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The Derivation of European Units


1. English units of weight were considered the best defined and stable of Europe. For this reason when Tito Livio Burattini, a younger contemporary of Galileo, advocated the adoption of a decimal metric system based on what he called a metro cattolico or metro universale, he thought that it could be based on English standards. Burattini was not the first to propose the adoption of a decimal metric system, but he was the first to advance a project that recived wide attention and was the one who first suggested the name meter for the basic unit of length. Since the cube of the English foot, the firkin, contains 1000 ounces averdepois of water, all that was necessary to establish a decimal metric system was to divide the English foot decimally. But there was no agreement on the exact length of the English foot.

English metrologists of the seventeenth century consider the correct value of the English foot that indicated by the iron yard of Guildhall, the seat of the municipal goverment of London. From various sources it can be inferred that the foot of Guildhall had a value of 304.919 mm.; it was the edge of a cube containing 1000 ounces averdepois rain water. Before 1742 A.D. foreign writers indicate a value of the English foot that corresponds to that of Guildhall. For instance, in 1740 A.D., Anders Celsius, who was connected with the measurement of the arc of meridian in Lappland and for this reason was particularly concerned with the exact value of lineal units, states that the English foot is 1027/1000 of Swedish foot, which means 304.916 mm., by a Swedish foot of 296.9. Taking the same value of the Swedish foot and Celsius’ statement that it is 1000/1094 of Paris foot, the latter would be 324.808 mm. (The meridian was measured in Lappland with the definitive Paris foot of 1732 A.D., which is 324.839 mm.). Robert Young in discussing Scottish measures in 1740 A.D., states that the English foot is 1351.7 Paris lines or 304.920 mm.

But there was also another value of the English foot which was sanctioned by royal authority. The royal standard was shorter than the standard of Guildhall and became the basis for the present American standard of 304.80 mm.

The cube of the American foot contains 998.9 ounces averdepois of distilled water at maximum density or of rain water at ordinary temperature. English metrologists were inclined to favor the standard of Guildhall because it fitted the usual logical structure of metric units; but it was difficult to defend the authority of a standard set by the municipal body of London against a royal standard, particularly after the policy of the Stuart kings aimed at restricting the power of the London municipal authorities. The conflict between the standards was reconciled by introducing a computation by ordinary water which is about 1/100 heavier than rain water. In relation to Scottish measures, metrologists speak of computation by clear running water of the Leith or of the Tay.

The great English scholar of ancient measures, Edward Bernard, writing in 1688 A.D., a century after the reforms of Qeen Elizabeth I, computed the Paris livre as 7560 grains, but the pound averdepois as 70091/3 grains. According to him the cube of the foot of Guildhall contains 1000.3058 ounces averdepois of fountain water. It would seem that before the reign of Elizabeth I, the principle that a firkin should contain 1000 ounces averdepois of rain water was preserved both in relation to the lineal standard of Guildhall and to the royal lineal standard. The cubit cubit of Guildhall contains 1000 ounces averdepois of 28.35 grams (62.5 pounds) of rain water; but there was also a smaller ounce averdepois which probably was related to the royal lineal standard. There are text that mention a relation 14:17 between pound Troy (of 12 ounces ) and pound averdepois, instead of the correct ratio 144:175. Since this pound Troy always remained 5760 grains the corresponding pound averdepois must have been a unit of 6994.286 grains. I have indicated that there are sample weights that correspond to this value. There are texts that compute the Paris livre as 63/64 of the pound Troy (of 16 ounces ). By the regular English grain of 0.06480 grams, 61/50 of Paris grain, the Paris livre should be 7554.102 grains and not 7560. The units established by Elizabeth at the Exhequer set as lineal standard the unit that I call roay[royal?], but computed the pound averdepois as 7000 regular grains.

English scholars of the seventeenth century tried to resolve these conflicts, but apparently they increased the confusion by using waters other than rain water in their calculations. Bernard states that the amphora anglicana, the firkin, is the cube of the iron rod of Guildhall, but then explains that in “fountain water” (aqua fontanae) it contains 62.52 pounds averdepois, (62.52 x 453 grams = 28,359 c.c.), 76.08 pounds Troy (76.08 x 377.368 grams = 28,405 c.c.), and 57.9 Paris livres (28,342 c.c.). This was the calculation performed at the beginning of the seventeenth century by Henry Briggs, the creator of the decimal logarithms and the first holder of the Saville chair of astronomy at Oxford. Bernard quotes this calculation as having been confirmed by John Wybard. But Wybard at the middle of the seventeenth century counted that 1000 ounces of pure rain or running water are 1725.56 cubic inches instead of 1728. For Wybard a pound averdepois was 6994.2848 grains. Bernard quotes Wybard as having obtained the same result as Briggs: 62.8 pounds averdepois of foundation water to the cubit foot. Bernard again quotes the experiment conducted by the scholars Caswell and Walker of Oxford which in his time would have confirmed the figure of 76.08 pounds Troy of fountain water. But a report of the Philosophical Society of Oxford mentioned that in 1685 A.D. there was found that “a vessel of wel season’d oak whose concave was an exact cubit foot” filled with “pump water” contains 62.8 pounds averdepois. The scales. however, were sensitive only up to 2 ounces. The report concludes that a cubic foot contains 1000 ounces of water. In 1696 A.D. Thomas Everard, a writer on metrology and an official of the Excise, found that according to the foot of the Exchequer a cubic foot contains 1000 ounces averdepois water.

Bernard was influenced by his studies of Arab measures in which there are units reduced as 62:62.5 in relation to the normal English ones, as I shall explain later. He states that the Hebrew ephah, cubic foot, contains 62 pounds averdepois of regular wine and 62.5 pounds of fountain water or heavy water. By Hebrew measures the Arab measures of Egypt are meant, following the calculations of Maimonides. And, in fact, Bernard states that the ephah contains 48 Arab modii, by which he means mudd; a double mudd contains 337 weight dirham of water, so that by assuming a weight dirham of 3.1104 grams (1/10 of English ounce Troy), the cubic foot would be 25.157 c.c. (62 regular pounds averdepois are 28.123 grams).

Bernard states that the Arab foot is the same as the English foot. But most likely the Arab foot he had in mind had a value of 304.401 mm. This foot seems to have been a royal standard in England in the Plantagenet period, but the royal standard of Elizabeth was close to the present American value of 304.80 mm. Bernard seems to refer to this earlier royal standard when he states that according to the constituta regum Angliae the cube of the English foot contains 989.3 ounces averdepois of light wine and adds that according to recent experiments, it contains 989.3 ounces of light wine or snow water. There must be some error; the statutes of the English kings may have spoken of 989.3 ounces of snow water, which has a density similar to rain water, and scholars of the age of Bernard may have tried to rationalize this figure by filling the cubic foot with light wine.

2. There is no problem about the principle that determined the standard of Guildhall, but it would be important to know when this standard came into existence. The origin of the royal standard is a more complex problem. Since the end of the last century the matter of English lineal standards has been treated as matter of picturesque folklore, rather than of scientific investigation. Several writers have imagined that the English foot or the English yard was determined by the length of the foot or of the arm of this or that king; they often specify the name of the king, without quoting any text in support of their assertions and without even ascertaining wheter the king in question was 7 feet tall, as he must have been in order to have feet measuring an English foot.The modern tendency has been that of making the value of the English foot progressively more vague. If Sir Charles M. Watson, writing in 1913, could say: “Prior to the thirteenth century, the length of the foot in England was uncertain,” Sir Charles Arden-Close, writing in 1947, felt free to state that the English foot and the English mile were established by an act of Queen Elizabeth I.

Because of the casual way in which the history of measures is treated today, no attention at all has been given to the legal text that fixes the value of the English foot. It is an Anglo-Saxon law ascribed by scholars either to King Athelstan (924-940 A.D.) or to King Ethelbert (978-1016 A.D.). Its text, translated into Latin, is included among the laws of King Henry I (1100-1135 A.D.). I have not found a single edition of Anglo-Saxon laws or commentary on the laws of England that bother to explain or mention this striking text. The law defines the extension of the King’s Peace, called grith or girth; it prescribes that it shall extend from the gate of the King’s residence for III milia and III furlang and IX aecera braêde and IX fôta and IX scaefta munda and 9 bere-corna, that is, 3 miles, 3 furlongs, 9 acres, 9 feet, 9 hanbreadths, and 9 barleycorns. The figure of 18,4251/3 feet is significant . The Persian parasang is equal to 18,000 artabic feet and was considered an hour of march; the English foot was considered a variation of the artabic foot. Hence, the meaning of the law must be that the girth extends for 18,000 English feet or an hour of march, and that 18,000 English feet are equal to 18,4251/3 feet of some other kind. The figure of 9 handbreaths and 9 barleycorns may be better explained by counting by fingers instead of inches: 4 fingers make a handbreadth and 6 grains make a finger. If the figure of 9 handbreadths and 9 barleycorns is 371/2 fingers, it may be assumed that it stands for 21/3 feet, that is, 371/3 fingers.

The unit considered the scientific foot in the Dark Ages was the geometric Roman foot of 297.761 mm. Multiplying this length by 18,4251/3 and dividing it by 18,000, there results an English foot of 304.7970 mm., which is most closeto the value legally adopted in 1824 A.D.

3. The fact that there was more than one possible value for the English foot, was conceived as a hindrance to the adoption of it as an international standard. Today we would say that one value or the other would be acceptable, provided an agreement is reached upon it. But this is not the way the problem was seen before the philosopher Hobbes criticized the notion of natural law and asserted the sovereign authority of the state, even when irrational and arbitrary. We know Hobbes as a philosopher and a political scientist, but he thought that his fame would rest on his mathematical work, which essentially is an effort to prove some patently absurd contentions in order to destroy the belief in the objective rationality of mathematics and geometry. It is significant that Bishop Richard Cumberland who as a philosopher tried to uphold the notion of natural law against Hobbes, is the author of a work (published in 1685) in which he defends the standard of Guildhall and defends the calculation of the foot as the edge of a cube of rain water: “The most exact and geometrical way of expressing the capacity of any vessel, or measure, is by expressing, in known terms, the solidity of a body which will precisely fill it; the fittest will be water, such as drops from the clouds.” To stress the superiority of this method he observes that by it the standard of the Roman foot and libra was preserved for nearly 2000 years. But the scientific superiority of the method and the accuracy of the standard of Guildhall were not sufficient to Bishop Cumberland; he tried to prove that English measures are derived from those of the Egyptians and that these in turn are those of the Hebrew patriarchs.

Since the beginning of the sixteenth century, it had been advocated that the Roman foot be adopted as a universal standard, but when numerous studies of the problem proved that there were several varieties of Roman foot, the plan to adopt the Roman foot was abandoned, even though the several authors had reached substantial agreement on the value of each variety. In 1514 Guillaume Budé. in the first comprehensive work on ancient measures to appear in print, had proclaimed:

Una fides, pondus, mensura sit idem
Et status illaesus totius orbis erit.

“Let there be one single faith, weight and measure, and the order of the entire globe shall be free from harm.” Budé, who later supported the Reformation, advocated a single religion together with a single system of measures. His assumption was that a system of measures must belong to the system of things that are considered objectively right. The Roman foot could not be adopted as standard, if it was impossible to prove that a specific value of the Roman foot was the correct one. The only exception to this view is represented by Sweden that in 1604 under King Charles IX introduced a new system of measures based on a Roman foot of 296.910 mm. But King Charles IX was a strongly authoritarian figure who had little respect for established institutions, religion included; he was trying to unify a kingdom that included such discrepant elements as Scandinavia and Lithuania. The older Germanic nations continued to use the trimmed wheat foot which had been their common standard since the period of the Barbaric invasions. But there an exact scientific definition of this standard was missing, until it was formulated for the Prussian state by the astronomer and geodesist Friedrich Wilhelm Bessel in 1838 A.D. The result was that scholars were uncertain about the exact value of the standard used by Tycho Brahe in his famous accurate astronomical observations, which were used by Galileo, and by Willebrord Snell in the first attempt to determine the length of the degree of meridian by triangulation.

The notion that an international scientific standard should have both an objective and a historical basis was so ingrained in the minds of scholars that serious consideration was given in the seventeenth century to the proposal of Melchisédech Thévenot, who organized the gathering of scholars out of which grew the Academic des Sciences, that the hexagonal cell built by bees be adopted as lineal standard. This strange proposal was supported in the following century by René de Réaumur with the added argument that the beehives of ancient Athens must have had cells of the same dimensions.

4. It was known that English measures were derived from those of the ancient Egyptians. This information apparently had been derived from the many Arabic treatises of metrology and from Talmudic commentators, among whom Maimonides was particularly authoritative, since his computations reflect the circumstance that this first calling in life was that of a jeweller at El Cairo. For this reason it was thought that the correct value of the English foot could be settled by the study of ancient Egyptian measures.

In 1635 John Greaves, mathematician and orientalist, resigned his chair of geometry at Gresham’s College to try to settle the matter of the length of the Roman foot. He left for Rome with accurate rules based on the foot of Guildhall divided into 1000 parts. But his diligent testing of ancient Roman monuments confirmed that there were several values of the Roman foot. He considered the pes Cossutianus of 967/1000 of English foot or 294.857 mm. to be the official one (my value for this foot is 294.722 mm.), but recognized the existence of the other varieties up to the one used in Rome at the time, the type called architettonico, which he estimated at 976/1000 of English foot or 297.602 mm. (My value for the geometric foot is 297.761 mm.). More than a century later Don Diego Revillas with an esattissimo piede inglese copied on the Graham rule of the Royal Society, given to him Martin Folkes, a member of this academy who had gone to Italy with a mission similar to Greaves’, computed the architettonico standard as 976.86/1000 of English foot. Assuming the foot of 1824 A.D., of which the Graham rule was a trifle shorter, the geometric foot would be 297.775 mm.

Since his Italian campaign could not settle the matter of the international standard, in 1639 Greaves landed in Egypt where he met with Burattini who had been there for two years trying to ascertain the value of the ancestor of the English foot. Burattini, as he says, had been measuring “pyramids, obelisks, sphinxes, mummies, the ruins of Alexandria, Lake Moeris, Egyptian, Persian, Greek and Roman buildings.” He also tried to establish astronomically geographical distances within Egypt. He had already twice gone through the process of measuring the Great Pyramid of Gizah, but since Greaves had the advantage of being provided with accurate rules, they repeated the operation together. It was the notion of Greaves that at least their operation would allow him to mark two points on this monument which would indicate an interval of which the English foot would be defined as a fraction.

There is an explanation for the apparently peculiar idea that the Great Pyramid should be the reference for the English foot. After several years of meditation over the texts brought me to the realization that the Egyptians computed geographic distances from the two base points, the Temple of Ammon at Thebes (32° 39’E 24° 43’N) and the ancient apex of the Delta (31° 13’E 30° 05’N), located where in Arab time there was El Cairo (now called Old Cairo), I noticed with great amazement that Girolamo Cardano in 1553 A.D. suggested that all measures of length and weight should be based on a mensura perpetua embodied in a geographical distance, represented by the Pyramids, the Labyrinth of Thebes, the city of El Cairo, and the Nile. The only element that does not belong to the ancient scheme is the Pyramids; perhaps the Pyramids of Gizah came to be considered in this context because they are located at one minute of degree from latitude 30°. But Vincenzo Scamozzi, writing in 1615 A.D. suggests that the Pyramids be taken as a standard of length.

After four or five months Greaves returned to England where he became Saville Professor of astronomy at Oxford, but he left his rules with Burattini who continued his campaign for two more years. The report about the dimension s of the Great Pyramid published by Greaves was interpreted by Newton, who concluded that the cubit employed was 1719/1000 of English foot or 524.1403 mm. The royal cubit of the Great Pyramid is of a special variety computed so as to be the edge of a cube containing 16,000 qedet of 9 grams, so that it has a theoretical value of 524.1483 mm. This figure proved that the cubit of the Great Pyramid is not related to the English foot. Newton hoped to be able to ascertain the value of the stadion used by Eratosthenes in his evaluation of the circumference of the earth, since he needed this datum before announcing the theory of gravitation. But there was another Egyptian royal cubit, and this is the basis of the stadion of Eratosthenes, for whom 300 cubits of 532.702 mm. form a stadion of which 250,000 are the circumference of the earth. The geographical data of Eratosthenes about distances in Egypt are grossly incorrect, but his final figure is correct because it was based on the traditional datum of 39,952 km. for m the circumference of the earth.

5. The official cubit of Egypt was that used to measure the flood of the Nile, the cubit called Nilometric in Greek texts. Up to around the middle of the first millennium B.C., this Nilometric cubit was the royal cubit of 525 mm. But from this moment on there appeared in Egypt another Nilometric cubit, also called royal cubit in Egyptian texts, which measures 532.702 mm. trimmed and 535.780 mm. natural . But the cubit of 525 mm. continued to be used for a long time since under Arab domination the famous qadi Yusuf ben Yaqub (died in 759 A.D.) set up a standard of this length (an unusual unit in Arab metrics) as Nilometric cubit. Since Arabic writers refer to the patriarch Joseph as Yusuf ben Yaqub, European writers compounded the confusion by understanding that the Nilometric cubit of 532.702 mm. had been established by Joseph, and since there was a tradition to the effect that the Pyramids of Gizah had been erected by Joseph as granaries, the irrelevant issue of the Pyramids was injected into the debate about the English foot. For instance, Newton in his discussion of the cubit of the Great Pyramid assumes that this was the cubit of Joseph; the origin of this conviction is revealed by his statement that Joseph established the Nilometers.

In 1838, before cuneiform documents became available, Böckh with brilliant insight gathered that there had been in the ancient world a unit that he calls Babylonian-Egyptian great cubit and which he computes as 236.07 Paris lines or 532.53 mm. This cubit was known in Hellenistic times as Ptolemaic (after the Greek kings of Egypt) and Philetairic (probably after the name of several rulers of the Kingdom of Pergamon in Asia Minor). Metrological, tables of the Roman period define it as 9/5 of Roman foot, that is, as 532.702 mm.

Petrie noticed that Egyptian measuring rods representing the royal cubit, beginning with about the middle of the first millennium B.C. are longer than the traditional length of 525 mm. He explained these rods that are usually reported as being 532 or 536 mm., as resulting from a gradual process of lengthening of all lineal units in the course of history. In the eighteenth century the mathematician Ruggiero Boscovich, in considering the varieties of Roman foot, had observed that in copying a standard the tendency is to err on the side of excess; but Newton earlier in order to explain the difference between the Egyptian foot, the length of which he had just determined, and the Roman foot, considered the possibility of a gradual shortening through time. Petrie followed the first suggestion in dealing with Egyptian units, and even calculated how much lineal units would be shortened in each century, but in order to explain why the Roman foot employed in Greek temples usually is of the geometric variety, whereas in Rome there is found the official Roman foot and the even shorter pes Cossutianus, assumed the existence of an equally inevitable process of gradual shortening.

Julius Oppert in the second half of the last century noticed that several buildings of Mesopotamia are constructed by a cubit of about 533 mm., He thought that this was the standard cubit of Mesopotamia; but the unit usually mentioned in cuneiform documents is the barley cubit (499.408 mm. trimmed and 506.250 mm. natural). However, in cuneiform tablets that date from the beginning of the scond millennium next to this cubit divided sexagesimally into 30 fingers, there is at times mentioned a cubit of 32 such fingers (532.702 mm. trimmed; 535.780 mm. natural). This cubit which is 16/15 of barley cubit, was adopted as their standard by the Assyrians when they established a universal empire, since it may be easily related to several units current in the ancient world. When the Assyrians conquered Egypt in 671 B.C., they brought this unit to that country, but in Egypt it was divided into 28 fingers, as the old Egyptian royal cubit.

A foot may be formed from this cubit of 28 fingers in two ways; by taking 2/3 the cubit or by taking 16 fingers or 16/28. The first type of foot occurs in France as pied de terre of Bordeaux which by its relation to other local units has been computed as 357.214 mm., and by the pied le comte of Franche-Comté, of which a standard consisting of rod of 7 feet is at the City Hall of Poligny and indicates according to report a foot of 357.8 mm. Two thirds of a natural great cubit of 535.870 mm. are a foot of 357.246 mm. The foot formed by taking 16 fingers comes to 304.401 mm. when trimmed. Apparently this value was known in England since a yard preserved at the Westgate Museum at Winchester is described as 0.04 inches short (0.339 mm. for a foot). This yard was constructed under King Henry I (1100-1135), and its extremities were countersigned with seals of Edward I and of Henry VII. Possibly this foot corresponds to the cubic foot mentioned by Bernard as containing 993.328 ounces of rain water. Counting by ordinary ounces and by rain water, this corresponds to a foot of 304.425 mm. This single instance indicates that it would be necessary to conduct an accurate survey of the English standards of length that may still be preserved here and there in museums, private collections, and churches. It would be necessary to measure accurately buildings, that is, in practice mainly churches of the Norman period. Hugh R. Watkins tested Norman buildings merely to establish that they were computed by units of 7 feet. Then it would be most important to know the dimensions of early Saxon churches, such as St. Peter and Paul, St. Pancras, St. Mary, St. Martin in Canterbury, and those of Reculver, Lyminge and Rochester in Kent, and that of Bradwell-juxta-Mare in Essex.

6. After Greaves’ departure, Burattini continued his campaign of measurements in Egypt for two more years. But as ill luck would have it, on his way of return Burattini was attacked by bandits in Hungary and robbed of his notes. All that remains of Burattini’s effort is hat was published in Œdipus Aegyptiacus of Father Athanasius Kircher on the basis of letters sent him by Burattini from Egypt: the dimensions of the obelisks of Heliopolis and Alexandria and some transcriptions of hieroglyphic texts.

Burattini who had settled in Poland, becoming secretary to the King and director of the mint, since he had been deprived of the data collected in Egypt, changed his plan for a metric system. In his Misura Universale published in Vilna in 1675, he proposed that the meter be defined as 1/3 of the length of the pendulum that beats the second. This made the proposed meter more than 13/12 of English foot. But he must not have considered the solution satisfactory since before his death in 1682 he was again in Egypt measuring the Great Pyramid for a fourth time.

When the Royal Society was established in 1682 A.D., one of the first tasks it set to itself was to establish a precise standard of length. Under the influence of Burratini’s project it planned to compute it by the pendulum. John Evelyn, who acted as secretary, in a trip to Italy gathered data on the value of the lineal units of Bologna, in order to interpret correctly the studies on the pendulum by the main antagonist of Galileo, Father Riccioli. The latter had said that the pendulum that beats the second is 31/3 ancient Roman feet, which are 1200/1495 of the foot of Bologna. Jonas Moore, who was the metrologist among the founding members of the Royal Society, computed the metrum catholicum as 1/3 of the length of the pendulum that beats the second at the latitude of London. It was 1089/1000 of the foot of Guildhall or 332.057 mm., so that the length of the pendulum would be 996.170 mm. (one of the current estimates of the second pendulum at the latitude of London, adjusted to sea level, is 994.138 mm.).

Eighty years after the foundation of the Royal Society the plan of defining a standard by the pendulum was abandoned. It had been discovered that the length of the pendulum not only is a function of the physical characteristics of the pendulum itself, but varies according to the latitude, the elevation above sea level, and presence of dense masses of land. In 1742 the Royal Society decided to construct an accurate rule based on the accepted length of the English foot. In this the Royal Society followed the lead of the Académie des Sciences. The French institution, which had been established in the same decade as the Royal Society, decided from the very beginning to establish as scientific standard a pied de roi of the proper length. But the issue of the proper length was not settled until 1732 A.D. when the engraver Langlois constructed for the Académie the Toise du Pérou (a toise is 6 pieds de roi) so called because it was used in the measurement of the length of degree at the Equator and in Lappland. Langlois computed the pied de roi as the edge of the cube that contains 70 Paris livres of water. This standard was employed to establish the Paris meter. In the operations of establishment of the Paris meter it was computed that the cube of a foot of the Toise du Pérou contains 70 livres, 141 grains of distilled water at maximum density (there are 9216 grains in a livre). This implies a difference of less than 1/46003 in the calculation of the standard of length. It is to be noted that the French Academy too tried from the very beginning to give an objective foundation to the pied de roi by using it to calculate the circumference of the earth. The uncertainties about the new pied de roi established in 1667 A.D. caused debates that lasted up to the middle of the eighteenth century on the question whether the earth was flattened or elongated at the Poles. The adoption of the Toise du Pérou by the Académie des Sciences made it possible to calculate with certainty the circumference of the earth. Only when the meter could be defined as 1/10,000,000 of the distance from the Equator to the Pole, it was considered that it had a sufficiently objective foundation to be adopted as an international standard.

George Graham, who was entrusted by the Royal Society with the project of constructing a rule, constructed a brass bar and marked upon it with two golden dots the length of a yard. Up to that time English standards usually consisted of a bar of brass or iron that was kept resting imbedded in a matrix of the same metal. The purpose of the matrix was to prevent the bending of the bar and to allow the reconstitution of the standard if the bar was broken, lost, or stolen. Furthermore, bars brought for testing could be easily examined by dropping them into the matrix. But it was not completely clear whether the standard was represented by the bar, by the matrix, or by the small space that of necessity had to be left between the two.

Graham marked with E (for English) on his rule a length intermediary between that of a yard kept at the Tower of London and a shorter yard kept at the Exchequer and marked with the seal of Elizabeth I. At the Exchequer the were, as there are still today, a yard of Henry VII (1490 A.D) and an ell and a yard of Elizabeth I (1588 A.D.). After Graham had marked his rule, it was objected that the standard of Elizabeth was to be preferred. As a result the Royal Society decided on a comparison of all the standards available in London, those of the Exchequer, that of the Tower, that of the Clockmakers’ Company, and those of Guildhall. As a result of this survey the length of the yard of Elizabeth was marked with Exch. on the rule of Graham. Taking this length as reference, the value of the other standards was found to be the following:

Matrix of the yard of Elizabeth

32.0102 inches

Rod of the ell of Elizabeth

45.0494 “

Rod of the yard of Henry VII

35.9929 “

Rod of the yard of the Tower

36.0111 “


When on April 22, 1743 the President of the Royal Society accompanied by six members presented himself at Guildhall to examine the standard, only the matrices were available. A matrix of yard exceed the Graham length E between 0.0434 and 0.0396 inches: a matrix of ell (45 inches) was compared with the rod of the ell of Elizabeth I and found to exceed it by 0.0444 and 0.0258 inches. Two figures were ascertained because the ends of the slots in the matrices were not at an exactly right angle.

The disappearance of the rods of Guildhall underscores the decline of the prestige of this standard. It would be important to know whether the rods of Guildhall were destroyed by accident or by an action of the royal authority.

7. In 1758 A.D. the Parliament appointed a committee “to enquire into the original standards of weight and measures of this kingdom.” It concentrated on the problem of the exact value of the yard, since the metrologist Harris, King’s Assay Master of the Mint, had expressed the opinion that “a lineal standard should be the standard of all measures of capacity.” The committee concluded that the correct yard should be indicated by the joint between the rod and the matrix of the Yard of Elizabeth. John Bird constructed a new rule in conformity with this decision. He chose a value intermediary between the marks E and Exch of the Graham rule. The proportions were the following:

Matrix of the yard of Elizabeth

36.0102 inches

Graham mark

E 36.007

Bird rule

36.0063

Graham mark Exch.

36.0000


Bird constructed two rules, one in 1758 and one in 1760, but the minimal difference between the two is of no concern here. In 1824 A.D. by act of Parliament the Bird standard became the Imperial Standard Yard. The act was so worded as to make the English foot officially a fraction of the length of the pendulum that beats the second, but when in 1834 A.D. a fire destroyed the Parliament building and with it the bar of the Imperial Standard Yard, it was proved impossible to reconstruct the standard by the pendulum and a new bar was constructed from the existing copies of the Bird rule.

In reality all these operations were ritualistic, and the records of the Royal Society are interesting only for the historian who wants to know which were the standards in existence before 1742 A.D. Ever since Newton based his calculations about gravitation on the report of Picard about the length of the degree of meridian, the pied de roi was accepted as the scientific standard. The controversies that developed between the age of Newton and the middle of the eighteenth century on the question whether the earth was lengthened or flattened at the Poles, were in great part the result of small discrepancies in the pied de roi employed. The famous “error of Picard” is explained by the circumstance that Picard had a standard which is 1/14,000 shorter than which began to be generally employed by scholars a few years later. The uncertainty should have ended when the Académie des Sciences adopted the standard of the rule constructed by Langlois, but the comedy of errors continued for a few more years: when in 1755 A.D. Father Boscovich proceeded to his triangulation of Italy that finally proved the polar flattening of the earth, he asked Jean Jacques de Mayran to send him an exact copy of the Toise du Pérou, but the latter, who had opposed the adoption of this standard by the Académie, since he had used a shorter standard in his calculations of the pendulum that beats the second in Paris, sent to Boscovich his own standard without informing him about the shift. But it was only in the nineteenth century that it was calculated that the standard of Boscovich was about 1/8500 shorter than the Toise du Pérou, just as the reason for the “error of Picard” was discovered only when it was an historical problem. Hence the small discrepancies in the value of the pied de roi established in 1677 A.D. did not prevent its general adoption as a scientific standard, even before the establishment of the Toise du Pérou. When Graham constructed his rule for the Royal Society, the Académie des Sciences sent to London a bar equal to half of the Toise du Pérou, and Graham marked this length on his bar. Graham calculated that the pied de roi relates as 1065.411/3: 1000 to the English foot. A copy of Graham bar was sent to Paris where Pierre Leminnier, official astronomer of the King of France and associated with the expedition of Lappland, computed the same ratio as 1065.4: 1000. In spite of the fact that these values must be based on Graham‘s mark E, since they indicate an English foot of 304.892 and 304.895 mm. respectively, from this moment on scholars began to calculate the English foot as a fraction of the Paris standard, rather than by any rule kept in England.

When the mètre des Archives was constructed on the basis of the Toise du Pérou, the English foot began to be defined in practice as a fraction of the Paris meter. This is the reason why, in spite of the Parliamentary standard of 1824 A.D. the custom had prevailed, particularly in the United States, to compute the English foot as 304.80 mm. This value was adopted by the British Board of Trade in 1885 A.D. All American legal definitions of the foot have been based on the Paris meter, even when computed by a fraction that makes the foot slightly different from 304.80 mm.

8. English scholars of the seventeenth century hoped that the problem of the exact value of English measures could be solved by knowing the metrics of ancient Egypt, but they could approach the problem only indirectly through data that are later than ancient civilization. Bishop Cumberland could read Coptic documents, Greaves studied oriental languages and wrote a Persian grammar. Bernard, who succeeded him on the Saville chair of astronomy, is better known through his contributions to the development of the study of Near Eastern languages than for those to his chosen field of the exact sciences. He spent several years travelling in the Orient, collecting manuscripts and learning about Arab measures from actual practice. He went closer to the ancient period by being the first to transcribe and publish Greek inscriptions from the caravan city of Palmyra and read in the original manuscript of the Bodleian Library the Syriac translation of the treatise on biblical measures by St. Epiphanios. This text, written in the fourth century A.D., of which the original Greek version is lost except for abstracts, is a most important source of information not so much about biblical measures as about the measures of the Eastern Roman Empire. Up to now nobody else has made use of this fundamental source of information, except through the frequent quotations that occurs in ancient and medieval writers. This work was considered not only scholarly but also practically useful, as indicated by the several Greek, Latin and Hebrew tables abstracted from it. But the prevailing modern opinion is that it is worthless nonsense. Paul de Lagarde tried to edit the Syriac version, but repeatedly missed the meanings of the next, justifying himself with the statement that St. Epiphanios ungewöhnlich dumm war. An edition that is linguistically satisfactory was finally prepared in 1935 by the Oriental Institute of the University of Chicago, but it is preceded by the warning that the content is the product of an “addle-headed old pedant.” A few years earlier the Mathematical Monthly had announced the completion of an edition with metrological commentary by a mathematician, but this more respectful treatment of the St. Epiphanios’ work did not succeed in finding its way to the press.

A new, more direct source of information on Egyptian measures has become available since the beginning of this century in the form of the Greek papyri found in Egypt, which date from the Ptolemaic and Roman periods. A high percentage of these papyri consists of legal transactions that specify in detail the nature of the units of measurement. Segrè has dedicated decades of his life to the interpretation of the metrics of these documents, but unfortunately he had no access to the Syriac version of St. Epiphanios; he tried to refer to it through the statements of Bernard, but was not sufficiently acquainted with English measures to understand them properly. In spite of his long diligence and his unusual intellectual gifts, Segrè came short of a final solution because he accepted a single erroneous datum. He was like myself first of all trained as a legal historian, but from the first day he began to initiate me to the reading of Greek papyri he warned me that metrology is a matter of mathematical constructions in which all data are interdependent and used to illustrate this point with examples drawn from his experience as cryptologist during the First World War. Nevertheless he failed to scrutinize carefully the accuracy of his initial datum.

I have reported that numismatics have not realized that the standard of the Athenian New Style coins and of the coins of the followers of Alexander (mina of 425.25 grams), is not the same as that of the earlier Athenian coins and of the coins of Alexander the Great (432 grams). Numismatists who assume that ancient standards could waver within a wide range, at best conclude that the later coins were somehow skimpier in weight. Segrè stated specifically that the coinage of the Ptolemies of Egypt, as that of the other followers of Alexander, was struck by a new standard, but did not investigate the numismatic data and accepted the figure of those numismatists who, confronted with two different standards, chose an intermediary figure. Segrè started with the figure of 428.54 grams for the Ptolemaic mina, assuming that this is the weight indicated by the coins. He accepted this figure because the texts indicate that Ptolemaic drachmai circulated at par with Athenian drachmai, but he did not know that the New Style drachma, too, is based on a mina of 425.25 grams. For this reason, he discarded the figure of 425 grams that is given by some scholars who are concerned only with the later Hellenistic coinages and with the coinage of Sassanid Persia. Starting with the figure of 428.54 grams he noticed that papyri count by a unit that is 5/6 of the mina, so that he set this unit at 357.1 grams.

The most important artaba of Hellenistic Egypt is composed of 80 litrai, just as the Roman quadrantal is divided into 80 librae. Concerning the libra Segrè discovered that it existed in two varieties related as 80:81, and insisted on the importance of this relation in the metrics of the papyri. But when he noticed that the careful study of the legal transactions documented by the papyri indicated that the Alexandrine libra is 108/100 of Roman libra, he took as a starting point the lower libra. If he had started from the higher libra of 328.05 grams, he would have arrived at 354.294 grams. The unit which is 5/6 of the Alexandrine mina is, by my reckoning, 5/6 of 425.25, so that it is 354.377 grams, which proves that it is nothing else but the Alexandrine libra.

Counting 80 litrai of 354.375 grams, there results an Alexandrine artaba of 28,350 c.c., which is exactly the English firkin of 1000 ounces averdepois, as it had been suspected by all the English scholars of the seventeenth century.

The Alexandrine artaba relates as 35:36 to the Persian artaba of 29,160 c.c. The papyri make clear that an artaba is the amount of wheat consumed in a month; but a distinction must be made between lunar calendar and solar calendar since units connected with the lunar calendar (81/80 350 days) are 35/36 of the units connected with the solar calendar (81/80 360 days). The importance of this relation 35:36 is indicated by Arab metrics in which next to a ratl (the term renders the Greek litra) of 140 weight dirham (the terms renders the Greek drachma or the Persian daric) there is a ratl of 144 weight dirham. In England, where the basic standard is the ounce averdepois, for trade with France and Germany merchants often used a pound of 7200 English grains (whereas the pound averdepois is 7000 grains), because the Carolingian system of measures is based on the artabic ounce (ounce of Cologne).

9. Once the value of the Alexandrine artaba is established, the next problem is that of computing the corresponding unit of length. Segrè properly stressed that the artabic foot is the edge of the Persian artaba, but could not clarify the issue of the foot related to the Alexandrine artaba. In his early writings he had declared about the Babylonian-Egyptian great cubit of 532.534 mm. computed by Böckh: “Böckh 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 later, since he had arrived at value of the Alexandrine artaba that is too high, in quoting Böckh’s value he put a question mark after it. He went back to the earlier opinion of Sagey that the Nilometric cubit of the Hellenistic and Arab period is composed of 28 artabic fingers and is 539.175 mm.

I maintain that the great cubit measures 532.702 mm., so that the normal foot corresponding to it is 304.401 mm. But as I have pointed out, the cube of this foot contains slightly less than 1000 ounces averdepois or 80 Alexandrine litrai.The exact edge of the Alexandrine artaba is represented by the English foot of Guildhall. It would be important to know whether the Egyptian rods indicate also the existence of a cubit slightly longer than 532.702 mm.; unfortunately Petrie who had the opportunity to examine a great number of them, confused the issue by imagining that the great cubit originated by a gradual lengthening of the Pharaonic royal cubit of 525 mm. But there is positive evidence that at least one solution adopted to solve the problem of the difference between the foot and the edge of the Alexandrine artaba was that of reducing the weight of the Alexandrine litra. Segrè himself observed that the value of the Ptolemaic mina and litra is indicated by the Arab dinar, but here again failed to examine critically the evidence gathered by others.

It would be necessary to analyze statistically the coins based in principle on a mina of 425.25 grams to ascertain whether this unit is at times reduced to 423.128 grams, so as to obtain a litra of 352.607 which is contained 80 times in the cube of 304.401 mm. But for the purpose of the present investigation a more direct evidence is given by the Arab coinpoises made of glass. These pieces may be more easily evaluated than the coins they are intended to test; they are bound to be more precise than the coins, although even here deviations are to be expected due to errors in measurement, and although even here a certain amount of wear and chipping of the pieces does occur. In the last years these Arab counterpoises have received a good deal of attention by collectors, so that soon an almost complete listing of the pieces in the main collections will be available and will make possible a statistical tabulation. But the data already gathered provide a specific indication.

Petrie found that three half dînâr coinpoises, made in the year 780 A.D., weigh respectively 32.662, 32.665, 32.667 English grains. Multiplying this value by 200 there results a specific indication of a mina of 423.30 grams. Petrie reports that a large set of coinpoises, cast in 765 A.D., vary from 32.51 to 32.67 grains. P. Casanova reports that in a collection of fifty dînâr coinpoises, all but five were between 4.21 and 4.23 grams; pieces of 4.25 grams and one piece of 4.28 grams should be ascribed to a normal error in measurement. Petrie concludes that the average of the specimens he examined indicates a dînâr of 65.3 grains or 4.231 grams. The recent treatise of Islamic metrology by Walther Hinz adopts the figure of 4.233 grams.

By examining the data already published, without proceeding however to a statistical analysis, I would observe that a large number of specimens are reported as having a value of 4.25 grams. I would tentatively explain this figure as indicating a separate value rather than as being the result of a normal probability distribution of errors around the figure of 4.233 grams. Hence I would say that possibly there were two values for the dînâr: a value of 4.2525 grams related to the correct ounce averdepois and a value of 4.213 grams adjusted to the foot of 304.401 mm. If I am correct, the frequency curve of the weights shall have two peaks.

10. Medieval treatises of Arab metrology appear complicated because their authors, being concerned with extreme precision, raise issues of minimal differences between the standards of one city and of another. They do not ignore differences similar to those that exist today between British and American units. But this concern with counting by camel’s hair should not obscure the essential simplicity of the Arab system.

Current treatments of Arab standards link the units with the libra and the solidus of Constantine, but this is a case of pretending knowledge by explaining obscura per obscurius, since our contemporary scholars do not have exact information about Byzantine units. It is true that the very earliest Arab coinage is based on the Byzantine solidus, but that very early, in 697 A.D. (year 77 of the Hegira) the standard was shifted. Decourdemanche recognizes that the new standard is based on the Attic and Sassanid drachma which he evaluates at 4.25 grams, but he links it with a non-existing Byzantine siliqua of 2.831/3 grams, which in reality is 1/10 of ounce averdepois or 2.835 grams. The Arabs abandoned the Byzantine units and adopted those of the Hellenistic kingdoms and of the Persian Empire.

A great clarification has been achieved by the recent work of Hinz. He states at the outset that there are two essential units: the monetary unit, represented by the dînâr or the dirham (which is 2/3 of it), and the unit used to weigh precious metals, the “weight dirham” (dirham al-kail). He estimates the monetary dirham at 2.82 grams or 43.52 English grains and the weight dirham at 3.125 grams or 48.225 grains. To this I have to add only the essential clarification that the dirham is 1/10 of ounce and that coins are weighed by an ounce averdepois, whereas bullion is weighed by ounces Troy. English writers of the seventeenth century were perfectly correct in considering Arab metric similar to theirs.

The only difficulty is represented by small variations in the Arab value of the averdepois and of the wheat ounce netto (ounce Troy). I have already reported that the monetary dirham may may be computed either as 1/10 of a regular ounce averdepois of 28.350 grams (437.5 English grains) or as 1/10 of a smaller ounce such that 1000 make the cube with an edge of 304.401 mm. Hinz computes the dirham as 2.82 grams (43.52 grains).

Concerning the weight dirham Bernard stated that he was not concerned with the subtleties of Arab metrologists that go beyond practical relevance (concedimus ergo Arabibus nulla invidia pilos suos camelinos, subtiliorem mensurandi normam quam utilem), but distinguished four main types:

 
 

a) 47.82 English grains, as silver dirham, common among the Turks, the Persians, and the Mongols.

b) 48 grains or 1/10 of ounce Troy, as common dirham of Syria.

c) 48.033 grains, as dirham of the Jewelers of El Cairo, which is 1/144 of the ratl of El Cairo.

d) 48.7014, as common dirham of Constantinople, of the Sultan Saladin, and heavier dirham of Aleppo.

Since the English grain used by Bernard was 7000/70091/3 smaller than the grain of 0.0648 grams, the dirham of El Cairo may be considered the one equal to the correct value of 3.1104 grams. Hinz reports that according to a metrological manual of the last century the dirham of Aleppo (fourth type of Bernard) is 3.167 grams. Hinz computes that the dirham of Anatolia under Osmanic rule was 3.086 grams; this value corresponds to the first type of Bernard. In 1854 A.D. the weight dirham of Egypt was officially computed as 3.0898 grams, but the procedure followed in order to arrive to this figure started from a highly questionable unit of length. H. Sauvaire thought that this figure should be taken as the starting point in computing Arab units, but he was justly criticized by Decourdemanche who preferred the heavier value of 3.148 grams. It is better to assume that the weight dirham was in principle 1/10 of 31.104 grams. In my opinion the ratl of 144 dirham (12 ounces of 12 dirham) is basically the pound averdepois of 453.6 grams. This unit is divided into 144 dirham of 3.150 grams of 48.61 normal English grains, corresponding to the fourth type of Bernard.

The English system is based in principle on a firkin containing 62.5 pounds averdepois of water, the Arab system is based on a cube with an edge of 304.401 mm. with a volume of 28.125 c.c. which is 62 pounds averdepois but 62.5 Arab ratl of 450 grams. The cube of 28,350 c.c. contains 63 ratl. Hence, Arab measures may be reduced as 62:62.5 =124:125 or 62.5:63 = 125:126. Hinz reports that E. W. Lane in 1836 A.D. stated that the Egyptian grain is 127/128 of English grain.

Arab metrics are connected with those of Hellenistic Egypt. The treatise of St. Epiphanios is essentially concerned with the jar containing 22 pints of wine and states that this jar is of Egyptian origin. Segrè stressed the fundamental role of the unit of 22 pints in the metrics of Hellenistic Egypt, but did not have access to the full text of St. Epiphanios which explains the significance of this unit. I have reported that what St. Epiphanios describes is a jar which was a volume of a wheat unit, but when filled with wine is filled only to the level of unit averdepois. This means that there were units for wine based on the ounce averdepois and units for wheat based on the wheat ounce, Segrè anchored his reconstruction of the metrics of Greek papyri on a modius xystus, “shaven peck,” of 21.6 or 22 pints, and on modius cumulatus, “heaped eck,” of 24 pints. The first is a unit averdepois and the second a wheat unit. The typical artaba of Hellenistic Egypt is connected with the first modius; the artaba is equal to 80 litrai, the unit by which coins are calculated. The same pattern is continued by Arab metrics, assuming a relation 11:12 between ounce averdepois and wheat ounce, but this ratio involves adjustments because by it either the first ounce is raised to 28.512 grams or the second is lowered to 20.927 grams.

Concerning Arab measures, Segrè properly calls to attention an Arabic papyrus containing an edict of Qarra ben Sarîk, who was govern or of Egypt from 708 to 713 A.D.

He was sent to that country after a famine and a period of financial disorder, taking the title not only of governor but also of director of finances; a great number of glass poises and seals for measures bear his name, suggesting that he was particularly concerned with measures. In his edict he banned for ever the artaba kail al-daimûs, which is the artaba metró demosió of Greek texts and prescribes that taxes paid in grain be measured by the artaba qanqal. The qanqal is the artaba metró kagkelló of Greek texts. What Segrè did not realize is that this artaba corresponds to the modius xystus. C. H. Becker, in editing the papyrus, has properly observed that the expression metró kagkelló has nothing to do with the Latin word cancellus, as it has been believed. But Becker is not entirely correct when he states that the quanqal is a Persian unit; the term is not Persian, but formed by a reduplication of the Semitic root qal which in Hebrew means “small, reduced, accursed” ; reduced measures are accursed in Hebrew and Islamic Law. In Akkadian qalqaltu means “survival in hunger.” In the Bible the daily gomor of manna is a level measure that cannot be heaped (Ex.16:18); because of this the Hebrews protest with Moses complaining about the lehem haqqeloqel, “The level ration of breadstuff” (Num.21:5).

From the edict of Qurra it appears most clearly that the artaba kail al-daimûs is larger and was obtained by heaping; he prescribes that henceforth the measure must be full, but without anything added to it. In my opinion he simply confirmed older practices which had been violated by abuse, since the artaba metró demosió that he prescribed used also to be called artaba metró thêsaurikó “by the standard of the state granary.” Segrè recognizes that the artaba metró thêsaurikó is a unit of 80 litrai. The artaba metró demosió was the Persian artaba of 29,160 c.c. From Arabic papyri it appears that the artaba is equal to 64 ratl; if the ratl is of 140 dirham there results a unit of 28,350 c.c., whereas if the ratl is of 144 dirham there results unit of 29,160 c.c. The edict of Qurra prohibited the use of the Persian artaba. Nothing is said about the unit of 1000 wheat ounces that in Hellenistic times was called artaba metró Athênaió, because it is identical with the Athenian metretês, the wheat talent.

11. Scholars of the seventeenth century hoped to explain English measures by the study of the Arab ones, but the connection between them is indirect. Russian measures are more closely linked with the English ones.

In the Viking period there was a unified area of trade that started from the Black Sea and through the rivers of Russia reached the Baltic Sea and continued through Denmark and Scandinavia to the North Sea including England and the Low Countries.

This area is characterized by the use of the ounce averdepois. The most important piece of evidence is provided by a set of weights found in the ruins of the Carolingian mint of Duurstede (Dorestatum ). This city near the mouth of the Rhine was the metropolis of North Sea trade until it was completely destroyed by a Viking raid around 837 A.D. The coins of Duurstede are found a are found in large quantities in Scandinavia, Poland and Lithuania. They were imitated in this area, particularly at the great trading center of Björkö (Birka), about 15 miles from Stockholm; these imitations even copied the legend DORSTAT, and it seems that the letter TAT, badly copied and misunderstood, were the origin of the coat of arms of the princes of Lithuania. When the discovery of weights of Duurstede was announced, there was a flurry of excitement among numismatists because it was hoped that they could provide information about a period for which certain metrological data are missing. But since scholars could not fit their units into any standard they expected, it was concluded that these pieces are not weights but merely lumps of lead used to test the coin punches. The bars are marked with the punches of Carolingian coins, but it was common ancient and medieval practice to use coin punches to countersign units of measure, both weights and vessels. The weights which are now at the Rijksmuseum van Oudheden of Leyden are multiples of the ounce averdepois. Usually old poises are most unreliable because their value is changed substantially by the formation of oxide, carbonate, or sulphate when they have been buried in damp soil, but these weights provide a trustworthy datum, since they are linked together in a set:

Reported weight

Ounces averdepois

Theoretical weight

284.0 grams

10

283.500 grams

226

8
226.800

70.5

21/2
70.875


Another piece of evidence is provided by the set of weights found at Riazan near Moscow. Their importance has been stressed by A. Mongait who dates them in the eleventh century A.D. and takes them as evidence of the existence of a pound of 397 grams. I would interpret them a based on a Chian-Thasian drachma of 3.96 or 3.966 grams.

Reported weight
Units
Theoretical weight

144.3 grams

36

142.776 grams

56.167

14

55.524

39.808

10

39.660

39.429

10

39.660

32.177

8
31.728

In 1956 the interpretation of Mongait was questioned by V. L. Ianin, in his survey of the monetary weights of pre-Tatar Russia, with the argument that only units related to the funt could have existed in that period. In 1948 B. A. Romanov, independently of Mongait, concluded that in the eleventh and twelfth century there had been in Russia a grivna, double ounce of 49.25 grams.

The native currency of Russia is constituted by bars of gold and silver; these bars were not intended to be of a fixed weight, but tended to conform more or less to a common unit, so by a table of averages it can be determined which unit was employed by those who cast them. The weights of these bars have been tabulated by N. Bauer. The pieces may be dated by the coins found with them; a few very early samples have been found with Roman coins. In my opinion the bars of the Kievan period seem to be based on the Roman libra. Bauer finds a specific indication of weight only in the thirteenth century; the bars of this and of the following two centuries indicate according to him a unit of 197 or 196 grams. This would be 7 ounces averdepois (198.35 grams) or half a Thasian mina.

In my opinion the original standard of the Black Sea area and of Russia was influenced by the Chian-Thasian standards. I have reported that in 411 B.C. the island of Thasos, which is located on the Thracian coast, near the present Greek-Turkish border, adopted a mina of 396.6 grams (14 ounces averdepois) and began to issue coins by drachma which is 1/100 of this mina. This monetary standard was connected with the use of wine jars which relate as 11:12 to the wheat unit. Thasian jars together with other Greek jars of the same type have been found in great quantity in the Black Sea area, whereas Greek wine was exchanged for wheat. A study by the Romanian numismatist Constantinul Moisil concludes that around 400 B.C. Thasian coins became most common in the Balkan Peninsula and particularly in Dacia, the lower Danube area. According to Moisil, in the Balkan area “the commercial penetration of Thasos lasted up to the beginning of the Roman Empire.” The Dacian coinage of the first and second century A.D. was an imitation of Thasian specimens.

For the period following the Fall of the Roman Empire evidence is provided by a tomb of a goldsmith excavated at Jutas in the Hungarian plain. It is reported that the skeleton reveals Mongolian traits; the goldsmith was one of the Avari who settled in Hungary. From Byzantine coins the tomb has been dated in the seventh century A.D.: in it, together with tools of the profession of goldsmith, there are scales and bronze square ounce weighing 28.67 grams. This unit has been explained as Byzantine solidi, but even though a small allowance must be made for an increase in weight because of oxidation of the bronze, the weight could not correspond to 6 solidi which by my estimate would be 26.66 grams. It is a matter of an ounce averdepois.

My conclusion is that the standards of the Black Sea area were determined by the system adopted by Thasos: wine units based on the ounce averdepois and wheat units based on the wheat ounce. Through Russia these units reached England. At Duurstede the ounce averdepois was still the monetary unit, but in England the ounce averdepois became a unit for merchandise whereas precious metals are measured by the wheat ounce (ounce Troy). This shift was influenced by the introduction of the artabic ounce or Cologne ounce as the monetary unit of the Carolingian Empire; this unit was called ounce Tower in England. But in Sweden there seems to have been an intermediary situation: up to the adoption of the French metric system in 1878 A.D. the “true pound,” skålpund, was a unit of 15 ounces averdepois, I presume that it should have been 425.25 grams, but in the ordering of Swedish measures of 1855 A.D. it was computed as 425.076 grams, being 8,848 Holland ass of 0.04807 grams. But the prontuary of weights of Horace Doursther in 1840 A.D. lists the skålpund as 425.54 grams and the Holland ass as 0.04786822 grams. The skålpund was a unit for general merchandise and for this reason was called viktualievikt; the corresponding unit for precious metals was the metallvikt, which in 1855 A.D. was fixed at exactly 12 ounces averdepois, that is, 340.20 grams or 7,078.7 Holland ass.

12. Because Russian measures were originally based on the ounce averdepois the foot of Russia is equal to the English foot, but, according to my interpretation, this foot was adapted to define units of different origin.

By ukaz of Czar Peter the Great the Russian foot must have the same value as the one adopted in England. The Law of 1918 A.D. that made Russian system of measures compulsory in the Soviet state, adopted values of the Russian units computed by Dmitri Mendeleev, as director of the Russian bureau of standards, on the basis of a foot of 304.80 mm., American value. The study of historical metrology not only of Russia, but of the entire Eurasian area, in the last decade has became a most active field of research in the Soviet Union and obviously receives strong impulsion by learned societies, in contrast with the increasing neglect and even hostility that has developed in the West since the death of Theodor Mommsen. But Soviet scholars have been so satisfied in finding that the Russian foot and the funt have not substantially changed since the earliest period of Russian history that they have neglected the investigation of the small variations that have occurred in the course of time; they assume that from the beginning the values were those of the Law of 1918 A.D.

Beginning with the middle of the eleventh century A.D. stone churches were built in Novgorod and Kiev; the churches of St. Sophia in Novgorod and in Kiev have been measured several times for the purpose of architectural interpretation, but a detailed analysis aiming at an exact determination of the lineal units is still wanting. The able metrologist Mongait examined the Cathedral of St. Sophia in Novgorod in taking advantage of the work of repair conducted after the Second World War, but he limited himself to report that there was employed a sajen’ of about 2.13 m., as we knew even before. He has promised a more detailed report, but if it has appeared I have failed in tracing it. The sarcophagus of King Iaroslav which is in the Church of St. Sophia in Kiev that he erected, is described as having the following dimensions:

2.36 m.

length

1.22

width

0.91

height without lid

1.61

height with lid

0.15

thickness of sides

Assuming a foot of 304.8 mm. the dimensions would be:

73/4 feet

= 2362.2 mm.

4

= 1219.2

3

= 914.4

51/3

= 1625.6

1/2

= 152.4


This proves that by a more accurate analysis of these monuments it would be possible to ascertain the exact value of the foot employed.

These figures prove that those metrologists who claim that the foot was not used in Russia before the age of Peter the Great are not correct. It is true, nevertheless, that the typical Russian computation is by a rod, the sajen’ (from the verb siagat’ “to stretch,” but linked also with the stuga “band,” “crosspiece,” as indicated by the Romanian equivalent stânjen) of 7 feet, divided into 3 arshin (from the Persian aras through a Turko-Tatar arish) of 28 inches. There is also a lokot’, “cubit,” equal, to 11/2 English feet. Russian units have preserved the septenary character that the great cubit had in Egypt.

The anchor point of the Russian units of volume and weight is the pud (the term is derived from the Latin pondus through a Norse intermediary), a cube with an edge of 10 English inches. The pud contains 40 funt of water (this term is also derived from pondus through the German Pfund). Another important unit is the chetverik of 64 funt.

The commission that fixed the Russian units in 1841 A.D. adopted as a unit of length the English Parliamentary standard of 1824 A.D., but took as standard funt that indicated by a sample deposited at the mint of St. Petersburg in 1747 by a committee that proceeded to a general revision of measures. It performed its calculations by using distilled water, as in the French metric system, but at ordinary temperature (13.5° Réaumur =17° Celsius). In this way the funt came to be 25.019 cubit inches of water. The volume is increased as 1000:1001.2 over what it would be at 4° Celsius. Similarly, a decree of 1835 A.D., as a result of the work of the commitee of 1747, had established that 8 funt be equal to 200.15 cubic inches of distalled water at 13.50 Réaumur. The reason for this procedure is that only by increasing the volume value of the funt over 25 cubic inches the chetverik comes to have a volume of 64 funt. In 1889 Mendeleev recalculated the funt and made it equal to 24 cubic inches by a foot of 304.80; by using water at less than maximum density he fixed the funt at 409.51241 grams. But the law of 1918 A.D. fixed the chetverik at more than 64 funt, making it 26,239 c.c. From this it appears that the chetverik was considered a unit of great stability.

N. T. Bielaiew (Beliaev) properly observes that the chetverik is the Roman quadrantal; I agree with him wit the qualification that it is a quadrantal of 80 librae of the geometric form. Hence, it should be 26,244 c.c. I may observe that the term chetverik may mean both “fourth” and “foursided,” so that it is a translation of the Latin quadrantal. The English term firkin too possibly meant originally the cube of the foot, rather than a “fourth” of some other unit.

L. V. Cherepnin in his treatise of Russian metrology, states that the Russian system was established in the Kievan period (tenth to twelfth century A.D.) taking as starting point a libra of 327.456 grams. But this figure is the erroneous figure of Letronne for the Roman libra slightly adjusted to make it fit with the funt of Mendeleev. I would count 4/5 of a geometric libra of 328.05 grams and arrive at funt of 410.0625. Sixty-four such funt would be a chetverik of 26,244 c.c. counting by rain water at ordinary temperature. By these standards the Russian foot would be 304.8955 mm., slightly less than the English foot of Guildhall.

But I am inclined to think that the value of the funt came to be adjusted to the libra of Byzantine coins, the reduced libra of 320 grams. The funt is divided into 96 zolotnik, “gold pieces,” divided in turn into 96 dolia, “parts.” I suggest that the dolia was computed as 1/100 of solidus or 0.0444 grams, which with a zolotnik of 4.24666 grams, such that 75 make a libra of 320 grams. Hence the funt was 409.60 grams.

A unit of 4.096 grams seems to be indicated by a set of Viking poises found in the Island of Gokstadt:

Reported weight

Units

Theoretical weight

819 grams

200

819.320 grams

57.25

14

57.352
32.65
8
32.773

32.4

8
32.773

24.38

6
24.579


If I am right in my inferences the pud as a unit of 16,384 c.c., indicated a foot of 304.7805 mm., which may be the origin of the value adopted by the Anglo-Saxon kings under Viking influence. According to a recent work of Ianin the Russian system was established in the last thirty years of the tenth century A.D., the period in which there was formulated the Anglo-Saxon rule about the grith.

There is evidence that the Russian system had parallels in that of the Low Countries. John Hasse, whose report on Russian measures written in 1555 A.D. was published in appendix to Richard Haklyut’s Principal Navigations, stated “the arshin I take to be as much as the Flanders ell, and the lokyt half an English yard”. The mathematician Leonhard Euleur wrote to the astronomer Joseph de la Lande that he had found that the Russian funt was 4/5 of the pound Troy of the Low Countries; I have computed the latter as 9216 grains of 0.05333 grams (1/6000 of 320 grams) or 491.52 grams. The zolotnik is 80 such grains.

It is possible to surmise which is the center from which the Russian system of measures originated and through which it was linked with the units of the Low Countries. In the Russian system there is a unit of 10 pud called berkovez after the city of Björkö, which was a great trading center between 800 and 1000 A. D. The commercial Law of the Baltic Sea called Bioerkö roetter remained influential for several centuries. It may be that Russian measures were those sanctioned by the “laws of Björkö.”

The berkovez was considered the standard load of pack animal (called last in England and in Scandinavia ) and was the naval ton (talentum navale) in the trade of the Baltic and the North Seas; it was called skeppund in Swedish and skippund in Danish. From Malines it can be gathered that what he calls skippound was used not only in several ports of the Baltic Sea but also at Amsterdam. A naval ton of 400 Russian funt is equal to 360 pounds Troy of the Low Countries. The berkovez may have been originally 500 geometric librae or 16,402.5 grams, equal to 400 funt of 410.0625 grams.

The pond Troy of the Low Countries had two values: one adjusted to the geometric Roman libra and one adjusted to the divisions of the reduced libra. The Troois pond of Amsterdam was equal to 11/2 geometric librae or 492.0750 grams. In 1799 A.D. a poise of the mint of Amsterdam was estimated at 492.16772 grams or 9266 Paris grains. But there was also another value, called Brabant weight in Amsterdam, which is the value of Troyes itself, of Brussels, of Bruges, and of Liège. The pound Troy is equal to 9216 grains based on a libra of 320 grams; the Russian funt of 409.6 grams is 4/5 of it and was equal to 9216 dolia ( 96 zolotnik of 96 dolia), the dolia being 96 x 72 of the libra of 320 grams.

In my opinion the theoretical value of the Brussels pound was 491.520 grams, but the poises of the mint of Brussels indicated a pound of 491.73 grams. There was a slight error to the excess in the poises of Amsterdam and Brussels, because the calculation was performed by a unit called as (aas in Dutch; ace in English) which should be 10/9 of the grain of 0.0432 grams or 0.0480 grams, but was 0.04804 as indicated by the Swedish data. Counting by the first value the pound Troy of Brussels equal to 10240 aces, would be 491.520 grams, whereas by the second value it was 491.929 grams. The calculation typical of Amsterdam, based on the geometrical libra, tended to be confused with that typical of Brussels, based on the divisions of the reduced libra of 320 grams: the tables of conversion issued in 1799 A.D. by the French administration of the Département de la Dyle (Brussels ) computed the pound at 9266 Paris grains, which is the value of Amsterdam. In my opinion this shift explains the structure of Russian measures and probably the existence of two values for the English foot: the value of Guildhall and the shorter royal value.

13. The official Roman mile (mille passuum) was composed of 5000 Roman feet (1000 paces) and was usually divided into 8 stadia of 600 artabic feet. But it is recognized that in the Roman Empire there was also a mile of 5400 Roman feet called miliarium and milion in Greek documents. The milion is equal to 3000 great cubits and is divided into 71/2 stadia of 600 feet which are 2/3 of great cubit (the foot which corresponds to the pied le Comte of France, to the piede Vicentino of Italy, and to half a Russian arshin’). This milion is connected with the great cubit or Ptolemaic Philetairic cubit. Since the great cubit is 32/30 of trimmed barley cubit, the milion is equal to 3200 barley cubits and to 4800 barley feet, so that it can be divided into 8 stadia of 600 barley feet. A bilingual inscription celebrating the construction of an aqueduct by the Roman administration of Egypt, under the praefectus Aegypti Julius Aquila in the fortieth year of the Emperor Augustus (10/11 A.D.). states in the Latin text that the length of the aqueduct is 25 miliaria but in the Greek text speaks of 200 stadia. These are stadia of 600 barley feet.

The English mile is the milion of the Roman Empire computed as 5400 Roman feet of the geometric variety. A mile of 5400 such feet would be 1607.9094 m., whereas the present English mile of 5280 English feet of 304.8 mm. is 1609.344 m.

The following are the division of the milion as used in the Roman Empire:

 
Milion
Stadion
Plethron
Great cubits
3000
375
37.5
Barley cubits
3200
400
40
Barley feet
4800
600
60
Roman feet
5400
675
67.5


The following are the divisions of the English mile:

 
Mile
Furlong
Chain or acre
English feet
5280
660
66

In the Anglo-Saxon translation of the Bible and in Bede furlan renders the term stadion.

Since in principle the English foot is 16/28 of great cubit, the English mile should be 5250 feet. The figure is increased to 5280 feet in order to obtain a computation by undecimal units.

Undecimal units were common in ancient and medieval metrics. A number of Greek buildings are computed by a rule that is 11/10 of a standard unit. The most important of the undecimal units is the pied de roi equal to 11/10 of Roman foot; in France the toise of 6 pieds de roi was was used together with the aune of 4 Roman feet. The undecimal computation was repeated by using a perche royale of 22 pieds de roi, but in Champagne and Orléanais the perche was 20 pieds roi.

The pieds de roi and units of the English type are combined together in Turkish metrics. There was an arshin which, like the Russian arshin, is twice a foot which is 7/6 of English foot. There was a rod called qasáb qabáni of 51/2 arshin which was divided into 6 Harshimite cubits which prove the double of the French pied de roi. The rod was also divided into 7 cubits beládi which are the standard of 555 mm., the double of the trimmed lesser foot.

The reason for combining a septenary unit with an undecimal unit is indicated by Renaissance and later manuals of architecture and of landsurveying: the square of the diameter is computed as 11/14 of the surface of the circle.

Undecimal units were particularly important in computing agrarian units. In the ancient and medieval world, agrarian units were arranged in a series in which each is the double of the preceding one and is assumed to be constructed on the diagonal of the preceding one. The diagonal was computed as 10/7 of the side. It was assumed that a square with a side of 70 units has a diagonal of 100 units; it was assumed that the square with a side of 100 units has a diagonal of 140 units. Hence the quadruple square has a side of 140 units and is exactly 4 times the square with a side of 70 units; but the figure of 100 units for the side of the intermediate square is approximate and it was corrected to 99 units (the exact figure is 98.995). The value of the square root of 2 was obtained by averaging the relation 7:10 between side and diagonal with the relation 10:7 between side and semidiagonal, so that the square root of 2 is 99/70 = 1.414286. In conclusion, as septenary units helped in problems involving the value of pi and the square root of 2, the same result was achieved also by the use of undecimal units.


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