Unifying The Cubits The Yard and The Rod

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…

The Method

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.

The Relationships

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.

In Conclusion

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.[8] 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…

Subscribe to feed

About E.M.Smith

A technical managerial sort interested in things from Stonehenge to computer science. My present "hot buttons' are the mythology of Climate Change and ancient metrology; but things change...
This entry was posted in Metrology, Stonehenge and tagged , , , , . Bookmark the permalink.

65 Responses to Unifying The Cubits The Yard and The Rod

  1. Hi EM
    Would they have had accurate enough chronometers when these relationships were linked to time? I would be sceptical. ;)

  2. E.M.Smith says:

    The “chronometer” would be as accurate as you care to make your sundial large and your markings fine.

    Using the solstice makes the Equation Of Time fairly accurate. After that it’s just the size of your sundial (star dial) and your precision of counting. A star behind a pole at 50 yards will transit in a very precise moment. A solar shadow on a 1 yard equatorial sundial with 1/8 inch division lines, less so.

    IMHO, that is WHY we have such a large Henge in Stonehenge, to get the precision / accuracy up.

  3. Peter Czerna says:

    Just FYI: the calcTool (very cool!) link has a quotation mark at the end of the URL which messes it up.

    Very interesting posts on time and measurement. I’m still struggling to get my head round them, though, because I keep stumbling over the relationship between time, pendulum swings and distance – e.g.:

    “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.”

    At least you kept the relativistic stuff out :)

  4. Peter Czerna says:

    OK – forget my last comment. I have worked my way back to ‘Making An English Foot’ and forwards from there…

  5. E.M.Smith says:

    @Peter Czerna:

    Thanks for the erratum. I’ve removed the offending quote.

    Just in case others could use the clarification too:

    A pendulum has a characteristic period (time to swing out and back) based on how long it is. If you have a given number of swings (the swing “count”) you also have a given amount of time and given length of pendulum.

    It is this relationship that makes it all ‘work’.

    So we have a simple time standard in the sun (via a sundial, and calibrated on a solstice… the Equation of Time is the corrections applied when you move to other times of the year when the earth has wobbled and nodded a bit out of whack).

    Given that “standard of time” if you make a pendulum that has a specific number of seconds for one swing out and back, it has a specific characteristic length.

    So there are two ways you could get this length standard. Just make a pendulum that swings a given “round” number of counts in your particular numbering system (by twenties if a Frenchman or Celt, but by 60s if a Babylonian, so that 360 count is attractive, or by 10’s if Germanic) and then chop it up in to shorter more usable lengths as per your particular preference. (Chop it by 1/10 if Germanic, but by 1/20 if Celtic or Celto-Roman?, and by 12 if an Egyptian).

    This gives you pendulums with characteristic swing cycles of 4.5 (old Rod) seconds, or 5 seconds for the Egyptian “dozen Royal Cubit length”, or 6 seconds for the newer “English Foot/Yard/ Roman Cubit” system.

    Then you can chop those long ropes into 6, 10 or 12 pieces to get the various cubits, yards, etc. Other cubits look to be based on other ‘divisions’ and the Rod looks to be undivided.

    So we have ONE time standard. Then we get somewhat different “base long length” pendulums based on what “count” of swings was used for the 1/2 hour or whole hour counting period. Then those long ropes can be chopped up a couple of ways for shorter units. Yet, in the end, all the units end up ‘oddly related’ for what looks like no good reason. But in reality, it’s the “time base” at their heart and the tendency to choose ’round counts’ and ‘factor rich cuts’ that leaves the relationship discoverable.

    That’s the thesis anyway.

    It may be that after some QA work it all just turns out to be an excercise in cherry picking, curve fitting, and too wide error bands…

    But I don’t think so. The errors look to be more systematic, and probably related to the Equation of Time being non-zero at the solstices and/or time drift since 2000 BC… add in some ‘string drag’ from a camel hair rope and stir…

    So I’m pretty sure that sometime way back when someone figured out the pendulum trick (prior to Galileo who rediscovered it and applied it much more eligantly) and the sundial trick and put them together. Then Bad Times Came and we forgot how to do this stuff. Perhaps in The Dark Ages…

    Hope that helps…

    Oh, and glad to see that someone else is also interested in time, distance, measuring and old stones… It’s what I’d spend all my time doing if I didn’t need to make money to eat and if the AGW folks would just give up and go away. It’s much more fun ;-)

    At any rate, I’ve now got the urge to make a 16.5 foot pendulum in my back yard and time it swinging on the Solstice … and hope the neighbors don’t call the cops and report a Pegan Druid Performing Rituals …

    Oh, and I need to get an old wheel and turn it into an equatorial sundial too…

  6. Peter Czerna says:

    Thanks for taking this trouble – it’s completely clear now. I share your interest, but lack your energy to dig into the subject as you do.

  7. E.M.Smith says:

    @Peter Czerna:

    It’s not so much energy as an inability to supress compulsive curiosity. So, for example, right now it’s about 1/2 past midnight. I’ve just found a connection to very ancient Mesopotamian units and The Meter (added text to the posting toward the end) and I’m “dead tired” as it’s been a very busy day (friends over, entertaining, etc.)

    Yet the eyes are wide open, the mind is alive with questions and they will not let go of me. So I could go lay in bed, staring at the cieling in the dark, with the questions that will not go away breeding more questions that will not go away… Or I can sit at the terminal dead tired until one of two things happen:

    1) I simply reach complete exhaustion and collapse in a muddle headed stupor and actually do fall asleep. Often about 4 AM to 5 AM when someone else wakes up.

    2) I answer ‘enough’ of the questions that I feel like I can marvel at their “connections” while drifting off to a happy satisified sleep.

    They jury is out as to which will happen tonight…

    I found the chart of Mesopotamian units very gratifying:

    http://upload.wikimedia.org/wikipedia/commons/7/71/Sumerian_Calendar_ISO_B0.svg

    and it includes some answers to things I needed to “figure out” for my Henge (like the exact details of the 19 year period…). But now I’m at risk of launching into the details of The Henge and doing ‘stone counts’ vs ‘Mesopotamian units’ to find how they relate. So I’m hoping to hold that one at bay, take comfort in having shown that the SI Meter is a bastard unit as they didn’t go with the 2 second pendulum when they ought to have done so; take comfort in having tied up a load of loose ends on cubits (several) vs the Rod, Yards, and feet… and be happy to have scratched an itch that first started to bother me about 4 th or 5 th grade when they first told us about ancient / old units and presented the Rod and Cubits… (I’d noticed the ‘almost but not quite’ relationship then and that loose end has bothered me ever since… that itch finally scratched a near 1/2 century later… )

    So can I bottle up The Hungry Mind and contain The Compulsion To Know and actually get to sleep? Or maybe I’ll just look at that Mesopotamian units chart for just a few more {seconds / minutes / hours / what do you mean it’s tomorrow already? }… And would you really call that “energy”? Or just burning the candle with a blowtorch?…

    It’s times like this that I sometimes apply the Benadryl … or a couple of well placed shots of Scotch…

    At any rate, very gratifiying to know that there is another soul who is interested, and that it “clicks”. Makes it kind of worth while…

  8. George says:

    But I don’t think so. The errors look to be more systematic, and probably related to the Equation of Time being non-zero at the solstices and/or time drift since 2000 BC… add in some ‘string drag’ from a camel hair rope and stir…

    Somewhere we should find a rock with a hole in it or a groove around it that was used as a weight for these pendulums.

    Also, the drift in count can be due to drift in swing. For example. Say one pulls back the weight to a standard location. Then one releases the weight just as the shadow on a sundial touches a point. Now as it swings, the distance of the swing will get progressively shorter. But lets say there is a reference rock under the swing path. Once the swing shortens to the point where it no longer surpasses that reference, it is given a little push to keep it swinging. The amount of push won’t always be exactly equal every time. Spring mechanisms had not yet been invented to give a standard amount of push to the pendulum every swing so a person would have to do it.

    So … one would meter out some length of rope and check its accuracy by counting the number of swings in one hour or some portion of an hour. Two people would be required. One to count swings and one to monitor the amplitude of the swing and give a little push now and then lest the pendulum stop.

    One might have a sack of stones and every 10 swings, drop a stone. That way one only must count to 10.

    You count the number of swings. As by this time they already know how much length to add or subtract to correct for a sortage/surplus of swings, they can now cut the rope to a “certified” length.

    The question is, though, why anyone would need that sort of precision for anything at that time. Nothing was built with any precision in the UK during that period. Land wasn’t bought and sold at that time. There wasn’t any concept of private property in that people had clearly demarcated plots of their own. So what would require any precision of repetition in determining measure? Why would they need to measure something one day and have someone else measure it later and come to the same result or measure two things and compare them?

    There was no commerce at the time, little trade, (if you are talking about the neolithic) nothing really of much consequence built that would require duplication at exactly the same scale. You could build a pyramid with a single master scale and build a different pyramid with a different one. The two might come out differently in absolute dimension but the ratios might be the same within a given pyramid.

    There would be no call for a standard unit of measure until one of two things happened:

    1. Something is sold or taxed by unit requiring measurement.

    2. Something is built which must be replicated elsewhere to the same dimension or repeated operations done.

    For example: One could make a whole fleet of boats from boards cut using a single set of standard rules. But you could not use boards cut at a different mill unless their standard rule was the same size as the other mill’s standard rule. There wasn’t much ship building going on in the UK at that time

    I am just having trouble figuring what might have required any sort of consistent measure at that period of time.

  9. George says:

    Uhm, you aren’t going to go all “366” on us, are you? :)

  10. Malaga View says:

    @ E.M. Smith
    I am just amazed how you have slotted things into place… shown the commonality… shown the connections. What really gets me is that you have demonstrated that the answers to these mysteries are actually in plain sight… but it needs an inquiring mind with a lateral thinking ability to see behind the curtain… and that is a rare gift. Thank you for sharing your rare gift.

    I might just be paranoid but I wonder about the origins of organisations such as The Royal Society and The Freemasons (as random examples)… perhaps they have two objectives: first to keep the secrets behind the curtain and secondly to create the curtain by obscuration and misdirection.

    @ George
    There is a third possibility: some people just have inquiring minds… they want to understand and discover… and there is long historical connection between religion and star gazers… however: I do tend to agree with your original two points… after all the only to certainties in life are death and taxes.

    The connections E.M. has highlighted… perhaps we should call it Metrological Archaeology… indicates that we really know so little about those ancient times… about civilisation before “the flood”… about the scattering of civilisations during “the flood”… and the interactions between civilisations after “the flood” stopped… so perhaps there were taxes in the UK at that time… perhaps we have to open our eyes a bit more to see the evidence… perhaps we are looking in the wrong places…. and perhaps we might have to wait for the sea level to decline in the next ice age before we can find more physical evidence… after all commerce and civilisations seem to flourish just above sea-level… life can be such a bitch when sea-levels are rising rapidly… but there are interesting times ahead… on the one hand there will be ice advancing towards Chicago and New York… on the other hand there will be new land masses revealed as the sea-level drops… and civilisations will be able to return home.

  11. Malaga View says:

    so perhaps there were taxes in the UK at that time

    More likely to have been a Tythe… perhaps paid at the Summer Solstice…

    http://en.wikipedia.org/wiki/Tythe

    A tithe is a one-tenth part of something, paid as a (usually) voluntary contribution or as a levy or tax-like payment usually to support a religious organization… historically tithes were required to be paid in kind, such as agricultural products (that grown of the land, or fruit of the tree).

  12. Malaga View says:

    Or maybe at the Winter Solstice…. hence the religious festival in late December associated with the giving of presents.

    I have often heard it said that successful religions expand by absorbing and adapting the customs of other competing religions… it is a business model that works… look at how Microsoft has grown through acquisition… perhaps we should see each new release of Windows as a Tythe… perhaps their pricing model is based upon one-tenth part of the average monthly salary :-)

  13. Myrddin Seren says:

    FWIW

    If you haven’t seen it already, a Wiki on

    http://en.wikipedia.org/wiki/366_geometry

    Knight and Butler also postulate use of the pendulum to effect standard measures.

    May be hocus-pocus, the maths beats me every time, but I love a good pre-historic enigma !!

  14. Jason Calley says:

    Complete unverified speculation follows:

    Why would anyone build henges with astronomical allignments, huge medicine wheels, and sundials, or count pendulum swings?

    IF (big “if”) there was in fact some type of crustal slip in the far past it would have caused tremendous death and damage. The easiest way to tell whether such a thing was starting again — and maybe prepare for the worst — would be to monitor solar and astronomical risings and settings.

    Similarly, if you already have a defined length, then counting pendulum swings compared to time tells you if the Earth’s rotational time has changed. After all, you can’t go around melting ice caps and shifting the crust without slowing the rotation at least a little.

    As I say, pure speculation on my part regarding ancient pre-history, although I am surprised that given “the recent huge rise in sea level” which the CAGW people speak of, no one seems to have detected a lengthening time for Earth’s rotation.

  15. pyromancer76 says:

    “It’s not so much energy as an inability to supress compulsive curiosity”. E.M., “compulsive curiosity” in someone with an unusual mind (and energy) is probably where much of the inventiveness (using truths) of the ages comes from, plus the practical purposes to which to put the knowledge — calculating seasons, star “movments”, crust slippage, repeating asteroid impacts, timing of massive floods, trade, taxes. Then, of course, there are those who for purposes of power, control, wealth (religious leaders, certainly) become jealous keepers of the knowledge.

    I remember your discussion of “the foot” and was immensely pleased that historical thinking could illuminate so much — no “stone” unturned. Why did they do it? Why might they have done it? And it takes the math to understand the common purposes in the differences. What I especially like is the assumption that from the “beginning” human minds (at least some of them) contain the same capacity for brilliance plus compulsive curiosity wherever there is time. More affluence and “warmth”, more time. From this perspective free enterprise and religious freedom are the two linch-pins for this kind of brilliance of mind to enhance human sociey. One of the reasons I “subscribe” is to do my tiny part to enhance your time for compulsive curiosity.

    Now what shall we do with your henge?

  16. DocMartyn says:

    “We also know they used 10, 60 and 20 base systems (the French still reflect this in their language with 80 being “4 twenties”.)

    Can we count to 14 properly?
    One
    Two
    Three
    Four
    Five
    Six
    Seven
    Eight
    Nine
    Ten
    TENTYONE
    TENTYTWO
    TENTYTHREE
    TENTYFOUR
    English numbers are based on a base 12 numbering system, which is why 11 and 12 have real names. The teens are infills for the transition between a base 12 and base 10 numbering system.
    A base 12 is much superior to base 10 for mast day to day calculations (being divisible by 2,3,4,5,6), but didn’t survive the introduction of the Arabic/Indian base 10 system.
    However, eleven and twelve are living fossils of the earlier counting system.

  17. George says:

    You are talking about the Neolithic Beaker Culture. About the only thing they made were pots and maybe some copper bracelets and knives.

  18. Malaga View says:

    E.M.Smith on 30 January 2011 at 7:11 pm
    The “chronometer” would be as accurate as you care to make your sundial large and your markings fine.

    Pendulums might even have been used as some form of “chronometer”….

    Foucault made his most famous pendulum when he suspended a 28 kg bob with a 67 meter long wire from the dome of the Panthéon, Paris. The plane of the pendulum’s swing rotated clockwise 11° per hour, making a full circle in 32.7 hours.

    http://en.wikipedia.org/wiki/Foucault_pendulum

    However, latitude does play a significant role and would have complicated matters…

    Diagrams are provided to illustrate a pendulum located at the North Pole, equator, and 45 degrees N to show how the rotation of Earth in relation to the pendulum is observed, or not, at these locations.

    http://en.wikipedia.org/wiki/Foucault_pendulum_vector_diagrams

  19. Kudos to you!:You are openly dealing with forbidden issues. Men should not find any relation whatsoever between measures and the universal laws, they could begin to realize things. That is why, since the “Illustration”/French revolution specific orders were given to “initiates” of the “freemasonry”, by their masters, with the purpose of “secularizing” the world, making men more managable, ductile and their “science” absolutely “settled”, then profits for the “white brotherhood” would be maximized….

  20. E.M.Smith says:

    Well, after “distracting” myself with a couple of hours more of reading “Liberal Fascism” (the book) about W.Wilson and FDR, I eventually got to sleep.

    Then I come back and find folks have been having a fine party without me!

    OK, some answers:

    Why?

    Well, first off and very clearly: Planting Calendar.

    I’ve been struggling with this in my own little garden in the nearly ideal climate of California and with pumped water on demand. If I’m not focused and on top of the calendar I suddenly discover it’s mid April (or, GASP! even May) and I don’t have any tomato starts to plant. So I’ve got to run off to the Nursery and buy some potted ones instead of setting out my own….

    For me, an inconvenience “fixed” with the application of $20 and a visit to someone who DID watch the calendar. For a subsistance agricultural society, the difference between life and death; starvation and plenty.

    So, just for grins, pick a 2 cubit x 2 cubit square of dirt in your back yard. Resolve to keep something edible growing in it as much of the year as possible. Then see just how soon you are up to your eyeballs in “calendaring”. WHEN does each particular plant do best? Turnips in winter, radishes in spring, corn / tomatoes mid summer, then barley or winter wheat as fall comes…. WHAT DAY EXACTLY do you start your tomatoes to have them ready to set out (and on which day do you do the setting?)

    Then ask yourself: How will I know just when that day arrives if I resolve NOT to use a modern pre-created calendar? Using only nature, some sticks and stones and string? Pretty darned quick you have the original Henge layed out. (Solstice alignments, monthly counting stones, lunar calendar tracker).

    From that point forward, it’s pretty easy to see an ‘escallation’ of technology as you start running into the minor ‘edge cases’ and errors. Leap Years. Lunar / Solar and Tropical / Siderial year differences. You are down to looking at exact days and even length of day pretty quickly.

    OK, Scond Need:

    Read the (translations) of the very earliest texts on the planet. The Cuneiform from Mesopotamia. It’s all about taxes and payments. Standard “rations” of grains and beer to be paid for particular work. Standard taxes to be collected. There have been needs for stadards of measure for as long as there has been society.

    FWIW, the more I’ve looked at that Mesopotamian Measures chart, the more convinced I am that it is a key foundational bit. The “second” they use is a “lunar second” not our modern “metric” or solar second. That alone may account for the difference (i.e. it’s not a precision thing, it’s a lunar vs metric second thing…) They clearly were highly atuned to the lunar calendar and had worked it out down to the seconds of precision. Same thing with the Maya. And once your living or dying depends on something, it can assume religious proportions; then there is no limit to have far folks will take an idea…

    OK, another more “hypothetical” use: At the medium time scale, you have planting calendars. These need precision down to the week, at least, and to the day is better (though only a little – I do remember my Dad telling me some Irish folk wisdom about potatoes had to be planted on a particular lunar day so the “eyes” didn’t “see the moon” and would grow. Wish I’d paid attention, as I think that was an echo of hundreds of years ago… now lost to me…) So folks do, and did, want daily precision in planting.

    But at the longer end? We know there are 11 year sunspot cycles and that these correlate with grain yields (several folks but done “compulsively” by Jevons…) and we know that there is a 19.x year “lunar cycle” that probably also has an impact. (What ‘resonance’ exists between them? I don’t know, but I’d bet the ancients did…) “Right Quick” you end up extending your Lunar calendar out to 19.x years (which I was VERY pleased to see in the Mesopotamian Measures graphic… there is a 19 counter at Stonehenge in the old stone horseshoe holes, but it has some ‘odd stuff’ near the open end of disturbed dirt – my thesis is that it’s a 19.x lunar counter…)

    Once you are doing that, it’s a modestly small leap to find that the constellations are slowly drifting (precession of the equinoxes) and realize that there is a Very Long Count as the Maya did. Did they realize that certain climate / weather patterns repeated on that time scale? Did they know about a 2x,xxx year cyclicality as did Milankovich? I would speculate that they did. Thus “The Age of Aquarius” – the Water Bearer, arriving right on top of The Grand Gatun marked by water flowing from the Water God in the Sky of Maya legend, right as we are getting massive flooding all over the tropical band.

    These folks were “primative” but they were NOT dumb.

    Frankly, take away my electricity and I’m not that much different from them. My “hut” is stick built, theirs a bit more sturdy made of stoneworks with a ‘renewable’ thatch roof… Solstice Celebration or “Christmas”, Vernal Equinox or “Spring Break”, what’s the difference?

    On the smaller end, once you start trying to make these things “line up” the inevitable small differences of lunar vs solar and changes from precession, nutation, obliquity, etc. start to nag. If you are hanging your whole food supply on something, and it’s slowly wandering on you, well, “it focuses the mind” and folks would start trying to nail it down. Looks like the Mesopotamians got to within seconds per 19.x year cycle.

    I would point out we’ve done the same thing. The Gregorian Calendar and drifting holidays et al are still reflected in the difference between Catholic Christmas and Russian Christmas… Wars have been fought over such things.

    Finally, there are eclipses. Once you’ve got a lunar calendar with solar count device, you start to notice that eclipses happen at certain ‘windows’. One of the uses of a Henge is as an eclipse predictor. We still enjoy eclipses today. (And I don’t think folks were really all afraid of them, more just in awe of the show. Just as we are.) So a Sun Stone and a Moon Stone in the Aubry ring holes moved on a regular schedule and you get an eclipse predictor.

    Oh, and never underestimate the power of “social niceties”. Sundials have been around forever. Since the first guy noticed a shadow moved during the day. All that has changed is the precision. So a Henge will pretty quickly get tied up in The Equation Of Time and making sure that if I pay “Robbie Rubble” for 1/2 days work, I get a real 1/2 day of work and my sundial is not ‘rigged’… Basically, desiging and ‘proving’ sundial accuracy is a non-trivial task once you realize you can be 14 minutes or so off during the year just based on trivia of planetary ‘wobble’ and tilt… and you don’t want to show up 14 minutes late for that lunch date with the lady in the next village over…

    Finally: Everybody loves a good party. IMHO, one of THE major reasons we have our “Christmas” parties today is the same reason they had them 4000 BC. It’s the dead of winter and we want to celebrate that it’s all going to be getting better from here. So all around Stonehenge we find evidence of burn pits and animal bones. The archaeologists want to turn this into Grand Religious Significance with High Priests. I’m pretty sure it is just “dead of winter, no farming to do, running out of stored hay, so time to slaughter some livestock and Party!” But someone needs to announce the official start of the party and the official end. The Winter Solstice. So it becomes:

    “Party At The HENGE!!! BYOB & B !!!” (Beef and Beer).

    And if that’s not enough reason, I can list a whole lot more. (For example, how do the Henge builders tell the stone cutters a few hundred miles away that they are cutting one stone a bit shorter than the others and it won’t stand up straight with a lintel on top; unless there is a unit of linear measure? You couldn’t just ‘pop over’ a few hundred miles on Saturday… (or even know it was Saturday..) without some standards for time and length. There are several different groups of stones with different relationships, so someone needs to keep them straight. Easiest is “this string… make it 2 of these by 3 of these…” and that string is the ROD and it’s subdivisions.)

    Per 366:

    I’ve been vaguely aware of it, but I’ve carefully avoided “Going There” as I don’t need the added complexity right now and don’t want the time sink. I’ll skeptically view it in more depth at some point, but I’m just not keen on it. 360 has been around since the start of known culture (Mesopotamia) and I just don’t see the 366 thing as needed. But someday I’m going to need to check on it as a ‘sanity check’ for having missed something that it might explain. Just not, I think, today…

    @Myrddin Seren:

    I’d not seen it. Maybe I will take just a peek today ;-)

    @Pyromancer76: I intend to use my Henge to run my planting calendar and tell me when to party; in the finest traditions of The Ancients ;-)

    @Jason Calley:

    I think there is an easier explanation. We’d just had a major ice melt event (tied to celestial changes ala Milankovitch) and potentially some “crustal shift” from an impact in the Younger Dryas ice fields of North America. That alone ought to be enough impetus to try to get a better grip on the cyclical and celestial driven “changes”. So you all pitch in to make an “Obervatory” and send the astronomy bright boys over there to try and make sense of it all. Oh, and to announce new party dates…

    @DocMartyn:

    Good point… I’ll have to watch a bit more closely for “dozens” and see if I can find a reference to historical ‘base 12’ in European counting systems.

    @Malaga View:

    Thanks! It just makes me happy to share and have someon else like the results…

    Per Masons et.al.: They had some function as a ‘secret society’ a few hundred years ago. Now? I don’t think so. I suspect they have been “depricated” in that regard. (In the 1800s there was some nastyness involving a guy being murdered who’s history escapes me at the moment as the coffee is still brewing ;-) and shortly after that, Masons kind of started a long slide into an ‘old guys club’ and not much else.)

    I do have a trivial bias here. My Father-In-Law was a Mason. That’s about all I know. He was a ‘3rd Degree’ but there are two lines of Masons; one where that’s just barely above “newbie” and one where it’s ‘near the top’ and I don’t even know which one he was in. At any rate, he was a great guy and after W.W.II sold cameras for a living. Not exactly your recipe for world domination… Every Mason I’ve ever met has been that way. Good folks who just like to get together with other good folks and do good things.

    Could they be The Knights Templar in hiding? Mountains of Gold in Swiss banks? Secretly stacking the deck of nations and the world for a new secular order? I suppose so. By definition such secret activites would be secrets. But if they are, they recruit from a very moral and good group and select for sound souls with a strong moral compass. If anyone were going to manipulate the world order, I’d rather it be them than The Evil Bastards who keep popping up from time to time on “the other side”… (King George, various French Kings et.al, Stalin, Hitler, Mao, …)

    But I’ve figured out for my own purposes that “investigation” down that line will be fruitless. There are thousands of folks compulsively doing it, and all they come up with are marginal conspiracy theories. Since I can’t be a ‘value add’ there, I just watch the TV shows from time to time. Maybe someday I’ll go ‘join up’ with the Masons and see if there’s anything I can do for them in the “antiquities and ancient measures” department ;-)

    Closing Note (as I prepare to visit Sir Coffee Pot):

    In reviewing the Mesopotamian Measures chart, I’ve realized that the “divide by 18” could also be a “Divide by 6” of a 1/3 sized pendulum. So I’m going to go back and look at a 2 second pendulum and see how close the “divide by 6” becomes then. It would be a minor “polish” to have both the M.Yard and Mesopotamian Cubit be a divide by 6, but of different pendulums. I also need to look just a tiny bit into the exact difference between a lunar second and a metric second. Never heard of a lunar second before… but it could tie up some significant loose ends…

    Well, off to Mr. Coffee…

  21. Malaga View says:

    @DocMartyn
    However, eleven and twelve are living fossils of the earlier counting system.

    Interesting… so that “dozen” and “gross” are also fossils.

    So what about Spanish?

    0 Cero
    1 Uno (a)
    2 Dos
    3 Tres
    4 Cuatro
    5 Cinco
    6 Seis
    7 Siete
    8 Ocho
    9 Nueve
    10 Diez
    11 Once
    12 Doce
    13 Trece
    14 Catorce
    15 Quince
    16 Dieciséis
    17 Diecisiete
    18 Dieciocho
    19 Diecinueve
    20 Veinte

    At first sight it looks like a hydrid: 0 to 10 are distinct words… so we have a ten base… but 11 to 15 might have been incorporated later as they have a common “ce” ending… that supports a hexadecimal system (which seems a bit strange – but links into base 60)… and the infill numbers then begin at 16.

    Hexadecimal or Base Ten? Any suggestions?

  22. Malaga View says:

    Thought I would have a look for a Latin Root… looks like a base 10 system – but with rather confused teenagers :-)

    1 – unus, una, unum
    2 – duo, duae, duo
    3 – tres, tres, tria
    4 – quattuor
    5 – quinque
    6 – sex
    7 – septem
    8 – octo
    9 – novem
    10 – decem
    11 – undecim
    12 – duodecim
    13 – tredecim
    14 – quattuordecim
    15 – quindecim
    16 – sedecim
    17 – septendecim
    18 – duodeviginti
    19 – undeviginti
    20 – viginti
    21 – viginti unus

  23. E.M.Smith says:

    @Myrddin Seren:

    Ok, I took a look.

    I do have a small interest in the potential for a 366 degree circle as there is that issue of the 366 day year if you count the rearrival day of a particular solar position (i.e. an ‘off by one’ type error of a sort where you count from “Sun here” to “Sun here again” and thus count the end point twice).

    But a major reason for why I went down this investigative path was all the hand waving about “Look!! I found a CORRELATION so it must be CAUSAL!!!” in a lot of the “ancient measurments” stuff.

    When I see things like the assertion that Freemasons were hard at work using Megalithic Yards in laying out Washington D.C.; yet we have the plans drawn in feet and miles, well, I’m driven away from “conspriacy” and toward “underlaying unifying nature”. And the result is this posting.

    Think of it as a kind of “emergent behaviour” centered approach. Everyone wants to put people in the rational center of causality. I prefere to think of them (us?) as passengers in an emergent process. We contribute, but do not control. (Even though we like to believe we control…) So in that view, we would look for a basic unification outside of human driven desires and motivations. We would look just at the facts, and find the natural thing that would drive a common outcome.

    For the mysterious Megalithic Yards showing up in modern construction, all it takes is that we used the same standard of time and the same pendulum method, but chopped things up into different lengths. Now we will find all sorts of “impossible connections” and multiples and sub multiples of things. Simply because at the core, there is a common mechanism. The rotation of the Earth.

    Then all the conspiracy theories and all the need for socially driven complex behaviors drops away. I measure my city and lay it out in miles, and it will automatically be a round number of Roman Cubits and of Megalithic yards (and of Babylonian Cubits and of Egyptian Royal Cubits and of Mesopotamian Cubits and of …)

    So I don’t need to chase down each of those rat holes of conspiratorial causality and / or secret relationships. I don’t need to ponder how Roman Cubits were mysteriously used in the design of the Golden Gate Bridge. I can just accept that Romans were as bright as we are and used a sundial and pendulum as we did. So things are ‘in proportion’ even if different.

    It really simplifies the whole world a great deal…

    But I’m going to be checking my Henge design against both a 360 degree and a 366 degree circle, for the simple reason that if I want to count from “Today is the Solstice” to the next “Today is the Solstice” I’ll count 366 inclusive. So I can see that being a potential…

    But I’m going to start with trying to fit the 360 based Mesopotamian system wherever it works…

  24. Malaga View says:

    @ E.M. Smith
    If anyone were going to manipulate the world order, I’d rather it be them than The Evil Bastards who keep popping up from time to time on “the other side”

    My train of thought wasn’t running along the lines of “world domination”…. after all my father was a Mason… and I think he was part of the long slide into an into an ‘old guys club’

    What I was thinking about was some of their imagey:

    There are lots of things to pick up on in this context… especially their three pillars based on the sun, the moon and man… and lots of tools and inventions hanging from ribbons… perhaps all “hanging off” the discovery of the pendulum.

    Masonic symbology has come down to us from the cuneiform scripts of the ancient Sumerians, circa 3000 B.C.. as well as the ancient Mesopotamians and Persians.
    http://www.masonic-lodge-of-education.com/freemason-symbols.html

  25. Jason Calley says:

    @ E.M. I took a look at the 366 thing too and was not very much convinced. When people start bringing in Moon circumferences, etc., I start to get suspicious.

    Having said that, there may be another simple way to get the 366 number. Everyone says that the year is 365.25 days long, but it really isn’t. It is 366.25 more or less rotations of the Earth for one trip around the Sun. Why? Because the earth rotation is only 23hr and 56 minutes, not 24 hours. If you were measuring rotation from the stars, not the Sun, you get the shorter figure.

    Just a thought, may be a factor in some of the length or time variants.

  26. boballab says:

    EM

    You postulated that the reason for the Henge was for agriculture. Well there is something in my mind that fits that postulation: The Mayan Civilization.

    The Mayan were probably the greatest Astronomers in Earth’s history and their prowess in that field is popularly well known:

    http://en.wikipedia.org/wiki/Maya_civilization#Astronomy

    What is less well known is that the Mayan as had pretty advance agriculture for the time:

    The ancient Maya had diverse and sophisticated methods of food production. It was formerly believed that shifting cultivation (swidden) agriculture provided most of their food but it is now thought that permanent raised fields, terracing, forest gardens, managed fallows, and wild harvesting were also crucial to supporting the large populations of the Classic period in some areas. Indeed, evidence of these different agricultural systems persist today: raised fields connected by canals can be seen on aerial photographs, contemporary rainforest species composition has significantly higher abundance of species of economic value to ancient Maya, and pollen records in lake sediments suggest that corn, manioc, sunflower seeds, cotton, and other crops have been cultivated in association with the deforestation in Mesoamerica since at least 2500 BC.

    http://en.wikipedia.org/wiki/Maya_civilization#Agriculture

    People popularly almost always attribute the Mayan’s advanced astronomy to their religious beliefs and don’t look any further. However part of their religious beliefs deals with the life cycle of Maize (Corn):

    The life-cycle of maize lies at the heart of Maya belief. This philosophy is demonstrated on the belief in the Maya maize god as a central religious figure. The Maya bodily ideal is also based on the form of this young deity, which is demonstrated in their artwork. The Maize God was also a model of courtly life for the Classical Maya.

    http://en.wikipedia.org/wiki/Maya_civilization#Religion

    So when you look at it from the perspective of your postulation not only was having that knowledge important to them to keep them fed, it was also part of their religious well being.

    I have always thought the ancients knew something that are modern “civilization” has forgotten.

    Or maybe this is all a wild co-incidence

  27. Paul Hanlon says:

    Fascinating. The way I was taught in school was that a cubit was from an adult’s elbow to the tip of their fingers and that a foot was the length of an adults foot, and I just accepted it.

    Now I see that it was far more sophisticated than that. Far more. My guess is that if people were at the point where knowing the part of a year they were in was important, they would have been already settled, and doing a significant amount of trade amongst themselves (isn’t trading the first oldest profession?), and the first thing that’s needed in order to trade honestly is an accurate and agreed way of measuring things.

    If any of the 366 goemetry is true, then we’d have to completely re-evaluate our opinion of our ancient forebears.

  28. boballab says:

    I forgot to add in their prowess at mathematics, even though it goes hand in hand with their Astronomy:

    In common with the other Mesoamerican civilizations, the Maya used a base 20 (vigesimal) and base 5 numbering system (see Maya numerals). Also, the preclassic Maya and their neighbors independently developed the concept of zero by 36 BC. Inscriptions show them on occasion working with sums up to the hundreds of millions and dates so large it would take several lines just to represent it. They produced extremely accurate astronomical observations; their charts of the movements of the moon and planets are equal or superior to those of any other civilization working from naked eye observation.[citation needed]

    In common with the other Mesoamerican civilizations, the Maya had measured the length of the solar year to a high degree of accuracy, far more accurately than that used in Europe as the basis of the Gregorian Calendar. They did not use this figure for the length of year in their calendars, however; the calendars they used were crude, being based on a year length of exactly 365 days, which means that the calendar falls out of step with the seasons by one day every four years. By comparison, the Julian calendar, used in Europe from Roman times until about the 16th Century, accumulated an error of only one day every 128 years. The modern Gregorian calendar is even more accurate, accumulating only a day’s error in approximately 3257 years

    http://en.wikipedia.org/wiki/Maya_civilization#Mathematics

  29. Serioso says:

    Every student of European history knows that Napolean’s scientists chose the length of the meter such that the distance between the equator and the North Pole was 10,000 kilometers, i.e., the circumference of the earth was supposed to be, by their definition, 40,000 km. But I wonder if these scientists were influenced by the fact that the period of a one-meter pendulum is almost exactly 2 seconds, and the period of a four-meter pendulum is almost exactly 4 seconds. Coincidence? Or is there some actual historical evidence that this coincidence influenced them? What say you, Chief?

  30. E.M.Smith says:

    Well this is going to be a pain…

    Trying to chase down a “lunar second” has lead to about a dozen different kinds of “scientific time”… the most interesting so far is the:

    http://en.wikipedia.org/wiki/Ephemeris_second

    So we have “Universal Time” and we have “SI Time” and here we have “Ephemeris Time”… Does anybody really know what time it is?… ;-)

    OK, one “good thing” from the ET page is that the note that Lunar based time is better. Good on you, Ancient Mesopotamians!

    Secondary realizations by lunar observations

    Although ephemeris time was defined in principle by the orbital motion of the Earth around the Sun, it was usually measured in practice by the orbital motion of the Moon around the Earth. These measurements can be considered as secondary realizations (in a metrological sense) of the primary definition of ET in terms of the solar motion, after a calibration of the mean motion of the Moon with respect to the mean motion of the Sun.

    Reasons for the use of lunar measurements were practically based: the Moon moves against the background of stars about 13 times as fast as the Sun’s corresponding rate of motion, and the accuracy of time determinations from lunar measurements is correspondingly greater.

    When ephemeris time was first adopted, time scales were still based on astronomical observation, as they always had been. The accuracy was limited by the accuracy of optical observation, and corrections of clocks and time signals were published in arrear.

    OK, so in about 1952 we caught up with Mesopotamia. Next stop: Babylonia! ;-)

    The part that is a bit of a bother to me, in that the old units of length are going to be measured in old units of time (probably earth rotation based, though maybe lunar for Mesopotamia) is this bit about the error offset between E.Time and Universal Time (that one presumes has an analog to Ancient Time ™ )

    The difference between ET and UT is called ΔT; it changes irregularly, but the long-term trend is parabolic, decreasing from ancient times until the nineteenth century, and increasing since then at a rate corresponding to an increase in the solar day length of 1.7 ms per century (see leap seconds).

    So we not only have a question of how the Ancient Time was figured, but we have a parabolic offset to deal with to get to ET from UT and an unknown offset from UT to AT… and all that happens before we even get to the question of SI Time… One can only hope that the errors reduce over time (i.e. move out to the 3rd or 4th decimal point about ET time…) and / or some of them offset.

    But I’m still looking for an easy way to compare a Mesopotamian Lunar Second with a Modern SI Second as used in the pendulum calculator… I know, nobody ever said it would be simple…

    And you’ve got to love this paragraph where we get

    “Barycentric Dynamical Time (TDB) or Terrestrial Time (TT)”

    tossed into the mix as well. I’m starting to need a spreadsheet just to keep all the different kinds of “time” straight… I’m beginning to think that sundial has a certain charm about it…. Sun. Stick. Rocks. Time.

    I note in passing that 360 and 60 a couple of times show up in Newcomb’s time forumla below…

    Redefinition of the second

    Successive definitions of the unit of ephemeris time are mentioned above (History). The value adopted for the 1956/1960 standard second:
    the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.

    was obtained from the linear time-coefficient in Newcomb’s expression for the solar mean longitude (above), taken and applied with the same meaning for the time as in formula (3) above. The relation with Newcomb’s coefficient can be seen from:
    1/31,556,925.9747 = 129602768.13 / (360×60×60×36525×86400).

    Caesium atomic clocks became operational in 1955, and quickly confirmed the evidence that the rotation of the earth fluctuated randomly. This confirmed the unsuitability of the mean solar second of Universal Time as a measure of time interval for the most precise purposes. After three years of comparisons with lunar observations, Markowitz et al. (1958) determined that the ephemeris second corresponded to 9,192,631,770 ± 20 cycles of the chosen cesium resonance.

    Following this, in 1967/68, the General Conference on Weights and Measures (CGPM) replaced the definition of the SI second by the following:

    The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

    Although this is an independent definition that does not refer to the older basis of ephemeris time, it uses the same quantity as the value of the ephemeris second measured by the cesium clock in 1958. This SI second referred to atomic time was later verified by Markowitz (1988) to be in agreement, within 1 part in 10^10, with the second of ephemeris time as determined from lunar observations.

    For practical purposes the length of the ephemeris second can be taken as equal to the length of the second of Barycentric Dynamical Time (TDB) or Terrestrial Time (TT) or its predecessor TDT.

    That last one, TDT, is also interesting…

    Revision of time scales

    In 1976 the IAU resolved that the theoretical basis for its current (1952) standard of Ephemeris Time was non-relativistic, and that therefore, beginning in 1984, Ephemeris Time would be replaced by two relativistic timescales intended to constitute dynamical timescales: Terrestrial Dynamical Time (TDT) and Barycentric Dynamical Time (TDB). Difficulties were recognized, which led to these being in turn superseded in the 1990s by time scales Terrestrial Time (TT), Geocentric Coordinate Time GCT(TCG) and Barycentric Coordinate Time BCT(TCB).

    You’d think by now that they would have got it worked out… but I do have to say that I’m just a tiny bit worried by the term “dynamical timescales”… I rather like my time scales a bit non-dynamical… but I’m old fashioned like that… ;-)

    I can tell that this whole ‘how long is a second’ thing is going to be a PITA with a bunch of “Pluto isn’t a real planet” potential in it…

    I think I’ll go sit in the garden and watch shadows move for a while…

  31. Malaga View says:

    @ Paul Hanlon
    Fascinating. The way I was taught in school was that a cubit was from an adult’s elbow to the tip of their fingers and that a foot was the length of an adults foot, and I just accepted it.

    You were not the only one to “just accepted it”… I really can’t understand why we were taught such nonsense… and now I can’t understand why archaeologists always seem to focus on religion and rituals… it really gets silly.

    Reading up on Stonehenge has been interesting… but why are they now calling the Sun-Stone the Heel Stone… and why is this under the heading of Folklore… rolls eyes…

    Folklore
    The “Heel Stone,” “Friar’s Heel” or “Sun-Stone”
    The Heel Stone lies just outside the main entrance to the henge, next to the present A344 road. It is a rough stone, 16 feet (4.9 m) above ground, leaning inwards towards the stone circle. It has been known by many names in the past, including “Friar’s Heel” and “Sun-stone”. Today it is uniformly referred to as the Heel Stone or Heelstone. When one stands within Stonehenge, facing north-east through the entrance towards the heel stone, one sees the sun rise above the stone at summer solstice.
    http://en.wikipedia.org/wiki/Stonehenge

    This posting has been a revelation… thanks to all.

  32. E.M.Smith says:

    After watching shadows for a while and feeding bunnies and having some sun on my face…

    To get a 1 mm / second resolution of shadow movement is going to take about a 86 meter sundial (or about 280 feet). (no, not precise, just a rough order of magnetude sizing).

    Thus, to get ‘seconds’ of accuracy, you will need a Henge…

    For lunar precision, you can watch the moon eclipse particular stars. Your “moon dial” becomes quite large, and so quite precise. But you need to count your pendulum for a whole lot of hours until the occultation happens again… Somehow I’ve got to get an “hour” or so out of stellar / lunar motion.

    Looks like a “moondial” is not going to help.. from the wiki:

    Moondial

    Moondials are time pieces similar to a sundial. The most basic moondial, which is identical to a sundial, is only accurate on the night of the full moon. Every night after it becomes an additional (on average)* 48 minutes slow, while every night preceding the full moon it is (again on average)* 48 minutes fast, assuming there is even enough light to take a reading by. Thus, one week to either side of the full moon the moondial will read 5 hours and 36 minutes before or after the proper time.

    More advanced moondials can include charts showing the exact calculations to get the correct time, as well as dials designed with latitude and longitude in mind.

    Moondials are very closely associated with lunar gardening (night-blooming plants) and some comprehensive gardening books may mention them.
    The Moon’s orbit is not circular, so it does not move around the Earth at a uniform rate. Thus while the average difference between moonrises is 48 minutes, the actual time can vary considerably (roughly 20min to 1hr50min depending on the time of year and the location of the Moon in its orbit). The time read by a moondial will also vary in a similar, though not so drastic manner.

    But maybe a nocturnal would do…

    http://en.wikipedia.org/wiki/Nocturnal_(instrument)

    There’s got to be a better way…

  33. E.M.Smith says:

    @Malaga View:

    Yeah, part of why I’m doing the Henge stuff is that the very first “report” I ever did (in something like 3rd grade?) was on Stonehenge. It was largely a list “Folks say maybe this, but nobody knows” and I’ve kept that mystery in mind ever since.

    Over the years, I’ve seen an enormous quantity of speculation and flat out fantasy come and go.

    The Heel Stone is pretty clearly just the Summer Solstice marker. It leans in so you can can clearly get a view as the solstice happens and the sun touches the ‘point’. There are also a set of what I think were called “A” posts (holes) around the same area that let you place a pole further out in the approach. Hypothetically (according to some folks) so that you can adjust for precession… See them on this diagram:

    One thesis is that stonehenge was abandoned when they realized that they needed to keep moving the (very large) stones every few hundred years… and when after a few thousand of not moving them it became inaccurate. There is evidence of ‘change over time’ in the particular ages of crap in particular holes and as stones were added at different times.

    If you just start from first principles (what do I need, how does this machine give it to me) it all makes a lot more sense. I won’t go into all of it here (or this will turn into the Stonehenge Posting thats to come ‘someday’ and in a not very organized way…) but the basics are just a time standard, through that a length and area and volume standard, and then a calendar. Finishing with a general purpose astronomical observatory AFTER the “funders needs” were met… The “million and one” times folks have found “alignments” is just more “correlation is causality” stuff. Start from “what do folks need” and it all fits much more clearly.

    Solstice. Equinox. Months. Hours minutes seconds. Rod and M. Yard. (volume and mass derived therefrom). When to plant, when to party, when to go ahead and start some seeds even if it IS cold and wet right now. Then the “priests” go back to trying to figure out why the Lunar Second is not exactly the same as the Solar Second until the next public calendar announcement…

    So “The Dog Star” rises and it’s “the Dog Days of Summer” and we have the beer making party. Later we’ve got the “coldest and darkest it’s going to be” mid winter festival and thin down the wintering herd as the need to ‘make it last’ is waning and have a big “we made it through the worst!” party each Christmas / Solstice / Yule / …

    And yes, it really IS that simple and it really DOESN’T have anything to do with fear of the moon, eclipses, human sacrafice, crazy religious zeal, or any of the other nonesense folks make up because they don’t want to sound mundane… Just like we don’t really “Pray” to “Father Christmas” and don’t really “Vernerate” “Old Saint Nick”… Just start from the notion that we’re pretty much the same kind of folks as we were 5000 years ago and things flow much more readily. It was the crazyness of the Dark Ages Church that has colored our thinking. IMHO, prior to that the pegans were much more “centered”.

    The wind just was, the rains just were, the sun and moon moved in regular motions and all was right with the world.

    Sidebar: FWIW, there is some fair evidence that folks understood they lived on the surface of a sphere long before. At least the time of the Ancient Greeks and probably all the way back to Mesopotamia. It’s in between then and now that folks got all nutty…

  34. DocMartyn says:

    With regard to stone henge. The softwear was run on top of the flat stones, which are the hardwear. There are holes bored into the flat stones, with a diameter big enough to hold a shaft; think Indian Jones looking for the chamber using a staff with a gem stone on it.
    The stone structures in Northern Ireland and in the Shetland/Orkneys are also astronomic, and were made by an adolescent population (their skeletons show late twenties is old age).
    These simpler structures are probably a better thing to decode, having simpler structures, but similar tooling.

    Old English twelf comes from Proto-Germanic twalif
    compound of twa (“two”) and lif (“left over”)
    as in TWAIN. Thirteen does not appear to be old English.

  35. E.M.Smith says:

    @Serioso:

    It’s recorded (and I think I quoted it above from the NIST page) that they WERE going to use a 2 second pendulum, then decided to go with the ‘very close’ polar arc subdivision… but then that got ‘slightly botched’ when they didn’t do the survey quite right…

    The had discovered that the standard gravitational field had some variations and thought that would screw up their standard… As we now know the continents move too, and spin changes will change oblateness just a tad, even their ‘solution’ has minor issues.

    So, had it been up to me, I’d have chosen a place (Say, Paris) and defined it as a 2 second pendulum there. Then just calibrated other places gravity as needed for alternative ‘standards’ creation centers. As it is, we had the “magic stick” in France anyway… but at least then they would have stayed in sync with ol’ Mesopotamia…

    So yes, they knew well the pendulum properties as a length standard. They just screwed up the solution to slight variations in gravity…

    @DocMartyn:

    Yeah, a lot of henges to be compared and contrasted. I’m patial to Stonehenge solely because I’ve put a lot of years into slowly soaking up bits of information in passing… but it does look like all the Henges have a consistent technological base. One of my favorites, BTW, is “woodhenge” nearby…

    Mostlike the Henge To Be in my back yard will be, and be christened:

    PVCpipeHenge

    ;-)

  36. Malaga View says:

    @ E.M. Smith
    If you just start from first principles (what do I need, how does this machine give it to me) it all makes a lot more sense.

    That sounds good to me… and it has achieved a lot for me in just one day :-)

    I have childhood memories of Stonehenge… but not so fun… the family had a good look round… but I had to stay in the car because I had whooping cough… so I will keep my fingers crossed for a Stonehenge posting…

    I have always had a thing about the word “Pagan”… it has always struck me as a really predujicial word… full of bigotry… like calling a CAGW skeptic a “denier”… so I just did a search to see how it was defined… and found this usage example:

    the Spanish conquistadores regarded the native peoples of the lands that they conquered as pagans who were uncivilized and inherently inferior
    http://www.merriam-webster.com/dictionary/pagan

    So I guess the crazyness of the Dark Ages Church still runs deep in our language and culture…

  37. E.M.Smith says:

    Well, as a Pegan-wanna-be it’s not a pejorative to me!

    I guess the wiki has a bit of clue to add on this one:

    In the Christian perspective the term has been used historically to encompass all non–Abrahamic religions. The term pagan is a Christian adaptation of the “gentile” of Judaism, and as such has an inherent Abrahamic bias, and pejorative connotations among monotheists,

    As I’m not really very monotheistic, I’m apparently not suffering from the “Abrahamic bias” nor have the “pejorative connotations”.

    “Druid” would be too specific and a generic “pre-Christian non-monotheists” is way to arcane / technical….

    So I’m just going to stick with Pegan. The wind moves because the wind force (or god or whatever) moves it. We live because of the life force. The sun rises because that is what “sun-ness” does… It works better for me than some cranky old guy with a beard in the sky who wants to count all the hairs on my head while demanding that I use free will but punishing me if I exercise it… (Yeah, I’ve got some Christianity “baggage issues” ;-) Then again, you grow up with a mix of Catholic, Amish, and Church of England in your family while attending a Bible Beater Southern Baptist Church for a dozen+ years and see if you don’t end up with a bit of ‘baggage’… )

    So yeah, “Monotheists” see it as a pejorative. Makes me glad not to be one of them…

    (For the terminally curious, I’d put myself as more of a Pegan-Budhist blend wanna-be with a very deep Christian background; but mostly agnostic.)

    Meanwhile, back at the cubit… I’m pretty sure there is some deeper detail of how the “lunar second” is constructed that is going to account for the (fairly consistent) discrepancy between counts and an atomic seconds pendulum. So I’m off to investigate out the Mesopotamians constructed their “second” on the theory that they taught others or that it was ‘the common way’ then.

  38. Serioso says:

    Sorry. EMS. You seem to have missed my query re Napolean, the meter, the second, and the circumference of the earth. What say you?

  39. Laurence M. Sheehan, PE says:

    I have studied much of this from insatiable idle curiosity (aside from majoring in physics). And I have traveled to see it for myself, to the extent possible. Turkey, Israel, Egypt and the Yucatan.

    Ancient engineers had to have accurate weights and measurement to design and oversee the constructions of those magnificent designs.

    As EM says, either know when to plant food plants or perish. In ancient Egypt, to know when the Nile would flood was a matter of survival.

    “Ancient Engineers” by L. Sprague De Camp is a good way to set off on a most interesting mental journey. The Earth Chronicles series by Zecharia Sitchin (putting aside the aliens from out of space part) gives an excellent accounting of ancient Sumer.

    So much to read, so little time, alas.

  40. George says:

    One thing that is not appreciated is that knowledge was power back then. Imagine … no written language, no books. Everything people knew had to be memorized and it had to be handed down accurately. Life was short and 85% of people didn’t live past 35 years of age. Practically all died by 50. It had to be handed down frequently and accurately without anything written.

    And maybe so is born rhyme and meter. The rhyme and meter act as a checksum to guard against changing a key portion of the information. It is a way of “pickling” information so as to preserve it in a way that can be more easily remembered. The Psalms probably had rhyme and meter in their original tongue, for example.

    Cro magnon had a brain about 10% bigger than ours. Maybe the reason is that if you forgot something back then, you were dead. Was it 3 moons or 4 moons I was supposed to meet up with that hunting party? Now you can just look it up. The book or the computer or someone else does the remembering for you. The more we know the less we know. The information age allows us to become stupid yet have instant information as long as the infrastructure supports the access.

    But without the access to the stored information, we are useless because we have forgotten how to remember. Those people knew how to remember.

    Now imagine if you were able, by swinging a rock on a string, able to tell someone how long the string was. Even a blind man could do it. That would seem like a miracle.

  41. @E.M.Smith
    on 31 January 2011 at 11:29 am
    Everyone knows that E.M. is the best at details, however we are all lost in the middle of an entanglement of measures. Then it seems, at least for planting and surviving, we must know all the measures, laws and rythms of nature. In the past these things were probably taught beginning in childhood.
    In the case of an armageddon we would have to learn it all again!: We depend on all the modern paraphernalia and gadgets, which though have “optimized”, in the end, markets and profits, have built a house of cards easy to tumble down. Not only that but, by not being learned in the basics, we have divorced knowledge from nature, which has taken us as far as conceiving phantom “black holes” but unable, at the same time, as E.M.Smith wisely observes, to know when to plant a tomato seed, when to plant food to eat!

  42. Then, it follows, that Universities should be the depositories of Universal knowledge, and not the bearers of confusion, self pride and self conceit.

  43. P.G. Sharrow says:

    Believe it or not, farmers know when to plant and harvest without the “help” of priests or other educated elites.
    Weights and measures is part of organization and government. Bureaucrats need or want precise measures of things, uniform throughout the countryside. Look at the effort used on the meter standard or the kilo standard. Real people can work with a stick and a rock. pg

  44. @P.G.Sharrow:
    That is true!: Real knowledge it is not hidden but rejected. Cosmic truths are to be understood with simple numbers and ratios.

  45. Steve says:

    Having grown up on a farm, I can tell you that agrarian peoples have other traditional ways of knowing when to plant. “Plant corn when the oak’s leaves are the size of a squirrel’s ear” and so forth. And you feel the soil – temperature and moisture. And so forth.

    Of course this is in the mid-latitudes, tropical regions might be different. But city people don’t understand just how in tune with nature and climate people are genetically hardwired to be.

    There is a theory that it has to do with navigation. As noted people always knew that the Earth was a sphere (the calumny that the medieval church did not is a fiction of I believe, Nathaniel Hawthorne. The Portuguese rejected Columbus because everyone knew the Earth was far too large for the smaller ships of the day to carry enough provisions to reach China by going west.

    Christianity (which dates Christmas starting with the known time frame of Zechariah’s division’s service in the Temple, NOT the solstice), consider that when humanity spread out from eastern Anatolia, they had a significant knowledge of navigation. There is speculation that there may have been considerably more sea-faring in neolithic and chalcolithic times than is commonly thought, for which more and more evidence accumulates, though I don’t think we can call that fact, yet. That hypothesis sees the royal cubits, advanced octal and hexadecimal math of the Sumerians and others, the locations of henges in more places than merely Europe, as tying into this need for accurate navigation.

    So, As an Anglo-American I happily continue with our advanced octal and hexadecimal math, rather than counting on my fingers like the French. (grin)

  46. P.G. Sharrow says:

    I always wondered how a “rod” was created and defined. 16 and a half feet is a real strange measurement. We set fence posts on a rod per post when designing field fencing for large tracts, 80 rods per quarter. A timed pendulum rod would be an earth defined unit of measurement. Every bit as good as a meter! Anyone with a sundial could create a well defined unit of measurement to measure the earth. A mile at 5280 feet is also lame. Now it is 320 rods! Well done sir and thank you. pg

  47. pyromancer76 says:

    EM, glad this thread has lasted and elongated (by what measure?). I am learning so much and am a fellow pegan-admirer as well as someone who appreciates the possibilities of monotheism, for good and ill. Questions I like about every culture: How well did they do “it” (“it” –adjust to or transform their chaotic Earth environment)? How many lives did they expand rather than lessen or end as they did “it”? Transparency and accountability. I like the emphasis on the mathematical truth telling.

    LM Sheehan, thanks for the reference to Ancient Engineers. This is the kind of history I thoroughly enjoy. Don’t know how I missed it during my teaching days since I very much admire engineers and always had my students learn something about economic history and technological advances as an essential part of their historical repertoire. Perhaps it was the time period. Well, E.M. is setting us right about how far back we can connect with our ancestors’ knowledge.

    (E.M., I find it your background fascinating in connection with the questions you raise and your indefatigabile investigations, especially your ability to ask “real” human questions. I also have some Southern Baptist background [those evangelical meetings were something!] and say a Buddhist thanks-for-our-food at mealtime with Grandchildren.)

  48. E.M.Smith says:

    Hmmm…. “Adjusting” the 10 yard pendulum with the Mesopotamian vs S.I. second offset (1.98 Mesopotamian Lunar Seconds per 2 S.I. seconds per the linked graph above) makes the error an order of magnetude smaller. It moves the error from the 1/100 second place to the 1/1000 second place in the pendulum (or makes it 6.0069735 seconds so the error from a pure 6 S.I. second pendulum is only 69/10000 second).

    While I suspect this “correction” may not be appropriate for all the various cubits and pendulums, there may in fact be some “mileage” from the notion that a “lunar second” was in use.

    Attempting to find more about this via Google has not been effective ( as “lunar second” pops all sorts of non-time things). The graphic has at least one error (that the S.I. meter is based on a pendulum… that was Proposed but not actually done; but the method used was to roughly reproduce it so I suppose they can be given a bit of leeway…) so I’m looking for confirmation that a “Mesopotamian Lunar Second” existed and that it was 1.98 S.I. seconds long.

    I’ve not been able to puzzle out their description of it either as it seems to give a 1/3600 of something, but that seems way short for any ‘rotation’ of anything. As near as I can tell they are saying it is “one lunar arc second of motion” so that would be of “one degree” if I’ve got it right ( 1/( 60 min x 60 sec) per degree). But trying to find the lunar orbital velocity in arc seconds hasn’t worked all that well either and I’m not feeling like deriving it from other units and inserting more fudge.

    At any rate, if anyone has ever heard of a ‘roughly 2 second’ long “lunar second’ of arc I’d love to have a pointer… or even just a pointer that gives the apparent motion of the moon in seconds of arc and I can back figure it.

    The thesis I’m headed toward is that the moon was used for calendars (VERY well attested all over the planet) with lunar months, Metonic cycles, and both lunar and solar years and their relative offsets all over the place. It would not surprise me at all to find that the sun (solstices) was used for the “planting calendar” and seasons while the moon was used for the “monthly / weekly” and thus the daily schedule. In that case, it would be easy to choose to use it as the ‘fast time standard’ and use it’s passage between 2 x 15 degree henge poles as the timer for your length pendulum. Folks would have been measuring the moon a lot anyway for all those lunar calendar adjustments… (Many calendars, like Hebrew and Arabic, physically watch the moon for the first visible bit of arc of the new moon to set the start of that particular month).

    Given that you have your “Calendar Guy” watching the moon every single month, wouldn’t be all that hard to imagine him passing the time making “standard pendulum ropes”…

    Also of note, I stumbled on a nice “equation of time” calculator on line…

    http://mb-soft.com/public3/equatime.html

    It will calculate to within about 0.01 seconds. They note more effects than I care to think about but also included two of particular interest:

    “And finally, there are various other interesting effects! For example, in 1246 AD, the Earth’s perihelion coincided with the Winter Solstice, and the Equation of Time curve was exactly symmetrical!”

    Which means that in each period of time from xxxx BC on through to now there is a slightly different Equation Of Time result to be applied…

    Also interesting is a ‘short form’ equation they provide. Why? Look at the formula. It has both 360 and 365 in it as a degrees / days computation. We’ve seen those two used together in the “English Foot” comparison to the equatorial distance around the planet. It may well provide clue as to why that fit is so accurate….

    The following sort of defeats the central point of this presentation of extreme accuracy, but it is clear that people may want to have a CRUDE formula to be able to calculate the Equation of Time, rather than having to use this very sophisticated set of equations. The following simple formula can generally give a value that is within about half a minute of being accurate (where the above Calculator determines a value that should be accurate to around 0.01 second).

    E = 9.87 * sin(2*B) – 7.53 * cos(B) – 1.5 * sin(B)

    where

    B = (360/365) * (N – 81)

    where

    N is the day number in the year, with January 1 meaning N = 1.

    The calculator used has to use degrees and not radians for this formula. This gives E, the Equation of Time, APPROXIMATELY, in decimal minutes.

  49. George says:

    “The moon’s motion across the sky can be measured in angular size: approximately 15 degrees every hour, or 15 arc-seconds per second. A one-mile-long line painted on the face of the moon would appear to us to be about one arc-second in length.”

    From http://en.wikipedia.org/wiki/Angular_diameter

  50. E.M.Smith says:

    @George: Yeah, I saw that one too. The problem is that word “approximately”. They used WAY too loose an approximation. 15 degree / hour is the solar rate. The moon has a slightly different rate as it is rotating us once / month as we are rotating the sun at 15 degrees / day.

    So what they are saying is that 2 is approximately equal to 2.000 and that’s not very informative when you are looking for 1.9xx vs 2.000 ….

    But “thanks for trying!”…

    NEXT! ;-)

  51. George says:

    I found another one that says 0.55″/sec

  52. George says:

    Are you looking for absolute motion or apparent motion? Because apparent motion from any particular place on Earth is going to vary even while orbital velocity remains constant.

  53. E.M.Smith says:

    @Adolfo:

    Part of my interest in “All Tech Old” is to recreate that ability of anyone, anywhere, to “bootstrap society” from sticks, stones, sun, moon, stars, and string…

    So one of the major intended uses of my Henge is just to do my own planting calendar. Perhaps even with dedicated little poles at the “plant tomatoes when the moon rises here” or “harvest corn when the sun sets there” points… Oh, and a sundial so I know when to have a proper cup of tea ;-)

    FWIW, it’s something that interestes me in the extreme: How could I; in a time of complete economic and social collapse, restart the infrastucture? I think I first got that “bug” back when learning about The Dark Ages and how it took several hundred years… I kept thinking “How STUPID. It ought to only take a year or two…”

    The exploration of “how to do it with ever less material goods surviving” leads to henges, sundials, monumental architecture with encoded directions and information… Oddly, a lot of the stuff we find scattered around the world. (Cue spooky music ;-)

    FWIW, one of my “Gee, that would be good to do” hypothetical projects is a “Laser Printer” that uses an industrial grade laser to cut information into stone like granite and diorite. Then you can put things like, oh, a calculus primer onto a large stone used in your Henge for your planting calendar. The farmers keep the Henge working (not wanting to lose the calendar feature) so the stone doesn’t get used to build a fort somewhere… then you wait for the particularly bright “calendar guy” to notice how calculus works….

    Had I a few million spare dollars, I’d be making a very large Henge somewhere with each stone “laser carved” with specific critical guides to technical understanding. Math. Physics. Chemistry. Astronomy. Weather. Geography. Construction tools (level, square), Archemides pump, Etc.

    I figure I could get about 4000 years of life etched into a proper set of stones. More if put somewhere like Arizona or Chile… but then you don’t have as many subsistence farmers to support the Henge…

    So yeah, every so often I stop grubbing after dollars and try to move that particular vision a foot further down the field…

    We have so much that we know, and the central part of it would not take that much space to present. It can take a thousand years to recreate the calculus, or just one to put it’s derivation on a large standing stone… for generations to come.

    Sigh. I need a “Druid Support Society” with grants given for visionaries with a Large Stone Project Vision ;-)

    So while I’m “fooling around” looking after cubits and feet and how they relate, I’m also thinking about how easy it would be to put a pictogram of a lunar rising over a 15 degree arc (with a circle divided into 1/4 and 1/8 and… to 360 for illustration of ‘the degree’) and a pendulum being made longer or shorter (with the one in the middle showing the exact count of 360 … then ) showing it divided into 1/6 parts to make the unit of measure, that would be engraved right below it. Both the standard and the method on one stone.

    Ah well, we have our volatile computer storage and our decaying books and our flood of trivia to hide the bits that are really important… So next time ‘around the wheel’ some archeologist can ponder the “Godess J-Lo” and why so many shiny disks were packaged with her name on them… and just how did we make those big long concrete elevated roadways that must have been used for Religious Processionals… and how did we get so many things to all come out the same sizes when we didn’t even have a physical length standard…

    (Length now being a unit of light, and the platinum bar in Paris unlikely to end up in a “dig” if the world really does collapse…)

    Yeah, a silly thing to ponder. How to set up for a ‘restart’ if all you have are sticks and stones. But it’s a very interesting problem. (Though I hope to never need the solutions…) And your point about why Universities don’t do anything even remotely like this is a rather good one…

    @P.G.Sharrow:

    Yes, “real farmers” do know when to plant. But in todays world, we have a lot of single crop farmers. Now make that a large area of families with gardens and needing to run them year round with a few dozen crops. Pretty quick you find your ‘real farmer’ looking at his calendar…

    So I don’t see a world where Priests are needed to tell me when to plant corn. But I do see one where a “priest” (aka the Calendar Guy) lets everyone know that their personal calendaring is going to be off by a couple of days and they ought to drift it over a few. (As regularly happens all through time and lunar and solar changes happen). And that is roughly what we see in the historical record. The sporadic “leap years” and shifts of holidays when they get too far from the actual seasons.

    Frankly, I’d expect that most stone age farmers would be able to set up their own “wood henge” of rough form pretty easy and only “check in” with the calendar guy for calibration if they wanted it. One fence pole here, one there, plant when the sun rises over that one, harvest toward the other one when ripe… BUT, if you are going to move a cart of goods 50 miles to The Party you would probably want to know exactly what days it was going to happen… showing up a day late would be a real bummer…

    @Pyromancer76:

    I’m just an information sponge. If I’m near it, it gets soaked up… then integrated. Learned a lot in S. Baptist meetings, then pretty quick was mostly learning by watching the people at the meetings rather than what the (repiticious!) guy at the front was saying. Yeah “It’s something!” all right ;-)

    Glad you like the discussion. BTW, I’ve got an interesting book you might like too:

    “Technology in the Ancient World” by Henry Hodges

    http://www.amazon.com/Technology-Ancient-World-Henry-Hodges/dp/0880298936

    Basicly an inventory of interesting old gadgets and technologies. Not hard core stuff like how to make Egyptian Blue, but more things like “how to cast a spear point” and “how Egyptians made boats”. Nice things to have as pictograms on a stone works ;-)

    Has some decent storytelling around it too.

    There may well be better books, but this was the one on the shelf when I walked by… and I’m happy I bought it.

    (Though I’d love to have a “How To” of All Things Egytian and All Things Roman including the chemistry… would have saved a thousand years or so re-inventing cement… so if anyone knows of such a title…)

  54. E.M.Smith says:

    Absolute. Then I can adjust it for elevation if I feel the need.

    My ‘working thesis’ is that folks were not that far from the equator when this stuff was first worked out (due to large Ice Sheet up north ;-) so I’m not worried about things more than about Mesopotamian elevation North…

    That 55″ is probably “good enough” as that makes it 60/55 for the correction factor. That gives 1.090909091 which is darned not as close as it looks at first blush to the 1.01010101 I got from the number in the Mesopotamian Units picture / poster. I didn’t notice the ‘off by one’ digit place at first glance as the leading ‘1’ of the repitant 10 is in the same place as the repitant 90… In fact, one is a correction of about 1/100 while the other is a correction of ‘just under 1/10’. Very different…

    I’ll re-run some lengths with that “lunar clock” and see just how close things get (but they might be closer to ideal…).

    At any rate, a ponter to the source of the 55″ and/or an absolute number too would be “nice to have”; but I feel pretty good about it with just another pair of eyes having taken a look. (Basically, I’m not just trusting some random wiki-poster thing…)

    But I’ve just “got a feeling” that this is the right path. There is something that happens when things start to line up. The history of lunar calendars. The tie to Mesopoamian systems. The consistent error offsets with S.I. seconds that reduce with Lunar Seconds. The “extra” factor of 2 in the divisions. All hinting at the same thing: They used a Lunar Second Of Arc, not a Solar Second of Time. And those are just a tiny bit different…

    the factor of 10 difference between the “lunar arc second” in the poster an the lunar arc transit rate leaves me back at wondering from where the poster got it’s Mosopotamian Second…

    At any rate, the numbers will tell the tail. But that will be after errands and lunch and the bank and bills and seeing just what does happen in Egypt now that the SHTF moment has arrived (but that’s in a different thread…)

  55. E.M.Smith says:

    I’ve edited the prior comment to add some bits in italics.

    It’s looking more like the Mesopotamian Lunar Second is different from the lunar arc motion, or that the poster is just “off” somehow with the 1.98 S.I. Second Mesopotamian Second…

    So I’m back to trying to figure out how you get a 1.97x second “Seconds pendulum”…

    Maybe it’s just that once you are out in the 1/100 place it’s subject to changes in the particular latitude, object tracked, and The Equation Of Time…

  56. E.M.Smith says:

    Hmmm… From that “equation of time” page:

    The calculated times of the beginnings of the seasons, corrected for Nutation and other effects, should be accurate to within a few seconds for most historical times. There are a LOT of effects which can cause the calculation of the apparent longitude of the Sun to be affected, and even one arc-second difference in that longitude results in a time difference of around 24 seconds! The Julian Day Number is given such that a more accurate value could be expressed if needed. HOWEVER, the calculated results here, involving the position of the Earth in its orbit regarding the start of seasons, have actually calculated the precise location of the BARYCENTER of the Earth-Moon system and not the actual location of the center of the Earth (or of any specific location on the surface of the Earth). For this reason, the precision of the timing of season beginnings, that is the Solstices and Equinoxes, can be affected by this factor. Therefore, depending on the Phase of the Moon (in other words, the relative positions of the Earth and Moon to our Barycenter), the values given above for the Season starts can be slightly off. Due to this effect of the Earth-Moon system, the (center of the) Earth can be as much as 2900 miles in front of or behind the barycenter. And then, depending on what time of day, a person on the surface of the Earth can be another 3950 miles ahead of or behind the Center of the Earth. At an Earth average orbital velocity of around 18.5 mi/sec, this can cause the Center of the Earth to have a variation of more than two minutes! MOST of the greater components of this effect are included in these calculations, but the Moon has a rather elliptic orbit which is also continuously having its perigee moving along! Some simple additional corrections regarding the Moon Phase and perigee changes should be pretty easy to add if better accuracy is required here. And if severe accuracy is needed, the VSOP87 program has several thousand mathematical components in that correction.

    So maybe being accurate to the 1/100 place is as good as it gets without a computer to sort out all the orbital mechanics details….

    I think I’m going to “let go” of worring about the 1/100 place for just a little while and let some other “project” float to the top while I let this one “steep” a bit more…

  57. E.M.Smith says:

    Serioso

    Sorry. EMS. You seem to have missed my query re Napolean, the meter, the second, and the circumference of the earth. What say you?

    Um, I thought I answered that in:

    https://chiefio.wordpress.com/2011/01/30/unifying-the-cubits-the-yard-and-the-rod/#comment-12321

    They had intended to use a pendulum, but due to concerns over the (then newly discovered) varations in local gravity went with something they thought would be more stable (not knowing about things like continental drift and spin rate changes…) and while it’s not stated, I suspect the exact arc to use and what divisions to use was chosen to make it what they thought would end up being a standard seconds pendulum… but botched it in the fine details…

    So it’s a very close pendulum, but not exactly right…

  58. Slacko says:

    @EMS
    Mesopotamian vs S.I. second offset (1.98 Mesopotamian Lunar Seconds per 2 S.I. seconds

    Hang about! Does that have any correlation to the 198 inches in a rod?

  59. E.M.Smith says:

    @Slacko:

    Hmm… I’m tempted to just say “No”… but I’ve got to admit that I really don’t know. Maybe there is an odd connection somewhere? One calculated one way and the other another yet they end up with the relationship leaking through? I’ve run into that in a few other things now (like Roman Cubits being one half yard…)

    Maybe once I find out exactly how to make a Mesopotamian Second I’ll be able to figure out the relationship…

  60. Henry Galt says:

    Ed,

    Sorry I have not had time to read all the post, or all the comments, but I shall return !!

    One of my good friends wrote a (I should say “the” but he is a modest fellow) book on the subject.

    http://secretacademy.com

    His contact is on there and I am certain you would hit it off.

    The site used to contain the meat in a more easily digestible form but as the byline says “We can’t give it away.”

  61. E.M.Smith says:

    I’ve put a brief description of an ancient Bolivian unit of measure that also looks like a pendulum division here:

    https://chiefio.wordpress.com/2009/06/06/chasing-the-greek-foot/#comment-26695

  62. Sera says:

    just a test- wtf is wrong with WordPress?

  63. E.M.Smith says:

    There are a lot of theories…

    The one I like the most is that we’ve had a Cosmic Ray Spike and various gear is finding out how sensitive it is… (memory cells are now small enough that a cosmic ray can cause a bit flip).

    Most likely is that that W.P. is “improving” things again (several folks have had ‘log in to use’ burps – including me! )

  64. E.M.Smith says:

    Oh, and it’s aways possible that you changed your login or IP number (like if you did a router reboot and it DHCPed a new IP number) and just went back to the moderation queue automagically until I can ‘OK’ it for the ‘new you’…

  65. Sera says:

    Apparently, WP changed my ‘handle’ from Sera to my full name (james glendinning) and I was having trouble logging in. I’ve never logged in before to post a comment (now it is mandatory, for some reason).

    Anywho… I took the telescope outside (lots of targets tonight), and while the mirror was cooling down, I decided to try the ‘sticks and strings’ method to see if a star really moved 15º in one hour. Actually, I knew it would, I just wanted to see it for myself. And I had some time to kill. So I am almost finished setting everything up when I realise “Crap, I could have used the telescope to do this!”. So I pointed her due south, slewed 7.5º east, and raised her up until I found a nearby star. After centering the star in the crosshairs, I then slewed exactly (via the computer) 15º west and went back inside for some black bean soup. In about one hour the star moved right into the middle of the crosshairs. So, confirmed that. And then it started raining.

    Since I am in one of those talkative moods, I thought you might want to know that ‘sera’ is actually Italian for evening/nighttime/vespers. It’s an astronomy thing, and I work at night. Thanks for keeping me company.

Comments are closed.