Unifying the Egyptian Royal Cubit, the Roman Cubit, the English Rod and The Yard, through the common method used in their creation.
These three units (Egyptian Royal, Roman, and the Yard that is something of a ‘double cubit’) can all be tied to a similar method of creation within reasonable error bands. They share a standard way of derivation from the rotation of the Earth.
I first ran into this in the Making an English Foot posting. (Actually, a clue had come up earlier in the Chasing The Greek Foot posting where we found that the Greek and Minoan foot were substantially identical to the English foot).
How could all those feet be ‘the same’ if not based on something more substantial. Something of a universal quantity?
That led to looking at the Rod and it’s odd relationship to the foot and the Megalithic Yard. There are 6 M.Yards per Rod, but 5.5 English Yards. Trying to turn that 5.5 length, or it’s double of 11, into some reasonable relationship kept being a bit of a pain. But I kept at it. That the M. Yard was a pretty even ‘6’ was not surprising. Multiples of 6 abound in old methods of measuring as it’s got both 2 and 3 as factors so makes simple fractional math easier.
As a reminder, the Rod (or Perch or Pole) is a very old traditional unit of measure in England. It is 5.5 yards long or 16.5 feet. It “didn’t add up” as it was a fractional relationship, but clearly it WAS a distinct and precise relationship. This argues for a more fundamental connection underneath. Buried in the dusts of time and technological change. But what?
Well, after a lot of “fooling around” (politely termed “research” in scientific circles) I think I’ve worked it out. There are still a couple of minor loose ends. One of them being that the actual length of the various cubits are subject to some variation. At best we have old rulers dug from the dirt, at worst, worn monuments that we measure and back extrapolate what was likely to be the common unit. Another being that our present “second” of time is a bit different from the solstice sundial “second”. So there is a significant “Q.A. Cycle” to be done on this theory before the Quality is really Assured…
It’s all about the sundial and the pendulum.
In the Equatorial Sundial posting, I noted that a 15 degree arc on an equatorial sundial is a one hour arc. Earlier I’d used a 15 degree arc on a Henge at higher northern latitudes to get an hour, and from that a rod pendulum. But using a sundial is just as good, takes much less space, and is usable during the daytime and at the equator. You will lose a bit of precision from the smaller diameter (unless you can put a 100 yard wheel on it’s edge ;-) but can get some of it back from more precise cutting of the marks and styles.
For now, assume you have an equatorial sundial with an hour marking, and that it is further divided into ½ hour and ¼ hour marks. (That is, 7.5 degree and 3.75 degree arcs).
Next, we make a pendulum. We want a very long one for greater precision. It is my belief that folks tended to make them about 2 to 3 “stories” tall as that was the intersection of what was relatively easy to make, yet accurate enough. These give swings of arc less than 5 degrees and so give fairly repeatable results (as that’s the range where degrees of swing changes don’t change the time of swing much at all). IMHO, that’s why we have things like The Rod being 16.5 feet (or about 2 stories). Hang a string from the top of the rafters in the barn, it’s about 2 stories.
OK, but back at the technical bits…
We took our hour and divided it into 60 minutes of 60 seconds. That’s 3600 seconds. Then took our Rod, and let it swing. It is a 4.5 second pendulum (4.49989 per this online pendulum calculator – that I used for all the following calculations as well.)
Why 4.5 seconds? Take 3600 and divide it by 2 a few times.
1800 seconds per half hour
900 seconds per quarter hour
450 seconds per 1/8 hour
wait a minute… that 450 second 1/8 hour would be 100 counts of a 4.5 second pendulum… so the Rod is just a 100 count 1/8 hour pendulum. Or it could easily be a “20 count” in the habit of the Celts and French of 200 on the 1/4 hour marks. “10 Twenties” on a 1/4 hour would be a very ‘reasonable’ thing to folks of that era and place. “Twenty Twenties” on a half hour mark even more so. That gives you a One Rod pendulum. Take your Rod, divide by 6 and you get the proposed Megalithic Yard.
So that chart becomes
1/2 hour – 400 count (Twenty Twenties)
1/4 hour – 200 count
1/8 hour – 100 count
Make a pendulum that counts 400 swings out and back in 1/2 hour, it will be one Rod long.
Perhaps the other units are some other counts of some other pendulums? That would provide a way to unify the foot / yard and the Rod, while still having a difference.
So I went off to
play with research different pendulums and multiples of lengths.
OK, the punch line is that it looks to me like the Egyptian Royal Cubit is the result of a 5 second pendulum being divided into 12 segments; while the English Yard is the result of a 6 second pendulum divided into 10 segments. (Each with factors of 5 and 6 for factor rich fractions, but in different parts of the relationship).
The 5 second pendulum has “counts” of 720 per hour or 360 per half hour. That 360 number is a common and attractive one to watch as it has many factors and is repeatedly used in old methods of measure (and still used by us in units of time and circles… We owe a h/t to the old Babylonians, so no telling how old this system of measures might be.)
So to make an Egyptian Royal Cubit of 52 cm, we can multiply by 12 and get a length of 627.6 cm and that gives a 5.02683 second pendulum. Since I doubt the ancients were able to measure 1/100 second increments, this is a pretty good fit. (Taking an exact 5 modern atomic second pendulum (620.92 cm), dividing by 12, gives a Theoretical Royal Cubit of 51.74333 cm, so we’re down in the 1/4 to 1/2 cm range on precision here. The estimates I’ve seen for the actual length of the Royal Cubit vary by that much. It would be interesting to know if there were any historical estimates of 51.7 ish cm range…)
OK, fair enough. Maybe it is, or maybe it isn’t, but a 5 second pendulum and divide by 12 gives a darned usable “cubit” from a 360 count on a 1/2 hour sundial.
One hour – 720 swings out and back
1/2 hour – 360
1/4 hour – 180
And The English Yard?
Make the pendulum 10 yards long ( that’s 914.4 cm more or less) and you get a pendulum of 6.06765 modern atomic seconds. Again, we’re down in the 1/100 seconds place. ( A “Theoretical Yard” based on an exact 6 second pendulum of 894.124 cm, would be 89.4124 cm or about 35.2 inches. 8/10 ths inch off the present standard. So they were about 8 inches off on their pendulum length out of 30 feet, or about 7/100 seconds per swing off. How many swings would that be? Depends on how long you count… How much does the difference from Sundial Time to present time count? Need to look into that…)
3600 seconds is 600 swings
1800 seconds is 300 swings
1200 seconds is 200 swings (a 20 minute or 1/3 hour period of counting)
900 seconds is 150 swings
At about a 1% error, that would be about 3 swings ‘error’ in 1/2 hour or 1.5 in a quarter hour. Easily inside the error bands of some poles, rope, rock on the end, and a bit of wind resistance and / or imprecision in the plumb bob center determination.
Heck, given how common “off by one” errors are in programming, I could easily see a bit of indecision over whether to count, or not, the final or initial part of a swing series on a quarter hour count giving exactly this result. i.e. start it swinging, count “one” on the return, but at the other end, wait for the final return to end the count, or not… now you have 1 cycle of slop either way. Which one would you choose? If you include the final swing, you need a slower pendulum to fit inside the 1/4 hour as it needs to actually score one lower on the real count, so you get a slightly longer pendulum… Yes, all the ways they might end up an average of 0.067 seconds per swing off are hypothetical. Frankly, I doubt if I could get it that accurate with sticks and string. But it would be an interesting exercise in precision to try it…
Another source of error is that you ought to measure from the center of mass of the plumb bob. If you measure from the very end, you get a slightly different length. Ditto if you measure from the top. So a ‘few inch’ bob on the end can account for all the error band. Clearly there are some details to polish about the exact method. Then there is that error from the Equation Of Time and the difference between a sundial second at the Solstice and an atomic second today that is used in our Hypothetical Pendulums.
This approach also clears up a nagging problem with The Foot. There are 36 inches in a yard. That’s just crying out for a Times Ten somewhere to make it 360 (that wonderful recurrent number). If you have 10 yards in your pendulum, then it is 360 inches long…
Now the foot becomes a back fit of an inch x 12 (to get all those lovely factors of 2, 3, 4, 6 for fractional math…) or just divide your pendulum into 30 x 1 foot lengths of 1/10 inch increments for 300 units of 1/10ths inches. (Reminiscent of that 300 count 1/2 hour) The 1/10 th inch was commonly used in the past for things like maps.
Now we can get rapidly to The Foot without a lot of funny divisions of The Rod. Make a 6 second pendulum. Divide into 360 inches. 1/10 of it is a yard. 12 inches is a foot.
The Roman Cubit
We could further speculate that since the Roman Cubit is almost exactly 1/2 yard, that it, too, might be based on a 6 second pendulum. Just divide by 20 in the style of the Celts…
For our Theoretical Roman Cubit, that would be a length of 44.7 cm which is not too far off from the 45.72 value often cited. 1 cm of ‘error band’. If we take 45.72 and multiply by 20, it gives a 914 cm pendulum that is not significantly different from the one for the English Yard. Whatever difference makes the English Yard a bit long also makes the Roman Cubit a bit long.
So IMHO the various “odd units” of the Egyptian Royal Cubit, the Roman Cubit, the English Yard, English Foot and English Rod; can all be unified via a common method of construction. The pendulum. The variations are the result of choosing a 4.5 second (round 400 / 200 / 100 count), a 5 second (round 360 count) or a 6 second (round 300 count) pendulum and then decisions about dividing it into 1/10 or 1/12 or 1/20 smaller units or leaving it undivided for the Rod or dividing the Rod by 6 for the Megalithic Yard.
Could all this be an accident of numbers and error bands? Certainly.
Do I think so? Not at all…
The ancients were not all that dumb and they knew about things like pendulums and the sundial. We have evidence for widespread use of standard units without the need for widespread distribution of unit standards. This is most easily explained by using a commonly available standard. The sun and stars.
We know they did fractional math and liked factor rich numbers due to that. The more common factor rich numbers being the dozen, 60 and 360. 3600 being 60 squared. We also know they used 10, 60 and 20 base systems (the French still reflect this in their language with 80 being “4 twenties”.)
It is not a large leap at all to say that different cultures chose a different counting base, but used the same technique, to arrive at very similar units, oddly related via the common base of time used in their construction.
As a “teaser” and showing where I’m looking now, the Babylonians had 2 kinds of cubit. The Lagash cubit of 496.1 mm and the “trade cubit” of 446.5 mm. These looks semi-random in our modern mm measures. Yet the Lagash cubit, if divided into a 6 second pendulum yields 18.023 and the “trade cubit” divided into the 6 second pendulum gives 20.025. That both are darned near ’round numbers’ is interesting (to put it mildly). That one is a factor of 3 x 6 while the other is 2 x 10 is also of interest. Perhaps the “domestic” cubit for folks who used math with 6 in it and a foreign cubit for folks who liked base 10? That both have an error term compared to present of about the same size and direction (in the 1/1000 second place) is highly suggestive…
Ooh. and this just in… In another reference to the Babylonian Cubit http://en.wikipedia.org/wiki/Ancient_Mesopotamian_units_of_measurement they give a slightly different length of 497 mm. This, divided into the 6 second pendulum gives 17.99042 which is very very close to 18. It’s sure looking to me like this ‘method’ has been around for a while… Interestingly, that page includes this quote:
Although not directly derived from it, there is a 1:2 proportional relationship between SI and Sumerian metrology. SI inherited the convention of the second as 1/86,400th of a solar day from Sumer thus, two Sumerian seconds are approximately one SI second. Moreover, because both systems use a seconds pendulum to create a unit of length, a meter is approximately two kuš3, a liter 2 sila3, and a kilogram is 2 ma-na.
I’d only point out that it need not have been a single “second” pendulum, but a supermultiple that would be a bit easier to make as a very accurate pendulum; though I could see that detail being ‘lost in the weeds’ of implementation by the standards keepers. Also, per the online calculator, a 2 second pendulum is 993.62 mm while a 1 second pendulum is 248.17 mm so there is some “not quite identity” in the actual pendulum to SI length matching. I also question that a seconds pendulum was ever actually used to make a SI length (though I’ve speculated on such a possible relationship; I’ve held it as a theoretical and slightly non-perfect relationship.) NIST indicates that it was proposed to use a pendulum, but that an alternative definition was chosen (then the implementation botched… giving our present ‘not quite right’ meter…). So there is some foundation for the idea of a meter as a 2 second period pendulum, but the reality ‘has error bars’ attached… http://physics.nist.gov/cuu/Units/meter.html It does leave me with another ‘error term’ to quantify. How much gravity actually varies from place to place.
The origins of the meter go back to at least the 18th century. At that time, there were two competing approaches to the definition of a standard unit of length. Some suggested defining the meter as the length of a pendulum having a half-period of one second; others suggested defining the meter as one ten-millionth of the length of the earth’s meridian along a quadrant (one fourth the circumference of the earth). In 1791, soon after the French Revolution, the French Academy of Sciences chose the meridian definition over the pendulum definition because the force of gravity varies slightly over the surface of the earth, affecting the period of the pendulum.
Thus, the meter was intended to equal 10-7 or one ten-millionth of the length of the meridian through Paris from pole to the equator. However, the first prototype was short by 0.2 millimeters because researchers miscalculated the flattening of the earth due to its rotation. Still this length became the standard.
If only they had chosen the 2 second pendulum we could have a standard of units that correctly mapped to the foot, cubit, rod, etc. and preserved our Mesopotamian heritage… Oh, for a truly rational unit like the Rod or the Roman Cubit ;-)
Some other Cubits:
The Persian Cubit gives a 10.057 factor when divided into the 4.5 second pendulum. This is particularly interesting when you realize that it is 500.1 mm long or almost exactly 1/2 meter. (The 1/2 meter gives 10.0589 from the division. Given that the meter is derived from the earth longitude, this raises the question of “Could some cubits be a longitudinal version of the equatorial measure?” and might the oblateness of the earth account for that error term. So on the ‘to do’ list is to compare equator to longitude lengths and how that compares to the error term above. No, I have no idea how one would get from equatorial time to longitudinal length in 2000 BC, but it could be fun to look into it ;-)
Also the Guard Cubit of 555.6 mm divided into the 4.5 second pendulum gives 9.052 while divided into the 6 second pendulum gives 16.093. Both interesting numbers as I could easily see dividing a string into 1/3 two times, or into 1/4 four times. It is also interesting that we are continuing the pattern of a slight overage of size in the 1/100 place. It is looking rather like a bit of a systematic error or offset. I think I really need to look more closely at that Equation Of Time as it relates to the modern time standard…