I just finished reading about this in John Harland's "Seamanship in the Age of Sail," a book, by the way, that should be in every enthusiast's library. It explains HOW almost everything on board was done through the ages... many drawings actually show these procedures in this book.
A heavy messenger was arranged in an endless loop around the capstan and lead forward to the anchor cable. Heavy deck blocks kept the messenger arranged. A series of Nippers (6' long lines) were attached to the cable (friction wraps), and the capstan pulled the messenger aft, bringing the cable along with it. As the cable came to the forward hatch, the closest nipper was detached, which allowed the cable to descend into the "cable tier." These nippers then passed forward to be attached again to the cable.The "Nippermen" walked along with their nippers, detaching them at the hatch and reattaching them to messenger and cable as the cable comes aboard at the hawsehole. Harland spends 4-5 pages explaining how this type of thing was done, along with many illustrations.
By the way, I thought I would contribute this as a way to pay back the many contributors to this forum... I've spent many enjoyable hours reading this forum, and am working on a first-time build of (what else) Syren. Fun stuff!
Thanks to all who contribute! - John