Ruthe

Steve, I've just found your latest update and had a quick scan of your document of the ancient's uses of the abacus. It will take a little while for me to give it the attention it deserves, but it appears from my rapid scan that you have done a fantastic job in investigating the methods used by the ancients. I will get back to you when I have had fully digested the details. I must admit my long absence from mathematics will require a little remedial effort first. I also saw your previous post and wondered why I never saw it when you first posted it. You asked which source I was referring to in questioning the 'expert' view that the symbol beside the lower slot in the first column which is similar to our present digit "2" is believed to be 1/3 of 1/12. Friedlein1 states it to be the symbol for a 'sextula' or 1/72 of an 'As' = 1/6 x 1/12 As or 1/6 of an uncia when used on a pocket/hand abacus. He also gives some examples of the use of the Salamis Tablet type or table top version of an abacus with the three separate subdivisions of an uncia and in this case shows the bottom position to have the symbol like a truncated union operator which he again indicates in his table to be the symbol for sextula. What he does not explain is if the value of 1/72 refers to the value of each counter in that position, or the total value of the two counters. It is only if each counter has a value of 1/144 that a continuous range of values from 1/144 to 11/144 or 1/12 x 1/12 to 11/12 x 1/12 can be represented. Thus the two counters would together have a value of 2/144 or 1/72 of an As. The entire volume of Friedlein's book can be viewed online at the University of Strasbourg and the table containing the symbols I noted are on webpage 171 and 172. 1Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und R

I am sorry to say I don't have an answer to your question, directly. However, just consider Roman legions and their capabilities. Caesar had his legion build two bridges over the Rhine in 55 and 53 BC. The first of these was between 140 and 400 metres long, 7 to 9 metres wide and was completed in only 10 days! So, who designed these bridges. Who calculated the amount of materials needed, the amount of work required and therefore the number of men required for the task? Somebody in that legion must have had a very practical knowledge of mathematics and geometry at the very least. Next consider Roman forts, towns and camps while on marches. Without mathematics they would not have been able to do any of these things. So that implies somebody within every legion had mathematical ability. This means that somewhere, somebody was able to teach these practical skills. I am amenable to the idea that the Romans did little to add to theoretical mathematics, but they must have had a very well developed knowledge of practical applications of mathematics. After all, how would they work out who owed what, particularly taxes, from the millions of people in the entire Roman empire? Have a look at Caesar's Rhine Bridges and Roman Military Engineering. They couldn't have done these things without extensive knowledge of mathematics!

Not sure about the roads. WHat did the British chariots run on? I seem to recal, burt ahve no reference to support it, that may Roman roads were built on top of pre-existing British roads. Can anyone comment one way or another?

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.

Could you please elaborate on this point? I'm very curious about it. (I'm sorry if I'm taking us far off topic now) Have a look at these previous threads in this forum first. The date and name for each is the date of the last post and the person who made it. Roman Numerals & Fractions. 6th July 2007 - 03:08 AM SKStephenson Roman Mathematics 5th July 2007 - 09:41 PM SKStephenson Twelve? 19th June 2007 - 05:31 PM Pantagathus Greek Numerical System and Mathematics 24th Feb 2007 - 07:30 PM Gaius Octavius Roman Mathematics 14th Feb 2007 - 02:18 PM Ruthe If you would like more specific detail, just ask.

I am not aware of any specific significance to the use of twelve by the Romans apart from one. The Romans were obviously practical in their approach to most things, and if something allowed them to solve problems with greater ease, they were likely to do so. Since twelve was a much better number for division due to its factors (2, 3, 4 and 6),and since halves , quarters and thirds were the first three fractions that were likely to be encountered in everyday activities, then the Romans used it as the basis for most calculations involving fractions. Unfortunately, due to human physiology and our two hands with a total of ten digits, most integer arithmetic was based on ten and not twelve, except of course for western weights and measures which had a long life with a component of twelve. Those were 12 ounces in a pound (before the English changed the avoirdupois to 16 ounces per pound), 12 inches in a foot and 12 pennies in a shilling, all directly traceable to Roman measures. Like the Romans, and despite the introduction of the Hindu Arabic number system and the later introduction of decimal fractions, these measures remained in use, simply because they were of practical value to the working man, the baker, the carpenter, the butcher, the accountant, the grocer and of course, the general public. There is no reason why the number system could not be extended to be a duodecimal system. A ten based system is not special, and in fact is less practical than a twelve based system. Too bad the Romans and then the French Acadamy didn't make that intellectual leap.

I am sorry to say that the reference given has one or two points with which I cannot agree. After clicking on the link 'Roman Numerals' you are presented with a page with several more links. There is a group of links that starts with 'How Roman Numerals Work'. Click this link to a yet another page. This has a very complete description of Roman numerals and how they were written including explanation of the rules for forming any required value. When you reach the section subtitled 'Fraction', the author makes a number of incorrect or incomplete statements. First, the table of fractions is far from complete and only shows one form of symbols for the truncated list of fractions, missing out any fractions less than 1/12, and alternative symbols for the ones he does list. He then states that "Other fractions could not be depicted in Roman numerals" which is clearly false since many other fractions existed and had specific names and symbols. Furthermore, Romans often indicated other fractional values for which they did not have a specific name or symbol by combing two or more of their duodecimally based fractions to provide a very close approximation to the required fraction. Secondly, in the subsection titled "Zero", the author correctly identifies that the Romans had no symbol for zero, but certainly had the concept in the use of an empty column in their abacus, but then implies they had no concept of a "place" value system when it is obvious that their decimally based bi-quinary abacus was the concrete embodiment of that very idea. What is accepted is that this knowledge was not transcribed to the Roman written format for numeric values but was easily transcribed directly from abacus to the Roman written format. In column one, beads in the bottom slot represented the same number of "I"s while the bead in the top slot in the counted position was a "V". For the second column these were "X"s and an "L" respectively, and so on for succesive columns giving "C"s and a "D", "M"s or a "(I)" and a "(X)" and so on. I recommend the majority of this information but with the provisos I have noted above.

All you ever need or want to know about calendars, including Roman calendars, 'kalends' and 'ides' can be found All you ever wanted to know about calendars. This does not explain why March 25th was the start of the new year until Britain switch from the Julian to Grtegorian calendars in 1752, but this entry in Wikipedia does. However, what is also not mentioned is why the UK tax year starts on April 5th. Because by the time the UK switched calendars the UK was 11 days out of step with the seasons, it was necessary to omit 11 days at the time the change took place. Thus in 1752 in the UK and its colonies September 2nd was followed by September 14th. Quite apart from the fact that many of the British population objected to the loss of those 11 days, even more urgent was the outcry from the British Treasury which could not countenance the loss of 11 days taxes for the entire tax paying population. So what did they do? They did not change the start of the tax year to coincide with the start of the new legal year on January 1st, but instead moved it 11 days later. Where does that take it? Well, to April 5th, which as every tax paying citizen of the UK knows is the current start of the tax year and has been since 1753! Hope these links and snippets answer your questions GA and others.

Exactly Gaius!!! The second column shows symbols for each fraction and shows different variations at different periods for the same fraction. As for the Greeks, I have no information on their practices. It could be they didn't use fractions. If I find anything I'll pass it on. P.S. Just so you don't have to go back to the previous thread I mentioned, I've included a link to a picture of a copy of a Roman pocket calculator ( a small portable abacus with two colums for fractions) that shows an example of the symbols for fractions. The second column from the right is for 'unciae' while the first column has three separate slots for fractions of an uncia. The accepted values are from top slot to bottom, 1/2, 1/4 and 1/3 of an uncia. But I claim the bottom slot with its two beads is for twelfths of an uncia. My argument for this is that if the bottom slot is used for thirds of an uncia, you could only represent the following fractions of an uncia: 0/12, 3/12 (1/4), 4/12 (1/3), 6/12 (1/2), 7/12 (1/4 + 1/3), 8/12 (2/3), 9/12 (3/4 = 1/2 + 1/4), 10/12 ( 1/2 + 1/3), 11/12 (1/4 + 1/3 + 1/3), 13/12 (1/2 + 1/4 + 1/3) and 17/12 ( 1/2 + 1/4 + 1/3 + 1/3) This arrangemant does not allow for representing 1/12, 2/12 (1/6) or 5/12 (1/4 + 1/12 + 1/12). If the bottom slot is used for twelfths of an uncia it would allow all other twelfths up to 11/12 to be represented. This is a far more logical progrssion and would allow these two columns to represent smaller duodecimal fractions by the simple expedient of a mental shift of the magnitude which is in effect the same as moving the decimal point in our 10 based system. Note that the decimal system is not unique since every base system allows for this same shift when multiplying or dividing by the base itself. The abacus is is a copy using diagrams and pictures and is on the site of the Online Abacus Museum. Note that the first two columns are decribed as ounces ans fractions of ounces, but in fact these are more generaly twelfths and fractions of twelfths. The Romans used he same names for weights that were parts of an 'As' and for fractions of a 'Pes' or foot. The Latin word 'uncia' is the origin of both 'inch' and 'ounce' ( the pound was previously 12 ounces before the introduction of the 16 ounce Avoirdupois pound).

The best description of Roman Fractions is given in Numbers by Graham Flegg. This lists each fraction with its name and symbol. The text that goes with this table is copied here. An excellent example of how completely these were integrated into Roman mathematics is shown by the references to two documents in this article in Speculum, the Journal of the Medieval Academy of America. A more direct example is given by The Aqueducts of Rome by Sextus Julius Frontinus. In particular, paragraphs 24-63 illustrate the degree to which these fractions were used in this one particular area of technology. I hope this helps your understanding.

According to Ifrah's The Universal History of Numbers, the Babylonians used a positional notation system to denote fractions, and the Egyptians used an eR-symbol to denote the k in k/n. Unfortunately, he doesn't say anything about the Roman and Greek notation for fractions, although fractions were certainly implicit in the abacus they used. Now, are you just being cruel? Look at the thread titled "Roman Mathematics" in this forum, last post on Aug 30th 2006. This thread answers the question of Roman fractions. The were very adept at calculations with fractions, though as you will see, their fractions used a base of 12, not 10 simply because it was a more practical basis as halves, quarters and thirds were more common in daily use than just halves and fifths in a base 10 system. Had the Romans or even the French adopted a base 12 number system we would have been better served by it than our base 10 system. Don't take my word for it, ask a mathematician.

But they were also known to charge interest at rates other than 12% and its multiples and factors. We have Latin terms for: Asses usurae, or one as per month for the use of one hundred = 12 per cent Deunces usurae 11 per cent Dextantes usurae 10 per cent Dodrantes usurae 9 per cent Besses usurae 8 per cent Septunces usurae 7 per cent Semisses usurae 6 per cent Quincunces usurae 5 per cent Trientes usurae 4 per cent Quadrantes usurae 3 per cent Sextantes usurae 2 per cent Unciae usurae 1 per cent I did not imply that other rates were not used, only that the standard was 12%. As for all the other rates, these were all elements of the duodecimal fractions used by the Romans. The first group below were used for both weights and lengths and a subset were used for fractional parts of capacities but with different names. ............................As=1........Uncia=1 As......................12/12.............12 Deunx................11/12.............11 Dextans..............10/12..(5/6)...10 Dodrans................9/12.(3/4).....9 Bes.......................8/12.(2/3).....8 Septunx.................7/12.............7 Semis....................6/12.(.1/2)....6 Quincunx...............5/12..............5 Triens....................4/12.(1/3).....4 Quadrans...............3/12.(1/4).....3 Sextans.................2/12.(1/6).....2 Sesuncia................1/8............1 1/2 Uncia.....................1/12.............1 In addition, there were also the following fractions. Semuncia].............1/24............1/2 Duella]...................1/36...........1/3 Sicilicus]................1/48............1/4 Sextula]................1/72............1/6 Drachma]..............1/96............1/8 Dimidio sextula]....1/144...........1/12 Tremissis]............1/216...........1/18 Scrupulus]............1/288...........1/24 Obulus]................1/576...........1/48 Bissiliqua]............1/864...........1/72 Cerates]..............1/1152..........1/96 Siliqua]................1/1728..........1/144 Calcus]................1/2304..........1/192 What I intended was to illustrate the Roman use of duodecimal fractions as part of the argument that they were far more capable practical mathematicians then previously portrayed. Simply because of their unusual and unwieldy written numeral system, it has been implied that they were incapable of performing complex calculations. This is an argument to refute that impression.

You may find a previous thread also titled "Roman Mathematics" currently at the bottom of page 3 of Romana Humanitas can provide some guidance on Roman interest calculations and the use of 'finger' and abacuses (I am informed that abacus is not a Roman word and so the plural is NOT 'abacii'). My post of August 14 2006 at 02:19Am in that thread and some of the following posts describe the use of a Roman Pocket Calculator (which was a form of abacus but streamlined for portability). It includes the following link to the site of Prof. Dr. J

I have very little idea of the rate of literacy in the Roman world, but I would like to hear views on the numracy of the same people and did this have any relationship to the rate of literacy?

From The Private Life of the Romans 427. Hours of the Day. The daylight itself was divided into twelve hours (hōrae); each was one-twelfth of the time between sunrise and sunset and varied therefore in length with the season of the year. The length of the day and hour at Rome at different times of the year is shown in the following table: Month and Length of Length of Month and Length of Length of Day Day Hour Day Day Hour Dec. 23 8