Calculating the coordinates of termini
Calculating modern survey coordinates for a centuriation grid |
The basis of modern surveys
Most states base topographic maps on square kilometre grids. Examples are
- Britain - Ordnance Survey (OS)
- France - IGN
- Italy - IGM.
These grids are the result of a variety of different transformations from the surface of the ellipsoidal Earth to a plane. The resulting distortions are small - indeed, according to Irwin Scollar,
- "All the common transformations preserve angles well enough and are of sufficiently low distortion so that linear distance measurements can be made on a map sheet of scales 1:25000 and larger."
- "For searches or distance measurement over a small region or if the search area is within a meridian or latitude strip, the Earth can be considered flat" (Scollar 1989: 265).
The sort of areas referred to here are
- England and Scotland, which lie within a meridian (north-south) strip
- Any one of three (east-west) zones of France, each covered by a Lambert projection (see below)
Relationship between Roman and modern surveys
From the above it is clear that even the largest known Roman survey lies in an area on a modern map which could be considered flat. What is more, the agrimensor himself took account of variations in ground level. He was instructed to reduce slopes to a plane surface (a procedure he knew as cultellatio ). Thus we have the problem of expressing a rectangular grids in the plane in terms of an x,y coordinate system.
The survey coordinates of any number of potential intersections of a hypothetical square or rectangular centuriation grid can be calculated, given the following parameters:
- The number of actus on each side of the centuries
- The survey coordinates of one intersection
- The size of the actus
- The grid orientation
We have the following formulae for the x (easting) and y (northing) of an intersection at i, j grid divisions from an origin at (a,b)
Where the grid is formed of formed of rectangular units of k actus in the direction of the kardines and d actus in the direction of the decumani, 
is the grid orientation and the actus is equal to m metres.
Implementation
A small Basic program has been used at UEA in the past, but this is not currently available.
It is also straightforward to set up a spreadsheet to calculate the pairs of grid references.
Advantages of using calculated coordinates
Coordinates may be calculated by small computer programs and printed as sets of coordinates starting from some point at a given displacement from the origin. Aternatively, a spreadsheet can perform the same task.
Several substantial benefits follow.
- Since effort is saved, large areas can be considered. The simple but repetitive calculations are otherwise daunting.
- A hypothetical grid may be located correctly, according to the hypothesis, on any map sheet. A local study at any scale, and at any distance from the origin, can use a representation of the grid which conforms precisely to its originally specified position, module and orientation.
- The dimensional instability of paper maps is overcome. Gerard Chouquer noted this problem in 1981, with reference to Italian IGM 1:25,000 maps. Even a single sheet may have slightly different scales along the two axes and it may be impossible to match two maps precisely at the edges, even if they have the same nominal scale. This can lead to imprecision in the use of large transparent overlays covering several joined map sheets. This adds to the inherent innaccuracies of computer-plotted grids.
- One possible source of the investigator's bias is removed. The calculated grid points provide a unique and accurate model, truly at right angles, and with an invariant module and orientation with respect to a modern kilometre square survey grid. It cannot be adjusted from place to place in order to obtain a better fit.
Last updated on 13 August 2009 by John Peterson
(e-mail j.peterson@uea.ac.uk)
