346 Napier’s bones

Multiplying with Napier’s bones

where Nis a number and Lis, what he called the “loga-

rithm” of that number. (Napier inserted factors of 107so

as to help avoid the appearance of decimals in computa-

tions.) Napier published his results in 1614 in his piece

Mirifici logarithmorum canonis descriptio (Description

of the marvelous rule of logarithms).

If we divide Napier’s formula through by 107we

obtain:

using the fact that , where eis

Euler’s number. Thus, up to a factor of 107, we see that

Napier’s logarithms are close to the logarithms of

today, but to the base 1/e. Under the advice of English

scholar H

ENRY

B

RIGGS

(1561–1630), Napier later

modified his method of logarithms to base 10.

Napier invented mechanical devices to assist in the

computation, multiplication, division, and the extrac-

tion of square and cube roots. He also made significant

contributions to the study of spherical trigonometry,

finding formulae for the ratio of sides of triangles

drawn on spheres. These formulae are today named in

his honor.

Napier died in Edinburgh, Scotland, on April 4,

1617.

See also

E

; N

APIER

’

S BONES

.

Napier’s bones (Napier’s rods) In 1614, Scottish

mathematician J

OHN

N

APIER

designed a set of gradu-

ated rods, now called Napier’s bones, that can be

used to convert all long multiplication problems to

easier, and more swiftly solved, problems of addition.

Each rod, made of bone or ivory, was engraved with

a column of numbers consisting of the multiples of

the digit inscribed at the head of the rod. Diagonal

lines were drawn to separate the tens and the units of

each multiple. A blank rod represented the multiples

of zero.

To compute a long multiplication, such as the

product 3717 ×25, one would line up four rods, one

for each digit 3, 7, 1, and 7, as shown:

Looking at the second and fifth rows (corresponding

to the digits 2 and 5), and adding along the diagonals in

the rows (this corresponds to keeping track of carried

digits), we see that 2 ×3,717 = 7,434 and 5 ×3,717 =

18,585. Adding a zero to the tail of the first product

gives 20 ×3,717 = 74,340, and so the product we seek is

the sum of the two numbers 74,340 and 18,585. This

can be swiftly computed with pen and paper.

There is one complication: in looking at any partic-

ular row, two numbers in a diagonal may sum to a

total with two digits. One must carry the first digit of

the sum to the next diagonal to the left. For example,

looking at the eighth row we see that 8 ×3,717 equals

29,736. (The 1 from the sum of 8 and 5 carries to the 6

in the next diagonal to the left.)

See also E

GYPTIAN MULTIPLICATION

; E

LIZABETHAN

MULTIPLICATION

;

FINGER MULTIPLICATION

;

MULTIPLICA

-

TION

; R

USSIAN MULTIPLICATION

.

Nash, John (1928– ) American Game theory, Topol-

ogy Born on June 13, 1928, in Bluefield, West Vir-

ginia, John Nash is remembered for his seminal work in