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Most people's thoughts go to powerboats when thinking fishing. It's possible that only an inflatable pool toy is a worse platform for catching fish than a sailboat. But for centuries, the sailboat was the only seagoing boat that could get you to where the fish were. So it must be possible.
Last summer, I had the opportunity to have a master show me how to catch fish while under sail. It boiled down to a few easy tricks of rigging, sailing, and timing. Since we had two weeks of glorious sailing to Hawaii to iron out the kinks, I'll save you the trouble and publish them here. Seriously, who wants to have to sail to Hawaii when you can just read the Internet.

The rig is the first place to start. You don't just throw some monofilament with a hook and a worm overboard. Since we didn't have a rod, we needed something to absorb the initial load. The lure is attached to thick monofilament which is tied to 1/8" line (we used blue for no real reason). Now, here's the important part, you need to tie a snubber into the line. A snubber is a rubber piece about 18 inches long. When the fish bites, it will absorb the shock to keep the lure from ripping out of its mouth. For the lures, we found the most success with cedar plugs and squid looking things that looked like what Mahi Mahi would eat (the bluish cedar plug looks like a flying fish, the red squid must look like a squid, not sure what the rainbow squid looked like but it worked). Personally, I found that giving names to your lures gave them personality that the fish could relate to; I went with Blue Meanie, Rainbow Warrior, Red Delicious and the like.

Another real key is how to sail the boat while fishing. You need to let out enough line to place the lure right behind your boat's wake; the fish will think that your lure is the slowpoke at the back of a large school of fish. The turbulent water also masks the fact that your fish is trying to eat a piece of wood or plastic with a hook in it, not an actual edible fish.
Now this is the tricky part, you do need to have the right speed. We found that fish would most likely bite when the boat was travelling around 8 knots (9 MPH). If your boat only goes faster or slower, I suggest packing extra cans of "Chicken of the Sea". Or just try to go 8 knots.

They say that timing is everything. Based on experience, fish bite at dawn and dusk. Probably not out of any sort of dining etiquette but more likely there is enough light then for them to see the lure but not enough light that they can figure out that it's not actually a meal. I think it's become apparent that fishing while sailing is all about a battle of wits with a fish. A fish with a brain about the size of a marble. But during my last trip to Hawaii, we won 9 times despite an almost imperceptible advantage in brain mass.
Easy mnemonic to remember it by, "Dusk or Dawn, yell fish on!"

Just as important as timing is location. It helps if you're sailing to Hawaii. On our last trip, we caught 9 Mahi Mahi, the first of which was about 400 miles from the Coast. But others in the same race caught fish much closer to shore and there was an occasional tuna thrown in. We found our fish-catching pace increased dramatically as we got into warmer waters.
After we caught our first fish on the fourth day out (on a Friday for you Catholics), our next one didn't come until the next Friday (hmm, Fish on Fridays). Ignoring the religious implications, I think it's apparent that our gear was set up for warmer water fish so we didn't catch many until we were in tropical waters. Fishing closer to San Francisco, use lures that are designed for Tuna.
And only catch what you are going to eat.