Mechanical calculator

A mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator.
 Description of the illustrated calculator:
 This is a pinwheel rotary type, operated by the hand crank at the right. Tips of the setting levers extend through the slots in the brass cover; pinwheels are inside. The result dials are inside the carriage at the front, which can be shifted sidewise. The right array of dials shows sums and differences, as well as the dividend for division. The other set of dials shows the number of cycles at each carriage position. This displayed number serves as a check on multiplication, and shows the quotient from division. "Butterfly" knobs clear the dials to zeros when rotated.
The mechanical calculator was invented in 1642^{[1]} and the first commercially successful device was manufactured from 1851. Machines with columns of keys were introduced in 1887 while 10 key calculators and electric motors appeared in 1902.^{[2]} The use of electric motors allowed for the design of very powerful machines during the first half of the 20th century. In 1961, A fullkeyboard machine like the comptometer, called the Anita, became the first mechanical calculator to receive an all electronic calculator engine, creating the link in between these two industries and marking the beginning of its decline. The Anita was the only fullkeyboard electronic calculator of any commercial significance. The last mechanical calculators were built in the middle of the 1970s.
The mechanical calculator was preceded by and competed with clerical aids such as abaci, Napier's bones and slide rules, and various books of mathematical tables. The true precursors to the mechanical calculator were machines made of toothed gears linked by carry mechanisms like odometers, astrolabes, clocks and pedometers.^{[3]} Stylusoperated adders with circular slots for the stylus, and sideby side wheels, as made by Sterling Plastics (USA), had an ingenious antiovershoot mechanism to ensure accurate carries.
Contents
History
Ancient history
Main article: AbacusDevices have been used to aid computation for thousands of years, using onetoone correspondence with our fingers.^{[4]} The earliest counting device was probably a form of tally stick. Later record keeping aids throughout the Fertile Crescent included clay shapes, which represented counts of items, probably livestock or grains, sealed in containers.^{[5]}
The first clerical aids were abathia, and were often constructed as a wooden frame with beads sliding on wires. Abathias were in use centuries before the adoption of the written Arabic numerals system and are still used by some merchants, fishermen and clerks in Africa, Asia, and elsewhere. As well, "counting boards", with parallel grooves to hold beads, were used like the abacus.
The counter abacus was devised by Egyptian mathematicians in Egypt in 2000 BC. It was used for arithmetic tasks. The Roman abacus was used in Babylonia as early as 2400 BC. Since then, many other forms of reckoning boards or tables have been invented. In a medieval counting house, a checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money (this is the origin of "Exchequer" as a term for a nation's treasury).
Other precursors to the mechanical calculator
A number of analog computers were constructed in ancient and medieval times to perform astronomical calculations. These include the Antikythera mechanism and other astrolabes from ancient Greece (c. 150100 BC), which are generally regarded as the first mechanical analog computers.^{[6]} Other early versions of mechanical devices used to perform some type of calculations include the planisphere and other mechanical computing devices invented by Abū Rayhān alBīrūnī (c. AD 1000); the equatorium and universal latitudeindependent astrolabe by Abū Ishāq Ibrāhīm alZarqālī (c. AD 1015); the astronomical analog computers of other medieval Muslim astronomers and engineers; and the astronomical clock tower of Su Song (c. AD 1090) during the Song Dynasty. The "castle clock", an astronomical clock invented by AlJazari in 1206, is considered to be the earliest programmable analog computer.^{[7]}
The 17th century
Overview
The 17th century was a turning point in the history of mechanical calculators. On one hand, it saw the invention of logarithms, logarithmic tables and the slide rule which, for their ease of use by scientists in multiplying and dividing, ruled over and impeded the use and development of mechanical calculators^{[8]} until the production release of the arithmometer in the mid 19th century, and, on the other hand, it saw the invention of the adding machine by Wilhelm Schickard^{[9]}, followed by Blaise Pascal^{[1]} and by the first fouroperations mechanical calculator by Gottfried Leibniz. He invented the Leibniz wheel and he was also the first to describe a pinwheel calculator^{[10]}. Neither of them was successful in commercializing their machines.
Logarithms and slide rules
Scottish mathematician and physicist John Napier noted multiplication and division of numbers could be performed by addition and subtraction, respectively, of logarithms of those numbers. While producing the first logarithmic tables Napier needed to perform many multiplications, and it was at this point that he designed Napier's bones, a device consisting of square rods positioned side by side, marked with digits, used for multiplication and division.^{[11]}
In 1622 William Oughtred invented the slide rule, which was revealed by his student Richard Delamain in 1630.^{[12]} Since real numbers can be represented as distances or intervals on a line, the slide rule allows multiplication and division operations to be carried out significantly faster than was previously possible.^{[13]} The devices were used by generations of engineers and other mathematically inclined professional workers, until the invention of the pocket calculator. The engineers in the Apollo program that sent a man to the moon made many of their calculations on slide rules, which were accurate to three or four significant figures.^{[14]}
Invention of the mechanical calculator
In 1642, Blaise Pascal invented the mechanical calculator while looking for a way to help his father who had been assigned the task of reorganizing the tax revenues of the French province of HauteNormandie.^{[15]} After three years of effort and 50 prototypes^{[16]} he introduced his machine to the public. He built twenty of these machines (called the Pascaline) in the following ten years.^{[17]} This machine could add and subtract two numbers directly and multiply and divide by repetition.
Gottfried Leibniz invented the first calculator that could perform all four arithmetic operations automatically while adding direct multiplication and division to the Pascaline ; it was called the Stepped Reckoner. He built it around 1672, but careful examination at the end of the 19th century showed a problem with the carry mechanism. It used his Leibniz wheels which, a century and a half later, will be at the heart of the arithmometer, the first calculator to be commercialized. He also was the first to describe a pinwheel calculator in 1685^{[10]}. Leibniz once said "It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used."^{[18]}
Unknown prototype
Wilhelm Schickard, a German polymath, designed a calculating clock in 1623; a fire destroyed it during its construction in 1624 and Schickard abandoned his project. Two sketches of it were discovered in 1957; too late to have any impact on the development of mechanical calculators.^{[19]}
The 18th century
Further information: pinwheel calculatorsFurther information: Leibniz wheelThe 18th century saw the first fully functional, four operations, mechanical calculators. Both pinwheel calculators and Leibniz wheel calculators were built with a few unsuccessful attempts at their commercialization.
The 19th century
Machines produced
 in 1820 Thomas de Colmar patented the Arithmometer. It was a true multiplication machine with a one digit multiplier (the millionaire calculator released 70 years later had a similar user interface). He spent the next 30 years and 300,000 Francs developing his machine.^{[20]} Notable was the calculator's internal mechanical product lookup table.
 in 1851, Thomas de Colmar simplified the arithmometer by removing the multiplier. This made it a simple adding machine, but thanks to its moving carriage used as an indexed accumulator, it still allowed for easy multiplication and division under operator control. The arithmometer was now adapted to the manufacturing capabilities of the time; Thomas can therefore manufacture consistently a sturdy and reliable machine.^{[21]} Manuals are printed and each machine is given a serial number. Its commercialization launches the mechanical calculator industry.^{[22]} Banks, insurance companies, government offices started to use the arithmometer in their daytoday operations.
 Dorr E. Felt, in the U.S., patented the Comptometer in 1886. It was the first successful keydriven adding and calculating machine. ["Keydriven" refers to the fact that just pressing the keys causes the result to be calculated, no separate lever or crank has to be operated. Other machines are sometimes called "keyset".] In 1887, he joined with Robert Tarrant to form the Felt & Tarrant Manufacturing Company.^{[23]} The comptometertype calculator was the first machine to receive an allelectronic calculator engine in 1961 (the ANITA mark VII released by Sumlock comptometer of the UK).
 in 1878 W.T. Odhner patented the Odhner Arithmometer which was a redesigned version of the Arithmometer with a pinwheel engine but with the same user interface. Odhner started manufacturing his machine in his Saint Petersburg workshop in 1890. Many companies, all over the world, manufactured clones of this machine and millions were sold well into the 1970s.^{[24]}
 In 1892 William S. Burroughs began commercial manufacture of his printing adding calculator^{[25]} Burroughs Corporation became one of the leading companies in the accounting machine and computer businesses.
 The "Millionaire" calculator was introduced in 1893. It allowed direct multiplication by any digit  "one turn of the crank for each figure in the multiplier". It contained a mechanical product lookup table, providing units and tens digits by differing lengths of posts. [1] Another direct multiplier was part of the MoonHopkins billing machine; that company was acquired by Burroughs in the early 20th century.
Prototypes and limited runs
 In 1822 Charles Babbage designed a mechanical calculator, called a difference engine, which was capable of holding and manipulating seven numbers of 31 decimal digits each. Babbage produced two designs for the difference engine and a further design for a more advanced mechanical programmable computer called an analytical engine. None of these designs were completely built by Babbage. In 1991 the London Science Museum followed Babbage's plans to build a working difference engine using the technology and materials available in the 19th century.
 In 1842, Timoleon Maurel invented the Arithmaurel, based on the Arithmometer, which could multiply two numbers by simply entering their values into the machine.
 In 1845 Izrael Abraham Staffel first exhibited a machine that was able to add, subtract, divide, multiply and obtain a square root.
 In 1853 Per Georg Scheutz completed a working difference engine based on Babbage's design. The machine was the size of a piano, and was demonstrated at the Exposition Universelle in Paris in 1855. It was used to create tables of logarithms.
 In 1872, Frank S. Baldwin in the U.S. invented a pinwheel calculator.
 In 1875 Martin Wiberg redesigned the Babbage/Scheutz difference engine and built a version that was the size of a sewing machine.
Operating the calculator at the beginning of this article
Although this is an old machine, nevertheless it represents how one operates any basic rotary calculator. Facits have a pinwheel cylinder that shifts internally, instead of a moving carriage, but the principles still hold.
 First, clear the result dials, and then move all setting levers to zero. Position the carriage appropriately. (Use the levers at the front.) The handcrank must be at home position, engaged with its positioning stop.
 To add, enter the number into the setting levers. Pull the crank handle to the right, and then toward you, so that it's going away from you when the handle is at the top. One turn will add the number into the accumulator dials, and the counter register to the left will show [1].
 Multiplication: If you continue turning, you'll multiply by the number of turns  you're adding repetitively. If you need to multiply by several digits, it's simplest to start with the rightmost multiplier digit, then shift the carriage to the right one position for the next digit.
 To subtract, pull the crank handle to the right, and push it away from you, so the handle is moving toward you when it's at its highest point. If you subtract more than the number in the accumulator dials, you'll get a complement, which you'll need to convert.
 Shortcut multiplication

 It's quite unnecessary to crank six or more times for a multiplier digit. Instead, you can shift the carriage one position to the right, add once, then back up the carriage and subtract, until the counter shows the correct digit. For instance, to multiply by 8, shift, add once, shift back, and subtract twice. (102 = 8) Thinking ahead, instead, you can subtract twice before adding; the calculator will keep track for you.
 Division

 Clear the machine and enter the dividend into the setting slides, starting at the left. Move the carriage to the right so the leftmost dividend digit aligns with the leftmost setting lever. Add once. Clear the counter. If you're lucky, the machine should have a counterreverse control that will make the counter increment for subtraction and decrement for addition. (Some later, better machines do this automatically for the first turn after entering the dividend and clearing the counter.)

 So, now you have the dividend in the accumulator, leftjustified.

 Change the setting levers to enter the divisor, again to the left.

 Be sure the counter is clear, and start subtracting. If the machine has a bell, you can crank mindlessly until the bell rings, then add once. Otherwise, you'll need to watch the accumulator contents to note (or anticipate) an "overdraft" (subtraction too many times); you have to correct it if it happens, by adding.

 Shift the carriage one position to the left, and resume subtracting.

 Repeat for each quotient digit, until you either reach the machine's limits or have enough digits. The counter cantains your quotient; the accumulator contains the remainder (if any).


 It's of some interest that essentiallyautomatic division generally appeared before automatic multiplication; as each quotient digit developed, overdraft was allowed to happen, and it triggered a single add cycle followed by a shift.

 Square root is possible, by the "fives method", but the description is rather more complicated. This type of machine, in particular, is quite good for this kind of calculation.
 Shortcut division

 This takes some thought, but can save time. By watching the accumulator, you can anticipate a large quotient digit, and in a fashion similar to shortcut multiplication, you can add and shift, instead of simply subtracting, to save cycles. (The Marchant calculator contains a multidigit analog magnitude comparator that prevents overdrafts! Changing from subtraction to addition and back is messy and slow in that machine.)
1900s to 1970s
Mechanical calculators reach their zenith
Two different classes of mechanisms had become established by this time, reciprocating and rotary. The former type of mechanism was operated typically by a limitedtravel hand crank; some internal detailed operations took place on the pull, and others on the release part of a complete cycle. The illustrated 1914 machine is this type; the crank is vertical, on its right side. Later on, some of these mechanisms were operated by electric motors and reduction gearing that operated a crank and connecting rod to convert rotary motion to reciprocating.
The latter, type, rotary, had at least one main shaft that made one [or more] continuous revolution[s], one addition or subtraction per turn. Numerous designs, notably European calculators, had handcranks, and locks to ensure that the cranks were returned to exact positions once a turn was complete.
The first half of the 20th century saw the gradual development of the mechanical calculator mechanism.
The Dalton addinglisting machine introduced in 1902 was the first of its type to use only ten keys, and became the first of many different models of "10key addlisters" manufactured by many companies.
In 1948 the miniature Curta calculator, which was held in one hand for operation, was introduced after being developed by Curt Herzstark in 1938. This was an extreme development of the steppedgear calculating mechanism. It subtracted by adding complements; between the teeth for addition were teeth for subtraction.
From the early 1900s through the 1960s, mechanical calculators dominated the desktop computing market (see History of computing hardware). Major suppliers in the USA included Friden, Monroe, and SCM/Marchant. (Some comments about European calculators follow below.) These devices were motordriven, and had movable carriages where results of calculations were displayed by dials. Nearly all keyboards were full — each digit that could be entered had its own column of nine keys, 1..9, plus a columnclear key, permitting entry of several digits at once. (See the illustration below of a Marchant Figurematic.)One could call this parallel entry, by way of contrast with tenkey serial entry that was commonplace in mechanical adding machines, and is now universal in electronic calculators. (Nearly all Friden calculators, as well as some rotary (German) Diehls had a tenkey auxiliary keyboard for entering the multiplier when doing multiplication.) Full keyboards generally had ten columns, although some lowercost machines had eight. Most machines made by the three companies mentioned did not print their results, although other companies, such as Olivetti, did make printing calculators.
In these machines, addition and subtraction were performed in a single operation, as on a conventional adding machine, but multiplication and division were accomplished by repeated mechanical additions and subtractions. Friden made a calculator that also provided square roots, basically by doing division, but with added mechanism that automatically incremented the number in the keyboard in a systematic fashion. The last of the mechanical calculators were likely to have shortcut multiplication, and some tenkey, serialentry types had decimalpoint keys. However, decimalpoint keys required significant internal added complexity, and were offered only in the last designs to be made. The Marchant (Model SKA) also offered square root calculation. Handheld mechanical calculators such as the 1948 Curta continued to be used until they were displaced by electronic calculators in the 1970s.
Typical European fouroperations machines use the Odhner mechanism, or variations of it. This kind of machine included the Original Odhner, Brunsviga and several following imitators, starting from Triumphator, Thales, Walther, Facit up to Toshiba. Although most of these were operated by handcranks, there were motordriven versions. Hamann calculators externally resembled pinwheel machines, but the setting lever positioned a cam that disengaged a drive pawl when the dial had moved far enough.
Although Dalton introduced in 1902 first tenkey printing adding (two operations, the other being subtraction) machine, these feature were not present in computing (four operations) machines for many decades. FacitT (1932) was the first 10key computing machine sold in large numbers. Olivetti Divisumma14 (1948) was the first computing machine with both printer and a 10key keyboard.
Fullkeyboard machines, including motordriven ones, were also built until the 1960s. Among the major manufacturers were MercedesEuklid, Archimedes, and MADAS in Europe; in the USA, Friden, Marchant, and Monroe were the principal makers of rotary calculators with carriages. Reciprocating calculators (most of which were adding machines, many with integral printers) were made by Remington Rand and Burroughs, among others. All of these were keyset. Felt & Tarrant made Comptometers, as well as Victor, which were keydriven.
The basic mechanism of the Friden and Monroe, described above, was a modified Leibniz wheel (better known, perhaps informally, in the USA as a "stepped drum" or "stepped reckoner"). The Friden had an elementary reversing drive between the body of the machine and the accumulator dials, so its main shaft always rotated in the same direction. The Swiss MADAS was similar. The Monroe, however, reversed direction of its main shaft to subtract.
The earliest Marchants were pinwheel machines, but most of them were remarkablysophisticated rotary types. They ran at 1,300 addition cycles per minute if you held down the [+] bar. Others were limited to 600 cycles per minute, because their accumulator dials started and stopped for every cycle; Marchant dials moved at a steady and proportional speed for continuing cycles. Most Marchants had a row of nine keys on the extreme right, as shown in the photo of the Figurematic. These simply made the machine add for the number of cycles corresponding to the number on the key, and then shifted the carriage one place. Even nine add cycles took only a short time.
In a Marchant, near the beginning of a cycle, the accumulator dials moved downward "into the dip", away from the openings in the cover. They engaged drive gears in the body of the machine, which rotated them at speeds proportional to the digit being fed to them, with added movement (reduced 10:1) from carries created by dials to their right. At the completion of the cycle, the dials would be misaligned like the pointers in a traditional watthour meter. However, as they came up out of the dip, a constantlead disc cam realigned them by way of a (limitedtravel) spurgear differential. As well, carries for lower orders were added in by another, planetary differential. (The machine shown has 39 differentials in its (20digit) accumulator!).
In any mechanical calculator, in effect, a gear, sector, or some similar device moves the accumulator by the number of gear teeth that corresponds to the digit being added or subtracted — three teeth changes the position by a count of three. The great majority of basic calculator mechanisms move the accumulator by starting, then moving at a constant speed, and stopping. In particular, stopping is critical, because to obtain fast operation, the accumulator needs to move quickly. Variants of Geneva drives typically block overshoot (which, of course, would create wrong results).
However, two different basic mechanisms, the MercedesEuklid and the Marchant, move the dials at speeds corresponding to the digit being added or subtracted; a [1] moves the accumulator the slowest, and a [9], the fastest. In the MercedesEuklid, a long slotted lever, pivoted at one end, moves nine racks ("straight gears") endwise by distances proportional to their distance from the lever's pivot. Each rack has a drive pin that's moved by the slot. The rack for [1] is closest to the pivot, of course. For each keyboard digit, a sliding selector gear, much like that in the Leibniz wheel, engages the rack that corresponds to the digit entered. Of course, the accumulator changes either on the forward or reverse stroke, but not both. This mechanism is notably simple and relatively easy to manufacture.
The Marchant, however, has, for every one of the its ten columns of keys, a nineratio "preselector transmission" with its output spur gear at the top of the machine's body; that gear engages the accumulator gearing. When one tries to work out the numbers of teeth in such a transmission, a straightforward approach leads one to consider a mechanism like that in mechanical gasoline pump registers, used to indicate the total price. However, this mechanism is seriously bulky, and utterly impractical for a calculator; 90tooth gears are likely to be found is the gas. pump. Practical gears in the computing parts of a calculator can't have 90 teeth. They would be either too big, or too delicate.
Given that nine ratios per column implies significant complexity, a Marchant contains a few hundred individual gears in all, many in its accumulator. Basically, the accumulator dial has to rotate 36 degrees (1/10 of a turn) for a [1], and 324 degrees (9/10 of a turn) for a [9], not allowing for incoming carries. At some point in the gearing, one tooth needs to pass for a [1], and nine teeth for a [9]. There's no way to develop the needed movement from a driveshaft that rotates one revolution per cycle with few gears having practical (relatively small) numbers of teeth.
The Marchant, therefore, has three driveshafts to feed the little transmissions. For one cycle, they rotate 1/2, 1/4, and 1/12 of a revolution. [2]. The 1/2turn shaft carries (for each column) gears with 12, 14, 16, and 18 teeth, corresponding to digits 6, 7, 8, and 9. The 1/4turn shaft carries (also, each column) gears with 12, 16, and 20 teeth, for 3, 4, and 5. Digits [1] and [2] are handled by 12 and 24tooth gears on the 1/12revolution shaft. Practical design places the 12threv. shaft more distant, so the 1/4turn shaft carries freelyrotating 24 and 12tooth idler gears. For subtraction, the driveshafts reversed direction.
In the early part of the cycle, one of five pendants moves offcenter to engage the appropriate drive gear for the selected digit. If possible, see John Wolff's Web site [3] for a superb collection of photos with some accompanying explanations. He has similar sets of photos for several other notable calculators.
Some machines had as many as 20 columns in their full keyboards. The monster in this field was the Duodecillion made by Burroughs for exhibit purposes.
For sterling currency, £/s/d (and even farthings), there were variations of the basic mechanisms, in particular with different numbers of gear teeth and accumulator dial positions. To accommodate shillings and pence, extra columns were added for the tens digit[s], 10 and 20 for shillings, and 10 for pence. Of course, these functioned as radix20 and radix12 mechanisms.
A variant of the Marchant, called the BinaryOctal Marchant, was a radix8 (octal) machine. It was sold to check very early vacuumtube (valve) binary computers for accuracy. (Back then, the mechanical calculator was much more reliable than a tube/valve computer.)
As well, there was a twin Marchant, comprising two pinwheel Marchants with a common drive crank and reversing gearbox.[4] The article at the link describes them and shows a twin Brunsviga (sidebyside machines). Twin machines were relatively rare, and apparently were used for surveying calculations (The CORDIC algorithm was invented later, but these machine might be able to execute it.) At least one triple machine (Brunsviga(?)) was made. It's likely that a given accumulator could be engaged with either half of the twin.
The Facit calculator, and one similar to it, are basically pinwheel machines, but the array of pinwheels moves sidewise, instead of the carriage. The pinwheels are biquinary; digits 1 through 4 cause the corresponding number of sliding pins ot extend from the surface; digits 5 through 9 also extend a fivetooth sector as well as the same pins for 6 through 9.
The keys operate cams that operate a swinging lever to first unlock the pinpositioning cam that's partof the pinwheel mechanism; further movement of the lever (by an amount determined by the key's cam) rotates the pinpositioning cam to extend the necessary number of pins. [5]
The end of an era
Mechanical calculators continued to be sold, though in rapidly decreasing numbers, into the early 1970s, with many of the manufacturers closing down or being taken over. Comptometer type calculators were often retained for much longer to be used for adding and listing duties, especially in accounting, since a trained and skilled operator could enter all the digits of a number in one movement of the hands on a Comptometer quicker than was possible serially with a 10key electronic calculator. In fact, it was quicker to enter larger digits in two strokes using only the lowernumbered keys; for instance, a 9 would be entered as 4 followed by 5. Some keydriven calculators had keys for every column, but only 1 through 5; they were correspondingly compact. The spread of the computer rather than the simple electronic calculator put an end to the Comptometer. Also, by the end of the 1970s, the slide rule had become obsolete.
See also
References
 ^ ^{a} ^{b} Jean Marguin (1994), p. 48
 ^ Ernst Martin p.23,133 (1925)
 ^ Jean Marguin p.3943 (1994)
 ^ Georges Ifrah notes that humans learned to count on their hands. Ifrah shows, for example, a picture of Boethius (who lived 480–524 or 525) reckoning on his fingers in Ifrah 2000, p. 48.
 ^ According to SchmandtBesserat 1981, these clay containers contained tokens, the total of which were the count of objects being transferred. The containers thus served as a bill of lading or an accounts book. In order to avoid breaking open the containers, marks were placed on the outside of the containers, for the count. Eventually (SchmandtBesserat estimates it took 4000 years) the marks on the outside of the containers were all that were needed to convey the count, and the clay containers evolved into clay tablets with marks for the count.
 ^ Lazos 1994
 ^ Ancient Discoveries, Episode 11: Ancient Robots. History Channel. http://www.youtube.com/watch?v=rxjbaQl0ad8. Retrieved 20080906
 ^ Scripta Mathematica, p.128 (1932)
 ^ Marguin, Jean (1994) (in fr). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante 16421942. Hermann. ISBN 9782705661663.
 ^ ^{a} ^{b} David Smith, p.173181 (1929)
 ^ A Spanish implementation of Napier's bones (1617), is documented in Montaner i Simon 1887, pp. 19–20.
 ^ Slide Rules
 ^ Kells, Kern & Bland 1943, p. 92
 ^ Another digit of accuracy was obtained with helical scales. Kells, Kern & Bland 1943, p. 82, as log(2) = 0.3010, or 4 places.
 ^ Maurice d'Ocagne (1893), p. 245 Copy of this book found on the CNAM site
 ^ (fr) La Machine d’arithmétique, Blaise Pascal, Wikisource
 ^ Guy Mourlevat, p. 12 (1988)
 ^ As quoted in Smith 1929, pp. 180–181
 ^ René Taton, p. 81 (1969)
 ^ L'ami des Sciences 1856, p.301 www.arithmometre.org (page consultée le 22092010)
 ^ Ifrah G., The Universal History of Numbers, vol 3, page 127, The Harvill Press, 2000
 ^ Chase G.C.: History of Mechanical Computing Machinery, Vol. 2, Number 3, July 1980, IEEE Annals of the History of Computing, p. 204
 ^ J.A.V. Turck, Origin of modern calculating machines, The Western Society of Engineers, 1921, p. 75
 ^ Trogemann G., Nitussov A.: Computing in Russia, GWVVieweg, 2001, ISBN 3528057572, p. 45
 ^ J.A.V. Turck, Origin of modern calculating machines, The Western Society of Engineers, 1921, p. 143
Sources
 Marguin, Jean (1994) (in fr). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante 16421942. Hermann. ISBN 9782705661663.
 Mourlevat, Guy (1988) (in fr). Les machines arithmétiques de Blaise Pascal. ClermontFerrand: La Française d'Edition et d'Imprimerie.
 Taton, René (1969) (in fr). Histoire du calcul. Que saisje ? n° 198. Presses universitaires de France.
 Turck, J.A.V. (1921). Origin of Modern Calculating Machines. The Western Society of Engineers. Reprinted by Arno Press, 1972 ISBN 0405047304.
 Ginsburg, Jekuthiel (2003). Scripta Mathematica (Septembre 1932Juin 1933). Kessinger Publishing, LLC. ISBN 9780766138353.
 Martin, Ernst (1992). The Charles Babbage Institute. ed. The Calculating Machines translation from Die Rechenmaschinen (1925). Cambridge, Massachusetts: The MIT Press.
 Smith, David Eugene (1929). A Source Book in Mathematics. New York and London: McGrawHill Book Company, Inc..
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