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Ancient Measurement Systems: Their fractional integration
John Neal

From the beginning of the 20th century the study of ancient metrology has faded into the background of academic research. Before this, it was a topic of lively debate among the scientific and archaeological communities. It was considered important to clearly define the ancient modules in order to interpret architectural intentions in the ancient monuments, understand itinerary distances, the statements of the classical writers and even biblical descriptions that abound with reference to measures.

Interest in the subject was briefly revived during the 1960s by the claim of Alexander Thom who asserted that the builders of the Megalithic structures had consistently used a common unit of measure throughout the British Isles and Brittany. The claim, to this day, has been neither confirmed nor disproved. This is entirely due to the fact that there is a prevailing ignorance of the subject of ancient metrology. The statisticians who number-crunched the Megalithic data, Thom, Broadbent, Kendall, Freeman and dozens of highly qualified authors, simply failed to recognise the modules that their analyses produced. None of them had made a detailed study of ancient systems and methods of mensuration.

At first glance, the subject seems formidable, causing one learned academic to exclaim that "[Ancient] metrology is not a science, it is a nightmare." It may seem so, there were a great many modules that were commonly used, various spans, feet, digit multiples such as pygme, remen, cubit and feet multiples -- step, yard, pace, fathom, pertica and various bracchia; intermediate measures of the various furlongs and stadia and the itineraries of miles, leagues, schoenus etc. However, the approach to the subject may be simplified by simply considering the basic measure of each system, which is invariably the foot. Augustus De Morgan gave a broad hint at this method of assessment in 1847, he stated:

There runs through all these national systems a certain resemblance in the measures of length; and, if a bundle of rods were made of foot rules, one from every nation, ancient and modern, there would not be a very unreasonable difference in the lengths of the sticks.

It was simply by comparing the foot lengths of the different national systems that a very elegant order was perceived, leading to the conclusion that all of these "national systems" formed a single organisation. It has become customary for us to name a unit from the society in which it is found to have been used, but most often, the bureaucracies have adopted particular units during the historical period. Obviously a universal system had been fragmented into these various cultures. The difficulties of research are compounded by the lack of agreement as to nomenclature, for example, what are universally known as Roman feet are often called Attic, and at one of its variations, Pelasgo.

Which brings us to the most confusing of all the aspects of metrology -- the variations. In all ancient societies, there is a quite broad variation throughout the modules, which has been wrongly regarded as slackness in the maintenance of standards. It would seem that the range of variations and the fractions by which they vary, are not merely similar from nation to nation, but identical. Once these fractions == and the simple mathematical reasons for them == had been established, it became possible to then classify these dissimilarities of the same module. The feet of the various national standards could then be compared at their correct relationships. By seeing the fractional integration through the basic foot structure, many modules could be discarded for comparative purposes, until very few Root feet remained. In fact, there are probably only twelve distinct feet from which all other "feet" are extrapolated. For example the Pythic foot is a half Saxon cubit, and many modules attributed to different cultures are in fact variations of the same basic foot, such as Saxon and Sumerian, or pied de roi and Persian. These feet in ascending order, in terms of the English foot are as follows:

Assyrian
.9ft -- When cubits achieve a length of 1.8ft such as the Assyrian cubit they are divisible by two, instead of the 1 ft division normally associated with the cubit length. Variations of this measure are distinctively known as Oscan, Italic and Mycenaean measure.
Iberian
.9142857ft -- This is the foot of 1/3rd of the Spanish vara, which survived as the standard of Spain from prehistory to the present.
Roman
.96ft -- Most who are interested in metrology would consider this value to be too short as a definition of the Roman foot, but examples survive as rulers very accurately at this length.
Common Egyptian
.979592ft -- One of the better-known measures, being six sevenths of the royal Egyptian foot.
English/Greek
1ft -- The English foot is one of the variations of what are accepted as Greek measure, variously called Olympian or Geographic.
Common Greek
1.028571ft -- This was a very widely used module recorded throughout Europe, it survived in England at least until the reforms of Edward I in 1305. It is also the half sacred Jewish cubit upon which Newton pondered and Berriman referred to as cubit A.
Persian
1.05ft -- Half the Persian cubit of Darius the Great. Reported in its variations throughout the Middle East, North Africa and Europe, survived as the Hashimi foot of the Arabian league and the pied de roi of the Franks.
Belgic
1.071428ft -- Develops into the Drusian foot or foot of the Tungri. Detectable in many Megalithic monuments.
Sumerian
1.097142ft -- Perhaps the most widely dispersed module of all, recorded throughout Europe, Asia and North Africa, commonly known as the Saxon or Northern foot.
Yard and full hand
1.111111ft -- This is the foot of the 40 inch yard widely used in mediaeval England until suppressed by statute in 1439. It is the basis of Punic measure and variables are recorded in Greek statuary from Asia Minor.
Royal Egyptian
1.142857ft -- The most discussed and scrutinised historical measurement. Examples of the above length are plentiful.
Russian
1.166666ft -- One half of the Russian arshin, one sixth of the sadzhen. Variants at one and one half of these feet as a cubit would be the Arabic black cubit, also the Egyptian cubit of the Nilometer.

Variants and variables in the above descriptions are in no wise arbitrary regional fluctuations but follow a distinct discipline. The extent of the variations covers a range of values that amounts to about one fortieth part. Immediately one can see one of the prime difficulties in the identification of ancient modules, because some of the distinct foot values are related by lesser fractions; the Roman is 48 to 49 of the common Egyptian and the common Egyptian is 49 to 50 of the Greek/English. They therefore overlap at certain of their variations, in the course of comparisons this often results in the lesser variation of a distinct measure -- that is essentially longer than the measure of comparison -- to be shorter in length than the greater variations of the lesser measure. Metrologists continually confuse the Belgic, Frankish and Saxon/Sumerian, the latter has also been appended Ptolemaic. But, the differences become distinctively identifiable at the lengths of the pertica, chain, furlong, stadium, mile etc.

It would appear from most of the empirical evidence that the full range of the variations in a single module, here given in terms of the variations of the Greek-English foot, (the English foot being one of the series of the Greek foot) are as follows:

Least
.98867
Reciprocal
.994318
Root
1
Standard
Canonical
1.0057143
Geographic
1.0114612
.990916 .996578 1.002272 1.008 1.01376

The above terminology is used as descriptive in the classification of the values. It was realised from the beginning that all of these variations were impossible to express in an ascending order. They must be tabulated in two rows, the fraction linking each of the variations across the rows is 175:176, and each of the values in the top row is linked to the value directly below as 440:441. "Root" prefixes the descriptive terminology from Least to Geographic in the top row and "Standard" in the bottom row. For example, 1.008 is Standard Canonical and 1.0114612 is Root Geographic etc.

As well as these values being measurements, they are also regarded as the formulae by which any other module is classified. That is, any of the listed feet could occupy the Root position in the above table, and all of its variants would be subject to the multiplications of the tabulated values. As an example, the Persian foot when subjected to this process:

Least
1.038102
Reciprocal
1.044034
Root
1.05
Standard
Canonical
1.056
Geographic
1.062034
1.040461 1.046406 1.052386 1.0584 1.064448

Thus, whichever of the measures shows a direct fractional link to the English foot, such as the one and one twentieth, as above, is Root, then the maximum value of 1.064448ft is both the Hashimi foot and the original pied de roi, both could be classified as a Standard Geographic Persian foot (1.05 x 1.01376). Or the given length of the Mycenaen foot at .910315ft could be classified as a Root Geographic Assyrian foot (.9 x 1.0114612ft) and so forth. Then, whenever one is making cultural comparisons of modules, the correct classification must be selected, Root Reciprocal to Root Reciprocal etc. otherwise one is looking at a compound fraction, i.e. the fraction separating the distinctive foot plus the fraction of the variation(s), which may then show no apparent rational relationship.

These fractional separations of the rows and columns have a practical purpose; they are designed to maintain integers in circular structures and artefacts such as storage and measuring vessels. If a diameter is multiple of four or a decimal, by using 22/7 as pi this results in a fractured number perimeter. Therefore 3.125 or 25/8 would be used to give an integer or rational fraction for the perimeter. Accuracy is maintained by using the longer version -- by the 176th part -- of the measure in the perimeter; this is because 25/8 differs from 22/7 as 175 to 176. Similarly, the fraction 441 to 440 maintains integrity of number in diameter and perimeter, but of different modules. If one has a canonical perimeter number such as 360 English feet, then the diameter will be exactly 100 royal Egyptian feet, but, the royal Egyptian foot that is directly related by a ratio 8 to 7 of the English at 1.142857ft (Root), is supplanted in the diameter by the foot that is the 440th part longer at 1.145454ft (Standard). Another example is to use as a diameter 100 Standard common Greek feet, then the perimeter is 360 Assyrian feet but of the Root classification -- the 440th part less. This is clearly indicative of the integrated nature of the original system, the purpose of which was the maintenance of integers in what would mostly be fractured numbers were a single standard measure used, which is what we have today. Ancient metrology is very simply based upon how number itself behaves.

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