Canonical grain weights as a key to ancient systems of weights and measures by Jon Bosak earlier work will be referred to in several places fur- Last revised 19 April 2014 ther on.) Equivalences like this are not uncommon in the traditional systems that survived into the 19th century. And in some cases, in particular, the ancient 1 Introduction Egyptian and Sumerian systems, we know the rela- tionship between measures of capacity and cubic vol- Historians have puzzled over the principles underly- umes (thus implying a connection with lengths) from ing ancient systems of measurement for more than a surviving texts. Therefore, the notion of an ancient century. Some have proposed various theories to ex- system that was, like our metric system, defined con- plain what we know based on conjectured relation- ceptually as an interrelationship between volume, ca- ships between cubic volumes of specific substances pacity, and weight3 based on a standard substance (usually water or grain) and specific units of the na- (water in the case of the metric system) does not in tive weight system. In other words, they posit a con- itself seem strange or unlikely. nection between standards of length and standards of The case for believing that such systems must have weight based on the density or weight per volume of existed becomes compelling when we think through a particular substance. what an ancient trader in grain would have needed to These interpretations have been out of favor for calculate in order to stay in business. Considerations many years, not, I think, because such definitional re- such a person would have needed to take into account lationships are unlikely but because, for several rea- include: sons I will review here, the proposed theories have been next to impossible to test with any rigor. Volume of grain in a cylindrical or rectangu- In this paper I introduce the concept of a canon- lar granary of a certain size ical weight of grain as a tool with which to resolve The same volume of grain expressed in ca- this problem, and then I draw on recent discoveries to pacity units4 propose theories for the ancient Egyptian, Sumerian, That same volume (or number of capacity Indus, and Chinese systems of weights and measures measures) of grain expressed in units of that are both historically motivated and unprecedent- weight at the origin edly precise. Weight that an animal can carry I also use this paper as the occasion to briefly dis- Percentage of the carrying capacity of a car- cuss subtractive weight sets and their possible role in avan that must be allocated to water and shaping some ancient weight standards.2 that portion of the grain that will be used as food for the animals Some considerations a priori Displacement of water by a sailing vessel, in cubic units, per load of grain, in capacity Theoretical weight/volume explanations for ancient units systems of measurement make for an intriguing ar- Capacity in units of sale at the delivery loca- chaeological “cold case” because the reasons for sus- tion corresponding to a weight of grain, pecting such systems to have existed are so com- or sacks or barrels of grain, or the capac- pelling. ity of a hold We start with the explicit relationship in many tra- Size of a granary in the receiving location ditional systems of measurement between a certain necessary to contain a shipment of grain, standard measure of capacity and a specified weight as calculated from dimensions expressed of water, wine, or grain. For example, in the ancient in length units current in that location system that the British gave up almost 200 years ago but that continues to be used in the United States, the Commercial transport and trade has always been a gallon is defined by the old English statute 12 Henry complicated business, and only the smart traders sur- 7 c. 5 (1496) as a measure holding 8 troy pounds of vive. It is a virtual certainty that the more success- wheat. (I have written a book on this subject and ful traders mastered some system for calculating an- related matters under the title The Old Measure; the swers to the questions above, and it is difficult to see 1 how they could have done this without some system any system of relationships must have been based of measurement that offered easy, standard (that is, on the simplest of ratios that workably repre- generally agreed upon) conversions between units of sented the average densities of actual trade goods length, weight, and capacity based on some assumed while at the same time providing the simplest standard density of a particular trade good. We do possible factors for conversion to the standards this now based on the density of water and refer the and systems used by major trading partners. density of everything else to that via the concept of specific gravity. I think that the ancients based their – We have a good understanding of the relationships systems to some extent on the density of water, too, among units of a certain kind of measure in ev- but (at least in the beginning) only so that weights of ery major ancient system. We are not in any real liquids and the volume displaced in shipping a certain doubt about how many digits were in an Egyp- weight by water could easily be calculated. tian cubit or how many shekels were in a Sume- The fundamental basis of any ancient system would rian mina; scholarship over several centuries has have been an assumed standard density for the ma- settled these details for most major ancient civi- jor trade good, grain. Based purely on considerations lizations. a priori, therefore, we have very good reason to be looking for the foundation of these ancient systems in – There is a fair archaeological consensus regarding the average bulk weight of the predominant grain or the most important ancient standards of length, in- grains. The question is whether we can use this ap- cluding the Sumerian cubit at 49.5 cm (19.488 proach to identify and explain the mathematical rela- inches), the Egyptian cubit at 20.62–20.63 inches tionships upon which each ancient system was based. (52.4 cm), the Roman foot at 11.65 inches (29.59 To be clear: the point of the inquiry is to understand cm), the Northern foot at 13.2 inches (33.53 the fundamentals upon which the ancients thought cm), and the common Greek foot at 12.44–12.45 their systems were founded; we are looking to recon- inches (31.6 cm). All of these ancient standards struct their notional framework, because that is the of length are well attested, and the last two were system. Their varying ability to actually implement still going strong in national systems of weights their own definitions (as discussed in more detail be- and measures as late as the 19th and 20th cen- low) is only of interest here as something to be over- turies.6 come in trying to answer the main question. This is what we have to work with: – Because they were usually made of pottery, intact ancient capacity measures are quite rare, leaving – We have the compelling reasons just enumerated for us with relatively little physical data about capac- believing that the kind of system we are looking ity standards. Fortunately, however, we have in for had to exist in order for successful large-scale the case of the ancient Egyptians and Sumerians competitive ancient trade to have taken place at all. textual attestation of relationships between stan- – Considering especially the calculations that go into dards of capacity and cubic volumes that allow figuring the sizes and displacement of sailing ves- the capacity standards to be calculated from the sels, we can see a strong likelihood that such sys- measures of length. tems would have been based on the density of the predominant trade grain and the density of water – Because they were generally made of metal or stone, (or wine, which is not the same but can often be we have a fair amount of physical data regard- considered so). ing ancient commercial weights, i.e., the weights used to measure trade goods—several thousand – We can be certain that the numerical relationships examples in all, including a few very elaborately in such a system were very simple. We must always wrought weights with inscriptions (sometimes in be careful to distinguish the abilities of the most multiple languages) stating that they are official expert users of the system—including the priests standards. There are many more examples if we or other officials who maintained the standards— include coins as indicators of weight standards, and the abilities of the everyday users of the sys- and some researchers7 have succeeded in deduc- tem. I believe that the greatest experts in any ing commercial weight standards from the evi- era were capable of precision that archaeologists dence provided by coins, but I believe that vary- have tended to underestimate (beginning with ing monetary policies, inflation, and seignorage the assumption that an ancient expert could not make coins too problematic a basis for this work. have been more capable than the archaeologist), Even leaving coins out of the picture, however, but this does not mean that mathematical abili- there are a reasonably large number of exam- ties at the other end of the spectrum went much ples of ancient commercial weights—vastly more above counting. As I have noted elsewhere,5 the of them than intact examples of ancient capacity ancients were notoriously bad with fractions, so measures.