3.    Calculation of Overall Error

The displacement of each point relative to its real position was determined for both the north-south axis (latitude) and for the east-west axis (longitude). Thus in each direction, the actual distance  from Kirkwall as determined from a modern Ordnance Survey map was subtracted from that indicated by the antique map. This yielded 35 estimates of error for both axes. In those rare cases where the map lacked a scale bar, a value was calculated by dividing the true distance from Dennis Head, North Ronaldsay, to Cantick Head, Hoy, by the corresponding distance on the map. An overall estimate of the inaccuracy of the map was then calculated from the 70 individual error values.

Table 1 lists the maps investigated together with their associated estimates of original overall error.

4.    Comparisons Between Maps

The  errors for each location were also used to compare maps by correlating the errors on one map with those of the corresponding points on a second map. The extent to which the errors matched between a pair of maps was expressed such that a value of 100% represented a perfect match with a value of zero suggesting no agreement between the errors on the two maps.

5.    Sources of Errors

Consideration of the antique maps suggested three possible sources of error which would apply to all points on a particular map. These were an error in the overall scale of the map, differential errors in estimation of longitude versus latitude and the fact that at least the older maps were mostly based on magnetic north rather than true north.

To examine these errors, the actual distances to each point from Kirkwall were divided by their distances on the old map. This provided 35 estimates of the best scaling of the map for both north-south and east-west directions. For each axis, an estimate of the overall best scale was then calculated from the average of the 35 individual estimates.

In addition, the true bearing of each point from Kirkwall as determined from the modern map was subtracted from the bearing on the old map and the average of all the values was used to correct values for any systematic error of rotation.

6.    Residual Error

After correction of the original data for differences in scaling and making allowance for the rotational error,  the residual errors for each point were determined and an estimate of the map’s overall residual error was calculated (Table 1).