## Description

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 crown-type 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 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.
1. Move the horizontal slat up to cover the the red row of digits.
2. 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.
3. 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:
1. 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.
2. 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.
3. Dial the subtrahend. This produces the subtrahend's nines complement.
4. Return the horizontal slat up to cover the nines complements row of digits.
5. Add the subtrahend's nines complement number with the minuend.
6. Finally, perform mentally an end-around 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 end-around 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.
1. Move the horizontal slat up to cover the the red row of digits.
2. 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.
3. Add the multiplicand as many times as the first digit from the right of the multiplier says.
4. 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.