Greek Numerical System and Mathematics

[Ruthe]

MPC, aren't those symbols in the 'Numbers' link above?

 

Ooops! Thanks GO (and Ruthe). This is great. I wonder where these fractional notations were found.

MPC

 

Sorry for not providing a reply earlier to your last comment, but try this for size. Once you are there, you can navigate to the next page to find the rest of the fraction symbols and if you wish can then peruse the whole book, if you can read German that is. For those that cannot, the phrase 'auf dem chernen abacus' in item 13 which refers to section 46 and likewise items 14 and 15 which refer to sections 48 and 63 respectively, means 'on the brass abacus'.

 

I would point out in item 14, that the three symbols that appear against the three slots in column 1 of the abacus are said by Friedlein to denote the values of 1/24, 1/48 and most importantly 1/72 of unity and since 1/12 of unity is the uncia, these symbols are 1/2, 1/4 and 1/6 of an uncia ( i.e. 1/2 x 1/12 = 1/24, 1/4 x 1/12 = 1/48 and 1/6 x 1/12 = 1/72) of a unit value, whether of an 'As' for weight or money, a 'Pes' or Roman foot and any other duodecimally divided Roman measure. Now since the lower slot of column 1 has a value of 1/72 (if Friedlein is to be believed), then it means that each of the two counters in that slot could have had only two possible values, either each was worth 1/144 so two counters = 2/144 or 1/72, or each counter had a value of 1/72 for a total of 2 x 1/72 = 1/36. This latter case leaves only the possibility this slot could denote a value of 1/72 if one counter is used or a value of 1/36 if both are counted. Thus this slot could only represent 1/6 x 1/12 (= 1/72) or 2/6 or (1/3) x 1/12 (= 1/36). It could not have been used to count 1/3 and 2/3 of an uncia (1/12).

 

The text in brackets below is now out of date as the site has been repaired. Nevertheless I have left the alternatives in place to allow a quick view of the symbols without the need to view the full online version of Friedlein's book.

 

[since I added this reply, I have found that the website I included no longer works when one clicks an image to see it in full size], so I have now added the following 2 links for two pages from Friedleins book, Pages 1-3 and Pages 4- 6 of the Tables at the end of the volume. Thus the symbols , and represented 1/2, 1/4 and 1/6 (not 1/3) of an uncia or 1/2 x 1/12 = 1/24, 1/4 x 1/12 = 1/48 and 1/12 or 2/12 x 1/12 = 1/144 and 2/144 or 1/72 of a unit value.

 

If you search the web for a Roman abacus you will find the majority of references state (without any supporting evidence) that this bottom slot in column 1 of the Roman bronze abacus was used to count 1/3 and 2/3 of an uncia(1/12). Since none of these give any evidence for their statement, can these authorities be reliable. Only Friedlein provides any such evidence and he also indicates his own sources to support his interpretations. There are even some references that make an argument that since the symbol for the top slot on some versions of the abacus looks similar to a version of the Hindu-Arabic digit 3 with a straight line at the top and the whole thing is rotated 180 degrees (see the second example in item 14 of Friedlein's table) that perhaps this is meant to be 1/3. Of course, what that individual fails to appreciate is that the Romans were very unlikely to have ever encountered these symbols, and that they were only introduced to Europe from some time in the 10th century in Spain and even later to the rest of Europe.

 

I have also noted elsewhere that if the counters in the bottom slot of column 1 had a value of 1/3 each, that in conjunction with the slots for 1/2 and 1/4, there would be no way to represent 1/12, 2/12 or 5/12 of an uncia and there would be the abilty to represent the redundant values of 13/12, 14/12 and 17/12. Only if each bead has a value of 1/144 (i.e. 1/12 x 1/12) is it possible to represent all duodecimal subdivisions of 1/12 (i.e. 1/12, 2/12, 3/12, 4/12, 5/12, 6/12, 7/12, 8/12, 9/12, 10/12 and 11/12) x an uncia. This is not surprising as anybody with knowledge of mathematics and even IT professionals who would be familiar with binary and possibly trinary number bases would recognize that with two slots with one counter each and one with two counters, you can represent 2 x 2 x 3 arrangements in a logical sequence from 0 to 11. After all, the factors of 12 are 2, 2 and 3.

 

To be truly complete, I should note that some versions of the Roman hand abacus had a single slot in the first column, but still with the three separate symbols and that it is probable there would have been the same total of 4 counters in that column. Some versions of the abacus also had the three slots with the same symbols in column 2 and the unciae were in column one. This cannot be thought to be surprising, after all there was bound to be some different versions of the abacus by different makers. After all, how many versions of electronic calculators are there today? Are they all the same arrangement of keys and how many even have the same number of keys? Knowing the Roman penchant (practicality?) for doing things the same way such as the layout of their towns, it would be more surprising to find a larger number of variations of the abacus.

Edited March 11, 2009 by Ruthe

[M. Porcius Cato]

Simply stunning. Thanks for this explanation Ruthe! I'm sure I'll have more questions later, but right now I'm just awestruck.