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P A R T  II
Chapter 3


The Structure of Linear Units


1. As I have reported, the opposition to the old school is so eeffective that the two most important metrological studies published in this century, that of Decourdemanche in 1909 and that of Oxé in 1942, have not only been ignored by ancient scholars, but not even reviewed in their professional journals. These works, written by authors of established reputation as a summation of decades of research, were dismissed in such a total fashion because they further emphasize the views of the old school by demostrating that the different talents are related according to strict and simple mathematical principles.

Oxé showed that cubit foot units, talents, are related so as to contain the same weight whether the filling is water, oil, wheat, or barley. According to his interpretation cubic foot units of weight and volume vary accordingly to the ratio 36:40:50(48):60. and all other units of weight and volume vary accordingly. Oxé implied that the variety of fillings was connected with the variaties of units of length, but did not enter into the problem of lengths. Decourdemanche discussed more specifically the relation between the several talents, determined by the variety of fillings, and the units of length; he stated that talents are related according to the specific gravity of water, oil, wheat, and barley, and that there are as many types of foot as there are talents, since the foot is the edge of a cube containing a talent.

Decourdemanche followed the line of other distinguished Arabists who moved from the study of Arab metrics to that of the ancient ones, since the two cannot be separated, just as one cannot separate the study of medieval metrics from that of the ancient ones. Oxé, as a specialist of Roman archeology, moved from the empirical data to a complete survey of Mesopotamian, Hebre, and Greco-Roman units. Both writers arrived at similar results starting from very different angles. But their most accurate demonstrations were dismissed, because it is the prevailing opinion that the ancients could not have had a system of measures so tightly constructed. Since the close of the last century the leading scholars of ancient history have flatly denied the results of metrological studies of the preceding four centuries, on the assumption that ancient culture was such that one cannot believe, in spite of any apparent evidence, that the ancients had a metric system built on mathematical principles; hence, studies such as those of Decourdemanche and Oxé, that present the ancient system as even more tinghtly knit than it had been assumed to be by earlier researchers, must be ignored. I have shown that if one wants to pinpoint the core of the controversy, the issue is whether akribeia was or was not a leading idea in Greek classical civilization.

It has been difficult to uphold against the rhetoric of the new school, that ancient measures were fixed with mathematical precision, as long as one could not explain mathematically the relation between Roman and Egyptian foot, the two varieties of the basic foot. Since I solved this problem, I have also been able to determine that the structure of ancient metrics is even more inter-connected according to numerical structures than Decourdemanche and Oxé had believed. Ancient measures were organized according to a scheme which, by its ingenious architecture and by its simplicity, could be compared with Mendeleyef’s periodical table of elements. This should give pause to those scholars who, in the name of preconceived notions of ancient culture, have rejected the results achieved by the old metrologists; my interpretation indicates a level of mathematization and of precision (akribeia) far superior to that assumed by any investigator of the old school.

2. If one starts with foot of 300 mm. (natural basic foot), which is divided into 16 fingers of 18.75mm. (natural basic fingers), all feet of the ancient world prove to be units of 15, 16, 17, or 18 basic fingers. Each of the four types of foot exists in a natural and in a trimmed version, related as 3Ã25: 3Ã24, or practically as 74:75 (exactly 73.9864:75) or 72:73 (exactly 72:72.9864). Hence, one can draw the following table, in which the figures in parentheses express the trimmed units calculated as 74/75 of the natural ones.

 
natural
trimmed

Foot of 15 basic fingers

281.250
277.4489
(277.5)

basic foot (16 fingers)

300.00
295.9454
(296.0)

foot of 17 basic fingers

318.75
314.4419
(314.5)

foot of 18 basic fingers

337.50
332.9384
(333.0)


From these units are derived corresponding cubits equal to 11/2 foot; occasionally one finds also cubits of 2 feet. These eight feet are the essential linear units of the ancient metric systems. The most important addition to this scheme is the artabic foot and cubit, which relate to the trimmed basic foot and cubit as 3Ã9:3Ã8, or practically as 25:24. The artabic cubic is considered equal to 25 trimmed basic fingers. Other additioons are the septenary units, of which the best known is the Egyptian royal cubit equal to 7/6 ofnatural basic cubit of 450 mm. I shall discuss the septenary units in a special section, since they are related with the squaring of the circle and the solution of the other irrational roots.

The cubes of the nine types of foot, the eight fundamental ones and the artabic foot, produce cubic units adjusted to the specific gravity of water, wheat and barley, according to the following principle: a cube with an edge of 15 basic fingers filled with water corresponds in weight to a cube of 16 basic fingers filled with wheat and to a cube of 17 basic fingers filled with barley; a cube with an edge of 16 basic fingers filled with water corresponds to a cube of 17 basic fingers filled with wheat and to a cube 18 basic fingers filled with barley. Since the second type of structuring of the units is the most important one, I shall adopt the following terminology:

Foot of 15 basic fingers

lesser foot

basic foot (16 fingers)

basic foot

foot of 17 basic fingers

wheat foot

foot of 18 basic fingers

barley foot

 

The values in mm. of the cubits are the following:

 
trimmed
natural

Lesser cubit

16.173

(416.25)
421.875
basic cubit
443.918
(444.00)
450.0

artabic cubit

461.698
(462.415)
---.----

wheat cubit

471.66
(471.75)
478.25

barley cubit

499.408
(499.5)
506.25


In parentheses are the values obtained by assuming the trimmed units to be 74/75 of the natural ones. In the case of the artabic cubit, the value in parenthesis is that obtained by assuming this unit to be 25/24 of basic cubit.

As I have said, the artaba is 3Ö9/3Ö8 of basic talent netto, and hence the artabic foot is 3Ö9/3Ö8 of trimmed basic foot; but the relation is often calculated as 25:24 which is obtained by constructing a reduced trimmed basic foot of 295.416 mm. (pes Statilianus) or by increasing the artabic foot to 308.276 mm. At times one made the natural wheat foot equal to 25/24 of artabic foot; this special foot of 320.633 mm. was known as Olympic foot, and it has been found that the stadion and other monuments of Olympia were in fact calculated by this unit.

Each of the feet and cubits so obtained is divided into 16 and 24 fingers respectively. The cubit considered standard in Mesopotamia is the barley cubit, composed of 27 basic fingers; in documents employing sexagesimal reckoning this unit is usually divided into 30 fingers but one finds occasionally the usual division into 24 fingers.

Later I shall discuss the relation between units of length and the structure of the units of volume and weight for water, oil, wheat and barley. But first I shall show how the numerical scheme I have formulated agrees with the classification of units of length arrived at by different investigators, using different approaches.

3. Sir Flinders Petrie agrees with the new school that length was not related to volume and weight, but did not accept the new rejection of the validity of Newton’s method. On the basis of this method he developed what he called inductive metrology; he formulated his method in Inductive Metrology (written in 1877), and applied it tirelessly, measuring all sorts of constructions and objects for the following half century. In his final statement of 1934, he classified units of length in four families. His classification is based on Egyptian evidence, but he shows the occurrence of the same units in other parts of the ancient world.

a) His first family corresponds to what I call basic units. He considers the main example of this family the Egyptian royal cubit which he calculates as 524 mm. granting however that it became 525 after the Vth Dynasty; he calculates the corresponding foot as 300 mm. He derives from it the Greco-Roman foot by a trimming of 1/100, instead of 1/75 as I calculate; but grants that the unit which was 297 mm. in Greece passed to Italy as a unit of 296.164, which was gradually reduced in Rome to 294.9.

Petrie states that these units are “the general Mediterranean standard of ancient times.”

He ascribes to the same family a foot of 316 mm. with variations up to 318.754 and down to 314.98; this is the unit I call wheat foot and calculate as 318.75 natural and as 314.44 trimmed. His reason for linking this unit with the basic one, is that a foot of 315 mm. can be conceived as 3/5 of an Egyptian royal cubit of 525 mm. Also Oppert and Hultsh ascribe significance to this relation, but simple numerical relations exist among all the units of length.

b) Petrie calls Northern foot a unit of 338 mm., which corresponds to the natural barley foot that I calculated as 337.50. He calls it Northern foot because it was used in the German provinces of the Roman Empire as pes Drusianus and because it continued to used in north-western Europe in medieval times. Latin authors state that the pes Drusianus was 18 fingers of Roman foot; from these statements Petrie calculates it as about 331.412 mm. but adds that ancient and medieval architectural remains indicate a foot of 332.74 mm., figure that agree with the value 332.94 I calculate for the trimmed wheat foot.

Petrie is still under influence of the notion that particular units of length have local origin; hence he tries to locate this unit in northern Europe, and adds that it was brought into Egypt by foreign invaders; but he contradicts himself by stating that it was used throught the ancient world “ as the most widespread standard we know.”

c) The third family is composed of the Syrian foot of 281.94 mm. He believes that it was spread through the Mediterranean by the Phenicians. This is what I call the natural lesser foot of 281.25 mm.

d) What I consider the trimmed lesser foot of 277.44 mm.,is classified by Petrie as the Eastern foot of 276.8 mm. He considers a particularly significant example of it the rule of 555 mm. cut on the mensa of Flaviopolis is Phrygia (Ushak in Turkey); this rule agrees to perfection with the value I calculate for the trimmed lesser foot. Even though he calls this unit the Eastern foot, he identifies with it the Oscan foot of Italy varying between 278.13 and 275.59 mm.: I have reported that the lesser foot occurs in a doubly trimmed version of 274.73 mm.

On purely geographycal grounds he classifies in this family scattered units that he thought he could not be other-wise fitted into his scheme. He mentioned a cubit of 507 mm. noted by Oppert in constructions of Mesopotamia; his is the natural barley cubit of 506.25 mm. that Petrie should have mentioned in his second family. Petrie reports also that in Mesopotamia Oppert found dimensions of 640 mm; these are units of two natural barley feet of 318.75 mm. The exactitude of my interpretation is indicated by the fact that Petrie considers the most reliable example of these units some mason’s marking on a wall of Abydos in Egypt theory spaced 638.30 mm.

In substance, I agree with the classification suggested by Petrie, expect that the elements of his fourth family can be subsumed under the other three. I would separate the wheat foot from the first family.

4. In this classification Petrie ignores the artabic foot, but mentions it in the similar classification he wrote for the Encyclopaedia Britannica. He mentions a Greek foot of 309 mm. adding that it occurs rather rarely in Egyptian metrology. Under the heading of Eastern foot Petrie mentions that Egyptian cubit rods of the Hellenistic age are longer than the earlier ones; he calculates the royal cubit of these rules as between 536.196 and 537.464. According to Petrie these rules should prove the alleged principle that units of length became longer in the course of time. But the well-known mathematician and philosopher of science Jacques Frederic Saigey (1799-1871) explained that these cubits are septenary units of 28 fingers like the old Egyptian royal cubit, but calculated by the artabic foot, instead of being calculated by the natural basic foot, According to a correct artabic foot of 307.796 mm., these cunits should measure 537.404. The insight of Saigey was confirmed when Greek papyri of Egypt revealed that one distinguished a of 4 *Iepatikov exoiviov a stadia from yewuetpikov exoiviov of *4 1/6 stadia; the lesser schoinos called geometric must be the older one, whereas the geometric one is 1/24 longer, exactly as Saigey had calculated. The relation 24:25 is formulated in two metrological tables contained in papyri; one papyrus (Oxyr. IV 669, 3) belongs to the age of Diocletian and the other (Lond. V, 1718) to the second half of the sixth century, but they reproduce older texts.

Lehmann-Haupt suggests that the increased Egyptian royal cubit was established by Ptolemy I, but it is more likely that it was introduced by the Persian conquest in order to coordinate Egyptian units with those already used in the rest of the Persian Empire. The quadrangular bronze cubit of Torino, which on one face is divided according to the longer cubit, bears an hieroglyphic inscription that may be older than the Hellenistic age.

Böckh noted that there is an Arab unit which goes back to a Babylonian-Egyptian great cubit. He takes as standard of it the cubit of the Nilometer of Rawdah, which was established in the year 97 of the Hegira, in the early period of the Arab occupation of Egypt; the cubit of this Nilometer has been found by the Napoleonic expedition to Egypt to have an average length of 540.7 mm. I have determined from cuneiform tablets that in Mesopotamia, next to the barley cubit divided sexagesimally into 30 fingers, one used occasionally a cubit divided sexagesimally into 30 fingers, one used occasionally a cubit of 32 fingers. The existence of a cubit of this length in Mesopotamia had been inferred by Oppert from the study of the dimensions of buildings. The longer Mesopotamian cubit measures 540.0 mm. natural and 532.701 mm. trimmed. According to Heron and other metrological writers of the Roman period, the Egyptian royal cubit relates to the Roman foot as 9:5; this ratio makes it a unit of 532.701 mm.

Böckh had the correct intuition when he spoke of a Babylonian-Egyptian great cubit. This cubit was probably introduced in Mesopotamia because it related both to the barley cubit, to the natural basic foot (Roman foot) and to the artabic foot; I have presented the hypothesis that in the theoretical calculations of Mesopotamian buildings one used the barley cubit which fits into sexagesimal reckoning, but that in the practical execution of the plans one used artabic feet. When the Persians conquered Egypt they may have used the same unit to coordinate the Egyptian royal cubit with the artabic foot, which was their official standard. From the metrological tables of Greek papyri and from the metrological texts of Heron and Didymos one gathers that the great cubit is particularly important in landsurveying. In the Roman Empire one finds a unit of surface called iugerum castrense (9/8 of normal Roman iugerum) which has a sude of 180 Roman feet of 53.2712 m. This unit is perfectly equal to the Mesopotamian acre, if calculated with a side of 100 cubits of 32 fingers of trimmed Mesopotamian cubit, and practically equal to the Egyptian aroura, if the side of 100 cubits is calculated by the longer royal cubit. One sees that the great cubit allows to coordinate units of different module.

Segrè is of the opinion that the “Nilometric cubit” of documents of the Roman period is composed of 7 artabic hands, which he calls Alexandrine; but observes that the markings of the several Nilometers of the Roman period are irregular and indicate only that the cubit which is the average may be between the older Egyptian royal cubit and the new one. But Segrè does not consider that 532.7 mm. is one of the values of the great Babylonian-Egyptian cubit, when calculated as an increased trimmed barley cubit. The cubit of some Nilometers, as that of Philai, and that of several Hellenistic rods, as those at the Cairo Museum, is about 532 mm. Segrè see some opposition between the explanation of the longer unit suggested by Böckh and that suggested by Saigey: “The black cubit of 540.7 mm., which is at least 15 mm. greater than the Egyptian royal cubit, could have been derived, according to Böckh, from the great Egyptian-Babylonian cubit; but in my opinion most likely it is derived from a cubit of 7 Alexandrine hands.” As I have stated, I do not see a conflict between the two explanations; there was an old Mesopotamian cubit of 32 fingers, which could be either 540.0 or 532.75 mm., that was identified with a cubit of 28 artabic fingers or 537.404 mm. On another occasion Segrè stated: “Böckh identifies the Philetairic-Ptolemaic with with a Persian measure common to the countries under the domination or the influence of Iraq, and in my opinion he is right.”

It has been determined that the Roman reckoning by miles of 5000 feet and 8 stadia goes back to Oriental models. It is an old Egyptian practice to calculate itinerary distances by units 5000 and 10,000 cubits. In the eastern part of the Roman Empire there was used a unit called mivlion or miliarium defined as equal to 5400 Roman feet and to 7½ stadia of 600 hybrid feet or 400 Egyptian royal cubits; in other words it is equal to 3000 increased royal cubits of 532.7 mm. But as I have pointed out this value of the cubit is not the Egyptian one but the one obtained by forming a cubit of 32 Mesopotamian sexagesimal fingers. The very reckoning by 7½ stadia suggests an adaptation from a reckoning by 8 stadia. In fact in a bilingual inscription celebrating the construction of an aqueduct by the Roman administration of Egypt, under the praefectus Aegypti of Julius Aquila in the fortieth year of Augustus (10/11 A. D.), the length of this work is said to be 25 miliaria in the Latin text and 200 stadia in the Greek text. The milion is therefore equal to 4800 trimmed barley feet or 3200 barley cubits. It is obvious that this unit cannot have been of Egyptian origin, but it was adapted to Egyptian units. The calculation as 3000 great cubits indicates the relation 30:32 of Mesopotamia.

According to the formulae of the metrological tables this milion relates to the Roman mile as 26:25; this ratio is correct if one calculates the great cubit by the value based on 32/30 of trimmed barley foot, 532.75 mm., but if one were to calculate by the value based on a cubit of 28 artabic feet or 537.404 mm., the ratio would be 25:24.

The existence of the milion has created endless difficulties for metrologists; Lehmann-Haupt, in his otherwise highly valuable article on “Stadion” in RE, finds himself forced to reject the data of the metrological tables as “bungling.” The problem has been complicated unnecessarily because Mommsen, when he first called to attention the existence of a “provincal” milion different from the Roman mile, identified it with a milion of 6000 feet mentioned in the Berytean Lawbook. This document is a text of Roman Law written under the influence of the school of of Berytos at the beginning of the sixth century A.D.; there remain Oriental translations of a Greek translation of the original Latin. Mommsen assumed that of necessity a text of Roman Law would calculate by Roman feet, but in the same breath he was noting the use of a milion different from the Roman mile.

The text excludes a Roman origin of the unit, since it states that it introduced when God gave intellect and wisdom to man, and markind erected cities, plotted fields, marked boundaries, and measured roads. This way of speaking suggests a Mesopotamian origin, since in Mesopotamia units of length are so connected with the origin of the kosmos. In my opinion this is calculated as 5000 barley feet, which are equal to 6000 lesser feet. The of the Berytean Lawbook text, retranslated into Latin reads: deterunt uiyov mille passus qui faciunt quingentas particas: pertica autem mensura, in qua sunt octo cubitus. The pertica of 12 lesser feet has the advantage of being equal to 10 barley feet; most commonly surveying rods measure ten feet or ten cubits. This milion is 9/8 Roman mile. A unit of 6000 lesser feet has also the great advantage of being easily related to the Persian parasang (18,000 artabic feet,), which is only a trifle shorter than 5550 m. or 20,000 trimmed lesser feet.

The misunderstanding about the three types of mile occurring in the Roman Empire, has created confusion in the study of Arab metrics, which usually provide a reliable starting point for the study of the ancient miles allows to solve a problem that has bedeviled scholars, since the revival of learning in Europe, that of the length of the Arab mile used in the calculation of the length of the degree of meridian performed by order of Calif al-Mamun around 825 A. D.

The great Arabist Carlo Alfonso Nallino dedicated to this problem his first published essay; this essay follows closely the approach of Boeckh and improves upon it, but fails to obtain preciser results because it accepts Mommen’s identification of the two types of milion. Possibly, if Nallino had written his essay after his masterly study of the Berytean Lawbook, he would have perceived this pitfall.

Nallino reports that Arabic rwiters usually calculate the degree as 75 miles, but often also give the figure of 66 2/3 miles; they ascribe both calculations to “the ancients” The value of 75 miles is oviously by Roman miles; as to the value of 66 2/3 miles, Nallino calculates it by the milion of 5400 Roman feet. But it is obvious that the second mile must be a milion of 5000 barley feet of the Berytean Lawbook. (9.8 of Roman mile).

By collating the different traditions, Nallino makes clear that the dimension obtained by the geodesists of al-Mamum is variously reproted as 56, 56½ 56 2/3 and 57 miles. The bulk of the tradition indicates that the datum was obtained by averaging two data obtained by the survey in the field. One would have averaged 56 2/3 miles obtaining 56 ¼, or one would averaged 56 ¼ miles obtaining 56 2/3. But on the issue of which field data were averaged, the tradition is most contradictory. Nillino does not call to attention the fact that the figure of 56 ¼ miles is merely 3/4 of the figure of 75 miles; the figure of 56 2/3 is derived from the round figure of 20.400 miles for the circumference of the earch. Possibly the tradition of the average was developed merely to explain why there were two figures in circulation; there may have been in circulation also round figures of 56 and 57 miles.

Two measurements were performed, one of a degree of meridian to the north of Palmyra in the direction of Raqqah on the Euphrates. In this case one aimed at measuring the interval between latitude 35 and latitude 36. It seems to me that most likely one operated on the basis of the segment of caravan route that moves almost directly north from Sukhenh to Hamman on the Euphrates, passing through the important ancient city of Rasapa (Sergoipolis). Guides for automobile drivers indicate a track of 130 km. A measurement of two degrees of meridian was performed in the desert of Singiar (areas of the ancient Singara); Nallino properly concludes that one measured or tried to measure between in the latitude 36 and latitude 34. Possibly one south from Singara to the Euphrates. This effort to measure exactly at degree 36 of latitude, the cardinal degree of ancient geographers, indicates that one was aware of the difference of length between degrees. However, one may have overestimated it, and the tradition of the average may indicate tha one originally intended to average a measurement between degrees 35 and 36 with a measurement between degrees 34 and 35. According to Ibn Khallikan, in the desert of Singiar, one fixed a starting point from which one measured one degree to the north and one degree to the south; the two results were found to be identical.

The purpose of the measurement was to verify the tradition of the books of the ancients (min kutub al-awa-il) that the degree was 66 2/3 miles; this figure was found to be exact, since some authors ascri e the figure of 66 2/3 to the calculations of al-Mamum. Since the calculation of the degree as 75 Roman miles is perfect, the geodesists of al-Mamum could not obtain any other result if they had succeded in being accurate.

As the length of the Arab mile the only possible uncertainty is whether 56 ¼ or 56 2/3 equal 75 Roman miles.

The Arab mile (mil) is a mile of 4000 cubits, like the mile of Berytean Lawbook. There miles make a parasang (farsakh). The Persian parasang has length of 5548.98 m. by the increased artabic foot. The Arab parasang relates to it as 15:16: it is equal to 4 Roman miles. As the Persian parasang calculated as 5550 m. is equal to 20,000 lesser feet; the Arab parasang is equal to 20,000 Roman feet. In the rabbinical system of measures the mile used to calculate the Sabbath way is also ¼ of parasang.

Already Boeckh and Saigey concluded that the Arab system of measures was a development of that of ancient Mesopotamia. Many important units are connected with Iraq and are called Babeli. The fixing of the canonical cubit is ascribed either to al-Mamum, Harun ar-Rashid or al-Mamum, that is, to great Abbasid Califs of the early period who are particularly associated with the center of learning of Baghdad. It is significant that al-Khazini, in order to calculate the volume and weight of the earth by the data of al-Mamun, adopts as standard the cubit used in the bazaar of Baghdad. But up to now ancient Mesopotamian units had not been determined with precision; particularly there has been great uncertainty about the cubit. One can see from my calculation of Mesopotamian weights that Arab units of weight are derived from them. I have determined that in cuneiform documents one usually reckons by the barley cubit, which is 499.4 mm. when trimmed.

The recent treaties of Arab metrology by Walther Hinz concludes that the canonical cubit has a length of about 498.75 mm. This is oviously a trimmed barley cubit. Hinz reports also that another important cubit has a length of aout 665.87 mm. But the sholar of ancient and Arab metrology, Mamum Bey, Astronomer Royal of Egypt (al-Falaki), from several data, such as buildings, itinerary distances, and units of volume, concluded that the canonical Arab cubit has a length of 493.2 mm. When in 1886 the French metric system was introduced in Egypt, the legal cubit was calculated by this figure, after it had been appoved by the doctors of the school of al-Azhar. Nallino agrees with the figure of Mahmud Bey. But he observed properly that several writers report that the Great Pyramid o Gizah according to the data of geometers has a side of 460 cubits of the had; since the Pyramid has a side of 440 Egyptian royal cubits of 524 mm., the “cubit of the hand” (dira al-yad), which must be the standard cubit, should be a unit of about 501.2mm.

Quite independently of the calculations made in Egypt, when in 1885 the metric system was introduced into Tunisia by the French authorities, the standards of the Tunis mint were taken to Paris for testing and the cubit was found to be 492.9. More distance evidence is provided of the area of Marseiles, Nimes Montpeller, the Rivera, and Genova by medieval European standard that indicate a barley footshorter than 333 mm., for instance the palmo (half cubit) of Genova had length calculated by sholars as 247.76 mm. (the official conversion rate at the moment of the adoption of the metric system was 248.283), 248.1951 coresponding to a cubit of 495.52 mm.

Without reference to these data, Decourdemanche concluded that the cubit which is at the basis of the system of measures of the Sassanid King Cosroes is 493.69 mm. This King esatblished an empire which corresponds in area to what shall be Moslem territory one century later and of which the main center was Baghdad.

It is likely that the trimmed barley cubit Ktesiphon near of ancient Mesopotamia was adopted as standard unit by Chosroes and by the Abbasid Califs of Baghdad, but from this unit of 499.4. taken as canonical, there was derived a trimmed unit (1/75 less) of 492.85 mm. One can assume that the cubit by which the Arab mile was reckoned was a unit of about 493 mm. A cubit of 493.3 would relate as 5:3 to the Roman foot and give a mile of 4000 cubits equal to 4/3 of Roman mile. It is possible that an adjustement of this sort from a mile of 6000 barley feet was introduced already in the Roma Empire. As I have already noted, an hour of march of 4 Roman miles is equal to three of these miles.

My conclusion is that the operations ardered by al-Mamun did not provide a datum different from that of 75 Roman miles to the degree. The purpose of the operations was that of varifying the figure of the ancients, which was found to be correct. Not one single writer finds a discrepancy between the datum of al-Mamun and that of 75 miles to a degree. The uncertainty about the figure of al-Mamun, whether it was 56, 56 2/3, or 57. reflects the fact that it simply confirmed the data already known. As I have said, some ascribe to al-Mamun the calculation of 66 2/3 miles to the degree.

There remains to clarify some problems of terminology. Arabic texts apply the terms of :black cubit: (ad-dira as-sawda) to this cubit; there is a legend that it was so called because it was measured by the arm of a black slave of al-Mamun. I suspect that in some Semitic language one confused the root of “black” (Aramaic with that of “small” ; in Akkadian document menst the adjective sahru is frequently applied to vessels, in opposition to rabu, large:. Hinz notes that the cubit of 665 m., a cubit of two barley feet by my interpretation, is called “great Hashimite cubit” (al-hashimiyyah al-qudra); the unit he calculates as 498.75 mm., that is the cubit of 1½ barley foot, is called “little Hashimite cubit (al-hashimiyyah as-sugra). This cubit which the Greek called “royal” that is, big, is small in relation to the great Babylonian-Egyptian cubit. Herodotos defines the cubit of the monuments of Babylon as royal cubit consisting of 27 fingers of a cubit of 24 called ; other Grek sources call the basic cubit koivavoridiotikov. Jakob van Gol, in his edition of the compendium of astronomy of al-Farghani (Alfraganus), quoted an Arabic manuscript in which the black cubit is described as a unit of 27 fingers. A manuscript cited by the historian of Arab sciences Louis Pierre Sedillot (Bull. Societe de geogr., I (1851), 229) states that the black cubit” at time is reckoned of 27 fingers, and at others as 25 2/3”. The second reckoning must be in relation to the wheat cubit; reckoning from a natural wheat cubit of 478.25 mm. one would derive a clack cubit of 499. 818 mm.

The text of al-Farghani mentioning the measure of al-Mamun specifiers “by the mile of 4000 black cubits.” The Latin translation made in the twelfth century by John of Seville reads per milliarium quod est 4000 cubitorum pert gradus aequales, secoundum quod sollicite probatum est.

In my opinion this translation is correct, in that the black cubit is a normal cubit in relation to the great cubit of 32/30 of barley cubit. The Hebrew translation by Jacob Anatoli defines the mile as being “by the middle cubit, the vulgar cubit” ; in the Mishnah this term is applied to the basic cubit. A Latin translation based on the Hebrew one, reads: milliare autem habet cubita 4000, prout cubitum accipitur in mensura media. Cubitum habet sex palmos communes. Here the terminology is exactly that of Greek writer, but the black cubit becames a unit of 24 basic fingers. The result was that scholars of the fifteenth and sixteenth century understood that the length of the degree was 56 2/3 Italian (6000 Roman feet) miles. And so did Columbus, quoting the authority of al-Frarghani.

5. Lehmann-Haupt approached the problem from a completely different point of view: he collated all the statement of ancient authors about the mutual relations of units of length. He organized the data around the information about ancient stadia, assuming correctly that in general there is a stadion for each module of foot.

He arrived at at classification which is extermely close to that of Petrie who had scanty concern with written evidence. As I ghave said, Lehmann-Haupt obliterated the distinction between trimmed and natural units, usually choosing intermediate values. He distinguished six types of foot.

a) The same of Babylonian-Persian-Pheidonian-Philetairic foot is given by Lehmann-Haupt to a foot varying between 330 and 332 mm. He is speaking of the trimmed barley foot, 332.938 mm.; this foot was taken as standard in Mesopotamia. Lehmann-Haupt gave paramount importance in fixing the value of the foot to the Sumerian statue of Gudea; he calculated the cubit of Gudea as 498 mm. at the most, whereas it is slightly longer. Furthermore, he followed some wrong deduction in interpreting the cuneiform stereonetric text that mentions the edge of the Mesopotamian pin, so that he fixed the barley cubit at 496 mm. The trimmed barley cubit is 499.408 mm.

b) Under the name of Attic-Roman foot he classifies the two versions of the basic foot. He confuses the Greco-Roman foot with the Egyptian and arrives at the intermediary figure of 297.7 mm.

c) He calls Oscan-Italian the foot I call the lesser one. His average figure of 275.7 (between 275 and 276.7) mm. agrees with my values of 277.449 for the trimmed lesser foot and of 274.7314 for the doubly trimmed one.

d) Under the name of Syrian foot, he classifies units of 248.1 (between 247.5 and 249), which in reality are merely the half of the trimmed barley cubit of 499.4

e) The name of Little Ptolemaic foot is given to the foot I call artabic. Lehmann-Haupt calculates it as 310 (between 309.4 and 15), whereas in my estimate it is a unit of 307.9 mm. His figure is too high, because this foot was calculated as 25/24 of Roman foot, and his estimate of the latter is too high(297.7 instead of 295.495 mm.). This foot is the most common unit of Egypt in Hellenistic times; but in my opinion it was almost certainly introduced as early as the Persian conquest, since it was the official standard of the Persian Empire.

f) Finally, Lehmann-Haupt gives the name of Phenician- Egyptian-Great Ptolemaic foot to foot of 354.3 (between 352.6 and 355.7).

I agree with Lehmann-Haupt’s classification, except that I consider the sixth type the same as the second and the first the same fourth. In my opinion his types should be reduced to five; he has completely left is that, as I have stated earlier, he could not interpret the evidence concerning the milion of 7½ stadia. In the effort to cope with it he arrived at the conclusion that the foot called Philetairic is a barley foot, whereas most metrologists recognize that it is a hybrid foot. Having lowered the figure applying to the Philetairic foot, as a result he lowered those refering to the wheat foot. His difficulty was increased by the circumstance that some Latin authors seem to have confused the stadium Olympicum of 625 artabic feet (600 wheat feet) with the artabic stadion of 600 artabic feet and 625 Roman feet. As a result he is forced to conclude that the length of the stadion of Olympia, 600 feet of 320. 45 mm., represents an anomaly that has no other parallels. But Lehmann-Haupt must grant that this foot is the edge of the cube containing 60 basic pints of 540 cc. This unit of 32,400 grams is usually known as centenarium (100 librae); Oxé in a special study has stressed its great importance in ancient metrics. This unit is often called Babylonian talent.

6. Decourdemanche was a distinguished Arabist and came to the study of ancient metrics from that of the Arab ones. A number of scholars of Arab metrics have dealt with the ancient ones, since the two cannot be separated, as one cannot separate the metrics of medieval Europe from those of the ancient world. Vazquez had noted that Arab metrogical treatises indicate that there are as many units of the talent class as there are feet, and tried to apply this knowledge to the study of ancient measures: an artabic foot of 308.6, which he calls ephah or talent of Rhodes a Babylonian foot of 318.8 mm. forms a Babylonian talent of 32.400 grams; the hybrid foot of 350 mm., which he calls Egyptian royal foot, forms the talent of Alexandria ; the cubit of 555 mm., which is the Arab cubit beladi, “of the land”, (which in my opinion is two trimmed lesser feet) also forms a cubit unit. Decourdemanche was able to clarify and simply this scheme. He classified four units of length as “ types” corresponding to four essential types” of talent. The “types” of foot are:

 
 

A. Babylonian foot of 319.6 mm.

B. Assyrian foot of 308.56 mm.

C. Egyptian pharaonic foot of 349 mm.

D. Ptolemaic foot of 277 mm.

The first type is the natural wheat foot of 318.75, according to my classification. The second in an artabic foot. Decourdemanche considers that the Roman foot, trimmed basic foot, is derived from it by taking 24/25. He realized, however, that this reckoning gives a Roman foot shorter than the official one; in substance he noted that the relation 24:25 between Roman foot and artabic foot is the reason for the existence of a pes Statilianus next to the correct Roman foot. The third type corresponds to the Egyptian foot, natural basic foot, and it derivatives. The fourth type is a trimmed lesser foot.

In substance, I agree with the classification of Decourdemanche, except in so far as he neglects the importance of the barley foot.

Decourdemanche recognized that Arab units known as Babeli are the old units of Mesopotamia, but he could not count on any satisfactory investigation of Mesopotamian units. I have shown that an accurate knowledge of the linear units of Mesopotamia allows to fix with certainty the length of the Arab canonical cubit from that of barley cubit. Decourdemanche calculates the cubit Babeli as 514.2 mm. and accordingly the foot, which should be a barley foot, as 342.8; as a result he confuses this foot with with the relied on investigations of cuneiform linear metric less inconclusive than those of Thureau-Dangin, he might have realized that the cubit of Cosroes, 493.69 mm., is a derivation of the cubit of 499.4 of Mesopotamia, and is the later cubit of al-Mamun.

7. My conclusion that all types of foot used in the ancient world are derived from the basic foot of 16 fingers, by forming units of 15, 17 and 18 basic fingers, with corresponding cubits, was almost reached by Petrie in analyzing the Egytpitan cubit rods. But he balked at concluding that all units lengrth were costructed by such a simple and organic scheme.

Egyptian rules are usually of the length of a royal cubit of 7 hands or 28 natural basic fingers, but they bear markings of lesseer units. These markings are of difficult interpretation since they are not placed in exact definable positions and because their positions are not the same on all rods. Since the beginning of the nineteenth century one has disputed whether these markings indicated units other than those that are directly related to the Egyptian royal cubit, such as the natural basic foot of 16 fingers (300 mm.), the septenary foot of 18 2/3 fingers (350 mm.), and the natural basic cubit of 24 finger (450 mm.).

The marking dsr, the arm holding the nhbt wand, is usually placed in correspondence with the fifteenth and the sixteenth fingers; but in some cases it is placed in correspondence with the sixteeth with the sixteenth fingers. The marking rmn, is placed in the area of the fifth hand , that is, from the seventheenth to the twentieth finger. The sign for” little cubit” is placed in the area of the sixth hads, that is, from the twenty-first to the twenty-fourth finger; the sign for the “royal cubit” is placed in correspondence with the seventh had.

Constantin Auguste Rodenbach dedicated three studies to the interpretation of these markings, without arriving at a solution ; but he grasped the general idea when he spoke of etalon prototype universel des measures de longueur. Petrie gave the correct answer when he concluded that the marking indicated “measures known in other countries” and found them to correspond to a Punic foot, a Roman foot, and a Northern foot (that is, units of 15, 16 and 18 basic fingers, according to my interpretation ). He states: “The various lengths are evidently other standards, approximately marked on the royal cubit; there could be no sence in marking such numbers as 15, 18, and 19 merely as such fractions of the great cubit.” What Petrie did not realize is that the other standard are eaxctly units of 15, 17, 18 and 18 2/3 fingers.

As I have said the rods ussually placed the marking dsr in correspondence with the fifteenth and the sixteenth finger, as in the case of the cubits Nizzoli and Raffaelli of the Louvre; but in some cases it placed in correspondence with the sixteenth finger exclusively, as it is clear in the Anastasi cubit of Leyden, formerly in Florence. The foot of 15 fingers is also marked by the fact that 15 fingers, beginning at one or the other end, according to the rods, are subdivided in fractions of increasing denominations,½, 1/3, ¼... 1/14, 1/15. In one case there are 16 finghers so divided, reaching the fraction 1/17; Lepsius thinks that thwe marked erred, but perhaps he intended to make a foot of 16 fingers. The sign rmn indicates feet longer than that that of the basic one: a wheat foot, a barley foot, and the hybrid foot. Vazquez though that the sign rmn referred only to the hybrid foot of 350 mm., 2/3 of 525 mm. But one can conclude that the sign refers to akll foot units longer than 16 fingers: the wheat foot of 17 fingers, the barley foot of 18, and the hybrid foot of 18 2/3. There were two kinds of cubit called royal in the ancient world, the Egyptian royal cubit of 28 fingers and the barley cubit of 27 fingers: the sign for royal cubit may refer to both. There is an Egyptian cubit of the Harris collection at the British Musaum that it is a double cubit of the length of 1048.9 mm. In spite of the circumstance that it is reported to have been found within the brick work of the Pylos of the Temple of Karnak, it has been considered a forgery. But a forger could not have known of my determination of the natural barley cubit (Babylonian royal cubit) as a unit of 506 .25 mm. The cubit is of white wood which may have skrunk considebly with time. The cubit is divided into 14 fingers, according to the usual Egyptian septenary reckoning. On the Drovetti cubit of the Museum of Torino the sign for “ royal cubit” is clearly placed in correspondence with the twenty-seventh and the twenty-eighth finger. In the same cubit and in the Drovetti cubit of the Louvrwe the sign for “little cubit” isd placed in correspondence wity the twenty-second and twenty-third finger; it seems to indicated the lesser cubit of 22½ fingers and the basic cubit of 24 fingers, thast is, the cubits corresponding to the feet of 15 and 16 fingers indicated by the sign dsr. But in other cubits, such as the Anastasi cubit of Leyden, the sign for “little cubit” covers the entire sixth hand and could refer also to the artabic cubit of 25 fingers.

The Egyptian cubit rods prove that one had habit of forming other units from as rule of the length of a royal cubit, by taking agiven number of basic fingers.

The practice of deriving all units from the length of the Egyptian cubit-rod, is illustrated by a quadrangular bronze cubit of Torino. Lepsius could not explain its subdivisions and for this reason considered it a forgery; but it reality it fits the essential structure of ancient measures of length. One face is inscribed with hierogly -phics that reproduce mechanically and without understanding an inscription was a standard one for cubits and and may refer to a specific sacred reference standard. Of the other three faces, one presents a regular royal cubit og about 525 mm.; but the cubit is divided into 6 hands and 24 fingers, as it accours at times. On another face the subdivisions leave a black one end, so that the cubit measures about 518 mm., it is the trimmed unit corresponding to the cubit of 28 natural fingers. This cubit is substantially divided as the preceding one: the the total length is divided into 12 half hands or 24 fingers; the fingers are divided into third, and at one end the fingers are further subdivided into fifteenths. The division up of the fifteenth of finger is normal in Egyptian cubit rods. The fourth face is divided into 27 fingers plus a 5 mm. of a finger that would continue beyond the length of the rod; this face intended to indicate the length of 28 artabic fingers, a length that is common in Hellenistic cubits of Egypt. By using the three faces of this rod, one could measure all the units of the ancient world.

More recently one has come to realize that a number of documents indicate the use of a unit equal to 1 1/3 royal cubit. In my opinion this mysterious unit is a cubit of two hybrid feet, that is, 37 1/3 natural basic fingers, 700 mm. The nama of this unit is nb, nebiu, which means “carrying indicates that the original unit of length was the carrying yoke; the term for cubit in Semitic languages and in Greek(-) means the arm of the carrying yoke, that is , the half ofg it. On Egyptian cubit rules, the position of the hybrid foot it indicated by the sign of the forearm rmn; the term means “cubit,” but it corresponds to the idea of “to carry” and it also means “half “, indicating that essentially it signifies the half of the carrying yoke.

My explanation of the unit nebiu is supported by a neglected specimen of the Metropolitan Museum of New York. This object is listed in the catalog as a a cubit of 27½ American inches (698.6 mm). It is a double hybrid foot (rmn) or a ‘carrying yoke” of 7000 mm. It consists of a simple round rod of plain wood divided by lines cut with a saw into 7 parts; the seventh at the meddle is further divided into two parts, so that the rod is divided at the center in two halves of 3½ sevenths.

8. A most striking confirmation of my theory that the unit of length were determined by a system of feet with a module of 15, 16, 16 2/3, 17 and 18 baisc fingers, comes from the work of Guilhermoz, the greatest expert on medieval measures. In his comprehensive article “De1’equivalence des anciennces measures”, he notes with a certain surprise thast medieval texts mention a foot of 15 and foot of 18 fingers of a /roman foot divided into 16 fingers, but concludes that the figures support this explanation of the relation among measures. In substance Guilhermoz is noting rthe existence of the lesser foot and of the barley foot next to the basic foot.

Very properly he puts the foot of 18 fingers in relation with the Mesopotamian cubit and with the Arab black cubit; he sees the relation between these units and the pes Drusianus. As a typical example of the foot of 18 fingers he citesd the palmo mercantile Of Rome (the medieval palmo is the half cubit, or 2/3 of foot) which in 1811, at the moment of the adoption of the metric system, was so calculated as to indicate a foot of 332 mm. I have calculated the trimmed barley cubit as 332.9384 mm. Guilhermoz notes that the use in Rome of a palmo architettonico (corresponding to a Rome foot) together with a palmo mercantile constitutes a joint use of a unit of 16 Roman fingers and of one of 18 finger which is common throughout Europe. I have mentioned the finding in Rome Germany of the end pieces of a Roman rod marked both with the Roman foot and which the pes Drusianus.

Guilhermoz sees the relation between the European foot of 18 fingers and the Sumerian cubit embodied in the Sumerian statue of Gudea. In general he stresses the continuity of measures through history. Certainly this is in marked contrast with the oposition taken today by most scholras of ancient Greece who wax indignant at the suggestion that Greek measures be related to Oriental ones.

Guilhermoz notes also the great importance in medieval Europe of units related as 25:24 to the Roman foot. He is aware of speaking of units that correspond to the ancient foot I call artabic.

Guilhermoz notes with some puzzlement the certain fact that in the Middle Ages thereoccurs a Roman foot shorter than the ordinary one. He takes as model of it thwe fact foot of Saint Hubert (so called because a standard of it was fixed onthe wall of thecathedral dedicated to this saint) evaluated at 294.779 mm. also belgium, there was an important foot called after the Church of Saint Lambert at Namur, evaluated at 294.763, and the City Hall of ons there is preserved the standard of the pied de Hainaut, found to be 293,43 mm. Similar units were used in the same general area of Europe, at Tried, Strasbourg, Noyon, and bar-le-Duc; they suggest a unit influenced by a trade route from the Alps to the Flandres. My remarked about the pes Cossutianus of 294.355 mm. explain completely the existence of this reduced Roman foot. Guilhermoz himself in order to explain the existence of this reduced Roman foot it the Middle Ages quotes evidence of ancient Roman foot “near to 294mm.,”

Finally, Guilhermoz reports the use in medieval Europe of of foot related as 6:5 to the Roman foot. This foot is usually calculated by constructing a pertica of 12 feet and dividing it into 10 feet. This foot is the hybrid foot, 2/3 of Egyptian royal cubit, according to the version of the Egyptian royal cubit, used in the Hellenistic age (28 artabic fingers). The use of this unit was generalized by its being prescribed as a fiscal unit in theLate Roman Empire. I have calculated that in the ancient world the increased hybrid foot was 355.135 mm., being 2/3 of an Egyptian royal cubit of 532.7. Metrological tables of the Roman period calculate this foot as 6/5 of Roman foot.

According to Guilhermoz an example of it is the pied de terre de Bordeaux which he calculates as 357.214 mm. Another example is the pied le comte, of Franche-Comte, measuring 357.8 mm.; it is very significant that the standard of this unit preservel at the City Hall of Poligny is bar of 7 feet. The original hybrid foot was a septenary unit.

The septenary character of the foot of 355 mm. allows to solve one of the greatest mysteries of metrological research, the origin of the English foot. The English foot is nothing but the non-septenary unit corresponding to the hybrid foot. From a foot of 355.131 mm. one would expect an English foot of 304,49, hence the length of the English foot is practically perfect.

My explanation of the origin of the English foot is confirmed by the circumstance that Peter the Great fixed the Russian sagene at 7 English feet. The sagene, which is the foundation of the Russian system of units, is equal to 3 arshin, a unit of Persian origin through a Turkish intermediary. It has already been pointed out that Russian units of weight go back to Persian standards and through them to a Mesopotamian origin. In discussing the structuring of the English basic units of weight and volume, I have observed that htey are nearer to the structure of Mesopotamian units than the units of France or Italy. It has been already suggested that the English system of measures is connected with a trade route that from Denmark, the Baltic, and Russia, leads to the Black Sea. Hence, when Peter the Great fixed the sagene at 7 English feet, he did not introduce any metric novelty, but merely adopted as reference a particularly stabilized, example. One wonders whether there is an influence on Peter the Great of the work, Metro universale,by Burattini which was published in Poland, since during the last part of his life, the author was a major figure of Polish politics; in this work he declares that the English reference standards are the most reliable of Europe.

Whereas it is reasonable to search the equivalents of English units along the trade route from Denmark to the Black Sea, one must keep in mind that the main legislation on measures dates from the Plantagenet period. Therefore it would be proper to consider the connections with the Plantagenet possessions in France; as far as I know this has never been done, but I have noticed a similarity between English measures and measures of Bordeaux and in general the units pied de table of Southern France. Hence, it may be significant to find in Bordeaux a foot equal to 7/6 of English foot.

In summation, Guilhermoz finds a structure of medieval measures that parallel that which I have established for the ancient world. The only difference is that he does not consider the existence of afoot 17 basic fingers, but he notes the frequent occurrence of a foot corresponding to this length. He reports that since at the beginning of th e seventeenth century Willebrod Snell used the pes Rhinlandicus in the calculation of the degree of latitude, the standard used by him was adopted as scientific foot by the scientists of central and northern Europe in whose countries one used a similar unit. The foot of Snell was called pes Phinlandicus after the district of Phinland of which Leyden is the main city; some metrologists arroneously speak of a foot of the Rhine. When Picard visited the observatory of Uranieburg in Denmark in order to ascertain the unit used by Tycho Brahe in his astronomical observations, he found that the foot of Leyden was 139.2 Paris lines and that of Denmark 140.36. When Louis Bonaparte, as King of Holland, introduced the metric system in that country, the foot of Leyden was calculated as 139.17 Paris lenes (313.946 mm.). A foot of this length wasthe official standard in the Swiss cantons of Luzern and Sankt Gallen, in Prussia, Denmark, and Norway in the eighteenth century; whereas Sweden under Gustavus Adolphus adopted the Roman foot. Guilhermoz considers the most precise example of it to be the foot of Besancon, evaluated officially at 314.7 mm. I have estimated the ancient trimmed wheat foot as 314.44 mm. On the cathedral of Modena in Italy in the sixteeenth century there was posted a pertica of 10 of these feet, measuring 3138.2897 mm.

Italy this foot was known as pie Liprando, apparently because it was cut on the tomb of the Lombard King Liutprand in Pavia. Unfortunately in Italy the history of the Longobard period is considfered a subject for Gothic novels rather than for serious scholarship; the studies of my techer G. P. Bognetti on Longobard institutions, including metrology, are a rather rare exception. For this reason the the investigation of the pie Liprand has been the occasion for picturesque speculations. Whereas practically nothing is done in Italy to preserve the record of medieval units, and reference standards that that still exist in churches and public buildings are gradually lost, scholars took the trouble to open the tomb of King Liutprand and a professor of medicine at the University of Pavia tried to deduct from the bones the length of tyhe foot. One was surprised when one discovered that the King was a man of normal stature; apparently one had taken at face value the northern Italian erxpression lombardan meaning somebody who is grande, grosso, e stupido.

The piede agrario of Pavia is a cubit of 417.954 mm., corresponding to a foot of 314.696. For the neighboring towns of Novara and Brescia and of Grema and Piacenza, there is reported a value of 417 and 470 respectivcely. The foot of Tyrol is 314 mm., but in the area of the Alps there must have been also a version wuith a slightly increased value, since the foot of Rovereto and Bolzano is 316 mm. The value of the foot of Vienna was evaluated at 316.0807 mm. when the metric system was adopted by Austria at the late date of 1871. The confusion of some Italian studies about the pie Librando, is to be explained in part by the circumstance that it used to be the foot of Milano, where one continued to use the same name when in 1612 this city adopted as unit the Roman foot, calculated as 297.458 mm. and that a similar phenomenon took place in Piedmont. King Charles Emmanuel I (1580-1630) tried to unify the French and Italian part of the possessions of the House of Savoy; in this process a standard of Torino of 470.95 mm. (corresponding to an ancient trimmed wheat cubit of 417.66 mm.) was substituted for by the standard of Lyons.

The distribution of the wheat foot includes an area extending from northern Italy, Switzerland, Austria, most of Germany, Holland , to Denmark and from there to Norway. This geographical distribution suggested to According to the system of Mesopotamia this libra would be But a study of the dimensions of a Viking ship of the fourth century A. D. discovered in Norway, crududes that it was build by a foot of 313.5 mm., and indicates that the same standard was used for other Vibing ships, The pied agrams of Pavia is a cubit of 417.954 mm. corres pouding to a foot of 314.696 mm. For the neighboring to was of Novara and Brescia there is reported a valume of 417 and 470 respectively Apparently at times the trimmed wheat foot was confused with the natural wheat foot or average with it , since from the same area there is reported that the foot of Parma is 318 mm. and that of Bolognais 317 *

related to the foot through a cube of 64 librae, wehich has as edge the trimmed wheat foot.

The research of Guilhermoz not only anticipates the acheme I have discovered in the inner structure of ancient units of length, but proves the continuity of standards through the millennia, whereas the new school contends that standards were freely changed through time at the discretion of each local ruler. The scheme outlined by Guilhermoz and completed by me, can helf to take medieval metrology out of the realm of folklore into that of science.

9. It is obvious that one of the most reliable data about linear units isd provided by ancient stadia, but only a few been surveyed by archeologists.

The only stadion for which there is a full report is that of Epidauros. The two terminal lines are clearly defined; the course is divided by alittle column every plethron (1/16 of stadion or 100 feet) and by a basin in the drainage gutter every half-plethron. The report indicates that the distance are effected by movements and settlements of the ground. The little column marking the plethron are at a distance from each other of 30. 21, 30.10, 30, 21, and 30.11 m. The length of the racing course is 181.30 m. at the North side and 180.94 at the South side; as trings appears today, the racer at the South end would have to run a distance of one foot more than his competitor at the North end. The archeological report does not state whether the distance were measured by streching a line on the ground or optically with a transit; if one increase the distances. The station was calculated by the natural basic foot possible in a version of 301 mm. ,slightly longer than the theoretical value of 300 mm.

The reports about the stadion of Delphoi indicate which care must be exercised in order to obtain reliable data. It is reported that a test performed by Convert and checked by a master mason gave a length of 178.35 m; but Theophile Homole with the help of an architect arrived at 177.55 m. The data of the stadion of Epidaros sugest that the two terminal lines may not be perfectly parallel in their present state. If the has been any settlement of the ground, by measuring with a line one does not obtain the same figure one obtains by triangulation. According to Convert’s figure the foot, a trimmed basic foot, woulsd have a value of 296. 28 mm., whereas according to Homolle’s figure the foot would be 295.81. The theoretical value of the trimmed basic foot is between these two values for the contemporary stadion of Herodes Atticus in Athens there is reported a length of 184.96 or 6000 artabic feet of 308.26 mm; it would seem that the artabic foot is calculated by the increased lengrth equal to 25/24 of trimmed basic foot, which I reckon as 308.276 mm.

A dimension in full agrement with the theoretical value of the trimmed wheat foot I have calculated, is reported for the stadion of Aphrodisias in Phrygia. It is said to be 188.70 m. or 600 feet of 314.5 mm.

The dimensions of the stadion of Olympia have been obtained by triangulation, since the central part of the course has not been excavated. Earlier reports gave the exact figure of 192.27 m; later ones that of 192½. The wildest speculations have been developed about the metrology of this stadion. A number of ancient sources state that the Olympic stadion has length of 625 feet; this means that the foot ny which it has been calculated relates as 25/24 to some other foot. The artabic foot can be calculated as 25/24 of trimmed basic foot, and correspondingly there is a stadion of 600 increased artabic feet equal to 625 trimmed basic feet. Similarly ther is a natural wheat foot which is calculated as 25/24 of artabic foot and hence 320.621 mm.; 6000 of these increased feet form a stadion, called Olympia stadion, equal to 625 artabic feet. the theoretical length of an Olympia stadion is 192.37 m. The increased natural wheat foot is used in other buildings of Olympia.

The data provided by these stadia indicate that one employed whichever module of foot was most desirable under the circumtances, and that local preferences had only a limited influence.

The necessity of relating the basic talent brutto of 27,000 grams to the artabic brought about the establisment of a peculiar calculation by the factor 11. the royal talent is equal to 80 basic pinds and to 88 reduced pinds, computed as 10/11 of 540 c.c., that is 490, 090 c.c. By this reckoning 55 reduces fints are a basic talent brutto of 27,000 c.c. and 60 are an artaba of 29,454, 5 c.c. A centenarium or wheat talent netto is eaxactly 66 such bints.

If the basic pint is computed as 55 basic sheqels of 9 grams, it is 495 c. cc or 11/12 of basic fond of 540 c. c.


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