Originally Posted by captain jack
Very good post. However, the Viking ship, as an example of a keel too shallow to sail upwind, was a bad choice. During the pre-Viking migration age, the Germanic tribes used longships that, to the best of our knowledge, had very shallow keels. The Anglo-saxon ship found at Sutton Hoo is a prime example. Although, in all fairness, due to the state of the find, we can't be sure it didn't have a deeper keel, like a Viking ship. It might have, but no one wants to jump to conclusions without evidence.
A modern replica ( although reduced in size ), of the vessel at Sutton Hoo, is the Sae Wylfing. This vessel will sail to wind at around 60 degrees, I think, but it's better reaching or running.
A deeper keel was the big innovation that the Vikings added to the Germanic longship design. The new, deeper keel allowed them to sail to wind, efficiently. A modern reproduction of the Gokstad ship, the Sigrid Storrada, will do 45 degrees to wind, at 5 mph, in moderate winds.
The abandonment of the longship design, in Northern Europe, was not due to performance issues. The ship that replaced them, the carrack, was easier to defend against the raiding attacks of the longships. The high sides, and even taller forecastle and poop deck, were hard to scale, if you were boarding from a low longship, and made a good platform for throwing rocks and things on the raiders.
This quirk, which made defense more important than performance, set ship design back until the 1800's, really.