Pascaline


The Pascaline is a mechanical calculator invented by Blaise Pascal in 1642. Pascal was led to develop a calculator by the laborious arithmetical calculations required by his father's work as the supervisor of taxes in Rouen, France. He designed the machine to add and subtract two numbers and to perform multiplication and division through repeated addition or subtraction.
There were three versions of his calculator:
one for accounting, one for surveying, and one for science.
The accounting version represented the livre which was the currency in France at the time. The next dial to the right represented sols where 20 sols make 1 livre. The next, and right-most dial, represented deniers where 12 deniers make 1 sol.
Pascal's calculator was especially successful in the design of its carry mechanism, which carries 1 to the next dial when the first dial changes from 9 to 0. His innovation made each digit independent of the state of the others, enabling multiple carries to rapidly cascade from one digit to another regardless of the machine's capacity. Pascal was also the first to shrink and adapt for his purpose a lantern gear, used in turret clocks and water wheels. This innovation allowed the device to resist the strength of any operator input with very little added friction.
Pascal designed the machine in 1642. After 50 prototypes, he presented the device to the public in 1645, dedicating it to Pierre Séguier, then chancellor of France. Pascal built around twenty more machines during the next decade, many of which improved on his original design. In 1649, King Louis XIV gave Pascal a royal privilege, which provided the exclusive right to design and manufacture calculating machines in France. Nine Pascal calculators presently exist; most are on display in European museums.
Many later calculators were either directly inspired by or shaped by the same historical influences that had led to Pascal's invention. Gottfried Leibniz invented his Leibniz wheels after 1671, after trying to add an automatic multiplication feature to the Pascaline. In 1820, Thomas de Colmar designed his arithmometer, the first mechanical calculator strong enough and reliable enough to be used daily in an office environment. It is not clear whether he ever saw Leibniz's device, but he either re-invented it or utilized Leibniz's invention of the step drum.

History

Blaise Pascal began to work on his calculator in 1642, when he was 18 years old. He had been assisting his father, who worked as a tax commissioner, and sought to produce a device which could reduce some of his workload. Pascal received a Royal Privilege in 1649 that granted him exclusive rights to make and sell calculating machines in France.
By 1654 he had sold about twenty machines, but the cost and complexity of the Pascaline was a barrier to further sales and production ceased in that year. By that time Pascal had moved on to the study of religion and philosophy, which resulted in both the Lettres provinciales and the Pensées.
The tercentenary celebration of Pascal's invention of the mechanical calculator occurred during World War II when France was occupied by Germany and therefore the main celebration was held in London, England. Speeches given during the event highlighted Pascal's practical achievements when he was already known in the field of pure mathematics, and his creative imagination, along with how ahead of their time both the machine and its inventor were.

Operation

The calculator had spoked metal wheel dials with the digits 0 through 9 displayed around the circumference of each wheel. To input a digit, the user placed a stylus in the corresponding space between the spokes and turned the dial clockwise until the metal stop was reached, similar to the way the rotary dial of a telephone is used. This displayed the number in the accumulator at the top of the calculator. Then, one simply dialed the second number to be added, causing the sum of both numbers to appear in the accumulator.
Each dial is associated with a one-digit display window located directly above it, which displays the value of the accumulator for this position. The complement of this digit, in the base of the wheel, is displayed just above this digit. A horizontal bar hides either all the complement numbers when it is slid to the top, or all the accumulator numbers when it is slid toward the center of the machine.
Since the gears of the calculator rotated in only one direction, only addition operations could be performed. However, to subtract one number from another, the method of [|nine's complement] was used. The only two differences between an addition and a subtraction are the position of the display bar and the way the first number is entered.
For a 10-digit wheel, the fixed outside wheel is numbered from 0 to 9. The numbers are inscribed in a decreasing manner clockwise going from the bottom left to the bottom right of the stop lever. To add a 5, one must insert a stylus between the spokes that surround the number 5 and rotate the wheel clockwise all the way to the stop lever. The number displayed on the corresponding display register will be increased by 5 and, if a carry transfer takes place, the display register to the left of it will be increased by 1. To add 50, use the tens input wheel, to add 500, use the hundreds input wheel, etc...
On all the wheels of all the known machines, except for the machine tardive, two adjacent spokes are marked; these marks differ from machine to machine. On the wheel pictured on the right, they are drilled dots, on the surveying machine they are carved; some are just scratches or marks made with a bit of varnish, some were even marked with little pieces of paper.
These marks are used to set the corresponding cylinder to its maximum number, ready to be re-zeroed. To do so, the operator inserts the stylus in between these two spokes and turns the wheel all the way to the stopping lever. This works because each wheel is directly linked to its corresponding display cylinder. To mark the spokes during manufacturing, one can move the cylinder so that its highest number is displayed and then mark the spoke under the stopping lever and the one to the right of it.
Four of the known machines have inner wheels of complements, which were used to enter the first operand in a subtraction. They are mounted at the center of each spoked metal wheel and turn with it. The wheel displayed in the picture above has an inner wheel of complements, but the numbers written on it are barely visible. On a decimal machine, the digits 0 through 9 are carved clockwise, with each digit positioned between two spokes so that the operator can directly inscribe its value in the window of complements by positioning his stylus in between them and turning the wheel clockwise all the way to the stop lever. The marks on two adjacent spokes flank the digit 0 inscribed on this wheel.
On four of the known machines, above each wheel, a small quotient wheel is mounted on the display bar. These quotient wheels, which are set by the operator, have numbers from 1 to 10 inscribed clockwise on their peripheries. Quotient wheels seem to have been used during a division to memorize the number of times the divisor was subtracted at each given index.

Inner mechanism

Pascal went through 50 prototypes before settling on his final design; we know that he started with some sort of calculating clock mechanism which apparently "works by springs and which has a very simple design", was used "many times" and remained in "operating order". Nevertheless, "while always improving on it" he found reason to try to make the whole system more reliable and robust. Eventually he adopted a component of very large clocks, shrinking and adapting for his purpose the robust gears that can be found in a turret clock mechanism called a lantern gear, itself derived from a water wheel mechanism. This could easily handle the strength of an operator input.
Pascal adapted a pawl and ratchet mechanism to his own turret wheel design; the pawl prevents the wheel from turning counterclockwise during an operator input, and also acts as a detent to precisely position the display wheel and the carry mechanism for the next digit when it is pushed up and lands into its next position. Because of this mechanism, each number displayed is perfectly centered in the display window and each digit is precisely positioned for the next operation. This mechanism would be moved six times if the operator dialed a six on its associated input wheel.

Carry mechanism

The sautoir is the centerpiece of the Pascaline's carry mechanism. In his "Avis nécessaire...", Pascal noted that a machine with 10,000 wheels would work as well as a machine with two wheels because each wheel is independent of the other. When it is time to propagate a carry, the sautoir, under the sole influence of gravity, is thrown toward the next wheel without any contact between the wheels. During its free fall the sautoir behaves like an acrobat jumping from one trapeze to the next without the trapezes touching each other. All the wheels have therefore the same size and weight independently of the capacity of the machine.
Pascal used gravity to arm the sautoirs. One must turn the wheel five steps from 4 to 9 in order to fully arm a sautoir, but the carry transfer will move the next wheel only one step. Thus, much extra energy is accumulated during the arming of a sautoir.
All the sautoirs are armed by either an operator input or a carry forward. To re-zero a 10,000-wheel machine, if one existed, the operator would have to set every wheel to its maximum and then add a 1 to the "unit" wheel. The carry would turn every input wheel one by one in a very rapid Domino effect fashion and all the display registers would be reset.
The carry transmission has three phases:
  1. The first phase happens when the display register goes from 4 to 9. The two carry pins lift the sautoir pushing on its protruding part marked. At the same time the kicking is pulled up, using a pin on the receiving wheel as guidance, but without effect on this wheel because of the top. During the first phase, the active wheel touches the one that will receive the carry through the sautoir, but it never moves it or modifies it and therefore the status of the receiving wheel has no impact whatsoever on the active wheel.
  2. The second phase starts when the display register goes from 9 to 0. The kicking pawl passes its guiding pin and its positions it above this pin ready to push back on it. The sautoir keeps on moving up and suddenly the second carry pin drops it. The sautoir falls of its own weight. During the second phase, the sautoir and the two wheels are completely disconnected.
  3. The kicking pushes the pin on the receiving wheel and starts turning it. The upper is moved to the next space. The operation stops when the protruding hits the. The upper positions the entire receiving mechanism in its proper place. During the third phase the sautoir, which no longer touches the active wheel, adds one to the receiving wheel.