The forms of the pieces in the illustration are, with the exception of two, all variations on a theme, with little consistency or distinctiveness where such is needed. However if we accept that both kings need to be on the board during a game, then the two pieces resembling a pear on a pedestal are them. One is in the hand of the Philosopher, hovering above d3, while the other is at c6. We can then exlude all that might be pawns. These are simply a turned base, tapering upwards, with a terminal ball. Some have a collar, while some are without. There are five on the board (g3, d4, c5, e8, f7) and two off the board, one to the right and one to the left, both partially hidden. This then leaves the queens, bishop, knight and rook. The two forms that do not conform to the majority are the fleur-de-lis on a base (a1) and the half sphere on a three-tiered base (b3). The fleur-de-lis most probably represents the rook, and the other probably the knight.
We are then left with three pieces off the board. Two, one on each side, are topped by small terminal balls on a tapering body above a wide flattened disc. One has a smaller flattened disc immediately below the first, sitting on a tapering base which sits on a pedestal. The other lacks the second disc and pedestal, but otherwise is similar enough in its significant upper features to suggest they are the same piece, probably the two queens. The third, pictured on the Philosopher's side, has a tapering base on a pedestal, surmounted by a wide, rounded and flattened waist, in turn surmounted by a tapering truncated point. This is probably the bishop. These interpretations are necessarily somewhat speculative and reflect the partial and inconsistent nature of the representation of the pieces in the illustration.