Blaise Pascal invented the mechanical calculator in 1642.[1][2] He conceived the idea while trying to help his father who had been assigned the task of reorganizing the tax revenues of the French province of Haute-Normandie ; first called Arithmetic Machine, Pascal's Calculator and later Pascaline, it could add and subtract directly and multiply and divide by repetition.
Pascal went through 50 prototypes before presenting his first machine to the public in 1645. He dedicated it to Pierre Séguier, the chancellor of France at the time.[3] He built around twenty more machines during the next decade, often improving on his original design. Nine machines have survived the centuries,[4] most of them being on display in European museums. In 1649 a royal privilege, signed by Louis XIV of France,[5] gave him the exclusivity of the design and manufacturing of calculating machines in France.
Its introduction launched the development of mechanical calculators in Europe first and then all over the world, development which culminated, three centuries later, in the invention of the microprocessor developed for a Busicom calculator in 1971.
The mechanical calculator industry owes a lot of its key machines and inventions to the pascaline. First Gottfried Leibniz invented his Leibniz wheels after 1671 while trying to add an automatic multiplication and division feature to the pascaline,[6] then Thomas de Colmar drew his inspiration from Pascal and Leibniz when he designed his arithmometer in 1820, and finally Dorr E. Felt substituted the input wheels of the pascaline by columns of keys to invent his comptometer around 1887. The pascaline was also constantly improved upon, especially with the machines of Dr. Roth around 1840, and then with some portable machines until the creation of the first electronic calculators.
Contents |
A short list of precursors to the mechanical calculator must include the Antikythera mechanism from around 100 BC, early mechanical clocks and geared astrolabes ; they were all made of toothed gears linked by some sort of carry mechanisms.
Some measuring instruments and automatons were also precursors to the calculating machine.
An odometer, instrument for measuring distances, was first described around 25 BC by the roman engineer Vitruvius in the tenth volume of his De architectura. It was made of a set of toothed gears linked by a carry mechanism ; the first one was driven by one of the chariot wheels and the last one dropped a small pebble in a bag for each Roman mile traveled.[7]
A Chinese text of the third century AD described a chariot equipped with a geared mechanism that operated two wooden figures. One would strike a drum for every Chinese Li traveled, the other one would strike a gong for every ten Li traveled.[8]
Around the end of the tenth century, the French monk Gerbert d'Aurillac, whose abacus taught the Hindu-Arabic numeral system to the Europeans,[9] brought back from Spain the drawings of a machine invented by the Moors that answered Yes or No to the questions it was asked (binary arithmetic) ; but its existence is contested.[10]
Again in the thirteenth century, the monks Albertus Magnus and Roger Bacon built talking heads made of earthware without any further development (Albertus Magnus complained that he had wasted forty years of his life when Thomas Aquinas, terrified by his machine, destroyed it[11]).
The Italian polymath Leonardo da Vinci drew an odometer before 1519.
In 1525, the French craftsman Jean Fernel built the first pedometer. It was made in the shape of a watch and had 4 dials (units, tens, hundreds, thousands) linked by a carry mechanism.[12]
From the introduction of the Pascaline and for more than three centuries Pascal was known as the inventor of the mechanical calculator, but then, in 1957, Dr. Franz Hammer, an expert in Johannes Kepler's work, challenged this fact by announcing that the drawings of a previously unknown calculating clock, predating Pascal's work by twenty years, had been found in two letters that Wilhelm Schickard had written to his friend Johannes Kepler in 1623 and 1624.[13] This machine was destroyed in a fire in 1624 as it was being built by Johann Pfister, a local craftsman, and Schickard abandoned his project the same year.[14]
Dr. von Freytag Loringhoff, a mathematics professor at the University of Tübingen built the first replica of Schickard's machine but he had to improve on the design of the carry mechanism:
This simple-looking device actually presents a host of problems to anyone attempting to construct an adding machine based on this principle. The major problem is caused by the fact that the single tooth must enter into the teeth of the intermediate wheel, rotate it 36 degrees (one tenth of a revolution), and exit from the teeth, all while only rotating 36 degrees itself. The most elementary solution to this problem consists of the intermediate wheel being, in effect, two different gears, one with long and one with short teeth together with a spring-loaded detente (much like the pointer used on the big wheel of the gambling game generally known as Crown and Anchor) which would allow the gears to stop only in specific locations. It is not known if Schickard used this mechanism, but it certainly works well on the reproductions constructed by von Freytag Loringhoff.—Michael R. Williams[15], History of Computing Technology, IEEE (1997)
Without this twentieth century improvement in the carry mechanism, which is not described in any of Schickard's letters or drawings, the replicas would not have worked. Another problem was found after the replicas were built:
... it is almost certain that Pascal would not have known of Schickard's machine ...
Pascal seems to have realized right from the start that the single-tooth gear, like that used by Schickard, would not do for a general carry mechanism. The single-tooth gear works fine if the carry is only going to be propagated a few places but, if the carry has to be propagated several places along the accumulator, the force needed to operate the machine would be of such magnitude that it would do damage to the delicate gear works.—Michael R. Williams[16], History of Computing Technology, IEEE (1997)
Schickard's machine used clock wheels which were made stronger and were therefore heavier, to prevent them from being damaged by the force of an operator input. Each digit used a display wheel, an input wheel and an intermediate wheel. During a carry transfer all these wheels meshed with the wheels of the digit receiving the carry. The cumulative inertia of all these wheels could "...potentially damage the machine if a carry needed to be propagated through the digits, for example like adding 1 to a number like 9,999".[17]
In Pascal's calculator each input wheel is totally independent from all the others. Pascal chose, for his machine, a method of re-zeroing that propagates a carry right through the machine.[18] It is the most demanding operation to execute for a mechanical calculator and proved, before each operation, that the Pascaline was fully functional. This is a testament to the quality of the Pascaline because none of the 18th century criticisms of the machine mentioned a problem with the carry mechanism and yet this feature was fully tested on all the machines, by their resets, all the time.[19]
Even though Schickard designed his machine twenty years earlier, Pascal is still the inventor of the mechanical calculator because the drawings of Schickard's calculating clock described a machine that was neither complete nor fully usable.
Just as the Wright brothers were credited for the first flight and therefore the invention of the airplane while Clément Ader flew 13 years before them, and Thomas Edison was credited with the invention of the incandescent light bulb while ten people had already worked on it before, Blaise Pascal is credited as the inventor of the mechanical calculator because he was the first person to present a machine that had all the parts required for its use, that had adequate solutions to all its challenges, a primitive machine, complete and ready to evolve.
Johannes Kepler, the recipient of Schickard's letters, died before Pascal's calculator was released to the public but he witnessed a similar duality of early precursor versus inventor with the invention of Logarithms:
...the accent in calculation led Justus Byrgius on the way to these very logarithms many years before Napier's system appeared; but ...instead of rearing up his child for the public benefit he deserted it in the birth.—Johannes Kepler[20], Rudolphine Tables (1627)
Discovered in the second half of the twentieth century, Schickard's calculating clock had no influence on the development of mechanical calculators.[21]
Besides being the first calculating machine made public during its time, the pascaline is also:
"...The first arithmetic machine presented to the public was from Blaise Pascal, born in Clermont, Auvergne on June 19, 1623 ; he invented it at the age of 19. Other machines have been designed since which, in the judgement of Mr of the Academy of Sciences, seem to have more practical advantages ; but Pascal's machine is the oldest one ; it could have served as model to all the others ; this is why we preferred it."[23]
"Mr de Roberval ... located in the college Maitres Gervais ... every morning until 8..."[24]
Pascal began to work on his calculator in 1642, when he was only 19 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 Pascal 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 gave us both the Lettres provinciales and the Pensées.
Pascal's genius and his machine have been celebrated all around the world for centuries.
Pascal, mad genius, born a century too early
Blaise Pascal was, simply, one of the greatest men that have ever lived. Having made the discovery of mathematics at the age of twelve, at sixteen he wrote a treatise on conic sections which is the herald of modern projective geometry. At nineteen he invented, constructed, and offered for sale the first calculating machine. He gave Pascal's Law to physics, proved the existence of the vacuum, and helped to establish the science of hydrodynamics. He created the mathematical theory of probability, in a discussion of the division of gamblers' stakes. His speculations were important in the early development of the infinitesimal calculus. After a night of religious revelation, when he was but thirty-one, he abandoned science, returning to it only to solve, as a diversion from the toothache, the problems of the cycloid. Espousing the theological principles of the Jansenists, he wrote, in their defense, the Lettres provinciales, a controversial weapon which has not yet lost its edge. His prose style, novel in its strong simplicity, determined the shape and character of the French literary language. He devised a new method of teaching reading. He organized the first omnibus line. In the lucid moments of cruel illness, he wrote his Pensées, in preparation for an apology for Christianity, thoughts which have affected the mental cast of three centuries, thoughts which still stir and work and grow in modern minds. He died at 39.—Morris Bishop[27], Pascal, The life of genius, New York, (1936)
...at the age of nineteen, when he invented the calculating machine - essentially a practical instrument - Pascal was already well known in the realm of pure mathematics. His perseverance with this invention helped one to appreciate more fully the character of Pascal. He had to fight not only ill-health but also the ignorance of his time, for his conception far out-stripped the mechanical experience and ability of those to whom the work was entrusted. It was not until Pascal had made more than fifty models that he achieved his final design.
The invention of the calculating machine illustrated Pascal's extraordinary creative imagination, allied with mathematical genius and precision, and tempered with critical penetration. These qualities were characteristic of the man throughout his life.—Prof. René Cassin[28], Pascal tercentenary celebration, London, (1942)
Pascal's invention of the calculating machine, just three hundred years ago, was made while he was a youth of nineteen. He was spurred to it by seeing the burden of arithmetical labor involved in his father's official work as supervisor of taxes at Rouen. He conceived the idea of doing the work mechanically, and developed a design appropriate for this purpose ; showing herein the same combination of pure science and mechanical genius that characterized his whole life. But it was one thing to conceive and design the machine, and another to get it made and put into use. Here were needed those practical gifts that he displayed later in his inventions....
In a sense, Pascal's invention was premature, in that the mechanical arts in his time were not sufficiently advanced to enable his machine to be made at an economic price, with the accuracy and strength needed for reasonably long use. This difficulty was not overcome until well on into the nineteenth century, by which time also a renewed stimulus to invention was given by the need for many kinds of calculation more intricate than those considered by Pascal.—S. Chapman[29], Pascal tercentenary celebration, London, (1942)
Pascalines came in both decimal and non-decimal varieties, both of which exist in museums today. They were designed to be used by scientists, accountants and surveyors. The simplest Pascaline had five dials ; later production variants had up to ten dials.
The contemporary French currency system used livres, sols and deniers with 20 sols to a livre and 12 deniers to a sol. Length was measured in toises, pieds, pouces and lignes with 6 pieds to a toise, 12 pouces to a pied and 12 lignes to a pouce. Therefore the pascaline needed wheels in base 6, 10, 12 and 20. Non decimal wheels were always located before the decimal part.
In an accounting machine (..10,10,20,12), the decimal part counted the number of livres (20 sols), sols (12 deniers) and deniers. In a surveyor's machine (..10,10,6,12,12), the decimal part counted the number of toises (6 pieds), pieds (12 pouces), pouces (12 lignes) and lignes. Scientific machines just had decimal wheels.
| Machine Type | All the other wheels | 4th wheel | 3rd wheel | 2nd wheel | 1st wheel |
|---|---|---|---|---|---|
| Decimal / Scientific |
base 10 Ten thousands... |
base 10 Thousands |
base 10 Hundreds |
base 10 Tens |
base 10 Units |
| Accounting | base 10 Hundreds... |
base 10 Tens |
base 10 Livres |
base 20 Sols |
base 12 Deniers |
| Surveying | base 10 Tens ... |
base 10 Toises |
base 6 Pieds |
base 12 Pouces |
base 12 Lignes |
| The decimal part of each machine is highlighted in yellow | |||||
The metric system was adopted in France on December 10, 1799 by which time Pascal's basic design had inspired other craftsmen, although with a similar lack of commercial success. Child prodigy Gottfried Wilhelm Leibniz devised a competing design, the Stepped Reckoner, in 1671 which could perform addition, subtraction, multiplication and division; Leibniz struggled for forty years to perfect his design and produce sufficiently reliable machines.
Calculating machines did not become commercially viable until the early 19th century, when Charles Xavier Thomas de Colmar's Arithmometer, itself using the key break through of Leibniz's design, was commercially successful.[30]
Most of the machines that have survived the centuries are of the accounting type. Seven of them are in European museums, one belongs to the IBM corporation and one is in private hands.
| Location |
Country |
Machine Name |
Type |
Wheels |
Configuration |
Notes |
|---|---|---|---|---|---|---|
| CNAM museum Paris |
France | Chancelier Séguier | Accounting | 8 | 6 x 10 + 20 + 12 | |
| CNAM museum Paris |
France | Christina, Queen of Sweden | Scientific | 6 | 6 x 10 | |
| CNAM museum Paris |
France | Louis Périer | Accounting | 8 | 6 x 10 + 20 + 12 | Louis Périer, Pascal's nephew, offered it to the Académie des sciences de Paris in 1711. |
| CNAM museum Paris |
France | Late (Tardive) | Accounting | 6 | 4 x 10 + 20 + 12 | This machine was assembled in the XVIIIth century with unused parts.[31] |
| musée Henri Lecoq Clermont-Ferrand |
France | Marguerite Périer | Scientific | 8 | 8 x 10 | Marguerite (1646–1733) was Pascal's goddaughter.[32] |
| Musée Henri Lecoq Clermont-Ferrand |
France | Chevalier Durant-Pascal | Accounting | 5 | 3 x 10 + 20 + 12 | This is the only known machine that came with a box. This is the smallest machine. Was it meant to be portable? |
| Mathematisch-Physikalischer salon, Dresden | Germany | Queen of Poland | Accounting | 10 | 8 x 10 + 20 + 12 | The second wheel from the right has a wheel with 10 spokes contained in a fixed wheel with 20 segments. This could be attributed to a bad restoration. |
| Léon Parcé collection | France | Surveying | 8 | 5 x 10 + 6 + 12 + 12 | This machine was bought as a broken music box in a French antique shop en 1942. | |
| IBM collection | USA | Accounting | 8 | 6 x 10 + 20 + 12 |
The calculator had spoked metal wheel dials, with the digit 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 until a metal stop at the bottom was reached, similar to the way a rotary telephone dial is used. This would display the number in the boxes at the top of the calculator. Then, one would simply redial the second number to be added, causing the sum of both numbers to appear in boxes at the top. Since the gears of the calculator only rotated in one direction, negative numbers could not be directly summed. To subtract one number from another, the method of nines' complement was used. To help the user, when a number was entered, its nines' complement appeared in a box above the box containing the original value entered[33].
For a 10 digit wheel (N), the fixed outside wheel is numbered from 0 to 9 (N-1). The numbers are inscribed in a decreasing manner clockwise going from the bottom left to the bottom right of the stopping lever. To add a 5, one must insert a stylus in between the spokes that surround the number 5 and rotate the wheel clockwise all the way to the stopping 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 fifty, use the tens input wheel (second dial from the right on a decimal machine), to add 500, use the hundreds input wheel, etc...
On all the wheels of all the known machines, except for the machine tardive[34], 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[35], some were even marked with little pieces of paper[36].
These marks are used to set the corresponding cylinder to its maximum number, ready to be re-zeroed. To do so, the operator must insert the stylus in between these two spokes and turn the wheel all the way to the stopping lever. This works because each wheel is directly linked to its corresponding display cylinder (it automatically turns by one during a carry operation) ; 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. They are mounted at the center of each spoked metal wheel and turn with it. The wheel displayed 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, each digit is positioned in 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 stopping lever.[37] The marks on two adjacent spokes surround 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 wheel, which are set by the operator, have numbers from 1 to 10 inscribed clockwise on their peripheries (even above a non decimal wheel). Quotient wheels seem to have been used during a division to memorize the number of time the divisor was subtracted at each given index.[38]
The sautoir is the center piece of the pascaline's carry mechanism. In his "Avis nécessaire...", Pascal wrote:
... as for the ease of use of this ... movement ... moving one thousand even ten thousand wheels if they were there, each one performing their motion perfectly, is as easy as moving only one (I don't know if, after the principle that I perfected to design this device, there is another one left to be found in nature)...[39]
A machine with 10,000 wheels would work as well as a machine with two wheels because each wheel is independent from the other. When it is time to propagate a carry, the sautoir, on the sole influence of gravity,[40] 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 (sautoir comes from the french verb sauter which means to jump). All the wheels (including gears and sautoir) 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 only move the next wheel one step. Therefore there is a lot of extra energy built up 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 it 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 animation on the right shows the three phases of a carry transmission.
The Pascaline is a direct adding machine (it doesn't have a crank handle) so the value of a number is added to the accumulator as it is being dialed in. By moving a display bar, the operator can either see the number stored in the calculator or the complement of its value. Subtractions are performed like additions by using some properties of 9's complement arithmetic.
The 9's complement of any one digit decimal number d is 9 - d. So the 9's complement of 4 is 5 and the 9's complement of 9 is 0. Similarly the 11's complement of 3 is 8.
In a decimal machine with n dials the 9's complement of a number A is:
and therefore the 9's complement of (A - B) is:
In other words, the 9's complement of the difference of two numbers is equal to the sum of the 9's complement of the minuend added to the subtrahend. The same principle is valid and can be used with numbers composed of digits of various bases (base 6, 12, 20) like in the surveying or the accounting machines.
This can also be extended to:
This principle applied to the pascaline:
| CP(A): | First the complement of the minuend is entered. The operator can either use the inner wheels of complements or dial the complement of the minuend directly. The display bar is shifted to show the complement's window so that the operator sees the direct number displayed because CP(CP(A))= A. |
| B: | Then the second number is dialed in and adds its value to the accumulator. |
| CP(A - B): | The result (A - B) is displayed in the complement window because CP(CP(A - B))= A - B. The last step can be repeated as long as the subtrahend is smaller than the minuend displayed in the accumulator,. |
The machine has to be re-zeroed before each new operation.
To reset his machine, the operator has to set all the wheels to their maximum, using the marks on two adjacent spokes, and then add 1 to the rightmost wheel.[18]
The method of re-zeroing that Pascal chose, which propagates a carry right through the machine, is the most demanding task for a mechanical calculator and proves, before each operation, that the machine is fully functional.
This is a testament to the quality of the Pascaline because none of the 18th century criticisms of the machine mentioned a problem with the carry mechanism and yet this feature was fully tested on all the machines, by their resets, all the time.[19]
| Re-zero | Set all the wheels to their maximum using the marks on two adjacent spokes. Every single wheel is ready for a carry transfer. |
|
||||||||||
| Add 1 to the right-most wheel. Each wheel sends its sautoir to the next one, the zeros appear one after another, like in a domino effect, from right to left. |
|
Additions are performed with the display bar moved closest to the edge of the machine, showing the direct value of the accumulator.
After re-zeroing the machine, numbers are dialed in one after the other.
The following table shows all the steps required to compute: 12,345 + 56,789 = 69,134
| Addition | The machine is at zero, the operator enters 12,345. |
|
||||||||||
| The operator enters the second operand: 56,789. If he starts with the rightmost number, the second wheel will go from 4 to 5, during the inscription of the 9, because of a carry transmission.... |
|
Subtractions are performed with the display bar moved closest to the center of the machine showing the complement value of the accumulator.
The accumulator contains CP( A ) during the first step and CP( A - B) after adding B. In displaying that data in the complement window, the operator sees CP( CP( A)) which is A and then CP(CP( A - B )) which is (A - B). It feels like an addition. The only two differences in between an addition and a subtraction are the position of the display bar (direct versus complement) and the way the first number is entered (direct versus complement).
The following table shows all the steps required to compute: 54,321 - 12,345 = 41,976
| Change display space | Move the display bar down to uncover the complement part of each result cylinder. From this point on, every number dialed into the machine adds its value to the accumulator and therefore decreases the total displayed in the complement window. |
|
||||||||||
| Subtraction | Enter the 9's complement of the minuend. The operator can either use the inner wheels of complements or dial the 9's complement of 54,321 (45,678) directly. |
|
||||||||||
| Dial the subtrahend (12,345) on the spoked metal wheels. This is an addition. The result, 41,976, is in the 9's complement window. |
|