Let us ask ourselves why it should be a 'deep' ditch; why it should be a long ditch; why it should be a broad ditch.
Wherever Cnut began his trench, whether at Rotherhithe or nearer the Bridge, he would have the same preliminary difficulties to encounter: that is to say, he would have to cut through the Embankment of the river at either end, and he would have to cut through the Causeway in the middle. In these cuttings he would perhaps have to take down two or three houses, huts, or cabins, all deserted, because the people had all run across the Bridge for safety at the first sight of the Danes, if there were any people at the time living in Southwark—which I doubt.
We may, further, take it for granted that Cnut had officers of sense and experience on whom he could depend for carrying out his canal in a workmanlike manner. A people who could build such perfect ships would certainly not waste time and labour in constructing a trench which would be any longer or deeper or wider than was absolutely necessary.
Now the shortest canal possible would be that in which he was just able to drag his vessels round without destroying the banks. In other words, if a circular canal began at C B, and if we drew an imaginary circle round the middle of the canal, what was required was that the chord D F, forming a tangent to the middle circle, should be at least as long as the longest vessel. Now (see diagram)—
AD² - AE² = DE².
If r is the radius, AD and 2a the breadth BC, and 2b the length of the chord DF—
r² - (r - a)² = b² ∴ r = (a² + b²)/2a.
This represents the length of the radius in terms of the length and breadth of the largest vessel in the fleet, and is therefore the smallest radius possible for getting the ships through. Now, the ship of Gokstad, already described, was undoubtedly one of the finest of the vessels used by Danes and Normans. The poets certainly speak of larger ships, but as a marvel. Nothing is said about Cnut bringing over ships of very great size. Now, that vessel was 66 feet in length, considering the keel, which is all we need consider; 16½ feet in breadth, and 4 feet in depth. She drew very little water; therefore a breadth of canal less than the breadth of the vessel was enough. Let us make the chord 70 feet in length, so that b = 35. Let us make the breadth of the canal 12 feet. Therefore 2a = 12 or a = 6 and r = 105 feet very nearly. Measuring, therefore, 105 feet on either side of London Bridge, we arrive at a possible commencement of Cnut's work. That is to say, if he made a semicircular canal, in that case the length of the canal would be 320 yards, which is certainly an improvement on four miles and a half, or even three miles and three-quarters.