Napier's Bones
Division &
Taking Square Roots

bones / strips / stick
multiples of a number
References / Resources
take a square root
big paper
smaller paper


    There are 3 kinds of Napier's rods or bones or strips.

  • Digit bones.
          More than one digit bone of the same kind is needed to represent a number with more than one of the same digit, 11 and 3345, for example.
  • Index bone.
          It's used to help align the strips and to identify rows on the digit strips.
  • Square root bone.
          Because taking a root requires 3 simultaneous computations, 3 column are needed on the bone. These are the square of the digit, the double of the digit, and the digit.



      Each bone depicts the multiples of the one-digit number at the top of the bone. See the 7 bone above.

      Multiples are used here to complete multiplication, division, or taking a root.

To obtain multiples of a multi-digit number:
1st: Arrange bones for the multi-digit number to the right of the index bone.
2nd: Recognize each row represents a multiple of that number -- the product of the number and the index number.
3rd: Write the sum of the digits in each diagonal of the row, one digit per diagonal.   Carry as needed. This sum is the product, a multiple of the number.


      With understanding of multiples and how Napier's Bones make listing of multiples of a number easy, multiplication, division, and the taking of a square root become tasks which may be completed by one who is not great at multiplication.

      With attention and repeated multiplication, division, and root taking, one may become more profficient at multiplication.

      The following pages and materials are now appropriate.

Instructions on the Operations
root extraction
Napier's Bones Digital Manipulatives
Napier's Bones big paper version
Napier's Bones small paper version digital manipulatives

      Many other manipulatives on many different topics are found at: MathTokens.Com.

Azzolino, Agnes "algorithm"
© 2005 at
(last modified 2005), visited 10 April 2009.
Azzolino, Agnes "Multiplication & Division"
© 2008 at
visited 10 April 2009.
Azzolino, Agnes "multiple"
© 2005 at
visited 10 April 2009.
Wikipedia®', Wikimedia Foundation, Inc.,
"Napier's bones - From Wikipedia, the free encyclopedia"
GNU Free Documentation License at
(last modified 16 March 2009, at 20:42 (UTC) ), visited 10 April 2009.

© 2009, October 2010, A. Azzolino
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