Category: Timeline Stories

The Machine Arithmetique of Lépine (sent for approval to l’Academie Royale des Sciences in 1725) was firstly described in the 1735 book of Jean Gallon Machines et inventions approuvées par l’Academie Royale des Sciences, depuis son établissment jusqu’a présent; avec leur description (see Gallon description of Lepine). The machine of Lépine was a solid, but simple adding device, with limited practical usefulness.

Jean Gaffin Gallon (1706-1775) was a colonel in the French army, and later the chief engineer for the port of Le Havre. In the early 1730s, the French Academie Royal des Sciences asked him to edit all the descriptions of the machines the Academie had approved. Six volumes were produced in Gallon’s lifetime, and a seventh volume appeared posthumously in 1777. This set of works contains both descriptions and engravings of all the inventions approved from the beginnings of the Academie in 1666 until 1735. The devices are described in chronological order. They cover all the areas then known in arts, sciences, engineering, and manufacturing. Notable among the many descriptions is one for Pascal’s adding machine and others by Perrault, Lepine, de Mean (which was really only a table upon which products could be looked up), and three by Hillerin de Boistissandeau.

We cannot be sure who actually made this calculator. Most probably the machine was devised (and sent for approval to the Academie Royale des Sciences) in the early 1720s by Philibert Depigny (1692-1727) (his surname was also spelled Lépine, Lespine, and L’Epine), a watchmaker and mechanic of King Louis XV of France, who used to make music boxes for the King, but the survived to our time devices were manufactured by his son—the famous and inventive French watchmaker Jean-Antoine Lépine (also King’s watchmaker, but of Louis XVI and Napoleon Bonaparte) many years later.

Jean-Antoine Lépine (born as Depigny), was born in Challex, a small village a few kilometers west of Geneva, on 18 November 1720, and later became one of the greatest watchmakers of all times, with numerous inventions in this area. He was fully capable to create such a simple calculating device, but in 1725 he was only five and he still lived in the house of his parents Philibert Depigny, and Marie, née Girod, in Challex, so obviously he was a little bit young for inventing machines 🙂 Jean-Antoine Lépine used to sign his clocks with the same inscription “LEPINE – INVENIT ET FECIT”, which can be found on the lid of one of the calculating devices, mentioned below.

The Machine Arithmetique of Lépine was obviously inspired by the famous Pascaline of Blaise Pascal, although in Lepine’s machine, carrying took place through the flex of spring and not, as in Pascal’s device, through the fall of a weight. It seems several working copies of the calculator had been manufactured because at least two of them are still preserved to our time. The smaller one, an 8-digit device, is kept now in the collection of CNAM Museum in Paris (see the upper image), while the bigger, 20-digit device (see the lower image) is in the National Museum of American History in Washington. The machine in Washington is inscribed “DE LEPINE – INVENIT ET FECIT – 1725”, and is marked inside the lid “reparé en 1844 par le Chr Thomas de Colmar” (repairs done in 1844 by Charles Xavier Thomas of Colmar). It is a device made of leather and wood (the case), and brass and steel (the body), with overall dimensions: 4.8 cm x 49.2 cm x 26.4 cm, while the machine in Paris is smaller: Length: 34 cm; Height: 6 cm; Width: 13 cm; Mass: 3.250 kg.

Let’s examine the variation of the device, described in the publication of the French Academy. The machine is a 12-digital adding device, which rightmost two dials are used for adding sols and deniers (french monetary units at this time, 1 sol is equal to 1/20 of the livre, 1 denier is 1/12 of the sol) and are divided into 12 and 20 parts. The next ten dials are decimal.

The wheels are arranged in two rows. The tens carry from the 6th (last of the lower row) wheel to the 7th (first in the upper row) dial is performed by means of a mechanical system (marked with L in the lower figure), which consists of two small rods (marked with 1), two springs (2) and a clamp (3). On every full revolution of the 6th wheel, one of the pins (which is prolonged) raises the lower clamp, and pushes the side clamp, which rotates the 7th wheel to 1/10 revolution. The two springs and the clamp sidelong are destined for fixing and supporting.

The 12 wheels in the upper part of the machine are not connected one to another and can be used for storing the intermediate results. In the lower figure can be seen the fixing mechanism of each wheel, which consists of a spring and a clamp (marked with P1 and P2) (the fixing mechanism of the lower wheels (which are destined for the showing of the result) is manufactured in the same manner). The rotating of these wheels can be done by means of clamps M (see the upper figure), which are connected with the axes of the wheels and are rotating by means of the stylus E. Over the windows, where can be seen inscribed over the periphery of the wheels digits, are inscribed the proper units—deniers, sols, units, tents, and so on to the billions, numbered clockwise.

The stylus E actually has two edges—short and long. The entering of numbers is done by rotating the dials, which are divided into 12, 20, 10 and so on parts. Bellow each one of these dials is mounted another dial, which has the same number divisions. Over the periphery of the lower disks are inscribed digits, which can be seen in the windows, situated over the disks and are used for showing the entered number.

If in the openings of the dial has been pushed the long edge of the stylus, then it will be rotated together with the lower disk. If it has been pushed the short edge of the stylus, then it will be rotated only the upper disk, but the lower disk, showing entered number, will remain static.

The result is presented in the two rows of windows, which are placed over the entering dials, where can be seen 12 smaller disks F. Transfer of the digits from the entering dials to these wheels is made by a system of clamps and springs. The tens carry is done by means of a similar system.

Over the periphery of the disks F, which are placed below entering dials and are showing the entered number, actually are inscribed two rows of digits. The first row is used during the adding and multiplication, and the second (its digits are a complement to 10 of the first row)—during the subtraction and division. Only one of the rows can be seen at a particular moment, and which row can be seen is determined by the clamp S, which can be seen on the left side of the figure. Rotating this clamp, we can show one or another row, depending on if we want to add or subtract. The resetting of the result to 0 can be done by means of clamps R, which are similar to clamps M, used for entering the numbers in the recording mechanism.

It seems the machine of Lépine possesses all the defects of Pascaline—the mechanism does not offer reliable transmission (carrying) of the units into the tens, the tens into the hundreds, etc.

In 1709 the young professor of astronomy, meteorology, and mathematics and Marquis of the Holy Roman Empire Giovanni Poleni published his first book—Miscellanea, a small collection of dissertations on physics (it was a sort of doctoral thesis to obtain the University appointment). Miscellanea includes dissertations on barometers, thermometers, and conical sections in sundials, as well as an illustrated treatise describing his arithmetical calculating machine, which proved to be the first calculating machine with gears with a variable number of teeth (so-called pin-wheel). It seems Poleni commenced his work on the calculating machine in 1707, having heard the news of Pascal’s and Leibniz’s calculators from scholars in person and from their writings and, though having no drawings nor technical description of the two previous inventions, decided to design and build an original one.

The second section of the Miscellanea describes the calculating machine of Poleni. The chapter opens with the following words:
“Having heard several times, either from the voice, or from the writings of scientists, which were made by the insight and the care of the most illustrious Pascal and Leibniz, two machines which are used for arithmetic multiplication, which I do not know the description of the mechanism, and I do not know if it was made manifest, I wanted and guess with thought and reflection to their construction, to build a new one that implements the same purpose.”

Then Poleni continues: “By a happy chance, I designed a machine with the use of which even a novice in the art of calculation, provided that knows the figures, can perform arithmetic operations with his own hand. So I am worried that it was made of wood, as I had planned and that, although initially built with poor precision, showed that this was achievable, rather than made. Therefore, I have recreated the machine from scratch, building it of the hardest wood, with all possible attention and the work done has not failed in vain.”

So, according to the inventor himself, two wooden copies of the machine were made, but unfortunately, none of them survived to the present time. The image below shows a replica of the machine from 1959.

A description of the Poleni’s machine appeared in 1727 in Jacob Leupold’s Theatrum arithmetico geometricum and in “Versuch einer Geschichte der Rechenmaschine” by Johann Bischoff, 1804.

One of the first biographers of Poleni, the Frenchman Jean de Fouchy Pajil Grandjean, claims in his 1762 book “Eloge de Jean POLENI, Marquis du St. Empire, (né 1683 mort 1761)”, that …having heard that Mr. Brawn, a famous mechanic in Vienna, presented a similar machine to the Emperor, Poleni destroyed his machine and no more wanted to rebuild it. The same story is repeated in the two later Poleni’s biographies—Pietro Cossali (1813), and Giuseppe Gennari (1839). Despite the fact, that Fouchy was in strict contact with Poleni (when alive) and knew him personally, this story is quite questionable, not only because it is not compatible with the gentle character of Poleni. Fouchy obviously mentioned the German mechanic, constructor, and optician Anton Braun, who worked in Vienna for the court and really presented his calculating machine (based on the pin-wheels of Poleni and Leibniz) with great success to the Emperor, but this happened as late as in 1727. It is possible Braun had gotten information about Poleni’s machine (we know that Braun worked under the supervision of the imperial engineer Johann Jacob Marinoni, who was in correspondence with Poleni and perhaps visited him in Venice), and had decided to use the idea of Poleni in his construction, the history of inventions is full of cases like that. In fact, if Poleni hadn’t managed for almost 20 years to manufacture and demonstrate a working copy of his machine, obviously he was not interested in this device at all and fully deserved to be outrun by others.

The machine of Poleni was quite big (with the dimensions of a big clock with a pendulum). Let’s examine its mechanism, looking at the illustration from the Theatrum arithmetico geometricum.

The mechanism is set in motion by means of the weight K, tied to the end of the rope l, which is winded over the cylinder t. The most important element of the machine—the so-called pin-wheel, invented probably (it is known that Leibniz has a similar design in one of his manuscripts and that Leibniz and Poleni had a correspondence) by Poleni, is placed to the right of the axis VY. Actually in the machine are placed 3 pin-wheels, which means, that the input mechanism has only 3 digital positions.

The pin-wheel of Poleni actually is a smooth wheel, with attached to it a sector with 9 blocks, illustrated in the right part of the figure. Each block contains a body (marked with 3 in the figure), a tooth (marked with 2), and a small rod (marked with 4). There is also a fourth element of the block—a small spring, which is not shown in the figure. In the assembled block (marked with 1), the tooth can be erected outside of the block, if the rod is pressed by the operator or hidden in the body of the block if the rod is released. Thus in each one of the three sectors with blocks can be set from 0 to 9 erected teeth. When the tooth is erected, it will engage with the calculating mechanism during the revolution of the calculating mechanism, otherwise—not.

The output mechanism consists of six dials, i.e. it has six digital positions. The tens-carry mechanism uses a one-tooth wheel and obviously has not been designed well, because it caused problems (it can happen that the machine stops in the middle of a tens-carry step due to adverse leverage forces.)

An additional crank was used for shifting the pin-wheel mechanism by one place at a time with respect to the transfer tooth wheels, thus allowing the multiplicand to be multiplied by the multiplier in higher places.

As a whole, despite the innovative idea of Poleni for the pin-wheel, which would become an extremely popular constructive element of calculating machines some 150 years later (see the machines of Baldwin and Odhner), his machine looks rather rough and ill-formed device, as compared to the machine of his successor—Anton Braun.

Biography of Giovanni Poleni

Giovanni Poleni was born on 23 August 1683, in Venice (most probably in the Palazzetto, still sitting projected onto the Grand Canal, see the photo below). He was the only son of Jacopo (Giacomo) Poleni (1655-1737) and Elisabetta Broiuola (or Brugnol, Brajola, or Brogiola). Jacopo Poleni was a well-educated “middle class” Venetian, a literate and a poet, but searching for glory, in early 1683 he entered as a volunteer in the army of Emperor Leopold I and in the same year’s autumn, he took part and displayed a lot of courage in the battles against Turks during the Siege of Vienna, and in Hungary. In June of 1685, the Emperor awarded him for his merits with the titles “Marquis, Count of St. Michael the Archangel, and Knight, along with his descendants”, a title, confirmed by the Senate in July 1686. So, only two years old, Giovanni Poleni became a Marquis, what a remarkable beginning of his life 🙂

As a young man, Giovanni showed brilliance in a wide variety of different subjects and it was clear that he was extraordinarily talented. His parents encouraged him to begin a judicial career but, after being introduced to mathematics and experimental science (following the works of René Descartes) by his father, it became clear that he had now found the subjects which gave him the most satisfaction.

In 1695 Poleni began his studies in the school of the Basilica Santa Maria della Salute in Venice, where he studied letters (under the guidance of Padre Stanislao Santinelli), philosophy and theology (under Padre Francesco Caro), civil and military architecture, drawing, painting, and perspective (under the painter Giuseppe Marcati). In 1707 Poleni began to deal with physics, carrying out the first scientific experiments. In the same year, he met Giovanni Battista Morgagni, the anatomist, regarded as the father of modern anatomical pathology, with whom Poleni started a friendship and collaboration that continued until his death.

At the end of 1709, Poleni accepted the chair of Astronomy and Meteorology at the University of Padua, and in February 1710, he moved with his family to Padua. For over 40 years he kept outstandingly accurate meteorological records which have proved invaluable to later scientists. In 1709 he published his first book—Miscellanea: de barometris et thermometris; de machina quadam arithmetica; de sectionibus conicis in horologiis solaribus describendis, a collection of dissertations on physics.

In 1715 Poleni became a professor of physics, in addition to holding the chair of astronomy and meteorology, and was invited by the Venetian Senate to investigate the problem of hydraulics relating to the irrigation of Lower Lombardy. He was appointed to the chair of mathematics at Padua in 1719 which had been vacated by Nicolaus Bernoulli. In 1717 he published a major treatise on hydraulics and hydrodynamics—De motu aquae mixto libri duo, which describes estuaries, ports, and rivers.

In 1710 Poleni was elected a fellow of the Royal Society of London, upon a proposal by Sir Isaac Newton. In 1715 he was honored by being proposed by Gottfried Leibniz, a correspondent of him, for election to the Berlin Academy of Science in 1715. In 1723 he was admitted to the Academy of the Institute of Sciences of Bologna. In 1724 he was appointed a partner of the newly founded Saint Petersburg Imperial Academy, starting his correspondence with Leonard Euler. In 1739, the prestigious Académie Royale des Sciences of Paris admitted him among its members.

Poleni observed the solar eclipse in 1724 and wrote a treatise on the topic which was published the following year. From 1739 he taught experimental physics since by this time he had a physics laboratory. It was the first physics laboratory to be established in an Italian university.

In 1729, Poleni built a tractional device that enabled logarithmic functions to be drawn. In 1743 he was invited by the Pope and made a significant contribution to the consolidation of San Pietro’s dome in Roma.

Poleni was appointed to the chair of Nautical Studies and Naval Construction at Padua in 1756. From this time he taught nautical sciences and shipbuilding. A music-lover, Poleni was the patron of Giuseppe Tartini (1692-1770)—a famous composer and violinist.

On 30 December 1708, obeying his father, Giovanni Poleni married Orsola Roberti (1686-1737), who was from a high-ranking and noble Bassano del Grappa family. The family had six children, five boys, and a girl. The firstborn was Jacopo (Giacomo), a very talented boy, born on 18 October 1709, and died in 1747. All of the boys of Poleni pursued a religious career, with the exception of the last, Eugenio (1717-1736), who was destined to give succession to the family but died only 18 years old, to the severe grief of Marquis Giovanni, who in 14 months lost three dearest people—his son, his wife, and his father. Poleni had three other sons: Bernardo, Luca, and Francesco, and one daughter—Elisabetta, who married the botanist professor Giulio Pontedera.

Giovanni Poleni, who since the 1720s had suffered from violent headaches due to an accident, died in Padua on 15 November 1761, after a few days of rapid worsening. The autopsy, performed by his colleague and friend Giambattista Morgagni, a famous anatomist, revealed that the death had been caused by an aortic aneurysm.

The great polymath Gottfried Leibniz and Charles Babbage are (to my mind) the two greatest persons in the history of computers and computing because only they managed to anticipate events in this area for centuries. In the second half of the 17^th century, Leibniz not only created the first mechanical calculator, suitable for addition, subtraction, and multiplication but also dreamed about the logical machine and binary calculator.

Leibniz’ Instrumentum Arithmeticum

Leibniz got the idea of a calculating machine at the end of the 1660s, seeing a pedometer device. The first mention of his Instrumentum Arithmeticum is from 1670, as the breakthrough happened in 1672 when he moved for several years to Paris, where he got access to the unpublished writings of the two greatest philosophers—Pascal and Descartes. Most probably in the same year, he became acquainted (reading Pascal’s Pensees) with Pascaline, which he decided to improve in order to be possible to make not only addition and subtraction but also multiplication and division.

At the very beginning, Leibniz tried to devise a mechanism, similar to Pascaline, but soon realized, that for multiplication and division, it is necessary to create a completely new mechanism, which will make it possible for the multiplicand (dividend) to be entered once and then by a repeating action (e.g. rotating of a handle) to get the result. Trying to find a proper mechanical resolution to this task Leibniz made several projects before inventing his famous stepped-drum mechanism (called also Leibniz gear).

The upper sketch is from Leibniz’s manuscript from 1685 (the full text is given below) and shows probably an early design for the calculating machine. There is an input mechanism, the lower circles, inscribed Rota multiplicantes, where must be entered the multiplier; there is a calculating mechanism, inscribed Rota multiplicanda, where must be entered the multiplicand; and there is a result mechanism, the top circles, inscribed Rota additionis, where one can see the result of multiplication. The movement from the input wheels to the calculating wheels is transferred by means of chains. The calculating mechanism seems based on pin-wheel, not on the stepped drum.

Something like a pin-wheel mechanism is described also in a sketch (see the upper drawing) from another Leibniz’s manuscript (written around 1670), which throws light on his initial idea for the calculating mechanism. The un-dated sketch is inscribed “Dens mobile d’une roue de Multiplication” (the moving teeth of a multiplier wheel). Interestingly, Leibniz’s pin-wheel mechanism was reinvented in 1709 by Giovanni Poleni, and improved later by Braun, Roth, and Staffel.

Obviously, the prototype and first designs of the calculator were based on one of the above-mentioned pin-wheel mechanisms, before the development of the stepped drum mechanism, which was successfully implemented into the survived to our time devices (the machine was under continuous development for more than 40 years and several copies were manufactured).

Starting to create the first prototype, Leibniz soon faced the same obstacles that Pascal had experienced—poor workmanship, unable to create the fine mechanics, required for the machine. He complained: “If only a craftsman could execute the instrument as I had thought the model.”

The first wooden 2-digital prototype of the Stepped Reckoner (this is a later name, actually Leibniz called his machine Instrumentum Arithmeticum), was ready soon, and at the end of 1672 and beginning of 1673, it was demonstrated to some of his colleagues at the French Academy of Sciences, as well as to the Minister of Finances Jean-Baptiste Colbert.

In January 1673 Leibniz was sent to London on a diplomatic mission, where he succeeded not only in meeting some English scientists and presenting his treatise called The Theory of Concrete Motion but also in demonstrating the prototype of his calculating machine to the Royal Society on 1 February 1673. Leibniz was recommended by Huygens, who called his machine a promising project in a letter to Henry Oldenburg, the secretary of the Royal Society. Oldenburg knew Leibniz as a friend of Boineburg and fellow countryman and was committed to helping Leibniz, who expected to make a splash in London with his calculating machine.

During the demonstration, Leibniz stated, that his arithmetic tool was invented for the purpose of mechanically performing all arithmetic operations reliably and quickly, especially multiplication. Leibniz explained it very well, but the demonstration was obviously not very successful, because the inventor admitted that the instrument wasn’t good enough and promised to improve it after returning to Paris. Nevertheless, the impression of Leibniz must have been very positive, because he was elected as a member of the Royal Society in April 1673. It is known also, that during his trip to London, Leibniz met Samuel Morland and saw his arithmetic engine.

Particularly unimpressed by the demonstration was the famous scientist and ingenious inventor Robert Hooke, who was the star of the Royal Society at the time when Leibniz came to show his machine. Hooke was infamous for engaging in brutal disputes (not always within the boundaries of fair debate) with his rivals, like Huygens and Newton. After looking carefully at all sides of the machine, and examining it in detail during the demonstration on 1 February 1673, Hooke expressed a desire to take it apart completely to examine its insides. Moreover, several days after the demonstration, Hooke attacked him in public, making derogatory comments about the machine and promising to construct his own superior and better-working calculating machine, which he would present to the Royal Society. Hooke stated that it seems to me so complicated with wheels, pinions, cantrights, springs, screws, stops, and truckles, that I could not perceive it ever to be of any great use… It could be only fit for great persons to purchase, and for great force to remove and manage, and for great wits to understand and comprehend. In contrast, Hooke announced that I have an instrument now making, which will perform the same effects (and) will not have a tenth part of the number of parts, and not take up a twentieth part of the room. Leibniz was not in London at that time to defend himself and had to hear about the attack from Oldenburg, who assured him that Hooke was quarrelsome and cantankerous, and urged him that the best course of action will be to finish his machine as quickly as possible.

In a letter of 26 March 1673, to one of his correspondents—Johann Friedrich, mentioning the presentation in London, Leibniz described the purpose of the arithmetic machine as making calculations easy, fast, and reliable. Leibniz also added that theoretically, the numbers calculated might be as large as desired, if the size of the machine was adjusted: a number consisting of a series of figures, as long as it may be (in proportion to the size of the machine).

Back in Paris, Leibniz hired a skillful mechanic—the local clock-maker Olivier, who was a fine craftsman, and he made the first metal (brass) prototype of the machine. It seems the first working properly device was ready as late as 1685 and didn’t manage to survive to the present day, as well as the second device, made 1686-1694. (Olivier used to work for Leibniz up to 1694. Later on Professor Rudolf Christian Wagner and the mechanic Levin from Helmstedt worked on the machine, and after 1715, the mathematician Gottfried Teuber and the mechanic Has in Leipzig did the same).

In 1675 the machine was presented to the French Academy of Sciences and was highly appreciated by the most prominent members of the Academy—Antoine Arnauld and Christian Huygens. Leibniz was so pleased by his invention, that he immediately informed some of his correspondents: e.g. Thomas Burnett, 1st Laird of Kemnay—I managed to build such arithmetic machine, which is completely different from the machine of Pascal, as it allows multiplication and division of huge numbers to be done momentarily, without using of consecutive adding or subtraction, and to another correspondent, the philosopher Gabriel Wagner—I managed to finish my arithmetical device. Nobody had seen such a device until now, because it is extremely original.

In 1676 Leibniz demonstrated the new machine again to the Royal Society in London. Let’s clarify, however, that this was a small device with several digital positions only. The full-scale workable machine will be ready as late as 1694.

It is unknown how many machines were manufactured by the order of Leibniz. It is known, however, that the great scientist was interested in this invention all his life and that he spent on his machine a very large sum at the time—some 24000 talers according to some historians, so it is supposed the number of the machines to be at least 10. One of the machines (probably the third manufactured device), produced 1690-1720, was stored in an attic of a building of the University of Göttingen sometime late in the 1770s, where it was completely forgotten. It remained there, unknown, until 1879 when a work crew happened across it in a corner while attempting to fix a leak in the roof. In 1894-1896 Arthur Burkhardt restored it in Glashütte, and it has been kept at the Niedersächsische Landesbibliothek for some time.

At present time exist two old machines, which probably are manufactured during Leibniz’s lifetime (around 1700) (in the Hannover Landesbibliothek and the Deutsches Museum in München), and several replicas (see one of them in the upper photo).

The mechanism of the machine is 67 cm long, 27 wide, and 17 cm high and is housed in a big oak case with dimensions 97/30/25 cm. Let’s examine what is the principle of the stepped-drum (see the nearby sketch).

The stepped-drum (marked with S in the sketch) is attached to a four-sided axis (M), which is a teeth-strip. This strip can be engaged with a gear-wheel (E), linked with the input disk (D), on which surface are inscribed digits from 0 to 9. When the operator rotates the input wheel and the digits are shown in the openings of the lid, then the stepped drum will be moved parallel with the axis of the 10-teeth wheel (F) of the main counter. When the drum is rotated to a full revolution, with the wheel (F) will be engaged a different number of teeth, according to the value of the movement, which is defined by the input disk and the wheel (F) will be rotated to the appropriate angle. Together with the wheel (F) will be rotated the linked to it digital disk (R), whose digits can be seen in the window (P) of the lid. During the next revolution of the drum to the counter will be transferred again the same number.

The input mechanism of the machine is 8-positional, i.e. it has 8 stepped drums, so after the input of the number using input wheels, rotating the front handle (which is connected to the main wheel (called by Leibniz Magna Rota), all digital drums will make 1 revolution each, adding the digits to the appropriate counters of the digital positions. The output (result) mechanism is 12-positional. The result (digits inscribed on the digital drums) can be seen in the 12 small windows in the upper unmovable part of the machine.

One of the main flaws of the Stepped Reckoner is that the tens carry mechanism is not fully automatic (at least this of the survived until now machine). Let’s see why. In the next sketch are shown mechanisms of two adjacent digital positions. The stepped drums are marked with 6, and the parts, that formed the tens carry mechanism, are marked with 10, 11, 12, 13, and 14.

When a carry must be done, the rod (7) will be engaged with the star-wheel (8) and will rotate the axis in a way, that the bigger star-wheel (11) will rotate the pinion (10). On the axis of this pinion is attached a rod (12), which will be rotated and will transfer the motion to the star-wheel (10) of the next digital position, and will increase its value by 1. So the carry was done. The transfer of the carry, however, will be stopped at this point, i.e. if the receiving wheel was at the 9 position, and during the carry, it has gone to 0 and another carry must be done, that will not happen. There is a workaround however because the pentagonal disks (14) are attached to the axis in such a way, that their upper sides are horizontal when the carry has been done, and with the edge upwards, when the carry has not been done (which is the case with the right disk in the sketch). If the upper side of the pentagonal disk is horizontal, it cannot be seen over the surface of the lid, and cannot be noticed by the operator, so manual carry is not needed. If however the edge can be seen over the surface of the lid, this will mean that the operator must rotate this disk, performing a manual carry.

It seems the problem with the tens carry mechanism was resolved by the German scientist Christian Gottlieb Kratzenstein, who in April 1765, presented to the Russian Academy of Sciences in Saint Petersburg with a perfected version of the calculating machine of Leibniz. Kratzenstein claimed that his machine solved the Leibniz machine’s problem with calculations over four digits and perfected the error that the machine is “prone to error when it is necessary to move a number from 9999 to 10000”, but the machine was not developed further.

The mechanism of the machine can be divided into two parts. The upper part is unmovable and was called by Leibniz Pars immobilis. The lower part is movable and is called Pars mobilis (see the sketch below).

In the Pars mobilis is placed the 8-positional setting mechanism with stepped drums, which can be moved leftwards and rightwards, so as to be engaged with different positions of the 12-positional unmovable calculating mechanism. Adding with the machine is simple—the first addend is entered directly in the result wheels (windows) (there is a mechanism for zero setting and entering numbers in the result wheels), the second addend is entered with the input wheels in the Pars mobilis, and then the forward handle (Magna rota) is rotated once. Subtraction can be made similarly, but all readings must be taken from the red subtractive digits of the result wheels, rather than the normal black additive digits. On multiplication, the multiplicand is entered employing the input wheels in the Pars mobilis, then Magna Rota must be rotated to so many revolutions, which number depends on the appropriate digit of the multiplier. If the multiplier is multi-digital, then Pars mobilis must be shifted leftwards with the aid of a crank and this action is to be repeated, until all digits of the multiplier are entered. The division is done by setting the dividend in the result windows and the divisor on the setup dials, then a turn of Magna rota is performed and the quotient may be read from the central plate of the large dial.

There is also a counter for the number of revolutions, placed in the lower part of the machine, which is necessary for multiplication and division—the large dial to the right of the small setting dials. This large dial consists of two wide rings and a central plate—the central plate and outer ring arc inscribed with digits, while the inner ring is colored black and perforated with ten holes. If for example, we want to multiply a number on the setting mechanism to 358, a pin is inserted into hole 8 of the black ring and the crank is turned, this turns the black ring until the pin strikes against a fixed stop between 0 and 9 positions. The result of the multiplication by 8 may then be seen in the windows. The next step requires that the setting mechanism be shifted by one place employing the crank (marked with K in the upper figure), the pin inserted into hole 5, and the crank turned, whereupon the multiplication by 58 is completed and may be read from the windows. Again the setting mechanism must be shifted by one place, the multiplication by 3 is carried out in the same manner, and now the result of the multiplication by 358 appears in the windows.

In 1685 Leibniz wrote a manuscript, describing his machine—Machina arithmetica in qua non aditio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur. As the 1685 design was based on wheels with a variable number of teeth, not on a stepped drum, obviously survived to our time devices are later work. In English, the manuscript sounds like this:
When, several years ago, I saw for the first time an Instrument that, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only counting but also addition and subtraction, multiplication and division could be accomplished by a suitably arranged machine easily, promptly, and with sure results.
The calculating box of Pascal was not known to me at that time. I believe it has not gained sufficient publicity. When I noticed, however, the mere name of a calculating machine in the preface of his “posthumous thoughts” (his arithmetical triangle I saw first in Paris) I immediately inquired about it in a letter to a Parisian friend. When I learned from him that such a machine exists I requested the most distinguished Carcavius by letter to give me an explanation of the work which it is capable of performing.
He replied that addition and subtraction are accomplished by it directly, the other (operations) in a round-about way by repeating additions and subtractions and performing still another calculation. I wrote back that I ventured to promise something more, namely, that multiplication could be performed by the machine as well as addition, and with the greatest speed and accuracy.
He replied that this would be desirable and encouraged me to present my plans before the illustrious King’s Academy of that place.
In the first place, it should be understood that there are two parts of the machine, one designed for addition (subtraction) and the other for multiplication (division), and that they should fit together.
The adding (subtracting) machine coincides completely with the calculating box of Pascal. Something, however, must be added for the sake of multiplication so that several and even all the wheels of addition could rotate without disturbing each other, and nevertheless, any one of them should precede the other in such a manner that after a single complete turn unity would be transferred into the next following. If this is not performed by the calculating box of Pascal it may be added to it without difficulty.
The multiplying machine will consist of two rows of wheels, equal ones and unequal ones. Hence the whole machine will have three kinds of wheels: the wheels of addition, the wheels of the multiplicand, and the wheels of the multiplier. The wheels of addition or the decadic wheels are now visible in Pascal’s adding box and are designated in the accompanying figure by the numbers 1, 10, 1(X), etc. Each of these wheels has ten fixed teeth.
The wheels that represent the multiplicand are all of the same size, equal to that of the wheels of addition, and are also provided with ten teeth which, however, are movable so that at one time there should protrude 5, at another 6 teeth, etc., according to whether the multiplicand is to be represented five times or six times, etc. For example, the multiplicand 365 consists of three digits 3, 6, and 5. Hence the same number of wheels is to be used.
On these wheels, the multiplicand will be set, if from the right wheel there protrude 5 teeth, from the middle wheel 6, and from the left wheel 3 teeth.
So that this could be performed quickly and easily a peculiar arrangement would be needed, the exposition of which would lead too far into details. The wheels of the multiplicand should now be adjoined to the wheels of addition in such a manner that the last corresponds to the last, the last but one to the last but one, and that before the last but one to that before the last but one, or 5 should correspond to 1, 6 to 10, and 3 to 100. In the addition box itself, there should be small openings with the number set as 0, 0, 0, etc., or zero. If after making such an arrangement we suppose that 365 be multiplied by one, the wheels 3, 6, and 5 must make one complete turn (but while one is being rotated all are being rotated because they are equal and are connected by cords as it will be made apparent subsequently) and their teeth now protruding will turn the same number of fixed teeth of the wheels 100, 10, 1 and thus the number 365 will be transferred to the addition box.
Assuming, however, that the number 365 is to be multiplied by an arbitrary multiplier (124) there arises the need for a third kind of wheels or the wheels of the multiplier. Let there be nine such wheels and while the wheels of the multiplicand are variable so that the same wheel can at one time represent 1 and at another time 9 according to whether there protrude less or more teeth, the wheels of the multiplier shall, on the contrary, be designated by fixed numbers, one for 9, one for 1, etc.
This is accomplished in the following manner: Each of the wheels of the multiplier is connected using a cord or a chain to a little pulley which is affixed to the corresponding wheel of the multiplicand: Thus the wheel of the multiplier will represent many units equal to the number of times the diameter of the multiplier-wheel contains the diameter of the corresponding pulley.
The pulley will turn namely this number of times while the wheel turns but once. Hence if the diameter of the wheel contains the diameter of the pulley four times the wheel will represent 4.
Thus at a single turn of the multiplier-wheel to which there corresponds a pulley having a quarter of its diameter the pulley will turn four times and with it also the multiplicand-wheel to which it (the pulley) is affixed. When, however, the multiplicand-wheel is turned four times its teeth will meet the corresponding wheel of addition four times, and hence the number of its units will be repeated as many times in the box of addition.
An example will clarify the matter best: Let 365 be multiplied by 124. In the first place, the entire number 365 must be multiplied by four. Turn the multiplier-wheel 4 by hand once; at the same time the corresponding pulley will turn four times (being as many times smaller) and with it, the wheel of the multiplicand 5, to which it is attached, will also turn four times. Since wheel 5 has five teeth protruding at every turn 5 teeth of the corresponding wheel of addition will turn once and hence in the addition box there will be produced four times 5 or 20 units.
The multiplicand-wheel 6 is connected with the multiplicand wheel 5 by another cord or chain and the multiplicand-wheel 3 is connected with wheel 6. As they are equal, whenever wheel 5 turns four times, at the same time wheel 6 by turning four times will give 24 tens (it namely catches the decadic addition wheel 10), and wheel 3 catching the addition-wheel 100 will give 12 hundred so that the sum of 1460 will be produced.
In this way 365 is multiplied by 4, which is the first operation. So that we may also multiply by 2 (or rather by 20) it is necessary to move the entire adding machine by one step so to say, SO that the multiplicand-wheel 5 and the multiplier-wheel 4 are under addition-wheel 10, while they were previously under 1, and in the same manner 6 and 2 under 100 and also 3 and 1 under 1000.
After this is done let the multiplier-wheel 2 be turned once: at the same time 5 and 6 and 3 will turn twice and 5 catching twice (the addition-wheel) 10 will give 10 tens, 6 catching 100 will give twelve hundred and 3 catching 1000 will give six thousand, together 7300. This number is being added at the very same turn to the previous result of 1460. To perform the third operation, the multiplication by 1 (or rather by 100), let the multiplication machine be moved again (of course the multiplicand-wheels together with the multiplier-wheels while the addition-wheels remain in their position) so that the wheels 5 and 4 are placed under 100 and in the same way 6 and 2 under 1000 and 3 and 1 under 10,000, If wheel 1 be turned once at the same time the wheels 3, 6, and 5 will turn once and thus add in the addition box that many units, namely, 36,500. As a product we obtain, therefore:

1,460
7,300
36,500
45,260

It should be noted here that for the sake of greater convenience, the pulleys should be affixed to the multiplicand-wheels in such a manner that the wheels must move when the pulleys move but that the pulleys do not need to move while the wheels are turned.
Otherwise, when one multiplier-wheel (e.g., 1) is turned and thus all the multiplicand-wheels moved, all the other multiplier wheels (e.g., 2 and 4) would necessarily move, which would increase the difficulty and perturb the motion.
It should be also noted that it does not make any difference in what order the multiplier-wheels 1, 2, 4, etc. be arranged but they could very well be placed in numerical order 1, 2, 3, 4, 5. For even then one is at liberty to decide which one to turn first and which afterward.
So that the multiplier-wheel, e.g., the one representing 9 or whose diameter is nine times as great as the diameter of the corresponding pulley, should not be too large we can make the pulley so much smaller preserving the same proportion between the pulley and the wheel.
So that no irregularity should follow the tension of the cords and the motion of pulleys tiny iron chains could be used in place of the cords and on the circumference of the wheels and pulleys where the chains would rest there should be put little brass teeth corresponding always to the individual links of the chain; or in place of cords, there could be teeth affixed to both the pulleys and the wheels so that the teeth of the multiplier-wheel would immediately catch the teeth of the pulley.
If we wanted to produce a more admirable machine it could be so arranged that it would not be necessary for the human hand to turn the wheels or to move the multiplication machine from operation to operation: Things could be arranged in the beginning so that everything should be done by the machine itself. This, however, would render the machine more costly and complicated and perhaps in no way better for practical use.
It remains for me to describe the method of dividing on the machine, which (task) I think no one has accomplished by a machine alone and without any mental labor whatever, especially where great numbers are concerned.
But whatever labor remains to be done in (the case of) our machine it could not be compared with that intricate labyrinth of the common division which is in the case of large numbers the most tedious (procedure) and (the one) most abundant in errors that can be conceived. Behold our method of division! Let the number 45,260 be divided by 124. Begin as usual and ask for the first simple quotient or how many times 452 contains 124.
It is very easy for anyone with a mediocre ability to estimate the correct quotient at first sight. Hence let 452 contain 124 thrice. Multiply the entire divisor by this simple quotient which can be easily accomplished by one simple turn of the wheel.
The product will be 372. Subtract this from 452. Combine the remainder 80 with the rest of the dividend 60. This gives 8060.
(But that will be effected by itself in the machine during the multiplication if we arrange in it the dividend in such a manner that whatever shall be produced by multiplication will be automatically deducted. The subtraction also takes place in the machine if we arrange in it the dividend in the beginning; the performed multiplications are then deducted from it and a new dividend is given by the machine itself without any mental labor whatever.)
Again divide this (8060) by 124 and ask how many times 806 contains 124. It will be clear to every beginner at first sight that it is contained six times. Multiply 124 by 6. (One turn of the multiplier wheel) gives 744. Subtract this result from 806, there remains 62. Combine this with the rest of the dividend, giving 620. Divide this third result again by 124. It is clear immediately that it is contained 5 times. Multiply 124 by 5; (this) gives 620. Deduct this from 620 and nothing remains; hence the quotient is 365.
The advantage of this division over the common division consists mostly in the fact (apart from infallibility) that in our method there are but few multiplications, namely as many as there are digits in the entire quotient or as many as there are simple quotients.
In common multiplication, a far greater number is needed, namely, as many as (are given by) the product of the number of digits of the quotient by the number of the digits of the divisor. Thus in the preceding example, our method required three multiplications, because the entire divisor, 124, had to be multiplied by the single digits of the quotient 365,—that is, three.
In the common method, however, single digits of the divisor are multiplied by single digits of the quotient and hence there are nine multiplications in the given example.
It also does not make any difference whether the few multiplications are large, but in the common method there are more and smaller ones; similarly one could say that also in the common method few multiplications but large ones could be done if the entire divisor be multiplied by an arbitrary number of the quotient.
But the answer is obvious, our single large multiplication being so easy, even easier than any of the other kind no matter how small. It is effected instantly by a simple turn of a single wheel and at that without any fear of error. On the other hand in the common method the larger the multiplication the more difficult it is and the more subject to errors. For that reason, it seemed to the teachers of arithmetic that in division there should be used many and small multiplications rather than one large one. It should be added that the largest part of the work already so trifling consists in the setting of the number to be multiplied, or to change according to the circumstances the number of the variable teeth on the multiplicand-wheels. In dividing, however, the multiplicand (namely the divisor) remains always the same, and only the multiplier (namely the simple quotient) changes without the necessity of moving the machine. Finally, it is to be added that our method does not require any work of subtraction; for while multiplying in the machine the subtraction is done automatically. From the above, it is apparent that the advantage of the machine becomes more conspicuous the larger the divisor.
It is sufficiently clear how many applications will be found for this machine, as the elimination of all errors and of almost all work from the calculations with numbers is of great utility to the government and science. It is well known with what enthusiasm the calculating rods of Napier, were accepted, the use of which, however, in the division is neither much quicker nor surer than the common calculation. For in his (Napier’s) multiplication, there is a need for continual additions, but the division is in no way faster than by the ordinary (method). Hence the calculating rods soon fell into disuse. But in our (machine) there is no work when multiplying and very little when dividing.
Pascal’s machine is an example of the most fortunate genius but while it facilitates only additions and subtractions, the difficulty of which is not very great in themselves, it commits the multiplication and division to a previous calculation so that it commended itself rather by refinement to the curious than as of practical use to people engaged in business affairs.
And now that we may give final praise to the machine we may say that it will be desirable to all who are engaged in computations which, it is well known, are the managers of financial affairs, the administrators of others’ estates, merchants, surveyors, geographers, navigators, astronomers, and (those connected with) any of the crafts that use mathematics.
But limiting ourselves to scientific uses, the old geometric and astronomic tables could be corrected and new ones constructed by the help of which we could measure all kinds of curves and figures, whether composed or decomposed and unnamed, with no less certainty than we are now able to treat the angles according to the work of Regiomontanus and the circle according to that of Ludolphus of Cologne, in the same manner as straight lines. If this could take place at least for the curves and figures that are most important and used most often, then after the establishment of tables not only for lines and polygons but also for ellipses, parabolas, hyperbolas, and other figures of major importance, whether described by motion or by points, it could be assumed that geometry would then be perfect for practical use.
Furthermore, although optical demonstration or astronomical observation or the composition of motions will bring us new figures, it will be easy for anyone to construct tables for himself so that he may conduct his investigations with little toil and with great accuracy; for it is known from the failures (of those) who attempted the quadrature of the circle that arithmetic is the surest custodian of geometrical exactness. Hence it will pay to undertake the work of extending as far as possible the major Pythagorean tables; the table of squares, cubes, and other powers; and the tables of combinations, variations, and progressions of all kinds, so as to facilitate the labor.
Also, astronomers surely will not have to continue to exercise the patience which is required for computation. It is this that deters them from computing or correcting tables, from the construction of Ephemerides, from working on hypotheses, and from discussions of observations with each other. For it is unworthy of excellent men to lose hours like slaves in the labor of calculation, which could be safely relegated to anyone else if the machine were used.
What I have said about the construction and future use (of the machine), should be sufficient, and I believe will become absolutely clear to the observers (when completed).
<<< End of translation >>>

The second description of Leibniz’s stepped-drum calculator, made by Leibniz himself, appeared in 1710, in Miscellanea Berolinensia, the journal of the Berlin Academy of Sciences. It was a 3-pages short description (see the images below), entitled Brevis descriptio Machinae Arithmeticae, cum Figura, and the internal mechanism of the machine is not described.

Leibniz did manage to create a machine, much better than the machine of Pascal. The Stepped Reckoner was not only suitable for multiplication and division but also much easier to operate. In 1675 during the demonstration of the machine to the French Academy of Sciences, one of the scientists noticed that “…using the machine of Leibniz even a boy can perform the most complicated calculations!”

In 1764, forty-eight years after Leibniz’s death, a Reckoner was turned over to a clockmaker in Göttingen for repair. The job wasn’t done, and the machine wound up in the attic of the University of Göttingen, where a leaky roof led to its rediscovery in 1879. Fourteen years later, the University gave the machine to the Arthur Burkhardt Company, one of the country’s leading calculator manufacturers, for repair and analysis. Burkhardt reported that, while the gadget worked in general, it failed to carry tens when the multiplier was a two- or three-digit number. As it was mentioned earlier, the carrying mechanism had been improperly designed. It’s unknown whether Leibniz has designed a machine without the above-mentioned flaw.

Leibniz’ Logical Machines

Gottfried Leibniz was one of the first men (after Ramon Llull), who dreamed of a logical (thinking) device.

In 1666 Leibniz published his first book (see the lower image), also his habilitation thesis in philosophy, Dissertatio De Arte Combinatoria (On the Art of Combinations, see the treatise), partly inspired by the Ars Magna of Ramon Llull (Leibniz was still a teenager when he encountered the works of Ramon Lull).

Though the design Leibniz places at the front of his book (see the lower figure) appears to be a very simple and even trivial diagram, compared to the copperplates of Kircher’s books, the entire text that follows speaks a new and different language. Leibniz was only 20, but he analyzed as an advanced mathematician the potential power and limits of the art of combination.

In contrast with Llull and Kircher, Leibniz was not at all interested in any esoteric applications of this method, but rather in a way of reproducing the totality of the universe within one science. After reading his very famous treatise on the monads (Monadology from 1714) (monads are something like atoms, situated in the metaphysical realm) as a model for the art of combination, his new, radical perspective is at once comprehensible.

In 1674 Leibniz described a machine for solving algebraic equations. A year later, he wrote comparing logical reasoning to a mechanism, thus pointing to the goal of reducing reasoning to a kind of calculation and of ultimately building a machine, capable of performing such calculations.

There is a letter written by Leibniz to Johann Friedrich, Duke of Hanover, in April 1679, which offers the whole ambitious program of the philosopher. In that letter, we find initially a confession about the source of the method of combination. But then Leibniz starts to criticize Llull and Kircher because, in his view, they did not go far enough in using this art of combination. Regarding his own idea of its use, he says:
“My invention contains the application of all reason, a judgment in each controversy, an analysis of all notions, a valuation of probability, a compass for navigating over the ocean of our experiences, an inventory of all things, a table of all thoughts, a microscope with which to prove the phenomena of the present and a telescope with which to preview those of the future, a general possibility to calculate everything. My invention is an innocent magic, a non-chimerical Cabbala, writing, which everyone can read and which everyone can very easily learn…”

It is quite a pathetic proclamation, but that was the style of the great philosopher 🙂 Leibniz apparently believed that he had invented a general problem-solver, like those in the computer sciences have always dreamed of. But of course, his whole super-ambitious program was not to be realized. Only some aspects of that proclamation were really transposed into useful applications. At first, Leibniz made a few essential steps toward the calculation of probability, which is obviously a very important problem for modern AI (artificial intelligence) applications. He then attempted to transcribe the whole art of combination into a system of formulas because he wanted to calculate every single part of the process, each step, and each result of an interval. Thus he used consequently his mathematical skills to produce a new kind of combination by transposing meanings into figures and values.

Even more—Leibniz was also one of the first men, who realized the importance of the binary numeral system (certainly, he is not the inventor of the binary system. It was known in ancient Egypt already in 18th century B.C., and in India about 300 B.C. In Europe, it was Thomas Harriot, who reinvented it.)

Leibniz discovered that computing processes can be done much easier with binary number coding (in his treatises De progressione Dyadica, dated 15 March 1679 (see the treatise) and Explication de l’Arithmetique Binaire from 1703). In these clear and lucid treatises (the lower figure is shown on the second page of the original manuscript “De Progressione Dyadica”), Leibniz analyzed the possibilities of the binary system and, demonstrated its four fundamental operations of calculation—addition, subtraction, multiplication, and division—he expressed the conviction that one day in future the machines would use this system.

Though hard to believe, in his 3-pages treatise De progressione Dyadica, Leibniz even outlines a calculating machine that works via the binary system: a machine without wheels or cylinders—just using balls, holes, sticks, and channels for the transport of the balls—This (binary) calculus could be implemented by a machine. The following method would certainly be very easy to implement. (A machine with) holes, which can be opened and closed. They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0. Through the opened gates small cubes or marbles are to fall into channels, through the others nothing falls. It (the gate array) is to be shifted from column to column as required for the multiplication. The channels should represent the columns, and no ball should be able to get from one channel to another except when the machine is put into motion. Then all the marbles run into the next channel, and whenever one falls into an open hole it is removed. Because it can be arranged that two always come out together, and otherwise they should not come out.

In a note, written later in his life, when he was reflecting on his works, Leibniz remembered the old program of the universal art of combination:
I thought again about my early plan of a new language or writing-system of reason, which could serve as a communication tool for all different nations… If we had such a universal tool, we could discuss the problems of the metaphysical or the questions of ethics in the same way as the problems and questions of mathematics or geometry. That was my aim: Every misunderstanding should be nothing more than a miscalculation (…), easily corrected by the grammatical laws of that new language. Thus, in the case of a controversial discussion, two philosophers could sit down at a table and just calculating, like two mathematicians, they could say, “Let us check it up…”.

Another remarkable idea of Leibniz, announced in his February 1678, essay “Lingua Generalis”, was connected closely with his binary calculus ideas. Leibniz spoke for his lingua generalis or lingua universalis as a universal language, aiming it as a lexicon of characters upon which the user might perform calculations that would yield true propositions automatically, and as a side-effect developing binary calculus.

Simon Marquis de Laplace wrote: Leibnitz saw in his binary arithmetic the image of Creation… He imagined that Unity represents God, and Zero the Void, that the Supreme Being drew all beings from the void, just as unity and zero expresses all numbers in his system of numeration. …

The simple adding device of the Frenchman Cesar Caze (1641-1719), which he called Nouvelle machine arithmétique and created around 1696, could be considered as one of the most basic calculating devices, which can be invented, a simplified version of the Abaque Rhabdologique of Claude Perrault.

Between 1704 and 1708 Caze corresponded on the topic of calculating devices, including his own, with Leibniz. The device was mentioned for the first time in May 1707, in the second issue of the French journal Nouvelles de la république des lettres, published in Amsterdam (see the nearby image). In the journal, in the paragraph for arithmetic, after mentioning the invention of binary arithmetic by Leibnitz, it is said that Mr. Caze created and demonstrated to the public a rather curious Machine. And that’s all!

In 1711 Caze managed to get a privilege (patent) for his calculating device.

There are several examples of the adding device of Caze, which survived to our time, all of them made in the first half of the 18th century. The example, shown in the photo below, was manufactured in 1720. There are three copies with different dimensions and materials used in the collection of CNAM, Paris. Dimensions of one of the CNAM devices are: 3 cm x 18.5 cm x 29.6 cm, weigh 310 g. IBM’s example is smaller: 27 x 17 cm, weight 80 g. Materials used are wood, cardboard, paper, brass, and textiles.

The device consists of movable rulers (bars) with inscribed digits, which can be seen in windows. The upper row of windows (over the rulers) is used during adding operations, while the lower row (its digits in fact are a complement to 10 of the digits in the upper windows), is used during subtraction. The constriction is so simplified, that the device even doesn’t have a tens carry mechanism.

The rightmost 3 rulers are used for adding sols and deniers (french monetary units at this time, 1 sol is equal to 1/20 of the livre, and 1 denier is 1/12 of the sol). The next 12 rulers are decimal and can be used for adding up to hundreds of billions. The rulers are moving by means of a wooden stylus.

The practical usefulness in calculations of such a simple device is quite questionable, but nevertheless, the machine of Caze gained some popularity at the beginning of the 17^th century under the bombast name Nouvelle machine arithmétique de Caze (The New Arithmetical Machine of Caze). Similar devices were invented many times during the next two centuries after Caze, for example, the device, invented in 1846 by Heinrich Kummer. The device of Kummer however is capable to perform carrying from one column to another.

Biography of Cesar Caze

The French Huguenot (Huguenots were a religious group of French Protestants) César Caze, sieur d’Harmonville et du Vernay, was born in Lyon, Rhone, in January 1641, as the first child in the family of Jean Caze (1608-1700) and Marie Huguetan (1601-1677). César had a sister – Isabeau.

Jean Caze was a Lyon Huguenot and wealthy bourgeois (born in Montpellier), titled as conseiller et maître d’hôtel du Roi et auditeur à la Chambre des Comptes de Montpellier(adviser and butler of the King and auditor at the Chamber of Auditors of Montpellier). Marie Huguetan was a daughter of the Lyon bookseller and bookbinder Jean Antoine Huguetan (1567-1650) and a sister of the lawyer Jean Huguetan (1599-1671), and the bookseller Jean Antoine Huguetan (1615-1681). Jean Caze and Marie Huguetan married on 4 April 1640, in Lyon. Marie was a widow from 1630.

César Caze had a younger brother—Jean Jacques Caze (born November 1644, in Lyon), and an elder brother (from the first marriage of his mother) Jean Antoine de La Motte.

On 3 April 1677, César Caze married in Charenton-le-Pont, Val-de-Marne, to the young Catherine de Monginot (1660–4 Sep. 1719), the daughter of Etienne de Monginot de la Salle, a well known Parisian doctor (b. 1627). The family had a son—Jean Caze (1682-1751), and a daughter, who died in infancy.

Since 1675 César Caze managed a tobacco farm, but in late 1682 he was forced, as many other French Huguenots, to escape from persecution (by the end of the 17th century, some 200000 Huguenots had been driven from France during a series of religious persecutions), to emigrate to the Netherlands. During the European religious wars in the 17^th century, many protestants fled to the Dutch Republic, England, and Switzerland, where they sought refuge. It seems the remaining part of his family, including his wife and his son, together with his father Jean, emigrated to Geneva, Switzerland.

In 1683 Case settled in Hague, and the following year he went to Amsterdam. However, his former business partners in France initiated a trial, and Caze was sentenced to pay the sum demanded. Sure of his right, Caze refused to pay (although his father and stepfather were rich men, so he could easily find the money requested), thus he was forced to serve in the prison of Leeuwarden for more than 12 years (from April 1688 to July 1700).

César Caze worked in Amsterdam mainly as a maker of glasses, telescope maker, and general technician, but remained known also for his calculating device, his scientific interests, and busy correspondence with Leibnitz and Huygens, as well as for his dissertation on the use and improvement of arithmetic from the beginning of 18^th century (Amsterdam, August 1711, The invention of calculating machines and a dissertation on the use and improvement of arithmetic).

A report from 1696 says that Caze “excelled in mathematics and other studies”. The Amsterdam burgomaster (mayor) Johannes Hude had employed Caze for many years for the city of Amsterdam.

It is known that as early as 1671, Caze had done experiments in Amsterdam with a machine calculating the speed of a ship. A few years later he was involved in the design of clocks and in 1688 he published a tract on balances, De l’usage des staters, ou romaines balances.

César Caze lived in Amsterdam until his death in 1719, separated from his family, which lived in Geneva.

Around 1670 the Parisian doctor of medicine and self-taught architect Claude Perrault devised a simple calculating device, called Abaque Rhabdologique. The device was firstly described in a small book—Recueil de plusieurs machines, de nouvelle invention… (Collection of several newly invented machines…) published in 1770 in Paris, which 22 pages of text and several pages of sketches (see the book digitized by Google) contain nine inventions of Perrault, between them two machines for escalating and moving burdens, a pendulum-controlled water clock, a pulley system to rotate the mirror of a reflecting telescope, etc.

Abaque Rhabdologique was described also in the journal of Academie Royale Des Sciences (Tome Premier, printed in 1735, see description of the Abaque Rhabdologique , pages 55-59). In this description is explained that the name of the machine Perrault derived from the mathematical practice of the ancients, who had used tablets (abacuses) to write numbers, and who had the capacity to perform many arithmetic operations by employing small rods marked with digits (rhabdology).

The device was devised probably between 1666 and 1675 (at that time Perrault was engaged mainly in architectural projects, designing the eastern facade of the Louvre, l’Observatoire de Paris, etc., so we can easily imagine that he needed some calculating tool). It is unknown whether a working copy of the device has been made by Perrault, and in the present day exist only several replicas.

In its contemporary reincarnations, the Abaque Rhabdologique is usually a small metal plate (30 cm x 12 cm x 0,7 cm) with a thickness of a finger and weight of some 1.15 kg. In the lower part of the lid is inscribed a multiplication table.

Over the plate are mounted seven small rules (marked with letters a, b, c, d, e, f, and g on the sketch), which can be moved upwards and downwards. The rules are graduated to 26 parts by deep cuts, and the edge of the pin, which actually moves the rules, can be pushed in these cuts. Between the cuts are drawn ascending and descending rows of digits, with four empty divisions between zeroes. Rule a represents the units column, rule b—the decimal column, and so on to rule g, which represents the millions. The rules are separated by thin plates, which have perforations at the bottom.

Near the bottom of each rule (with the exception of the rule for units), to the right side, there is a rule with 11 notches (marked with L), and the distance between notches is equal to the distance between digits, marked on the rules. From the other side of the rule with notches by means of springs are attached the hooks M. Due to the separating thin plates, the hook will be hidden in the body of the rule till the moment, when the hook will become symmetrical toward the opening in the plate. At this moment, the spring will push out the hook, which will pass the opening and will clutch to the notch of the lower rule, and will move it one division downwards, making a carry to the next column.

On the front lid of the device ABCD are placed two long horizontal windows EF and GH. When the rulers are moving up or down, in these windows are seen the digits on the plates, and at every moment the sum of the digits of a particular ruler in upper and lower windows is always equal to 10. The window GH is used during adding operations, while the window EF is used during subtraction.

Between the windows are made 7 narrow vertical channels I-K, which are divided into 10 and marked with digits.

For entering a digit, in the particular cut of a ruler, which can be seen in the vertical channel, must be put a stylus, and then the ruler must be moved until the stylus touched the bottom edge of the channel. After this action, the number, which has been entered, will be shown simultaneously in both windows.

If to an entered number, for example, 7, must be added 6, we have to perform the same action. During the movement of ruler a to the bottom of the device, hook M will enter into cohesion with the cogs of ruler b and will move it one division downwards. As a result of this in the decimal column will appear 1. In order to get the proper digit in the units column (which in this example must be 3), we have (without pulling the stylus out of the cut) to move the ruler upwards, until the stylus touched the bottom edge of the channel.

During the performing of subtraction, the actions of the operator are analogous, but the result must be read not in the lower, but in the upper window. If the minuend contains one or more zeroes, the result of the operation must be corrected.

Let’s see the original description of the device (pages 17 through 20 of the book Recueil de plusieurs machines, de nouvelle invention… are translated in the following section):
An excerpt from the book Recueil de plusieurs machines… (pp. 17-20)
I call this machine Rhabdological Abacus because the Ancients called abacuses small tables or boards on which they wrote arithmetical numerals and because they called rhabdology the ability to perform various arithmetical operations by means of several small rods marked with digits.
The machine that I propose does about the same thing. It is an abacus or small board about one finger thick, one foot long and half a foot wide. It is carved and made of thin ivory or copper plates, enclosing small rules marked with figures. In the cover plate, marked ABCD, two long and narrow windows in which the figures are displayed have been cut out, one EF at the top and one GH at the bottom. These windows are about three inches apart and the area between them has cut-out grooves IK, ending at about fives lines of the windows and distant also about five lines from each other.
Under the cover plate, several small rules a, b, c, d, e, f, g, lying side by side, can slide up and down: they are about 4 lines wide and seven and a half inches long: their length is divided into 26 equally spaced parts by engraved crossing lines. These lines are deep enough to hold in position the tip of a stylus used to move them. Twenty-two figures have been marked in the spaces between the engraved lines, eleven upwards and also eleven downwards: this is done in a way that four spacings are left empty between each series of figures. Thus we find, beginning from above, 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 0. and continuing downward, after having left four empty spacings, 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. 0.
Between the grooves, the cover plate is marked upwards with the nine digits 1. 2. 3. 4. 5. 6. 7. 8. 9. keeping the same spacing as on the rules.
When the rules are lowered or lifted, the digits show one by one in the windows, but in a way that two digits of the same rule appearing in the two windows always sum up to 10. That’s to say, if figure 9 shows in the upper window, figure 1 will show in the bottom window, and if there is 6 in one window there will be 4 in the other one.
These rules, lying side by side, represent the decimal order; the first rule at the right, marked N above the upper window EF, being for the units; the second one, marked D, being for the tens; the third, marked C, for the hundreds, etc. They are separated by very thin blades, which are interrupted by the value of three spacings; the middle of this interruption being opposite the bottom window. Each rule has LL rack-type notches at one side of the bottom, each notch being opposite the eleven figures, and at the other side a pawl M, to pull downwards the adjoining left-hand rule. To ensure that the pawl does not lower the rule it has to pull, more than one spacing, as is necessary, it needs to enter into the rule and stay hidden there, without being able to get out, until it is opposite the bottom window. Furthermore, it has to retract as soon as the rule it is pulling has moved a value of one spacing. This is done thanks to two features: the first one is that a spring N shoves the pawl outwards; the second one is that the interruption of the blades separating the rules, enables the pawl to get out and engage into the notches of the rack. This engagement is possible only opposite the interruption and when the rule slides up or down; at the places where the blades are not interrupted, the pawl stays enclosed and is not in a state to engage.
To use the machine, one puts the tip of a stylus in one of the grooves, opposite one of the digits marked from top to bottom between the grooves. Pressing the stylus in the engraving between the digits, one slides it until it reaches the bottom of the groove: the same number chosen by the stylus will then appear in one of the windows, the bottom window being for addition and multiplication and the upper window for subtraction.
For instance, if one wants to obtain the number 8, one lowers this number in the window as previously explained: but if one wants to add 7, instead of this number, a digit 1 would show in the window as being the number of tens and nothing would show at the units location. In this case, without removing the tip of the stylus from the engraving where it presses, one has to slide it upwards to the top of the groove, and the digit 5 will then appear in the window at the units position. Thus, each time that the rule has been lowered as much as possible, nothing or only a 0 would show in the window, the stylus would have to be slid upwards.
For subtraction, one needs to set in the top window the number from which another one has to be subtracted.
For instance to subtract 34 from 123, one needs to put the stylus on the 4 of the units location and pull it to the bottom and then pull in the same way the 3 of the tens location. The number 123, which showed in the window, will then be replaced by the number 89.
It must be pointed out that when the number from which another one is subtracted contains one or several 0s, one unit needs to be subtracted from the remaining number, that is to say the one after the 0 toward the left.
For instance, if one wants to subtract 92 from 150, the machine will give 68 instead of 58, but the latter will be obtained if we subtract one from the 6 appearing at the tens location, and after the 0 of 150 which is of the units order. The same applies when there are several 0s.
For instance, if one wants to subtract 264 from 1500, the machine will give 1346 instead of 1236, but the latter will be obtained by subtracting one unit from 4, because of the first 0, and another unit from 3, because of the second 0.
For multiplication, one needs to do the same as for addition. For instance, if one wants to multiply 15 by 15, one needs to mark five times 5 which is 25 in the lower window by setting a 5 in the units location and a two in the tens location; finally mark one time 5 in the tens location and one time 1 in the hundreds location: this will give the result 225.
End of the excerpt

The simple and ingenious idea of the calculator of Claude Perrault was a step aside from the common development of mechanical calculating devices, which are based on the gear-wheels. This same idea will be applied by many inventors later in several cheap, simple, and reliable calculating devices, such as the adding devices of César Caze and Heinrich Kummer, and even in more sophisticated devices as the multi-column adding machine, designed in 1891 by Peter J. Landin of Minneapolis (US Patent No. 482312), which will be later produced in several countries in great quantities and many varieties, e. g. popular Comptator in Germany.

Biography of Claude Perrault

Claude Perrault was born in Paris on 25 September 1613, in the wealthy bourgeois family of a Parisian barrister—Pierre Perrault (1570-1652), and his wife Pâquette Le Clerc (or Leclerc) (d. 1657). Perrault was a numerous, talented, versatile, and close-knit Parisian family. Its founder, Pierre Perrault, born to a royal embroiderer from Tours, moved in 1592 only 22 years old to Paris and developed a career as an avocat (barrister) in the city.

After their marriage on 27 January 1607, Pierre Perrault and Pâquette Le Clerc had seven children, but two of them died young, so Claude had four remaining brothers. The eldest brother—Jean Perrault (1609-1669), was (just like his father) a lawyer and advocate at the Parliament, and later was in the service of Henri II de Bourbon, Prince de Condé. Pierre Perrault (1611-1680), who was also a lawyer and Receiver General of Finances in Paris (1654-1664), later became a famous scientist (he developed the concept of the hydrological cycle, and together with Edme Mariotte, he was primarily responsible for making hydrology an experimental science). Nicolas Perrault (1624-1662), was an amateur mathematician and doctor of theology in Sorbonne, known for his denunciation of the Jesuits, and who was excluded from the Sorbonne for Jansenism and for defending Antoine Arnauld. However, the genuine celebrity of the Parisian branch of the Perrault family was found to be the youngest brother—Charles Perrault (1628-1703), who became the world-renowned author of Tales of Mother Goose. The only daughter of the family—Marie, died at thirteen, and François, a twin of Charles, died in July 1628, only 6 months.

As boys, the Perrault brothers collaborated in such things as writing mock-heroic verses, and in adult life, each brother aided the career of the other. Claude Perrault was educated at the prestigious Collège de Beauvais (later he was followed by his younger brother Charles), one of the leading schools of France, educating pupils whose parents were prominent in the French establishment. Pierre Perrault wanted his son to study medicine, anatomy, and mathematics, so in 1634 Claude enrolled at the University of Paris to study medicine. He received a bachelor’s degree in 1639, and two years later, on 19 December 1641, he received a master’s degree (doctor of medicine).

After graduation, Perrault started his career as a physician and later on became a leader of a group of anatomists, who undertook dissections and descriptions of various animals. He proposed two theories, concerning the circulation of sap in plants and embryonic growth from preformed germs. These theories were highly influential in his lifetime and for many years thereafter. In 1681 Perrault began publishing an all-embracing natural philosophy, which comprehended his research in anatomy, various aspects of animal and plant physiology, and acoustics. In his longest essay, he explained sound as an agitation of the air, rather than by the concept of sound waves.

After twenty years of practicing medicine, around 1660 Claude Perrault turned his attention to architecture and now he is best known as one of the architects of the eastern facade of the Louvre (see the photo below), known as the Colonnade, built between 1665 and 1680 and cited everywhere as an example of the classicist phase of the French baroque style. Perrault’s architectural career was actually inspired by the translation into the French language he had started of the ten books of Vitruvius (published in 1673, with the help of Jean Baptiste Colbert), the only surviving Roman work on architecture.

When King Louis XIV decided to rebuild the Louvre in the 1660s, Perrault collaborated with the famous architects Louis Le Vau, Charles Le Brun, and Francois d’Orbay to submit a worthy design for the competition, and his design was selected (not without the support of his brother Charles, who was at that time the First Commissioner of Royal Buildings, and who was promoted to this position from his brother Pierre, the Receiver General of Finances for Paris from 1654 until 1664).

The Colonnade was begun in 1668 and was almost completed in 1680, by which time Louis XIV had abandoned the Louvre and focused his attention on the Palace of Versailles. Nevertheless, the Colonnade may justly be regarded as the masterpiece of French architecture, and the finest edifice that exists in Paris.

Perrault’s architecture projects include also several other buildings in Paris like: l’Observatoire de Paris; the church of St-Benoît-le-Bétourné; the church of St-Geneviève; the altar in the Church of the Little Fathers; the triumphal arch on Rue St-Antoine, started in 1670 (Perrault’s design was preferred to competing designs of Le Brun and Le Vau, but was only partly executed in stone, and when the arch was taken down in the 19th century, it was found that the ingenious master had devised a means of so interlocking the stones, without mortar, that it had become an inseparable mass); a house for Louis XIV’s prime minister— Jean Baptiste Colbert, in Sceaux in 1673.

Although Claude Perrault stopped practicing medicine around 1661, he continued to treat family, friends, and the poor. At that time, besides the calculating device, he designed several other machines, which he occasionally displayed to the Academy: a pendulum-controlled water clock, a pulley system to rotate the mirror of a reflecting telescope, a machine to produce roper for ships, a machine for testing projectiles, machines to overcome the effects of friction. Many of his machines were used in the Louvre and by 1691 at Les Invalides. Perrault also wrote an essay on ancient music, to show its inferiority to that of his own day.

After Colbert’s death in 1683, the position of the Perrault family gradually declined. In the middle 1680s, Claude Perrault’s house was among those torn down to make room for the Place des Victoires, and he seems to have spent his last years writing his essays (like Essais de Physique, and his own attempt to apply a modern approach to beauty, his architectural treatise, Ordonnance des cinq espèces de colonnes selon la méthode des Anciens), living at his brother Charles’ house.

Claude Perrault became a founding member of the French Academy of Sciences (Académie des Sciences) when it was founded in 1666. He remained a keen academician until his death and died as a genuine researcher in Paris on 9 October 1688, of an infection, caught during a dissection of a camel in the Botanical Garden of Paris.

Doctor Illuminatus Ramon Llull is an amazing figure in the field of philosophy during the Middle Ages, and one of the first people who tried to make logical deductions in a mechanical, rather than a mental way. His method was an early attempt to use logical means to produce knowledge. He demonstrated in an extremely elementary (but nevertheless workable) way that human thought can be described and even imitated by a device. This was a small step toward the thinking machine. But every journey, even the longest, begins with a tiny step.

Around 1275, Llull designed a method, based on something like a logical machine, which he first described in full in his Ars magna generalis ultima. Ars brevis (“The Ultimate General Art”, published in 1305). This was a method of combining religious and philosophical attributes selected from a number of lists. It was intended as a debating tool for winning Muslims to the Christian faith through logic and reason. Llull’s inspiration for the Ars magna is thought to have come from observing a device called a zairja, which was used by medieval Arab astrologers to calculate ideas by mechanical means. It used the 28 letters of the Arabic alphabet to signify 28 categories of philosophic thought. By combining number values associated with the letters and categories, new paths of insight and thought were created.

The radical innovation Llull introduced in the realm of logic is, in fact, the construction and the use of a machine made of paper to combine elements of thinking, i.e. elements of language. With the help of connected geometrical figures, following a precisely defined framework of rules, Llull tried to produce all the possible declarations of which the human mind could think. These declarations or statements were nevertheless represented only by a series of signs, that is, chains of letters.

Let’s take a look at the hardware of Llull’s machine:

This doesn’t look very impressive, does it!? It is just three circles and some kind of alphabet.

This hardware consists of three circular paper disks fixed on an axis on which they can be turned. The paper disks contain a limited number of letters—a special lullistic alphabet. When the circles are turned, step by step, all possible combinations of these letters are produced. So what is the trick of this machine? Let’s look at the next figure:

Every single letter, from B to K, represents not merely itself, but several strictly defined and placed meanings. By writing the letters from B to K as key terms heading a table, a series of different sentences can be easily constructed. For example, B=Bonitas, C=Magnitudo, D=Duratio, E=Potestas, F=Sapientia, G=Voluntas, H=Virtus, I=Veritas and K=Gloria. This is, initially, the paper circle called the Prima Figura. The next strictly defined table of words can be produced on the next circle, perhaps as seen on the Secunda Figura (shown below), where we find categories and relations of thinking.

Hence the machine allows all the words to be combined by turning the circles step by step. In this manner, it is possible to connect every word with every other word placed in a position of a table—depending only on the construction of the individual tables. Imagine how Llull could play this out: bearing in mind the inscribed words of the Prima Figura. These nine words are none other than the attributes of God. Combined with a table of nine questions, it is possible to construct the skeleton of the “Proofs of God.” The machine shows all possible statements and declarations about God.

Examining the next design from the treatise, the so-called Tabula Generalis, we can find a very interesting table. Behind these series of letters are the hidden words of the utilized tables. These columns of letters are supposed to represent, very precisely, neither more nor less than, the totality of human wisdom;-), but actually the letters of the lullistic alphabet contain a rich potential of meanings. The connection to a certain site in each table allows each letter to represent unlimited words of unlimited fields.

So, this Tabula Generalis must have been something like a subset of the truth table of the logic, described from the notions and defined in the circles. The logic itself consists of all possible reasonable combinations of the notions. While we have no rules, algorithms, or programs to tell which combination is right or wrong, we could have a truth table (One may just wonder “Who knows how to compose such a table, maybe God?”). And if we can include in the proper circle each human-defined notion, and we have a proper entry in the table, then we have what? Bingo! We have the Universal Wiseacre.

Llull’s ideas would be developed further by Giovanni de la Fontana and Nicholas of Cusa in the 15^th century (in his work De coniecturis Nicholas developed his method ars generalis coniecturandi, in which he describes a way of making conjectures, illustrated by wheel charts and symbols that much resemble those of Llull), Giordano Bruno in the 16^th century (Bruno used the rotating figures of the Lullist system as instruments of a system of artificial memory, and attempted to apply Lullian mnemotechnics to different modes of rhetorical discourse), and by Athanasius Kircher and Gottfried Leibniz in 17^th century. That’s why Llull is sometimes considered a pioneer of computation theory, especially given his influence on Leibniz.

Biography of Ramon Llull

Ramon Llull was born in 1232 (or beginning of 1233) into a prosperous Catalan-Aragonese family in Palma, the capital of the just-formed Kingdom of Majorca. He was the only son of Ramon Amat (sobriquet Llull), who had come from Catalonia with the conquering armies of James I three years before, and Isabel d’Erill, who came to Majorca in 1231. Ramon and Isabel were members of bourgeois middle-class families in Barcelona, who in 1229 encouraged and financed the efforts of King James I of Aragon to conquer the island of Majorca, at that time under Muslim dominion, in exchange for land and privileges. Following the triumph over the Moors, they received lands and moved to the island.

Very little is known about his youth, but Ramon obviously was a restless and clever lad, who spent the first 3 decades of his life as a typical courtier, serving King James I and from the middle 1250s as a seneschal (the administrative head of the royal household) to the future King James II of Majorca.

In 1257 Llull married Blanca Picany (several years younger wealthy relative of the King), and they soon had two children, Domènec (Domingo) and Magdalena. Although he formed a family, he continued the licentious and wasteful life of a troubadour up to 1263, when he had a religious epiphany, and decided to dedicate his life to God. He sold his possessions, reserving a small portion for his wife and children, and made a pilgrimage to Santiago de Compostela. Upon his return to Mallorca, he dedicated the rest of his life to learning, translating, teaching, writing, and traveling through Europe and North Africa.

Llull was an extremely prolific author, and starting from the early 1270s he wrote a total of more than 250 works in Catalan, Latin, and Arabic, in a variety of styles and genres, and often translating from one language to the others. The romantic novel Blanquerna is widely considered the first major work of literature written in Catalan, and possibly the first European novel.

In 1314, at the age of 82, Llull traveled to North Africa to promote catechesis but was stoned by an angry crowd of Muslims in Tunis. Genoese merchants took him back to Mallorca, where he died at home sometime in the first quarter of 1316.

In 1842 a certain Francis Marston of Aston, Parish of Hopesay, County of Salop, England, obtained a Letters Patent №9235 for Apparatus for Making Calculations. No details are known about this machine, most probably it remained only on paper.

The device of Marston combines two columns of figures arranged in two circles, the one being fixed and the other revolving, and also combined therewith a pound, shilling, and pence table, for determining the amounts when the numbers ascertained are required to be reduced to pounds, shillings, or pence.

Biography of Francis Marston

Very little is known about the inventor of this calculating device. Francis Marston, Gentleman of Diddlebury, Shropshire, was born on 17 October 1789, in Aston on Clun, Shropshire, the first son of local Squire John Marston (1757-1831) of Cheney Longville, Shropshire, and Mary Carter of Sibdon (died 10 Dec 1806). Francis had an elder sister – Ann (born 7 Aug 1788, died several weeks later), and a younger brother – Richard (3 Nov 1792 – 12 Nov 1866), and a sister – Elizabeth (31 Aug 1795 – 18 Apr 1882). Marston family had close ties to this part of Shropshire and were major land owners for two centuries or more.

Francis Marston was a landowner and farmer with much experience of inclosure, who used to serve also as a Magistrate of the county of Shropshire. On 5 June 1824, he married to Elizabeth Jones (1794—2 Feb 1883) of Hope, Edvin Loach, Worcestershire. The couple had 5 children, 4 sons and 1 daughter – Richard (17 Oct 1825 – 26 Jun 1892), and Alfred (9 May 1834 – 20 Mar 1896).

Francis Marston died on 6 June 1850.

George Brown (1650-1730) is a Scottish arithmetician and dissenting minister, known primarily for the invention of an arithmetical instrument, called Rotula Arithmetica—a very simple and not very successful calculating device for addition, subtraction, multiplication, and division.

It seems Brown invented Rotula Arithmetica sometime in the 1690s, as in 1698 he was given the sole privilege (something like a patent) to frame, make and sell his instrument for the space of 14 years…

Brown described his instrument in the book An Account of the Rotula Arithmetica, published in 1700 in Edinburgh (see below the title page of the book). In the book, he described also his new method of teaching the simple rules of arithmetic. In the same year, 1700 Brown published also another book, called A Specie Book, to be used in conjunction with the Rotula Arithmetica. The book contains currency tables because many of the coins were not Scots-minted silver, but foreign currency, legal tender in Scotland, at values fixed by the Privy Council and Parliament.

Several copies of the instrument were made, but it is doubtful that Brown himself made the rotula, he probably employed an engraver. The instrument in the lower photo is inscribed for the right Honble Patrick Earle of Marchmont, Lord High Chancelour of Scotland, Feb 28. 1699, as Brown was clearly searching for an influential patron (Sir Patrick Hume (1641-1724) was a Scottish statesman).

Rotula Arithmetica consists of two parts: a circular plain (movable plate), moving upon a center-pin; and a ring (fixed plate), whose circles are described from the same center. The outermost circular band of the movable, and the innermost of the fixed, are divided into a hundred equal parts, and these parts are numbered 0, 1, 2, 3. etc. Upon the ring, there is a small circle having its circumference divided into ten equal parts, furnished with a needle that shifts one part at every revolution of the movable. With this simple instrument, the calculations can be done by moving the plate around the axes and accounting the numbers, as the four arithmetical operations can be performed not only for integers but also for decimals finite and infinite.

Biography of George Brown

Not a whole lot is known about George Brown. Born in 1650, Brown attended Aberdeen’s Kings College, matriculating in 1664 and graduating in 1668. Then he worked as a teacher of mathematics in Edinburgh. Later on, he worked as a minister in Stranraer, schoolmaster of Fordyce School, Banffshire, and from 1680 schoolmaster in Kilmaures, Ayrshire.

George Brown was a good mathematician, but a poor minister. He wasn’t zealous in prayers and was frequently charged for exercising discipline and marrying without proclamation. He was banished from Edinburgh from 1692 until 1698, for “… he had not prayed for their said majesties and in the terms aforesaid and having refused to do the same in time coming…”.

Brown is the author of several other mathematical books, popularizing the decimal notation, considered to be important works at the time. The last of them, Arithmetica Infinita; or The Accurate Accomptant’s Best Companion contriv’d and calculated by the Reverend George Brown, published in 1717-1718 in London, was endorsed by the famous Scottish mathematician John Keill.

George Brown died in 1730 in London.

In the 1650s the Italian scientist Tito Livio Burattini created a calculating device (so-called ciclografo), which in 1658 (or even before) he donated to Ferdinando II de’ Medici, Grand Duke of Tuscany (there are two letters from the Italian scientist Giovanni Alfonso Borelli, dated November and December 1658, which mentioned istrumento o cassettina numeraria (instrument or casket for numbers) sent by Burattini to the Grand Duke.)

The Grand Duke was obsessed with new technology and had several hygrometers, barometers, thermometers, and telescopes installed in his palace. Burattini apparently knew him very well because he served the Polish court and in 1655-1657 he took part in several diplomatic missions in Austria (Vienna) and Italy (Florence, Bologna), spending some time as a guest of the Grand Duke. It is known also that while in Florence, Burattini designed a water clock for the Grand Duke and later made several microscopes and telescope lenses for Duke’s brother, Cardinal Leopoldo de’ Medici. The Grand Duke obviously highly appreciated Burattini, because in August 1657, returning from his mission in Florence, Burattini brought with him to Poland many gifts of the Grand Duke, “quelques gentilesses de mécanique”.

As it was mentioned in the article for Pascaline of Blaise Pascal, there was at least one Pascaline at the disposal of the Polish court. In the second half of the 1640s Pierre Des Noyers, secretary of the Queen of Poland Maria Luisa Gonzaga, had come into possession of a specimen of Pascaline, and had lent it to King Władysław IV Vasa. It seems that the King had literally fallen in love with this wonderful device, and he never wanted to return the specimen borrowed from Des Noyers. He even ordered two more specimens to the French mathematician Gilles de Roberval, custodian of Pascal’s discovery in Paris. Des Noyers also explained the Polish monetary system to Roberval, in the belief that Pascaline could also be used in Poland for calculating army wages. Obviously, while in the Polish court in Krakow, Burattini had the opportunity to observe the work of Pascaline. Thus in the 1650s, he decided to build a similar device (like Pascal’s contrivance) himself.

Presently the machine, attributed to Burattini (see the photo below) is kept in Florence, Italy, in the Istituto e Museo di Storia della Scienza.

The device (complete with a wooden case) consists of a thin sheet of brass with a length of 20 cm, upon which surface are mounted 18 disks. All the disks are connected 2 by 2, which means, that every upper (bigger) disk is connected to the lower (smaller) disk. By that means, the carrying of numbers can be done only from an upper (bigger) disk to the lower (smaller) one, but not between different digital positions of a number. The number to be transferred to the next digit is displayed on a marker disc above each digit. The tens carry is then transferred manually by turning the next digit by the corresponding number of tens carry and resetting the marker.

The lower six pairs of disks are decimal (10 graduations—from 0 to 9), while the upper pairs of disks are graduated from 1 to 12, from 1 to 19, and from 1 to 7 respectively (from the left to the right), in order to be used for monetary calculations (Italy had no unified currency in the 17th century since it has been for centuries divided into many city-states, but for example according to the Venice and Tuscany monetary systems: 1 Ducato=7 Lire, 1 Lira=20 Soldi, 1 Soldo=12 Denari, etc.)

Recently a new version (and it seems rather well-grounded) for the above-mentioned device, attributed to Burattini, was proposed by the historian Vanessa Ratcliff. Exploring Samuel Morland and his calculating machines, she not only noticed the well-known fact, that Burattini’s machine is quite similar to one of the devices of Morland but also examined at some length the inventory information for Burattini’s machine to make the conclusion, that the present machine was not the original one of Burattini.

Yet the first note for the machine (from Borelli in November 1658) mentioned the device as “casket”, not as “plate” or “sheet”. There are also several inventory records for the Medicean scientific collection (from 1660, 1704, 1738), which described the machine of Burattini as an eight wheels device, with a size of about 43 x 12 cm. However, in the catalogue from 1779, the machine is described as Una macchinetta forse aritmetica di due lastre di ottone centinate che racchiudono 18 cerchi tra grandi e piccoli, numerati, imperniati, e da muoversi a mena dito. La macchinetta ha la faccia dorata, ed è lunga nel più pollici 7.3… (a small machine, probably arithmetic, made of two ribbed brass plates that enclose 18 large and small circles, numbered, hinged and to be operated with fingers. The machine has a golden face and is 7.3 inches long…)

So it seems an entirely different machine, not only by appearance but also by dimensions (21 cm long, with 18 wheels, while the first machine was 43 cm long and had 8 wheels.) Interestingly enough, the new description fits perfectly with the present object from the Florence Museum, but not with the description of the original machine of Burattini. So what happened?

The most probable version is the following: Burattini did make in the 1650s a Pascaline-like calculator (first descriptions of the device fit quite well with the 8-wheel Pascaline), which he donated to the Grand Duke. Sometime between 1738 and 1779, the machine sank into obscurity (it is known that in 1746 almost the whole Medicean scientific collection was sent to Vienna, while the Florence collection was enriched with many pieces from Lorrainese Chamber of Physics of Lunéville, under the supervision of the famous french mechanic Phillippe Vayringe (1684-1746), as the maker of one of the machines. Probably during this period the original device was lost or sent to Vienna, while the present device was included in the Florence collection.)

So it seems the present machine is a later device (very similar to the money adder of Morland), made by an unknown maker and mistakenly attributed to Burattini, while the original machine of Burattini, unfortunately, had been lost.

Literature:
1) Ratcliff, J. R., ‘Samuel Morland and his Calculating Machines c.1666: the Early Career of a Courtier–Inventor in Restoration London’. Brit. J. Hist. Science, 40(2), 2007, pp. 159–179.
2) Hénin S., Early Italian Computing Machines and their Inventors, in: A. Tatnall (ed.), Reflections on the History of Computing, Springer, 2012, pp. 204-230.

Biography of Tito Livio Burattini

Tito Livio Burattini (known also under his polonized names Tytus Liwiusz Boratini, Boratyni, or Buratin), was born on 8 March 1617, in an old and wealthy family of the local rural nobility in Agordo, a small mining town in the Republic of Venice (now in the province of Belluno, Italy). The family was involved in mining production and owned many lands in the area and even a house in Venice.

The paternal grandfather of Tito Livio Burattini—Niccolò, was knighted by the Holy Roman Emperor Rudolf II in 1591, together with his brothers Tito Livio, Girolamo, and Giovanni, thus allowing the family to add to their names the title “da Susino” (because the family originated and had property also in Susin di Sospirolo, a small town some 20 km south of Agordo).

Burattini was baptized under the name Tito Livio Niccolò (to remember his grandfather) and had a younger brother—Filippo (Filip) (born in 1620). Burattini’s father’s name was also Tito Livio (he died in 1665). Burattini’s mother was Isabella (or Elisabetta) Milo (1590-1695), also from a local noble family.

The native house of Burattini is still preserved in the central square of Agordo (see the nearby images).

In Agordo is still preserved also another house of the Burattini family with their coat of arms over the front gate (see the image below).

The primary school in Agordo now is called Tito Livio Burattini (see the photo below):

Almost nothing is known about Burattini’s childhood. As a boy, he was interested in the problems of mechanics, and in the middle of the 1630s, he studied at the Universities of Padua and Venice, winning a comprehensive knowledge of mathematics and physical sciences, architecture, and others.

In 1635 a terrible fire destroyed the town of Agordo, but it seems Burattini had left his hometown several years before this disaster.

Burattini early became a traveling scholar and in 1637 he went abroad to Egypt (just like the inventor of the Sector, Fabrizio Mordente), where he stayed until 1641, devoting himself to the study of Oriental languages ​​and the discovery of Egyptian antiquities, visiting and measuring pyramids and obelisks, exploring the Nile and its periodic flooding. Burattini even worked for some time as an assistant of the English mathematician and astronomer John Greaves (1602-1652) (see the nearby image) with his famous work on the pyramids, crowned by his important book Pyramidographia (1646). In 1639-1640 they measured several pyramids (especially the Great Pyramid of Giza), obelisks and monuments, trying to classify them, and drew up plans of several towns, including Alexandria, Memphis, and Heliopolis. In his notebooks, describing his exploration of the Great Pyramid of Giza, Greaves noted his work with Burattini, moreover, a part of his notes are in Italian, which comes to demonstrate the close collaboration between them.

After returning to Europe in 1641, Burattini settled for some time in Germany, but in 1642 was invited to serve at Polish Royal Court in Krakow. He accepted the proposal and settled in Poland for several years. Here he found good friends like Stanislaw Pudlowski (a pupil of Galileo and Professor at Jagiellonian University), Johannes Hevelius (a prominent Polish astronomer), Girolamo Pinocci (1613-1676), and Pierre Des Noyers, a King’s Secretaries, and others, and worked together with them on various scientific topics.

In 1645 Burattini returned for some time to Italy, then traveled again to Egypt, before settling permanently in Poland in 1647, this time together with his younger brother Filippo. The new Polish queen Marie Louise de Gonzague was a high-ranking and keen patron of sciences and arts and invited many European scientists to settle in Poland.

Burattini lived in Poland up to his death (leaving that country only occasionally and for short periods), serving 4 Polish Kings—Władysław IV, Jan II Kazimierz, Michał Korybut, and Jan III Sobieski as an architect, engineer, mechanic, diplomat, etc. He wrote several books, carried out experiments in optics and astronomy, manufactured lenses for microscopes and telescopes, constructed devices of various types, designed several important buildings, performed a couple of diplomatic missions ordered by his patron Queen Marie Louise Gonzaga, took part in military missions and battles, etc.

In 1650 Burattini was appointed as the Regis Poloniae Architectus (Polish Royal Architect) and directed the construction of the royal palace at Krakowskie Przedmieście in Warsaw, the Palace of Andrzej Morsztyn, the Church of the Discalced Carmelites, etc. He carried out also restoration works at Ujazdowski Castle in Warsaw (see the photo below). In Ujazdowie he arranged the first Polish astronomical observatory, in which he discovered the spots on Venus in 1665. In 1660 Burattini was appointed as a financial secretary of the Royal Court.

In 1647 Burattini presented to the Polish King Władysław IV a treatise entitled Dragon Volant (Flying Dragon, see the sketch below) with drawings of a complex ornithopter. The King showed particular interest and despite the difficult for Poland wartime, ordered a working model to be produced. In the same 1647 a small 150 cm model, carrying a cat as a passenger was demonstrated before the Polish Court. Burattini was granted 500 talers from the Royal Treasury for the construction of a full-size machine.

The machine was ready in May 1648 and had four pairs of wings, mounted in tandem and a large folding parachute. The machine had a crew of three, and obviously, no one can suggest that it really flew, despite the fact, that Burattini even maintained, that he would fly from Warsaw to Constantinople (some 1700 km) in 12 hours:-) Despite this funny statement, most historians believe, that Dragon Volant is the most important milestone in the development of “heavier-than-air” flying machines between Leonardo Da Vinci at the end of the 15th century and Sir George Cayley in the early 1800s.

The news of the flying models constructed by Burattini and about plans of implementation of the machine itself circulated among many European countries. What remains today is a treatise by Burattini Il volare non e impossibile, and two drawings of the flying dragon, one of which was sent to be assessed by Blaise Pascal.

In 1655-1657 Burattini took part in several diplomatic missions in Austria (Vienna) and Italy (Florence, Bologna). Returning to Poland in 1657, Burattini (together with his brother Filippo) participated in the Polish-Swedish War battles under the command of general Stefan Czarniecki, with the rank of captain, commanding a company of infantry recruited at his own expense.

The year 1658 was a very successful one for Burattini. In compensation for his service to the Polish King, on 1 May 1658 he leased the crown vineyard in Cracow, in August he was granted a diploma and a nobleman title, and in November 1658 he opened a mint in Ujazdów, which struck small copper coins referred to as borattines. The production of the coins triggered a violent campaign against him; he was accused of making huge profits from the mint operation (40% of the coin value) and of adding glass to the coins which made them brittle. In 1662 Burattini was brought to the Treasury Commission which however found him innocent and consented to prolong the lease. Apart from the mint in Ujazdów, Burattini opened another mint in Brest-Litovsk. In 1668 Burattini faced new charges of abuse and bribery in favor of the candidacy of Prince de Condé, but he managed again to refute them. In order to pay back the enormous debt (circa 1.5 million zlotys) due to Burattini by the Polish state, in 1678 he was appointed administrator of the silver mint in Cracow.

In 1660 Burattini married to Teresa Bronisława Opacka (17.09.1640-03.10.1701), the young daughter of the prominent Polish nobleman Zygmunt Opacki h. Prus (III) (1587-1654). They were to have six children. In the same 1660 was born their first child—the daughter Ludwika Izabela Boratini, followed by four sons—Aleksander, Franciszek, Kazimierz Karol, and Zygmunt, and another daughter—Barbara.

In the 1660s Burattini was in his prime. In the early 1660s, he designed a giant 60-foot focal telescope, described in a letter to Pierre Des Noyers in September 1665. Later he also designed an ingenious wire micrometer to be inserted in the focal plane of telescopes for measuring angular distances. In 1665 Burattini bought the village Jelonek, south of Warsaw. In 1666 he built a bridge over the Vistula for the army. After a short difficult period in his life in the late 1660s, following the death of his patron Queen Marie Louise Gonzaga in 1667, Burattini again got in favor of the Polish Crown and in September 1671, just before Second Polish–Ottoman War, he was appointed as a commander of the fortress of Warsaw. In 1678 he was again awarded a silver mint administration in Krakow.

In his famous book Misura Universale, published in 1675, Tito Livio Burattini first suggested the name meter as the name for a unit of length. He chose the word meter after metron, a Greek word for measure. Burattini’s meter was a universal unit of measurement, based on the length of a pendulum, beating one second. He named this unit metro catholico, which simply means universal measure. Burattini actually was not the first to propose the adoption of a decimal metric system, but he was the first to advance a project that received wide attention and was the one, who first suggested the name meter for the basic unit of length.

Burattini was a known scientist for the time and hold a busy correspondence with some other famous scientists of his time like Ismael Boulliau, Johannes Hevelius, Athanasius Kircher, Marin Cureau de la Chambre, and many others.

In Poland, Tito Livio Burattini managed to establish himself not only as a scientist but also as a businessman and diplomat, to become a rich and powerful man, and to begin a family. His end however was miserable—he died poor and sick on 17 November 1681, in Vilnius.

One of the most important events in the life of the modest deacon of Nürtingen, Wilhelm Schickard, was his meeting in October 1617, with the great astronomer Johann Kepler. Obviously, during this meeting, Kepler immediately recognized the massive intellect of the young Wilhelm and encouraged his occasions with sciences, which led to the creation of the first mechanical calculator in the world (Schickard referred to it as Rechen Uhr—calculating meter or calculating clock).

It was not a casual meeting. Kepler, just like Schickard, had studied theology at Tübinger Stift (Kepler lived in Tübingen from 1589 till 1594) and worked as a Lutheran minister some 20 years before him, before devoting his life to mathematics and astronomy. Kepler visited Tübingen during one of his journeys in Württemberg, to see his old friend Michael Maestlin (1550-1631) (a famous German astronomer and mathematician, who used to be a mentor of Johann Kepler and just like Schickard and Kepler, was Magister of theology at Tübinger Stift from 1571 and worked some time as a Lutheran deacon) and others. It seems Schickard was recommended to Kepler just by Maestlin, who was Schickard’s teacher and precursor in the chair of astronomy. Maestlin probably was some kind of a patron for Schickard also (as he used to be for Kepler), because at that time there was no academic appointment without patronage.

Kepler wrote in his diary about his first impressions of Schickard—”In Nürtingen I met also an excellent talent, a math-loving young man, Wilhelm, a very industrious mechanic and lover of oriental languages.” From this moment on, Schickard entered into a close friendship and busy correspondence with Kepler until his death, made science investigations for him, and took care of Kepler’s son—Ludwig, who was a student in Tübingen and received a Master of Arts degree in 1629, created by Kepler’s request figures and copper plates, and helped for the printing of Kepler’s renown books.

Kepler was a great admirer of the logarithms of Napier. When in 1617 he first saw a copy of Napier’s book on logarithms, he didn’t fully understand them. He wrote to Schickard saying that some Scottish nobleman had come up with a way of turning all multiplications and divisions into additions and subtractions but later remarked that he doubted it would work properly. About a year later he reconsidered the concept and became so enthusiastic, that he wrote to Napier, and dedicated him his Ephemerides.

Unfortunately, the calculating machine, designed by Schickard around 1623, didn’t manage to survive to the present day. Only three documents about this machine have been found till now—two letters from Schickard to Kepler, and a sketch of the machine with instructions to the mechanic.

The two letters were discovered by a famous biographer of Kepler—Max Caspar, who worked in 1935 in the archive of Kepler, kept in the Pulkovo Observatory, near S. Peterburg, Russia (Kepler’s manuscripts were bought by order of the Empress of Russia Екатерина II Великая (Catherine the Great) in 1774). While searching through a copy of Kepler’s Rudolphine Tables, Caspar found a slip of paper, that had seemingly been used as a bookmark. It was this slip of paper that contained Schickard’s original drawings of the machine (from the second letter to Kepler). Later Max Caspar stumbled upon the other pages of the two letters.
In the 1950s another biographer of Kepler—Dr. Franz Hammer (1898-1969), made a connection between the two letters from Pulkovo and a sketch of a machine (along with instructions to the mechanic Johann Pfister), described in Schickard’s manuscripts (Schickard sketchbook), kept in Württembergischen Landesbibliothek in Stuttgart (see the figures below).

Caspar and Hammer however were not the first men, who noticed the machine of Schickard. Who was the first?
In 1718 one of the first biographers of Kepler—the German philosopher, theologian, and mathematician Michael Gottlieb Hansch (1683-1749), published a book of letters of Kepler, which includes the two letters from Schickard to Kepler. There is even a marginal note of the publisher Schickardi machina arithmetica in the second letter, obviously on the calculating machine.
In 1787, in the book “An account of the life, writings, and inventions of John Napier, of Merchiston”, the author—David Erskine, Earl of Buchan, mentioned that …Shickartus in a letter to Kepler, written in the year 1623, informs him that he had lately constructed a machine consisting of eleven entire and six mutilated little wheels, by which he performed the four arithmetical operations.
In 1899 Stuttgart’s surveying magazine Stuttgarter Zeitschrift für Vermessungswesen published an old article for the topography in Württemberg, Germany, written many years ago and probably published in other editions, by the famous German scientist Johann Gottlieb Friedrich von Bohnenberger (1765–1831). In this article, the name of Schickard is mentioned several times, not only concerning his important contribution to the field of topography but it is mentioned also that …it is strange, that nobody admitted, that Schickard invented a calculating machine. In 1624 he ordered a copy for Kepler, but it was destroyed in a night fire. Bohnenberger (known mainly as the inventor of the gyroscope effect), just like Schickard, studied and later was appointed a professor of mathematics and astronomy at the University of Tübingen in 1798.
In 1912 the yearly German magazine Nachrichten des Württembergischen Vermessungstechnischen Vereins published the sketch and the notes of the machine from the Württembergischen Landesbibliothek. The author of the article A. Georgi was however probably not aware of the two letters of Schickard, but only with the note of Bohnenberger. He even claimed, that Leibnitz was aware of the machine of Schickard and accused him of plagiarism, which is unbelievable.
In April 1957, Hammer announced his discovery during the conference about the history of mathematics in Oberwolfach, Germany. From this moment on, gradually it was made known to the general public, that namely Schickard, but not Blaise Pascal, is the inventor of the first mechanical calculating machine.

In 1960 Mr. Bruno v. Freytag Löringhoff (1912-1996), a professor of philosophy at the University of Tübingen, created the first replica of Schickard’s machine.

The first letter—Wilhelm Schickard to Kepler in Linz, 20. September 1623, includes (letters are written in the Latin language, which was the international language of science and scholarship in Central and Western Europe until the 17th century):

…Porro quod tu logistice, idem ego mechanice nuper tentavi, et machinam extruxi, undecim integris et sex mutilatis rotulis constantem, quae datos numeros statim άώτομάτος computet, addat, subtrahat, multiplicet, dividatque. Rideres clare, si praesens cerneres, quomodo sinistros denarium, vel centenarium supergressos, sua sponte coacervet, aut inter subtrahendum ab eis aliquid suffuretur…

In English, it sounds like—I have tried to discover a mechanical way for performing calculations, which you have done manually till now. I constructed a machine, that includes eleven full and six partial pinion wheels, which can calculate automatically, to add, subtract, multiply, and divide. You would rest satisfied if you could see how the machine accumulates and shifts to the left tens and hundreds, and makes the opposite shift during a subtraction…

From 1612 to 1626, Kepler lived in Linz, Austria, where he worked as a mathematics teacher and as an astrologer. In this period (1623), he was completing his famous Tabulae Rudolphinae and certainly needed such a calculating instrument. He must have written back asking for a copy of the machine for himself, because the second letter, dated 25 February 1624, includes a description of the machine with two drawings and bad news about a fire, which destroyed the machine:

…Arithmeticum organum alias delineabo accuratius, nunc et festinate hoc habe, aaa sunt capitella cylindrorum erectorum, quibus multiplicationes digitorum inscriptae, et prominent, quantum ijs opus est, per fenestellas bbb ductiles, ddd intus habent affixas rotulas 10 dentium, sic contextas, vt mota qualibet dextra decies, proxima sinistra semel; aut illâ 100 vicibus circumactâ, tertia semel etc. promoveatur. Et quidem in eandem partem; quod vt praestarem, intermediâ consimilj h opus fuit.
(A marginal note) Quaelibet intermedia omnes sinistras movet, debitâ proportione; nullam verò dextram, quod singularj cauitione indiguit. (End of the note) Quotus eorum prominet per foramina ccc in scamno medio, tandem in pavimento inferiorj e vertebras et f similiter foramina pro apparitione numerorum notat, quibus inter operandum usus est. Sed ista sic tumultuariè scribj nequeunt, facilius ex autopsiâ cognoscentur. Et curaveram tibj jam exemplar confierj apud Joh. Pfisterum nostratem, sed illud semiperfectum, vna cum alijs quibusdam meis, praecipuè aliquot tabellis aeneis conflagravit ante triduum, in incendio noctu et ex improsivo ibj coorto, quod Mütschlinus referre amplius sciet. Harum jacturam admodum aegre fero, praesertum nunc quando non vacat alia reficere tam cito.

In English—…I will describe the computer more precisely some other time, now I don’t have enough time: aaa are the upper faces of vertical cylinders (see the upper figure), whose side surfaces are inscribed with multiplication tables. The digits of these tables can be looked out of the windows bbb of a sliding plate. From the inner side of the machine to the disks ddd are attached wheels with 10 cogs, and each wheel is clutched with a similar wheel in a manner that, provided some of the right wheels spin round ten revolutions, the left wheel will make one revolution, or provided the first wheel spins round 100 revolutions, the third wheel to the left will make one revolution. For the revolutions of the wheels to be in the same direction, intermediate wheels h are necessary.
(A marginal note) Each intermediate wheel moves to the left needed carry, but not to the right, which made special caution measures necessary.

The digits, inscribed upon each wheel, can be looked out of the windows ccc of the middle bank. At the end of the lower bank are arranged rotating heads eee, used for the recording of numbers, which are the result of the calculations, and their digits can be looked out of the windows fff. I have already ordered a copy for you to our Johann Pfister, together with some other things for me, especially some copper plates, but when the work was half finished, yesterday night a fire burst out and everything burnt out, as Maestlin informed you. I take this loss very heavily because there is no time for its replacement.

Schickard obviously was not satisfied with the work of the mechanic, involved in the production of the device, because the note to Pfister begins (old German language is used):
Rechen Uhr betreffs.
1. Die zän seind gar vngleich und vnfleißig…
(which means in English, kindly translated by Mr. Stephan Weiss, www.mechrech.info):
Concerning Calculating Clock,
1. The teeth are inequally made and don’t work. Sometimes more than a tenth part is driven, sometimes less. 20 teeth would be better.
2. The front eccentric smooth disk drags a little, it should be turned.
3. (NB) The single tooth (note: for the carry of tens) should not be placed in the middle between two others. Should it touch right onto a numeral tooth, it will push the number forward twice.
(NB) 4. Only the 0, and also the 9 should move the left number, the first when subtracting, and the next when adding.

This is why the numbers must be written in this way:
1. Start on the right with disc 1, turn right, where the disc starts to engage, write 9 on top, then turn to the left, where it starts to move and write 0 on top. The rest is self-explanatory.
2. Where the teeth are unevenly spaced, first place hidden points, then take the middle between the two.
3. The front holes should be right in front of the numbers.
NB: To annotate the arithmetic wheels. When a right wheel is driving its left wheel, on the right wheel it should read 9 on top before the transfer, and the other numbers should be written to the left.

That’s the whole information, survived up to the present for the Calculating Clock of Schickard. It seems the prototype of the machine, mentioned in the first letter, was rather successful, that’s why Schickard ordered the next copy for Kepler. It is unknown whether another copy was ever created, and how many devices are made or ordered by the inventor. It is out of the question, however, that such a device has not been delivered to Kepler. Most probably, only two machines were produced, the prototype, mentioned in the first letter, which was in the home of Schickard and disappeared after his death, and the second, made for Kepler, which was destroyed during the fire.

Let’s examine the structure and the functioning of the device. The Calculating Clock is composed of 3 main parts:

• A multiplying device.
• A mechanism for recording intermediate results.
• A decimal 6-digit adding device.

The multiplying device is composed of 6 vertical cylinders with inscribed numbers of Napier’s rods (see the photo nearby).

From the front side, the cylinders are covered with 9 narrow plates with windows, which can be moved leftwards and rightwards. After entering the multiplicand by rotating the cylinders through the knobs on the upper side of the box, using the opening of the windows of plates can be made consecutive multiplying first by units of the multiplier, then by tens, and so on. The intermediate products can be added by adding devices.

The mechanism for the recording of intermediate results of calculations is composed of 6 rotating using small knobs disks with peripheries inscribed with digits, which can be seen in the small windows in the lower row (see the photo below). These disks are not connected to the calculating mechanism and don’t have a tens carry mechanism.

The adding device is composed of six basic axes in a row. On each axis is mounted a smooth disk with ten openings (marked with 1 in the lower photo), a cylinder with inscribed digits (marked with 3), and a pinion wheel with 10 teeth (marked with 2), over which is a fixed pinion-wheel with 1 tooth (which are used for tens carry). On the other 5 axes are mounted pinion wheels with 10 teeth (marked with 4).

The smooth disks are used for entering the numbers and resetting the machine. The digits on the inscribed cylinders can be seen in the upper row of windows and are used for reading the results of adding and subtracting operations. Over each of the 10-teeth disks on the basic axes is mounted a one-tooth disk, in such a manner, that for each full revolution of the 10-teeth disk, 1-tooth disk enters once in contact with the proper intermediate disk and rotates it to 1/10 revolution. This is the mechanism of tens carry and it is not original. The use of an analog train of gear wheels (linked so that each time one wheel completes a revolution the next wheel turns one-tenth of a revolution, thus recording a carry) is very ancient and even appears in the works of Heron of Alexandria.

The axes can be rotated in both directions, so the machine can be used not only for addition but for direct subtraction too (no need to use the arithmetical operation complement to 9, as it was the case with Pascaline). Due to the intermediate disks, all smooth disks are rotated in the same direction.

The machine has also an indicator for overflow—a small bell, which rings if the leftmost pinion wheel rotates from 9 to 0.

Let’s make a simple multiplication with the machine, for example, 524 x 48. First, we have to rotate the rightmost cylinder to 4, the next cylinder to 2, and the third from right to 5 (the multiplicand is 524). Then we have to open the windows on the 8th row (units of the multiplier are 8) and we will see in the windows the first intermediate result (4192). We have to enter the 4192 in the calculating mechanism. Then we have to open the windows on the 4th (tens of multiplier are 4) row and to see the second intermediate result—20960, which we have to enter into the calculating mechanism, and we will have the result—25152.

As described by Schickard mechanism presented two eventual faults. First, the inventor didn’t describe a means for fixing the intermediate disks, which is certainly necessary. As you can see in the photos, the technicians of Mr. Freytag Löringhoff have provided such a mechanism (the small disks below the intermediate disks). The second problem is friction. At the beginning of the 17^th century, the turret lathes had not been invented yet, so the pinion wheels had to be produced manually and with great precision, otherwise, the friction in case of full carrying (for example when to 999999 must be added 1) will be enormous and the machine will be hard for operating and easy to break. Schickard obviously had faced such problems, and that’s why his machine had only six main axes, despite the vital necessity of Kepler to work with big numbers for his astronomical calculations.

Biography of Wilhelm Schickard

Wilhelm Schickard was born in the morning at half past seven on 22 April 1592, in Herrenberg, Germany. Herrenberg is a small town, located in the area of Württemberg in the southern part of Germany, some 15 km from one of the oldest university centers in Europe—Tübingen, which University was founded in 1477.

Wilhelm was the first child in the family of Lukas Schickard (1560-1602), a carpenter and master builder from Herrenberg, who married in 1590 to Margarethe Gmelin-Schickard (1567-1634), a daughter of Wilhelm Gmelin (1541-1612), a Lutheran pastor from Gärtringen (a small town near Herrenberg) and Magdalena Rieger (1540-1580). Wilhelm had a younger brother—Lukas and a sister.

The Schickards is a well-known Herrenberg family, which was originally from the German region Siegerland (in region Nordrhein-Westfalen) but had moved south at the beginning of the 16th century. The great-grandfather of Wilhelm—Heinrich Schickard (1464-1540) from Siegen, called Heinrich Schickhardt der Ältere or Heinrich der Schnitzer, was a famous woodcarver and sculptor, whose wood-works (stalls from 1517) are still preserved in the church Stiftskirche Herrenberg. He was the founder of the Herrenberg’s Schickards family, moving in 1503 from Siegen, Siegerland, to Herrenberg. A brother of Lukas Schickard and uncle of Wilhelm is Heinrich Schickard (1558-1635)—one of the most prominent German architects of the Renaissance. The other uncle of Wilhelm is Philipp Schickard (1562-1635), a well-known in his lifetime theologian.

Wilhelm, a precocious child, started his education in 1599 in a Latin school in Herrenberg. After the death of his father Lukas in September 1602, his uncle Philipp, who served as a priest in Güglingen, took care of him, and in 1603 Wilhelm attended a Latin school there. In 1606 another uncle of his, Wilhelm Gmelin, took young Wilhelm to the church school in Bebenhausen Monastery, near Tübingen, where he was a teacher.

The school in Bebenhausen was associated with the Protestant theological seminary Tübinger Stift, in Tübingen, so in March 1607 the young Wilhelm entered a bachelor program of the Stift, held in Bebenhausen. In April 1609, he received his bachelor’s degree. In Bebenhausen Wilhelm studied not only languages and theology but also mathematics and astronomy.
In January 1610, Wilhelm went to the Tübinger Stift to study for his master’s degree.

Tübinger Stift is a hall of residence and teaching of the Protestant Church in Württemberg. It was founded in 1536 by Duke Ulrich for Württemberg born students, who want to be ministers or teachers. They receive a scholarship that consists of boarding, lodging, and further support (students receive for their personal needs six guilders per year cash). This was very important for Wilhelm because his family apparently was short of money and could not support him. After the death of his father in 1602, in 1605 his mother Margarethe married a second time to Bernhard Sick—a pastor in Mönsheim, who also died several years later, in 1609.
Besides Schickard, other famous students of Tübinger Stift are Nikodemus Frischlin (1547-1590), a famous humanist, mathematician, and astronomer from the 16th century; the great astronomer Johannes Kepler (1571-1630); the famous poet Friedrich Hölderlin (1770-1843); the great philosopher Georg Hegel (1770-1831) and others.

After receiving his master’s degree in July 1611, Wilhelm continued studying theology and Hebrew language in Tübingen until 1614, working at the same time as a private teacher of mathematics and oriental languages, and even worked some time as a vicarius in 1613. In September 1614 he took his last theological examination and started his church service as a Protestant deacon in Nürtingen, a town, located some 30 km northwest of Tübingen.
On 24 January, 1615, Wilhelm married to Sabine Mack from Kircheim. They were to have 9 children, but (as it was common in these times), only 4 survived by 1632: Ursula Margaretha (born 03.03.1618), Judith (b. 27.09.1620), Theophil (b. 3.11.1625) and Sabina (b. 1628).

Schickard served as a deacon till the summer of 1619. The church duties left him plenty of time for his studies. He continued his work on old languages, translations, and wrote several treatises, for example in 1615 he sent to Michael Maestlin an extensive manuscript on optics. During this time he developed also his artistic skills, creating several portraits, and astronomical tools, he had even a copper press.

In 1618 Schickard applied for and in August 1619, he was appointed as a professor of Hebrew language at the University of Tübingen, recommended by Herzog Friedrich of Württemberg. The young professor created his own method for presenting material, together with some wise auxiliary means, and taught also other ancient languages. Schickard learned also Arabic and Turkish. His Horolgium Hebraeum, a textbook for learning Hebrew in 24 hourly lessons, went through countless editions during the next 2 centuries. Actually, Schickard was a remarkable polyglot. Besides German, Latin, Arabic, Turkish, and some ancient languages like Hebrew, Aramaic, Chaldean, and Syriac, he knew also French, Dutch, etc.

His efforts to improve the teaching of his subject show remarkable innovation. He strongly believed that, as the professor, it was part of his job to make it easier for his students to learn Hebrew. One of his inventions to assist his students was the Hebraea Rota. This mechanical device displayed the conjugation of Hebrew verbs by having two rotating discs laid on top of each other, the respective forms of conjugation appearing in the window. Besides Horologium Hebraeum, in 1627 he wrote another textbook—the Hebräischen Trichter, for German students of Hebrew. However, his research was broad and, in addition to Hebrew, included astronomy, mathematics, and surveying. In astronomy, he invented a conic projection for star maps in the Astroscopium. His star maps of 1623 consist of cones cut along the meridian of a solstice with the pole at the center and apex of the cone. Schickard also made significant advances in map-making, writing a very important treatise in 1629, showing how to produce maps that were far more accurate than those that were currently available. His most famous work on cartography was Kurze Anweisung, wie künstliche Landtafeln auss rechtem Grund zu machen, published in 1629.

In 1631 Schickard was appointed professor of astronomy, mathematics, and geodesy at the University, because he had already significant achievements and publications in these areas, taking the chair from the famous German astronomer and mathematician Michael Maestlin, who died the same year. He lectured on architecture, fortification, and hydraulics. He also undertook land surveying of the duchy of Württemberg, which involved the first use of Willebrord Snell’s triangulation method in geodesic measurements. As a professor of astronomy, Schickard lectured on the topic and undertook research into the motion of the moon. He published Ephemeris Lunaris in 1631, which allowed the position of the moon to be determined at any time. We should note that, at a time when the Church was trying to insist that the Earth was at the center of the universe, Schickard was a staunch supporter of the heliocentric system. In 1633 he was appointed dean of the philosophical faculty.

An important role in the life of Schickard played the great astronomer Johann Kepler. After their first meeting in the autumn of 1617 (Kepler was passing through Tübingen on his way to Leonberg, the Württemberg town where his mother had been accused of being a witch), they had a busy correspondence and several other meetings (in 1621 for a week, later on for 3 weeks). Kepler used not only Wilhelm’s talent for mechanics but also his artistic skills. In 1618 Schickard built a tool for comet watching for Kepler. Later on, Schickard took care of Kepler’s son—Ludwig, who was a student in Tübingen.

Schickard agreed also to draw and engrave the figures of the second part of the Epitome Astronomiae Copernicanae of Kepler on woodblocks, yet Krüger (Kepler’s publisher), always ready to interfere with Kepler’s plans, stipulated that the carving had to be done in Augsburg. Schickard sent thirty-seven woodblocks for books 4 and 5 to Augsburg towards the end of December 1617. Schickard engraved also the figures for the last two books (the carving was done by one of his cousins).

Wilhelm also proposed to Kepler the development of a mechanical means of calculating ephemerides and created the first hand planetarium. Schickard created also, probably by request from Kepler, an original instrument for astronomy calculations (see the photos below). Kepler showed his gratitude, sending him several of his works, two of which are still preserved in the University Library in Tübingen.

In 1631 the life of Schickard and his family was under threat from the battles of the Thirty Years’ War, which approached Tübingen. Before the Battle of Tübingen in 1631, he fled with the entire family to Austria and returned after several weeks. In 1632 the family again fled to Austria. In June 1634, hoping for quieter times, he bought a new home in Tübingen, suitable for astronomical observations. His hopes were vain although. After the battle of Nördlingen in August 1634, the Catholic forces occupied Württemberg, bringing violence, famine, and plague with them. Schickard buried his most important notes and manuscripts, to save them from plunder. These partly survived, but Schickard’s family did not. In September 1634, in sacking Herrenberg, the soldiers from the Catholic forces beat Schickard’s mother—Margarethe, who died a lingering death of her wounds. In the next January of 1635 was killed by soldiers his uncle—the architect Heinrich Schickard.

At the end of 1634 died from plague Schickard’s eldest daughter—Ursula Margaretha, a girl of unusual intellectual attainment and promise. Then died his wife Sabine and the two youngest daughters—Judith and Sabina, two servants and a student, who lived in his house. Schickard survived this outbreak, but the following summer the plague returned, taking with it in September his sister, who was living in his house. Schickard and his only surviving child—9-year-old son Theophilus, fled to the village Dußlingen, near Tübingen, having the intention to emigrate to Geneva, Switzerland. However on 4 October 1635, fearing that his house and especially his library would be plundered, he returned to Tübingen. On 18 October he became sick of the plague and died on 23 October 1635. His little son followed him after a day.

Besides Kepler, Schickard also corresponded with some other famous scientists of his time—mathematician Ismael Boulliau (1605-1694), philosophers Pierre Gassendi (1592–1655) and Hugo Grotius (1583-1645), astronomers Johann Brengger, Nicolas-Claude de Peiresc (1580-1637) and John Bainbridge (1582-1643), and many others.

Wilhelm Schickard was one of the most reputable scientists in Germany of his time. The opinions of this universal genius from his contemporaries are—the best astronomer in Germany after Kepler’s death (Bernegger), the foremost Hebraist after the death of the elder Buxtorf (Grotius), one of the great geniuses of the century (Peiresc). However, like many other geniuses with wide interests, Schickard was in danger of stretching himself too thin. He succeeded in finishing only a small part of his projects and books, being struck down in his prime.

Books, written by Wilhelm Schickard:
• Cometenbeschreibung, Handschrift, 1619
• Hebräisches Rad, 1621
• Astroscopium, 1623
• Horologium Hebraeum, 1623
• Lichtkugel, 1624
• Der Hebräische Trichter, 1627
• Kurze Anweisung, wie künstliche Landtafeln aus rechtem Grund zu machen, 1629
• Ephemeris Lunaris, 1631

Biography of Johann Pfister
Who was the mentioned in the second letter mechanic Joh. (Johann or Johannes) Pfister, who was involved not only in the production of Schickard’s calculating machines but also in other projects, for example in preparing metal plates for his and Kepler’s books?

The Pfister is a well-known for the time Tübinger family, known primarily as book-binders and Universitätspedells at the University of Tübingen (Universitätspedell was a relatively prominent position, responsible for arresting and detaining students in the karzer and functioning as a prosecutor at the university court).

First was Hans Pfister, the Older (1523-1607), the grandfather of Johannes Pfister, who was not only a member of Tübinger’s book-binders guild but served also a long time in the University of Tübingen as a watch and Universitätspedell.

Hans Pfister’s son—Hans (Conrad) Pfister, Jr. (b. 1560) succeeded his father and worked as a book-binder, seal-engraver, Universitätspedell, and schoolmaster in Tübingen. He married in 1578 Anna Ruckaberle (1563-1624), the family had ten children, and one of them was Johannes Pfister.

Johann(es) Pfister was born on 15 January 1582, in Tübingen. He succeeded in the family trade and worked as a bookbinder and printer, as well as an engraver and mechanic. He must have been a decent painter also because an interesting painting from 1620 survived to our time (see the upper painting of Pfister). Pfister married in 1606 to Rosina Steininger, a daughter of the Lutheran scholar Gall Steininger.