Napier's Rods

Date: 03/05/99 at 11:38:48
From: phyllis rodin
Subject: Napier's Rods

I have been asked to do a math workshop on Napier's rods and  I need a 
picture of them and some basic information. 

Please help!  

Date: 03/05/99 at 11:56:04
From: Doctor Peterson
Subject: Re: Napier's Rods

There is a nice site about Napier's rods at

   http://www.cee.hw.ac.uk/~greg/calculators/napier/   

Here is an explanation from our Dr. Math archives:

  http://mathforum.org/dr.math/problems/crystal1.20.97.html   

The basic idea is that each "bone" represents a column of the 
multiplication table, like this (the 8 bone):

    +--+
    |0/|
    |/8| 8*1 = 8
    +--+
    |1/|
    |/6| 8*2 = 16
    +--+
    |2/|
    |/4| 8*3 = 24
    +--+
    |3/|
    |/2| 8*4 = 32
    +--+
    |4/|
    |/0| 8*5 = 40
    +--+
    |4/|
    |/8| 8*6 = 48
    +--+
    |5/|
    |/6| 8*7 = 56
    +--+
    |6/|
    |/4| 8*8 = 64
    +--+
    |7/|
    |/2| 8*9 = 72
    +--+

To multiply a number by a single digit, you put together the bones for 
its digits, then add along the diagonals in the row for the digit you 
are multiplying by. For example, if I line up the bones for 287, row 6 
will be

        2  8  7
      |  |  |  |
      +--+--+--+
      |1/|4/|4/|
    6 |/2|/8|/2|
      +--+--+--+
      |  |  |  |

and you add like this

         +--+--+--+
         |1/|4/|4/|
        /|/2|/8|/2|
       / +--+--+--+
      /  /  /  /
     1  6 12  2 (carry the one)
    ------------
     1  7  2  2

so 287 * 6 = 1722.

To multiply by a multiple-digit number, you would use the bones to get 
each row of the normal multiplication method:

     287
    * 26
    ----
    1722 <-- what we did above
    574  <-- the same, using row 2 instead of 6
    ----
    7462

As you can see, Napier's bones mostly just save you from having to 
memorize the multiplication table; you have to do all the rest of the 
work yourself. They are little more than a mechanization of the 
"lattice method" for multiplication, which is not limited to single-
digit multipliers. (You can look that up in our archives, if you are 
not familiar with it.) Oddly, Napier also invented the logarithm, which 
was the basis of the slide rule, a much more effective tool for 
multiplication; but his bones, which have nothing to do with 
logarithms, were as popular in his day as the slide rule was in mine.

I think that should get you started.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/15/2001 at 21:30:42
From: Doctor Jodi
Subject: Re: Math history

There's a great page on various methods of multiplication, including 
Napier's Bones, at 

   Napier's rods, abacus and slide rule - Gerold Beckers
   http://www.geo.tudelft.nl/mgp/people/gerold/indnap.htm   

For some historical information, see 

   1600 AD  John Napier and Napier's Bones - A History of Computers
   http://www.maxmon.com/1600ad.htm   

For instructions on making Napier's bones, see 

   Napier's Bones - Rokeby High School, Rokeby, Tasmania
   http://www.tased.edu.au/schools/rokebyh/curric/infotech/stage1/assign2/napier.htm   

This last page is part of a great site about pre-20th century computing.

- Doctor Jodi, The Math Forum
  http://mathforum.org/dr.math/