S. , “Astronomical foresights used by Megalithic Man”, Archaeoastronomy, no. 2 (1980), S90–94.

See RBAR, Section 4.3, where it is reassessed as part of the Level 2 (lower precision) hypothesis not involving horizon foresights.

RBAR, 183; see also McCreery T. , “Megalithic lunar observatories—a critique, Part I”, Kronos, v (1) (1979), 47–63.

Thom and Thom , op. cit. (ref. 123), S93, Table 1.

MRBB, 124, Fig. 10.2.

TBAR, 24.

MLO, 39.

Morrison , op. cit. (ref. 90).

See, e.g., AA2.

Morrison , op. cit. (ref. 90), S67 and S72.

Morrison , op. cit. (ref. 90), S67–69 and S72; note that his “Δ” sometimes differs from the Thoms' in sign, and also that it includes a term dδ₂ (equivalent to the Thoms' c₂) which, following the Thoms, we consider separately below.

Morrison , op. cit. (ref. 90), S74.

Morrison , op. cit. (ref. 90), S73–74.

See JHA, 174 for an explanation, and AA2, S86–87 for latest estimates of the value of the graze.

Morrison , op. cit. (ref. 90), S67.

Morrison , op. cit. (ref. 90), S69–72.

MLO, ch. 8.

AA2, S85–86. The Thoms' c₁ and c₂ are equivalent respectively to Morrison's dδ₁ and dδ₂ (op. cit. (ref. 90), S69).

Morrison , op. cit. (ref. 90), S73. In calculating dp, Morrison took ø = 57°; even for Parc-y-Meirw (ø = 52°) an error of only about 0′·01 is introduced.

In JHA, 175, the Thoms consider the alternative case and give it equal weight, but in the analysis of the 42 lines, the possibility of eliminating some less likely alternatives is included in the analysis (AA2, S86–87).

These values are not dissimilar to those assumed by the Thoms (see, e.g., JHA, 175). We find that differences in mean barometric pressure for different times of year and for different heights of site are negligible.

Larger errors are to be expected for one or two foresights at altitudes below 0°, but here the refraction correction for given conditions itself becomes much more uncertain anyway.

Morrison , op. cit. (ref. 90), S69. The variation is produced by a complex interaction of terms: See Danby J. M. A. , Fundamentals of celestial mechanics (New York, 1962), 282.

Morrison , op. cit. (ref. 90), S74. For a sinusoidal variation, the r.m.s. is √ times the amplitude of the variation.

Morrison , op. cit. (ref. 90), S74. Up to ±0′·2 of the variation in both cases is sinusoidal with period half of 179 years, but this is a negligible perturbation on the main variation. Variations in altitude given by Morrison's formulae have been multiplied by 0·925 to bring them to changes in declination.

Maximum values are deduced from the formula for ε_t r.m.s. values are deduced by assuming a uniform distribution over the given dates, hence they are 1/√3 times the maximum variation.

From Bessel's formulae (ref. 89), a variation of, say, ±20 mb at 10°C gives a variation in the altitude correction of about ±0′·8 at h = 0°, ±0′·6 at h = 1° and ±0′·2 at h = 5°; say ±0′·5 on average. A variation of, say, ±10°C at 1005 mb gives a variation of about ±0′·8 on average. Thus we estimate a possible overall variation of up to ±1′·3 in altitude, or ±1′·2 in declination. The r.m.s. value is estimated by assuming a uniform distribution over this range.

Fluctuations under (ii) and (iii) are related, running in phase for upper limb observations and out of phase for lower limb ones: See Morrison, op. cit. (ref. 90), S74. The other variations are mutually independent.

At about five of the ten relevant standstills occurring during one cycle, any event will occur at the alternative possible month and time of day and hence be unobservable. Even observing all five possible occurrences, given the need for extrapolation, requires runs of good weather just at the appropriate times. For a fuller discussion see RBAR, 194–5 and references therein.

Combining terms (i), (vii) and (viii) gives a variation of up to ±2′·6 with an r.m.s. of 0′·9, which reduces after five observations to 0′·4. In addition, the (almost) linear decrease in ε_t over 179 years leads (unless it is recognized as such) to a further effective variation of ±0′·6 with r.m.s. 0′·3. Hence the total quoted. The quoted uncertainties in Q in Table VII include, in addition to this, the uncertainty in term (v) due to our lack of knowledge about the exact date of construction of any sightline.

See, e.g., AA2, S85–88. Hypothesis (4) corresponds to the case c_s = 0.

MSB, ch. 10; MLO, ch. 7.

AA2, S81, Table 1.

AA2, S83.

That varying corrections for graze from line to line have been employed is clear from the phrase (our italics) “The value of the [mean] graze, or if we have already used a graze, the correction to this, is then given by …” (AA2, S83).

JHA, 178, Fig. 4; AA2, S82, Fig. 2.

AA2, S84.

AA2, S82, Fig. 2.

Freeman P. R. and Elmore W. , “A test for the significance of astronomical alignments”, Archaeoastronomy, no. 1 (1979), S86–96, p. S89.

Ibid., S89–90.

TBAR, 19.

TBAR, 24.