Since these metricians derived their weight standards by variously dividing a carefully measured cubic foot of water, and since we have reasonable data on these ancient weights, we can calculate from such fossil artifacts the standard length of the pendulum used in ancient times. Livio Stecchini determined that the standard Roman libra was 324 grams and that there were 80 of them to the Roman talent comprising a cubic Roman foot, giving us a Roman talent of 25,920 grams. If we make the assumption—to be justified later—that the Roman foot is 6/5ths of the sidereal-second pendulum we can calculate a standard pendulum length equal to the cube root of 15,000 cc.
Stecchini did all of his work on Roman weights in the metric system, thus we can suspect that rounding error played a part, however, we can also make a similar derivation from the venerable English avoirdupois ounce which is not dependent upon the meter. The avoirdupois ounce (modem value) is 28.34952313 grams, the avoirdupois pound is 453.5923701 grams. The conversion between pound and libra was always via an English pound of 7000 grains and a Roman libra of 5000 grains. Therefore the Roman talent derived from English units is 80 x 323.9945501 grams = 25,919.56401 grams, giving the standard pendulum length as the cube root of 14,999.75 cc.
The cube root of 15,000 cc gives a pendulum of 24.6621 cm or 0.246621 meters.
Can we place a pendulum of this length and period at any reasonable ancient site? Keith R Johnson has made a very plausible case for the niche in the Queen’s Room of the Great Pyramid as housing for one such pendulum. Assuming that the Great Pyramid is on the 30th parallel and located 100 meters above sea level, at a gravity of 9.7930417 m/sec2, we get the following values for the length of the pendulum that beats exactly 86,400 times in one sidereal day:
L = 0.246708 m, for a swing angle of 0 degrees
L = 0.246621 m, for a swing angle of 3.0374 degrees
L = 0.246482 m, for a swing angle of 4.9 degrees
L = 0.246370 m, for a swing angle of 6 degrees
(Do not make the mistake of assuming that gravity measured in meters per second squared is the same as gravity measured in meters per sidereal-second squared.)
Other sites should be investigated. Persian measures are a tad short and suggest that their calibration was done in the mountains around Persepolis, at an altitude of 1890 meters.
Back Derivation of the Standard Foot
The Roman talent of 25,920 grams was divided into 80 libras of 324 grams, the libra broken into 12 unciae of 27 grams, the uncia broken into 3 shekels of 9 grams. For fine work in precious metals the libra was shaved into 5000 grains.
More conventionally, the sides of this cubic talent of water could be divided integrally to give a variety of weights:
(1) 12 by 12 by 12 = 1728 cubes of 15 grams
(2) 6 by 6 by 6 = 216 cubes of 120 grams
(3) 4 by 4 by 4 = 64 cubes of 405 grams*
(4) 3 by 3 by 3 = 27 cubes of 960 grams
(5) 2 by 2 by 2 = 8 cubes of 3240 grams, ten libra
*This weight, called the mina of the Heraion, is equal to a water filled cube whose sides are three tenths the length of a one sidereal-second pendulum (three Roman inches), or is equal to the wheat filled standard pint ration of 486 cc. It occurs repeatedly in ancient measures—64 in the Roman talent, 70 in the English talent where it is 404.9931876 grams, 72 in the Egyptian artaba of 29,160 grams (1000 ounces Tower), et cetera.
These kinds of divisions were convenient for a civilization that computed in fractions with an arithmetic so clumsy that they had been forced to invent a calculation-free geometry just to get their surveying and geography done in reasonable time. Note that 1000 cubes of 15 grams pile up to make a cube whose side is the one sidereal-second pendulum length at the Great Pyramid;
Thus the Roman foot can be calculated at 6/5ths of the standard one sidereal-second pendulum.
The Calibration Latitude and the Reference Latitude
It is known from many documents that the ancient navigators calculated 75 Roman miles to a degree of latitude. Since there are 5000 Roman feet in a mile and 360 degrees in a circumference, there are 135 million Roman feet in the circumference of the Earth*, and 375,000 Roman feet in a degree, 6250 in a minute of arc. Multiply these numbers by 6/5 and we get 162 million sidereal-second lengths in the Earth’s circumference, 450,000 sidereal-second lengths to the degree, and 7500 sidereal-second lengths to the minute of arc.
What kind of numbers does this give us? Multiplying the pendulum length of 0.246621 meters by 450,000 gives us 110,979.5 meters, which is the number of meters to a degree in the notorious Stecchini latitude at 37°36\ which Stecchini claimed was the reference latitude by which the ancients measured the circumference of the Earth. (There are 110,572 meters in a degree of latitude at the equator, 111,697 meters per degree of latitude at the pole, and 111 ,322 meters per degree of longitude at the equator.) Stecchini, himself, did not believe that the ancients used pendulums, extracting his numbers from old weights and measures, ancient building surveys, and by reading widely among old economic documents, as well as by reading Aristotle, Herodotus, Sumerian inscriptions, Egyptian tomb walls, et cetera.
*Earth: one of the ancient names that the early Solurthians gave to the planet Rith.
Note here that the number 1.62 is an approximation to the golden section, 1.61803398... If we were to send out surveyors to measure the length of the degree at 30 degrees of latitude in terms of a standard foot of 0.246621 meters we would get, not 450,000 sidereal-second lengths per degree but 449,454 lengths, which, when multiplied by 360, gives us 161,803,400 such lengths for the circumference (of a sphere tangent to the oblate spheroid) of the Earth at the Great Pyramid. This might be coincidence. Or it might be that the Egyptian astronomer-priests chose the very strange number, 86,400 seconds, to make the circumference divisible by the golden section. The Fibonacci numbers, which Fibonacci learned while his father worked as a diplomat in the Middle East, were known in antiquity and provide a very easy tool for computing the golden section if you are limited to an arithmetic heavily dependent upon fractions.
Two critical latitudes emerge once the sidereal clock is set at 86,400 seconds and the circumference at 162 million pendulum lengths.
The Calibration Latitude: To define the length of the pendulum whose period is one sidereal-second we must specify at which latitude our apparatus functions and at what altitude (usually ground level)—while at the same time being careful to confine the pendulum to a predetermined swing angle. At the Great Pyramid the sidereal-second foot cannot be greater than 0.2467 meters, corresponding to a swing angle of zero. At a swing angle of 5 degrees its length would decrease to 0.24647 meters. The evidence is good that in the later period it was calibrated close to 0.246621 meters using a swing angle of 3.04 degrees. The Egyptian Royal cubit of the King’s Chamber of the Great Pyramid, as measured by Petrie, would have been calculated at a swing angle of 4.9 degrees as the 7/6th part of a pendulum with a sidereal cycle of 64,000.
The Reference Latitude: Once calibrated, the sidereal-second foot defines a mythical reference latitude at which a sphere with a circumference of 162 million feet will be tangent to the inside surface of our flattened planet.
Example 1: For a calibration latitude of 30 degrees, altitude 100 meters, and a 3.04 degree swing angle, we get a foot of 0.246621 meters and a reference latitude of 37°36'
Example 2: Other regimes are possible. For instance, a pendulum set up just north of the Black Sea at a calibration latitude of 45°24' altitude of 100 meters, with a normal swing angle of 3.04 degrees will have a length of 0.246956 meters and a reference latitude of 45°24'—this being the one latitude at which the calibration and reference latitudes are identical. Stecchini, without invoking pendulums, claims that there is evidence of a very early Egyptian survey team that worked at this latitude from the mouth of the Danube, cutting across the Crimea, and ending at the foot of the Caucasus.