# The Origin of Metrics

1. When writing appears for the first time in human history in the form of cuneiform writing in Mesopotamia (about 2900 B.C.), the system of measures which is at the basis of all metric systems of the ancient world and of China had been already conceived and formalized. This system continued substantially the same among the Arabs and in medieval Europe, including Russia. Present English measures are the last survival of a system of measures that is as old as human civilization. The French metric system was the first break in the continuity of measures in millennia.

This system of measures was probably organized when agriculture was first developed in the area that goes from Syria to Iran around 6000 B.C. As soon as agriculture became the main source of food supply, it became necessary to calculate how the available stock may be distributed between one crop and the next. In case of famines, it was necessary to know how to stretch supplies. Throughout the ancient world the rule was that an adult free male consumes two basic pints (540 c.c.) of wheat a day. A man consumes in a month half of the normal weight transported by a man, which is a talent. Women and slaves often received half of the basic ration. During famines half a pint of wheat (about 700 calories) was considered the minimum survival ration. These rations appear as standard in Oriental, Greek and Latin documents up to the Carolingian age. The only unit of volume that has a natural basis is the half-pint of 270 c.c. which is described as the contents of the two hands cupped together, (ophenayim in Hebrew). This handful is the origin of the standard volume for a cup or glass: in Greek kotyle means the half-pint measure, the two hands cupped together, and a cup. This suggests that the very first measuring was that of grains by handfuls.

2. The earliest documents of Mesopotamia and Egypt indicate that the system was based on a foot of 300 mm. This unit is usually known as Egyptian foot; because it was the standard of Egypt from predynastic times into the first millennium B. C. Its value was first determined by Newton from the dimensions of the Great Pyramid of Gizah and was verified with certainty at the beginning of the nineteeth century when Egyptian studies developed as a result of the Napoleonic expedition to Egypt. This foot was divided into 16 fingers—the division of feet into 12 inches (unciae) dates from the Roman period. To the foot there corresponds a cubit of 450 mm. divided into 24 fingers. Newton was able also to infer, on the basis of his knowledge of Talmudic literature, that in Egypt there was used a special cubit which is 7/6 of the regular cubit of 450 mm. Later the discovery of measuring rods confirmed the existence of this cubit of 525 mm., divided into 28 fingers; it is called royal cubit in Egyptian texts.

The royal cubit is an example of septenary units; septenary units were most common in the ancient and medieval world, but usually they were represented by rods of 7 feet or 7 cubits. The rod of 7 cubits is common in late cuneiform texts and is mentioned in the Bible. A recent example is the rod of 7 Swedish feet (Roman feet) used in Sweden up to the introduction of the French metric system. Septenary units were used to solve in a simple practical way problems involving p, the square root of 2, and the square root of 3; the value of was computed as 31/7, that of the square root of 2 as 1.40, and that of the square root of 3 as 7/4. Hugh R. Watkins has shown that in England several buildings of the Norman period are calculated by units of 7 English feet; he quotes this as evidence of Russian influence carried by the Vikings, since the Russian sajen’ (the legal unit up to the Soviet Revolution) is equal to 7 English feet; but the computation of architectural dimensions by septenary units is common in all areas in all periods. The first evidence of it precedes the origin of writing by one two centuries.

3. Units of weight were determined by filling the cube of the foot or of the cubit with rain water at ordinary temperature; this filling proves to have the same density as the distilled water at maximum density adopted by the French metric sytem. The impurities in the rain water compensate for the higher temperature. The calculation of weights by rain water at ordinary temperature was employed up to the eighteenth century, when distilled water began to be substituted.

The cube of the foot has a volume of 27,000 c.c. and a weight of 27,000 grams. It was the basic talent, that is, the weight that a man could carry at each end of a carrying yoke. It was divided into 1000 ounces, known as Roman unciae of 27 grams.

At the turn of the century, Friedrich Hultsch in the summation of his lifelong research on ancient measures, concluded that all weights of the ancient world may be calculated by a unit of 9 grams which I call basic sheqel, but he called by its Egyptian name of qedet. Hultsch, however, could not determine exactly the value of the basic sheqel and chose values that waver between 9.0 and 9.1125 grams. In 1906 Jean Adolphe Decourdemanche, at the end of his treatise on ancient and Arab measures, observed that his definition of the units of volume and weight is correct with the proviso that they may vary as 80:81. In 1928 Angelo Segrè discovered the theoretical reason for this discrepancy: the cube of the cubit (which was considered as the basic load that can be carried by an ass or by a man for a short distance) is equal to 33/8 talents or cubes of the foot, since 13:(1½)3 =1:33/8, but this relation is often simplified so that the cube load is equal to 31/3 talents of 10 thirds of talent (called modius in Latin). By making 31/3 equal to 33/8, there results the discrepancy 80:81. This discovery of Segrè must be linked with Hultsch’ investigation: the cube of the foot is divided into 3000 basic sheqels of 9 grams, but the cube of the cubit, which is 91,125 c.c, is divided into 10,000 basic sheqels of 9.1125 grams. The Romanian scholar Mihail Sutzu, at the end of a lifelong investigation of all existing ancient sample weights, concluded in 1930 that all weights are mathematically related to a qedet of 9.0 grams that constitutes the bazele fundamentale ale metrologiei ponderale din antichitate.

The discrepancy 80:81 is connected with the calendar. The Egyptian calendar consisted of 360 days of 12 months of 30 days plus 5 supplementary days. Scholars have been surprised in noticing that accounts of temples reckon by years of 360 and not of 365 days. The explanation of this is that for the sake of easier reckoning food supplies were counted by a year of 360 days, but by increasing the units as 81:80 they would last for 364.5 days. In the case of Mesopotamia, for the same reason, scholars have noted the occurrence of the year of 360 days, but not that of the five suplementary days. The last five days of the year or the first five were considered an extra boon connected with gift giving. This practice survives in our custom of gift giving either on Christmas or on the Epiphany.

4. C. F. Lehmann-Haupt dedicated half a century of research to proving his initial discovery, announced in 1888, that most ancient and medieval units of volume and weight exist in two varieties related as 24:25. This discovery was further developed in 1940 by August Oxé in a survey of all ancient units of volume and weight, based on decades of research. Oxé explained why there are units related as 24:25, which he called respectively netto and brutto. First of all, the existence of units arranged as

 24 48 60 72 96 144 25 50 62.5 75 100 150

allows to shift from duodecimal to decimal reckoning. From the point of view of computation, decimal units are preferable, but it is much easier to divide a cube duodecimally. For instance, the basic talent netto of 27,000 c.c. is easilly divided arithmetically into 50 basic pints of 540 c.c.; but for the sake of constructing cubes equal to a pint, it is better to count by a basic talent netto of 25,920 c.c. (Athenian monetary talent, Roman quadrantal) which is equal to 48 basic pints.

Oxé determined that the units of volume were connected with the specific gravity of water, wheat and barley. The units for wheat and barley were increased in volume so as to weigh as much as the corresponding water unit. The relation between a water unit and a wheat unit is calculated either as 5:6 or as 4:5; similarly the relation between a wheat unit and a barley unit is either 5:6 or 4:5. If it is assumed that the barley unit is 1½ water units, the wheat unit between them may be 5/4 of the length lower unit and 5/6 of the upper unit, or conversely be 6/4 of the lower unit and 4/5 of the upper unit. This is another reason for the existence of units related as 24:25, since 6/5:5/4 = 24:25.

Carrying Oxé’s research to its logical conclusion, I have determined that as there are units of volume and weight related as 24:25, correspondingly there are units of length related as <sup>3</sup>24:<sup>3</sup>25: For instance, next to the foot of 300 mm., which I call natural basic foot, there is a foot of 295.9454 mm., which I call trimmed basic foot and which is the Roman foot. This solves the problem of the mathematical relation between Roman and Egyptian foot, a problem that has bedeviled not only metrologists, but some of the best mathematical minds of Europe, since Newton determined the value of the Egyptian foot.

I have detrmined that the ancient metric system was conceived by a most ingenious pattern. The basic foot, edge of the talent of water, was divided into 16 fingers; a cube with an edge of 17 fingers contained a volume of wheat weighing a basic talent; a cube with an edge of 18 fingers contained a volume of barley weighing a basic talent.

 There were the following types of foot: Trimmed Natural Basic (16 fingers) 295.9454 mm. 300.0 mm. Wheat (17 basic figers) 314.442 318.75 Barley (18 basic fingers) 332.939 337.50

There were as many talents as there were feet. The specific gravity of wheat was computed either as 0.75 or as 0.8. Starting from a basic talent netto of 25,920 c.c. of water or grams, there was a wheat talent netto increased by 1/5, that is, 31,104.0 c.c. (the cube of the foot of 314.442 mm. is 31,090 c.c.) and a wheat talent brutto increased by 1/4, that is 32,400 c.c.(the cube of the foot of 318.75 mm. is 32,385.5 c.c.). The specific gravity of barley was computed either as 0.66 or as 0.7. A barley talent with a volume of 1½ basic talent netto is 38,880 c.c. A barley talent equal in volume of 1½ basic talent netto is 38,880 c.c. (the cube of the foot of 337.50 mm. is 38,443.4 c.c.). A barley talent equal in volume to 10/7 of basic talent netto is 37,028.6 c.c. (the cube of the foot of 332.939 mm. is 36,905.6 c.c.). One type of multiple or the other was used according to the grain locally used. Other units were obtained by filling the wheat and barley talents with water. The system allowed all sorts of combinations that were used for other substances and for purposes other than the calculations of specific gravity.

5. The study of weights indicates that the units have not changed by a grain in millennia. The evidence in this matter is positive, since it is provided by very heavy sample weights for which the margin of error is less.

The official standards of reference were kept in the most important temples and were sent from one country to another. For instance according to Hebrew tradition, the length standard of the Temple of Jerusalem came from Thebes in Egypt. In the Second Temple the standards of length were kept next to an image of the palace of Persian Kings at Susa probably to indicate that they were a copy of those kept there. Some of these official standards have been actually found by excavators, as for instance a huge bronze weight from the Temple of Apollo at Miletos that the Persians, upon conquering this Greek city, took to their capital, Susa. Bernal Haussoulier in the archeological report states that it weighs 93.7 kg, but according to the curator at the Loure Héron de Villefosse, it weighs 93.5 kg. I suspect that an accurate weiging, which would have to take into account the effect of oxidation, would result in a figure close to 93,312 grams, that is, 3 wheat talents netto or 3000 English ounces Troy. A Persian bronze weight in the form of a lion, also found in Susa and exhibited as an art piece at the Louvre Museum, was found by Héron de Villefosse to weigh 121.2 kg; probably it represented 300 units of 405 grams., equal to 121,500 grams. But, even though Héron de Villefosse was competent in ancient metrology (his library on the subject is now at Columbia University as the Dale Collection), his test was not as accurate as that performed by Colonel de la Chaussée who arrived at 121,543 grams.

The weight of the lion of Susa is confirmed by a bronze lion with an archaic Aramaic inscription reading “exact according to the silver unit” found at Abydos in Egypt. At first it was announced that this lion, which is at the British Museum, weighs 56 pounds, 9 ounces; since the beginning of the century some of the best names of metrology and ancient history exerted their ingenuity in trying to explain this figure; until Lehmann-Haupt in 1938 asked the British Museum for a verification . He obtained the answer that the weight is 68 pounds, 2 ounces (30,901 grams) with a small piece missing. Unfortunately Lehmann-Haupt’s communication could not be published until 1956, because of the antisemitic legislation of Germany. This stresses the fact that the greatest difficulty in metrological research is more the inacuracy of archeological reports than the subject matter itself.

Another important sample weight was found in the Palace of Knossos in Crete. It is a beautiful piece of pink alabaster with an octopus sculptured upon it. It has been reported that it is “almost exactly” 29 kg. It represents an artaba of 29,160 grams. In my opinion the figures of Cretan economic texts indicate that artabic units were used in that civilization.

I have classified and evaluated all these official standards, but for many of them adequate reports are missing and in some cases no report is available at all. But only a few are sufficient to establish the value of ancient standards. The official reference standard that was considered most authoritative in Greece was found in 1894 A.D. in the excavation of the Temple of Hera near Argos. But the man who unearthed the Greek equivalent of our Paris standards, Charles Waldstein, when he tried to insist on the importance of his discovery met such unfriendly reception among fellow archeologists that the continuation of his academic career in the United States became impossible and he was no longer entrusted with the direction of any archeological campaign. The object consists of an iron bar weighing 180 times 405 grams; the unit of 405 grams is confirmed by a set of 180 pieces of 405 grams each, found with it. The bar is bevelled at one end, because it indicates not only the unit of weight but also both the length of the natural basic foot and the length of the trimmed basic foot. The bar is 4 feet long, because this would be the perimeter of a cube, with the volume of a talent. A similar set was kept in front of the Temple of Apollo at Delphoi; I have determined that this set was sent by King Psammetichos I of Egypt around 650 B.C.

A reference bar made of bronze, cast in the shape of a steelyeard, was excavated in the Sumerian Temple of Nippur; it was found with a tablet mentioning King Gudea who ruled around 2050 B. C., but the bar could be much older. The nature of this object, which is now in storage at the Museum of Antiquities in Constantinople, has not in been recognized. It represents 96 times a weight of 432 grams. The weight is certain because on the bar there are marked the lengths corresponding to the edge of cubes containing standard multiples of the unit of 432 grams. The total length of the markings is four times a foot of 274.731 mm. which cubed contains 48 x 432 grams of water, that is, half the weight of the bar. This is foot is called Italic or Oscan foot by archeologists. At Ushak in Turkey (the ancient Flaviopolis) there was divcovered a stone reference standard of the Hellenistic age in which there were cut holes corresponding to standard units; the size of the holes was not reported, even though it can be inferred from their inscribed names, but it was reported that on the stone there is marked the length of 555 mm. (twice the length of the Nippur bar). The foot of 274.731 mm. is obtained by taking 15 fingers of Roman foot and reducing it by the usual relation 324:325; without this reduction the cube would contain 50 minai of 432 grams.

The Greek iron bar indicates the weight of 405 grams (45 basic sheqels of 9 grams), because a cube with an edge of 4 fingers (a handbreadth) of a trimmed basic foot has a volume of 405 c.c. The standard reduced pint is 486 c.c.; to each pint there corresponds a mina which is 5/6 or 4/5 of it, representing the weight of the pint filled with wheat, so that 405 grams is the mina corresponding to a pint of 486 c.c. The Sumerican bronze bar indicates a mina of 432 grams (48 basic sheqels of 9 grams ) which is 4/5 of the basic pint of 540 c.c. A basic talent netto (Roman quadrantal) contains 60 minai of 432 grams and 64 minai of 405 grams. The first fraction is convenient arithmetically; the second fraction is convenient in the geometric division of a cube.

6. The evidence povided by sample weights may be confirmed by the study of coins, but this requires particular caution, because of necessity coins deviate somehow from the norm at minting and also because ancient coins are usually worn and often chemically altered by burial in damp soil. However, when there are large samples of the same type of coin, these disturbing factors may be taken into account by a proper statistical analysis; but numismatists have preferred to apply fanciful methods of their own invention instead of following orthodox statistical methods. As a result most numismatists believe that the standard of the mint was changed for each series of coins, whereas mints kept the same exact standard for centuries. Many important works on ancient coins are concerned with explaining the political reasons for these imaginary shifts in standard.

The only study of ancient coins that is statistically correct is that of Greek coins, mostly Athenian, to which Henry Kercher, a private scholar of Cincinnati, dedicated twelve years; unfortunately the manuscript of Kercher was not judged fit for printing by the American Numismatic Society who upholds the so-called humanistic, non-mathematical, approach to measurements. Among other points, Kercher proved statistically that the more valuable coins were tested at the mint by scales calibrated to register 1/3 of an English grain; this is a level of precision that is considered satisfactory today in the gold and silver trade. It seems that this was the limit of the best scales of the English mint in the eighteenth century A. D., even though Newton, as director of the mint, tried to introduce even higher precision.

The proper analysis of coins confirms the data provided by sample weights. The Attic drachma is certainly 1/100 of 432 grams; the official Roman libra is 3/4 of this Athenian monetary mina and is 324 grams. This libra divided into 5000 units gives the English grain 0.0648 grams of which 7000 make a pound averdepois of 453.60 grams (Athenian weight mina). The exact knowledge of these units is most important because 60 Athenian monetary minai or 80 Roman librae make a basic talent netto, a cube with an edge of a trimmed basic foot (Roman foot). This unit was called pes quadratus or quadrantal because the standards, such as one kept at the Capitol, were actually constructed as cubes. This unit is filled with water; the corresponding unit for wheat, containing 100 librae in volume, was called centenarium and was considered the cube of the natural wheat foot.

At the beginning of the last century Antoine Jean Letronne from the weight of Roman republican gold coins calculated the libra as 327.42 grams; later it was found that these coins were the product of forgers who wished to satisfy the demand of collectors for Roman gold coins in a period when it had not yet been ascertained that before Caesar’s time gold was struck only occasionally in Rome. In spite of this the figure of Letronne has become the Rock of Gibraltar in the study of ancient coin weights; the standard of coins in the entire Mediterranean area is computed by it. The efforts that have been occasionally made to question it in the course of a century and a half have never found response, whereas great ingenuity has been displayed in developing peculiar theories that force the data to agree with Letronne’s figure.

Letronne himself reported that he tested 1350 silver denarii of the Roman Republic selected from among the heaviest specimens and came to an average of 325.55 grams; he accepted the higher figure indicated by the gold coins, which now we know were forgeries, because he assumed that gold pieces would have been tested more carefully and because a similar figure is indicated by medieval coins; but the most common libra of medieval Europe is the geometric Roman libra which is increased as 81:80, being 328.05 grams. This was the libra of the medieval city of Rome. To this libra there corresponds a Roman foot of 297.761 mm., which is the one usually employed in the construction of Greek temples: this unit was considered the scientific Roman foot beginning with the Dark Ages. It was called geometric foot and it became the standard of the city of Rome in medieval times. Before the French Revolution it was usually computed as 132 Paris lines, that is 297.7695 mm.

There was also a Roman libra obtained by reducing the official libra as 80:81; this libra of 320 grams corresponds to a Roman foot of 294.722 mm., which Renaissance scholars called pes Cossutianus and found employed in many ancient Roman buildings. It was also common in medieval France. The libra corresponding to this foot was used by Emperor Constantine the Great to issue a new gold coin, the solidus equal to 1/72 of libra. The coinage of this solidus was continued after the fall of the Western Roman Empire not only by the Byzantine mints, but also by the mints of Europe under barbaric control. Since this coin enjoyed exceptional popularity extending as far as Russia and Scandinavia, it was used as a standard for precious metals through the Dark Ages and beyond. For this reason it is most important to calculate exactly the value of the solidus. Recently Howard L. Adelson has broken with the usual practice of numismatists by publishing accurate statistical tabulations of the weights of Roman and Byzantine solidi; but having gone so far on the way of scientific statistical analysis, in the final interpretation he disregarded them and calculated the solidus as 1/72 of the imaginary libra of 327.42 grams mentioned by Letronne. Adelson’s tabulation indicates a value around 4.445 grams, whereas my value (1/72 of 320 grams) is 4.444 grams.

 There were three types of Roman foot with three corresponding types of libra: geometric 297.761 mm. 328.050 grams official 295.945 mm. 324.000 grams reduced 294.722 mm. 320.000 grams

7.
Because of the problem of calculating food distributions by the month and by the year there was developed a consistently sexagesimal system of units. The starting point was the wheat talent of 60 basic pints. The fact that 6 volumes of wheat weigh as much as 5 volumes of water, and 6 volumes of barley weigh as much as 5 volumes of wheat also contributed to the development of sexagesimal computation. Sexagesimal computation developed also because the easiest way to divide a cube is to divide it into 8 smaller cubes; by repeating the operation for each of the resulting cubes, there are obtained 64 little cubes. This procedure occurs in the English system of measures in which the cube of the foot is divided into 8 gallons and each gallon is divided into 8 pints (English wine pint of George II and earlier). This procedure was most common in Egypt where the units remained essentially decimal, or duodecimal, whereas in Mesopotamia there was adopted not only the full-fledged sexagesimal system of measures but also a sexagesimal system of reckoning.

In the field of measures sexagesimal computation resulted in the development of the artaba. Whereas normally a cubit trimmed cubit or load (called imêru in Akkadian and homer in Hebrew; the term means both “load” and “ass,” since the ass is the typical carrying animal) is divided into 33/4 or 31/3 talents netto, it is divided into 3 artabai which are therefore 9/8 of basic talents netto (29,160 c.c.). The artaba is divided into 60 reduced pints of 486 c.c. (9/10 of basic pint). The system has the advantage that the wheat ration may be computed directly as an artaba a month, and a load becomes the ration for three months. The reduction of the pint is compensated by shaking the measure, since by shaking the grain contents may be increased by about 10%. Hence, the normal daily ration in the artabic system is 2 reduced pints of wheat a day.

The edge of the artaba is the artabic foot (called Greek foot by metrologists of the last century). Since the artaba is 9/8 of quadrantal, the artabic foot should be equal to 39/38 of official Roman foot, and in fact there is an artabic foot of 307.796 mm. But in practice the artabic foot was calculated as 25/24 of Roman foot, that is, 308.276 mm. From the dimensions of Greek temples Böckh computed the artabic foot as 136.658 Paris lines, that is, 308.277 mm.

The artabic foot was used to calculate itinerary distances. The reason for this is that a stadion of 100 artabic feet corresponds exactly to a second of degree latitude. Taking 600 stadia of 600 feet each there results the length of the degree of meridian at latitude 36°, which was considered the basic latitude by ancient geographers. This is the axis of the inhabited world: it is the exact axis of the Mediterranean from Gibraltar to Rhodes, it passes along the southern coast of Asia Minor, it joins the Mediterranean to the nearest point of the Euphrates where transportation by water can start again, it constitutes the traditional border between southern and northern China. A degree is 110.980 m.; by our modern theoretical geoid this is the value of the degre of meridian at latitude 37°. Considering the elevation above sea level, the goedetic survey conducted in Mesopotamia at latitude 36° was perfect.

Newton did not trust the traditiopnal datum of 75 Roman miles to a degree (there are 8 artabic stadia in a mile), also because he was not certain about the exact value of the Roman foot, and waited for the calculation of Picard. But when exact data began to be gathered at the beginning of the eighteenth century, it was found that the ancient figures were at least as accurate as Picard’s calculation. A number of famous scholars examined the problem of ancient reports about the dimensions of the earth and all of them from the beginning of eighteenth century up to the first part of the nineteenth, concluded that all ancient reports agree on the same value using different types of stadion. There were as many types of stadion as there were types of foot. It was also observed that the calculation must be pre-Greek, since the Greeks did not have the organization to proceed to a geodetic survey. Beginning with the romantic period all these studies were rejected a priori by people who were not competent in the field of ancient measures, on the assumption that the ancients had to be primitive. With the texts available today and with more accurate values of the units I have obtained, I have ascertained that the Egyptians by the beginning of the Middle Kingdom could determine the latitude of a place with a precision greater than a minute of degree. Before Greek times they had calculated intervals of longitude over a space of several degrees with almost the same precision. Furthermore, there is absolutely clear evidence that the Egytians knew that the degrees of meridian are longer as one proceeds to the north. The only issue in which they differed from the modern view was that they concluded that the earth was elongated at the poles, the degrees of meridian being somewhat stretched towards the Equator. The low estimates of ancient science result from the circumstance that modern investigators cannot read properly documents involving measurements. For instance, Father F. X. Kugler, the specialist of Mesopotamian astronomy, states that the angular unit of 2’’ in cuneiform texts may not have had any practical application, since angles of that nature were beyond the physical possibility of observation. Actually the angle of yearly precession of the celestial pole around the pole of the ecliptic was directly observed as being 0° 0’ 50” with an error in the range of .01” by the Fourth Dynasty of Egypt. There is a mass of Egyptian, Mesopotamian and Greek texts that state or apply to other problems the calculation that the celestial pole makes a full circle around the pole of the ecliptic in 25,920 years (50” a year). I mention here these conclusions that I will present in future publications in order to indicate what kind of precision in units of measurement is presupposed by these results.

The artabic foot and the artaba were adopted as standard units by the Persians when they established an universal empire in the second half of the sixth century B.C. The term artaba is a Persian word meaning the “great measure” ; it can be applied to any cubic foot unit, but more specifically is applied to the unit of 29,160 c.c., the cube of the correct artabic foot of 307.796 mm. One hundred such artabic feet were considered the correct length of the second of meridian at the latitude of Thebes, the geographical center of Egypt.

The Temple of Ammon at Thebes at latitude 25° 43’ N was considered, and is, located at 2/7 of the distance between the Equator and the pole. Ancient geographers divided the space between the Equator and the Pole into 7 zones. Egyptologists have vainly tried to explain why the Greeks gave the name of Thebai to the city called Wast by the Egyptians; the explanation is provided by the Hebrew word thibbun meaning “navel”. From the Bible (Jud. 9:37) we learn that “a navel of the earth” was located at Mt. Gerizim where there was originally the sacred center of the Hebrews before it was moved to Jerusalem; the Samaritans never accepted such a shift, and geographically they were right, since the claim of Jerusalem to be the navel of the earth was not correct. The eastern gate of the Second Temple, where the standards of length were located, was called Gate of Susa, but Susa was located at the latitude of Mt. Gerizim which is 32° 11’ N. The sanctuary of Mt. Gerizim was located at a latitude that is 2½ sevenths from the Equator. Egyptian benchmarks had the shape of the “navel” found at the Temple of Delphoi in Greece. These “navels” had the shape of a hemisphere with the meridians and parallels marked upon them; at times they are half a sphere and at times they are elongated at the Pole. The sanctuary of Delphoi was considered a “navel of the earth,” as being located at 3/7 of the distance from the Equator to the Pole. This would correspond to a latitude 38° 34’ N; the Temple of Delphoi is actually located at a latitude 38°29’ N, which is the latitude of Sardis, the capital of Lydia, also considered a basic anchor point. There was a confusion of 5 minutes in the calculations, due to the fact that the axis of the inhabited world at times was computed at latitude 36° N and at times at latitude 36° 05’ N, which makes it 6° to the North of one of two Egyptian anchor points, the original apex of the Nile Delta at latitude 30° 05’ N on the axis of Egypt which is 31° 13’ E. Susa was computed as being 17° to the East of this point; it is at latitude 48° 15’ E. When the Assyrians established their religious capital at Nimrud in 875 B.C. they chose a point that was 6° to the North and 12° to the East of this Egyptian anchor point.

The Roman mile was equal to 8 stadia of 600 artabic feet of the longer variety; the Persian parasang, an hour of march, was equal to 18,000 such feet; so that there are 20 parasangs to a degree of latitude. It would be desirable to test the interval between Roman mile stones to ascertain whether on the southern shore of the Mediterranean, the Roman mile was shorter and equal to 8 x 600 artabic feet of the correct variety. A large number of Roman mile stones may be found still in place in Libya.

8. The artaba is divided into 60 reduced pints of 486 c.c. (9/10 of basic pint ); this pint is in turn divided into 60 reduced sheqels of 8.10 grams. (9/10 of basic sheqel). This sheqel is the typical sheqel of cuneiform texts, because these texts compute sexagesimally.

The Roman ounce of 27 grams is equal to 31/3 sheqels of 8.1 grams. But another important multiple is the ounce averdepois of 28.350 grams equal to 31/2 sheqels.

The artaba contains 3600 reduced sheqels, but by a discrepancy 35:36 there was also established another artaba equal to 3500 reduced sheqels. This special artaba, that I shall call Alexandrine, is equal to 28,350 c.c. and is 35/36 of the typical artaba that I shall call Persian.

As I have said, to the reduced pint of 486 c.c. there corresponds mina of 405 grams. The Alexandrine artaba is divided into 70 minai of 405 c.c.,whereas it takes 72 such pints to make a Persian artaba.

The cube of the English foot, called the firkin, contains 100 ounces averdepois and 62.5 pounds averdepois of 16 ounces, each pound being 453.60 grams. The Alexandrine artaba is the English firkin.

The units averdepois are connected with the factor 7. These units are connected with the week of 7 days, the lunar months of 28 days and the lunar years of 354 days. As sexagesimal units are adjusted to a solar year of 360 days, with the discrepancy 80:81 taking care of the excess of days over 360, the units averdepois are adjusted to a lunar yera of 350 days with the discrepancy 80:81 taking care of the excess of days over 350.

Unfortunately archeologists do not expect to find weights in sites older than the origin of writing, with the result that they do not report on objects that could be weights. A laudable exception is provided by the excavations of Tepe Gawra, near the present Iraqi oil center of Mosul. The lowest strata of Tepe Gawra represent the very first steps in the transition from village to urban life and may be date more than 1000 years before the origin of writing. Stratum VII of the excavation corresponds to the origin of writing; weights found at this level and at the higher levels indicate the units mentioned in cuneiform mathematical texts, as I have interpreted them. But the weights found at the lower levels beginning with Stratum VIII indicate units calculated by the ounce averdepois. Miss Virginia Gross, who reported on the weights of the more recent strata, excluded from her list an ovoid of basalt weighing 1368 grams, because in her opinion it did not conform to any standard one could expect, but she did mention incidentally its weight because it has the typical appearance of many Mesopotamian weights. In my opinion it represents 48 ounces averdepois (3 pounds averdepois ) which would be exactly 1360.08 grams [1366.875]. The weights of the earlier strata have been described by Arthur J. Tobler; the early sample having been found in Stratum XVIII which dates about 1000 years before the origin of writing and precedes the discovery of the potter’s wheel. I explain them as based on a unit which is 1/6 of an ounce averdepois, except for the specimens that seem to be based on a unit which is 1/6 of a geometric Roman ounce of 27.3375 grams.

 Specimen Stratum Reported Weight Units of 4.55625 Units of 4.7461 Theoretically (grams) Crouching animal XV 24.00 5 23.730 No. 1 IX 66.81 14 66.445 2 IX 45.70 10 45.562 3 XIII? 47.45 10 47.461 4 XVIII 11.43 2½ 11.391 5 IX 3.56 3/4 3.560 6 XI 30.79 6½ 30.850 7 X 19.36 4 1/4 19.364 8 XIII 5.83 1 1/4 5.933 9 XII 8.71 1 5/6 8.701

In the following literate period Mesopotamia, the ounce averdepois is not common but a clear sample of the ounce averdepois, dating between 2500 and 2400 B.C. is provided by a sample in stone now at the British Museum: it is reported as weighing 680.485, whereas 24 ounces averdepois is 680.400 grams. This stone is marked with the name of Dudu, a scribe of whom we have a statue and about whow we know that he was high priest under King Entemena of the city of Lagash in Sumeria. Since important Sumerian standards of volume and weight that have come down to us are marked with the name either of Dudu or King Entemena, it seems that the scribe Dudu was particularly concerned with setting exact standards of measure.

9. Around the middle of the seventh century B C., coins were invented on the coast of Asia Minor. According to the aecheological evidence I have gathered they were developed at Ephesos from objects in cuneiform documents and have been found in Greece in sites dating from the end of the second millennium B. C. The first coins were of electrum and appear to have been struck as fractions of a mina of 425.25 grams, which is equal to 15 ounces averdepois. This mina is certainly the standard by which Persian gold darics were struck a century later.

The earlier coins of silver which are also the earliest coins of Greece proper were struck in the island of Aigina around 600 B. C. ; rtoughly two decades later silver coins began to be struck in Athens on the same standard. The problem of Grrek monetary standard is considered most obsure for the mentioned reason that mathematists follow facciful statistical methods in averaging the weights of coins, but also because in this century specialists of Greek epigraphy have come to consider below the dignity of classical studies to investigate the monetary sapects of Greek financiakl and economic inscriptions, even though these Greek. inscription. Among other things these inncriptions throw on a fundamental problem that has been totally ignored: the relative value of coins in relation to uncoined silver or to foreign coins accepted at par. There were two rates; a rate 20:21 and a rate 15:16.

The coins of Aigina and the earliest coins of Ahtens were computed on the basis of a double Roman lebra of 648 grams, The coins were struck by a drachma which is 1/100 of 20/21 of 648 grams and hence of 617.143 grams. This figures is confirmed by the statistical analysis of coins. The relation 20:21 was practical because 100 coins bought 105 coins not accepted at pasr of the equivalent weight of uncoined silver, but the unit of 617.143 grams had the disadvantage of not corresponding to any established standard. After a few series Athens changed to a more practical standard; this was the result of the reforms of the lawgiver Solon that are generally dated in 594 c.c. because od lack of insight into Greek chronological methods, but probably have to be dated in 581 B. C.

The metric system os Solon is based on a mina of 453.60 grams. which is exactly the English pound averdepois (16 ounces averdepois), but coins, were struck as 1/100 of a special menetary mina is 16 Roman ounces of 27 grams, that is, 432 grams. The earlier Aigenetan monetary mina is 10/7 of the new Athenian monetary mina. The weight mina of 453.60 grama is 7/5 Roman libra or 7/10 of the earlier weigght mina of 648 grams . The rate 20:21 between coins and uncoined silver was preserved.

It is possible that the English grain by which the pound averdepois is 2000 grains and the libra is 5000 grains came to existence at this time. The English system counts 24 grains to a pennyweight of which 20 make a shilling or ounce; but in the Plantagenet period there was also a computation by 32 grains to a pennyweight. In the Arab metrics there are two grains that probably are 0.0648 and 0.0432 grams as the English grains. When the French metric system was adopted in Egypt in 1924 A. D., the first grain was computed by the round figure of 0.0650 grams. An English grain of 0.0432 grams is 1 1/100 of Ahtemian monetary drachma.

Athenian coins of the new standard became the most widely accepted in the Mediteranean area and their standard was adopted by the most important economic centers as for instance Syracuse in Sicily. After the defeat of the Persian at Salamis in 480 B. C., Ahtens established an empire that included many important economic centers of the Aegean Sea. A fundamental point of the imperical policy was to impose the use of athenian weight standards. An exception was made for the island of Chios, the main economic activity of which was the export of high quality wine. When the Athenian defeat in the expedition aginst Syracuse in 411 B. C., the island of Thasos, even before seceding from the empire, obtained as concession to adopted the standard of Chois. Thasos too was also an important exporter of high-quality wine, and the numismatic, epigraphic and literary evidence indicates that the Chian-Thasian monetary standard was connected with the export of wine. The same standard was adopted by other wine exporting centers that revolted against Athenian domination. An inscription indicates that the Thasian drachma was computed as 1/100 of a mina, which is 11/12 of Athenian monetary mina, that is 396 grams. But probably the unit was intended to be 14 ounces averdepois, taht is, 396.90 grams.

The calculation by ounces averdepis obviously had advantages, since the gold coins of the Persian Empire were issued by a mina of 425.25 grams (15 ounces averdepois), and this unit was adopted by the followers of Alexander the Great for their silver coinage, even though Alexander himself had employed the Athenian mina of 432 grams. finally, Athens itself in 196 B.C. began to strike coins by the mina of 425,25 grams; numismatists have not recognized that the issuance of what they call New Style coins is marekd by this shift in the standard.

Possibly the later Athenian standard may indicate the adoption of the less generous ratio of 15:16 between coins and uncoined silver. One hundred New Style drachmain (425.25 grams) would buy an Athenain weight mina (453.6 grams) of silver.

But the adoption of the New Style mina appears connected with wine trade. Many numismatists claim that their task is to be concerned with coin devices and not with coin weights, but often devices cannot be explained if the weight standards are ignored. In spite of the great attention given to New Style coins, numismatists have glossed over the fundamental revolution in the device of the Athenian coins represented by these coins. In the New Style coins the device is no longer the Athenian owl, but this owl with a wsine jar at its feet. This new device is explained by the circumstance that in the earliest Chian coins the device is a sphinx, but when the standard of 396 grams was adopted an amphora surmounted by a bunch of grapes was placed in front of the sphinx. On Thasian coins based on the mina of 432 grams the device is a satyr abducting a nymph; after the adoption of the Chian standard of 396 grams the larger units have a completely new device, the head of Dionysos, god of wine, but the smaller denominations present a satyr holding a wine kantharos instead of the nymph, with a wine jar on the reverse.

There is Hebrew and Greek textual evidence to the effect that high-quality wines were packed in jars, the volume of which was computed by the wheat talent netto of 31,104 c.c. This is confirmed by the recent tests of the volume of Thasian wine jars. Hebrew wine hars of the royal period are based on the same standard. The reason for using a when unit for wine, is that the jars were not filled completely, because an air space at the top improves the quality of wine: when the jars are shaken the air is forced through the wine, thereby increasing the oxidation, which is the essential element in the process of wine aging. This practice is discussed in Talmudic literature, because the action of not filling the jar completely was questioned as a failure to give an honest measure. Greek papyri and Greek metrological texts indicate that these jars were computed by a special fraction of eleven. The reason for this is that the actual contents of the jar were computed by ounces averdepois: 11/10 of 28.35 grams is 31.185 grams, which is about a wheat ounce of 31.104. This explains why the units averdepois were connected with the wine trade.

This epxlains also the device of the earliest Athenian coins that are of Aiginetan weight: the device is an amphora. The Aiginetan drachma of 6.171 grams is about 1/5 of a wheat ounce: 1/5 of 31.104 is 6.228. This early Athenian drachma was adjusted to the full volume of the wine jars. The Aiginetan and Athenian pre-Solonian mina is equal to 21.77 ounces averdepois instead of 22.

10. I have shown that an ounce averdepois, which is the unit for which we have the oldest archeological evidence, was the most important unit in the pre-Roman monetary systems.

The measures of Egypt under the rule of the Ptolemies and the following rule of the Romans, are documented by the Greek papyri found in Egypt. According to my interpretation, the main unit was the Alexandrine artaba which is 35/36 of the standard artaba that I call Persian artaba. The Alexandrine artaba is equal to 80 Alexandrine litrai, which are 5/6 of the mina of 425.25 grams (15 ounces averdepois). The Alexandrine artaba is the English firkin, the cube of the English foot, containing 1000 ounces averdepois.

I shall discuss this matter in detail in later pages, because the methods followed in calculating the units in Egypt and later in Viking Russia is such that as a result the English foot came to have two values: one a foot of 304.91 mm. (foot of Guildhall), which is exactly the edge of a cube of 1000 ounces averdepois of rain water at ordinary temperature; the other a foot of 304.79 mm. (foot of the Exchequer), which corresponds to a cube of slightly less than 1000 ounces of river water. The second value determined the official definition of the English and Russian foot promulgated in the eighteenth and nineteenth centuries. By ukaz of Czar Peter the Great the value of the Russian foot must be that adopted in England. The law of 1918 that made the French metric system compulsory in the Soviet Union calculated Russian units of volume and weight by an English foot of 304.80 mm, which is the value adopted in the United States.

11. Medieval metrics are considered a glaring chaos in which nothing is just or fit. But by considering the ancient values of the several types of oucne I have established they will appear rather simple. This is particularly true in the case of English weights which, instead of being a hopelessly complicated matter, are a structure of elementary claierty and delightful harmony.

It was the general practice in Europe to use one type of ounce for gold and silver and another type for the other commodities that are sold by weight (the avers de pois). In Italian texts a distinction is made betwee a peso d’oro or peso d’argento and peso da mercatantia. Similarly in ancient Athens next to the units I have described and which were used for coined and uncoined silver, there were units called emporika “merchants.’”

In England commodities were weighed by the pound of 16 ounces averdepois or 453,60 grams. A firkin contains 62,5 pounds averdepois of water (25/24 * 60 = 62.5). The pound averdepois was computed as 7000 grains of which 5000 make a Roman official libra of 324 grams.

For gold and silver there was used the ounce called Troy after the city of Troyes where there were held the foires de Champagne, the great international market of the Middle Ages. The ounce Troy of England is equal to 480 grains or 31,040 grams; it is exactly the ancient wheat ounce.

There was also a unit called ounce Tower, after the Tower of London, where there was one of the important mints of the realm. The ounce Tower is 450 grains or 29.160 grams; it is exactly the ancient artabic ounce.

The reason for the use of the ounce Tower is that in England gold and silver coins related to bullion as 15:16. I have pointed out that the same rate applied to silver in Greece. Whether the same rate applied to gold in Greece I cannot state at this moment, because, when years ago, I was laboriously piecing together the evidence provided by financial inscriptions of Greek temples, it was objected to me by numismatists and epigraphists that the rate 15:16 I had ascertained for silver was preposterously too high. As a result I hesitated in pursuing the evidence concerning gold, for which a lower rate may be reasonably expected; but now on the strength of the light provided by the English data, I shall proceed to a new survey of the Greek texts.

The matter of English standards has been obscured by numismatists who usually insist that both in ancient Greece and in England the charge for the cost of minting or seignorage was about 1%. This figure not only is economically unrealistic, but not supported by a single text. Numismatists like to introduce this imaginary rate in order to deny the existence of the mentioned adjustment 80:81 in the weights. According to them the weights were not fixed with mathematical precision. Many numismatists have argued that the cost of minting was taken care of not by a formal relation, but by a little cheating here and there at the mint. Gunnar Mickwitz has tried to prove this by an elaborate statistical analysis of the coinage of the late Roman Empire; even though this sutdy has been widely acclained, his interpretation of the data may be frankly called burlesque by the usual standards of statistical science.

The matter of English monetary standards is clearly explained in the famous work of Gerard Malines Consuetudo vel Lex Mercatoria, or, The Antient Law-Merchant (first edition 1622): “There has been used from the beginning (in the Mint) both Troy and Tower weight, each containing twelve ounces in pound weight; by which Troy weigh the Merchants bought their gold and silver abroad, and by the same did deliver to the Kings Mint, receiving counterpeaze but Tower weight for Troy, which was the Prince Prerogative, gayning thereby full three quarters of an ounce in the Exchange of each pound wieght converted into monies...”

12. The resistance of numismatists to the acceptance of the existence of rates such as 15:16 or 20:21 between coined and uncoined metal, reflect an old common superstition that there is something unjust in establishing a rate of this sort. As a result of such feelings King Henry VIII in 1527 abolished the rate 15:16 between gold coins and uncoined gold and by the same bill abolished for ever the units Tower, prescribing that only units Troy be used for gold and silver. But the study by Sir John Craig of the records of the London mint, proves that in the period before 1666 the mint still charged 15 pence to the pound (1/16) for coining gold and charged even more, 20 pence (1/12) for coining silver. In 1662 there had been introduced a mechanical press; it was thought that, since thereby the minting costs were reduced, the unpopular differential between coined and uncoined metal could be abolished. It was decided that henceforth the cost of mining had to be paid out of a custom duty on imported liquor. Later it was found necessary to add the produce of a new tax on the windows of buildings. The abolition of seignorage had disastrous economic results for England. Its aftermath were commercial crises, riots, and a halt of cash payments by the Bank of England. According to Gresham’s Law, if bullion and coin circulates at the same rate, coins will be drawn out of circulation. The matter was complicated by another prejudice, namely that there is something immoral or dishonest in clipping coins.. The amount of clipping in coins is determined by the reality of economic relations; in some situations the clipping of coins may be in the general interest, in that it prevents the flight abroad of coins. This is indicated also by the German phrase kippen und wippen, which means to clip coints, but also to weigh them accurately. The amount of clipping of coins of a given type in a given period is a fundamental datum that numismatists could provide to economic history, but it has been ignored, because quite understandably numismatists have a professional bias in favor of well struck and well preserved coins. The evil consequences of the abolition of the signorage in England were compounded by the circumstance that the new mechanical striking of coins produced coins that were better finished, more uniform, and with a clear design and rim that defeated the clipper. As a result the new coins tended to be saved, exported, or melted as soon as they were put in circulation.

In the hope that he could put an end to the monetary crisis, Newton was made Warden of the Mint, located at the Tower of London, in 1696. Three years later he was promoted to Master of the Mint. As Sir John Craig observes in his essay Newton at the Mint, the great scientist introduced a statistical way of argument in monetary questions. Newton noted that his opponents did not use figures and followed “a general way of arguing without coming to a reckoning.” The same objection could be raised today to most of what is written in the history of monetary standards. Newton understood that whether the money was of paper or of gold and silver was not the important point; what was important was the quantity in circulation: there should be enough money in circulation to provide “market money and workmen’s wages.” An increase in the amount favored full employment, but could lead to an expansion in unnecessary imports and a flight of the money abroad. A high rate of interest would indicate that there was a shortage in the amount of money in circulation, whereas a low rate of interset would indicate an excess. In this field the insight of Newton was far ahead of his times.

13. There was an area in which Newton’s bias as a scientist worked against sound monetary policy. He insisted on the eact weighing of the coins. The custom was to allow a certain tolerance in the weighing, called the remedy; the practice had been established to count almost all the permitted remedy in favor of those who brought bullion for minting. Newton insisted on exact weighing, but there was a specific reason for the earlier procedure.

The monetary reforms introduced by Pepin the Short and his son Charlemagne, are considered an obscure subject in spite of the great number of investigations devoted to them. This has occurred because investigators have neglected the most fundamental text: a capitular of Pepin the Short issued in 754/755, prescribing that the minter return 21 solidi in coins for each 22 solidi of bullion. This relation determined the system of weights of the Carolingian Empire and later of the Kingdom of France.

The monetary ounce introduced by the Carolingian reforms was the artabic ounce, the same as the ounce Tower of England. On the Continent it was called Cologne ounce, because the city of Cologne was the seat of one of the most important mints of the Carolingian Empire. To this monetary ounce there corresponded an ounce Troy for uncoined gold and silver, but whereas the English ounce Troy was 16/15 of the ounce Tower, the ounce Troy of Paris was 22/21 of the Cologne ounce. Malines in the treatise mentioned earlier recommends that the English units Troy be reduced to the level of the units Troy of the Continent, because the higher weight for the English units Troy had as a result that merchants did not import bullion into England. It appears, therefore, that the practice of the London mint to which Newton objected aimed at obtaining indirectly what Malines recommended.

14. The relation of value between coined and uncoined metal is the most important factor in the determination of monetary weights in the ancient and in the medieval world, but it has been totally ignored. I know only one exception: the numismatist J. G. Milne has recognized that in ancient Athens there was a relation 20:21 between coined and uncoined silver. As an example of the difficulties that have followed I may mention the disputes over the meaning of a passage of the treatise Economics, traditionally ascribed to Aristotle: the Athenian tyrant Hippias (end of sixth century B.C.) is charged with the following abuse: “He declared the standard of Athenian coins no longer acceptable. Having set a rate, he ordered that the coins be deposited with him. When the Athenians complied, expecting to have silver struck according to a new type, he issued the same money.” Numismatists and historians of Greece have developed all sorts of theories about modifications of the Athenian standard in order to explain this passage. But the numismatist C. T. Seltman observes that all these “elaborate theories” have to be rejected because the study of Athenian coins does not reveal any shift in the standard at this time. But Seltman in turn develops a theory that is not only elaborate, but totally gratuitious: Hippias, because of “a strong personal grudge against the noble families” wanted to withdraw all the coins issued by them; on Hippias’ proposal “the idea of an actual debasement was mooted at a meeting, presumably of financial advisers, held to consider the issue of a new denomination; but sound finance won the day and Hippias re-issued the same silver on the same standard.” In the Aristotelian text, which is our only source of information, there is nothing to justify the erection of this castle of cards by Seltman. What Hippias did or was charged with having done, is made clear by a similar event in English history. King Edward I accused the Jews of having clipped the coins of the realm as a substitute for usury. A series of dramatic trials and executions was staged, which culminated with the total expulsion of the Jews from England in 1290. The campaign against the Jews justified the proclamation on Easter 1279 (Easter has been the traditional date for the launching of persecutions against the Jews, up to the Russian pogroms at the beginning of this century) of an order that all clipped coins be brought to the mint to be exchanged, charging a fee of 14 to 14½ pence to the pound (slightly less than 1/16).

I suspect that probably the coins of England were extensively clipped in the time of King Edward I; but this had nothing to do with Jewish greed. The reforms of Pepin the Short had an immediate echo in England, where before 800 A.D. the English mints began to issue silver pennies of 22½ grains, calculated as 20 to a shilling or ounce Tower. The Carolingian system computed coins by artabic ounce. But on August 15, 1266 King Louis IX of France introduced a new silver coin imitating the special coinage issued since the age of Pepin by the city of Tours. I have ascertained that the Tours ounce is an ounce averdepois. The general adoption by France of coins based on the ounce averdepois was imitated throughout Europe by the issuance of pieces called Turnosen in Germany and Tornesi in Italy: England reduced the weight of the penny of 22½ grains only in the reign of Edward III (1327-1343). Hence it is most likely that in the age of King Edward I there was an epidemic of coin clipping in England. A study of the records of the trials of the Jews would enlighten us on this point.

15. Since I have clarified the structure of the English system of weights and the fundamental relation of the continental systems, it is possible to ascertain the level of precision achieved in the definition of standards. The treatise of Malines lists the divisions of the Troy pound, the only unit for preciourse metals after Henry VIII: the extreme minuteness of the subdivisions of the Troy grain are striking: “and every graine twenty mites, every mite twenty and foure droicts, every droict twentie periods, every period twentie and foure blanks, although superfluous (but in the divisionof the subtile assay).” These subdivisions are so extremely small that one must wonder how this could be performed in practice. This is partly explained by Malines’ report on the divisions of the grain on the European continent: “and then the graines subdivided of paper, making 1/8, 1/18 and 1/32 part of a graine which concurreth the most with our Assay weight.”

I understand that 1/32 of a grain would be considered today beyond the ability of the usual weight-master in the gold and silver trade. But the concern with fine subdivisions is indicated by the circumstance that in Greece there were in circulation coins as small as 1/48 of Athenian drachma, that is, 0,09 grams (less than 1½ English grain). Sir Flinders Petrie found that there are issues of Arab silver half dinar of the eighth century A.D. in which the specimens differ from each other by 0.004 grams.

There is an indication that already in the ancient Orient the corn husk was employed as a weight smaller than the grain. In Greece the amount added as correction for coins below par was called kollybos. In modern Greece and in Romania coliva is a paste of wheat grains covered with sugar; in ancient Greece kollybos is also a sort of cake. In Semitic languages the same root (Hebrew khalap) indicates a type of vegetable covering that may grow and come off, so that it may have originally meant “husk.” In Akkadian alappanu means “husked wheat.” Since the same root also means “exact” in Semitic languages, there is an indication of what was considered the standard of exactness.

16. The livre of Paris of 16 ounces was considered a unit Troy. The ounce of Paris should have been 22/21 of artabic ounce, but its weight was adjusted by a very minor fraction, so that the livre of 9612 grains can be calculated by a grain such that 6100 make exactly a Roman official libra of 324 grams. The Paris grain is 50/61 of English grain. Originally the grain used in France was calculated as 1/6000 of the reduced Roman libra of 320 grains. Since the relation between the two librae is 80:81, the official libra should have contained 6075 grains of 0.05333 grams. This grain was used to calculate the pound Troy of the Low Countries, so that when a sample of two pounds was sent from the Brussels mint to Paris for verification in 1529 it was found in excess by 48 grains to the pound; in 1732 an entire set was sent from Burssels to Paris for verification and found to be on the average in excess by 42 grains to the pound. In my opinion the correct weight of the Low Countries pound Troy is 491.520 grams, so that the excess should have been 38 grains. The pound Troy of the Low Countries was 6/5 of the Russian funt, which according to my reckoning should be 409.60 grams, being 96 x 96 times a tenth of Byzantine solidus of 4.444 grams (1/72 of reduced libra of 320 grams). The law of 1918 fixed the Russian funt as 409.5124 grams, following a recalculation performed by Dmitri Mendeleev, as director of the Russian Bureau of Standards, counting from the cube of the English foot of 304.80 mm.

The standard of Paris was embodied in a set of weights kept at the Hôtel des Monnayes (now preserved at the Conservatoire des Arts et Métiers) known as pile de Charlemagne, even though it may have been constructed at the earliest at the beginning of the thirteenth century. This set of weights was employed to compute the Paris kilogram. at that time the Paris livre was defined as 489.50584 grams, with a grain of 0.0531148 grams. By these figures 6100 grains are 324.00028 grams. This indicates how precise was the computation of the Roman libra and of the Paris grain.

In order to make the grain of the Paris livre equal to 1/6100 of official Roman libra, the weight of the Cologne ounce was increased from 450 English grains to 451. William Clarke, writing in 1767, reports that the Cologne ounce, which should be 450 English grains as the ounce Tower, was 451 grains and a document of of 1238 A.D. lists a Strasbourg ounce of 451 grains. A test conducted in 1829 computed the mark of Cologne as 233.8123 grams, whereas 8 exact artabic ounces would be 233.280 grams. Slightly different values were used by other German mints; for instance, the lowest value was that f 233.280 grams. Slightly different values were used by other German mints; for instance, the owest value was that of 233.612 grams at Bonn. In 1838 the German states agreed to strike coins by a unified mark of Cologne that was computed in French metric grams as 233.855. But the pound of Aragon composed of 12 ounces equal to Cologne ounces, was estimated in the last century as being exactly 350 grams, which correspond exactly to the ancient artabic ounce of 29.160 grams (450 English grains).

I may take the occaion to mention an ordonnance of King Alphonsus XI of Castille, issued following a meeting of the Cortes in 1348, which illustrates what were the typical practices of medieval Europe: oro e plata e todo byllon que se pesen por el marco de Colonna ... et cobre et fierro et estanco e plomo et azoque e miel e gera e azeyte e lana e los otros averes se venden a peso que se pesen por el marco de Tria. In this case the units of Cologne are adopted for uncoined gold and silver, whereas for the averes a peso the units of Troy are adopted.

17. English measures are based on a grain such that 5000 make an official Roman libra and 7000 make a pound averdepois. The only deviation from this computation occurs in metrological writers of the seventeenth century who compute the pound averdepois at 7008 grains, but their texts indicate that they arrived at this figure in order to substitute the round figure of 438 grains for the correct one of 437.5 for the ounce. This result must have been obtained by reducing the grain since the Paris livre is computed in these texts as 7560 English grains of 15 8/9 ounces Troy, that is, 63/64 of the English pound Troy of 7680 grains, whereas calculating exactly the Paris livre should have been 7554.1 English grains. As a result in some texts of the eighteenth century the value of the pound averdepois is computed by the intermediary figure of 7004 grains. These computations had an effect on the Imperial Standard Troy Pound (12 ounces or 5760 grains) made official by act of Parliament in 1824. This standard was based on a sample constructed for the Parliamentary commission that investigated weights in 1752 and indicates a pound averdepois of 7000 grains equal to 453.59243 grams. But the more correct value of 453,60 grams divided into 7000 grains of 0.06480 grams has been generally adopted both in Great Britain and in the United States.

In 1825 it was found that the Troy pound of the mint constructed in 1707, when Newton was in charge, was 1½ grains lighter than the Imperial Standard Troy Pound. Coins continued to be struck by the lighter standard until Gladstone in 1842 caused the mint standards to be made uniform with the Imperial Standard. As a result the weight of the Sovereign (123.274 grains) was augmented by I/30 of grain.

18. Because the units of England were so closely interconnected and were linked also with the units of the Continent, the exact preservation of standards was achieved by rather crude means. In England there is no evidence of the Byzantine practice, which was fully developed by the Arabs, of constructing reference weights of glass. It has been found that small Arab sample weights for the testing of coins, made in 780 A.D., differ from each other by 0.003 grams. At the London mint a set of Troy weights was constructed in 1707, replacing a set constructed in 1588. Newton found by comparing the new set with a set of the Exchquer, that the total weight of the mint set was 12 grains heavier on one pan of the scales and 6 grains lighter on the other pan. A new comparison of the two sets was performed in 1818, and it was found that the total set of the mint was exceeded by 30 grains, even though the pound pieces were identical in the two sets.

The Royal Society caused a standard pound to be constructed in 1742 and then had it compared with the sets kept at the Exchequer, which are still in existence, and with the sets kept at the Tower, at Guildhall, and by several corporations of London. I may quopte some of the figures of the comparisons performed by the Exchequer, because they are representative of the results obtained in general for all weights:
 14 pound piece made in 1582, pound of 6998.35 grains 7 pound piece made in 1588, pound of 7000.7   grains 1 pound piece made in 1588, pound of 7002.0   grains

These three pieces were part of a single set. A pound piece of another set was 6995.5 grains. This last piece confirms my contention that at times the grain was reduced in order to make the English Troy pound of 16 ounces equal to 64/63 of the Paris livre.