When a sailing ship is pursuing another upwind, why does the distance gained to windward matter so much?
In one of the Hornblower books,
"We're weathering on her" said Hornblower; his eye was aching with staring through the glass, and his arms even were weary with holding the telescope to his eye, but in the excitement of the chase he could not relax.
"Not as much as I'd like" growled Bolton.
"Hands to the mainbrace!" roared Pellew at that moment.
It was a matter of the most vital concern to trim the sails so as to lie as close as possible to the wind; a hundred yards gained to windward would count as much as a mile gained in a stern chase. Pellew was looking up at the sails, back at the fleeting wake, across at the French ship, gauging the strength of the wind, estimating the strain on the rigging, doing everything that a lifetime of experience could suggest to close the gap between the two ships. Pellew's next order sent all hands to run out the guns on the weather side; that would in part counteract the heel and give the Indefatigable more grip upon the water.
Source Emphasis mine.
Why is the distance gained to windward so important in this type of situation?
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For the same reason it matters in any other chase: distance gained in a pursuit brings you closer to the objective.
Sailing upwind is like climbing up hill, it's slower going as opposed to being on flat ground or going down hill. Small gains upwind at slower speeds are as the book indicates: they count the same as larger gains at higher speeds. The quote quote above isn't indicating that windward gains are especially important, it's drawing attention to the fact that relatively shorter gains windward are equally as valuable as greater gains in a stern chase.
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Windward progress is so valuable in a tactical situation like a warship chase because the windward position is the more maneuverable one. If you can overcome your quarry's distance-made-good-to-windward, you can completely overcome any evasions they may attempt.
This is a different situation than non-violent, course-specified racing, which has its own reasons why velocity-made-good is valuable, and cruising/passagemaking, which again has different reasons why velocity-made-good is valuable. I'm not describing what those reasons are because the question specifies the Hornblower situation.
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The ship to windward is "ahead"
Nothing is entirely linear in this game, but the key is that the further off the wind you're sailing, the faster you can go (this is not true of modern racers), up to an effective limit of the same speed as the wind (also not true of modern racers). The ship to windward has the option to turn off the wind and make up that ground at will, though it's usually better to maintain the windward position until a tactically appropriate moment.
Being the ship to windward also gives you a greater ability to control the situation, consider two ships running close hauled and parallel, both would like to perform a boarding action, however only the ship to windward is in a position to make this decision. The ship to leeward can only turn further from the wind and not practically towards the windward vessel.
If the vessels are running close to each other there's a more direct effect, the sails of the windward vessel will be shadowing the leeward vessel from the wind, leaving them slower and less maneuverable.
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