Concept One: the carry
Pascal designed the first mechanical adding machine in 1642. The most important feature of his adding machine was the carry. The carry involves raising the value of the next digit when the current digit passes from 9 to 0. The carry could easily have been implemented with a long tooth designed to rotate the next gear as the current gear passes from 9 to 0. However if fifty gears were all lined up in the 9 position every gear's long tooth would be ready to turn the next gear. A person would have to apply the force necessary to turn fifty gears to turn the first gear from 9 to 0.
Pascal's solution was have each increment of a gear raise a weight by a small amount. When the gear passed from 9 to 0 the weight was released. It was the falling of the weight that turned the next gear producing the carry. Pascal emphatically claimed with this method "it is just as easy to move one thousand or ten thousand dials, all at the same one time".
Concept Two: the multiplicand teeth
In 1671 Gottfried Wilhelm, Freiherr von Leibniz constructed an improvement to Pascal's machine which could perform multiplication. His machine worked with variable toothed multiplicand gears and variable sized multiplier discs. He built his machine on top of Pascal's assuming Pascal's carry mechanism would take care of all caries.
The multiplicand gears are set to have the number of teeth corresponding to the digit they represent. This means that as each multiplicand gear rotates once, the number of teeth it has will catch the corresponding gear in Pascal's adding machine incrementing it the digit the multiplicand gear represents. So if the multiplicand was set to 23, a two and a three toothed gear, turning each gear once will produce 23 on Pascal's machine. Turning each gear twice would produce 46 or 2 * 23.
Concept Three: the multiplier ratio
Each multiplier disc has a different size corresponding to the digit it represents. The 1 multiplier disc has a diameter the same size as the pulley disc it is connected to. Thus every time it is rotated the pulley disc will rotate once. The 2 multiplier disc is has a diameter twice the size of the pulley disc. The 3 disc has diameter three times the size and so on.
Multiplication is performed by connecting one of the multiplier discs to the multiplicand gears. If the 2 multiplier disc is chosen it will rotate the multiplicand gear twice for every time it rotates. Every time the multiplicand gear rotates it will produce its result on Pascal's machine. Thus if the multiplicand gear is 3, having three teeth, the multiplier disc will rotate it twice incrementing Pascal's addition gear three times for each rotation. The result will be six.
For the multiplier it is the ratio between the multiplicand disc and the pulley disc that creates the effect of multiplication.
Directions:
1. Press 'Start' Button.
2. Select 'hand' tool with the mouse.
3. Click on numbers to increment them.
4. Click on gears to rotate them.
5. Multiplication:
- Select 'Arrow' tool.
- Drag a disc from the bottom to in front of one of the pulleys in the middle.
- Select 'Hand Tool'.
- Click on attached disc to perform a multiply.
- Click on left and right arrows to shift the multiplicand's position.
6. Select the 'hammer' tool and click the word 'counter' to reset the machine.
Addition
Addition is performed by simply incrementing each of the digits at the top to the first number you would like to add. If the number was 23 you would click the tens digit twice and the ones digit three times. Then increment each place by the number of the place of the second number you would like to add. If this number was 35 you would click the tens digit (currently showing a 2) three more times and the ones place (currently showing a 3) five more times. This can be repeated for as many numbers you are adding. The role of the machine is simply to take care of the carries for you.
Multiplication
Set the variable toothed gears in the middle to the first number you would like to multiply (the multiplicand). Shift the multiplicand to the right, if necessary, so it is under the ones and tens place of the digits across the top. Select the disc at the bottom corresponding to the ones place of the second number you are multiplying (the multiplier), and connect it to one of the multiplicand pulleys. Perform the multiply. Connect the disc to the other multiplicand pulley and perform it's multiply. Return the disc.
Shift the multiplicand to the left. Perform both multiplies with the tens place multiplier disc. Repeat for as many places as are in the multiplier.