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)--
If _r_ is the radius, AD and 2_a_ the breadth BC, and 2_b_ the length of the chord DF--
_r_² - (_r_ - _a_)² = _b_² ∴ _r_ = (_a_² + _b_²)/2_a_.
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 2_a_ = 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.
There is, however, more to consider. Why should Cnut make a semicircle when an arc would serve his turn? All he had to do was to draw an arc of a circle with the radius just found, to clear any obstacles in the way of approach to the Bridge, and use that arc for his canal. This is most certainly what he did: I am quite certain he adopted this method, because it was the only sensible thing to do. He would thus get off with a canal about fifty yards long, of which the only difficulty would be the cutting through the Embankment and the Causeway.