I will be honoured. It is a pleasure to correspond with others who enjoy the manuals and developing a true understanding of the tactics - not just how thing could be done, but why certain things were actually done.
I have gone back to Frederic's original post. In that, he makes it clear that he was studying the British manual and attempting to understand why the square from column at quarter distance was employed instead of close columns.
The thread has demonstrated that certain armies, such as Prussia abandoned the open square, but for reasons driven by column structure.
He asks two questions still to be adequately answered.
1) "If a solid square could do all this [meaning proved adequate defence against cavalry], why bother moving in quarter distance columns at all? Why not move forward simply in close columns and form solid square if threatened by cavalry?"
Here are my thoughts on this question. The British used column at quarter distance because they could. Before a battalion was sent on foreign service it had to be found fit for that service. This means that British battalions tended to benefit from prolonged periods of training and were very carefully inspected before going into action. No Continental army had that benefit. It is doubtful that a Prussian landwehr battalion could develop the skill in maintaining accurate intervals given the limited time between being called out and pressed into action. I suspect that many of the regular units could (such as the Guards, Grenadiers and Regiments 1-11) simply because they would have had more time to drill. However, there would have been no reason to do so. The zug kolonne was a manouver column only, not a tactical column, used as an interim formation between road columns and the angriffekolonne. When in zug kolonne, there would be adequate depth to form square from column at quarter distance, but there would be no incentive to spend time on training for this when such a column could close up just as easily.
Also because of the prolonged training, British infantry were highly disciplined. I doubt that they aimed much better than anyone else (given the lack of sights and time on firing ranges), but they could hold fire better and they could reload more quickly. This means that generating fire was very valuable. A square formed from column at quarter distance could provide one company's worth of fire in all four directions. In contrast a close column could provide twice as much firepower to the front and rear, but virtually none to the flanks.
2) "Wouldn't moving in close columns be easier across uneven terrain compared to moving in quarter distance columns?"
My thoughts on this question are, yes, it would be. However, the extent of practice that the British infantry had before ever deploying overcame this limitation. The critical skill was to ensure that exact intervals were maintained. As with any other skill, practice makes perfect and the British were very well rehearsed. For an army relying on infantry with less time to perfect such a skill, a close column would be better.
Consider that in 1814, many French formations that were newly raised could not operate in any formation other than column (such as the Young Guard under Ney at Craonne). If threatened by cavalry, they would have no choice but to close up. Given these battalions had literally weeks of training, I think their performance was creditworthy.
My concluding thoughts are that column at quarter distance is the better formation for two rank infantry IF the infantry battalion is highly trained. Also, that any form of open square is superior to a closed column, but only if the battalion is skilled enough to complete the manouver AND it can form at least three ranks on all faces.
Imagine now a hypothetical question. We know how the foreign service British infantry battalions fought. What if by some miracle Napoleon should have been able to invade the UK, would the various home service units have been able to perform to the standard we consider "normal" for British infantry? I submit not. If Britain had been as vulnerable to attack as was Prussia in 1813, we would have seen many British battalions forming closed columns when threatened by cavalry.