CRS and QGIS georeferencing for Cassini projection

Question

I have scans of OS 6" series maps which I understand are Cassini projection. According to an article my Mugnier, Photographic & remote sensing, Oct 2003 they had the typical scale factor equal to unity, no false origin, and a single triangulation station as the projection origin. From this I take it that the proj4 string should be something along the lines of:

 +proj=cass +lat_0=53.9619966389 +lon_0=-1.0804806111 +x_0=0 +y_0=0 +units=m +no_defs

where lat_0 and lon_0 values are the coordinates of the origin, in this case York Minster. If I then use the QGIS georeferencer with a linear transform with control points taken from the printed coordinates of the corners the input and output set to this CRS I expect it to generate a world file for the scan with the coordinates are the top left of the sheet, x & y rotations of zero and x and y pixel values which are equal except that y is negative. In fact the pixel values are quite different so that when QGIS opens the file the proportions are grossly distorted. Clearly something in my understanding of this is wrong, but what?

These are points very near the corners of the printed map. The actual scan file is somewhat larger as it includes the entire sheet. Projections are not printed. The 6" county series had different projections for each county or, I think, in some cases for small groups of counties. For Yorkshire this was, as above, centred on York Minster. This particular sheet is 272.

-1.72916666666666652,53.72888888888888914
-1.87361111111111112,53.52027777777777828
-1.72833333333333328,53.52111111111111086

Answer 1

Your corner coordinates are in degrees. If you want the result in the Cassini projection, you have to enter Cassini coordinates (in meters, relative to the anchor point). You can calculate those with cs2cs.

I took the image from http://maps.nls.uk/view/102345139 and read out those corner coordinates (converted to decimal degrees):

-1.8747 53.5781
-1.7290 53.5789
-1.8737 53.5202
-1.7285 53.5211

Put them in a file named Yorkdeg.txt, and write a batch file with this content:

cs2cs +init=epsg:4277 +to +proj=cass +ellps=airy +lat_0=53.9619966389 +lon_0=-1.0804806111 +x_0=0 +y_0=0 +towgs84=446.448,-125.157,542.06,0.15,0.247,0.842,-20.489 +units=m +no_defs -f "%%.2f" <Yorkdeg.txt >Yorkm.txt

The resulting coordinates are:

-52600.52   -42430.79 -0.00
-42950.41   -42439.52 -0.00
-52606.04   -48874.84 -0.00
-42975.81   -48872.07 -0.00

Using those values in the QGIS georeferencer with the same projection as custom CRS for the target CRS, the result fits good to modern Openstreetmap background around Holmfirth:

Alternatively, you can import the corner coordinates in degrees into QGIS as delimited text using EPSG:4277 (which is the closest to what the surveyors used as lat/lon), and set the project CRS to the custom CRS mentioned above. Then use from map canvas in the georeferencer to grab the cassini coordinates of the corner points.