Published (Lanchester, 1916): two fleets fire at each other, every ship able to reach any enemy ship, and each side loses strength in proportion to the number of enemy ships still firing: dA/dt = −F, dF/dt = −rA, so rA² − F² never changes and fighting strength goes as the square of numbers. Published (Nelson's memorandum of 9 October 1805, as Lanchester reads it): 40 British against 46, the allied line cut in half, 32 British on the rear 23 and 8 to hold the van. Published (the battle): 27 British ships of the line against 33.
Ours: the van fights only the ships sent to hold it until the moment it joins, and from then on everything left fights everything left. Time is in exchange times: the time one ship, firing unopposed, takes to destroy one enemy ship. "On the day" cuts the allied line in the middle, which is our assumption, not a record: Collingwood broke it about a third from the rear and Nelson just ahead of the centre. Every number on this page is computed here from the closed-form solution of the equations; the post checks it against Python.