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Napier's Bones
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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Update: June 2012
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