[As per usual I’ll probably revise this over time, changelog is at the bottom.]

This post introduces a new way of dividing the circle. We already have the normal 360° method inherited from the Babylonians, as well as the French misguided attempt to use 400 Gradians, and even binary degrees that divide a circle into 256 parts.

As I delve into the Egyptian way of measuring things, it seems possible that they divided the circle into 336 parts, which I will call Zeps (from Zep Tepi, the first times). I don’t think the classical Egyptians used this, but maybe the people from Zep Tepi who most likely built the Great Pyramid, and came up with the cubit and related measures, did. Reasons why I think so will become clearer during this article.

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The British/Imperial inch as we know it today has a long and varied history, stretching back to ancient times, as does the foot. Different countries and empires and ages had different values, which you can read about here, here and here.

The inch is currently “defined” at 2.54 cm “exactly”, but there is no record of them measuring any king’s thumb to get that figure, so I’m guessing other factors convinced them to pick that value.

But as it happens, it is probably a good choice, because there’s a nice relationship between the Royal Cubit and the inch, via the Egyptian digit.

If we take, as I believe was the original case, a cubit as being 1/6 of the circumference of a circle with diameter one metre, then a digit will be one 28th of that.

Then we simply multiply by e/2, where e is the base of the natural logarithm.

That gives us 0.0254 m, or one inch.

From the digit to the inch.

We can simplify that down to the following, noting that both 12 and 28 were important numbers to the Egyptians.

pi and e inch formula

The problem:

There is a lot of confusion about the actual lengths that Egyptians used to measure things.

For example, the Royal cubit is given as 0.51m to 0.55 m, which is quite a range. The cubit is quoted at “about 18 inches” or 44 or 45 cm, while the poor remen jumps between 0.3701 and 0.3750 m.

Then some sources claim that a cubit is 24 digits and the royal cubit 28 digits, while the best example of an actual cubit rod, in the Turin museum, clearly shows that the digits in the cubit are shorter than the extra four needed to make the royal cubit.

So why so much confusion?

Possible causes:

Having played around a lot with the numbers, in the process discovering or rediscovering assorted formulas both for the royal cubit and the parts of the cubit, with varying degrees of accuracy, some ideas bubbled up.

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The strange system the pyramid builders and/or the ancient Egyptians used for measuring has been bothering me for a while. On the one hand, we have the logical divisions of the cubit, into nice 15cm chunks and smaller, while on the other hand we have the royal cubit as π/6 which is a horrible irrational number.

So on the face of it it seems odd that there should be some relationship between a cubit at 0.4500m and a royal cubit at 0.5236+ m. The difference is 0.07356+, and trying ratios does not get us very far.

But my intuition was insisting that there must be some easy connection between the two. So I played around a bit, and noticed that cubit/royal cubit was around 0.8594, which rang a bell, as 0.5236 x phi is 0.8472.

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