User talk:Gurdjieff/Sandbox

To Gurdjieff:

I have a long interest in Ancient Metrology and found your "sandbox" while I was doing some research. Your classification of Sumerian time caught my attention because I have been exploring the possibility that the ancients used pendulums to establish their length measures. They don't seem to have left any written description of such a method, but they have left a record of lengths which all have swing periods when used as pendulums.

The example you give in your diagram, which is essentially Stecchini's 499.4mm cubit, is an easy pendulum to construct. It is not, as Lehmann-Haupt supposed, half of a two SOLAR-second pendulum -- the one second "half-swing" of the grandfather clock. The two SIDEREAL-second pendulum, however, is EXACTLY two of Stecchini's Arabic cubits in length at Mediterranean latitudes. The Sumerian length of 499.4mm, to which Stecchini refers, is a length exactly (by pendulum theory) 81/80 times larger than Stecchini's Arabic cubit. The 499.5mm value seems to be constructed as the tenth part of a pendulum which swings with a frequency of 19,200 cycles per sidereal day -- ie with a period of 9/2 sidereal-seconds -- which, by the way, is almost exactly 5 meters in length at the location of Corfu which is at the 3/7 sacred latitude, 38.57deg, where, incidently g = 9.800(in meters/(solar-second)^2).

[We have to reduce this value -- by 365.2421/366.2421 squared -- to get g in meters/(sidereal-sec)^2] to use in any sidereal calculations involving gravity.

To make a rough calculation of the length of the 4.5 sidereal-second pendulum at the location where g=9.800 (and with zero swing angle, and zero elevation) I use the pendulum formula:

[9.8/1.005483316]*[4.5/4pi]^2. Using this value and dividing by ten we get 0.4999384m. At lower latitudes -- or higher elevations, or greater swing angles -- the value will, of course, be smaller. For instance the Royal cubit and the Sumerian cubit are related (by pendulum theory) as 21 Sumerian cubits / 20 Royal cubits so that there must be 440*21/20 = 462 Sumerian cubits in the base of the Great Pyramid. 230.364m (Cole report) divided by 462 = 0.49862m which is the length of the Sumerian cubit at Giza (30deg lat), elevation 100m, derived from a Giza pendulum with a swing angle of seven degrees.

There are many examples of ancient measures which have plausible pendulum definitions. The Roman cubit is the fifth part of a three sidereal second pendulum. The remen is the 6th part of the same pendulum, and the Oscan foot is the eighth part. (Stecchini's Uzbec cubit, being two Oscan feet is, of course, the 4th part. Note that the Uzbec cubit is 5/4 of the Roman cubit.)

Two megalithic yards has a sidereal frequency of 33,333.33. The Megalithic foot is the sixth part of this pendulum. Two-thirds of this pendulum is the length of the Nippur Bar which Stecchini noted was divided into 64 fingers. (I have checked this statistically and it checks out on the nose.) It is also obvious, for statistical reasons, that the bar is about 3-4mm short on the round, damaged end. Stecchini estimates the finger indicated by the markings on the Nippur bar to have length 17.17mm which makes the Nippur foot to be the 10.8inch Assyrian foot (Khorsabad) of Oppert. The best statistics that I've used suggests Stecchini was wrong and that the Nippur finger has a longer length very close to 17.268mm which makes it too small to be an Oscan foot but very close to a megalithic foot divided into 16 fingers. 48 such fingers make a megalithic yard and 96 such fingers swing with a sidereal frequency of 33,333.33 cycles per sidereal day. Which is intriguing because it would mean that the Nippur priests were fooling around with dividing the sidereal day into 100,000 parts which requires different instrumentation than the 86,400 day or the 77,760 "English" day. It is kind of amusing to speculate about the Sumerians using BOTH our 86,400 second day and the 100,000 "seconds" day of the French revolution!

The English foot has a frequency of 77,760 cycles per sid-day at reasonable Mediterranean latitudes. Dividing by 3 we get a frequency of 25,920 cycles which will give us a pendulum of length 9 English feet and incidentally of ten Assyrian feet of 10.8 inches, see Oppert's work at Khorsabad. A frequency of 25,920 cycles is, of course, the Jewish time interval, the heleq of 3.3333 seconds, still in use today. The heleq pendulum is, as I have mentioned, 9 English feet long and ten Assyrian feet long.

The question is how would the ancients have calibrated their pendulums? Which is why your analysis of Sumerian time interested me. How would you calibrate a pendulum so that it would swing exactly 19,200 = 2^8*3*5^2 times in a sidereal day? What you need is a method to measure a celestial angle and to take as your temporal unit the time it takes a point source star to cross that angle.

In one Sumerian sidereal hour, this pendulum must swing exactly 800 times. If we can divides this hour angle into 20 three minute parts, the pendulum must make a full swing exactly 40 times in those three minutes -- which should be easy for a priest to count, and very accurate because a star is a point source and will appear and disappear instantly as it passes across the three minute slot.

The pendulum will then be exactly ten Sumerian cubits long.

My java pendulum simulation (the associated data now woefully out of date) is still a reasonably accurate representation of latitude, elevation, and swing angle variables and is at donaldkingsbury.com and you might want to fool around with it. I'd love to chat with you about Sumerian astronomical instrumentation.

70.48.79.215 (talk) 23:47, 26 March 2009 (UTC)Donbury