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Name: Phyllis
Status: student
Age: 50s
Location: N/A
Country: N/A
Date: N/A 


Question:
I have seen a reference to calculating square roots of numbers by using Napier's bones (rods), but can find no explanation of how it is done. I have searched the Internet, and looked in every math book in our local public library. Please help! I am taking a college Liberal Arts II math class and we need the answer. Thank you.



Replies:
"Napier's bones" is just a slide rule. On a slide rule, numbers are represented by their logarithms, so that operations like multiplication are reduced to addition; log(ab) = log(a) + log(b). Similarly, finding exponents reduces to multiplication: log(a^b) = b log(a). To take the square root of a number is equivalent to raising it to the power of 0.5, so that log(a^0.5) = 0.5 log(a). So, to find the square root of A using a slide rule, you just find the number that is half the distance between 1 and A on the logarithmic scale.

This sounds quite mysterious. You should be able to find the rules for exponents in a high school math textbook. Learning how to use a slide rule is a little tricky at first, but it's actually fairly easy. Not quite as easy as using a calculator, though, which is why calculators have pretty much replaced slide rules. (If you're actually in your 50's, as your profile indicates, you probably have more experience using a slide rule than I do.)

Richard Barrans Jr., Ph.D.
Chemical Separations Group
Chemistry Division CHM/200
Argonne National Laboratory
9700 South Cass Avenue
Argonne, IL 60439


See http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Napier.html and references included. Napier's Bones are an early form of the slide rule.

Tim Mooney


Perhaps the confusion is that Napier also proposed using logarithms to do multiplication. However, Napier's bones are a different invention by this guy. It's more a clever multiplication table and has no relation with either logarithms or slide rules. For example, it does exact multiplication (unlike slide rules) and nowhere are logarithms or any other transformation involved during the operation.

More information on Napier's bones for multiplication as well as a later improvement by a couple of Frenchman called the Genaille-Lucas rulers is at:

http://pages.cpsc.ucalgary.ca/~williams/History_web_site/time%201500_1800/N apier's%20bones.htm

also see
http://www.geo.tudelft.nl/mgp/people/gerold/indnap.htm

One answer to the original query on obtaining square roots via Napier's bones is at:

http://www.qnet.fi/abehr/Achim/Calculators_Napier_rods2.html

Guess I'm a somewhat idle busybody but hey -- the archive looks pretty neat, so this is my 0.02 to add to it :-)

Regards,

KB Sriram



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