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The Origin of English Measures

As I have explained, the natural basic foot of the ancient world (300 mm.) was the standard unit of Egypt. The only peculiarity of Egyptian practice was that from the Egyptian foot there was derived a normal cubit of 450 mm.(called “little cubit” in the inscriptions) and a “royal cubit” of 525 mm. This septenary unit (7 hands instead of 6, 28 fingers instead of 24) was the official standard of Pharaonic Egypt.

Jacques Frederic Saigey, the mathematican who in 1820 founded with Francois Raspail the Annales des sciences d’observation, has the perceptivity to notice that in Egypt there was also used a royal cubit composed of 28 artabic fingers (538.625 mm. by my figures).

In 1838, before cuneiform documents became available, Boeckh with brilliant insight gathered that this unit is of Mesopotamian origin and is continued by an Arab unit used at Baghdad. He calculated it as 236.07 Paris lines or 532.53 mm. He spoke of a Babylonian-Egyptian great cubit which is called Philetairic or Ptolemaic during the Hellenistic age. Metrological tables of the Roman period make this cubit equal to 9/5 of Roman foot or 532.702 mm. Segre states: “Boeckh identifies the Philetairic-Ptolemaic cubit with a Persian measure common to the countries under the domination and influence of Iraq, and in my opinion he is right.” But at some other point he observes that the value calculated by Boeckh does not seem correct, because metrological tables contained in Greek papyri from Egypt make clear that the cubit was calculated as 28 artabic fingers, as Saigey had correctly guessed. In my opinion both Saigey and Boeckh were right.

From the dimension of monuments of Mesopotamia, Oppert gathered that there had been used a cubit of about 533 mm. From cuneiform tablets as early as the Old-Babylonian period, ?I have determined that whereas the typical cubit of Mesopotamia is the barley cubit divided sexagesimally into 30 fingers (instead of 24), there is used at times a great cubit of 32 such sexagesimal fingers . This longer cubit (32/30 or 16/15) measures 532.702 mm. when trimmed and 540 mm. when natural. The first figure is exactly 9/5 of a Roman foot.

I have determined at least one reason why this great cubit was used in Mesopotamia. There were used two agrarian units of surface: one decimal, the acre with a side of 100 cubits ; and one sexagesimal, the iku, with a side of 180 cubits; by making the cubit 32/30 of normal Mesopotamian cubit the acre becomes exactly 1/3 of iku. This corresponds to the process by which the basic talent (which is 3 3/8 of basic load) is increased by 1/8 to make the artaba which is 1/3 of basic load. The weight of wheat necessary to sow an iku is a basic load; that necessary to sow an acre calculated by the great cubit is an artaba.

I have already explained that this great cubit has the advantage of forming in its trimmed form a cube (kurru) which contains 300 pints or qu of 503.885 c.c., that is, to all practical purposes Mesopotamian pints of 504 c.c.

The great cubit is not only related perfectly to the Roman foot, but is related also to the trimmed lesser foot. There was in common use a cubit composed of two trimmed lesser feet (cubit of the mensa of Ushak); this unit of 554.8978 mm. is 1/10,000 of a Persian parasang (18,000 increased artabic feet or 5,548.96.).

In his cosmological studies Laplace noted the occurrence of a unit of 555mm. in ancient itinerary and geographical computations. This cubit is the basis of the units called Pontic in ancient metrological texts: the cube of the cubit (170.860 c.c.) is divided into 6 units of 28,473 c.c., which are almost exactly an English firkin or artaba of 80 Alexandrine litrai (cube of the English foot). As I have said, the English foot is the foot corresponding to the Babylonian-Egyptian great cubit of 532.702 mm. Hence the English foot should be 16/28 of a unit which is 9/5 of Roman foot, that is, 26/25 of Roman foot. This would make it 304.4 mm.

To the natural Babylonian-Egyptian great cubit of 540 mm. there corresponds a foot of 308.572 mm. which is almost an increased artabic foot of 308.276 mm.

The Babylonian-Egyptian great cubit constitutes a compromise among most of the important units and therefore fitted the needs of the fiscal adminisation of empires from the Assyrian to the Roman.

The fiscal unit of the Late Roman Empire is the iugerum castrense (9/8 of standard iugerum). Whereas to the standard iugerum there corresponds as amount of seed 5 standard Roman modii of 16 sextarii (each modius is 1/3 of quadrantal), to the iugerum castrence there corresponds 5 modii castrenses of 18 sextarii (each modius is 1/3 of artabic). But if the relation between quadrantal and artabic was calculated as 10:11, as the relation between Roman libra and Alexandrine litra, the modius castrensis would be 17.6 sextarii or 1/3 of the English firkin (artaba metrw/ qhsaurikw’ /).

The data about the modius castrensis are somehow uncertain and it is possible that the name was applied to the third of any heaped version of the Roman quadrantal. whether the heaping was equal to 1/8, 1/10, or 1/11. Hence, when texts of the Late Roman Empire speak of modius castrensis they may mean the third of the English firkin.

Petrie noted that the Egyptian rods representing a royal cubit are longer during the Hellenistic age and found them to vary between 536.196 and 537.464 mm. From this empirical datum he suggested the explanation that the cubit of 525 mm. became longer with the passing of time. But some Nilometers, as that of Philai, and some rods, as those of the Cairo Museum, indicate a cubit of 532-533 mm. The mentioned bronze rod of the Museum of Torino indicates on one face a cubit of 524 mm. and on another face a cubit of about 532 mm.; I have explained the mathematical reason for which the Egyptian royal cubit of 525 mm. at times is calculated as 524.104 mm. Segre calculates the Nilometric cubit of the Roman and Arab period between 539.175 and 540.7 mm.

The empirical data can be explained by the following considerations. The Mesopotamian great cubit could be either 532.70 or 540 mm. By choosing an intermediary figure there is formed a great cubit which is 536.351 mm., which is almost 28 artabic fingers or 538.625 mm.

2. It is agreed that the Roman mile composed of 5000 feet and divided into 8 stadia of 625 feet (equal toa stadion of 600 artabic feet) goes back to Oriental models. On one side, the stadion of 600 artabic feet is the stadion of the Persian Empire, and, on the other, the calculation of itinerary distances by units of 5000 or 10,000 cubits is typical of Pharaonic Egypt.

In the eastern part of the Roman Empire there was used a unit called milion or miliarum defined as equal to 5400 Roman feet and to 7½ stadia or 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 great 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 600 barley feet 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 the great cubit is calculated by the value of 32/30 of trimmed barley cubit, but if it is assumed that the great cubit is 28 fingers of artabic cubit the ratio is 24:25. The existence of the milion has created endless difficulties for metrologists; Lehmann-Haupt finds himself obligated to reject the figures of the metrological tables of the Roman period as “bungling”. The difficulty can be removed , since I have explained that the great cubit was calculated either as 32/30 of barley (Mesopotamian) cubit or as 28 artabic fingers; in the first case the milion relates as 26:25 to the Roman mile and in the second case it relates as 25:24. As I have explained, the relation between Roman foot and artabic foot was correctly 25:26, but was most often calculated as 24:25 for the sake of convenience in reckoning.

The problem which is rather simple has become complex because Mommsen found in the Berytean Lawbook a mention of milion of 6000 feet and thought that this “ provincial” milion was the same as the one mentioned above. But it is actually a third and different unit.

The Berytean Lawbook is a text of Roman Law written under the influence of the school 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 was introduced when God gave intellect and wisdom to man, and mankind 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 milion is calculated as 500 barley feet, which are equal to 6000 lesser feet. The text of the Berytean Lawbook retranslated into Latin reads: dederunt ** mille passus qui faciunt quingentas 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 million is 9/8 of 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 Persian parasang, an hour of march, can be calculated either as 18,000 artabic feet or 5540 m. or as 18.000 increased artabic feet (25/24 of Roman foot) or 5548,97.

In conclusion, there were used in the Roman Empire three types of mile:

-standard Roman mile, 1470 m. depce 110,979

-milion of 5400 Roman feet, 1598m. 1598,106

-milion of 5000 barley feet, 1665 m.

These figures, however, were sujected to several small adjustments. The second unit is equal to 3000 great cubits. If the great cubit is calculated by the value of 536.418 mm. (Royal cubit corresponding to the artabic “by the Roman usage”) the milion becomes 1609.254 m., which is exactly the English mile (present official value 1609. 3426 m.).

The longer milion allows to explain a problem that has made the despair of metrologists of the Renaissance and has at times confused the calculations of Boeckh and Hultsch. Renaissance metrologists had noted that the Roman foot used in the city of Rome at their time and the one by which one calculated the miles of the Italian roads is longer than the ancient Roman foot. This foot was called geometric fot in the Middle Ages. Before the French Revolution it was rougly calculated as 132 Paris lines (297.779 mm.). I have calculated it as 297.1731 mm. Renaissance scholars realized that this longer foot is connected with the fact that the Roman libra of their times is heavier; they traced back the origin of this libra to standards connected with the re-establishment occasioned by the destruction of the Temple of Juno Moneta by the partisans of the Emperor Vitellius. I have determined that the medieval Roman libra and the unit called libra Vespasianica by Renaissance scholars, was a unit of 328.05 grams (instead of 324). and goes back to a

If one calculates the Roman quadrantal as 80 librae of 328.05 grams, its edge is the geometric folt of 297.1734 By calculating, the milion as 5400 of such feet, the value of this unit is 1604.7 m., which is practically the milion equal to the English mile. But I shall show that even this little difference of a meter in a mile is taken into account in calculations of the Anglo-Saxon period of England.

3. The misunderstanding about the types of mile occurring in the Roman Empire has created confusion in the study of Arab metrics, which usually provide a good starting point for the study of ancient ones. This confusion has bedeviled scholars since the revival of learning in Europe, because they could not understand properly the reports about the calculations of the length of the degree of meridian performed by order of Calif al- Mamun around 825 C.E.

The great Arabis Carlo Alfonso Nallino dedicated his first published essay to the problem of the calculation of the degre by al Mamun; but in sin spite of his skill he was confusing the milion of the 5000 barley feet with the more common milion of 3000 great cubits. He did not realize the existence of the first milion even in his masterly study of the Barytean Lawbook.

Nallino reports tha Arabic writers usually calculate the degreeeas 75 miles, but often also give the figure of 66 2/3 miles; they asscribe both calculations to “the ancients”. The value of 75 miles is obviously by Roman miles; as to the value of 66 2/3 miles, Nallino calculates it by the milion of 54000 Roman feet. But it is obvious that the second mile must be a milion of 5000 barley feet of the Berytean Lawbbok. (9/8 of Roman mile).

The figure 66 2/3 miles si exactly 8/9 of 75, and the relation between a Roman mile of 5000 trimmed basic feet and a milion of 5000 trimmed barley feet is exactly 8:9.

The purpose of the measurement of al-Mamun was to test the exactness of the datum of 66 2/3 miles “according to the books of the ancients” (min kutub al-awa-il) Since some authors ascribe the figure of 66 2/3 miles to the geodesists of al-Mamun, it appears that the figure was confirmed. Since the datum of 75 Roman miles to the degree is correct, the Arab geodesists could not have obtained a different figure if they had suceeeded in being accurate.

But the operations were performed in Arab miles. The Arab mile (mil) is a mile of 4000 cubits or 6000 feet. If these feet were trimmed barley feet, the Arab mile would relate as 6:5 to the milion of the Berytean Lawbook, but it proves that the Arab mile is somewhat shorter than 6000 trimmed barley feet. If the Arab mile had been 6/5 of the milion, the geodesists of al-Mamun would have obtained a result of 55.55 Arab miles to the degree. By collating the different traditions , Nallino gathers that the result al-Mamun’s survey is variously reported a 56, 56¼, 56 2/3 and 57 miles. In my opinion the significant figure is 56 ¼ which is exactly 3/4 of the datum of 75 Roman miles. The figures 56 and 57 miles are round figures; the figure of 56 2/3 may have been obtained by averaging the accurate datum of 56¼ miles with that of 75 miles or by dividing the round figure of 20,400 miles for the circumference of the earth. The figure of 56¼ miles indicates that the Arab mile was calculated as 4/3 of Roman mile; in fact, the Arab parasang (farsakh) is composed of 3 Arab miles and is considered equal to 20,000 lesser feet, the Arab parasang is equal to 20,000 Roman feet.

On principle the Arab mile should be equal to 6000 trimmed barley feet, but the barley foot must have been considered as 10/9 of Roman foot, which makes it 328.8286 mm. instead of 332.9387. This allows to calculate the Arab mile by the convenient relation 4;3 to the Roman mile.

According to this reckoning the Arab cubit should be 493.2424 mm. And in fact in the last century , the scholars of ancient and Arab metrology, Mahmud 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 into Egypt, the legal cubit was calculated by this figure, after it had been approved by the doctors of the school of al-Azhar. Quite independent of this computation is the datum obtained at the moment of the introduction of the French metric system into Tunisia in 1885; the French authorities took to Paris the standard that used to be kept at the Tunis mint and there it was founds to be 492.9m.

Without reference to these data, Decourdemanche concluded that the system of weight established by Cosroes, the Sassanid King of Persia, must have been based on a cubit of 493.69 mm. Cosroes was the head of an empire of which the capital was Ktesiphon near Baghdad; about one century later the territory of his empire was obserbed by the empire of the Abassid Califs of Baghdad. Hence, it seems that the canonical Arab cubit is the cubit of King Cosroes. The fixing of the canonical Arab cubit is ascribed either to al-Mansur, Harun ar-Rashid, or al-Mamun, that is to the Califs of the early Abassid 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, interpreted the measurement of the degree performed by al-Mamun taking as reference the cubit used in the bazaar of Baghdad.

King Cosroes may have taken into account the Roman foot and computed the trimmed barley foot as equal to 10/9 of Roman foot, but he must have considered the Persian parasang. This unit is equal to 18,000 artabic feet or 5,548.971 m. by the artabic foot computed as 25/24 of Roman foot. If the Arab parasang was computed as 16/15 of the Persian porasang, as Nallino indicates, the canonical Arab foot comes to be 10/9 of Roman foot, as I have calculated.

The recent treatise of Arab metrology by Walther Hinz concludes that the canonical Arab cubit is about 498.75 mm. He also notes the existence of a cubit of 665 mm., which is obviously a cubit of 2 feet of the preceding cubit. In my opinion the unit traced by Hinz is essentially the original trimmed barley cubit of 499.4 mm. The this unit was used is indicated by the statement of several Arab writers that the Great Pyramid of Gizah has a side of 460 “cubits of the hand” ; if they computed exactly, this cubit would be about 501.2 mm. Considering the medieval use of the term pes manualis, the term “ cubit of the hand” (dira al-yad) means a unit with something added. It is possible that the original trimmed barley cubit came to be considered a the natural unit corresponding to the canonical Arab cubit of 493.2424 mm.

Hinz calculates his value of the Arab cubit from the units of volume and weight. What appears from the computations is that one took as starting point a mina of 495 grams and calculated from it the cubic cubit as 250 minai. The mina is the old reduced mina of Mesopotamia (the mina sent to Charlemagne by al-Mamun ), and the cubit was calculated from it as in cuneiforn mathematical tablets, but without the small adjustments that one finds in these texts. As a result the cubit cubit was computed as 123,750 grams (495 grams x 250), producing a cubit which is 498.328 mm.

The Arab mile, however, is calculated by the main Arab cubit of 493.2424. This cubit is called “black cubit” (ad-dira as-sawda) in the texts; there is a legend to the effect that it god its name because one took as standard the cubit of a black slave belonging to al-Mamum. Scholars quote this childish explanation because they have not found a better one. But I suspect that in some Semitic language one confuced the root of “black” (Aramaic **) with that of “small”. In Akkadian documents the adjective sahru is frequently applied to vessels, in opposition to rabu is frequently applied to vessels, in opposition to rabu, “large”. In Moslem Persia the “little mina”, which is actually a derivation of the Euboic mina and is 5/6 of the mina of al-Mamun, is called mann-e sar i. Today this mina is calculated as 5/6 of kilograms; it is a double mina and the relation 5/6 established today with the kilogram the old Mesopotamian relation 5:6 between the Euboic mina and the normal reduced mina. By the present practice of Persia, the old mina of Mesopotamia and the mina of al-Mamun would be 500 grams instead of 495.

The canonical Arab cubit was called “small”, to distinguish it from the cubit formed by taking 2 feet instead of 1½ . Hinz notes that the cubit of 665 mm., which by my interpretation is a cubit of barley feet, is called “great Hashimite cubit” (al-hashimiyyah al-qudra), whereas the unit of 498.75 mm., which by my interpretation is equal to 1½ barley cubits, is called “little Hashimite cubit” (al-hashimiyyh as-sugra). The clarification of this terminology is important in order to dispel a misunderstanding that had a great impact on the thinking of Columbus.

Jakob van Gol, in his edition of the compendium of astronomy of al-Farghani (Alfraganus), quotes an Arab manuscript in which the “black cubit “ is described as a unit of 27 fingers. This definition is the same as that used by Herodotos who defines the royal cubit of Babylon (barley cubit) a unit as 27 fingers of a cubit called *** “common, standard”. The text of al-Farghani mentioning the calculation of the degree by al-Mamun specifies “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 per gradus aequales, secundum quod sollicite probatum est. Here the cubit of al-Mamun is called aequalis, which is perfectly correct according to Arab practice, meaning the “little cubit” of 1½ feet, in opposition to the larger and less usual cubit of 2 feet. The Hebrew translation by Jacob Anatoli of Naples defines the mile of al-Mamun as *** “by the meddle cubit, by the vulgar cubit.” But a Latin translation of the Hebrew one, understood the term according to the practice of the Talmud, in which the common cubit is the basic cubit, as in Greek terminology. As a result this translation reads: milliare autem habet cubita 4000, prout cubitum accipitur in mensura media. Cubitum habet six palmos communes. This Latin texts may have used a marginal note in which the cubit was correcty described as a cubit of 6 hands or 1½ feet; but the translation in mensura media is such that scholars of the fifteenth and sixteenth century understood that the length of the degree was 56 2/3 miles of 4000 basic cubits or 6000 Roman feet. Columbus too understood in this way the statement of al-Farghani.

4. The origin of the English foot—304.799978 mm. by the Act of 1824; 304.80 by the American standard —has been a deep mystery.

Some writers resort to the myth of an English king who had a foot of this length, assuming thereby a king of rather gigantic stature. A step in the right direction was made by Bishop Richard Cumberland, the founder of English utilitarian philosophy, at the turn of the eighteenth century, when he noticed a similarity between the structure of the English units and that of the units of Hellenistic Egypt. But, as I have reported, his remarks were immediately distorted, giving rise to two Anglo-Israel. The English foot would be the foot used for the construction of the Great Pyramid of Gizah; according to notions that have their basis in statements of Philo of Alexandria and in gnostic beliefs, the foot of the Great Pyramid would have been established by Abel, Abraham, Moses, or some other biblical figure acting under divine inspiration. I have recounted how these beliefs influenced the lore of Masonic societies and account, for instance, for the Great Seal of the United States. Pyramidism has been used to prove that the Anglo-Saxon races are the descendants of the Ten Lost Tribes, the true Israel. I have reported that the argument that English measures are divinely ordinary and are those prescribed by the Bible, has a major role in blocking the efforts to introduce the Frenth metric system in Great Britian and the United States. I have traced the first pyramidite book back to 1704; ever since the flood of pyramidite literature has continued unabated up to the present day. The last major effort appeared in 1953 as the Historical Metrology of A. E. Berriman. I have reported that pyramidism continues to have influence on some Egyptologists, even though there have been major Egyptologists who have tried to prove directly its lack of any scientific value. The new school of metrology, which is now absolutely prevailing in ancient studies, is an autgrowth of pyramidism.

Petrie who because interested in Egyptology and metrology, because of his original interest in pyramidism, in one of his better moments connected the English foot with a pes Drusianus used in the German provinces of the Roman Empire. Roman authors indicate that in the German provinces, there was used a pes Drusianus equal to 18 fingers of Roman foot; this foot must have got his name from Nero Claudius Drusus, the adopted son of Emperor Augustus and brother of the Emperor Tiberius, who established Roman rule in Germany. This foot is obviously a barley foot. The structure of English measures in which the rod is 16/½ feet and the chain (width of the acre) is 66 feet, indicate that the English foot was conceived as 10/11 of another unit which is the pes Drusianus. Because of this, one has tried to delve into the Germanic past to trace the origin of English measures. But, if the tendency of modern scholarship is to expect not too much rigor or consistency in Greek or Roman practices pertaining to measures, most lovers of Germanic antiquities see the world of the Germanic tribes as a misty land of fairy tales where even metrology would have the colorful and erratic quality of a dream sequence. Personally

I am most inclined to consider a German origin for the English units, with the proviso that units came to the Germans either from Asia Minor or from the Persian Empire. In a future study I shall show that the immediate antecedents of the runi alphabet are the alphabets of Asia Minor, such as the Lydian alphabets and measures, it is possible that the English measures came through the trade route that leads from the Black Sea to the Baltic and to Denmark and Scandinavia. I owe to my teather Claudius von Scherin the habit of interpreting Germanin antiquities by assuming a normal amount of rational behavior. My studies of the runes, which finally can be traced back to a Mesopotamian model, will destroy the captivating romances that have been written on the subject, but on the specific matter of measures, it can be shown that at least the Anglo-Saxon rulers of England were about as rational ass their Roman predecessors.

The following table makes clear that the English mile and its subdivisions are derived from the milion as used in the Roman Empire.

Great cubits
Barley cubits
Barley feet
Roman feet
English feet

The correspondence among the units is indicated also by Anglo-Saxon texts, such as Beda’s in which stadion is rendered by furlang.

The correspondence between English mile and milion is perfect if the barley cubit is calculated at a value intermediary between the trimmed and the natural version; in other words, the English foot corresponds to the great cubit calculated as 536.351 mm; the cubit of any Egyptian rods of the Hellenistic age. According to these figures the English foot should be 304.7448 mm.

There is a peculiar break in the multiples of the English foot at the level of the furlong. It has been observed that this break indicates that the English foot is related as 11:10 to some other unit. Petrie has concluded that the English foot was calculated as 10/11 of pes Drusianus, but this unit is a barley foot. If the barley foot is calculated at a value intermediary between the trimmed barley foot of 332.9386 mm. and the natural barley foot of 337.5 mm., there results the English foot. The Romans adopted the pes Drusianus in their Germanic provinces obviously for the reason that the barley foot was used there before the Roman occupation. It would be necessary to determine whether the barley foot used in Germany hah a value intermediary between the natural barley foot and the trimmed one; if it were so, then we would be sure that the English foot is of Germanic origin. The mather could be settled by the study of the dimensions of military camps and other settlements. There have been found in a Roman camp the two bronze ends of a measuring rod with markings by the Roman foot and by the barley foo, but the markings are too inaccurate to allow us to settle the matter. Equally important would be to investigate English architectural remains to determine whether the pes Drusianus or the English foot were used in England in Roman times. But those archeological reports I have examined are so vague as to measurements that I have not been able to determine even the units used in early Anglo-Saxon times. In relation to this last problem, it would be necessary to examine the metrology of the remains of the earliest churches the metrology of the Anglo-Saxon period, as St. Peter and Paul, St. Pancras, St. Mary, St. Martin in Canterbury, and those of Reculver, Lyminge and Rochester in Kent, and of Bradwell-Juxta-Mare in Essex.

If archeological reports seem to be inconclusive in relation to the metric problem (but they may be more meaningful to somebody more familiar with this particular material than myself ), there is a written text of the Anglo-Saxon period which is so precise as to fulfill the dreams of any metrologist.

There is a statute of Henry I (1100–1135 A.D.) that prescribes that the King’s peace shall extend in all directions from the gate of his residence for a distance fo 3 miles, 3 furlongs, 9 acres, 9 acres, 9 feet, 9 hand breadths and 9 barleycorns. This enactment is the restatement of an Anglo-Saxon regulation ascribed by scholars either to King Athelstan (924–940 A.D.) or to King Ethelbert (978–1016 A.D. ); this rule specifies that the girth or grith shall extend for III milia and III furlang and IX aecera braede and IX fota and IX scaefta munda and IX bere-corna. Commentaries on Anglo-Saxon laws do not even try to explain this text, since it is assumed that what is merely attractively bizarre can be expected in Germanic institutions. I have found only one writer on English metrology that mentions this text at ll. In my opinion the text is clear and provides precious information. The total length of the grith is 18.426¼ feet. Since the parasang, which is an hour of march, is 18,000 artabic feet, and their value is expressed in geometric feet, since the geometric foot was considered the scientific foot of Middle Ages. According to geometric foot of 297.761 mm., the grith extends for 5486.1211 m; dividing this figure by 18,000 there results an English foot of 304.786 mm.

It is easy to see how the figure of the Anglo-Saxon regulation was calculated. If the milion is considered as composed of 3000 Egyptian royal cubits, it contains 5275 Egyptian feet. If it is calculated as 3200 barley cubits (3000 barley cubits increased as 32:30) , it contains 4800 barley feet, so that, by assuming a relation 10:11 between English foot and barley foot, it contains 5280 English feet. The last figure is the one used in calculating the division of the English mile; but in the Anglo-Saxon regulation a milion of 5400 geometric the equivalent of Egyptian feet. Calculating exactly the regulation should have made the King’s grith equal to 18,426,54 geometric feet, but the figure 18,426.25 was chosen in order to obtain the mnemonic formula: 3 miles, 3 furlongs, 9 acres, 9 feet, 9 handbreadths, and 9 barleycorns. This makes the English foot equal to 304 786 mm.

The amazing result is that the English foot has not changed more than 2 hundredths of millimeter since Anglo-Saxon times. Some of the rods considered in establishing the standard of 1824 had a value quite close to that of the Anglo-Saxon standard.

This positive fact by itself destroys the notion that so-called primitive people do not have precisely fixed standards. Even during the barbaric ages of Europe, standards were kept with the utmost precision.

5. The study of the value of the English standard of length has been confused by the dogma of the new school of metrology that standards were variable up to modern times. Sir Charles Arden-Close in 1947 expressed the opinion that the English mile was established by an act of Queen Elizabeth I; several writers have declared that the present English foot dates from the age of this queen. Sir Charles M. Watson in 1913 stated: ‘Prior to the thirteenth century, the length of the foot in England was uncertain, and there appears to have been several measures in use, varying from the Roman foot of 11.66 English inches to the Belgic foot of 13.12 English inches to the Belgic is the is the pes Drusianus of Germany, which according to some writers was brought to England by the Belgae, the Celtic tribe that Caesar found in Southern England.

In 1877, before the birth of the new school, H. W. Chisholm, who was Warden of the standards, expressed a sounder view: “There can be but little doubt that our imperial yard is substantially the same length as the old Saxon yard.” In my opinion the qualification “Substantially” could be removed from this statement.

When the Royal Society was established in 1662., one of the first tasks it set to itself was that of establishing a scientific standard of length. The French Academie des Sciences and Academie des Inscriptions established in the same decade, also applied themselves to this same task. In France a solution of the problem was attempted by establishing in 1667 a new pied de roi related by a simple fraction to the Roman foot; but in England it was preferred to set a standard by the length of the pendulum that beats the second. In the first half of the century Tito Livio Burattini had suggested the adoption of a new basis called meter (meter cattolico ) for a decimal system of measures; he thought that the calculations should be performed by the English standard which he considered the most accurately defined in Europe, but that standard should be the old Egyptians one (which is actually the basic foot of the ancient world). While engaged in measuring Egyptian monuments, he was Joined by Greaves who had brought with him an accurate copy of the standard of Guildhall, the seat of the municipal goverment of London. Greaves and Burattini measured together the Great Pyramid of Gizah, the side new standard. When Greaves departed from Egypt he left with Burattini his rule, which was a bar of10 feet divided decimally into thousandths of foot, and the latter continued his survey. But while returning to Europe Burattini was attacked by bandits in Hungary and robbed of his notes. After this he settled in Poland, where he became a major political figure, and from there announced his project of meter, substituting the length of the pendulum for the Egyptian standard. As a result of this, the Royal Society thought of adopting as standard the length of the Society, tried to implement Burattini’s plan by calculating a metrum catholicum equal to 1/3 of the length fo the pendulum that beats the second at the latitude of London; he announced that the length was 4/3 of the palmo of Genova. The palmo is half a cubit, and the cubit of Genova is a slightly shortened trimmed barley cubit. At the moment of the adoption of the French metric system the palmo was estimated at 248.283 mm. (cubit of 496.566); Bernard in 1688 estimated the foot of Genova at 1089/1000 of English foot of 1824 A. D. In the same period of time another member of the Royal Society, John Evelyn, in a trip to Italy recorded the length of the units of Bologna and Florence in order to evaluate exactly the studies of Galileo and Riccioli concerning the pendulum. The project of building a new international system of measures on the length of the pendulum was attractive, because thereby in one unit there could be coordinated time, length, volume, and weight. But in the early part of the eighteenth century it was realized that the length of the pendulum is affected by the latitude, the elevation above sea level, the presence of large land masses, and other factors, so that it does not provides a reliable reference. But the influence of the first plan of the Royal Society lingered on. Jefferson blocked the adoption of the French metric system in the United States because it was not based on the pendulum; he was not as well informed about scientific developments as he believel to be. The British act of 1824 in theory defined the English foot by the pendulum, but in fact made reference to a copy of the Bird bar of the Royal Society.

It was only in 1742 that the Royal Society decided to construct a standard adequate for the scientific needs of the time, by marking most accuratly on a bar the length indicated by the best public standards available. Traditional English standards consisted of a brass rod encased into a matrix or bed also of brass since the rod had to be lifted , of necessity there was some play between the rod and the matrix indicated the right length. Apparently this method allowed to test bars rapidly by droppin, them into the matrix, but it must have cused the bars, which were the standard concretely applied, to be on the scanty side. In 1642 the Royal Society adopted the better method of using long bar and marking on it with two dots the length of the yard. George Graham who constructed the standard, apparently was expected to mark the length of a yard kept at the Tower of London which was considered the most reliable one; but the marked with E (for “English”) a length intermediary between the yard of the Tower and the shorter value of a yard with the seal of Elizabeth I kept at the Exchequer. At the Exchequer there were, as there are still today, a yard of Henry VIII (1490 A.D.) and ell and a yard of Elizabeth I (1588 A. D. ). The objection was raised that only the standard of Elizabeth I could be considered legally valied. For this reason in 1743 the Royal Society proceeded to a survey of the standards available in London, such as that of the Tower, those of Guildhall, and that of the data Clockmakers Company. Following this survey, of which the data are preserved, Graham marked on his bar a length called Exch corresponding to the length of the rod in the Yard of Elizabeth. In 1758 and 1760 John Bird constructed two new bars; the only point significant for historical metrology is that he considered that the joint between the rod and the matrix of the yard of Elizabeth I; he marked a length intermediary between E and Exch, but closer to the former. This became the official standard by the act of 1824. The British Board of Trade and American practice, sanctioned by act of Congress in1928, choes the value of 304.80 mm. which can be more easily related to the French metric system.

The yard of Elizabeth happened to be one of the shortest reference available. The very ell of Elizabeth would have given a longer standard. If the standard of the Tower had been adopted, the present English standard would have been identical with the Anglo-Saxon standard I have calculated.

By the survey of 1643 this is the relative value of the standards:
-rod of the yard of Elizabeth I
-matrix fo the same
-rod of the ell of Elizabeth I
-rod of the yard of Henry VII
-rod of the yard of the Tower

As I have stated, the standard of 1824 A. D. is intermediary between the rod and the matrix of Elizabeth.

The oldest English standard of which I have found mentioned in print is a yard preserved at the Westgate Museum at Winchester. This yard was constructed under Henry I (1110-1135), but is two extremities were countersigned with a seal under Edward I and Henry VII. This yard is described as 0.04 inches short (0.33 mm. for a foot), but a certain amount of wear must be taken into account).

A perfection comparable to that of medern scientific standards should not be expected from these bars; the preservation of the standard was better assured by the sample weights. The limit of perfection achieved y the lineal standards is indicated by those of Guildhall. By 1743 the rods of Guildhall had disappeared and there remained only the matrices. This indicates a decline in the importance of the standard of Guildhall; this decline may be a result of the policy of the Stuart king aiming at a reduction of the political power of the municipal body of London. On April 22, 1743, the President of the Royal Society accompanied by six members, represented himself at Guildhall where he examined the standards with the assistance of the municipal officials who were in charge of them. A matrix of yard (36 inches) exceeded the Graham length E between o.o434 and 0.0396 inches; a matrix of ell (45 inches) was compared with the length of the ell of Elizabeth I an found to exceed it by 0.0444 and 0.0258 inches. Two figures are given because the hole in the matrices was not perfectly square; the figures indicate that an approximation of about 0.02 inches (o.5 mm.) for an ell (0.125 mm. for a foot ) was considered adequate. interpreting the Egyptian royal cubit as a septenary unit.

Newton based his calculation on Greaves’ report about the dimensions of the so-called King’s Chamber inside of Great Pyramid of Gizah (V Dynasty). He assumed correctly that the chamber had been calculated as 20 x 10 cubits, and with this assumption arrived at a cubit equal to 1719/1000 of English foot. Assuming that the foot of Guildhall, by which these calculations were performed, was equal to the American foot, the cubit of the Great Pyramid would be 523.951 mm. Modern calculations of the dimensions of the Great Pyramid indicate the use of a cubit which is very close to 524 mm. Petrie calculated with extreme care the dimensions of the King’s Chamber and arrived at a cubit of 524.053 + 0/10 mm.; in performing the computation he deducted from the actual dimensions the spacing of the blocks which he considered the result of earthquakes.

There seems to be a conflict between the value of the cubit of the Great Pyramid and that of 525 mm. indicated by most measuring rods and by many other monuments. The value of 524 mm. occurs in the Complex of Zoser (first king of the IV Dynasty), which is the first large stone construction of Egypt; its architect seems to have been the famous Imhotep. On the other side, the measurement of one side of Pyramid of Newserre (VI Dinasty) at Abydos indicates the use of a cubit of 525.07 mm. Petrie tried to explain the cubit of 524 mm. by asserting that the cubit, like all units of length , became longer in the course of time; but he contradicted himself by noticing the occurrence of a cubit of 525 mm. in the constructions of predynastic Egypt. Saigey reports that an Egyptian palette of the Louvre Museum is calculated by a cubit of 525mm. Palettes are the most important artistic and historical remains of the predynastic period; they consist of slabs of slate illustrated with carvings. Since the carvings are distributed in bands and the figures are often in mathematical proportion , by studying the palettes, of which there are large collections, by studying the palettes, of which the are large collections, we could learn a good deal about the metric of the predynastic period.

There is a metrological explanation for the existence of a slightly shortened royal cubit. The royal cubit is derived from an ordinary cubit of 450 mm., corresponding to a foot of 300 mm. The cube of the foot is the talent of 27,000 grams (basic talent brutto) which is divided into 3000 basic sheqels of 9 grams. But the basic sheqels can be calculated also by dividing by 10,000 the cube of the cubit of 450 mm. (basic load brutto), so that the basic shequel may also be 9.1125 grams, with a discrepancy komma (80;81). In Egypt the basic sheqel, called qedet or qet ( Kite in Coptic), was calcuated also by dividing by 16,000 the cube of the royal cubit. In this way there was obtained a value that is intermediary between the two mentioned above: the cube of the royal cubit of 525 mm. gives a basic sheqel of 9.043944 grams.

Petrie, who was more concerned with the value of the qedet than with any other metrological problem, by examining hudreds of sample weights, arrived at the conclusion that it had a value varying between 139 and 141 English grains (9.00 and 9.1368 grams) with several pieces indicating specifically a value of 140 grains (9.0729 grains). He concluded that within the range he had indicated the sample weights could be subdivided into types; but he did not try to calculate by a unit more exact than the English grain. However, among the samples he considered particularly important, there are some that indicate a qedet of 139.5 grains (9.03960 grams).

A similar value of the basic sheqel is indicated by the units of Mesopotamia. The normal mina of Mesopotamia has a value of 495 grams, corresponding to 55 basic sheqels of 9 grams. But this mina was calculated also as half of a cube with an edge of 6 sexagesimal fingers; this mina (stereometric mina netto) was 498.226 grams or, with an adjustment in relation to other units, 497.664 grams. These values indicate a basic sheqel of 9.0586 and of 9.0484 grams. Hence, both in Egypt and in Mesopotamia, there occurs a basic sheqel computed as intermediary between the values of 9.0 and 9.1125.

If the cube of the royal cubit is calculated as equal to 16,000 sheqels of 9 grams, the cubit becomes 524.104 mm., that is, the cubit of the Great Pyramid. If the qedet is calculated by the maximum value of 9.1125 grams, the cubit cubit is 145,800 cc., corresponding to a cubit of 526.3230 mm. This value of the cubit cubit is indicated by computations in which it is equal to 5 artabai (the artaba is 29,160 c.c.). There are buildings and measuring rods that document the use also of this longer royal cubit. For this reason some scholars have calculated the royal cubit around 526 mm; Brugsch calculated it as 526.86 mm. This cuibt is indicated, for instance, by a measuring rod of the Anastasi collection that embodies a cubit reported as 526.5 mm.

2. If there is some doubt about the exact value of the Egyptian royal cubit, practically all scholars agree that the Egyptian foot had a value of 300 mm. But metrologists have not been able to exploit to the full this fundamental datum, because they have not succeeded in solving a problem that came to the fore the moment Newton calculated the length of the Egyptian royal cubit. It was clear to Newton and it has been clear to those who followed him that the Roman foot of about 296 mm. is a modification of the Egyptian foot, but nobody has been able to determine the principle linking them. The problem is vital also because most metrologists agree that the Egyptian foot of 300 mm. is the basic foot of the ancient world.

Newton instead of offering a mathematical explanation, suggested that the Roman foot is shorter because lineal units tend to become shorter in the course of time. Newton reasoned on the basis of a value of the Egyptian foot that made it too short and used reports about the value of the Roman foot that made it too long, so that he thought that the Roman foot is merely an inaccurate copy of the Egyptian foot.

If Newton assumed that copies of rules tend to become shorter in the course of time, another genius of mathematical physics advanced the opposite opinion. Father Boscovich in reporting with Father Maire abut their triangulation of Italy based on the Appian Way, in order to explain why the Roman foot used in calculating Italian roads in medieval and modern times was longer than the ancient Roman foot, suggested that lineal units become longer through repeated copying. The empirical truth is that generally a rule in which the length is indicated by the two ends of the bar, tends to be shorter, whereas a rule which consists of a longer bar with the unit length indicated by marking two lines, tends to be longer. But the cumulative effect of repeated copying considered by Newton and Boscovich does not apply in the long run, because lineal units are recalculated from the units of weight, and it is most easy to copy units of weight with an approximation such that the units of length are not affected within the limits of approximation that can be acheived without optical instruments.

The hypothesis of Newton has been accepted uncritically by some metrologists of the second half of the nineteenth century and used by them to avoid the task of solving most vital, but difficult, problems.

When the first reports about Mesopotamian archeology became available, Brandis tried to prove what had been surmised by Böeckh, that all units derive from Mesopotamia. He did creditable work in relation to the units of volume and weight, but in dealing with the derivation of the units of length, he constructed an infantile theory. On the basis of one of Oppert’s early studies, he assumed that in Mesopotamia there had been used a foot of about 320 mm., which would have been gradually shortened into an artabic foot of 308 mm. used in Greece, which in turn became the Roman foot of about 296 mm., when it passed into Italy. The passing of time and geographical migration would cause a shortening of units. Uzielli tried to explain this assumption of Brandis by suggesting that the ropes used as standard in Athens became shorter in the more humid climate of Rome. This laughable explanation submitted by a metrologist who is not an incompetent one, manifests the absurdity of the theory of progressive shortening.

Brandis had found that units of weight and volume are derived from a single model, and assumed that units of length too must be derived from a single model. But in explaining the derivation of the latter, he constructed a theory that would make units of length totally uncertain and as a result also the units of weight and volume. This most unsatisfactory method of explanation was accepted by Hultsch. According to him, the Egyptian royal cubit is identical with the Mesopotamian cubit; from these cubits of 525 mm. there is formed a foot of 315 mm. by irregularly taking 3/5 of cubit. This foot of 315 mm. would appear in Greece in the form of a foot of 320.6 or 321 mm at Olympia, or 318 mm. at the Temple of Zeus at Nemeia and at the Temple of Apollo Didymaios at Miletos, of 314.5 at the Heraion of Samos, of 314.3 at the Temple of Apollo Epikourios at Bassai. Anticipating what I will explain later, I can say that Hultsch noticed the occurrence of the wheat foot which is 314.44 trimmed and 318.75 natural; this foot occurs at Olympia also in a special form called Olympic foot in which it is calculated as 25/24 of artabic foot or 321.107 mm.

According to Hultsch, this foot would have become a foot of about 308 mm. (which is actually the artabic foot) and finally a foot of 297.5 of which he finds evidence at Olympia. (This is actually the geometric foot of 297.50 mm. which came to be considered the scientific Roman foot in the Middle Ages and accounts for the mile used in measuring Italian roads, the mile that caused Boscovich’s puzzlement). The Roman foot would be the next step in development. About the value of the Roman foot Hultsch cannot decide himself between a foot of almost 296 mm. and a foot of 295.5; actually he noticed the occurrence of the normal Roman foot of 295.9454 mm. and of a slight variation calculated as 24/25 of artabic foot, that is, 295.416 mm. Hultsch grants that the roman foot appears as a stable entity from the earliest times to the Empire; but in order to preserve the doctrine of progressive shortening, he claims that it became 294.2 in the age of Septimius Severus and Diocletian. In reality most of the monuments of this period are calculated by the usual Roman foot; for instance, the Senate Hall was reconstructed by Emperor Diocletian by using a foot of 296 mm. What Hultsch noticed was the occurrence of a special Roman foot (pes Cossutianus) equal to 294.395 and used to calculate with extreme precision the value of ¹. This foot is 3/¹ of increased artabic foot.

Petrie adopted the views of Hultsch for what concerns the Roman foot, except for the figures that are rounded ones. An Egyptian foot of 300 mm. passed into Greece as 297, but was imported into Italy as 296, to become 294.6 in Rome. But when he comes to the problem that Boscovich had in mind, that of explaining why the foot of the Italian miles is slightly longer than the ancient Roman foot, Petrie incorporates Boscovich’s hypothesis and claims that “units of length become longer by repeated copying.” He even tried to calculate how much ancient and English measures increased in each century.

Petrie used the theory of progressive lengthening in order to argue that the length of the Egyptian royal cubit was determined by the length of the pendulum that swings 100,000 times in a day at latitude 30° (latitude of Memphis and the Pyramids of Gizah). This pendulum of 740.57 mm. is the diagonal of a square, the side of which is 523.62 mm. This would be the original Egyptian cubit which would have become 524 mm. by the time of the construction of the Great Pyramid and 525 mm. later; the Hellenistic rods which indicate a much longer cubit and are based on a different standard (as Saigey and Böeckh have noted) would be according to Petrie the result of this process of progressive lengthening.

Petrie had his weak moments because he never succeeded in completely sloughing off the pyramidite beliefs of his father. But, in a more reasonable mood, he observed not only that the royal cubit of 525 mm. was used in predynastic Egypt, but reported repeatedly that he had met with Egyptian constructions and with Egyptian rules calculated by the Roman foot. Unfortunately he never provided a detailed numerical report; but, according to what I shall explain below, it is most likely that this particular subvariety of Egyptian foot known as Roman foot, was used in Egypt. A clear piece of evidence is a rectangular bronze measuring rod of the Museum of Torino; on one face there is a hieroglyphic inscription which is rather ignorantly copied from an older standard inscription found on another rod of the same museum; beginning with the Sai’te Dynasty, hieroglyphic writing becomes a gradually less competent mechanical imitation of older models. On one of the three other faces there is marked a standard Egyptian royal cubit of about 524 mm; on the second face there is marked a cubit of about 524 mm; on the second face there is marked a cubit of about 532 mm. which is the Babylonian Egyptian cubit and which is common in Egypt in the Hellenistic age, but may have been introduced by the Persians (this unit is the septenary cubit corresponding to the present English foot); on the third face the marking indicates a cubit which is reported as being about 1 cm. shorter than that of 524 mm. and which most likely is the septenary cubit corresponding to the Roman foot. Such a cubit would be 517.905 mm. Lepsius thought that this rod was a forgery, but Segrè who wrote his treatise on metrology in Torino and examined the rod repeatedly defends its authenticity, even though he cannot account for its markings.

3. The preposterous belief that ancient units of length could be determined by the length of the pendulum was accepted by a scholar who, perhaps was not as brilliant and intelligent as Petrie, but usually was scientifically rigorous and alien from any flight of mystical imagination: Lehmann-Haupt.

In the age of Petrie and Lehmann-Haupt, Orientalists were divided between those who believed that civilization had been born in Egypt and those who, on the basis of the new archeological evidence, claimed that it had been born in Mesopotamia. Lehmann-Haupt held the second view and was correct; he was also convinced that the metric system had been conceived in Mesopotamia, but could not prove that the units of Mesopotamia were the oldest ones. The Egyptologist Brugsch had shown in polemics with Lehmann-Haupt that for structural reasons the Egyptian units are more archaic and that those of Mesopotamia are derivative. I have explained this phenomenon by considering that in the case of diffusion archaic institutions may be preserved outside the center of diffusion, in which the original stimulus may continue to be effective and create new forms that obliterate the archaic ones. But Lehmann-Haupt could not use this explanation which I have learned from the more subtle contemporary anthropology studies. He felt that, unless he found a most solid argument, he would have to bow to Brugsch and, hence, also accept Petrie’s contention that civilization and the metric system had been born in Egypt. If Petrie claimed that the Egyptian cubit was based on a pendulum that swings 100,000 times in a day, Lehmann-Haupt could claim that the Mesopotamian cubit was based on a more reasonable pendulum, a pendulum that beats the second. He calculated that at latitude 30°, which is not only the latitude of Memphis in Egypt but also the latitude of the mouth of the Tigris and Euphrates, the pendulum is 992.35 mm.

At times, Lehmann-Haupt presented the theory of the pendulum as a mere possibility, but the fact is that he distorted all other data in order to make the figures agree with the length of the pendulum. He claimed that Hermann von Helmholtz had approved of his theory, but actualy all that the great physicist had said was that Lehmann-Haupt’s argument was arithmetically correct. Helmholtz could not know that the empirical data had been distorted.

Lehmann-Haupt concluded correctly that coins indicate an Attic mina of 432 grams and a Roman libra of 324; but continued to consider as standard the mina of 436.66 and the libra of 327.45, erroneously indicated by Böeckh. In order to explain the difference, he introduced the concept of Schlagschatz or difference in weight that defrays the cost of minting. By introducing this problem he made an important contribution to metrology and to numismatics; but he stooped short of the goal by not noticing that the Schlagschatz accounts for the relation 20:21 between the Attic monetary mina of 432 and the Attic weight mina of 453.6 grams (English pound avoirdupois). Nevertheless it was he who correctly interpreted Aristotle to mean that in Athens there were weight units and monetary units related as 21:20. He clung to a mina of 436.66 in order to calculate the Mesopotamian mina as 491.17 grams (6/5 of Attic monetary mina) because he assumed that 250 such minai would make a cubit of which the edge is half the length of the pendulum. He was able to solve the key problem in the interpretation of cuneiform stereometric texts by noticing that the double mina is calculated as a cubit with an edge of 1/5 of cubit, but was not able to exploit this discovery, with the result that the interpetation of these texts suffered up to today.

The notion of the pendulum became such an idée fixe with Lehmann-Haupt, that when Mesopotamian scholars began to agitate the problem of the origin of sexagesimal numeration, he asserted that the computation by 360 came into existence because this is the number of times the length of a pendulum fits into a stadion. Hence, the pendulum that beats the second would have preceded the adoption of sexagesimal computation.

Lehmann-Haupt composed a masterly study of the ancient units of length indicating how they were all related by simple fractions, but, having started with a wrong value of the Mesopotamian cubit as 496.165 mm. instead of 499.4, arrived at an Egyptian foot of 297.7 (instead of 300), which he had to identify with the Roman foot. He distorted all other lineal data accordingly.

Lehmann-Haupt obliterated the distinction between Egyptian and Roman foot and thereby brought the problem back to the point where it was in the time of Newton. But Lehmann-Haupt had in his hands the key to the solution of the problem. He considered the greatest achievement of his scholarly life to have remarked that most ancient and medieval units of volume and wieght exist in two varieties related as 24:25. Oxé has shown how this distinction between what he called netto and brutto units is central to the structure of ancient metrics.

If there are units of volume and weight related as 24:25, there must be units of length related as 3Ã24:3Ã25. If the Egyptian foot of 300 mm. corresponds to a talent of 27,000 c.c., divided into 50 basic pints of 540 cc., the corresponding unit netto of 48 pints is the Roman quadrantal, or Attic monetary talents (60 Attic monetary minai or 80 Roman librae) of 25,920 c.c., so that the Roman foot must be 295.9454 mm., a figure that agrees with the most accurate evaluations of the empirical data. This is the simple explanation of a problem that has bedeviled metrologists for so long.

The distinction between Roman and Egyptian foot corresponds to the distinction between netto and brutto units of volume and weight. They relate as 3Ã24:3Ã25; this ratio could be calculated as 73:74; calculating exactly, it would be 73:74.00013.

Since the Roman foot is divided into 16 fingers or 12 inches, the relation could be calculated in practice by the ratio of 72:73, which would make the Egyptian foot equal to 300.056 mm.

Roman and Egyptian foot, together with the artabic foot, were the fundamental lineal units of the ancient world. Roman foot and artabic foot are related as 3Ã8:3Ã9 or as 24:25. roman and Egyptian foot are related as 3Ã24:3Ã25.

6. Since I have traced the origin of the English foot and determined thereby the units with which it is connected, the history of English units of volume and weight does not present any major difficulty.

A fundamental unit of the ancient world , as Hultsch has demonstrated, was the cube of the Egyptian foot, which is the basic talent brutto of 27,000 grams. Oxé calls it the talent of 1000 ounces, since it was equal to 1000 Roman ounces of 27 grams. Since the English foot it an increased Egyptian foot, according to the practice of the Hellenistic age, English units of volume and weight are the ancient basic ones adjusted according to the increase of the foot.

The main unit of the English system is the cubit foot, the firkin, of 1000 ounces. The ounce is increased over the Roman one according to the foot, being 26.350 grams by the present definition. The cubit foot is calculated as 62.5 pounds avoirdupois of 16 ounces, the ounce being 437.5 grains. At times, he ounce has been calculated by the round figure of 438 grains (as in the writings of Greaves) making the pound avoirdupois equal to 7008 grains, instead of 7000.

By the present calculation of the pound, one thousand ounces are a trifle more than a cubit foot, because when the units were regulated in 1824, one did not take into account the relation of the pound to the foot; one was concerned with making the pound avoirdupois, considered the only pound, equal to 7000 grains. The legislation of 1824 was colored by a deliberate hostility to the French metric system and, hence one did not define the units beginning from the foot. The foot was defined by the pendulum. But the legislation was based mainly on the reports of a committee of the House of Commons appointed in 1758 “to enquire into the original standards of weights and measures in this kingdom.” On that occasion the metrologist Harris, King’s Assay Master of the Mint, having been consulted as an expert, declared that “a lineal standard should be the standard of all measures of capacity.”

Thomas Everard, a writer on metrology and an official of the excise, in 1696 found that according to the standards of the Exchequer a cubit foot was exactly 1000 ounces of water. The same result was obtained by a group of Oxford scholars in 1685. The only disagreement is represented by John Wybard who at the middle of the seventeenth century found 1000 ounces of pure rain or running water to be 1725.56 cubic inches, instead of 1728; but later writers, such as Jonas Moor and Everard, object that Wybard calculated the ounce by the rough relation 14:17 between the pound Troy and pound avoirdupois (the exact relation is 144:175 ). Hence a detailed study of the exact value of English weights would allow to determine exactly the small variations through history of the English foot; this length in turn should be tested on the monuments.

Since the cube of the Roman foot is 24/25 of the cube of the Egyptian foot and the theoretical English foot is 36.35 of Roman foot, the cube of the English foot should be 62.5354 pounds avoirdupois. Through the history of English measures one has calculated this cube as 62.5 pounds. The contemporary English definition of the units by which a cubit inch of distilled water at temperature 62 Fahrenheit, the barometer being at 30 inches, must weigh 252.326 grains, of which 7000 go to the pound, makes the cubit foot equal to 62.2885 pounds of water. The cube of the American foot by the same reckoning corresponds to 62.428 pounds of water. This proves the amazing stability of English units through more than a millennium of history. The reckonings that occur in the English system, in turn, had already been formulated in the Roman Empire, and probably earlier.

The present pound avoirdupois is 453.6 grams; The history of the submultiples of the English cubit foot has been influenced by the effort to reduce the number of pounds or pints in a cubit foot to either 60 or 64. The pound usually has been kept stable around the figure of 62.5 pounds to the cubit foot, but the pint, which should be equal to the pound, has been calculated as 1/60 of 1/64 of cubit foot. For this reason there are even today several types of pint. In the American system there are 59.8442 liquid pints in a cubit foot. The submultiples of the English foot have been influenced by their confusion with the very similar submultiples of the artaba (cube of the artabic foot of 307.796 mm.) The artaba is 9/8 of basic talent netto (cube of the Roman foot) and 27/25 of basic talent brutto (cube of the Egyptian foot). If the English cubit foot (cube of the increased Egyptian foot) is calculated as 62.4 basic minai or pounds avoirdupois, it is 26/25 of the basic talent netto. The artaba contains 64.8

In the Roman Empire, the basic talent netto (cube of the Roman foot, quadrantal of 80 Roman librae) was divided into 3 Roman modii of 16 sextarii or basic pints, whereas the artaba was divided into 3 modii castrenses or 18 sextarii or 20 Alexandrine sextarii (reduced pints). A Roman iugerum was sown with 5 Roman modii, whereas a iugerum castrense was sown with 5 modii castrenses; I have shown that the English foot is calculated as 1/175 of the side of the iugerum castrense.

One of the basic problems of ancient metrics was that of reconciliating a division of the cubic units by 60 with a division by 64. From the point of view of sexagesimal and decimal computation, it is easier to divide a cube by 60, but from the point of view of geometric construction it is easier to divide a cube into 8 smaller cubes and in turn divide these into 8 smaller cubes. This second procedure is exemplified by the division of the English cubic foot into 8 gallons of 8 pints or 8 pounds of water (wine). The gallon calculated as 1/8 of cubic foot would be 221 cubic inches. The gallon of Guild Hall was 224 cubic inches, but most often the gallon has been so calculated as to make the cubic foot equal to 60 pints. The American gallon of 231 cubic inches is based on a statute of Queen Anne defining the wine gallon as a cylinder with a diameter of 7 inches and a height of 6 inches (230.9070 cubic inches). A gallon such that 60 pints make a cubit foot, should be 230.40 cubic inches. The wine gallon of Edward I (1272-1307) appears calculated as 230½ cubic inches.

Because cubic units could be divided either by 60 or by 64 there were in the ancient world units discrepant by a diesis (I shall show that the interval diesis of the musical scales derives from the units of volume). I have shown that the relation 15:16 is particularly important in Mesopotamia; the regular occurrence of this discrepancy in medieval measures has been stressed by several metrologists. A typical embodiment of this discrepancy was the marc sterling, which is 16/15 of Roman libra. In France this unit was called livre de La Rochelle, after the most important French harbor on the Atlantic; in England it was called Tower pound, because it was used as the monetary standard up to 1527 (when it was abolished by Henry VIII), and at the Tower of London there was located one of the principal mints. This unit was fixed in England at 5400 grains; 16/15 of an ancient Roman libra would be 5333.3 grains, The actual weight of the Tower pound is exactly related as 16/15 to the Roman libra increased by a komma, the libra of 328.05 grams that I have calculated as the medieval Roman libra. the ancient Roman libra of 324 is exactly 5000 grains.

Today there are slightly discrepant definitions of the grain, because the bill of 1824 made the pound avoirdupois equal to 7000 grains, whereas before it had been 7004. One usually reckons the grain as 0.648 grams, even though it is legally defined also as 7000/7004 of this weight.

The pound Troy is based on a Roman ounce increased by 1/8; it should be 30.3750. The present ounce Troy is 31.104 grams (480 grains); considering the medieval libra of 328.05 grams, one would expect an ounce Troy of 30.7547. The last weight well corresponds to the Continental values of the once de Troyes; but in England, where it was essentially a foreign unit, the pound Troy was fixed at 5760 grains, that is, 16/15 of the Tower pound, in the great reordering of weights of 1587, under Elizabeth I. The pound Troy is theoretically 3/4 of the Alexandrine sextarius of 486 grams of water, which became the Carolingian libra (Paris livre). Since there are 60 Alexandrine sextarii in an artaba, the artaba contains 80 pounds Troy; hence, the pound Troy has the same relation to the artaba that the Roman libra has to the basic talent netto, which is 8/9 of artaba and contains 80 librae.

In England one identified the difference between Troy weights and avoirdupois weights, with the difference between units of volume filled with wheat and filled with water (wine). This relation should be 4:5, but for the sake of convenience, in England as in the ancient world, this relation could be calculated as 5:6. The pound Troy is related to the pound avoirdupois as 144:175 in order to make the Troy pound 16/15 of the Tower pound; a ratio 4:5 between pound Troy and pound avoirdupois would be 140:175, and a ratio 5:6 would be 144:172.88. I have reported that Wybard calculated the ration as 14:17.

The pound Troy is important today only because it defines the grain, the unit used for precious metals. According to the reform of Elizabeth I, the pound Troy is divided into 12 ounces of 20 pennyweights; the pennyweight is divided into 24 grains, whereas it used to be divided into 32. Numismatists have developed farfetched theories about the metrology of the pennyweight, the English denarius. The explanation is most simple: in the Carolingian system 240 denarii are to be struck from a pound of silver; in the English system 240 denarii are equal to a pound Troy.

A most important unit is the Winchester bushel, which originally was equal to 64 sextarii or 4 Roman modii (34.560 c.c.). This unit may have been legally established when Winchester was the political center of the Anglo-Saxon kingdom. The Winchester bushel is today the official unit of the United States (35.2393 c.c.), defined as 2150.42 cubic inches or 77.62701 pounds avoirdupois of water, being a cylinder with a diameter of 18½ inches and a height of 8 inches. In 1696 when a bill was pending concerning an excise duty on malt, it was considered proper to define the bushel. Everard conducted a test before some members of the House of Commons and found that samples with the seal of Henry VII kept at the Exchequer indicated a bushel of 2145.6 cubic inches. Hence, at the time of Henry VII the bushel was still exactly the ancient unit of 4 modii. Everard suggested that the bushel be made of the size of 18½ inches of diameter with a height of 8, for the sake of round figures. As a result the Winchester bushel, now the American bushel, became a unit of 2150.42 cubic inches.

The present Imperial bushel of 36,368 c.c. is the ancient artaba of 29.160 c.c. increased by ¼ (36.450 c.c.), according to the specific gravity of wheat. In the ancient world wheat units were increased by ¼ or by 1/5 in some calculations. In the United States the Winchester bushel when heaped must be 1¼ struck bushels. The Imperial bushel is 11/5 Winchester bushels. A statute of Henry VII defines the bushel as containing 8 gallons of 8 pounds Troy of wheat; hence a gallon (8 pounds avoirdupois) contains 10 pints of wheat. Local practices preserve even more than legislation the procedures and the units of the ancient world. For instance, I have found that the Clerk of the Peace of Bedford reported in 1854, on occasion of a Parliamentary inquiry on measures: “wheat is usually sold by the load of five bushels, each bushel containing eight gallons; the average weight per bushel is 62 lb averdepois; sometimes the seller guarantees a given weight per bushel... and makes good the deficit, if any.”

The stability of English measures indicates that ancient techniques allow to preserve correct standards through the centuries, and the millennia, provided there is a sufficiently effective government. In England the most important official....

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