A common way is to install the shrouds in pairs, port and starboard, from fore to aft, and tying the upper deadeye at the correct location on the shroud line using the jig similar to the one below. Then the deadeyes are lashed. This will avoid getting tangles and wrestling with the deadeye lashings, and is quite easy once you do a few.
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Wrap the shroud round the deadeye and set the desired tension. Secure the shroud like to the deadeye with a bit of CA that was applied to the groove of the deadeye.
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Sieze the shroud at the top of the deadeye. Use thin needles and a needle nosed pliers to make maneuvering of the siezing line easier, and alligator clips to hold the end of the shroud out of the way above. Ensure no gap between the deadeye and the loop of the shroud remains. Secure the seize with diluted PVA glue.
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Bind the bitter end of the shroud to the standing part as shown, apply diluted PVA to the siezes, the cut the shroud end to length.
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Remove the jig and lash the deadeyes. If you were good at making the shroud tensions even, then adjusting the distance between the deadeyes to make the top deadeyes line up in a perfectly straight, horizontal row will not pull your mast out of alignment. Using poly material for the shrouds gives you more room for error in the tension because it is stretchy. Cotton or linen do not stretch easily, so care should be taken in making the tension even from port to starboard. Periodically check your mast alignment to be sure you are not over tensioning the shrouds and distorting the masts. There should be a slight tension on each shroud, and zero slack.
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Tie off your deadeye lashings in the manner befitting the time period and method for your ship. There are variations.
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When viewed from the side, the shrouds should ALWAYS line up perfectly with the angle of the chainplates below the chainwales, since the chainplates transfer the tension of the shrouds to the hull, and no side pressure is applied to the chainwale. If there is interference with gun ports, the location of the shroud should be shifted fore or aft to ensure the gun has a clear field of fire.
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