# Appendix 1 – Methodology

In order to allow the comparison of the size of buildings through time, I have written formulae for the two principal shapes of buildings found in the Western Isles, round and circular. These formulae allow:

## 1) An estimation of the internal area of the building

For longhouses, I have used a simple length multiplied by breadth formula taking no account of internal divisions, furniture etc. since the evidence is, in almost all cases, too scanty. Similarly, I have not taken into account the possibility of lofts in some buildings, particularly those of the Norse period. This multiplication gives only an approximate answer since many of the buildings are irregular and not strictly rectangular.

For the circular buildings, I have used the simple formula for the area of a circle, pi multiplied by the square of the radius. This approximates the shape of most buildings, although some are quite irregular. For cellular buildings, I added the main cell area to that of the main annexe cell and did not take account of any other annexe cells. For the Atlantic roundhouses, I did not factor in the area within cells and galleries, principally because of the difficulty in doing so. In those with archaeological evidence for a scarcement ledge, I have presented areas with or without a second floor of the same size as the first. For wheelhouses, I estimated the size of piers and their number and subtracted this from their area, since many of the piers make a significant difference. For those where I could not source a measurement of the size of the piers, I used an average taken from the other examples of approximately 1 square metre for each pier.

## 2) An estimation of the roof surface area of the buildings

This will be relevant to the amount of roofing material such as turf and thatch that was used and also to the amount of timber/rope used as sarking, and possibly to the amount of timber in total. For all the buildings I have assumed a roof pitch of 45°. This is based on the photographic evidence contained within this article and also specific references to the usual roof pitch on the islands, and on the Scottish mainland (Gerald 2000). There are certainly illustrations of buildings, and particularly byre ends, with shallower pitches but some of these are the result of partial collapse and evidence generally suggests 45° is a reasonable average. This figure also makes it easier to model the roofs mathematically.

For the rectilinear buildings, all of which show evidence for a piended (hipped) end, I used a formula based on the area of two rectangles plus the area of a cone. This approximates the two straight sides of the building, and the two semi-circular hipped ends. The actual equations are detailed below. In each case, I assumed that the roof rested on the interior wall of the building, as in the Lewis blackhouse. In some cases, it is possible that the roof lay towards the outer edge but in most historic examples, it lay in, say, the inner half of the wall.

For the circular buildings, I calculated the roof area as the area of a cone. For Atlantic roundhouses I used the inner diameter to define the size of the cone. For the cellular buildings I calculated the roof area both as a rectilinear roof, and also as conical roofs. For the rectilinear, I used an approximate width that would cover both buildings without over complicating the mathematics. For the wheelhouses, I used an equation adding the area of the conical roof over the central area to the area of the outer corbelled part of the building. The later was calculated as a flat circle, subtracting the central circle from the total area of the internal space. This part of the roof would have had a slight pitch, similar to the wall tops in post-medieval buildings. The effect on the calculated area is minimal, but makes the calculation much more simple.

## 3) A calculation of the maximum rafter length with a 45° pitch

This used the width of the longhouse, or the diameter of the circular buildings, to calculate the long side of a right-angled triangle. Each length is from the inner edge of the main wall of the buildings to the apex of the roof.

For each type of building, I have drawn together the available dimensional data and presented it in tabular form. I have combined the post-medieval and medieval buildings in one table (Table 1) because the medieval sample is so small and relatively similar. Further, the post-medieval examples include both textual references and the results of survey, equally weighted. Further, individual buildings such as those at Arnol are given equal weight to survey averages from settlements at Gortein and Crubisdale, and areas such as Ardveenish, all in Barra. This is to counteract against a bias towards Barra and, further, specifically to take account of the fact that there are significant numbers of Lewisian large multiple roof structures of a similar size to those at Arnol. It would not be representative to use them as signal buildings as compared to, say, 42 smaller buildings in Ardveenish. These methodological problems are, to some extent, offset by Graph 2 (Comparing individual structures by time period) where individual buildings are compared across the temporal spectrum.

Wheelhouses examples were drawn principally from Crawford's synthesis (2002) and their measurements were corroborated with the NMRS and individual site reports when necessary. Broch examples were drawn principally from Armit's distribution map (1996, 110) and then interrogated using the NMRS and the GIS facility on CANMORE, the RCAHMS website.