When we explored Fibonacci's Liber Abaci on Thursday, I was intrigued by chapter 13: On the Method Elchataym and How with It Nearly All Problems of Mathematics Are Solved. What a title! I couldn't stop wondering about this mysterious method of elchataym and how it could possibly be so universal. I read through a translated snippet of the chapter online and also studied some articles which discuss elchataym in modern math terms. 

Elchataym comes from the Arabic "al-khata'ayn", meaning "the two errors". In Liber Abaci, Fibonacci introduces elchataym as "the method of double false position". This method was first used before his time by Chinese and Arab mathematicians. Fibonacci's description of the method went right over my head, each of the eight times that I read it. I decided to start with the basics and find out what double false position actually means.

It seems that elchataym is a form of "guess and check". An article titled "False Position in Leonardo of Pisa's Liber Abbaci" by John Hannah (University of Canterbury, New Zealand) says that to use double false position, "Leonardo uses two guesses at the value of the unknown, and then compares them to see how much closer he is to the target value" (Hannah 11). Fibonacci seems to approximate his answers by adjusting the values for x and y until he reaches the number he is looking for (or one very close to it).

A quote from the translation of chapter 13 of Liber Abaci shows Fibonacci's thought process regarding elchataym:
“For the one pound that I increased in the second position, I approached more closely to the true value by 3 ounces of silver. How much should I increase the second position so that I approach more closely by another 3 ounces?”

Multiple sources defined double false position as a method that we now call linear interpolation. This term was also new to me, so I researched it. Wikipedia defines linear interpolation as "a method of curve fitting using linear polynomials." This definition opened my eyes a bit. I read on to find out that for two known points with coordinates (x0,y0) and (x1,y1), the formula for linear interpolation is
This writing will be left open-ended because I need to do further research in order to gain a better understanding of elchataym and all of its wonder.

Sources:
http://en.wikipedia.org/wiki/Linear_interpolation

http://link.springer.com/chapter/10.1007%2F978-1-4613-0079-3_14#

http://en.wikipedia.org/wiki/False_position_method

"False Position in Leonardo of Pisa's Liber Abbaci", John Hannah (University
of Canterbury, New Zealand)


 


06/15/2013 6:02am

Very neat post. Would be well worth the follow up. Could use more tying up, as you note. (Consolidation) 4.5/5

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