An interesting suggestion from Bob Graf reads:
The brass tub in Solomon's temple was a thick-sided vessel, and the measurement
of ten cubits referred to the outer diameter, while the measurement of thirty cubits referred to the inner circumference. The thickness of the annulus was recorded as a hand-breadth. If one considers a hand breadth to be 4 inches, and uses a figure of 17.75 for a cubit, the value of p in the equation:
(10 - 30/p)/2 17.75 = 4
is p = 355/113 .
I don't think the Hebrews calculated the values recorded, merely observed them. The true value of π would give slightly different values for a hand-breadth and a cubit. I think this fact is more interesting than the improper imputation of 3 as the 'Biblical' value of π.
We note that the value 355/113 as an approximation for π was first noted by Tsu Ch'ung Chi (429-501 AD)
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