Math in 12th c England

[Alisha Marie Klapheke]

When did merchants in medieval England begin using Arabic numerals? And what did they do to keep their account books before that time period? I am sadly ignorant on this point and I can't seem to find clear info on this. Help!

[annis]

It seems that Roman numerals were still being used until later than you'd think, and were certainly still being used in 12th century England, along with the tally stick (http://en.wikipedia.org/wiki/Tally_stick) system for the less literate. By the 1300s, shopkeepers and moneylenders from Japan to England were using the abacus to track sales and calculate interest. The abacus was made up of beads on parallel wires that represented units of one, 10, 100, 1,000, etc. This allowed people to add or subtract large numbers without losing their place.

Arabic numerals were introduced from the east into Italy from the 10th century, and in the 12th century an Italian merchant and mathematician known as Leonardo Fibonacci studied the Arabic numeric system and wrote several books explaining its use. However it was only during the Renaissance period, when humanists translated the mathematical works of the ancients and disseminated them through the newly developed printing press, that the Arabic numeric system became widely used in Europe. The earliest evidence of use of Arabic numerals in England is in the 15th century.

The other change of significance for merchants was the development of the double bookkeeping system (http://www.investopedia.com/articles/08/accounting-history.asp) during the fifteenth century for recording transactions.

[Alisha Marie Klapheke]

Thanks, Annis. I'm going to take a look at those links. I think the tally mark info is what I need.

Also, do know anything about the finger counting technique used by merchants? I don't know when exactly and to what extent it was used; I only read an article that displayed pictures of the various hand positions and how merchants used it to establish place value with only ten fingers.

[Ash]

Not sure if its relevant, but its possible in American Sign Language to count to 10 on one hand (6 is thumb and pinky, 7 is thumb and ring finger, 8 is thumb and middle finger, 9 is thumb and index finger, 0 is 'o' shaped hand.) So its possible the merchants in that time used one hand i to count out, the other hand used for placement maybe?

[annis]

I hadn't heard of this finger counting technique, but I've just found this article which explains how it works. It does seem to bear some resemblance to Ash's American sign language approach.

Medieval Finger Counting
http://orion.math.iastate.edu/mathnight/activities/modules/count/countleft.pdf

[Margaret]

This is pretty fascinating! It feels so strange to think that people didn't have the concept of zero as a numeral, although of course they did have a concept of "nothing." I wonder if it changed the way people thought in any way to have the concept of zero (other than the obvious ability to calculate large sums more easily).

[annis]

I'm intrigued by this finger-counting system, which it turns out is very old, and was used in both the east and the west with some small differences. There's always something new to learn- fascinating! The Romans used it as a universal commercial sign language, handy when dealing with traders of the many different nations which made up the Roman Empire. The Venerable Bede mentions it, so it was still being used in what we used to call the Dark Ages, and in fact in Europe and England it only really faded from use when the Arabic numeric system became common. It was pretty much finished by the 16th century.

If you want to follow it up, there's quite a bit of history in Number Words, Number Symbols (http://books.google.co.nz/books?id=BFJHzSIj2u0C&printsec=frontcover&dq=number+words+and+number+symbols&source=bl&ots=EE7phSAT4-&sig=SoRsYUHNSQZgAaRUXMbZtIM5vzs&hl=en&ei=OWtwTZOvPIz4sAOVy6nFCw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCkQ6AEwAg#v=onepage&q&f=false) by Karl Menninger. My link is to the Google Book version and from there you can link to the chapter on Finger Counting from the Contents, and there is also a chapter on Tally Sticks.

[Alisha Marie Klapheke]

Thanks for the info Annis. Yes I do think the system is similar to American Sign. I'm going to check out that Number Words, Number Symbols article now. I hope they mention something about the church's stance on varying types of mathematics. I know some new ideas were thought of as an evil type of magic. In my WIP, an intelligent, dangerously forward thinking third son training in the priesthood secretly teaches some mathematical concepts to a young woman whom he has befriended. She later uses her skills while working at an inn. I think I'll include the tally sticks in another area--with a textile merchant scene. This is fun!

[annis]

The Church definitely played a part in the repression of the use of Arabic numerals, and I saw this mentioned somewhere, but of course can't find the reference now! It was a contributing factor in the surprising amount of time it took before the Arabic numeric system became widely used in the West. The concept of "zero" was particularly hard to for people to deal with, I think, and may have seemed somehow sorcerous. However it has been pointed out that professional abacus users were equally responsible for resistance to Arabic numerals, if not more so. They felt threatened by the new system, as it seemed to make their skills redundant. The use of Arabic numerals (or more strictly Hindu-Arabic numerals) was known in North-Eastern Spain from the 10th century, and Italian merchants who moved freely through the Byzantine Empire knew of them. Fibonnaci was the son of an Italian merchant based in North Africa, and he wrote about them in the 12th century. Fibonnaci, btw, was a favourite with the enlightened Holy Roman Emperor Frederick II, who was considered a heretic and arch-enemy of the Church by various popes. Anything coming out of the east was automatically suspect and tainted with the possibility of heresy according to Church understanding.

Apparently there was yet another method of number-notation used in the Middle Ages, a type of shorthand or cipher used by monks to record information. The medieval monks frequently used shorthand/abbreviations in written work as well. It's very interesting to look at a copy of the Domesday Book in its original form. Translators had their work cut out to make sense out of what would have been commonplace abbreviations when it was written.

[Kveto from Prague]

Annis, you really know everything:-)

its interesting. I do recall (probably while I was half asleep in algebra class) that the concept of zero is a relatively recent (last 1000 years or so) and revolutionary idea that really aided in the spread of arabic numerals.

can you imagine whiting down a telephone number in roman numerals?

[Ken]

The use of Arabic numerals (or more strictly Hindu-Arabic numerals) was known in North-Eastern Spain from the 10th century, and Italian merchants who moved freely through the Byzantine Empire knew of them. Fibonnaci was the son of an Italian merchant based in North Africa, and he wrote about them in the 12th century. Fibonnaci, btw, was a favourite with the enlightened Holy Roman Emperor Frederick II, who was considered a heretic and arch-enemy of the Church by various popes. Anything coming out of the east was automatically suspect and tainted with the possibility of heresy according to Church understanding.

Annis, I have studied some of Fibinacci’s works particularly in relation to his sequence of numbers that provide the ‘Golden Ratio;’ a ratio that medieval (and earlier) builders new well and used when deciding on the form of their buildings.
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 ….. where by definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two.

Now, if one draws a line divided into two parts a and b, with total length = a + b, then a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b. This is shown graphically in Wikipedia or other publications on line, making it easier to understand.
The golden ratio is derived from this geometric relationship that arises from this and can be expressed algebraically as:
a+b/a = a/b = Φ
This equation has one positive solution in the set of algebraic irrational numbers:
Φ = 1/2 + √5/2 = 1.6180339887
So the Golden Ratio is 1.6180339887 …….

Now, if you take the Fibonacci sequence above and, ignoring the lesser numbers divide each number by its preceding number you will get the following:

6765/10946 = 1.618033996 ….. a close approximation to the golden ratio!

At least since the Renaissance, many artists and architects have proportioned their works to approximate to the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.