## 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.

…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:

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!