Presented at the Sigma Xi SOUTH WEST Regional Conference Jan 5 2013

Roland A. Boucher

MS Yale 55, Retired, Orange County CA Chapter Sigma Xi

*Only Five Standards of Length Were Needed to Measure the Ancient World*

• All standards could be reproduced with an accuracy of one mm or less

• All standards were logical variations of the original one second pendulum

• All pendulums were timed from astronomical observation

• These five standards were used to measure most of the Ancient World

All standards were developed through use of a Pendulum

All standards were related to the polar circumference of the Earth

*The Metric System and the Measurement Standards of Sumeria in 3000 BCE*

The original definition for the Meter was the length of a one second pendulum, measured in the Earths gravitational field at 45degrees North Latitude. Its length was 993.7 mm in today’s metric system.

When studying the standards of measurement in Ancient Sumeria it became obvious that they had invented the metric system over 5000 years before the French proposed it. The standard length, the double “Cubit” or “Step”, was 994 mm. Their standardvolume was a cube 1/10 Step on edge (a Sumerian liter). Their standard of weight was this same volume of rain water. For short lengths they divided their “STEP” into 60 parts. Long distances were measured in “Cables” of 360 Sumerian Steps of 994 mmeach which was also divided into 1000 Sumerian feet. The “Step” was defined as the length of a one-second pendulum just as the French were later to propose.

**Details of the Sumerian Measurement Standards of Length**

The Sumerian Double Cubit = 994 mm ( length of pendulum which beat 240 times in 1/360 solar day).

The Sumerian Cable = 360 double Cubits = 357.84 meters which also defined 1000 Sumerian feet.

The Sumerian foot = 357.84 mm. This Sumerian foot spread across the world from France to China

In ancient China circa 1100 BCE the Zhou Dynasty established that the Royal chi = 358.2 mm.

In ancient France the town of Bordeaux established the Pied de Terre at 357.2 mm

*The 994 mm Length of this Mesopotamian Standard can be Accurately Reproduced*

**The Cable and the Foot can be related to the Polar Circumference of the Earth**

The Earth According to WGS 84:

The average minute of latitude (nautical mile) = 1852.216 meters.

The average degree of latitude = 111.13296 km.

The Polar Circumference of the Earth = 40,007.863 km = 21,602.518 US ? nautical miles.

The kingdom of Sumeria was aligned with the Tigris and Euphrates rivers in a north-south orientation. The elevation of the stars in the heavens would not go unnoticed nor would the fact that this elevation changed by one degree for every 310 cable lengths that the observer moved in a north-south direction. 310 Cable lengths = 310,000 Sumerian feet = 110930.4 meters.

**The Division of the Earth’s Circumference into Minutes and Seconds**

The Polar circumference of the Earth = 60 x360 = 21600 minutes (one Geodetic nautical mile).

The Polar circumference of the Earth = 6000 x 21600 = 129,600,000 seconds (100 Geodetic Feet).

Therefore 360 Cables = 360,000 Sumerian feet = 128.8224 km or 1.15917 degrees of latitude

360×360 cables = 129.6 million Sumerian feet = 1.15917 times the Polar Circumference of the Earth.

Also 100 Sumerian feet = 1.159 arc seconds of the Polar Circumference of the Earth.

The Sumerian pendulum was timed by the motion of the Sun. When the Egyptians and others used the motion of stars or theplanet Venus to time their pendulums, they took this opportunity to select an alternate division of the day to produce a standard of length more closely aligned to the division of the polar circumference into degrees, minutes and later into seconds of arc.

*Looking to the Stars — The Egyptian Standards of Length*

The Sun subtends an angle of nearly 1/2 degrees in the sky while the stars are mere pinpoints of light. Although the stars provide a more accurate reference with which to time a standards pendulum, they rotate through the heavens at a faster rate than the Sun, arriving nearly 4 minutes (one Gesh) earlier each night. Accordingly the Egyptians divided the day into 366 parts and timed their pendulum through 366 beats in this time.

Details of Egyptian Measurement Standards

The Egyptian pendulum length = 409.84 mm (The pendulum beat 366 times in 1/366 star day).

The length of the Egyptian Cable = 366 x2 Pendulum lengths = 1000 Egyptian feet = 300 meters.

The Egyptian foot = 300 mm. This standard Egyptian foot spread across the Mediterranean and was established as the Phoenician foot = 300 mm; the Fuss of the Canton of Aargau = 300 mm; the and Reichsfuss of Baden in Germany = 300 mm.

Additional Egyptian Standards of Length

The Reman = 375 mm ( 20 Digits of 18.75 mm each or 20/16 of an Egyptian foot).

The Standard Cubit = 450 mm (24 Digits of 18.75 mm each or 24/16 of an Egyptian foot).

The Royal Cubit = 525* mm (28 Digits of 18.75 mm each or 28/16 of an Egyptian foot)(*28.28 Digits).

The New Standards are a Much Better Fit to the Polar Circumference of the Earth

One Egyptian mile = 5000 Reman = 1875 meters = 1.0123 arc minutes on the Polar Circumference

21,600 Egyptian miles = 40,500 km, just 1.23 percent larger than the Polar Circumference

100 Egyptian feet = 30 meters or 0.97213 arc-seconds on the Polar Circumference of the Earth.

*Looking to Venus — The Minoan Standards of Length*

The Planet Venus is closer to the Sun than the Earth and orbits it in 244 days. By viewing Venus when it is in opposition its motion cancels out some of the apparent motion caused by spinning Earth. The result is a division of the day which is close to1/365.25 day when timed through 1/366 of the Earth’s circumference thus adding approximately one second to the period which would have been provided by a star as a reference.

Details of Minoan Measurement Standards

The Minoan Pendulum length = 414.75 mm ( pendulum which beat 366 times in 1/366 Venus day).

The length of the Minoan Cable = 366 x 2 pendulum lengths = 1000 Minoan feet = 303.6 meters.

The Minoan foot = 303.6 mm This Minoan foot spread to both Europe and Japan.

It was established as theJapanese Shaku = 303.0 mm

The;Stadtschuh = 304.0 mm in the Canton of Basel

and the Fuss = 303.0 mm in Linz Austria.

This New standard was a Better Fit Without the Need to Introduce an Auxiliary Length

Therefore we find that 360 Cables = 360,000 Minoan feet = 109.296 km or 0.98347 degrees of latitude, and 360×360 cables = 1296 x105 Minoan feet = 0.98347 times the Polar Circumference of the Earth. Also 100 Minoan feet = 0.98347 arc seconds of the Polar Circumference of the Earth.

One Minoan mile of 6000 feet = 1821.60 meters or 30.6 meters short of the Geodetic Nautical Mile.

This is an error of only 1.653 percent short was achieved almost 4000 year ago.

*Looking to Venus and the 360 degree circle — The Final Solution*

The final solution came about by combining the time period of the Minoan Venus timing method with a return to the original Sumerian 360 beats. The new cable was made up of 360 x 2 pendulum lengths. We shall call this new standard the ‘Greek“Attic” foot.

**Details of Greek “Attic” Measurement Standard**

The “Attic” Pendulum length = 428.61 mm (Pendulum which beat 360 times in 1/366 Venus day).

The length of the Attic Cable = 360 x 2 Pendulum lengths = 1000 Attic feet = 308.6 meters.

The Greek Attic foot = 308.6 mm.

The most outstanding example of this standard is found in the 100 attic foot width of the Parthenon in Athens Greece. This cable is essentially 1/6 of a nautical mile with the length of 360 cables equal to one degree in latitude.

The Greeks created the Stadia of 600 Attic feet making the length of 600 Stadia equal to one degree in latitude.

The New Standards Were Almost Perfect Fits to the Polar Circumference of the Earth

100 Attic feet = 30.86 meters = 0.999667 arc seconds of the Polar Circumference of the Earth.

Note: The WGS 84 gives the length of one average second of arc at 30.87027 meters.

The 100 foot wide Parthenon was measured at 30.897 meters. an error of 2.67 cm.

*The Roman Empire and the Spread of 24/25 of the Near Perfect Solution*

The Greek “Attic” Stadia was Adopted by the Roman Empire as a 625 Foot Roman Stadia

The Greek Attic Stadia = 185.16 meters = Roman Stadia = 625 new Roman feet.

The Roman foot = 24/25 Greek attic feet = 296.2 mm.

The Roman mile = 5000 feet = 1481.280 meters.

Quedlinburg und Leipzig, G. Basse. 102

75 Roman miles = 375000 feet = 0.999667 degrees in the Polar Circumference of the Earth.

Roman Navigators could determine their position with an error of only 3.3 km when calculating their latitude.

*The Curious Case of the Chinese Market Foot and the British Furlong*

If the original Sumerian pendulum had been timed through 360 beats instead of 240, or if the Greek Attic pendulum had been timed using the Sun rather than the planet Venus, a pendulum length of 441.8 mm and a cable length of 318.08 meters would result.

*The Chinese Market Foot*

This pendulum length, which may not be related to the Earth’s circumference, appears to have been used to created the Chinese Market Foot or Shin Ch’ih (318 mm) which was adopted by the Zhou Dynasty in China.

It also became the standard foot in the cities of Bern and Innsbruck Austria as well as the Doric Foot (322 mm) in Greece, and the Luwain pous (323 mm) in Anatolia.

*The British Furlong, Foot, Mile, and Nautical Mile*

It appears that the Doric Foot may have taken a curious part in the development of the modern British Foot. The length of the Furlong (201.2 meters), an early Anglo-Saxon land measurement, is 625 Doric feet. This Furlong became the standard for land measurement in early England.

In 1592 Queen Elizabeth the First created the British Statute Mile while maintaining the exact

length of the Furlong. She declared the new British Foot to be 1/660 of a Furlong (304.8 mm) and the British Mile to be 8 Furlongs or 5280 feet. Later the British Admiralty would declare the Admiralty Mile (nautical mile) to be 6080 British feet.

References

1 Robson, E 2008. Mathematics in Ancient Iraq, Princeton University Press 2008 Table A.3, 294

2 Margenau, Watson & Montgomery. Physics Principles and Applications, New York McGraw-Hill 1949. 178-180

3 Bronwell, A. Advanced Mathematics in Physics and Engineering, New York McGraw-Hill 1953. 137-139

4 Janhke E, & Emde F. Table of Functions, New York Dover Publications Fourth Edition 1945 Table V. Complete elliptical integrals 85

5 WGS 84 Gravity of Earth http://en.wikipedia.org/wiki/Gravity_of_Earth

6 Earth according to WGS 84 http://home.online.no/~sigurdhu/Grid_1deg.htm

7 Berriman, A.E Historical Metrology, New York ,E.P. Dutton & CO 1953.

8 Katz, Victor J. (editor), Imhausen, Annette et.al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007

9 Petrie, Sir W.M.F. Inductive Metrology London,H. Saunders, 1877 section 21- 39

10 Graham, J. W. The Palaces of Crete Princeton University Press 1962. 224

11 United Nations. Department of Economic and Social Affairs. World Weights and Measures Handbook of statistics series M 21 English Rev 1 62

12 Knight C. & Butler A., Civilization One, Watkins Publishing London 2004. 18,30

13 Kollerstrom, N. Greek Foot The Acropolis Width and Ancient Geodesy http://www.dioi.org/kn/stade.pdf) 2005

14 P. Guilhiermoz Bibliothèque de l’école des chartes De l’équivalence des anciennes mesures. A propos d’une publication récente Volume 74 278

15 Noback, Christian, Friedrich Eduard (1851) (in German). Vollständiges taschenbuch der Münz-, Maass- und Gewichts-Verhältnisse etc. aller Länder und HandelsplätzeComprehensive pocketbook of money, weights and measures for all counties and trading centres]. I. Leipzig: F. А. Вrockhaus. Retrieved October 24, 2011. 101

16 Niemann, Friedrich (1830) Vollständiges Handbuch der Münzen, Masse, und Gewichte aller Länder der Erde fur Kaufleute, Banquiers . in alphabetischer Ordnung.