About Pascaline
Description
Pascal's Adder or Pascaline, or Pascale or Pascalene,
is 36 cm long, 13 cm wide and 8 cm high. As one can figure, Pascaline has
the size of a shoe box and that's why it can be easily carried. The surface
is metal. There are eight windows on the top of it. Inside each of the
windows, one can see a small drum with the result digits designed on it.
Each drum has two rows of numbers. One black row under one red row. The
black row of numbers is dealing with addition and the red row is dealing
with subtraction. Every time that an addition is taking place one horizontal
slat is covering the red row of the numbers. The black row of numbers is
being covered when one wants to make a subtraction. You will be able to
see how is this happening later in this document. In front of the windows
there are eight setting mechanisms. Those mechanisms help the user enter
the number that he wants to compute. They have the appearance of a wheel.
The spokes of the wheels are turning around the axle of the wheel. The
periphery of the wheel is attached on the machine and it is inscribed with
the digits. The first wheel from the right has twelve spokes, the second
has twenty, and the rest of the wheels have ten spokes each. The first
wheel serves for adding the derniers, the second for adding the
sols and the rest of the wheels to add the livres (at that
time the system of English currency was still in use in France). If someone
wants to add numbers which serve purposes other than the national currency
of France, then he has to use only the six left wheels of the machine.
Although Pascaline was designed for adding French livers, Pascal also designed
machines that they don't have setting mechanisms for the derniers and
the sols, machines that are adding five digit numbers, six digit
numbers and eight digit numbers. The reader is able to interact
with a Pascaline able to add six digit numbers without derniers
and sols.
Depending on the place value of the amount to be added, the wheels
were to be moved by as many tooth positions as corresponded to the value
of the digit concerned. This turned the gears in the interior of the machine
and caused the resultant sum to appear in the windows. According to today's
view, the principle, whose discovery was unquestionably Pascal's achievement,
was not properly carried through. Nevertheless, his invention must be regarded
as the basis of innumerable adding machine systems of a later date. Even
today a few Pascal's machines are in existence. That's why we are in position
to know how a Pascaline is working. The first written document about Pascaline
can be found in Diderot's Encyclopaedie (1751).
Inner Workings
Each wheel of the top of pascaline has one axle that has one horizontal
crown type gear. The gear is transmitting the rotation of the wheel to
a vertical crown type gear. That gear is attached to an horizontal axle
that has two additional crowntype gears. One that is the main gear and
the third gear that is a little smaller than the other two and it is linking
the base axle to the the numbered drums. The three gears are in a row from
the front to the back. . The mechanism also included a weighted ratchet
between the main gears that pushes the next in the row gear whenever there
is a carry from the previous gear.
Addition
Addition is performed easily. You just have to follow the next steps. All
the next examples are using a six digit machine without derniers and
sols.

Move the horizontal slat up to cover the the red row of digits.

Eliminate the value from a preceding operation that may still remain, by
rotating with the hand the numbered drums inside the small windows of the
machine to zero.

Dial the numbers in, and the result will appear in the windows on the surface
of the machine.
Example:
Lets say that you want to add number 20 and number 81.
To dial 20, you just have to put your finger into the space between
the spokes next to digit 2 and before digit 3 of the second wheel, and
rotate the wheel until your finger strikes against the fixed stop on the
bottom of the wheel. This rotation transmits the value of two into the
second window from the right. So now the machines is displaying number
000020.
To dial 81, put your finger into the space between the spokes next
to digit 8 and before digit 9 of the second wheel and rotate it. After
the second drum will reach number 9 the gears inside Pascaline will carry
to the next drum one unit and the third drum of the machine will rotate
by one tenth. So after the end of the dialing of number 8 (the first digit
of the second number to be added) the machine is displaying number 100.
Now put your finger into the space between the spokes next to digit 1 and
before the digit 2 of the first wheel and rotate it the same way you did
before. Now the machine is displaying number 000101 which is the
final result of the addition you wanted to make.
Subtraction
Pascaline was really good only for basic addition. Subtraction was a rather
tedious procedure. As Pascal designed the device, the gears could only
rotate in one direction. That meant that subtraction couldn't be performed
by rotating the wheel of the machine to the opposite direction, but it
had to be carried out by a roundabout method known as nines complements,
which is performed by rotating the wheels to the same direction. The nines
complements method is an ancient trick that performs subtraction into a
form of addition.
To perform a subtraction with that method you have to follow the next
steps:

Move the horizontal slat down to cover the the black row of digits. Now
a new set of numbers is revealed on the drums  the nines complements.

Eliminate the value from a preceding operation that may still remain, by
rotating with the hand the numbered drums inside the small windows of the
machine to zero, the same way you did in addition.

Dial the subtrahend. This produces the subtrahend's nines complement.

Return the horizontal slat up to cover the nines complements row of digits.

Add the subtrahend's nines complement number with the minuend.

Finally, perform mentally an endaround carry, adding the leftmost
digit of the result of the last addition, to the number that arises from
the other digits of the number (except the first one).
You are probably confused with the previous procedure. Read the following
example to understand better subtraction with nines complements method.
Example:
Say you want to subtract 20 from 50. Pull down the horizontal
slat to reveal the nines complements digits. Clear the previous result
as you did before in addition. Dial in (as you did in addition) 20.
This produces a nines complement of 79, which is the difference
between 20 and 99. Return the horizontal slat to its original
position and add 50 and 79. This produces number 129.
Now you mentally perform the endaround carry, adding the first
digit of 129, which is 1, to the number that arises from
the other digits of 129 (except 1), which is 29. This
is resulting 30 which is the correct result of the subtraction that
you wanted to perform.
Multiplication
Even multiplication is possible with Pascaline with an efficient way. The
following procedure will show you how you can achieve that.

Move the horizontal slat up to cover the the red row of digits.

Eliminate the value from a preceding operation that may still remain, by
rotating with the hand the numbered drums inside the small windows of the
machine to zero.

Add the multiplicand as many times as the first digit from the right of
the multiplier says.

Repeat step 3 as many times as the digits of the multiplier are, using
each time the next wheel in the row from the right as you move on to the
next digit.
Example:
Say you want to multiply number 21, 23 times. Enter 21,
3 times starting from the first wheel from the right. Then Enter
21, 2 times starting with the second wheel of the machine
from the right. The correct answer is now displayed in the windows of the
machine.
Division
As for division, Pascaline accomplished it, maddeningly enough, by repeated
subtraction.
Conclusion
Pascaline's mechanism was very promising but in practice was very complicated
and the weighted ratchets have a tendency to jam.
Except that the ratchets didn't let the gears to rotate in both directions,
so subtraction had to be implemented with nines complements trick, which
is not so elegant way to perform a subtraction.
In addition as you can see from the previous examples, Pascaline is
not a calculating machine, but actually just an adding machine. In fact
all similar machines with stylus setting mechanisms should be called adding
machines rather than calculating machines.
Despite all that Pascaline was a historical achievement and it is considered
today as the first calculating device that the human race has discovered.