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1. Already before counting the beads on the strings in Flamsteed's picture it is possible to see that the 3 strings of the upper bowl appear to have different lengths:

We should count the beads to find out what structure (if any) Flamsteed may have had in mind. I invite the Reader to count for himself, and to facilitate his task the following tables will be useful:

Upper bowl:
L C R Sum
       
Lower bowl:
L C R Sum
       

With L, C and R, I mean left, central, respectively right string as seen at the 2 'bells' from which they are leading down to the bowls.

One problem is what to do where we can suspect there is a bead hidden behind another string which is crossing above. My instinct says it is not advisable to count beads which cannot be seen.