Christian Ludwig Gersten

Smallness of mind is the cause of stubbornness, and we do not credit readily what is beyond our view.
Francois de la Rochefoucauld

The prototype of Gersten's machine kept in Darmstadt until 1943
The prototype of Gersten’s machine kept in Darmstadt until 1943

In 1735 Christian Ludwig Gersten (1701-1762), a professor of mathematics at the University of Gießen, traveled to London, to be elected a fellow of the Royal Society and to present to the Society his project of an arithmetical machine. In his description, he claims, that the machine had been designed 12 or 13 years before, most probably in 1722.

The machine was described in the scientific journal Philosophical Transactions of the Royal Society Nr. 438 for the months of July, August, and September 1735 (pages 79–97). In his 19-page detailed presentation, named Description and Use of an Arithmetical Machine… (see description of the machine of Gersten) Gersten mentioned, that he has been acquainted with the machines of Morland, Leibniz, Poleni, and Leupold, and learned that Charles Pascal invented one, and was inspired from them (especially from Leibniz) to create his own calculating device.

A sketch of Gersten's machine from the presentation to the Royal Society
A sketch of Gersten’s machine from the presentation to the Royal Society

The first prototype of the machine was wooden, and after the device had been shewed to some Patrons and Friends of Gersten, they encouraged him to continue the development. The second copy was made from brass and at the end of 1725, Gersten presented it to the Landgrave of Hessen Darmstadt Ludwig V and his son, Prince Georg II (who became Landgrave in 1726). The Landgrave admired the inventor and encouraged him to try to popularize it. Besides the presentation to the English Royal Society, the description of the machine was sent and approved by the French Academie Royale des Sciences, and in 1735 the description was published in the first book of six volume set of Jean Gaffin Gallon—Machines et inventions approuvées par l’Academie Royale des Sciences, depuis son établissment jusqu’a présent; avec leur description.

In 1736 a copy (probably the only prototype) of the machine of Gersten was donated to the Landgrave of Hesse-Darmstadt and was kept in the Castle of Darmstadt until WWII (see the first figure), when it was destroyed by bombs in 1943.  Fortunately, to the present day managed to survive a replica of the device from 1920, made by order of Dr. Franz Trinks, the founder of Rechenmaschinenfabrikation in Braunschweig which is preserved in the Braunschweigisches Landesmuseum (see the third figure).

Let’s examine the arrangement of the machine in the sketch from the presentation to the Royal Society (see the sketch in the second figure).

A copy of Gersten's machine in Braunschweigisches Landesmuseum
A copy of Gersten’s machine in Braunschweigisches Landesmuseum

The dimensions of the machine are—36 cm in width, 37 cm in height, and 7 cm in thickness. It has a six positional input mechanism and seven positional display mechanism (from the design of the machine is evident, that it can be easily extended with additional positional modules). In part 1 of the upper figure are shown the first three digital positions—for units (marked with AA), for tens (marked with BB), and for hundreds (CC). Every position has two sliders, the first of them (the upper one) Gersten named operator, the second one—determinator. The determinator indicates what number will be entered in the calculating mechanism, while the entering is done by moving the operator.

The main element in the calculating mechanism of the machine are ratchet-wheels. They are used in the digital dials for entering numbers, and for tens carry during the calculations.

At the end of his presentation, the inventor acknowledged, that the major constructive problem of the machine is significant friction between the parts of the mechanism when must be done tens carry for two or more digital positions.

The machine of Gersten is not as elaborate and easy to operate, as the machines of the other early inventors, but nevertheless, it is well-designed and fully capable of calculations device. Interestingly, there is a letter
from Johann Müller from 9 February 1784 to his friend Lichtenberg in Göttingen, in which he writes the following about Gersten: “Unexpectedly, I found the calculating machine of the former local Prof. Gersten in the castle in Darmstadt, of which I hereby provide you with a description and drawing. Maybe you can make some use of it. It is strange that the thought of whether not appropriated in perfection is what I give you on my wooden models to be the more comfortable and faster installation of the given digits.”

Biography of Christian Ludwig Gersten

Johänn Christian Ludwig Gersten (or Gerstein) was born on 7 February 1701, in Gießen, a town in the German federal state of Hessen. He was the son of Johann Justus Gerst(en) (1666-1712), and Maria Margaretha Uhrban (1666-1741). Johann Gerst was a lawyer, Fürstlich-Schwarzburgischer Kanzler und Konsistorial-Präsident in Sondershausen (a town in Thuringia), who married Maria Uhrban in 1692, and they had seven children: Carola Henrica Eleonore (1693-1720), Johann Christian (b. 1695), Juliana Charlotte (1698-1726), Christian Ludwig (1701-1762), Friedrich Ludwig (1703-1705), Christine Frederike (b. 1706), and Johann Maria Lucia.

From 1718 Gersten studied law and mathematics at the University of Gießen and at the beginning of the 1730s he traveled to London, England, to improve his mathematical knowledge. In London, in 1733 he became a fellow of the English Royal Society. In May 1733, he returned back to Gießen, to accept the position of a professor of mathematics at the University of his hometown, which he held until 29 Apr 1745.

As a scientist, Gersten is primarily known for his dissertation work, published later as a book, describing a series of experiments, using the barometer, titled “Tentamina Systematis Novi ad Mutationes Barometri ex Natura elateris Aerei demonstrandas, cui adjecta sub finem Dissertatio Roris decidui errorem Antiquum et vulgarem per Observationes et Experimenta Nova excutiens” (see the nearby image). Gersten as a matter of fact was the first scientist to find out, based on observations, that dew did not fall from the heavens, but ascended from the earth, especially from plants. He is also the author of several other books in the fields of astronomy, physics, and mathematics.

As we already mentioned, in 1735 Gersten traveled again to London, to present to the Royal Society his project of an arithmetical machine.

It seems Gersten was quite an active person not only in the field of science but also in public affairs, which later led him into severe troubles. Gersten was admired in 1725 for his calculating machine by Landgrave of Hesse-Darmstadt Ludwig V, and by his son, Prince Georg II, who became Landgrave in 1726. In the early 1740s, Gersten was deposed from his office after refusing to submit to the sentence of a court of law, in a lawsuit with his brother-in-law Johann Friedrich Wahl (1693–1755), a Professor and from 1735 Rektor der Universität Gießen, and in 1744 he wrote a petition to the Landgrave Georg II, which was considered as offensive. The Landgrave accused Gersten and initiated a trial against him. Having a presentiment what will be the result, Gersten decided to escape justice and in July 1744 he left the town.

Castle Marxburg bey Braubach, where Gersten was imprisoned for 12 years
Castle Marxburg bey Braubach, where Gersten was imprisoned for 12 years

Initially, he settled in Altona bei Hamburg, and informed the University officials, that he was not going to return to Gießen and lost his position in April 1745. Then he departed to St. Peterburg, Russia, maybe to some other places also, but finally decided to return to Germany. And then his disrespect to the Landgrave and the attempt to escape from justice finally got him into big trouble. In September 1748 he was arrested in Frankfurt am Main by request from Darmstadt, and condemned to lifetime detention at the castle Marxburg bey Braubach (see the nearby image).

Gersten never recognized his mistakes and repeatedly addressed the court with offensive petitions. Interestingly, despite his problems with justice, in 1748 he was partly restored as a professor at the University of Gießen and was allowed during his stay in the castle to teach young people in mathematics. Gersten was eventually released as late as 1760.

Christian Ludwig Gersten died on 13 August 1762, in Frankfurt am Main.

Hillerin de Boistissandeau

Imitation is not just the sincerest form of flattery – it’s the sincerest form of learning.
George Bernard Shaw

In 1735 was published the first book of six volume set of Jean Gaffin Gallon—Machines et inventions approuvées par l’Academie Royale des Sciences, depuis son établissment jusqu’a présent; avec leur description, in which are described several calculating machines. In one of the books (Tome 4) was described the machine arithmetique of the French mathematician and inventor Hillerin de Boistissandeau, sent for approval to the Academie in 1730.

The machine of Hillerin is quite similar to the earlier Lépine’s machine (a solid, but simple adding device, with limited practical usefulness), so obviously Boistissandeau was strongly influenced by Lépine. Actually in the publication (see desciption of Hillerin’s machines) there are three separate articles, which described three different machines, but the second and the third are essentially improvements of the first one.

It seems at least one copy of the device was manufactured, preserved now in the collection of CNAM, Paris. The dimensions of the device are: height: 6.00 cm, width: 17.00 cm, length: 30.50 cm. The weight is 3.250 kg. Materials used are: wood, leather, iron, brass, and textile.

The machine of Boistissandeau has six digital positions, appropriate to the french monetary system in the 17th century. The rightmost position is for deniers, that’s why its wheel is divided into 12 parts, the next is for sols (the wheel is divided into 20 parts), remaining wheels are decimal (wheels are divided into 10 parts).

Entering and resetting the numbers is done by means of a stylus with two edges—short and long, which can be stuck into openings of the input wheels. When the short edge of the stylus is used, then will be rotated only the upper (input) disk and it can be set to 0. When the long edge of the stylus is stuck, then will be rotated not only the upper (input) disk, but also the lower disk of the calculating mechanism, and the number will be entered into it. The small wheels, which are placed over the big wheels are working as revolution counters and are used during a division operation. In these wheels is stored the quotient of the division, while below in the main mechanism is stored the remainder.

Arithmetical machine of Hillerin de Boistissandeau, first machine (in the lower half is shown the lid, in the upper part—back mirrored internal view)
The arithmetical machine of Hillerin de Boistissandeau, first machine (in the lower half is shown the lid, in the upper part—back mirrored internal view)

The five wheels (marked with w in the upper figure), which are placed between the six small wheels-revolution-counters, can be rotated by means of a special rod. These wheels are not connected to the calculating mechanism and are used as temporary storage for intermediate results.

Bellow, each input wheel is placed a small window, in which can be seen digits, inscribed on the wheel below. Here can be seen the result of the calculations. Over the digital wheel actually are inscribed two rows of digits. The first row is used during adding and multiplication, and the second—during subtraction and division. Which row of digits can be seen is determined by the rods (marked with B-B on the figure), which are moving a plate, covering one of the rows. Over the input wheels are placed small axles, which are stopping the styles during the rotation of the input wheels, and actually are determining which digit will be entered.

Below the digital disk, which digits can be seen in the windows below the input disk, are mounted on the same axes 3 disks. The first disk is a pinion-wheel, which can have 10, or 12 (according to the digital position of the wheel) teeth. This wheel takes the carry from the neighboring position (this is the reason that the rightmost position doesn’t have such a wheel). After this wheel is placed one-tooth wheel, which during each revolution will be engaged with the wheel of the higher position and will rotate it to 1/10 (or 1/12) revolution. The carrying is not direct but by means of an intermediate pinion-wheel, which is placed in the same plane, as the one-tooth wheel of the previous digital position and with pinion-wheel of the next digital position.

The lowest wheel actually is a ratch-wheel, and together with a rod and spring forms a system, destined for fixing the position of the digital positions mechanism.

The small wheels-revolution-counters also have openings, by means of which can be rotated, thus resetting the wheels. Bellow these wheels are placed ratch-wheels, which are fixed by means of rod and spring. The transfer of rotation from the main counter-wheels to these wheels is done by means of a special tooth on the big wheels, which once during a full revolution pulls a rod, which rotates to one tooth the ratch-wheel of the appropriate revolution-counter. Their digits are seen in the windows over each wheel.

 Arithmetical machine of Hillerin de Boistissandeau, part of the second machine
The arithmetical machine of Hillerin de Boistissandeau, part of the second machine

The major fault of the first machine of Boistissandeau is its tens carry mechanism. The friction is huge and if it is necessary to propagate several consecutive carrying operations, then must be applied a vast effort and the mechanism can be blocked or even broken. That’s the reason in the second variant of the machine (see the nearby figure), Boistissandeau changed the tens carry mechanism.

Here the tens carry mechanism is made by partial pinion-wheels (actually a rod with an attached sector with several teeth on the edge). These rods are moved by means of a spring, which is connected to the mechanism of the junior position. This system already has the advantages of the tens carry mechanism of the Pascaline, and allows the carrying to be done without any effort. A similar mechanism is used for the wheels-revolution-counters. Another new element is the possibility to add mechanisms of new digital positions to the machine and to change the mechanisms of the digital positions, which allows the machine to be used for counting in the different numeral systems (monetary systems, weights, lengths, etc.)

 Arithmetical machine of Hillerin de Boistissandeau, part of the third machine
The arithmetical machine of Hillerin de Boistissandeau, part of the third machine

Boistissandeau proposed also a third variant of his arithmetical machine (see the nearby figure), which again has improved the tens carry mechanism by lengthening the rod, which is stretching the spring, and decreasing the diameter of the pinion-wheel, which takes the carry, so it is required smaller effort during the carry.

Biography of Hillerin de Boistissandeau

The French mathematician and inventor Jean-Baptiste-Laurent de Hillerin, Seigneur de Boistissandeau and Seigneur de Jumaville, was born on 24 (baptized on 26) July 1704, in Paris, the first child in the noble family of Jean-Baptiste de Hillerin de Boistissandeau (13 Nov 1664-14 May 1732), conseiller au Parlement de Bretagne et du Roi, and Marie-Catherine Moreau de Villiers (born 7 Dec 1679). The family had also a daughter—Marie Catherine Armande (6 July 1709-23 Sep 1742).

René Antoine de Réaumur, a friend of de Boistissandeau
René Antoine de Réaumur (1683-1757), a friend of Hillerin de Boistissandeau

The young Hillerin de Boistissandeau was energetic and passionate about the study of science. Abbé Robert, a former parish priest of Bourges, was his tutor. Growing up in the family Château du Boistissandeau (Castle of Boistissandeau) from the late 16th century (see the lower image), in the vicinity of the town Les Herbiers, western France, he made many developments around the castle and in particular construction of large raised orangeries with attics, granaries, and rooms to house servants. He was interested in physics and mathematics, and later in instrument making. Besides his calculating machine, de Boistissandeau was known as an inventor of various instruments, including a regulateur de parquet (a clock, used as a reference to other clocks, so that everyone is at the same time) in 1732, a surveyor and odometer in 1744, a portable barometer in 1758, etc.

Hillerin de Boistissandeau was a Counselor of the Parliament of Bretagne and was appointed adviser to the Board of Auditors of Paris by 1744. Chevalier Hillerin de Boistissandeau was also a Gentilhomme ordianire de la chambre du Roi and from 1744 membre correspondant de l’Académie Royale des Sciences (section of physics and mathematics) in Paris. He was a friend of the prominent french scientist René Antoine Ferchault de Réaumur, who made frequent visits to the Castle of Boistissandeau and even had a reserved room on the first floor of the castle, equipped as a small laboratory.

Hillerin de Boistissandeau married on 17 August 1735, in Paris, to Marie Agathe Bouret de Beuvron (24 Jan 1711-31 January 1794), daughter of Guillaume Bouret de Beuvron and Mathilde le Camus. The pair had eight children—six daughters and two sons (Marie Catherine (1736-1794), Antoinette Henriette (1737-1794), Louis François (1738-1799), Marie Agathe (1740-1794), Catherine (1742-1774), Julie (1744-1790), Louise Adélaïde (1750-1790), and Armande (1752-1820).

Castle of Boistissandeau (Château du Boistissandeau), a postcard from the beginning of 20th century
Castle of Boistissandeau (Château du Boistissandeau), a postcard from the beginning of the 20th century

Seigneur de Boistissandeau died on 15 February 1779, in the Castle of Boistissandeau, happy to live not long enough to see the French Revolution. During the massacres of War in the Vendée, in 1793 his castle was looted and partly destroyed by the Republican army. Moreover, on 31 January 1794, a group of hussars-republicans raided the castle and killed his widow, the 83 years old paralyzed Marie Agathe, together with her daughters Antoinette Henriette and Marie Agathe.

Anton Braun

He who wants to get to the source must swim against the current.
Stanisław Jerzy Lec

When in 1724 the German mechanic Anton Braun (1686-1728) got an appointment as a mechanic and optician of the imperial court in Vienna, Austria, he started to design a calculating machine for the purposes of the court. Braun finished his work in 1727, producing a calculating machine of very good design and workmanship. When in the same year he presented the machine to the Holy Roman Emperor Karl VI, he was so impressed, that later appointed him as Imperial instrument maker, and granted him a diamond chain (with the portrait of the Emperor) and a huge sum of money—10000 guilders.

It seems at the same time when Braun designed and made the calculator, and presented it to the Emperor (we will call it the first machine of Braun), he devised a similar device, but with a different calculating mechanism (let’s call it the second machine of Braun). Obviously, Braun knew the machines of previous inventors, like Poleni (the first machine’s calculating mechanism is based on the pin-wheel of Leibniz and Poleni), and Leupold (the second machine’s calculating mechanism is based on the switching latch of Leupold, and the appearance of both machines is similar to Leupold’s device.)

There is an interesting story, connected with the first machine of Braun. One of the 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. 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. It is possible Braun to had gotten information about Poleni’s machine (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 so decided to use the idea of Poleni in his construction, the history of inventions is full of cases like that. In fact, if Poleni didn’t manage 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 first calculating machine of Anton Braun is quite big (almost 40 cm diameter and over 20 cm height) and a fancy device, finely decorated and looking like a Renaissance table clock cylinder, made of gold, steel, silver, and brass.

The first calculating machine of Anton Braun from 1727 (© Technischen Museum, Wien)
The first calculating machine of Anton Braun from 1727 (© Kunsthistorischen Museum Wien)


The example on the image is not the original one, made by Anton Braun in the 1720s, but a copy, made in 1766 by his son—Anton Braun the Younger (1708-1776), who just like his father was a skillful optician and watchmaker (the case was made by famous Munich sculptor Johann Baptist Straub).

The calculating mechanism was based on the pin-wheel (or the sprocket wheel), invented by Leibniz and Poleni. The machine’s six-place setting mechanism is in the form of six circular segments arranged in a circle on the top, with nine sliders each (for digits 1 to 9), which move the relevant pins radially outwards on the pin-wheels below. Turning the crank adds the entered number to the result mechanism (12-digits with complementary numbers shown), and the result is shown in the windows along the periphery of the cover (the silver-plated part). The setting mechanism can be rotated with respect to the result mechanism so that both multiplication and division are possible. The machine also featured a single-digit revolution counter.

The second calculating machine of Anton Braun (see the image below) is a much smaller device, similar in appearance to the first machine (round shape, crank in the middle, concentrically arranged numerical windows, and magnificent decorations), but its calculating mechanism is almost identical to the Leupold’s machine and it is based on a ratchet-wheel. This machine probably was only begun in the workshop of Braun, but after his early death in 1728, it was finished as late as 1736 by his son and by the famous French mechanic Phillippe Vayringe (1684-1746), who was hired by the Emperor to fix the machines, kept in his collection. The only surviving example of the machine (on its lid is engraved Braun invenit, Vayringae fecit) (Invented by Braun, manufactured by Vayringe) is now in the exposition of Deutsches Museum, Munich.

The calculating machine of Leupold-Braun-Vayringe from 1736 (© Deutsches Museum, Munich)
The calculating machine of Leupold-Braun-Vayringe from 1736 (© Deutsches Museum, Munich)

The second calculating machine of Braun is commonly named Leupold-Braun-Vayringe machine, due to the fact, that the idea of the calculating mechanism was proposed by Leupold, the construction was made by Braun, while the actual manufacturing was made by Vayringe. It is believed that Braun had already gotten to know and realized Leupold’s construction in detail before his volume Theatrum arithmetico-geometricum was published in 1727. Leupold himself reported that he had been dealing with calculating machines “for more than 20 years”, that he “had released four to five types” and that he “could show their workings to different friends”.

The machine featured a single central so-called adapting segment, which allowed the number of special, complicated parts to be greatly reduced. Below the setting mechanism is placed a set of vertical cylinders, each with nine rods of different lengths rising from its top. For example, if digit nine was entered, the shortest rod was rotated to the outside, and then one full turn of the crank turned the central adapting segment once around the central axle. It consisted of a disc with various steps as well as a segment with nine cogs. When it was turned once round, it passed the setting cylinders, on each of which a certain rod pushed the corresponding step outwards, whereupon the cog-segment of the adapting segment engaged a cog-wheel of the result mechanism and thus rotated the numbered disc to the correct digit in the corresponding window. Thus, the smaller the entered digit was, the later the adapting segment engaged, and fewer cogs were moved. Multiplication was done by repeated revolutions of the crank, as a place-shift mechanism enables multiplying with multi-digit multipliers. Subtraction (and division) were done using the 9-complements of digits.

A copy of the calculating machine of Braun-Vayringe from 1736 with a glass lid
A copy of the calculating machine of Leupold-Braun-Vayringe from 1736 with a glass lid (© Deutsches Museum, Munich)

Even though the tens-carry mechanism of the machine did not function properly in every place, the idea of a central adapting segment was a great innovation that found extensive use in several brilliant mechanical calculators some 200 years later on, like the magnificent Curta of Herzstark, even though it used a stepped drum as the central element.

The collection of Deutsches Museum, Munich (the world’s largest museum of science and technology), contains not only the original of Leupold-Braun-Vayringe machine but also a very beautiful modern replica with a transparent glass lid (see the nearby photo).

Biography of Anton Braun

Anton (spelled also Antoni and Antonius) Braun was born on 22 October 1686, in Möhringen an der Donau (bei Tuttlingen), a small town on the upper Danube, in Baden-Württemberg, Germany, in an old Bürger family, mentioned to live in Möhringen as long ago as in 1491. Anton was the first child from the second marriage of Hans Jacobus Braun (born 25 July 1651) and his wife Franziska Riestler (Hans Jacobus Braun had three daughters and two sons from his first marriage). Anton Braun had a younger brother, Johann Georg (b. 27 May 1688), who also became an optician and instrument maker, but could never reach the technical brilliance of his elder brother.

Hans Jacobus Braun used to work as a mechanic and watchmaker, so obviously, Anton learned the basics of mechanics in his parental home.

Anton Braun probably married young in his hometown, because in 1708 was born his son, Anton Braun Jr. (der Jüngere) (1708-23.10.1776), who also became a skillful instrument maker, optician, and watchmaker as his father, and made the copy of one of his calculating machines, still preserved in Technischen Museum, Wien.

At some point, Braun left his hometown to go to Vienna, most probably to study at the University of Vienna. There on 19 April 1712, Braun, designated as “University optician and mathematician” married in Cathedral St. Stephan, to Maria Magdalena Steinin (the daughter of Georg Stein, the postmaster at Ettlingen in Swabia, and his wife Maria Eva).

Johann Jakob Marinoni (1676-1775)
Johann Jakob Marinoni (1676-1775)

Braun probably left Vienna soon after his marriage, because he established mechanical workshops in Prague and Milan in the following years. During this period, Braun became one of the most prominent instrument makers of his time and was highly appreciated by the Imperial Engineer and professor at the University of Vienna—Johann Jakob Marinoni (see the nearby portrait). Braun was warmly recommended by Marinoni and worked for him in the years 1719-1722 as a surveyor in cadastral surveying in the duchy of Milan.

Anton Braun returned to Vienna in 1723, and the next year he was appointed to the prestigious position of Kammeropticus und Mathematicus at the Austrian court, due to his outstanding precision mechanical, and mathematical skills. Three years later, he sat down as a candidate for the post of Imperial instrument maker (Kaiserlicher Compaß- und Instrumentenmacher). And he won (there were four candidates), presenting to the Emperor his advanced calculating machine, which he constructed in 1724 and which was already in use at the imperial court.

Braun apparently got in favor of the Holy Roman Emperor Karl VI, because he was not only appointed as an imperial instrument maker but was also granted a 12-diamond chain (value of 500 guilders), occupied with the portrait of the Emperor (kept now in the Museum in Rathaus Möhringen) and a huge sum of money—10000 guilders. The 10000 Gulden were never paid out, however, because of war-related expenditures and financial difficulties of the Viennese Court under Empress Maria Theresia, the daughter of Karl VI. Nevertheless, Braun bequeathed half of his assets to his hometown, so 6000 guilders were used for charity and the construction of a hospital.

A circular sundial made by Anton Braun in 1719 is kept now in the collection of the Adler Planetarium in Chicago.

Unfortunately, Braun’s tireless zeal and restless activity used up his physical powers early, and he died too young (41 years old) from a long-running lung disease, on 20 April 1728, in Vienna. His brother Johann Georg also got the support of Marinoni and succeeded Anton’s position as Kaiserlicher Optikus.

Jacob Leupold

“One man’s “magic” is another man’s engineering. “Supernatural” is a null word.
Robert A. Heinlein

Jacob Leupold
Jacob Leupold (1674–1727)

In 1727, after the death of the German scientist and engineer Jacob Leupold was published the 8th volume of his encyclopedia Theatrum Machinarium. This volume, entitled Theatrum arithmetico-geometricum is the best-illustrated work on calculation and measurement published during the 18th century. It describes and illustrates the calculating devices and machines of Kircher (Schott), Grillet, Leibniz, and Poleni, along with Napier’s rods and several calculation tables (interestingly, Leupold missed the Pascaline, the most famous calculating machine of the time). It also discusses and illustrates the various analog devices available, including slide rules and sectors, and other calculating and measuring rules, as well as systems of computing using the fingers. In this volume, Leupold not only mentioned that “he was interested in calculating machines more than 20 years”, and “he had four or five types brought out”, and that “he had been able to show the effect to different friends”, but also described an original calculating machine, designed by himself.

Obviously an example of the above-mentioned calculating machine of Leupold has never been manufactured. In fact, Leupold defined himself as Mathematico und Mechanico (i.e. Mathematician and Mechanic, what we today would call Engineer) because he did not produce machines in his pursuits, but he studied them, corrected the errors he found in those designed by others, and designed machines and solutions of his own.

The Mechanical Calculator of Jacob Leupold

The calculating machine of Leupold, front view
The calculating machine of Leupold, front view

The calculating machine of Leupold has a circular shape which will very soon become very popular in the world of mechanical calculators. Yet in 1727 the calculating mechanism and the form of Leupold’s machine will be used by the German mechanic Anton Braun, who designed a similar device. Several machines, which appeared to be similar (not only) externally to Leupold’s will be devised later by Philipp Matthäus Hahn, Johann Helfrich Müller, Johann Reichold, and others.

The numbers are entered into the calculator by means of six small digital wheels, placed around the handle in the middle of the lid. Rotating this handle to a full revolution in a counterclockwise direction will transfer entered in the input mechanism number to the six dials (digital positions), placed in the outer ring (marked with A on the nearby figure), which present units, tens, hundreds, etc. The resulting mechanism has nine positions—the nine dials F, mounted on the outside ring (marked with C in the figure). The dials F have two graduations—the first is used during addition and multiplication, the second—during subtraction and division. The digits of the result are pointed by special pointer-arrows, which are rotating around the middle of the axes.

On the same axes, but inside of the machine (see the figure below), are placed 10-teeth stop-wheels G, M, I…, which can be rotated only in one direction (clockwise), and can be fixed by means of the small rod D and spring-rod C.

The calculating machine of Leupold, internal view
The calculating machine of Leupold, internal view

The motion from the input (six small dials) to the result (nine bigger wheels in ring C) is transferred by means of the 9-teeth sector-teeth-strip N, which can be rotated around axis w. On the strip N, on the plane, perpendicular to it is attached a thin plate X, which is illustrated separately in the upper right part of the figure. The left side-ward surface of the plate is straight, the right is formed as nine steps of the same height. The function of this plate is to raise the teeth strip, which will engage with the result mechanism and will rotate it. That is done in this way:
Below each one of the entering wheels is placed a ring with a spiral inclined upper surface A, which is shown in the upper left part of the figure. Each position of the input mechanism has an eccentric rod (marked with m, n, and o), which has a bulge in its upper part, contacting the plate X, and another bulge in its lower part, contacting to the spiral surface of the ring. Rotating one of the six entering wheels, thus rotating the appropriate ring, we actually will raise or take down these eccentrics in a plane, perpendicular to the plane of the teeth-strip and parallel to the plane of the plate X in this way, that upper bulge will raise and fix in the upper position this plate for a different distance, and in this way, a different number of teeth will be engaged with the top-wheels G, M, I. Thus, the number of the teeth, to which will be rotated on each of these wheels, is determined by the length of the engaging way of the eccentric with the plate X.

The invented by Leupold mechanism became quite popular at the end of the XIX century in several calculating machines, e.g. machines of Dietzschold, Büttner, and Pallweber. It will be called Schaltklinke in German, in English it will be called switching latch, intermittent contact, adjustable pawl, and selectable ratchet.

The calculating machine of Leupold, the tens carry mechanism
The tens carry mechanism of the calculating machine of Leupold

In the middle of the machine is placed an auxiliary counter, which scale can be seen around the handle. This counter is destined to count the revolutions of the handle.

The tens carry mechanism (see the nearby figure) is made as follows:
On the same axes, on which are fixed the stop-wheels G, M, I…, are fixed also 10-teeth wheels O, P, Q… Between them are placed intermediate wheels K, L, M… Each intermediate wheel has an attached finger, fixed with a spring (d, e, g, j…), and in odd-numbered wheels, this finger is placed over the wheel of units, while in even-numbered wheels this finger is placed below it. In one revolution of the units-wheel Q the finger de will rotate to 1/10 revolution the wheel of the tens P, and thus the tens carry will be performed. The tens carry from the other wheels will be performed in much the same manner.

The Calculating Drum of Jacob Leupold

In the Theatrum arithmetico-geometricum of Leupold is described also a calculating tool, the so-called Rechenscheiben (calculating drum), based on Napier’s rods (see the lower figure). Let’s examine its construction.

Leupold's Rechenscheiben
Leupold’s Rechenscheiben, the drawing in Theatrum arithmetico-geometricum

The Rechenscheiben (calculating drum) of Leupold (in the nearby image you can see a later replica of die Rechenscheiben, from the exposition of Arithmeum Museum, Bonn) consists of 11 ten-sided disks, mounted on common axes (marked with F in the drawing). The cylindrical housing is broken through so that only two levels of the number series (two sides of the disks) are visible.

The right side disk is fixed, while the remaining ten disks can be rotated with a hand. On the surface of the device are placed 10 round openings (marked with d), which are used for fixing the angular position of the disks, by means of pins (c), pushed in the appropriate opening, as it is shown on the right side of the figure. On the surface of each of the 10 sides of the rotating disk are inscribed the digits of the same Napier’s rod, while on the side of the fixed disk, with face to the operator, is inscribed a column of digits from 1 to 9. The multiplicand is entered by rotating the proper disks and fixing them by means of the pins against the fixed column of digits (multiplicands) of the fixed disk.

Leupold also mentioned another variant of the device. The modification consists of replacing the disks with rings the width of a number strip. The rings are pushed onto a wooden cylinder and can be adjusted independently of each other.

Biography of Jacob Leupold

Jacob Leupold
Jacob Leupold (1674–1727), engraving from 1726, author Martin Bernigeroth

Jacob Leupold was born on 22 July 1674, in Planitz, a village near Zwickau (then in Electorate of Saxony), to George Leupold (1647-1707), a skilled mechanic (cabinet-maker, turner, sculptor, and watchmaker) from Johanngeorgenstadt (Erzgebirge), and Magdalena Leupold.

Since early childhood Jacob scrutinized his father’s work, developing an interest in various mechanical things and as he described it later, …I had not only the opportunity of seeing how different things have been made, but also manual work made me strong. Due to his “ailing physical condition,” it did not seem advisable to let the young Leupold learn a trade, so George Leupold sent him to Zwickau’s Latin school. In 1693 Leupold started to study theology at the University of Jena, but he did not give up his interest in mechanical things—he also attended lectures of the well-known astronomer and mathematician Erhard Weigel. In 1694 Jacob left Jena, since he obviously could not afford the registration fee, and switched to the University of Wittenberg, but soon left it for the same reason—lack of money.

Finally, in 1696 Jacob was enrolled for free at the University of Leipzig, where he apprenticed to an instrument maker, and in 1698 he started to produce globes, quadrants, sundials, and measuring and drawing instruments. He was so successful that he broke off his studies to set up a mechanical workshop. He was helping to design and build many instruments needed for experimental physics studies and his interests had fully changed from theology to mechanics and mathematics.

In 1701 Leupold got a position as a hospital warden in George Military Hospital, thus obtaining not only regular income but also enough free time to dedicate himself to mechanics. In the same 1701, he married Anna Elisabeth, and they had three sons and three daughters, who all died young except for one daughter. Unfortunately, Anna Elisabeth also died young in March 1713.

In 1704 Leupold became ill from an unknown illness, possibly a stroke, which affected his memory and hearing. In his discussions on the capacity of humans for labor, he notes that he was able to lift and move far more weight before his illness.

In 1714 Leupold resigned from the hospital and managed his instrument shop with several assistants. The shop produced both musical and scientific instruments. During this time, he was associated with the University of Leipzig as a mechanic.

In 1715 Leupold was appointed a member of the Berlin Academy of Sciences. This position quickly attracted two other plum appointments: Commissioner of Mines to Saxony and Counselor of Commerce to Prussia.

Leupold's Theatrium Machinarum Generale from 1724
Leupold’s Theatrium Machinarum Generale from 1724

Leupold is also credited as an early inventor of air pumps. He designed his first pump in 1705, and in 1707 he published the book Antlia pneumatica illustrata. In 1711 following the advice of its president Wilhelm Leibniz, the Prussian Academy of Sciences acquired Leupold’s pump.

In 1720 Leupold started to work on the manuscript of his famous encyclopedia Theatrium machinarum, a nine-volume series on machine design and technology, published between 1724 and 1739. It was the first systematic analysis of mechanical engineering in the world. Leupold’s intended audience was not highly educated elites, but rather the common mechanic, that’s why the book is written in German, not in Latin (the dominant language of science and the universal means of communication in Western Europe up to the end of the 17th century). His work is addressed …not to the learned and experienced mathematicians who are already, or should be, better acquainted with them… [and most of whom] have studied mechanics more as a subject of curiosity and a hobby, than with any view of service to the public. The people we had in mind were rather the mechanic, handicraftsman, and the like, who, without education or knowledge of foreign languages have no access to many sources of information…

In the 1720s Leupold rejected several offers by the Russian tsar Peter the Great to move to St. Petersburg.

At the beginning of January 1727, when the eighth volume of the Theatrum Machinarum was nearing completion, Leupold fell gravely ill and died on 12 January 1727, at the age of only 52.


Creative minds don’t follow rules, they follow will.
Amit Kalantri

Jean-Antoine Lépine
Jean-Antoine Lépine (1720-1814)

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 Arithmetical Machine of Lépine
The Arithmetical Machine of Lépine (© CNAM, Paris)

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.

The Arithmetical Machine of Lépine
The Arithmetical Machine of Lépine (© National Museum of American History in Washington)

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 Arithmetical Machine of de Lépine, front view
The Arithmetical Machine of Lépine, front view

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.

Arithmetical machine of de Lépine, back view
The Arithmetical Machine of Lépine, back view

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.

Giovanni Poleni

When a great genius appears in the world you may know him by this sign; that the dunces are all in confederacy against him.
Jonathan Swift

Giovanni Poleni
Giovanni Poleni (1683-1761)

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.”

Poleni Miscellanea
The calculating machine of Poleni (the drawing in Miscellanea)

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 replica of the calculating machine of Poleni (© Museo Nazionale della Scienza e della Tecnica, Milan)
A replica (made in 1959) of the calculating machine of Poleni (© Museo Leonardo da Vinci, Milan)

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 calculating machine of Giovanni Poleni, a part of illustration from the Theatrum arithmetico geometricum
The calculating machine of Giovanni Poleni, a part of 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.

The calculating machine of Giovanni Poleni, the whole illustration from the Theatrum arithmetico geometricum
The calculating machine of Giovanni Poleni, the whole illustration from the Theatrum arithmetico geometricum

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 🙂

Palazzetto Pisani in Venice Poleni
Palazzetto Pisani in Venice, a property of the Poleni family until 1751, when Giovanni Poleni sold it to Andrea Pisani

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.

University of Padova
The University of Padova is one of the world’s oldest centers of higher learning, founded in 1222

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.

Giovanni Battista Morgagni, а long standing friend, colleague, and doctor of Giovanni Poleni
Giovanni Battista Morgagni (1682–1771), а long-standing friend, colleague, and doctor of Giovanni Poleni

Gottfried Leibniz

Genius is one percent inspiration and ninety-nine percent perspiration.
Thomas Alva Edison

Gottfried Wilhelm (von) Leibniz (1646-1716), portrait from Johann Friedrich Wentzel (circa 1700)

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 17th 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).

A sketch is from a Leibniz's manuscript from 1672
A sketch is from Leibniz’s manuscript from 1685

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.

Leibniz pin wheel, circa 1670
Leibniz pinwheel, circa 1670

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.

One of the old machines of Leibniz
One of the old machines of Leibniz

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.

A replica of the Stepped Reckoner of Leibniz form 1923 (original is in the Hannover Landesbibliothek)
A replica of the Stepped Reckoner from 1923 (the original is in the Hannover Landesbibliothek)

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 stepped-drum mechanism
The stepped-drum mechanism

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.

Stepped Reckoner without the cover (image from Leporello album on investigation in Glashütte, 1894)
Stepped Reckoner without the cover (image from Leporello album on investigation in Glashütte, 1894)

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.

The tens carry mechanism (© Aspray, W., Computing Before Computers)
The tens carry mechanism (© Aspray, Computing Before Computers)

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).

An outside sketch (based on the drawing from Theatrum arithmetico-geometricum of Leupold)
An outside sketch (based on the drawing from Theatrum arithmetico-geometricum of Leupold)

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.

Upper view sketch of Stepped Reckoner
Upper view sketch of Stepped Reckoner

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:


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.

Brevis descriptio Machinae Arithmeticae, cum Figura
Brevis descriptio Machinae Arithmeticae, cum Figura

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

Dissertatio De Arte Combinatoria of Leibniz
Dissertatio De Arte Combinatoria of Leibniz

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.

Leibniz Circle
Leibniz Circle from Dissertatio De Arte Combinatoria

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 system (certainly, he is not the inventor of the binary system. Pingala, an otherwise-obscure Indian author, wrote about 300 B.C. Chandahsastra, or Science of Meters. In this treatise, the author used a binary logic system (short-long, in this case, rather than 0-1) to explore meter in poetry.)

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 (in the lower figure is shown 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.

The second page of De Progressione Dyadica
The second page of De Progressione Dyadica

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.

Cesar Caze

Simplicity is the ultimate sophistication.
Leonardo da Vinci

Nouvelles de la république des lettres
Nouvelles de la République des Lettres, May 1707

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 New Arithmetical Machine of Caze (© IBM Europe collection)
The New Arithmetical Machine of Caze (© IBM Europe collection)

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 17th 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.

The New Arithmetical Machine of Caze (© CNAM, Paris)
The New Arithmetical Machine of Caze (© CNAM, Paris)

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 17th 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 18th 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.

Claude Perrault

Many a zero thinks it is the ellipse on which the Earth travels.
Stanisław Jerzy Lec

Claude Perrault in 1656
Claude Perrault (1613-1688), a portrait from 1656, attributed to Philippe de Champaigne (© Musée du Louvre)

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.

A replica of the Perrault's calculating device (© Musée des Arts et Métiers)
A replica of the Perrault’s calculator (© Musée des Arts et Métiers)

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.

The sketch of the Abaque Rhabdologique
The sketch of Abaque Rhabdologique from the original description

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

Perrault Portrait By Gerard Edelinck
Portrait of Claude Perrault by Gerard Edelinck

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).

Portrait de l'Architecte Claude Perrault, Musée de l'Histoire de la Médecine, Paris
Portrait de l’Architecte Claude Perrault, Musée de l’Histoire de la Médecine, Paris

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).

Eastern facade of the Louvre, known as the Colonnade, designed mainly by Claude Perrault
Eastern facade of the Louvre, known as the Colonnade, designed with the participation of Claude Perrault

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.

The triumphal arch on Rue St-Antoine, designed by Claude Perrault (Source: Le Magasin pittoresque, 1847)
The triumphal arch on Rue St-Antoine, designed by Claude Perrault (Source: Le Magasin Pittoresque, 1847)

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.

Ramon Llull

И рад бы в рай, да дверь то где?
Николай Некрасов

Ramon Llull
Ramon Llull (1232-1315)

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:

Llull's machine
Llull’s logical 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 15th 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 16th 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 17th 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.