[SIGCIS-Members] History of Rome. How did they calculate?

herbert.bruderer at bluewin.ch herbert.bruderer at bluewin.ch
Fri Feb 22 01:19:04 PST 2019


Hello Allan,
Thank you for your comment.
The terminology is indeed a bit confusing, especially in English and German. In French and Italian there are different terms for the two basic forms of the abacus.
a) counting boards, e.g. Salamis tablet, medieval table abacus: lines, pebbles, calculi, counters (no rods, no beads),
b) bead frames, e.g. Chinese, Japanese and Russian abacus, school abacus (rods, beads), Roman hand abacus (slots/grooves, buttons).
The Antikythera mechanism is generally considered as (analog) astronomical calculating machine.
Best,
Herbert

----Ursprüngliche Nachricht----
Von : allan.olley at utoronto.ca
Datum : 21/02/2019 - 18:03 (MZ)
An : rgj at dcs.bbk.ac.uk
Cc : herbert.bruderer at bluewin.ch, members at sigcis.org, members at lists.sigcis.org
Betreff : Re: [SIGCIS-Members] History of Rome. How did they calculate?

Hello,
 	I think there may be some terminological confusion  (or maybe I 
am confused), but I sense that the terms table abacus, counting table and 
counting board all refer to basically the same sort of material object, 
but some in this conversation may think (?) they refer to different 
things. Anyway counting boards from the Roman era are definitely extant as 
one of the links in Herbert's original stories includes images of the 
Salamis tablet which is a Babylonian counting board from 300 BCE and the 
implication is other such objects from the period survive. However 
if I understand the claims correctly most Roman counting boards were made 
of wood and so did not survive the intervening millenia. 
https://www.ee.ryerson.ca/~elf/abacus/history.html
 	One extant calculating aid that I know of is the written 
mathematical table, Ptolemy's Almaghest contains a table of chords, which 
would instantiate trigonometric relations and so allow one to do various 
trigonometric calculations as needed for Ptolemaic geometric astronomy 
(so would be analogous to the role of tables of functions like sine, 
cosine and tangent in modern mathematics). In addition to looking at how 
these tables were used one might look at how they were calculated (any 
tell tale errors, how rounding and allied operations were used etc.) to 
suggest how the calculations were performed. I am sorry to say I do not 
know enough about the literature to say whether someone has already done 
such investigations. Also the relationship between scientific calculation 
and mercantile calculations need not be very strong. I would point out 
that in ancient astronomy one mostly deals with degrees, minutes of arc 
(60 minutes to a degree) and seconds of arc (60 seconds to a muinute), so 
fractions are dealt with using such units. This gives Roman astronomy a 
sexagismal (base-60) character taken from the Babylonians and not seen I 
imagine in the way merchants would count. The ancient Babylonians also 
left us with lots of mathematical tables, as I recall there have been 
investigations into how and why these were calculated that might give 
insight into ancient arithmetic techniques.
 	One other ancient device of interst is the Antikytheria 
mechanism (which models some motions of the heavenly bodies relating 
them to the calendar). I hesitate to say it is a calculating device since 
it seems to me more like a device created to agree aproximately with 
astronomical calculations. If you want a good starting point for this and 
its place in Roman era scientific culture I would suggest Alex Jones 
recent book a Portable Cosmos. Although again this takes us far from how 
Roman era merchants did arithmetic....


-- 
Yours Truly,
Allan Olley, PhD

http://individual.utoronto.ca/fofound/

On Thu, 21 Feb 2019, Roger Johnson wrote:

> Dear Herbert
>
> I think there is a small table abacus in a Roman fast food outlet on the counter in Pompeii
>
> Roger
>
>
>
> Sent from my Samsung device
>
>
> -------- Original message --------
> From: herbert.bruderer at bluewin.ch
> Date: 21/02/2019 11:00 (GMT+00:00)
> To: Roger Johnson <rgj at dcs.bbk.ac.uk>
> Cc: members at sigcis.org, members at lists.sigcis.org
> Subject: Re: RE: [SIGCIS-Members] History of Rome. How did they calculate?
>
> Dear Roger
>
> Thank you very much for your helpful comment!
> In 1970/1971 I was teaching at two London state grammar schools (later transformed to comprehensive schools) and recollect the British currency and D day (decimal day).
>
> A video of the Deutsches Museum, Munich, shows that all four basic operations can be done very quickly on a school abacus:
> https://www.youtube.com/watch?v=jwabVzlobZI
>
> I think there was no real fractional arithmetic, the Roman number system was not suitable for this purpose.
>
> Thus the right hand column on an abacus was, as I understand it, often just for counting quarters.
> What about the 1/2 and 1/3 slots?
>
> I have never been convinced that the hand abacus was used for largescale multiplication. What was used were the table abacus.
> I believe that you are right. Unfortunately, to my knowledge no Roman table abacus has survived.
>
> Best wishes,
> Herbert
>
> ----Ursprüngliche Nachricht----
> Von : rgj at dcs.bbk.ac.uk
> Datum : 21/02/2019 - 11:02 (MZ)
> An : herbert.bruderer at bluewin.ch, members at sigcis.org
> Cc : members at lists.sigcis.org
> Betreff : RE: [SIGCIS-Members] History of Rome. How did they calculate?
>
> Dear Herbert
>
> It is at least 10 years since I looked at this topic n detail and I am writing this from memory without going back to original sources although I hope I can still find them amongst y papers if needed.
>
> You ask about the righthand most column on an abacus. My recollection is that Romans did not use the concept of fractions as we do today where we teach schoolchildren to add 5/8 to 7/12 etc. The Romans understood subdivisions of a unit such as a coin or of a unit of area. Until 1971 in the UK we had sterling based on pounds shillings and pence, denoted (following Roman practice) by the letters L (£), s and d. The penny was divided into 4 farthings (1/4 of a penny). I remember as a boy that 4 farthings made a penny, 12 pennies made a shilling and 20 shillings made a pound. The different currency units were essentially treated arithmetically as just having different bases. 12 pennies was one shilling etc. They were not treated as fractions. In particular adding half a shilling and three-quarters of a penny would have made no sense so arithmetic on fractions was not needed.
>
> (If readers in France or USA think this is all very odd – please remember the same system operated in France until the Revolution and it continued in the USA into the very early 19th century – around 1790 the value of a pound differed between states and I have exchange rate tables between a pound in Rhode Island, a Boston pound and from memory the first of the states adopting the US dollar. Some Pascal adding machines are for currency based on LSD and some for arithmetic).
>
> Thus the right hand column on an abacus was, as I understand it, often just for counting quarters. I also recollect that surviving artefacts and pictures suggest several different arrangements on different devices when dealing with less than the unit. Possibly for different applications.
>
> I have never been convinced that the hand abacus was used for largescale multiplication. What was used were the table abacus. I would commend Mike Williams’ chapter for a short summary of this interesting field. Barnard in his massive tomb on “The Casting Counter and the Counting Board” shows how easy it is to do multiplication of integers, even large ones, using counting tables. They seem to have been used in the view of the parties to the transactions and having demonstrated one to students it is very easy to cheat by sleight of hand. Hence the king (and the taxpayer) may well have watched as his court officials calculated and collected the taxes for his Exchequer !
>
> On the inability of Roman numbers to handle fractions. Specifically I believe there is a good argument that it is not coincidental that the spread of early banking such as developed by the Lombards and the spread of Indo-Arabic numbers took place across Europe from south (in Italy) to the north (Britain) over several centuries. Personally I believe that interest calculations on loans could well have been a major driver on changing the commercial sector from Roman to Indo-Arabic numbers. In the 18th century in the UK dates were often written in Roman notation while money items were in Indo-Arabic in the same document.
>
> Another interesting aspect is demonstrated by the Early English Books Online project which looks at early printed books in the UK. The book by Robert Recorde on arithmetic which Mike Williams illustrates is an early example of hundreds of such books published mainly in the 17th century to teach arithmetic and multiplication in particular. As long as the table abacus was sufficient for performing calculations all multiplication could be done without any knowledge of multiplication tables because the table abacus relied on left shifts to multiply by 5 and 10 and multiple addition for up to 4. With practice it is easy and quick. However as soon as someone wishes to charge a merchant 10% per year interest on  a loan that runs for 240 days we are all in trouble – enter Indo-Arabic and decimal fractions.
>
> For similar reasons, pre-printed pocket size sets of tables usually called Ready Reckoners, were printed in millions in Europe and USA and were carried by merchants so that they could look up what 127 articles at 3 pence and 3 farthings should cost. Again this reflected partly ease of use but also a lack of ability to do multiplication.
>
> In passing it was the special software and printing features needed for sterling that acted as a major barrier to the advance of the US computer companies into the British “Empire” and its “sterling zone”. COBOL-60 had a sterling currency field – although my commercial coding career started one year too late to have used it!
>
> Herbert – far too long so my apologies
>
> Good wishes
>
> Roger Johnson
>
> From: Members <members-bounces at lists.sigcis.org> On Behalf Of herbert.bruderer at bluewin.ch
> Sent: 20 February 2019 20:36
> To: members at sigcis.org
> Cc: members at lists.sigcis.org
> Subject: [SIGCIS-Members] History of Rome. How did they calculate?
>
>
> How did the Romans calculate?
>
> https://cacm.acm.org/blogs/blog-cacm/234881-how-did-the-romans-calculate/fulltext
>
>
>
> Hi all:
>
> Does anyone know how the Romans calculated with the Roman numeral system?
>
> What is the meaning of the right-most slot of the Roman hand abacus?
>
> Do you have Roman hand abaci (original devices or replicas) in your collection?
>
> Thank you very much for your help.
>
> Best wishes,
>
> Herbert Bruderer
>
>
>
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