[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.
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
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?
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.
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.
I’ve rearranged the formulas in Cubit parts and more, updated to be sorted by the general method used. This shows the patterns between the different divisions better.
2018-12-16: Added some formulas for Royal Cubit and Great Span (half royal cubit)
2018-12-18: Added formulas based directly on π, and also √2/π
2019-01-15: Added formulas for “double pole”, standardised a lot of the formulas to use same denominator, to better show link between formula and number of digits, added page for formulas based on e/π∛3, added tau-equivalent formulas to pi page, added inch, and assorted minor tweaks and corrections.
While playing around with the calculator, I stumbled onto the very close identity, where ln(4) plus royal cubit is very close to inverse of the royal cubit.
A bit of rearranging of this identity, replacing the royal cubit by x, allows us to solve for x using the high-school quadratic formula, and thus get a very close approximation for the royal cubit… there are five zeroes after the decimal point if we subtract the royal cubit from the answer.
Here’s the maths: (click to enlarge)
Getting the royal cubit from a simple quadratic equation.
The difference between x above and ₢ (as π/6) comes out at -0.00000727677051.
As previously shown (here and here), φe/8.4 is a good approximation (0.5236) for the Royal Cubit.
But, you may ask, what’s so special about 8.4?
Well the 8.4 is actually the simplified version of the numbers that came out when deriving the relationship. But we can go in the other direction as well, and explode 8.4 into various components and equivalent fractions.
We can do some exposition as follows:
The nautical mile has a chequered history. The basic idea was to have it set to one degree of arc of the circumference, but that depends on what latitude you are at. You can read Wiki’s take for some background.
Anyway, yes, that royal cubit can be related to a good approximation of the nautical mile, given that the exact definition has varied over time. It’s currently “defined” at 1852 metres.
While playing around with various formulas for the Royal Cubit ,I stumbled across this formula:
Which looks pretty innocuous until you see that for practical purposes, as on a ruler, it’s effectively equal to 1. The actual value is about 1.00000970264.
This allows us to rewrite pi, phi and e in terms of the other two, as follows:
[Note: for older version and discussion, see No body parts were harmed producing this post. ]
[Note 2: for newer version, see 5 Different methods of getting the 10 parts of a cubit. ]
2018-11-26: My gut had been insisting for a long time that there was another formula involving pi, phi and e involved with these cubit parts, and eventually today I found it. Not only is it incredibly simple (in basic form) but also incredibly accurate. The basic form is simply (phi x e)/pi = φe/π, which comes out at 1.400013584. If we divide this by 7, we get 0.2000 m, and from that can scale to 0.3000 m, and from that, to the other subdivisions. Have updated the formulas in the PDF attached. I’m particularly chuffed with the formula for the foot…. 🙂
2018-11-30: Added formulas based on √(π²+φ²)/e
2018-12-02: Added formulas for great span and remen based on φπ/e construction.]
While playing around with my calculator in recent times, manipulating pi, phi, e, and the cubit e.t.c., I sometimes ended up with an answer of 1.199981615 which is extremely close to 1.2, and I thought that was interesting. Eventually I took note of the formula to get there, and then realised that 1.2 m is 4 Egyptian feet … and then the penny dropped.
From the foot, we can double to the pole, or half to the double handbreadth. A bit of playing around revealed reasonably easy adjustments to give other fractions of the cubit, all based off the same ratio, which is simply pi over (phi squared) = π/φ².
So here’s an updated list of cubit parts, plus some other measurements like assorted feet, in PDF form. Posting screengrabs from LibreOffice Math is getting tedious 🙂