The Japanese *shaku* (30.30cm) and the Minoan foot (30.48 cm) are easy to calibrate, although the result on Okinawa may differ from the result on Crete due to the slightly different gravitational acceleration, which varies a little by latitude and altitude.

According to the computational-knowledge engine, WolframAlpha, the gravity at Knossos is 9.79753 meters per solar second, squared (m/s^2) at 57m altitude, and the gravity at Okinawa is 9.78849 m/s^2 at 86m altitude; I’ll use the latter calculation for my pendulum simulation. But first I must calibrate my pendulum.

[Ed. note: When superscript is unavailable, ^ may be used for exponential notation.]

The ancients, as far as I know, were limited in their use of solar time via the sundial, which is not a very accurate clock because (a) the solar second (sol-sec), unlike my manufactured second, varies over the course of a year, and (b) the large sun produces a fuzzy shadow, which makes difficult the computation of precise seconds. Sidereal time is a different matter because (a) the sidereal second (sid-sec) does not vary over the course of a year, and (b) since stars appear as points, the timing of a star as it crosses an angular slot of arc is very accurate.

So, to calibrate a pendulum using sidereal time, I need a long base-line radius with an angular slot of night sky on the circumference of circular arc. I then count the swing cycles of my pendulum as a star crosses my slot. In one swing cycle, the pendulum begins at center (equilibrium), swings to its highest point, returns through equilibrium to the opposite point, and returns to equilibrium. The count gives me the period (P) of my pendulum, which, when squared, is proportional to its length (L). By varying the count, I can vary the length, which, later, can be accurately duplicated by using the same count.

The *shaku* is determined by a nice count of 45000 (2^3 x 3^2 x 5^4) swings per sidereal day (86400 sid-sec).

So, to simulate a pendulum on Okinawa, I must first convert gravity (g) from the given units of m/sol-sec^2 to sidereal gravity in units of m/sid-sec^2. The conversion factor is about (366.25/365.25)^2 or, more accurately, 1.005483315. Thus, I convert the gravity at Okinawa from 9.78849m/sol-sec^2 to 9.73511 m/sid-sec^2.

Also, I am not going to spend a whole sidereal day counting (which is slave labor even for graduate students!), so I will divide the arc of the sky with an angular slot that uses a convenient integer such as 360 (360 arc sec = 1° of sky): 45000/360 = 125 swings as the star passes through 1° of sky. If I count more than 125 swings, my pendulum must be lengthened; if I count less than 125 swings, my pendulum must be shortened.

At both Okinawa and Knossos, P = 86400/45000 = 1.92.

At Okinawa:

g = 9.78849m/sol-sec^2 or 9.73511m/sid-sec^2

Therefore,

L = P2 x (g / 4 x pi^2) = 1.922 x 9.73511 / (4 x 3.14159^2) = 0.9090412m. Dividing by three, I get the Okinawan *shaku* of 0.3030137m. A small swing angle will make it a hair’s breadth shorter.

At Knossos,

g = 9.79753m / sec^2 or 9.74410m / sid-sec^2.

Therefore,

L = P^2 x (g / 4 x pi^2) = 1.92^2 x 9.7441 / (4 x 3.14159^2) = 0.9098807m. Dividing by three, I get a Cretan foot of 0.303294m.

Since the sidereal day is universal to all cultures, these two similar feet may be independent discoveries. However, cultural contact cannot be ruled out. The Minoans, were, after all, a really great sea people before that volcano did them in, and every culture includes crazy adventurers who sometimes accomplish amazing things. Genetic markers might just provide traces of such adventurers!

]]>The one-second pendulum is based upon the one-second interval, or beat, between two passes through the vertical axis. A pendulum must complete this cycle ten times in ten seconds to be considered a one-second pendulum. The length of this pendulum–-which can be created with a metal washer, string or dental floss, and a pencil–-should measure approximately 39 inches, or 994 mm (0.994 meters), from the bottom edge of the pencil to the center of the washer. The beat can be shortened by knotting the string.

The cable is derived by multiplying the length of the one-second pendulum by 360. The Sumerians wrote 360 as 1000. The French did not use a sexagesimal numbering system so wrote 1,000 as 10x10x10. In each case, the length of the cable was the standard, so that each foot equaled 1/1000 of the cable.

Ancient scholars modified, by small increments, the length of the pendulum and the multiplication factor that was used to establish the cable. They used numbers that made sense to astronomers. There may be 360 degrees in a circle, but there are 366 sidereal days in a year. The Minoans used 365.25, which is recognized as the solar year.

Consequently, all cable lengths were approximately 1/360 of a degree or 1/6 of a nautical mile, which is defined as the length of one minute of latitude at 50 degrees north. The WGS84 defines the nautical mile as 1853.82 meters. One sixth of this distance is 308.97 meters, or 1000 Greek feet.

There was considerable variation among the ancient Phoenician, Minoan, Sumerian, and Greek cables, all of which were approximately 1/6 nautical mile. The length of the Phoenician cable equalled 300.00 meters (366 x 819.67 mm), and the length of six Phoenician cables equalled 0.97097 of one nautical mile. The length of the Minoan cable equalled 303.6 meters (366 x 829.51 mm), and the length of six Minoan cables equalled .98262 of one nautical mile. The length of the Minoan pendulum can be found in the megalithic yard, which has been determined to be between 829.0 and 830.2 mm. The length of the Greek cable equalled 308.97 meters (360 x 858.25 mm), and the length of six Greek cables equalled 1.0000 of one nautical mile. The length of the Sumerian cable equalled 357.84 meters (360 x 994 mm), and the length of six Sumerian cables equalled 1.15817 of one nautical mile.

The British cable is a modern measurement that may have been influenced by the furlong (660 British feet) or by the Greek stadia (600 Greek feet). In any case, it is defined as 1/10 of a nautical mile. Since the ancient Sumerian, Minoan, Phoenician, and Greek cables were all approximately 1/6 nautical mile, they are not related to the British cable except in name.

]]>This is great information. I’m just trying to wrap my mind around the historical implications.

However, I have some questions:

Does the one-second pendulum refer to a distance interval (i.e. arc second) or to a time interval? Also, I understand that the British Imperial cable equals 1/10 of a nautical mile. How does it compare to the Minoan and the Sumerian cables?

]]>In Japanese units of measurement, the *Table of Lengths* shows that one *shaku* equals .3030 meters and that one *ri* equals 12,960 *shaku* or 3,926.88 (3,927) meters.

One hundred *shaku*, or 30.30 meters, equal approximately one arc second. There are 360 x 3,600, or 1,296,000, arc seconds in 360 degrees. Therefore, one *ri* equals 129.6 arc seconds or 1/10,000 of the earth’s circumference.

In Li (unit), the value of the *li* during the Western Zhou Dynasty (1045-771 BCE) equalled 358 meters. This value represents 360 units of a one-second pendulum that is exactly 994.44 mm. This latter measurement is nearly identical to the 994 mm found in ancient Sumerian records and can easily be replicated.

In conclusion, the Japanese *ri* makes use of a measurement that is directly related to the Minoan cable and that represents a decimal division of the earth’s circumference, while the Chinese *li* replicates the Sumerian cable.

You stated that

“[m]aybe it’s not so surprising that sea-faring nations a half world apart used the same navigational standards so long ago.”

My main premise is not that the Minoan and the Japanese civilizations are concurrent cultures but that the Japanese culture is an extension of the Minoan culture. While I’m still researching this premise, preliminary evidence suggests that the foundation of the advanced Japanese culture appeared sometime around 1200-1000 BCE, about 200 to 400 years after the “collapse” of the Minoan civilization. Japanese lore suggests, perhaps, two waves of this culture, whether by sea or by land. Evidence to support my premise is forthcoming on an indefinite timetable.

]]>The ancient Sumerians, in the third millennium BCE, created a system of measurements that was based on an imaginary one-second pendulum, which was almost one meter long. Distances were measured in multiples of this standard. One of these distances was a little longer than 1/6 of a nautical mile, and is known as an arc minute, while the next distance was about 30 times the arc minute, or five nautical miles. The accuracy of the Sumerian standard was improved throughout the centuries.

Consequently, the Minoan foot was precisely derived from the length of an imaginary pendulum that swings 366 times in the period that the earth rotates 1/366 of its circumference, as measured on a line of sight with Venus. The length of this imaginary pendulum is doubled and then multiplied by 366 to produce a length which is approximately 1/6 of an arc minute. The Minoan foot is 1/1000 of this distance.

In a similar manner, using the same formula and taking into account the difference in gravity between two locations–the origination and the destination–as well as allowing for the properties of an actual pendulum, the error is well within the measuring ability of ancient civilizations.

Today, the arc second and the arc minute are shown on every navigational chart used on both the sea and in the air. Maybe it’s not so surprising that sea-faring nations a half world apart used the same navigational standards so long ago.

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Roland A. Boucher: Independent Scholar, Engineering

March 14, 2012

This may not seem important to the layman but, to a navigator, it meant that 360 of these basic units was one degree and that 360 degrees was all of the way around the world on a meridian through Japan. Taking a trip on this meridian would take us southward through Australia and Antarctica, then northward though Brazil, the Atlantic Ocean, Canada, Iceland, and the north pole, and then southward through Siberia and back to Japan. The length of this journey taken over 3000 years ago would have been 99 percent of what we know the true distance to be. I would like to thank Gretchen Leonhardt, on her website, “Kanashi”, for bringing this ancient Japanese measurement to the attention of scholars throughout the world.