The position of this trench has been the subject of much discussion. I submit the following as a reasonable solution of the question:—
Why should it have been a long canal? The conditions of the work were exactly the same whatever place should be selected, viz. the Danes would have to dig through the river embankment on both sides of the bridge. They would also have to dig through the causeway. In the latter part of the work certainly, and in the former part probably, they would have to remove buildings of some kind. The continual wars (800-1000) with the Danes make it quite certain that Southwark must then have been in a very deserted and ruinous condition.
Why should Cnut make his canal a single foot longer than was necessary? We may assume that he was not so foolish. Now the shortest canal possible would be that in which he could just drag his vessels round. In other words, if a circular canal began at CB, and if we draw an imaginary circle GEG round the middle of the canal, it is evident that the chord DF, forming a tangent to the middle circle, should be at least as long as the longest vessel. I take the middle of the canal as the deepest part: there would be no time to construct a canal with vertical sides.
Now (see diagram)
AD2 = AE2 + DE2.
If r is the radius AB or AD and 2a the breadth of the canal and 2b the length of the chord DF,
r2 = (r - a)2 + b2;
[therefore] 2ar = a2 + b2;
[therefore] r = (a2 + b2)/2a.
This represents the length of the radius in terms of the length of the largest vessel and the breadth of the canal, and is therefore the smallest radius possible for getting the ships through. Now the great ship found in Norway in the year 1880 is undoubtedly one of the finest of the vessels used by Danes and Norsemen. The poets speak of larger ships, but as a marvel. Nothing is said about Cnut having ships of very great size. This vessel was 68 feet in length, 16 feet in breadth, and 4 feet in depth. She drew very little water; therefore a breadth of canal equal to the breadth of the vessel would be more than enough. Let us make the chord 70 feet in length, and the breadth of the canal 16 feet. Then
2b = 70, or b = 35,