Roman Mathematics

[M. Porcius Cato]

One of Cicero's letters to Atticus (15 Feb 50 BCE) discusses the interest rates at which the Salamanians in Cyprus were being loaned money by friends of Brutus. What makes this discussion interesting is the last line below, which I bet will delight our Primus Pilus to no end. From Cicero's letter:

 

Everybody in court, exclaimed that Scaptius was the greatest knave in the world for mot being content with twelve per cent. plus the compound interest: others said that he was the greatest fool. In my opinion he was more knave than fool. For either he was content with twelve per cent. on a good security, or he hoped for forty-eight per cent. with a bad one. 12 That is my case; and if Brutus is not satisfied with it, I cannot see why I should regard him as a friend: I am sure that his uncle at any rate will accept it, especially as a senatorial decree has just been passed--I think since you left town--in the matter of money-lenders, that twelve per cent. simple interest was to be the rate. What a wide difference this implies you will certainly be able to reckon, if I know your fingers.

 

Now here's a math puzzler for you: how could Atticus quickly calculate Scaptius' claim under the two different interest rates (12% and 48%) ON HIS FINGERS? Now don't be lazy and give up--I assure you it can be done fairly easily. I'm wondering if anyone with, say, the expertise of an accountant, knows a fast and frugal heuristic for solving this puzzle.

 

(Totally off-topic needle to Clodius: Note who Cicero mentions as the potential savior of our Cypriot debtors?? Far from being guilty of extortion, Brutus' uncle--i.e., M. Porcius Cato--is mentioned by Cicero as the very one who will stand up for the Cypriot's lawful rights! OK, digression completed.)

[Primus Pilus]

One of Cicero's letters to Atticus (15 Feb 50 BCE) discusses the interest rates at which the Salamanians in Cyprus were being loaned money by friends of Brutus. What makes this discussion interesting is the last line below, which I bet will delight our Primus Pilus to no end. From Cicero's letter:

 

Everybody in court, exclaimed that Scaptius was the greatest knave in the world for mot being content with twelve per cent. plus the compound interest: others said that he was the greatest fool. In my opinion he was more knave than fool. For either he was content with twelve per cent. on a good security, or he hoped for forty-eight per cent. with a bad one. 12 That is my case; and if Brutus is not satisfied with it, I cannot see why I should regard him as a friend: I am sure that his uncle at any rate will accept it, especially as a senatorial decree has just been passed--I think since you left town--in the matter of money-lenders, that twelve per cent. simple interest was to be the rate. What a wide difference this implies you will certainly be able to reckon, if I know your fingers.

 

Now here's a math puzzler for you: how could Atticus quickly calculate Scaptius' claim under the two different interest rates (12% and 48%) ON HIS FINGERS? Now don't be lazy and give up--I assure you it can be done fairly easily. I'm wondering if anyone with, say, the expertise of an accountant, knows a fast and frugal heuristic for solving this puzzle.

 

(Totally off-topic needle to Clodius: Note who Cicero mentions as the potential savior of our Cypriot debtors?? Far from being guilty of extortion, Brutus' uncle--i.e., M. Porcius Cato--is mentioned by Cicero as the very one who will stand up for the Cypriot's lawful rights! OK, digression completed.)

 

I'm a bit perplexed by what Cicero is saying here. Does he mean to compare 12% compound interest with 48% simple, or is he suggesting a straight 48% with 12% simple. The cost of simple or compound interest is directly affected by time, so such a thing is a key component of the calculation.

 

For instance, let's say someone borrowed 10,000 denarii payable over 10 years.

At 48% flat, the borrower would owe (of have paid) 4,800 interest plus the principal for a total of $14,800.

 

At 12% simple (INT - Principal x rate x time) (10 month roman years, or 100 months in total, for interest purposes) the borrower would owe 10,000 denarii in interest for a total of 20,000 due to the lender. (This transaction forms the mathematical basis of the Roman borrowing system... interest = principal over a 10 year period).

 

As such, the transaction is entirely dependent upon time to determine which method of calculation is better for either borrower or lender.

 

On the same 10 year loan, if we are using simple interest vs. compound interest the cost would be something such as...

At 48% simple (I = PxRxT) (10 month Roman year), the borrower would owe 40,000 in interest for a total due (P & I) of 50,000

At 12% compound [Actual = Principal x (1 + rate)10] the borrower would only owe 21,000 of interest or a total of 31,000. However, I do believe that such compounding had been outlawed in Cicero's time and I believe the first comparative example above is more applicable.

 

Clearly time is the key component in factoring the most favorable results in any of these transactions. For a rather confusing description of how the Roman calculated interest... Fenus... William Smith Dictionary

 

(Personally, I am far too reliant on 10-keys and computers for such calculations anymore, but the concept is fairly easy once the brain shakes off the cobwebs. For the ancients, such things were a matter of course for mathematicians.)

[M. Porcius Cato]

According to a subsequent letter by Cicero, the loan was due in six years, with interest compounded annually. This struck me as an awfully convenient term for a loan, given the rule of thumb I have in mind.

 

EDIT: I'll link to the letter shortly.

Edited February 11, 2007 by M. Porcius Cato

[caldrail]

Also many people involved in such deals already knew the result beforehand, they'd done this calculation previously. It wouldn't suprise me if some hadn't learned whole series of figures much like we used to with the 'times table'.

[M. Porcius Cato]

Also many people involved in such deals already knew the result beforehand, they'd done this calculation previously. It wouldn't suprise me if some hadn't learned whole series of figures much like we used to with the 'times table'.

 

This is a really good idea, but when one mentally retrieves a product from a memorized 'times table', no fingers are involved. So, if Atticus is memorizing anything, it must be an ordinal series that can be counted out on his fingers.

 

My first guess (when I posted this) was that the Romans were using a trick for estimating the outcomes of compound interest, which I know only as the 6/12 rule. The rule of thumb of is: at 6%, you can double your money in 12 years; at 12%, you can double your money in 6 years; and so on. (As I say, this is a rule of thumb. The numbers aren't exactly correct, but it's darned useful if you're haggling in the marketplace--and it would be just fine for Atticus.) If this is correct, Atticus could quickly estimate that Brutus' friends would double their money in 6 years at 12% interest.

 

The problem is that the rule (by itself) doesn't really work that well for 48% since (1) the rule of thumb breaks down for very low (0-2%) and very high (>36%) interest rates, and (2) all it would tell Atticus is that Brutus' friends could double their money before the end of 2 years. What it doesn't tell us--and this is what Atticus is supposed to be counting on his fingers--is that Brutus' friends would have increased their money 10-fold after 6 years at 48%.

 

Which leads me to think that Caldrail's suggestion might be onto something: maybe Roman businessmen simply memorized a series of financial outcomes for conventional rates of interest (normally fixed at 12% in the provinces, btw). If this is right, they could have memorized that after 6 years, every 10k drachma loaned would yield approximately 12k at 3%, 14k at 6%, 20k at 12%, 36k at 24%, 64k at 36%, and 108k at 48% (which is, btw, a typical log function).

 

Still, I'm not entirely satisfied with this solution either. Try memorizing this table for yourself and see how quickly and accurately you can guess what the difference would be between two different rates. I think it would be an awfully unwieldy strategy, and I'm not sure how you'd use your fingers either.

Edited February 12, 2007 by M. Porcius Cato

[The Augusta]

Phew - gents! Well, all I know is that Atticus' fingers must have been a lot better than mine.

 

But more seriously, did Cicero mean to imply that Atticus would count on his fingers, or was he referring to Atticus' speedy fingers on an abacus? Just a thought that struck me.

[Gaius Paulinus Maximus]

Phew - gents! Well, all I know is that Atticus' fingers must have been a lot better than mine.

 

But more seriously, did Cicero mean to imply that Atticus would count on his fingers, or was he referring to Atticus' speedy fingers on an abacus? Just a thought that struck me.

 

Ahhh very clever Augusta, you could be onto something there. It certainly seems more plausible than actually doing the math on your fingers, I'm not saying that it can't be done because PP and MPC have almost proved it possible.

 

Didn't some Romans (more than likely moneylenders) carry some sort of portable pocket abacus, I think I may have seen a picture somewhere in Perinax's gallery?? Surely if anyone had speedy fingers than it would have been these guys.

[Pertinax]

Here is the pocket calculator:

 

http://www.unrv.com/forum/index.php?act=mo...=si&img=907

[Primus Pilus]

But more seriously, did Cicero mean to imply that Atticus would count on his fingers, or was he referring to Atticus' speedy fingers on an abacus? Just a thought that struck me.

 

That's entirely possible. After all, accountants are often referred to as bean-counters... but how many actually count beans? Bean-counting is actually a rather negative connotation meaning that an accountant is too busy counting to notice the bigger financial picture. Perhaps Cicero was just using a common figure of speech when the method of calculation really had little to do with Atticus' actual fingers?

 

Augusta now has me wondering if perhaps the word for finger (digitus) was mistranslated or improperly understood in relation to its original context. Perhaps another word was used rather than digitus or perhaps using one's fingers had simply become a form of slang to substitute for more proper mathematics terminology.

 

If I could find the original Latin it would help but I have had no luck finding Ad Atticum in the original. If the word Cicero used is digitus or the proper derivative thereof, than its unlikely I could prove that Cicero was using a form of slang, so I suppose I am left only with a suspicion anyway.

[M. Porcius Cato]

If I could find the original Latin it would help but I have had no luck finding Ad Atticum in the original. If the word Cicero used is digitus or the proper derivative thereof, than its unlikely I could prove that Cicero was using a form of slang, so I suppose I am left only with a suspicion anyway.

 

The Latin you requested:

<alii> nihil impudentius Scaptio qui centesimis cum anatocismo contentus <non> esset, alii nihil stultius. mihi autem impudens magis quam stultus videbatur; nam aut bono nomine centesimis contentus non erat aut non bono quaternas centesimas sperabat. habes meam causam.

 

quae si Bruto non probatur, nescio cur illum amemus. sed avunculo eius certe probabitur, praesertim cum senatus consultum modo factum sit, puto, postquam tu es profectus, in creditorum causa ut centesimae perpetuo faenore ducerentur. hoc quid intersit, si tuos digitos novi, certe habes subductum. (emphasis added for the cheap seats in the back)

 

For more of Cicero's letters to Atticus (in Latin), see here.

[Gaius Octavius]

Let me throw a few questions in. Are we speaking here of the present value of 1 or its future value? Is the interest due at the end of the first period added to the loan and compounded into the next period, i.e, a $1,000 loan at 10% yields interest of $100 at the end of the first period. If not paid and thus 'compounded' into the next period, the interest at the end of that period would be $110. Is the interest discounted up front or paid at maturity or periodically. Does the loan have call provisions?

 

[M. Porcius Cato]

BTW, what was the Latin idiom for calculating on an abacus? I'm starting to come around to this view.

[Primus Pilus]

hoc quid intersit, si tuos digitos novi, certe habes subductum.[/i]

 

I am hardly an expert, but this translates as originally posted. Seems we have little to go on with Cicero's actual choice of words anyway. There are any number of variations of the word numero that could have been used to simply mean count. He could have used abacus or calculus (the small pebbles used for counting) or some other mathematics term (Cicero was clearly educated enough to be aware of the various methods) but yet he chose the word finger. Seems we could easily take him at face value, but yet I am still left wondering if he was just being humorous or deferent to the math skills of his friend.

 

Still we know that Romans kept meticulous books on such things and interest rates and how they functioned, when payments were made, etc. was largely a standardized process. I'm skimming through Jean Andreau's "Banking and Business in the Roman World" (especially chapter 9 on interest) but it doesn't mention specifics on how an individual would do or refer to the actual math. I'm still skimming through, but this work is a complete analysis of Roman "banking" rather than an indexed dictionary of terminology and functions. Alas.

[M. Porcius Cato]

As promised, here is an excerpt from Cicero's letter spelling out the length of the loan:

 

Now for the case of the Salaminians, which I see came upon you also as a novelty, as it did upon me. For Brutus never told me that the money was his own. Nay, I have his own document containing the words, "The Salaminians owe my friends M. Scaptius and P. Matinius a sum of money." He recommends them to me: he even adds, as though by way of a spur to me, that he had gone surety for them to a large amount. I had succeeded in arranging that they should pay with interest for six years at the rate of twelve per cent., and added yearly to the capital sum. 2 But Scaptius demanded forty-eight per cent. I was afraid, if he got that, [p. 136] you yourself would cease to have any affection for me.

 

It's interesting that Cicero arranged for a loan that perfectly fit the 6/12 rule for doubling one's money.

[Gaius Octavius]

Re compounding periods. On a two year loan, the compunding period may be annual, semi-annual or monthly, etc. Think of a 0% bond. It is the compounding periods that are critical in determining the sum returned.