Thanks! A very early computer with floating point arithmetic was the (binary, tape-controlled) Zuse Z4 relay machine (1945). It was in operation at ETH Zurich from 1950 to 1955 and is on display at the Deutsches Museum in Munich. Herbert ----Ursprüngliche Nachricht---- Von : mike@willegal.net Datum : 21/02/2019 - 15:45 (MZ) An : members@sigcis.org, members@lists.sigcis.org Betreff : Re: [SIGCIS-Members] History of Rome. How did they calculate? Floating point is the most common way to represent fractional numbers in computers. I’d like to point out that for various reasons, usually related to performance, a few modern computer platforms don’t support floating point operations. For the occasion where floating point is not supported and integer resolution is not sufficient, one common fall back is to use fixed point math. This is very similar to just dividing an integer number by some factor of 2. This is essentially the same as dividing currency by some fixed units (Pounds, Shillings, Pence) and just counting the total number of smaller units. Regards, Mike Willegal

On Feb 21, 2019, at 9:02 AM, Roger Johnson <rgj@dcs.bbk.ac.uk> 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@bluewin.ch Date: 21/02/2019 11:00 (GMT+00:00) To: Roger Johnson <rgj@dcs.bbk.ac.uk> Cc: members@sigcis.org, members@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@dcs.bbk.ac.uk Datum : 21/02/2019 - 11:02 (MZ) An : herbert.bruderer@bluewin.ch, members@sigcis.org Cc : members@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@lists.sigcis.org> On Behalf Of herbert.bruderer@bluewin.ch Sent: 20 February 2019 20:36 To: members@sigcis.org Cc: members@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/ful...

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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_______________________________________________ This email is relayed from members at sigcis.org, the email discussion list of SHOT SIGCIS. Opinions expressed here are those of the member posting and are not reviewed, edited, or endorsed by SIGCIS. The list archives are at http://lists.sigcis.org/pipermail/members-sigcis.org/ and you can change your subscription options at http://lists.sigcis.org/listinfo.cgi/members-sigcis.org