The early seventeenth century was a time of much gnashing of teeth
in matters of arithmetic. Methods to do long division and
multiplication were not widely known and prone to errors. Moreover, in
the years following Nicolaus
"we're not in the middle anymore" Copernicus (1543) and
culminating in Isaac
"the force binds the universe inversely as the square of the
distance" Newton (1687) science had transformed itself from a
metaphysical discussion between philosophers to a quantitative,
experimental and mathematical discipline. And in doing so, generated
an increasing demand for quick and efficient ways to do
arithmetic.

Enter John
Napier, the Laird of Merchiston. Basically, your average eccentric
Scottish lord with a castle in a large estate located in what is now
Edinburgh. Today there's even a Napier's University centered
around where he lived (scroll down for pictures of Merchiston
Tower.)

He obsessed about religion and wrote a book called *A Plaine
Discovery of the Revelation of St John* where he discovered
patterns in the numbers in the Book of Revelations. What did the
patterns say? That the Pope was the Antichrist (Johnny himself was a
fervent Protestant and Calvinist.) Oh, and that the world
would end sometime between 1688 and 1700, so his theological
talents seem more modest than his mathematical ones.

For all his obsessions, he sounds like a canny laddie. A story
relates that he told his servants his black cockerel was going to
discover which of them had been stealing from him. They were told to
go one by one into a dim room and stroke the bird, and it would crow
when the guilty one touched it. The bird remained silent, but he
identified the thief anyway. He'd secretly put soot on the bird, and
the guilty one was the only one who emerged with clean hands, being
too afraid to touch the bird.

Napier toiled away at another hobby, mathematics, primarily working
to find better ways to do computations. Training in arithmetic being
about as spotty as it is today, he'd come up with a cute trick using
rods
inscribed with numbers to reduce multiplication and division to
addition and subtraction. It became very popular for a while, and kept
making encore appearances through the nineteenth century.

He wrote a best-seller on it in 1617 calling it the art of *Rabdology*, a
word he made up from the Greek for rod
(ραβδoς) and reckoning
(λoγoς.) Everyone simply called it Napier's bones
because of the way the rods often made of ivory looked.

All of which is interesting, but it gets even better. For, tucked
away as a throwaway section at the end of Rabdology, he described
nothing less than a primitive mechanical binary calculator.

Take a chessboard or a similar checkered grid and some counters,
and you've got yourself a binary adder, multiplier, divider, square
root finder.

Here's how you add binary numbers. Place the numbers you want to
add, one on each row. The counters are going to stand for the 1's in
the binary number. For example, 18 which is 10010 in binary, has a
counter on the second and fifth squares from the right.

`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|...|...|...|.X.|...|...|.X.|...| 18

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|.X.|.X.|...|.X.|.X.| 27

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|...|.X.|.X.|...|.X.| 13

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

Move the counters vertically down to the bottom row like you would
move a rook in chess.
`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...| 18

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...| 27

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...| 13

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|.X.|.X.|...|.X.|.X.|

|...|...|...|.X.|.X.|.X.|.X.|.X.|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

Okay. Now if you've got two (or more) counters on a square, remove two
counters and place one counter to its left. For example,
remove two counters from the rightmost square
`
+---+---+---+---+---+---+---+---+`

|...|...|...|.X.|.X.|...|.X.|.**X**.|

|...|...|...|.X.|.X.|.X.|.X.|.**X**.|

|...|...|...|...|...|...|..**<--**..|

+---+---+---+---+---+---+---+---+

and put one counter to its left giving
`
+---+---+---+---+---+---+---+---+`

|...|...|...|.X.|.X.|...|.X.|...|

|...|...|...|.X.|.X.|.X.|.X.|...|

|...|...|...|...|...|...|.**X**.|...|

+---+---+---+---+---+---+---+---+

That's easy enough. Let's take two from the last but one square on the
left and put one to its left.
`
+---+---+---+---+---+---+---+---+`

|...|...|...|.X.|.X.|.X.|...|...|

|...|...|...|.X.|.X.|.X.|.X.|...|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

You get the idea. What you want to do is end up with no more than one
counter on each square. I won't bother showing all the intermediate
moves, and here's what happens when you finish up.
`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|...|...|.X.|.X.|.X.|...|.X.|...|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

And there's your answer -- 111010, or binary speak
for 58 = 18 + 27 + 13.
Okay, that was nice. How about multiplication? Mark one number
along the right edge of the board, and the other number along the
bottom. Let's try 18 times 13 or 10010*1101

`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|...|...|...|.X.|...|...|.X.|...| 1

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|.X.|...|...|.X.|...| 1

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|...|...|...|...|...| 0

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|...|...|...|...|...|

|...|...|...|.X.|...|...|.X.|...| 1

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

..............1...0...0...1...0

Also place counters along every "intersection" of the
columns and rows that have a 1; you can see how they should be placed
above.
Now move the counters "diagonally" to the bottom row like you would
move a bishop.

`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|...|...|...|./.|...|...|./.|...|

|...|...|...|/..|...|...|/..|...|

+---+---+---+---+---+---+---+---+

|...|...|../|...|...|../|...|...|

|...|...|./.|./.|...|./.|./.|...|

|...|...|/..|/..|...|/..|/..|...|

+---+---+---+---+---+---+---+---+

|...|../|../|...|../|../|...|...|

|...|./.|./.|...|./.|./.|...|...|

|...|/..|/..|...|/..|/..|...|...|

+---+---+---+---+---+---+---+---+

|...|...|...|.X.|...|...|...|...|

|.X.|.X.|...|.X.|.X.|...|.X.|...|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

And now do the same thing as in addition. Replace two counters on a
square with one to its left. There's only one square to fix up, so we
end up with
`
+---+---+---+---+---+---+---+---+`

|...|...|...|...|...|...|...|...|

|.X.|.X.|.X.|...|.X.|...|.X.|...|

|...|...|...|...|...|...|...|...|

+---+---+---+---+---+---+---+---+

and also the answer, 11101010 which is 234 = 18*13.
Pretty nifty.

Why does this work? And how do you do division and square roots?
The best technical information I've read is in a book by Martin
Gardner called "Knotted
doughnuts and other mathematical entertainments" (ISBN 0716717948.)
Not only does he explain the chessboard calculator, but he shows other
neat ideas like using it to multiply negabinary
numbers and so on. Napier's algorithms themselves are related to
an old multiplication and division technique often called Russian
Peasant arithmetic. There's also an English
translation of Rabdology which includes an interesting
introduction to the historical context of Napier's work.

However I found surprisingly few good technical resources about
this on the web. You can play with an online
chessboard calculator, and I've slowly been adding to a Wikipedia
article on location
arithmetic, the term Napier used for his invention. There is also
a very
large (117Mb) PDF scan of Rabdology in the original Latin.

All said and done, Napier's chessboard calculator never really took
off. Maybe everyone figured that the fuss of converting stuff in and
out of binary wasn't really worth it, I don't know. In fact his own
work on logarithms quickly led to the extensive use of log tables for
arithmetic and the development of the slide rule, all of which
sidelined even the popular Napier's bones. The chessboard calculator
and binary arithmetic seem to have been about three centuries ahead of
their time.