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 🙂
Continues from Part 1.
The second lemma needs to be deduced from reading these pages from chapter two of Christopher Milbourne’s Illustrated History of Magic published in 1975. (Dear publisher: please treat this as free advertising for your wonderful book.). Yes it’s all relevant, and it starts off with a story about Khufu. Click to enlarge for easier reading.
Further to my thoughts (which covers some ground covered by others already, so they were not all new ideas) about why the great pyramid is not a tomb, let me speculate on what I think it really was.
But before we get there, it is possible that it may be a tomb, but not in the parts that we know about. There may be another entrance on the west wall (i.e. the one away from the Sphinx), with a separate tunnel and room system like the Bent Pyramid. This possibility is strengthened by the fact that there are different stone types used in the lower courses, which make a triangular pattern, with the current known entrance at the apex of the one on the northern side.
[Note: the formulas below have been superseded by Cubit parts and more, updated ]
It is the mainstream academic consensus that ancient people based their measurements on body parts. As discussed in Some thoughts on the cubit (and foot) , I don’t think that’s necessarily so, despite at least two (the Persian and Roman feet) at one point being based on the length of the foot of their leader (or his over-sized statue).
I still think the names that these measurements became known as (cubit, foot, palm, etc) were backnyms to create a easy-to-use-and-remember name for a length.
Even something simple like measuring the width of a hand is not exact because fingers move, flesh compresses, etc., and fingers are not all the same width.
So… here’s a little exercise to show that we can get the same list of parts of the cubit just with maths. In some cases two different methods produce values on either side of the target.
Just the British mile.
Which, bless them, they defined as being 8 furlongs long. Yeah, what the heck is a furlong… you need to be a British horsey type to be familiar with such things. For the rest of us, a furlong is 660 feet. So that makes a mile 8 x 660 = 5280 feet.
In the early hours of this morning I had an embarrassing “light bulb” moment. As previously shown, a foot is simply ⅕ of a Grand Metre ( 1 + π/6). This morning, the flip side of that hit me … that the Grand Metre (especially in the slightly rounded for practical purposes form of 1.524m) is exactly five feet. I suppose we could call that a fiveet.