King Solomon’s Bath

In 1 Kings 7 (King James version), we read the description of Solomon’s molten sea:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

bath

…which I take is some kind of bath. Superficially, it seems to state that \pi=3, since the circumference of a circle of diameter d given by \pi d=2\pi r. But what if “round about” doesn’t strictly means “perfect circle”?.

What if Solomon’s bath is an ellipse? A ellipse is described by the equation:

\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1,

which is a scaled circle. The parameters a and b control the shape of the ellipse:

ellipse

The circumference of the ellipse is

\displaystyle C\approx \pi(a+b) \left( 3 \frac{(a-b)^2}{(a+b)^2\left( \sqrt{-3\frac{(a-b)^2}{(a+b)^2}+4} +10 \right)} +1 \right) .

Solving the above for C=30, with one of the axes fixed, say a=5 (because it has “ten cubits from one brim to the other”), we find:

b\approx{}4.543845.

Solomon’s bath is only 10% longer than wide. Not an outrageous ellipse.

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So it could be that the Bible’s description of Solomon’s ritual bath is mostly accurate, and “round about” only means “rounded”. Most likely, however, is that the idea was that \pi=3. The Egyptians, for example, while they didn’t seem to even have the concept of a constant such as \pi, computed the area of a circle using \frac{64}{81}th of a square with a side the same length as the diameter (that’s in the Rhind Mathematical Papyrus, problem 48). That’s already much better estimation for \pi; it implies \pi\approx 3.16, that’s less than 1% error!

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