How Slide Rules Work

Date: 10/11/95 at 12:31:34
From: Anonymous
Subject: Napier's bones

I am trying to complete an essay on John Napier's
(Scottish Mathematician 15th, 16th century) contributions
to math. One concept that I am having a hard time getting detailed
information on is called Napier's Bones. He developed a method
for multiplication using ivory rods with numbers marked off on them
and by putting the appropriate rods beside each other he could read
the answer. I'd like to find out how these rods (markings) were
constructed and what math principles were used in determining the 
answers. Any sources that I have looked at don't reveal how he
accomplished this. I have touched on his work with logarithms
and have seen how you can solve multiplication of complex numbers
by simply adding the logarithms of those numbers (same base)
and wonder if Napier's Bones is just an extension of this concept.
Thanks for any help!

Stephen Powers, 
Fredericton High School
Grade 10
NB Canada

Date: 10/11/95 at 16:42:0
From: Doctor Andrew
Subject: Re: Napier's bones

It sounds like Napier's Bones are an early slide rule.  I'll explain
a bit about how a slide rule works.  You can probably verify that
this is what Napier's Bones are.

It is a property of logarithms that:

log (ab) = log a + log b

So by spacing numbers logarithmically on a bone or piece of wood, you 
can do multiplication. In fact, I can do it right here. I'll use
logarithms in base 2 so I can deal with smaller numbers. What I've done
is space the numbers so that the distance between two numbers is the 
difference of their logarithms in base 2 (log base 2 is written lg).

So lg 2 - lg 1 = 1 - 0,  lg 4 - lg 2 = 2 - 1 = 1, lg 8 - lg 4 = 3 - 2 = 1 
etc. The spacing between other numbers, such as 3 and 2 isn't integral
so I can't write them here without placing them wrong. All the numbers
below are spaced evenly and are right justified.

So here are my bones:

1    2    4    8   16   32   64  128  256  512

1    2    4    8   16   32   64  128  256  512

So now I want to multiply 8 by some number.  So, I slide the 1 on the 
top bone to the 8 on the other bone like this:

------------>  1    2    4    8   16   32   64  128  256  512

1    2    4    8   16   32   64  128  256  512

So 1 * 8 = 8, 2 * 8 = 16, 4 * 8 = 32 and so on. Pretty cool, don't you 
think? Almost as good as a calculator. Now we really can write other 
numbers like 3, or 2.5 on these bones, but I can't position them 
correctly here. This is one way people did arithmetic quickly before 
calculators.

I hope this helps. If you have any questions about logarithms please 
send them to us. I recommend looking up Napier in a Math history book 
or even an Encyclopedia, but you might have better luck looking up 
slide rule in a the Encyclopedia, or even finding a book about slide 
rules. Good luck!

-Doctor Andrew,  The Geometry Forum

Date: 2/24/97 at 14:05:14
From: Andrew Lister
Subject: Re: Napier's bones

I was searching for an image of "Napier's Bones" on the web, and
came up with your answer to a 10th grader's query, but I fear
that you didn't actually answer his question properly.  However,
as the original query was dated sometime in 1995, I'm a trifle
late to help him. I don't know whether you are still interested,
but my recollection (from a mathematics class in primary school
in England, longer ago than I care to remember - Oh, OK, about
30 years) is as follows.  Ideally, you need a graphic of this,
but I don't have one ready, and I don't know how to send it to
you anyhow. Let me know if you are at all interested, and I'll
see waht I can do. In the meantime, I'll just have to try to
describe them as best I can.

Each "bone" has the proportions of a typical 12" ruler; say, 2cm
wide by 22 cm long. This is etched with lines dividing it into a
series of 11 squares. All but the first are also etched with a
diagonal line dividing the square into two triangles. When the
bone is held long side vertical, the diagonals run from top right
to bottom left of each square.

The first square (at the top or the bottom) has a single large
digit. This will be one of the digits of the multiplicand. The
squares below give the times table for that digit, writing the
"tens" digit above the line and the "units" digit below the line.

For example, the rod for 7 would read something like this:

SEVEN

0 / 7

1 / 4

2 / 1

2 / 8

3 / 5

4 / 2

4 / 9

5 / 6

6 / 3


To multiply, select the rods corresponding to the digits of the
multiplicand, and arrange them in normal decimal order. If you
want to demonstrate this to kids, a great number to use is
12345679. Make sure that they are all lined up with tops and
bottoms in line.

Ask for the child's favourite number (between 1 and 9). Read off
the result from the appropriate row of the bones. [It is
convenient to have an extra bone to indicate the multiplier. This
would be similar to the standard bones, but would not have the
diagonals, and instead be marked with X 1 2 3 4 5 6 7 8 9 in
large characters.]

To actually read off the calculation,  [this is the tough bit to
describe] starting from the right, pairs of digits in the same
diagonal stripe on adjacent rods are added. In other words, the
digit above the diagonal (of the reading row) of one rod is added
to the digit below the diagonal of the rod immediately to its
left. If the sum is greater than ten, then it is necessary to
carry 1 to the next sum.
Sounds complicated, but anyone who can add two digits can easily
learn to do it. (But you really have to draw it to see how...)

To TRY to finish my example:

     1     2     3     4     5     6     7     9

x7  0/7   1/4   2/1   2/8   3/5   4/2   4/9   6/3

 0   7     4     1     8     5     2     9     3
    +1    +2    +2    +3    +4    +4    +6
    __    __    __    __    __    __    __    __
     8     6     3     1     9     6     5     3
carry           +1                +1
    __    __    __    __    __    __    __    __
     8     6     4     1     9     7     5     3

Funnily enough, this is the same answer I get with my calculator.

Now, if you have followed me this far, and don't already know
where I am going, you may be wondering why we got the child to
pick a number. Well....depending upon their enthusiasm or
patience, you get them to write down this intermediate result and
now do it all over again, but this time just multiply by the
"magic number" 3. Due to its special mathemagical properties this
gives you, after some deft manipulation of the bones, er...

259259259.

Hmmm, perhaps I forgot something. Or perhaps the magic is running
at half power. Try again

259259259 x 3 = 777777777! Ah ha!

I hope that this is of interest...