P A R T  V

APPENDIX

# Basic Structure

1. The metric system is based on a foot of 300 mm. which is called Egyptian foot by metrologists; I call it natural basic foot, but I shall refer to it as Egyptian foot for the sake of simplicity of expression. The cube of this foot constitutes the basic talent. The cube was filled with rain water at ordinary temperature; this filling proves to have the same density as distilled water at the temperature of 4° Celsius, because the impurities in the rain water compensate for the higher temperature. The calculation of metric units by rain water at ordinary temperature was the usual practice up until the eighteenth century A.D., when the calculation by distilled water began to be employed. The reason for adopting distilled water at maximum density in establishing the French metric system was in part that this water does not have (within the limits of the precision achieved up to that time) a density different from that of the medium traditionally employed.

Units computed by the specific gravity of water were usually called wine units, since they were normally used to carry wine. But even though the texts speak of calculation pondo vini, they specify that the exact value is obtained by using rain water; it is also specified that the rain water must be collected after it has rained for some time, obviously so that excessive dust be not included. Ordinary wine was lighter than water, whereas syrupy wines had a higher density. The usual high-quality wines had about the same density as water.

The cubic foot filled with water is a unit of 27,000 grams which is considered a talent. The talent is the weight that a man can ordinarily carry at each end of a carrying yoke. If a man, instead of carrying two talents balanced on a yoke, carries a sack, it is assumed that the ordinary sack weighs two talents.

The basic talent of 27,000 grams is divided into 1000 ounces of 27 grams. The ounce of 27 grams is the Roman uncia (1/12 of libra). Every talent is divided into 1000 parts called ounces. The cube of 27,000 c.c. was equal to 100 basic cups of 270 c.c. or to 50 basic pints of 540 c.c.

The reason for computing the units in terms of the cargo carried on a yoke was not only practical, but also ideological since the balance derived from the carrying yoke. The balance could not have been invented unless the concept of equilibrium had been already discovered. The concept of equilibrium was suggested by the circumstance that when two loads are carried on a yoke, it is expedient to have them equivalent to each other.

The akkadian term for talent is beltu from the verb abulu, "to carry a yoke." the Greek term talanton refers to swaying motion; from this motion there is derived that of "weight" and of "balance." The mythological figures of atlas, Tantalos, and talamon indicate the asociation of the term talent with the carryng of burdens. The Hebrew term is kikkar from the root KKR meaning "to oscillate, to move in a circle." The Hebrew kokkar as well as the Greek talanton, refers also to the pans of a balance. The Hebrew and the Greek term are used to describe a ritualistic object consisting of a small disk representing a scale pan; I have determined that coins developed from this object. The meaning of the Akkadian gaggaru is related to this last meaning of the Hebrew kikkar.

Since the yoke was an important element of the system of measures, its length was adopted as a standard unit of length. The length of the horn of a yoke is related to the human arm from the fist to the elbow, because the yoke must be such that a man can easily grasp one of the ropes descending from the extremities in order to steady the cargo. The length of each horn of the yoke is the cubit or ell. A yoke was equal to 2 cubits and to 3 feet, so that the cubit is equal to 1½ feet. The cubit is divided into 24 fingers, so that the foot is equal to 16 fingers. Four fingers constitute a hand; there are 4 hands to a foot and 6 to a cubit. It would seem that it was only in Rome that the hand came to be divided into 3 unciae or inches (interpreted as width of the thumb) instead of 4 digiti or fingers, making the foot divided into 12 inches. The division of the foot into 12 inches results from the division of the cubit into 16 inches. The division by 16 is the result of a series of bisections, division by 2, 4, 8, 16. In Mesopotamia there occurs a division of the cubit into 32 fingers and this division is most common in Arab metrics. But the typical method of subdivision of lineal units is that of a foot of 16 fingers.

The existence of a cubit of 32 fingers is related with the occurrence of cubits of 2 feet; cubits of 2 feet occur in Mesopotamia and are particularly common in Arab metrics. At times the cubit is divided into 2 halves; the half-cubit is the Hebrew zereth and is a most common unit of medieval metrics in which it is called palmus. Medieval learned texts distinguish the palmus major of 8 inches (half cubit) from the palmus minor, which is the hand of 3 inches.

The Greek term pechys refers to the forearm, the cubit as a measure, the horn of a yoke and the beam of a balance. A reason for selecting the forearm or cubit as the basic unit of length is also that this is the length of each loop when a rope is coiled around the palm and the elbow.

2. The cubit of 450 mm., corresponding to the foot of 300 mm., was the common cubit of Pharaonic Egypt (natural basic cubit in my terminology). The cube of this cubit is a unit of 91,125 c.c. or grams which was considered the standard load of an ass or the load that a man can carry for a short distance. The ass load of 91,125 grams was divided into 10,000 units which I call basic sheqels. The sheqel was the unit by which there were measured precious metals used as means of exchange. The basic sheqel was 9.1125 grams, but it was also calculated as 9 grams, that is, as 1/ 3000 of the basic talent of 27,000 grams (1/3 of an ounce). There is a discrepancy 80:81 between the two forms of basic sheqel, with the result that often ancient units of weight and of volume vary as 80:81. For instance, the typical Roman libra is 324 grams (36 basic sheqels or 12 Roman ounces), but there was also a libra of 328.050 grams which I call geometric libra, equal to 36 basic sheqels of 9.1125 grams. I call the discrepancy 80:81 by the name of discrepancy komma.

The cube of the cubit is the standard ass load. In Akkadian emëru means both the ass and its load; the original meaning of the term is " heap." The uassoretic text of the Old Testament distín guishes the term háinór, "ass," from the unit of measure hómer, but I shall show that originally the two terms must have been confused with each other in Hebrew usage. St. Epiphanios describes the hömer as the volume of barley that "a young man can lift and place on an ass.'° In many languages the same term applies to the ass and to its load. In Greek papyri the ass is called gomarion from gomos "load;" in modern Greek it is called gomari. In Italian it is called somaro (English sumpter), from the Greek sagma, "pack saddle." In German it is called Saumtier, a term corresponding to the Italian bestia da soma, "pack animal." Saum means "burden" and also refers to a large unit of volume. In Italian salma is both a large unit of weight or volume and a corpse carried on a stretcher. In Greek kanthélion is the "pack saddle," whereas kanthélios is the "pack ass." Possibly this term is related to the Aramaic root KNS, "to gather, to heap." But the terms seems to occur also in Fenno-Ugrian languages; Finnish kanta" to carry, to bear, " Ostyak kantìm, 'ito carry on one's back, " Cheremissían kande and Mordvinian kando- "to carry, to bring." In Greek kanthé ion means ''rafter" and seems to refer rather to the wooden frame across the baek of an ass to which the panniers are attached than to the panniers themselves. For this reason consideration should be given to the following Fenno-Ugrian terms: Ostyak kant, '°horizontal beam resting on two posts, " Mordvinian kando, "wind-fallen tree, " Finnish kanta, `Jstump," Livonian kand, `'tree trunk." In Latin canterius means "ass, mule," but it means also `9spar, crosspiece for supporting a vine." In Low Latin there occurs a term cantarium referring to a unit of volume (Arabic gintár) which may be a result of a confusion óf canterius with centenarium, a unit of 100 librae. The term be comes kynterkyn and kilderkin ín English. In Turkish kantar means " balance;" from this there is derived the Aumanian cantari, "to weigh."

The terms for load are transferable from one pack animal to another. A papyrus speaks of kanthelia kamelika, “camel panniers.” When I speak of load I refer to the typical load, the ass load, cube of the cubit. When the use of the barrel spread through the Roman Empire from Celtic territory, the standard barrel was made equal to the ass load and such it has remained ever since.

The load is called h-r, "sack ," in Egyptian; bastagma, ballantion, lemma in Greek; follis in Latin. The ass load is called gur in Sumerian. The Akkadian term karu is probably derived from the Sumerian. The Hebrew term is kor, randered into Greek as koros. Septuagint translates homer with kor, when he states that the kor "gets its name from the fundamental idea of a heap." In Hebrew kar means " camel saddle," and in Arabic kur means "saddle " and kara, " to carry on one's back."

The camel load, considered equal to two ass loads was also called by equivalents of the Greek koros. Acording to St. Epiphanios the Hebrew kor refers to a camel load. The "camel saddle " is kor in Hebrew and karrar in Arabic.There is also a unit double the camel load which is considered the cart load: it is called carrus or carrum in Latin.The name carrus for the two-wheeled transport wagon was believed by the Romans to be of Celtic origin. But them term voccurs in Greek papyri of Egypt, and an inscription of the caravan city of Palmyra, in the Syrian desert, speaks of gomos karrikos, "cart load ." These terms may have been derived from a Semitic language, rather than from the Latin. The Celtic terms may have a Semitic origin. Septuagin and the Vulgate interpret the term kirkaroth of Isaiah 66:20 as "cart " whereas modern scholars prefer to interpret it as "dromedary."

The Celtic term could have a Semitic origin, being finally derived from the Sumerian gur. The Sumerian term seem to refer to any measure of grain from the cubit foot to the cubic cobit range. There is also a homophone gur refering to a very large quantity making a granaryful. Sumerian homophones are words that are the same, or about the same, in sound, but are represented by different caneiform symbols.

In Diocletian’s “Edict on Prices” the list of charges for transportation (XVII, 3-5) indicates that cart load is 1,200 librae, a camel load 600 librae, and an ass load 300 librae or 97,200 grams. The figures given for the price of wood (XIV, 9-11) seem to be in contradiction with those just mentioned:

 cart of wood fully loaded 1200 librae 150 denarii camel load of wood 400 50 mule load of wood 300 30 ass load of wood 200 ....

The explanation of the difference is that the specific gravity of wood was considered equal to that of barley, that is, 0.66. In the case of the camel and of the ass, in spite of the lower specific gravity of wood in relation to water, the cargo was not increased over the usual volume, whereas the cart is fully loaded (gegomôménê). The reason for not loading the camel and the ass to the full is that a greater volume, even without greater weight, is more difficult for an animal to carry. The Mishnah (B.M. 6,5) prescribes:

If a man hired an ass to carry wheat and he used to carry (a like weight of) barley, (if the ass was injured) he is liable; if (he hired it) to carry grain and he used to carry (a like weight of) chopped straw, he is liable since the greater bulk is more difficult to carry. What increase in weight renders him liable? Symmachos says in the name of Rabbi Meir: one se’ah for a camel and three qabh for an ass.
The se’ah or saton is equal to 6 qabh.

3. The basic talent is equal to 50 basic pints and to 100 basic cups. This division is convenient in reckoning, but is most impractical if the cubic foot is to be divided physically into fractions. The most normal division of the cubic foot is into 8 parts, which is obtained by dividing each side of the cube in half. The English cubic foot is divided into 8 gallons. The unit corresponding to the gallon is the chous in Greek and congius in Latin. If the resulting 8 small cubes are divided again by the same process into 8 smaller cubes there result 64 little, cubes. Each of these cubes is about 1½ cups: by introducing a discrepancy 24:25, which, call discrepancy leimma, the cubic foot becomes equal to 96 cups. But calculating exactly from the cup of 270 c.c, 96 cups are 25,920 c.c. This talent is 24:25 of the talent of 27,000 c.c. and is the Roman quadrantal or amphora and the Athenian monetary talent called Euboic talent by ancient authors. Oxé has introduced the name of unit brutto for the talent of 27,000 grams or c.c. and the name of unit netto for the talent of 25,920 grams or cc. Lehman-Haupt discovered that most units of volume and weight exist in a variety brutto and a variety netto, related as 25:24. Multiples of basic units could be arranged as

 12.5 25 50 62.5 75 l00 125 150 or as 12 24 48 60 72 96 120 144

and the two series were considered practically equivalent. In my terminology the difference between the units brutto and the units netto is the discrepancy leimma.

The talent of 27,000 c.c. or grams is the typical talent of Pharoanic Egypt, whereas the talent of 25,920 c.c. or grams is the typical talent of Greece and Rome, being equal to 80 librae. I call the former by the name of basic talent brutto and the latter by that of basic talent netto.

4. The basic talent was computed as a cube filled with water, but the original and most important function of the units was to measure wheat and barley. Hence, there were two other talents that correspond to the volume of wheat or barley weighing a basic talent. It was established that a barley talent should have a volume of 1½ basic talents: there was a barley talent brutto of 40,500 c.c. equal to 1½ basic talents brutto and a barley talent netto of 38,880 c.c. equal to 1½ basic talents netto. This implies a conventional specific gravity of 0.66 for barley.

The specific gravity of wheat was assumed to be intermediary between that of water and of barley. It was assumed that the wheat unit may be either 11/5 or 1¼ of the water unit. Beginning with the basic talent netto of 25,920 c.c. (80 librae) there is obtained a wheat talent netto of 31,104 c.c. or 96 librae (which I call talent Troy, since it is exactly equal to 1000 English ounces Troy) and a wheat talent brutto of 32,400 c.c., which was called centenarium since it is equal to 100 librae. The centenarium is also 11/5 of the basic talent brutto of 27,000 c.c. There was also a wheat talent of 33,750 c.c. equal to 1¼ basic talent brutto: but this unit, that I shall call kantar, is less important than the others.

Each of wheat and barley units could be filled with water and become a unit of weight.

The units for wheat may have a volume that is either 6/5 or 5/4 that of the corresponding water units, thanks to the discrepancy leimma, since 6/5:5/4 = 24:25. Since the barley unit 1½ times the water unit, the wheat unit could be ¼ more than the water unit and unit and 1/5 less than the barley unit, or conversely 1/5 more than the water unit and ¼ less than the barley unit. If the counting starts with 2 water units netto and ends with a barley unit brutto, there are two increases of ¼. If there are two increases of 1/5, the counting begins with a water unit brutto and ends with a barley unit netto. We have 1¼ + (¼ . 1¼) = 1½ + 1/24 and 11/5 + (1/5 . 11/5) = 1½ – 1/25

The existence of conventional specific gravities is related to the fact that scales were seldom used, since they were cumbersome, too costly, and too complicated for the ordinary illiterate man. Scales were essential only for the exchange of precious metals and here too coined money was invented in order to dispense with weighing, as Aristotle points out. Everything was sold by weight, but on the basis of a measurement by volume converted into weight on the basis of specific gravity. Segrè notices with surprise that even meat was measured by volume in Hellenistic Egypt. Decourdemanche sums up the matter as follows:

Everything was sold by weight. Now, barley, oil, honey, wheat, and flour were without any doubt the object of important trade. In order to avoid the trouble of measuring these wares, the ancients had the idea of testing their weight. For this reason they attributed to barley, honey, wheat, flour and oil a conventional density. This was fixed for barley and honey at 2/3 of the weight of water, for wheat at 80%, for oil at 90%, for wheat flour at 25/27, and for barley flour at ¾.

These last figures are based on the statements of Arab metrologists and are substantially correct for classical antiquity.

Still today the United States government for customs purposes assumes that a bushel in volume is the equivalent to so many pounds in weight according to a table set for the several kinds of grains. Similar rules have been adopted by some states of the Union with values that do not always agree with the Federal ones. The American standard bushel contains 77.6274 pounds averdepois of water and is assumed to contain 47–50 pounds of barley and 60 pounds of wheat. The English bushel contains 80 pounds of water and is assumed to contain 60 pounds of English wheat, between 60 and 63 pounds of foreign wheat, and 50 pounds of barley, English or French. Adjustments for the discrepancy between the assumed specific gravity and the actual one can be made by heaping. In 1854 the House of Commons published a Return from each County of England and Wales of the Different Measures, and their Capacities in Gallons, and the Different Weights and their Contents in Pounds, under which Wheat, Barley, Oats, and Flour are Sold. The Clerk of the Peace of Bedford County reported:

Wheat is usually sold by the load, of five bushels imperial measure, each bushel containing eight gallons; the average weight per bushel is 62 lbs. avoirdupois; sometimes the seller guarantees a given weight per bushel according to his own estimate, and makes good the deficit if any, on the aggregate quantities sold.

Similar practices were followed in the ancient world in that heaping could be used to compensate for a specific gravity inferior to the norm.

The general practice of Europe up to the end of the last century has been to sell grains by volume, taking the specific gravity into account. As in ancient times it was assumed that high specific gravity is an indication of good quality. A French encyclopedia states that good wheat poured with precaution and gently should have a density of 0.795; when the wheat is dropped and pressed the density should be 0.84. It also states that grains of desirable quality should not have a density below the following figures:

 Winter wheat 7.5 to 8.2 Spring wheat 7.2 to 7.7 Winter barley 6.8 to 7.2 Spring barley 6.8 to 7.4

5. The crowning of the metric system was achieved by the discovery that if starting from the basic foot of 16 fingers, there are calculated feet of 15, 17 and 18 basic fingers, these feet (which are in turn each divided into 16 fingers) constitute the edges of cubes which are each 1/5 more than the lesser one:

Given cubes that are

 153= 3375 163 = 4096 173 = 4913 183 = 5832

each is almost 6/5 of the lower one, since

 6/5 3375 = 4050 6/5 4096 = 4915 6/5 4913 = 5896

As a result there were in the metric system four basic types of foot related as 15:16:17:18. Since there are two basic feet, the Roman foot of 295.945 mm. and the Egyptian foot of 300 mm., related to each other as cube root of 24 to cube root of 25, there were 4 couples of feet. I call the two terms of each couple trimmed foot and natural foot.

 trimmed natural Foot of 15 basic fingers 277.4489 mm. 281.250 mm. Foot of 16 basic fingers 295.9454 300.00 Foot of 17 basic fingers 314.4419 318.750 Foot of 18 basic fingers 332.9384 337.500

The trimmed feet may be computed as 74:75 of the natural feet, obtaining the following rounded values:

 Foot of 15 basic fingers 277.5 mm. Foot of 16 basic fingers 296.0 Foot of 17 basic fingers 314.5 Foot of 18 basic fingers 333.0

The exact relation cube root of 24 to cube root of 25 is 73.9864:75.

The foot of 15 fingers is called Italic or Oscan by metrologists; I shall call it lesser foot. It forms the edge of a talent that I call little talent. The little talent is the weight of a basic talent filled with wheat. The feet of 16 basic fingers, either trimmed or natural, are the Roman and the Egyptian foot. I shall call the foot of 17 basic fingers by the name of wheat foot, and the foot of 18 fingers by the name of barley foot.

There are as many talents as there are types of foot. This is the volume of the talents computed in cubic fingers of the trimmed basic foot (Roman cubic fingers ):

 netto brutto Cube of 15 basic talents little talent 3375 3515.625 Cube of 16 basic fingers basic talent 4096 4266.650 Cube of 17 basic fingers wheat talen 4913 5117.708 Cube of 18 basic fingers barley talent 5832 6075.000

The Roman cubic finger is 6.3281255 c.c.

The calculation of the talents by cubes of feet related as 15:16:17:18, does not fit with absolute perfection with the intended proportions among the talents, but the differences are minimal. The following table presents the correct volume of the talents in Roman cubic fingers, followed by their difference from the cubes of the feet:

 little talent neto 3276 -99 basic talent netto 4096 — wheat talent netto 4915.2 +2.2 wheat talent brutto 5120.2 +2.0 barley talent netto 6144 +69

Expressing the values in grams or c.c., this is the ordering of the most important talents, calculated according to their proper relations:

 little talent 21,600 4/5 basic talent netto 25,900 1 basic talent brutto 27,000 25/24 wheat talent netto 31,104 1 1/5 wheat talent brutto 32,400 1¼ barley talent netto 38,880 1 ½

The wheat talent, computed as the cube with an edge of 17 basic fingers, comes out practically perfect. The wheat talent netto (talent Troy) is 31,104 c.c., whereas the cube of the trimmed wheat foot of 314.4419 mm. is 31,090.05. The wheat talent brutto (centenarium) is 32,400 c.c., whereas the cube of the natural wheat foot is 32,385.5 c.c. The difference is so small that, at the present state of the empirical evidence, its influence can be noticed only in the study of Arab measures and possibly of English measures.

The little talent should be 21,600 c.c., but the cube of the trimmed lesser foot is 21,357.4 c.c. I will show that this difference possibly affects the calculation of monetary standards in Greece and in the Hellenistic world. The Athenian monetary standard is the Euboic mina of 432 grams, which is 1/60 of Euboic talent (basic talent netto); 50 such minai make a little talent of 21,600 grams.

There was also a lesser foot of 274.7314 mm. which I call doubly trimmed lesser foot, which is the edge of the cube of 48 Euboic minai, 4/5 of Euboic talent. The value of this foot is documented with certainty by the Sumerian bronze bar of Nippur.

The greatest irregularity in the system is caused by the fact that the cube of the natural barley foot corresponds to the barley talent netto and not to the barley talent brutto. The barley talent netto is 38,880 c.c., whereas the cube of the natural barley talent is 38.4434 c.c. This difference is taken into account by using the discrepancy komma or 80:81.

The barley talent is equal to 120 librae. The libra of the coinage of Constantine which became the Byzantine libra is not the usual libra of 324 grams, but a libra reduced of 1/81 to 320 grams: 120 such librae are 38,400 grams. But most usually the calculations of the barley units is made not from the foot, but from the cubit. The cubic barley cubit is computed as 31/3 cubic barley feet (whereas a cubic cubit should be 33/8 cubic feet), with a resulting increase of 1/80; the cube of the natural barley cubit becomes 405 librae instead of 400, or 400 geometric librae. In Mesopotamia where the barley cubit is the basic unit of length, weights are related to the geometric libra of 328.05 grams.

To each of the eight types of foot there corresponds a cubit so that there are the following cubits:

 trimmed natural lesser cubit 416.173 mm. 421.875 mm. basic cubit 443.918 450.000 wheat cubit 471.663 478.250 barley cubit 499.408 506.250

The cubit is the edge of the load. The basic load netto is 87,480 c.c. or grams and the basic load brutto is 919,125 c.c. or grams. The basic load netto is 162 basic pints of 540 c.c., but commonly it is calculated as 160 basic pints with a discrepancy komma. Similarly the basic load netto is often computed as 90,000 c.c. or 10,000 basic sheqels of 9 grams. The barley load brutto is usually computed as 240 basic pints instead of 243, that is, as 129,600 c.c. instead of 131,220 c.c. The cube of the natural barley cubit of 506.250 mm. is 129,746 c.c. This is the load of barley that weighs a basic load.

6. Since the cubit is equal to 12 feet, each load is equal to 338 talents, since (112) - 338. But often the relation 1:38 between talent and load is simplified into a relation 1:33, with a resulting discrepancy 80:81, since 338: 313 = 80:81. I have already mentioned the occurrence of this discrepancy komma. For instance, if a basic load netto of 87,480 c.c. is divided into 313 parts there results a quadrantal (basic talent netto) of geometric form of 26,244 c.c. (80 geometric librae of 328.05 c.c.). This unit occurs among other instances as the Russian chetverik, which at the moment of the adoption of the French metric system in 1918 A.D. was computed as 26,239 c.c.

The division of the load into 33 talents is particularly convenient when the talent is divided into 3 pecks or sata, making the load equal to 10 pecks. The Roman modius, for instance, is 3 of quadrantal and also 1lO of the corresponding load, if the quadrantal is of geometric form.

The relation between talent and load can be further simplified by dividing the load into 3 parts. The basic load netto divided into 3 parts constitutes the Persian artaba of 29,160 c.c. or grams. The Persian term artaba, meaning “great , could be used to refer to any unit of the dubic foot type, but it is used by metrologists to refer to the unit of 29,260 c.c. or grams which was the official unit of the Persian Empire. The artaba is equal to 98 basic talents netto. The artaba is 90 librae.

The edge of the artaba is a special foot that I call artabic foot. The artabic foot was 307.796 mm. with an artabic cubit of 461.695 mm., but it was customary to compute the artabic foot as of Roman foot or 308.276 mm. (cubit of 462.414 mm.).

7. The foot usually employed in Pharaonic Egypt is the natural basic foot of 300 mm., to which there corresponds a cubit of 450 mm. But usually Egyptian monuments and measuring rods of the Pharaonic period, indicate a cubit of 28 fingers-called Egyptian royal cubit by metrologists instead of 24; a cubit of 7 hands (instead of 6) has several geometric advantages that I shall describe, but a unit of 525 mm. is convenient also for other reasons.

The cube of 525 mm. is 144,703 c.c. which is almost exactly 5 artabai (5 . 29,160 c.c. = 145,800 c.c.). The Egyptian royal foot had 3 different values: it could be 525 mm. as I have indicated, or also be 524.1483 mm. or 526.564 mm. The reason for this is that the cube of the royal cubit was reckoned as 16,000 basic sheqels, called qedet in Egyptian. The qedet, which was the fundamental weight of Egypt, could be either 9 grams (13000 of the cube with an edge of 300 mm.) or 9.1125 grams (1O 1OOO of the cube with an edge of 450 mm.) As a result there were used in Egypt two qedet differing of a komma. the royal cubit of 524.1483 mm. (cubit of the Great Pyramid of Gizah) forms a cube of 16,000 qedet of 9 grams, whereas the royal cubit of 526.564 mm. forms a cube of 16,000 qedet of 9.1125 grams. There was in Egypt also a qedet of 9.0439 grams equal to 1 16,000 of the cube with an edge of 525 mm. The cube of qedet of 9.1125 grams is 145,800 c.c. or exactly 5 artabai.

The Egyptian royal cubit of 525 mm. could also be divided into 24 fingers, giving origin to a royal foot of 16 fingers or 350 mm. Counting from the royal cubit of 526.564 mm. the royal foot is 351.043 mm., which has a cube which is 112 artabai. A unit of 112 artabai is 43,740 c.c., corresponds to a cube with an edge of 352.068 mm. The royal foot could not be used for exact calculations of the artabic units, but it would be used in practice to obtain a unit of 112 artabai, that is, to obtain a cube containing barley weighing an artaba.

8. I have made occasional references to the occurrence of discrepancies. Discrepancies occur because of the conflict between the decimal divisions of a cube and the possible physical divisions of it. In dividing a cube physically the sides may be divided conveniently into 2 or 3 parts. If each side of a cube is divided into 2 parts, there results 8 smaller cubes; if it is divided into 3 parts there result 27 smaller cubes. As I have said the simplest physical division of a cube is obtained by dividing it into 8 small cubes, divided again in the same manner so as to obtain 64 smaller cubes. This is one of the important reasons for the use of sexagesimal units in the metric system: by using units adjusted according to the discrepancy 15:l6, which I call discrepancy diesis, a division by 64 may be considered equivalent to a division by 60. As I have said, if one takes 23 of the resulting 64 cubes, there results a division by 96, which is as close as one can get to a division by 100. For this reason there is the discrepancy leimma or 24:25, by which a division by 96 is considered equal to a division by 100. A similar result may be obtained by dividing a cube into 8 smaller cubes and then dividing each of these cubes into 3 parts; there result 24 cubes which may be considered equal to 215 of the original cube because of the discrepancy leimma.

By the trisection of all the sides a cube is divided into 27 cubes. If each resulting cube is divided into halves there result 54 units, which is the number of basic pints in an artaba. If each resulting cube is divided into thirds there result 81 units which can be considered equal to 180 of the original cube because of the discrepancy komma. The typical application of this division is that of basic talent netto into 80 librae; 81 librae are a basic talent netto of geometric form. A cube can also be divided into 9 parts, by trisecting two of the sides; by repeating the operation there results a division by 81.

A cube may also be divided into 6 parts, by bisecting a side and trisecting another. By combining a division by 6 with a division by 9 there results a division by 48 which is considered equal to a division by 50, because of the discrepancy leimma. Finally, a division by 8 combined with a division by 9, produces 72 cubes which are considered equal to cubes for the purpose of reckoning. There occurs here a discrepancy 35:36, which I call tetartemorion. If a unit of 72 parts is duplicated there results a unit of 144 parts which may be considered equal to 150 parts, because of the discrepancy leimma.

The discrepancies that commonly occur in the metric system are:

discrepancy diesis or 15:16
discrepancy leimma or 24:25
discrepancy tetartemorion or 35:36
discrepancy komma or 80:81
I have given to the discrepancies names derived from the accidentals of musical scales, because there is a close correlation between units of measures and ancient musical scales.This is made clear by Chinese musical treatises. Actually I have found that the reading of the Greek musical treatises or of the similar Chinese ones, which must have been derived from a common source, is the best preparation for the understanding of the arithmetic of ancient metrology.

The arithmetic of discrepancies is essential to the understanding of the development of problems in cuneiform mathematical texts.It is disputed among musicologists whether musical scales have a physiological or conventional origin. The evidence I have gathered indicates that musical scales were derived from the arrangement of the units of volume. The ancients used to arrange the units of public reference standards in a series, in ascending or descending order. The relation among the contents of the basic units of volume appears to have been adopted as determining the basic tetrachord. If the fundamental talents are arranged in a series, we have:

barley talent netto 72 pints
wheat talent brutto 60
artaba 54
basic talent brutto 50
basic talent netto 48

The series

barley talent netto
wheat talent brutto
artaba
basic talent netto

provides a chromatic tetrachord, whereas the series

barley talent netto
artaba
basic talent brutto
basic talent netto

provides an enharmonic tetrachord. Two tetrachords form an octave.

It would seem that the basic elements of musical scales spread from a single center together with the metric system.I shall discuss the question of musical scales in a special article, but since in this work I shall discuss at length the weight standard of the Heraion of Argos, which is at the basis or the units used in the Peloponnese, I may point out that this standard indicates a talent of 24,300 grams corresponding to 45 basic pints.

The following arrangement of units, wheat

talent brutto 60 pints
artaba 54
basic talent netto 48
talent of the Heraion 45

corresponds to the ordering of the notes according to the Dorian mode.

The interval between a barley unit and a water unit is a fifth; that between a barley unit and wheat unit or between a wheat unit and a water unit is either a major third or a minor third; that between the artaba and the basic talent netto is a tone.

The earliest Chinese musical treatise, probably written around the third century B.C. and ascribed to Kuan-tzu who lived in the seventh century, lists 5 pitch pipes arranged as following.

4/3 81 = 108
2/3 108 = 72
4/3 72 - 96
2/3 96 = 6

The relation could be interpreted as being modeled on the following units of volume:

112 artabai 108 minai of 405 c.c.(15 Roman ounces)
barley talent brutto 96
wheat talent brutto
(geometric form) 81
artaba 72
basic talent netto 64

The unit of 1½ artabai would correspond to the cube of the Egyptian royal foot.

The discrepancy 80:81 that occurs in relation to the wheat talent brutto or centenarium (100 Roman librae, either normal or of geometric form), is the main preoccupation of Chinese musicians, because to maintain the pure relationship between Yang and Yin, according to the Chinese concept of cosmic order it is necessary to keep the interval of the fifth correct and at the same time keep the octave correct. This can be achieved only by a deviation of a komma or 80:81. Various solutions were adopted in the course of history.

Chinese legal texts define the basic unit of volume as the capacity of a flute that gives a specific note.

For the present work I have not proceeded to a detailed investigation of Chinese measures, but a preliminary survey indicates that the fundamental talent is the picul, which is equal to the (Persian) artaba of 29,160 c.c. As in the Mediterranean area the artaba is usually divided into 60 reduced pints of 486 c.c., so in China the picul is divided into 120 catty of 283 grams or c.c. The picul is the monthly ration of rice, as in Egypt the artaba is the monthly ration of wheat. The lineal unit ts’un is 210 of artabic cubit. According to Kibi no Mabi who in the eighth century A.D. brought to Japan a Chinese treatise of music together with a set of copper pitch pipes, the fundamental pipe huang-chung (Yellow Bell) has a length of 9 ts’un and a diameter of 0.3 ts’un. Kibi no Mabi quotes Chen Hsuan: other Chinese authors mention the length of the huang-chung but not its diameter. The volume of the pipe appears to be 1360 of Alexandrine artaba or English firkin (cube of the English foot): the Alexandrine artaba is 3536 of Persian artaba, that is, 1000 ounces averdepois or 28,350 c.c. The volume of the pipe should be 78-750 c-c- or 3536 of 81 c-c- (15 of 405 c-c- or 13 of catty). Computing by the artabic cubit of 462.414 mm., the pipe is described as being 78.160 c.c. However, the English foot which, as the edge of a cube of 28,350 c.c., should be 304.919 mm. (standard of Guildhall ), in Mesopotamia and Egypt was often described as 3635 of Roman foot or 304.401 mm.; these two values were occasion for debate among English metrologists of the Renaissance period. If the shorter English foot was employed in China, the huang-chung pipe could have been 78.349 c.c.

9. In music the accidentals, which correspond to what I call discrepancies in the system of measures, serve to adjust the notes of a scale to the notes of a different scale and to approach an even-tempered scale when the chords are tuned merely by intervals of fifths or fourths. The pentatonic scale, which is diffused all over the world, begins with C, tunes up a fifth to a, a fourth down to D, a fifth up to A, a fourth down to E, adding a C an octave above. A scale can be obtained also merely by the use of fifths, as in the following scale:

 do re mi fa sol la si do 1 98 81/64 4/3 3/2 27/16 243/128 2

The fifth of 9/8 is 9/8 * 3/2= 27/16 or la
The fifth of la is 27/18 * 3/2 = 81/32 or mi
The fifth of mi is 81/32 * 3/2 = 243/128 or si

The diatonic scale is obtained by the following series of increases of frequency:

9/8 ; 10/9 ; 16/15 ; 9/8 ; 10/9 ; 8/9 ; 16/15

Comparing the vibrations per second in an even-tempered scale and in a diatonic scale, the following discrepancies appear.

 A B C# D E F# G# A Even-tempered 220 246.94 277.18 293.66 329.63 369.99 415.30 440 Diatonic 220 247.50 275 293.33 330 366.66 412.50 440 Discrepancy -0.56 +2.18 +0.33 -0.37 +3.33 +2.80

The diatonic scale corresponds to the following distribution of the units of volume.

 A basic talent netto 25,920 c.c. B artaba 29,160 C wheat talent brutto 32,400 D 80 Euboic mina 34,600 E barley talent netto 38,880 F 100 Euboic minai (2 little talents brutto) 43,200 G 100 ruduced pints 48,600 A 2 basic talents netto 51,840

This is the scale used by Ptolemy. Didymos had used a scale in which the intervals are transposed in the fifth and sixth position:

 9: l0: 16: 9: 10: 8: 16 8 9 15 8 9 9 15

In such a case F corresponds to 43,740 c.c., that is, half a basic load netto. The difference between the F of Ptolemy and that of Didymos is a komma.
In the system of measures the discrepancies have the purpose of reconciling the convenient physical divisions of a cube with decimal or sexagesimal reckoning. The discrepancy leimma allows one to divide by 48 (6 X 8) and 96 and to reckon by 50 and 100. The discrepancy diesis allows one to divide by 64 (8 X 8) and to reckon by 60. The discrepancy komma allows one to divide by 81 (9 X 9) and to reckon by 80. The discrepancy tetartemorion allows one to divide by 72 (8 X 9) and to reckon by 70.

10. The basic pint is 540 c.c. This unit is also used as a weight, called Babylonian silver mina by metrologists (20 Roman ounces). The reduced pint is 486 c.c. or 109 of it (18 Roman ounces).By filling the basic pint with wheat there is obtained a mina, Euboic mina, of 432 grams (45 of 540). The Euboic mina is obtained by dividing by 60 the basic talent netto (Euboic talent)- Dividing it by 64 there is obtained a mina of 405 grams (mina of the Heraion). This mina is 34 of the basic pint of 540 c.c. It is equal to 15 Roman ounces, whereas the Euboic mina of 432 grams is 16 ounces, so that the difference between the two is a diesis. The mina of 405 grams is 1 64 of Euboic talent, whereas the Euboic mina is 1 60. The mina of 405 grams is an important reference point in computation, because it can be calculated directly and with absolute exactness as a cube with an edge of 4 Roman fingers (a hand). The barley talent brutto (112 basic talent brutto) of 40,500 c.c. is 100 minai.

The artaba is equal to 72 minai. There is also a unit equal to 70 minai, that is, a tetartemorion less than the artaba: this unit is the English firkin (cube of the English foot) of 28, 350 c.c Whereas the artaba is divided into 1,000 artabic ounces of 29,120 grams (Cologne ounce in medieval Europe, ounce Tower in England), the firkin is divided into 1000 ounces averdepois of 28,350 grams (Alexandrine ounces in Hellenistic times) which are a tetartemorion less than an artabic ounce.

In Mesopotamia the usual cubit is the barley cubit, which is generally divided into 30 fingers in order to fit into sexagesimal reckoning. But there is also a cubit equal to 32 such sexagesimal fingers. Counting from the trimmed barley cubit of 499.408 mm., this cubit is 532.702 mm. It is called Babylonian-Egyptian great cubit by Boeckh. In the course of the first millennium B.C., probably after the Assyrian conquest of Egypt, the great cubit became the standard cubit of Egypt, replacing the royal cubit of 525 mm. Metrological texts of the Hellenistic period call it Ptolemaic or Philetairic cubit and describe it as 95 of Roman foot. In fact, 3/2 * 18/16 * 16/15 = 9/5.

One reason for having this great cubit is that the system of cubes determined by the feet of 15, 16, 17, and 18 basic fingers, fails to provide a barley talent equal to 65 of the wheat talent netto (talent Troy), a talent of 37,324.8 c.c. or 115.2 librae. The cube of the great cubit is 151,165 c.c., which divided by 4 makes a talent of 37,791 c.c. that is exactly 115.2 librae of geometric form (115.2 328.05 = 37,791.36).

In Egypt this cubit was divided into 28 fingers as the older royal cubit. The corresponding foot of 16 fingers is a foot of 304.401 mm., but the foot corresponding to the edge of a firkin of 28,350 c.c. is 304.919 mm. A cube with an edge of 304.401 mm. would be 28,205.8 c.c. This small difference among the possible values of the English foot and of the firkin, is reflected in wavering of the standards of medieval England.

11. I have pointed out that the fundamental units were so conceived so as to be each 14 more than the lesser one. This increase is connected with the process of heaping grain measures. The typical vessel is as high as it is wide, whether it is a cube or a cylinder. If the heap is as high as the vessel it increases the contents by l3: if it is half as high it increases it by 16. It was assumed that the typical heap is equal to 34 of the width of the vessel, so that the contents is increased of 14 by heaping.

The American struck bushel (Alexandrine medimnos) is 114 firkins (cubes of the English foot, Alexandrine artabai), and the American heaped bushel (Alexandrine talent) is 114 struck bushels.

When vessels are not as high as they are wide, they are most commonly twice as high as they are wide. Contemporary American and French regulations about the practical construction of measuring vessels, prescribe that they be as high as they are wide in some cases or twice as high in some other cases. When the vessels are twice as high, the heaping increases the contents by 18, which is the difference between the basic talent netto and the artaba.

The contents of grain measures was affected by shaking: modern experience indicated that the contents of a grain vessel may be increased about 10% by shaking. The ancient considered that the shaking of a vessel (mensura conferta, metron sesaleumenon) had the same effect as a moderate heaping. An artaba of 29,160 c.c. when shaken is equal to a centenarium of 32,400 c.c., since they relate as 9:10. For this reason both the artaba and the centenarium were used as the monthly ration of wheat. The centenarium is divided into 60 basic pints of 540 c.c., it was the monthly ration of wheat in Athens, where the centenarium is called metretes. In those areas that counted by artabai the monthly ration of wheat was an artaba divided into 60 reduced pints of 486 c.c.

The existence of units of volume reduced as 9:10, was explained by assuming that the cube that determines the units of weight is not filled with water or wine (calculation pondo vini), but with oil (calculation pondo olei,

The most important application of the calculation pondo vini is the mentioned reduced pint of 486 c.c. It was convenient to assume that oil has a specific gravity 0.9, but the edible oils used in the ancient world, such as olive oil, sesame oil, palm oil, have a specific gravity that is closer to 9.166 or 1l12 of the density of water. For this reason the reduced pint is often computed as 495 c.c. or 11 of 540 c.c.

The physician Galen explains that in Rome there was used an oil libra (libra olei) that consisted of a vessel marked with divisional lines corresponding to 12 unciae. The vessel filled up to 11 ounces is a wine measure: filled up to 12 ounces it is an oil measure. But the oil weight is obtained by counting up to the tenth line, in terms of wine filling. The libra olei or, must have been an Alexandrine litra of 354.375 c.c. (1212 ounces averdepois or 12 ounces plus a leimma), which is roughly 1211 of the Roman libra of 324 c.c. (1211 = 324 = 353.454 c.c.) The ounces appear to be 2524 of ounce averdepois of 29.4545 grams and are 1211 of Roman ounce of 27 grams. The counces are approximately artabic ounces increased of a komma (81 80 29.160 = 29.524 grams).

The calculation pondo olei not only determines the existence of units of volume and weight that are smaller, but also of units of volume that are larger. If a man can carry 2 jars of wine, he can carry oil vessels that are slightly larger. Since wine jars were not usually filled through the neck, the practice was usually to fill the jar through the neck when the contents was oil. At times the wine unit is increased of 9 for oil, but at other times it is increased of 111. The firkin is considered the oil unit corresponding to the wine amphora of 25,920 C.C.9 since 1112 28,350 c.c. = 25,985 c.c. At times in practice the oil unit is computed as 111O of the wine unit.

In the Alexandrine system of measures, which is the system of measures used in Egypt when the Babylonian-Egyptian great cubit became the standard unit of length, the basic unit is the firkin (cube of the English foot) called Alexandrine artaba. This artaba is divided into 80 litrai of 354.375 c.c. There is also a unit of 100 litrai which is the Alexandrine medimnos (equal to the American bushel) and is 114 Alexandrine artabai. In Athens the Panathenaic amphoras used for oil had the same value: they were intended to be the oil unit corresponding to the Attic wine amphoreus which had tne volume of a centenarium or Attic matretes: 1211 of 32,400 c.c. is 35,345 c.c. and hence almost 35,437.5 c.c.

The difference between wine and oil unit is connected with the shift between divisions by 3 and division by 313. The Alexandrine medimnos is divided into 3 modii cumulati, “heaped pecks,” of 24 reduced pints and into 313 modii xysti, “level pecks.”

The modius xystus is equal to 21.6 reduced pints or 910 of 24, but it is also calculated as 22 reduced pints or 910 of 24. The difference between modius comulatus and modius xystus is considered that amount of heaping that is equivalent to the effect of shaking, but it corresponds also to the difference between a wine measure and an oil measure. In Alexandrine measures the difference between modius xystus and modius cumulatus constitutes a separate unit called cumulus, “heap.’ The modius xystus is considered equal to 20 basic pints and the modius cumulatus to 22 basic pints; 3 modii xysti are 60 basic pints or a centenarium of 32,400 c.c., whereas 3 modii cumulati re 66 basic pints or 35,640 c.c., slightly more than Alexandrine medimnos of 35,429.4 c.c.

12. Next to the basic pint (2 basic cups) of 540 c.c. there is a reduced pint of 486 or 495 c.c. The difference between the two is related with the difference between oil units and water uits and between shaken and unshaken units. A reduced pint when shaken contains as much as a basic pint. A basic pint filled with oil weighs as much as a reduced pint filled with wine. A basic pint may be considered a heaped reduced pint.

The pint was called sextarius is Latin. The term appears to mean ;’one sixth,’ being 16 of congius: but this is a popular etymology, since the terms come from the Greek xestes. The Latin term for the cup unit is hemina (when it is not cotyla from the Greek kotyle), being derived from a Dorian Greek term that means ‘half unit.’’ The Greek term or is derived from the verb xeo, “to scrape down, as St. Epiphanios observes. The term must have originally referred to a pint that is level or shaven, that is xyston. It would seem, therefore, that the xestes or sextarius was originally the reduced pint. In Roman times the reduced pint is called Alexandrine sextarius, whereas the basic pint is called Italic sextarius.

The advantage of the reduced pint is that 180 of them make a basic load netto (180 486 c.c. = 87,480 c.c.). This calculation fits sexagesimal reckoning. a basic load netto is 360 cups. But the calculation is usually by pints, because the artaba is equal to 60 reduced pints.

Since the reduced pint fits particularly well into sexagesimal reckoning, it is the standard pint and the mina of Mesopotamia, called sila and qa in Sumerian and qu in Akkadian. Since Egyptians measure by artabai, the reduced pint is also the standard unit of volume of Egypt, callad henu (hin in Coptic and inion or hinion in Greek). In medieval Europe it became the livre of Paris, the unit that was used in computing the kilogram of the French metric system.

To reduced pint there corresponds a reduced basic squel of 8.10 grams. A reduced basic pint is divided into 66 reduced basic sheqels; 50 reduced basic sheqels make the mina of 405 grams.

Dividing by 60 a basic talent netto (Euboic talent) there results the mina, of 432 grams or Euboic mina. Dividing it by 64 there results the mina of 405 grams.

When the artaba is divided by 60 there results, as I have said, the reduced basic pint: when it is divided by 64 there results a pint of 455.625 c.c., which occurs as value of the Egyptian hin. This last unit is identified with the pound averdepois of 16 ounces averdepois or 453.60 grams. The pound averdepois is 75 of libra. The pound averdepois has the advantage of being almost exactly1200 of basic talent brutto, since 200 453.60 = 90,720 grams. The basic talent brutto may be computed either as 90,000 or as 91,125 grams. In Egypt the unit of.10,000 qedet may be 90,439 grams.

In the ancient world the most common multiple of ounce averdepois was the Alexandrine mina of 15 ounces or 425.250 grams. This unit happens to be rather close to the Euboic mina of 432 grams or 16 Roman ounces. In the classical period Athens issued coins by the Euboic mina, but in the second century B.C. shifted to the Alexandrine mina.

The Alexandrine mina of 425.250 grams fits both into the units of Pharaonic Egypt and into the units of later Egypt, since the royal foot of 350 mm. (3 of Egyptian royal cubit of 525 mm.) has a cube of 42,875 c.c. If the Egyptian royal cubit is computed as 524.148 mm., the cube of the royal foot is 426.667 c.c.

The Euboic mina of 432 grams is the weight of the basic pint filled with wheat (45). The mina of 405 grams may be considered the weight of the reduced pint filled with wheat (56). The Roman libra of 324 grams is the weight of the reduced pint filled with barley: it is the weight of the mina of 405 c.c. filled with wheat (45).

The following table indicates how the reduced pint and the mina of 405 grams or c.c. relate to the cubes of the units of length.

 Volume in c.c. Weight in grams Unit of Pint Mina length Unit of volume or weight in mm. 25.920 64 296.0 Cube of trimmed basic foot Basic talent netto 29.160 72 307.8 Cube of artabic foot Artaba 32.400 80 318.75 Cube of natural wheat foot Wheat talent brutto 36.450 90 333 Cube of trimmed barley foot 38.880 96 337.5 Cube of natural barley foot Barley talent netto 72.900 150 416.25 Cube of trimmed lesser cubit 87.480 180 444 Cube of natural lesser cubit Basic load netto-3 artabai 97.200 240 462 Cube of artabic cubit 3 wheat talents brutto 129.600 320 506.25 Cube of natural barley cubit Koros brutto 145.800 300 525 Cube of Egyptian royal cubit Great measure

The most significant approximation in this reckoning occurs in the case of the cube of the trimmed lesser cubit (exactly 417.1734 mm.) which is 72,082 c.c. But the great measure (cube of the Egyptian royal cubit) which is the double of this cube, may be computed both as 16,000 basic sheqels or qedet or 9 c.c., that is, as 144,000 c.c., and as 16,000 basic sheqels of 9.1125 C.C.9 that is, as 145,800.

13. Reducing the Euboic mina by ll0, a there is obtained a mina of 388.80 grams which is 1l00 of barley talent netto. This unit is the Rhodian mina, divided into 100 Rhodian drachmai which today are English apothecaries’ drams. At present the English apothecaries’ ounce is officially defined as 27.34375 grains, that is, as 1.77185 grams, which is exactly 1200 of Alexandrine litra. There are 16 drams in an ounce averdepois. The present English grain is 15000 of Roman libra or 0.06480 grams (17000 of pound averdepois). Up to the seventeenth century there was in England also a grain which was 23 of the present one and which was 1 10,000 of Attic drachma which was 1-100 of Euboic mina of 432 grams. In Arab metrics there are also two types of grain, a grain of 0.0648 grams and a grain of 0.0432 grams. When the French metric system was adopted by Egypt in 1924 A.D., the first type of grain was defined by the round figure of 0.0650 grams.

The Rhodian drachma came into existence because some Greek exporters of high quality wine, took the Euboic mina of 432 c.c. as an oil unit. The corresponding wine unit is 388.80 c.c. In the Athenian League all members were compelled to measure by Euboic minai, the standard of Athens; but the island of Chios, exporter of the most famous Greek wine, was granted the special privilege of reckoning by a mina of 388.80 c.c. or grams. After the Athenian defeat in the Peloponnesian War) the allied who revolted against Athens adopted the mina of Chios. This standard was adopted by all the exporters of high quality wine and after 400 B.C., known as Rhodian, it become the most widespread monetary standard after the Athenian. Among these exporters of high quality wine there were Kos and Knidos, the seats of the two most important Greek medical schools. For this reason, the Rhodian drachma became the English apothecaries’ dram.