# The Descent of Egyptian Rulers

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.

1. The cubit is a very ancient measure. The biblical story of Noah quotes the dimensions of the ark in cubits, but the Hebrews appropriated the story of Noah from the Babylonians, whose Epic of Gilgamesh is much older, and his ark was also dimensioned in cubits. So clearly a very old measurement. The Egyptians, who appear to be the source, claim they got it from their gods, but the truth is probably simpler.

2. Given that the origins are very old, the original “standard” must have been copied many times. This inevitably introduces copy-errors, and if later versions are copied from these, then the error multiplies.

3. If the original was wood, then it will be subject to change over time. Further, if the original is REALLY old, perhaps from a wetter climate, and things are now drier, then there may be even more shrinkage or other changes.

4. The gentlemen who did the most famous measurements back in the 1800s (Smyth and Petrie) did their job with British equipment. Google won’t show me what they used, but I’m assuming probably wooden Imperial rulers or yard-sticks, perhaps graduated in 16ths of an inch, or possibly 32 or 64ths. If only in 16ths, then the base accuracy is 1.5875 mm which means any distance less than 1.5mm was “interpolated”. Which may have been okay back in 1880 but now we deal with 1/10ths of a mm ….

5. Further to (4), we have no idea how accurate their equipment was… standard school-room issue, or precise scientific equipment? And if wooden and made in a cold damp climate like England, would it still be exact in the hot, dry Egyptian desert?

6. Then once the measurements get into print, further corruption takes place, where a very reasonable royal cubit length of  20.62 inches (already suffering from rounding errors) is suddenly quoted as 0.525 m instead of the actual 0.523748m

Regardless of the above, it is clear from the history that the values of the cubit did change over the centuries. This is not unexpected, given that Egyptian culture lasted thousands of years, and their language and writing changed over that time as well.

Possible / Probable explanation:

So while playing with the numbers, I’ve come up with a possible/probable explanation for the changes, over and above copy-errors (though these may have played a role too).

While the cubit and cubit parts are given names related to body parts, I don’t think they were originally based on body parts. You’ll see why in a moment. However, as centuries pass, these origins get lost and people assume that a “hand” must have some exact relation to a physical hand. And if, over the centuries, people have gotten smaller or bigger, then someone may feel inclined to “adjust” the measurements to fit the anatomy better. That’s one aspect of what may have happened.

Another aspect is that what I think were the original divisions, do not divide nicely into each other. So some clever priest / Wise Guy / Smart Alec / Egotistical King (take your pick) likely decided at some point to tweak the measurements a little, to make everything relate to a length of 0.0750 m, (7.5cm, about 3 inches), which made for a more harmonious relationship between the various cubit parts. However it broke the relationship between the cubit and royal cubit, forcing them to use one digit size for the cubit, and a slightly larger one for the remaining piece between cubit and royal cubit.

Before we start, here is what Wikipedia claims current values are. I’ve added the alternate remen, and more correct royal cubit.

DigitsNameMetres
1
Digit0.01875
4Palm0.0750
5Hand0.0938
6Fist0.1125
12Small span0.2250
14Great span0.2600
16Foot0.3000
20Remen0.3750
0.3702 (alternate)
24Cubit0.4500
28Royal cubit0.5230 (Wiki)
0.5250 (Wiki)
0.5236 (actual)
32Pole0.6000

So this is what I think happened… we start as always with the metre (explanation for THAT coming at some point), and a circle with diameter one metre. That makes the circumference π metres (3.14159+).

We then simply divide it as follows:

DigitsNameMetresFormula
1Digit0.0187π / 168
4Palm0.0748π / 42
6Fist0.1122π / 28
12Small span0.2244π / 14
14Great span0.2618π / 12
Remen0.3702π / 6√2
24Cubit0.4488π / 7
28Royal cubit0.5236π / 6

Experts will immediately notice that the Hand, Foot and Pole are missing. That is because they are not simple divisions of π, but more complex fractions. So I’m in two minds as to whether they existed “at the beginning” or were later additions.

The Hand is 5 digits, so that would be 5π/168, while the Foot is 2π/21 (i.e. π/10.5) and the Pole double that, at 4π/21 or π/5.25. Not as elegant as something like π/14.

The expert will also note that the cubit is 24 times the digit length, and likewise the royal cubit is 28 of the same digit. None of those new-fangled “alternative digits” here.

We can also do a Nby-rod of 36 digits (or 3 small span) as 3π/14 which has its own charm.

So I think the above is where we started. An alternative methodology, which may have been concurrent, or developed shortly afterwards (I say so because it also handles the Hand very nicely) is based on π/√2. We know the ancients had a great of love of both those numbers.

That arrangement looks like this:

DigitsNameMetresFormula
1Digit0.0185π/120√2
4Palm0.0740π/30√2
5Hand0.0926π/24√2
6Fist0.1111π/20√2
12Small span0.2221π/10√2
14Great span0.2618π/2√18√2 = π/12
20Remen0.3702π/6√2
24Cubit0.4443π/5√2
28Royal cubit0.5236π/√18√2 = π/6

The observant will once again notice that the Foot and Pole are missing. Their formulas are given by 2π/15√2 and 4π/15√2 respectively. Also, this methodology breaks the “same digit” relationship between the cubit and the royal cubit.

It was some time after this that I think they decided to standardise on the 7.5cm distance, for the palm, double handbreadth, small span, foot, remen, cubit and pole, giving us the modern versions at 7.5, 15, 22.5, 30, 37.5, 45 and 60 cm respectively.

Possibly they switched to using ratios based on π/φ², which is not only accurate for this purpose, but again uses their favourite numbers. I’ve expressed the ratios as parts of 64, to show how nicely they line up with the digit counts.

DigitsNameMetresFormula
1Digit0.018751π/64φ²
4Palm0.07504π/64φ²
5Hand0.09385π/64φ²
6Fist0.11256π/64φ²
12Small span0.225012π/64φ²
Great span0.2618(π - φ²) /2
16Foot0.300016π/64φ²
20Remen0.375020π/64φ²
24Cubit0.450024π/64φ²
Royal cubit0.5236π - φ²
32Pole0.600032π/64φ²
36Nby-rod0.675036π/64φ²
64"Double pole"1.200064π/64φ² = π/φ²

The observant will notice that the royal cubit and great span (half royal cubit) no longer divide nicely into digits. Wikipedia fudges this. But this is the end result of adaptations over time.

Thus is the mystery of the confusion nicely solved.