Napiers bones

THIS PPT IS ABOUT NAPIER'S BONES

1. Napier’s BonesNapier’s Bones
An Adventure inAn Adventure in
1717
thth
Century Arithmetic Century Arithmetic
byby
YASHNAYASHNA

2. John NapierJohn Napier
John Napier, a 16
th
Century
Scottish scholar, contributed
a host of mathematical
discoveries, among them the
world’s first calculator.
John Napier (1550 – 1617)

3. He is credited with creating the first
computing machine, logarithms and
was the first to describe the
systematic use of the decimal point.
Other contributions include a
mnemonic for formulas used in
solving spherical triangles and two
formulas known as Napier's
analogies.
““In computing tables, these large numbers may again be made still larger In computing tables, these large numbers may again be made still larger
by placing a period after the number and adding ciphers. ... In numbers by placing a period after the number and adding ciphers. ... In numbers
distinguished thus by a period in their midst, whatever is written after distinguished thus by a period in their midst, whatever is written after
the period is a fraction, the denominator of which is unity with as many the period is a fraction, the denominator of which is unity with as many
ciphers after it as there are figures after the period.” ciphers after it as there are figures after the period.”

4. High tech in the 17
th
century, was what
we’d now call basic astronomical
arithmetic calculations, all done by hand.
It took Johannes Kepler (1571-1630) nearly
1000 large pages of dense arithmetic do
discover the laws of planetary motion!
Johannes Kepler (1571-1630)
A typical page from one of
Kepler’s notebooks

5. Napier’s BonesNapier’s Bones
In 1617, the last year of his life,
Napier invented a tool called
“Napier's Bones” which reduces the
effort it takes to multiply numbers.
This was a time when few people
could multiply beyond 5 (x) 5.
“Seeing there is nothing that is so troublesome to
mathematical practice, nor that doth more molest and
hinder calculators, than the multiplications, divisions...
I began therefore to consider in my mind by what
certain and ready art I might remove those hindrances.”

6. Napier’s bones were called that
because they were often made of
bone, ivory, silver, or wood. The
were were universally popular
and common until the late 1800s.
Sometimes the Napier tables
were engraved on rods in a case
so that numbers could be
“dialed in”.

7. Napier’s bones make multiplication Napier’s bones make multiplication
and division easier. Multiplication and division easier. Multiplication
and division are reduced to simple and division are reduced to simple
addition, although a pencil and addition, although a pencil and
paper are required.paper are required.
This boxed set has ten rods, allowing This boxed set has ten rods, allowing
computations up to 100,000,000. The computations up to 100,000,000. The
left (or “index”)rod is fixed to the left (or “index”)rod is fixed to the
case. It is numbered from 1 to 9.case. It is numbered from 1 to 9.
The movable rods are numbered at The movable rods are numbered at
the top. The numbers down them the top. The numbers down them
rods show the product of the number rods show the product of the number
at the top times the corresponding at the top times the corresponding
numbers on the index rod.numbers on the index rod.
Here the “3” rod shows three times Here the “3” rod shows three times
each of the numbers on the index rod.each of the numbers on the index rod.

8. The bones are easy to use.
Multiplication and division are
set up the same way.
Set the problem up by laying
down rods corresponding to
the number being multiplied or
divided.
This setup shows the number
3579 which we will show being
multiplied by 43.

9. This is the problem
shown on our “paper
bones.”
The “3”, “5”, “7” and
“9” strips are set up
next to the index.

10. Using a pad of paper, we write down the individual products of 40
and 3 times each digit of 3579. The results are:
Multiply by 40:
4 Times : 3 5 7 9
is 12 20 28 36
Adjust carries1 4 3 1 6
---------------------------------------------------------
Shift to the right one decimal place and multiply by 3:
3Times 3 5 7 9
is 9 15 21 27
Adjust carries 1 0 7 3 7
---------------------------------------------------------
Add results1 5 3 8 9 7
(Adjust carries if necessary)

11. Divide75159 by 3579:
Set the rods of the divisor 3579,
giving tables of 1 (x) 3579 to 9 (x)
3579 or 10 (x) 3579 to 90 (x) 3579.
First digit: 10 (x) 3579 to 90 (x) 3579
Row 2: 20 (x) 71580
Row 3: 30 (x) 107370 meaning the
dividend is between 20 and 30.
The first digit is 2. Subtract:
75159
(-)71580
3579 <- The remainder is
now the dividend.
Second Digit: The table now used as
1 (x) 3579 to 9 (x) 3579
Row 1: 3579, so the second digit is 1,
and the solution is 21.