## Aa As

at major standstill of the Moon. This is an application of the principle of parallax discussed in §3.1.3: The larger the distance to the foresight, d, the larger the "baseline" shift, AS, required to follow a given shift in angle, AA.

Thom's (1971/1973/1978, Fig. 1.1, p. 14) (see our Figure 6.33 for a similar idea) illustrates clearly that if the observer stepped step back on successive evenings a fixed number of paces, before moving laterally to determine the view to the Moon's set point against the distant foresight and to set a stake or stone, then the Moon's change in declination, scaled by some factor, would be illustrated by the curve on the ground created by the placements of the stakes or stones. Indeed, extending the arc between the two markers closest to the moment of extreme declination would give the precise declination (if the scaling factor were known) as well as the moment when this extreme declination occurred.

Whether this purely geometric determination was ever carried out is completely unknown; that is, at present, no one has come forth with evidence of stones so placed at any relevant site, but the process is transparently clear, at least to us, millennia later! What is not so clear is whether the geometric interests of megalithic circle builders extended to the determination of the arcane motions of the Moon, and whether any class of them living in precarious times (see Burl 1987 for vivid accounts of life at Stonehenge!) really had the leisure to study a practical form of analytical geometry, contemplate the meaning and consequences of these motions, and engage in the painstaking work of making both the observations and the determinations. For the time being, we skip over these questions to explore with Thom further methods of determinations—at least how scientists of today transported back to ancient Britain in a Connecticut Yankee sort-of-way could have done so.

Thom (1971/1973/1978, pp. 86-90) related the lateral distances required to observe the shifts in azimuth to the geometric grids found at Merrivale, Mid-Clyth, and elsewhere. He defined a quantity G as that value of AS corresponding to the change in the extreme declination exactly half of a lunar day before or after the extreme. Because the Moon is not likely to be setting at exactly the same moment that it reached its extreme declination, G is not directly measurable. Yet Thom claimed that the value of G was nevertheless recorded at some megalithic sites. Here is how he thought this could happen: If AA is expressed in arc-minutes and d is given in kms (1000 m), then because there are 3438 arc-minutes/radian, the expression

describes the change in position, in meters, in half a day at major standstill. The corresponding shift at minor standstill is

8100 AA AA

3438 A5 A5

The difference ratio, AA/A8, is evaluated from (6.11). Recall that G represents the theoretical baseline shift of the observer in half a lunar day before or after the peak in order to preserve the aspect of the Moon over the same distant foresight. If the stakes are at the same spot, the distance to the peak position would be close to a distance G from the stakes.25 From (6.12), over a whole lunar day, the change in declination will vary by a factor of 4 over the 1/2 day change (12.3' or 8.1' at major or minor standstill, respectively) given above. The corresponding motion of the observer would be 4G, which may be a large distance.26 Thom's suggested method of using this quantity was for the megalithic observer to place a stake each night at the location from which the alignment was seen against the selected and presumably distinctive foresight on the distant horizon. If two stakes were separated by a distance 4G, the position at the peak must lie close to one of the two stakes (stake positions on subsequent days will easily indicate which—if there is enough pacing distance at the observing site). A method of graphical extrapolation in position (corresponding to an interpolation in time) could have been used to obtain the position of maximum declination. Thom (1971/1973/1978, pp. 86-90; Fig. 8.2 illustrates the process). He describes two methods and Wood (1978, pp. 114ff) elaborates on them. The methods involve the use of triangles and sectors, such as those found at Mid Clyth and Merrivale.

Thom (1971/1973/1978, p. 87) shows that if observations of the lunar maximum were made exactly one lunar day

25 The quantity G can also be understood geometrically as a sagitta. Given the stepping technique, and the tracing of a parabolic arc on the ground, it is the line between a point on the arc midway between the two stakes and the center of the straight line between the two stakes.

26 One of the questions raised about the site of Kintraw, discussed above, was whether the ledge provided sufficient space to recreate the alignment within a day.

Figure 6.34. The triangle method of determining the time correction to the extreme declination of the Moon, showing relationships among the quantities G, p, and h: See text for details. Drawing by E.F. Milone.

Figure 6.35. The sector method of determining the time correction to the extreme declination of the Moon, showing relationships among the quantities G, p, and h: See text for details. Drawing by E.F. Milone.

apart, and if the alignment-giving backsight locations were marked by stakes or some other means, then the distance between the midpoint of the stakes and the point from which the maximum-declination Moon would have set, would be given by

Figure 6.35. The sector method of determining the time correction to the extreme declination of the Moon, showing relationships among the quantities G, p, and h: See text for details. Drawing by E.F. Milone.

where 2p is the separation of the stakes. By dividing both sides of (6.17) by p, we get h/p = p/4G, from which h can be found by setting up a right triangle of adjacent side 4G and opposite side p, and laying off a distance p along the adjacent side (see Figure 6.34).

The length of a perpendicular from this point to the hypotenuse then defines a similar right triangle, and the opposite side of this small triangle is h. Because the ratios of like sides of similar triangles are equal, the ratio h/p = p/4G follows. If the leftmost stake represents the Moon's position prior to the maximum, and the rightmost a lunar day later, after the maximum, then by moving the distance G + h to the right, the position of the Moon at true maximum would be determined. Thom (1971/1973/1978, p. 88) suggests that this is the method that was used in Argyllshire.

In a second analytic method, p is the arc of a sector of radius 4G (see Figure 6.35); if one moves in, along the radius, a distance p, and defines a second arc of length, say, x, at the radius (4G - p), then the difference between p and this arc length, p - x, is h.

We demonstrate this as follows: If 0 is the angle subtended by the arcs, then the relation between 0, p, and 4G is (4G)0 = p, so that 0 = p/4G (expressed in radian measure). Then, because x = (4G - p)0, x = p - p2/4G, and h = p - x follows. Thom suggests that a variety of this method was used at Caithness.

When p > G, that is, when the stake separation (2p) is greater than 2G, so that one of the stakes lies fairly close to the maximum, one can define a quantity m = 2G - p. Then, setting m2/4G = G - p + p2/4G = G - p + h = yL, so that

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