Like in the lands of Sumer and ancient Egypt, accounting and writing evolved hand-in-hand and thus numeric characters were intergral part of ancient writing systems. Similar in concept to the Egyptian numbers, but not in shape, the Minoan-Mycenaean system was also decimal. The sign of '1' was a simple vertical stroke. Higher numbers consisted of multiple strokes: two for '2', three for '3', etc. In case of discretely countable entities (e.g. people or animals), these signs denote plain numbers, while in the case of goods like grain, figs or oil, the situation is less trivial (more about this later). Different magnitudes have different characters. The sign for '10' is a simple horizontal stroke. Typtically, this precedes the lesser digits, as Minoan and Mycenaean scribes also made some limited use of positional notation. Number '100' is written as an empty circle. A 'radiant circle' denotes '1000'. There is even a sign for '10,000', a radiant circle filled with a horizontal stroke inside. There are too few examples in Linear A to ascertain the latter values, but since '1' and '10' look exactly the same as in Linear B, it is rather trivial to identify the rest of the integer signs as well. Unlike the rather complex measurement systems of ancient Mesopotamia, Minoans did not have separate counting systems for different goods. The same integer numbers could be applied to items measured by wet volume (e.g. wine, olive-oil), dry volume (grain, figs) or weight (metals). I spent quite some time in the past months, eyeing the Cretan tablets eagerly for any small difference between the ways separate types of goods were catalogued. Yet I failed to find any.

To understand how Cretans were able to use essentially the same measure system for fluids and solids we should examine how later Hellenes did define their measurement units. It seems that everything revolved around a base unit, the quantity of water a single amphore could hold. The volume of an "average" Greek amphore differed from place-to-place, but was roughly about 19 to 36 litres. This is suspiciously close to the Minoan volume unit restored based on a vessel from Zakros (ZA Zb 3). An inscription mentioning 32 units runs below its rim; the actual volume of the pithos is slightly above 1000 litres. If it was meant to be full, the Minoan volume unit is around 32 litres; if not, it is somewhat smaller (29-31 L). Equally, this integer unit could also denote a weight of 29-32 kilograms, since the density of water (or wine) is 1.0 g/cm

^{3}. Ancient Greeks also defined their mass units based on this hypothetical standard amphore: the weight of its water content equalled one talent. The talent was further sub-divided into 60 minas, following the ancient Mesopotamian tradition of 60-based systems. For a Minoan scribe, it must have been trivial to switch between specimen numbers, volumes and weights using this simple and easy-to-learn system of counting by amphores.

While the system of integers is simple, nice and succint, the fractional units are unexpectedly complex. Both the earlier Minoans and later Mycenaeans used a high number of fractional signs to express quantities less than their (rather large) base units. Yet here the two systems are rather divergent: Mycenaean fractional units are usually specialized to measure different goods by either volume or weight, while this is not the case for Minoan fractions. But the two systems are not as radically different as one might think - we shall see this soon enough. Since ancient peoples had no knowledge of irrational numbers, all non-integer quantities typically denote simple, rational fractions (e.g. 1/2, 3/4, 2/5, etc.). Any quantity that was smaller than the unit, was to be approximated by a sum of these fractions. Egyptians preferred the use of fractions of the form

*1/n*(

*n*= any natural number). One does not need to be a mathematician to prove that if we defined

*1/n*-type signs for all

*n*<

*N*, using a sufficiently high

*N*, all quantities can be approximated arbitrarily well with this system. So we only need a finite number of fractional signs, if it is our objective to measure with a fixed error margin.

The most common fraction seen in Linear A is named

**'J'**, graphically looking like a "lesser than" sign. This is - obviously - the one for 1/2. There are plenty of tablets with totals (e.g. HT9, HT104, etc.) to demonstrate that. The second most common fraction is labelled

**'E'**, it looks like an almost exact mirror of 'J'. This has (you could have guessed it) the value 1/4. The next symbol in this series is a bit more tricky. Sign

**'F'**looks like 'E' with a horizontal line crossing it in the middle, thus symbolically cutting it in half. To show that this one is 1/8, we also have direct proof: on tablet HT8, there are two fraction 'F's following two well-established names (QA-*310-I and PA-JA-RE) thus they need to add up to 'E' in order to make that calculation work.

With a little luck, we can even identify the sign

**'H'**as 1/6. Tablet HT123+124 features *122 and *308 quantities for each name, being nicely proportional (3:1) to each other for practically all lines. You can see this on my reconstruction above. Based upon the considerably high quantities, *122 (= RA3, "LAI") is probably OLIV (olives), and not CROC (saffron), the only other Linear B logogram similar to it; but there is no hint on the identity of *308. Due to the fragmentary nature of this tablet, it is not the easiest thing to reconstruct the exact numbers that once stood in each entry; yet the end totals give enough constraints (in the form of an equation system) to fix most illegible fields. To make this work, one has to propose, however, that the scribe sometimes disregarded the positional notations when making calculations (there is ample evidence that the same happened on side B of the very same document). If we also assume that the scribe made no errors upon totalling, there is only one correct solution, that you can see on my facsimile. Interestingly enough, the tablet - if read properly - yields evidence on the existence of a sign with 1/12 value. As the sign 'X' (Godart & Olivier) on the end of the second line is more plausibly read 1+'A', this strongly suggests that sign 'A' had the value 1/12.

Finding out the exact value of other fractions is more tricky. For example, there are the 'B' and 'D' series of fractions - both reasonably common. Obviously, one would expect fractions with a higher value (lower denominator) to be more prominetly featured than quantities below 1/10. This would mean that the fractions 1/3 and 1/5 be the most common. As there are not many ancient numeral systems putting a stress on 1/7 and 1/9 fractions, these ones were probably rare in Mycenaean and Minoan contexts - if existent at all. This leaves us on a quest to find out how to match the 'D' and 'B' series to derivatives of 1/3 and 1/5.

We have very little information on the true value of the

**'D'**sign (looks like the Latin letter 'S') and the related doubling

**'DD'**. John Younger tried to pursue the idea that these are exclusively dry weight measures - based on their close similarity to the Mycenaean 'M' sign used for metals or wool; but none of the tablets support Younger's conclusions. 'D' and 'DD' are seen with every imaginable type of goods, even with olive-oil. This indicates that it was a generic fraction, not exclusive to - say - cereals. Others (Dieter Rumple) suggested the value 1/5 - again, without any convincing proof. That idea solely relies on a single tablet (HT115), where a quadrupling of the base fraction unit (DDDD) can be read. As for me, after a thorough research, I came to the conclusion that the value of 'D' is probably 1/3. 'DD' would therefore be 2/3. As for 'DDDD' (4/3), one must see that similar overshoots are rare, but definitely do exist on several tablets (JJ=2/2 [PH9, PH22], EE=2/4 [PH12, PH13]). Now, if 'D' is indeed simply 1/3, it can also nicely explain the graphical image of fraction 'H' = 1/6. Numeral 'H' looks as if it were carrying the upper half of 'D', divided by a line below. Similar to the way 'F' = 1/8 was built, its graphical image actually shows (1/3) / 2 = 1/6 ! Linear B documents have a similar sign to represent a (wet volume) unit ('S'), that might indicate a quantity 1/6 instead of the previously proposed 1/3 (by Chadwick). It probably evolved from a mirrored version of Linear A fraction 'H'; similarly to the way the Mycenaean fraction 'V' seems to correspond to the Minoan 'L' family of fractions.

The

**'B'**-series of fractional numerals is definitely more complicated than the 'D'-series. 'B' looks like a plain cross.

**'BB'**is a simple doubling of 'B', but there are also a couple of signs ('K', 'A', 'X', 'W') that resemble the shape of 'B'. How many of these are related to the base sign 'B' is still an open question. Fortunately, this time we do have some solid evidence on our hands for the value of 'B', albeit meager in quantity. Tablet KH7 appear to contain food rations shared among a fixed number of people. The rather strict proportinality straightforwardly implies that 'B' is in fact 1/5. While different tablets seem to employ slightly different food shares per person, we can probably accept that on the same table, rations are more likely to be fixed. Interestingly enough, two closely related Haghia Triada tablets (HT16 and HT20) - both mentioning a list of animal products - appear to list goods proportionally to one another (WA:*188+KU = 1:2), if but only if 'B' = 0.19 ~ 0.2 = 1/5! 'B' definitely follows both 'J' [KH5, KH6, KH17] and 'E' [KH9] on the tablets, thus (from the positional notation) suggesting that it is indeed < 1/4.

Sign

**'K'**is one probably related to 'B'. Its appearance is indistinguishable from the Linear B fraction 'T', of a value 1/10. Graphically, it is just the lower half of sign 'B' (1/5). Earlier, this sign was assumed to have a value as small as 1/16, based on a flawed piece of 'evidence': the graffito on the wall of a house at the Haghia Triada site (HT Zd 155). It was interpreted by some authors (Pope, Olivier, Stieglitz) as a geometric series. I examined that piece eagerly, but much to my dismay, it is just another piece of 'wishful thinking'. The "inscription" is nothing but a maze of purely vertical and horizontal lines. There is probably no fraction 'K' there at all, and the two last signs (that were even transliterated by J. Younger as TA-JA) are nothing but speculations (note that these signs consist exclusively of vertical and horizontal lines). So I must reject any identification of fraction 'K' as the half of sign 'F', and rather insist on inferring its value from the Mycenaean counterpart (very likely 1/10).

Quantities

**'A'**,

**'X'**, and

**'W'**are pretty mysterious. All these appear to be rather small. (Note that the reading 'ABB' on KH 86 is apperently faulty, the entire number is likely just a single 'X'.) Although I previously pursued the idea that sign 'A' might be related to the 'B' series (1/5 based), now I uncovered pieces of evidence (see the restoration of tablet HT123-124) that it could plausibly be just 1/12. While fraction 'A' has no Mycenaean counterpart, fraction 'X' corresponds to a Linear B weight unit 'N', used for metals or wool. Since it would not make much sense to propose sign 'X' to be another 1/6 quantity, this leaves it unidentified for now. No matter how well this would fit the proportion M:N = (2/3) / (1/6) = 4, as in Linear B. Fraction 'W' is only seen on a small number of Khania tablets (KH12, KH60, KH61, KH77), offering no clue for its value.

The smallest units ever used on Linear A are - no doubt - the so-called

**'L'**-series of signs. This is the only instance where signs are supplemented with counting-strokes on their right side in Linear A (this was customary in Linear B for all fractions, though). The very base unit (that has no counting stroke) is called 'L', but it was seldom employed. It is much more commonly used with 2 strokes

**(L2**), 4 strokes (

**L4**) or even 6 strokes (

**L6**). L5 is never found, and L3 is rare as well (the only unambiguous example stems from a single Middle Minoan Knossos tablet). Again, we have no direct proof for their values. This opens up the place for wild speculations. Formally, the fact that L3 and L5 are disfavoured, would suggest that the denominator of L-fractions (if common) was divisible by both 3 and 5. The smallest such number is 15. The problem here is that L2 very commonly follows fraction 'K' on the Khania tablets, that would make little sense in terms of magnitudes (as 2/15 > 1/10, obviously). On the other hand, the L-series of fractions are apparently related to the Linear B quantity 'V', sometimes termed 'choinix', based on the similarity of 'Z' and 'V' fractions to the classical Greek

*κοτύλη and χοῖνιξ volume units. The problem there is that their proportion to the base unit ('1') is not fixed, but different for fluids and solids, giving no candidate number for the denominator of Linear A fraction 'L'. (You can see a sketch of the Mycenaean systems above.) John Younger attempted to use Khania tablet KH7 (that you could also see a few paragraphs before), to fix the value of L2 at 3/20. But this complex value would make little sense. Also, the numbers in the penultimate entry reconstructed by Godart are probably slightly faulty (it likely mentions 37 people, not 38). Yet the next line may feature the text KI-RO (badly damaged). If so, then L2 and L6 must ad up to 1/10. If they have the same base denominator (that is probable), then L = 1/80. This would also offer an alternative explanation why examples with an odd number of strokes were rare: because it was used primarily to express 1/40 units (L2 = 1/40, L4 = 2/40, L6 = 3/40, according to this theory). Note that L2 is reasonably common with grain portions (especially at Khania). In Linear B, the daily meal of an average worker was three 'kotylai' (Z3), that might be corresponding to exactly unit 'L' in the Minoan system.*

Other fractions still exist in Linear A contexts: they are still unresolved, and will probably remain so. Such ones are fraction

**'Y'**(seen only at Phaistos [PH26]) and fraction**'Ω'**(seen as a vessel-qualifier at Malia [MA10]). There might have been graphically distinct fractional signs for all numbers of a form*1/n*for sufficiently high integers - say*n*< 20, but numbers that were rarely used (as they were of a prime*n*or they did not match the procession of any quantity) were unlikely to appear in regular accounting. While fraction Y might have been something like 1/7, with the lack of evidence, it will essentially remain undeciphered in the end.
Thus we have seen that - while there are still many unsolved problems in ancient Aegean metrology, some questions may be answered with pure logic and deduction. This also applies to the Mycenaean units. Encouraged by the Minoan discoveries, I made slight changes to the metric procession of Linear B dry and wet units. You can see this on the table above. With the different densities of water and (ground) grain taken into consideration, this system now perfectly meets all expectations of precise metrics. The only thing that needed changing was unit 'S' (now proposed to be 1/6 instead of 1/3). While Kim Raymoure's Linear B database offered no example of S3, S4 or S5, this only means that 'S' ≤ 1/3. And this is what we have to contend for the time being. Unless we are willing to bravely attempt to fill in the gaps in our current knowledge.

Glad to see you're back. There's an absence of creative discussion online on the subject of the Minoans and that's a shame.

ReplyDeleteI have only an idle comment about the fractional series, noticing that it roughly follows the Fibonacci sequence (FS) consisting of 1, 1, 2, 3, 5, 8, 13, 21, etc. (nb. each number in this series is the sum of the two previous numbers of the sequence).

Of course, since I've been reading a lot about this mathematics lately on the side, perhaps it's just an accidental coincidence. However FS shows up curiously throughout nature too and it wouldn't be so odd if an ancient mathematical system unconsciously reflected that.