Answer: Peonies are propagated by dividing the crown (the underground mass of the plants). If the plant in question has been in the same garden spot for many years, the root mass is probably quite a large mass. I'd continue excavating around the original plant until the entire mass is unearthed. Eventually you'll be able to cut the mass into more manageable clumps. Once you've gotten this far, cut the clumps with a sharp knife into pieces with at least 3 "eyes" each. The "eyes" are growing points on the crown. If you can't plant the divisions immediately, place them in a box of moistened sawdust or peat moss so they won't dry up during transport to their new homes. Transplant the divisions at least 3' apart. The eyes must be no deeper than 1-2" below the soil surface, or they won't grow. It's hard to explain these parts and processes without illustrations, so if it sounds confusing, find a book on plant propagation, perennials, or peonies, and that should help. Another suggestion is to try collecting seeds from this wonderful old plant.
According to the American Horticultural Society's book, "Plant Propagation," peonies are propagated only by division or seeds. Seeds are doubly dormant, meaning they need two periods of chill. First, they are sown in pots and left outdoors to be exposed to winter cold. The first summer roots will develop, but they need a second winter of cold before shoots appear. Plants can take five years to reach flowering size.
Hope you're successful in propagating the plant!
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