George Brown

I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use.
Galileo Galilei

George Brown's An Account of the Rotula Arithmetica
George Brown’s An Account of the Rotula Arithmetica

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

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

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

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

The brass plate of George Brown's Rotula Arithmetica
The brass plate of George Brown’s Rotula Arithmetica (© National Museums Scotland)

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

Biography of George Brown

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

Arithmetica Infinita; or The Accurate Accomptant's Best Companion contriv'd and calculated by the Reverend George Brown
Arithmetica Infinita; or The Accurate Accomptant’s Best Companion contriv’d and calculated by the Reverend George Brown. The author’s engraved portrait is on the page opposite the title page (© National Museums Scotland)

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

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

George Brown died in 1730 in London.

Tito Livio Burattini

It is so pleasant to come across people more stupid than ourselves. We love them at once for being so.
Jerome K. Jerome

Tito Livio Burattini
Tito Livio Burattini (1617-1681)

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

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

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

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

The calculating machine (Ciclografo) of Tito Livio Burattini from 1658
The calculating machine (Ciclografo) of Tito Livio Burattini from 1658 (© Museo Galileo, Firenze)

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

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

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

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

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

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

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

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

Biography of Tito Livio Burattini

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

The native house of Burattini in Agordo
The native house of Burattini in Agordo

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

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

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

The plaque on the house of Burattini
The plaque on the house of Burattini

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

The second house of Burattini in Agordo, with their coat of arms
The second house of the Burattini family in Agordo, with their coat of arms

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

The Primary School "Tito Livio Burattini" in Agordo
The Primary School “Tito Livio Burattini” in Agordo

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

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

John Greaves (1602-1652)
John Greaves (1602-1652)

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

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

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

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

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

Ujazdowski Castle in Warsaw, Poland (Photo: Marek and Ewa Wojciechowscy)
Ujazdowski Castle in Warsaw, Poland (Photo: Marek and Ewa Wojciechowscy)

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

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

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

A sketch from the treatise Dragon Volant by Burattini
A sketch from the treatise Dragon Volant by Burattini

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

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

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

Burattini's most famous book—Misura Universale, published in 1675
Burattini’s most famous book—Misura Universale, published in 1675

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

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

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

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

Wilhelm Schickard

You can never cross the ocean unless you have the courage to lose sight of the shore.
Cristoforo Colombo

Wilhelm Schickard, portrait from 1632
Wilhelm Schickard (Wilhelmus Schickart), Professor of Hebrew, Astronomy, and Mathematics – a portrait from 1632

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

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

Johannes Kepler in 1610
Johannes Kepler in 1610

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

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

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

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

The sketch from the Württembergischen Landesbibliothek in Stuttgart
The sketch from the Württembergischen Landesbibliothek in Stuttgart
The instructions to the mechanician from the Württembergischen Landesbibliothek in Stuttgart
The instructions to the mechanic from Württembergischen Landesbibliothek

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

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

A replica of the Schickard's machine, created by Bruno v. Freytag Löringhoff in 1960 (© Universität Tübingen)
A replica of Schickard’s machine, created by Bruno v. Freytag Löringhoff in 1960 (© Universität Tübingen)

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

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

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

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

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

The first drawing from the second letter to Kepler
The first drawing from the second letter to Kepler

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

The small second drawing from the second letter to Kepler
The second drawing from the second letter

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

Schickard obviously was not satisfied with the work of the mechanic, involved in the production of the device, because the note to Pfister begins (old German language is used):
Rechen Uhr betreffs.
1. Die zän seind gar vngleich und vnfleißig…

(which means in English, kindly translated by Mr. Stephan Weiss,
Concerning Calculating Clock,
1. The teeth are inequally made and don’t work. Sometimes more than a tenth part is driven, sometimes less. 20 teeth would be better.
2. The front eccentric smooth disk drags a little, it should be turned.
3. (NB) The single tooth (note: for the carry of tens) should not be placed in the middle between two others. Should it touch right onto a numeral tooth, it will push the number forward twice.
(NB) 4. Only the 0, and also the 9 should move the left number, the first when subtracting, and the next when adding.

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

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

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

  • A multiplying device.
  • A mechanism for recording intermediate results.
  • A decimal 6-digit adding device.
A view to Napier's rods in the replica, created by Bruno v. Freytag Löringhoff (© Universität Tübingen)
A view of Napier’s rods in the replica (© Universität Tübingen)

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

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

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

A view to the mechanism for recording of intermediate results of v. Freytag Löringhoff, (© Universität Tübingen)
A view to the mechanism for the recording of intermediate results (© Universität Tübingen)

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

A view to the wheels mechanism of the Schickard's machine of v. Freytag Löringhoff (© Universität Tübingen)
A view of the wheel mechanism of the machine (© Universität Tübingen)

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

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

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

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

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

Biography of Wilhelm Schickard

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

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

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

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

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

Tübinger Stift nowadays
Tübinger Stift nowadays

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

Stadtkirche St. Laurentius in Nürtingen, where Schickard served in 1614-1619
Stadtkirche St. Laurentius in Nürtingen, where Schickard served in 1614-1619

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

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

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

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

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

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

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

A page from Kepler's <em>Harmonices Mundi Libri V</em>, with figures, made by hand of Schickard
A page from Kepler’s Harmonices Mundi Libri V, with figures, made by the hand of Schickard

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

A partial reconstruction of the Schickard's planetarium, Stadmuseum Tübingen (© Universität Tübingen)
A partial reconstruction of the Schickard’s planetarium (© Stadmuseum Tübingen)

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

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

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

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

    Books, written by Wilhelm Schickard:

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

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

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

A raft on the Neckar River in Tübingen, painting by Johannes Pfister from 1620
A raft on the Neckar River in Tübingen, painting by Johannes Pfister from 1620

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

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

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

Claude Dechales

There are many of us that are willing to do great things for the Lord, but few of us are willing to do little things.
Dwight L. Moody

The title page of book L' art de naviger demontré... of Claude Dechales
The title page of book L’ art de naviger demontré… of Claude Dechales

The French Jesuit mathematician Claude François Milliet Dechales (1621–1678) was best remembered for his book Cursus seu mundus mathematicus, a complete course of mathematics, published in Lyon in 1674. Dechales published also several other books, and in one of them, devoted to the principles of navigation—L’ art de naviger demontré par principes & confirmé par plusieurs observations tirées de l’experience, published in Paris in 1677, he described a small counting device with a ratchet wheel as input and analog display indicators.

The counting device of Dechales is described in a section for the calculation of distances, on pages 191, 192, and 193 (see page 192 below). Its purpose is to measure the force of the wind.

Page 192 from the book L' art de naviger demontré... of Claude Dechales
Page 192 from the book L’ art de naviger demontré… of Claude Dechales

The horizontal wheel, one or two feet in diameter (marked with C on the upper drawing), composed of small cone-shaped wings, rotates according to the force of the wind. The motion from wheel C is transferred to the inner counting wheels D, G, K, etc., connected to display indicators N, O, P.

Biography of Claude Dechales

Claude François Milliet Dechales (aka Milliet de Challes, or Deschales), was born in 1621 in Chambéry, Savoy. He was the youngest son (of three) of Hector Milliet, Baron de Challes and d’Arvillars (1568-1642), and his second wife Madeleine de Montchenu (1586-1651).

Claude’s grandfather, Louis Milliet (1527-1599) was a jurisconsult, first president of the Senate of Savoy, Grand Chancellor, and ambassador. He obtained the title of Baron in 1569, buying the seigneuries of Faverges, then of Challes.

Claude’s father, Hector Milliet, became Baron de Challes in 1618, then he acquired the lordship of Arvillars in 1628, and took the name Milliet d’Arvillars. He was the first president of the Chamber of Accounts of Savoy, then the first president of the Sovereign Senate of Savoy and ambassador.

Claude’s elder brothers Jean Louis (1613-1675) and Sylvestre (1616-1685) inherited from his father noble titles Marquis de Challes and Marquis d’Arvillars, but Claude choose a religious and scholarly career. At the age of 15 (21 September 1636), he entered the Jesuit Order. By the time Dechales entered the Order, it contained over 16000 men. The main task of the Jesuits was education, but the next most important task was missionary work throughout the world.

After spending some 10 years in education (as a full Jesuit he would have had both the equivalent of a B.A. within the order, and a doctorate in theology), in the middle 1640s Dechales was sent as a Jesuit missionary to the Ottoman Empire, where he taught letters in the schools of his order for nine years. After his return to France around 1656, Dechales lectured at Jesuit colleges, first in Paris where for four years he taught at the Collège de Clermont. Then he taught philosophy, mathematics, and theology at College de la Trinite de Lyon and was a rector at Chambéry. From Chambéry he went to Marseilles, where King Louis XIV appointed him Royal Professor of Hydrography. In Marseilles he taught navigation, military engineering, and other applications of mathematics. From Marseilles, he moved to the College of Turin, in Piedmont, where he was appointed professor of mathematics.

Claude Dechales was known for his friendliness and pedagogical competence in the teaching of mathematics and for his ability to teach and write clear explanations of complex technical topics, not for his research ability, advanced mathematical imagination, or originality of his work. His correspondence with Hevelius, Huygens, and Cardinal Bona, among others, survives.

Claude François Milliet Dechales died on 28 March 1678 in Turin, Italy, where he taught mathematics at the local college.

Leonardo da Vinci

This article was created with the expert advice of my correspondent Mr. Silvio Hénin, Milan, Italy
Georgi Dalakov

Leonardo da Vinci (1452-1519)
Leonardo da Vinci (1452-1519)

In Leonardo’s manuscript Codex Madrid I, compiled by the genius in 1493, when he served at the Castle of Milan under Duke Ludovico il Moro, there is a sketch, picturing a mechanism, which is very likely to be designed for calculating purposes.

Leonardo da Vinci is probably the most diversely talented person ever to have lived. He was a remarkable painter, engineer, anatomist, architect, sculptor, musician, etc. During his life, Leonardo produced thousands of pages of perfectly illustrated notes, sketches, and designs. A part of these pages (only half of almost 13000 original pages), as impressive and innovative, as his artistic work, managed to survive to our time. They are collected now in 20 notebooks (so-called codices), comprising some 6000 pages. These codices decorate expositions of many museums (the only codex, that is in private hands is the Codex Leicester, a small volume of only 36 pages, which was purchased in 1994 by Bill Gates for $30.8 million).

Some of the manuscripts were so thoroughly lost, that they weren’t found again until the last few decades! In February 1965, an amazing discovery was made by Dr. Julius Piccus, Professor of Romance Languages ​​in Boston, working in the Biblioteca Nacional de España (National Library of Spain) in Madrid, who was searching for medieval Spanish ballads and troubadours. Instead of ballads, searching in some cabinets, he stumbled upon two unknown collections of Leonardo’s manuscripts, bound in red Moroccan leather, which were known by the Spanish librarians, but were not described yet (the public officials stated that the manuscripts “weren’t lost, but just misplaced”, because of an error in the catalog).

These manuscripts were in the collection of the 16th-century Italian sculptor Pompeo Leoni (1533-1608) and were taken by him to Spain, to be offered to the Spanish King Felipe II in the 1590s. For some reason, Leoni kept the collection in his house, and after his death, it was divided into several parts, which went to England, France, and Italy, but few remained in Spain and in 1642 were donated to King Felipe IV, to become later part of the Royal Library (later Biblioteca Nacional) in Madrid. These manuscripts (almost 700 pages, on subjects such as architecture, geometry, music, mechanics, navigation, and maps) are now referred to as Madrid Manuscripts or Codex Madrid I and Codex Madrid II.

Page 36 back (folio 36v) of Codex Madrid I (the upper drawing is interesting for us, as the lower is picturing a multiple pulley mechanism)
Page 36 back (folio 36v) of Codex Madrid I (the upper drawing is interesting for us, as the lower pictures a multiple pulley mechanism)

Codex Madrid I (composed of 192 pages with a size of 21/15 cm) is an engineer’s delight treatise on mechanics, full of perfectly drawn and laid out gadgets, gears, and inventions. Interestingly, this codex was not put together after Leonardo’s death (which is for example what happened with the miscellaneous contents of the Codex Atlanticus, the largest collection of Leonardo’s manuscripts), but it was in fact Leonardo himself who put it together and it has survived almost intact, except for 16 pages, which have been torn out and seem to have been lost. Codex Madrid I can be called the first and most complete treatise in the history of Renaissance mechanics, and in it, there is a sketch, picturing a mechanism, which is very likely to be designed for calculating purposes.

The gear wheels in the figure are numerated as follows: the small wheels are numerated with 1, while the bigger wheels are numerated with 10 (take into account, that in this case Leonardo, as in many of his writings, writes laterally inverted from right to left!).

The text bellow the figure is written using the strange 15th century Italian language of Leonardo, again inverted, and it is not very informative – Questo modo è ssimile a cquello delle lieve che qui è a risscontro in pari numero di asste. E non ci fo altro divario sennonchè quessto, per essere fatto con rote dentate colle sue rochette, esso ha continuatione nel suo moto, della qual cosa lo sstrumento delle lieve senplici n’è privato.

The language of Leonardo is never easy to understand because it is indeed quite cryptic and his writings are more private notes, than a clear explanation for readers. A rough translation of the text below the figure is: This manner is similar to that of the levers, although different, because, being this made of gears with their pinions, it can move continuously, while the levers cannot.

Dr. Roberto Guatelli, demonstrating one of his models (Leonardo's flyer spindle), a photo from the journal Popular Science, October 1949 (pp. 164-165)
Dr. Roberto Guatelli, demonstrating one of his models (of Leonardo’s flyer spindle), a photo from the journal Popular Science, October 1949 (p. 164)

The Italian engineer, Dr. Roberto Ambrogio Guatelli (born 4 Sep. 1909 in Binago, near Milan-died Sep. 1993), a science and engineering graduate from the University of Milan, was a world-renowned expert of Leonardo da Vinci, who specialized in building working replicas of Leonardo’s machines since the early 1930s. When in the late 1930s his collection (some 200 models) went on exhibit, Italian dictator Mussolini was so pleased with this great example of Italy’s engineering heritage, that he decided to fund a traveling exhibition to impress the whole world.

In 1940 Guatelli, emerging as one of the leading model makers, traveled to New York for an exhibition at the Museum of Science and Industry in the Rockefeller Center. When World War II broke out, Guatelli was forced to flee to Japan with his models, but there he was interned, and the models blew to bits in a bombing of a warehouse, in which they were stored. After his return to Italy, he was invited to show again his models in the USA, so he began rebuilding his models and created about 60 new devices.

In 1947 Guatelli had an exhibition at the Ford Museum in Dearborn, Michigan. Eleanor Roosevelt, who was visiting the museum introduced Guatelli to Thomas Watson Sr. who was also visiting. Watson recognized Guatelli’s genius and hired him in 1951 to rebuild more of his models for IBM which then toured them as a corporate-sponsored exhibition. IBM placed a workshop at Guatelli’s disposal and organized traveling tours of his replicas (see the upper image of Guatelli, demonstrating one of his models), which were displayed at museums, schools, offices, labs, galleries, etc.

In 1967, shortly after the discovery of the Madrid Manuscripts, Guatelli went to the library of the Massachusetts Institute of Technology (MIT), to examine its copy of the manuscripts. When seeing the page with the above-mentioned sketch, he remembered seeing a similar drawing in Codex Atlanticus. In fact, Codex Atlanticus contains a lot of gear-wheel transmissions, see for example the lower sketch.

A part of page 01r recto (front) of Codex Atlanticus
A part of page 01r recto (front) of Codex Atlanticus

The left and the middle drawings are of hodometers, devices for measuring distance, known for many centuries (it seems the first odometer was described by Heron of Alexandria, who used an analogue train of gear wheels (linked so that each time one wheel completes a revolution the next wheel turns one-tenth of a revolution, thus recording a carry), while the right drawing is of a ratio mechanism. Leonardo certainly was not the inventor of gear-wheels and the hodometer, and he knew very well at least the fundamental treatise of Vitruvius—De architectura, written about 15 BC. In Book X, Chapter IX of this treatise describes one hodometer with gear wheels and a carry mechanism.

Using the two above-mentioned drawings, in 1968 Dr. Guatelli built a hypothetical replica of Leonardo’s calculating device (see the lower photo). It was displayed in the IBM exhibition, as the text beside the replica was:
Device for Calculation: An early version of today’s complicated calculator, Leonardo’s mechanism maintains a constant ratio of ten to one in each of its 13-digit-registering wheels. For each complete revolution of the first handle, the unit wheel is turned slightly to register a new digit ranging from zero to nine. Consistent with the ten-to-one ratio, the tenth revolution of the first handle causes the unit wheel to complete its first revolution and register zero, which in turn drives the decimal wheel from zero to one. Each additional wheel marking hundreds, thousands, etc., operates on the same ratio. Slight refinements were made on Leonardo’s original sketch to give the viewer a clearer picture of how each of the 13 wheels can be independently operated and yet maintain the ten-to-one ratio. Leonardo’s sketch shows weights to demonstrate the equability of the machine.

The replica of the Leonardo's machine, built by Dr. Guatelli
The replica of Leonardo’s machine, built by Dr. Guatelli

After a year the controversy regarding the replica had grown and an interesting Academic trial was then held at the Massachusetts Technological University, in order to ascertain the reliability of the replica.

Amongst others were present I. Bernard Cohen (1914-2003), professor of the history of science at Harvard University—consultant for the IBM collection, and Dr. Bern Dibner (1897-1988)—an engineer, industrialist, and historian of science and technology.

The objectors claimed that Leonardo’s drawing was not of a calculator but actually represented a ratio machine. One revolution of the first shaft would give rise to 10 revolutions of the second shaft, and 10 to the power of 13 at the last shaft. And what is more, such a machine could not be built, due to the enormous amount of friction that would result. Dr. Guatelli managed to build a replica, but the technology in the 15th century was so primitive, compared to ours, that a working machine could not be built.

Leonardo Codex Madrid It was stated that Dr. Guatelli had used his own intuition and imagination to go beyond the statements of Leonardo. The vote was a tie, but nonetheless, IBM decided to remove the controversial replica from its display.

Sadly, it seems the objectors were right because such mechanisms were quite popular several centuries ago. There is another picture on the same Codex Madrid I (f. 51v), which describes a similar mechanism (see the nearby image). Leonardo’s text below the picture says: Quando la rota d dà una volta intera, la rota ne dà 10 e lla rota b ne dà 100, e così la rota a ne dà 1000. Onde, per l’oposito, si scanbia la forza, come detto nella 2a del sesto di sopra, che dov’è magiore il moto, lì è magiore forza. (When wheel makes one full turn, wheel c makes 10 turns and wheel makes 100, and so wheel a makes 1000. Thus, the force changes the other way round, as said in the 2nd paragraph above, that where higher is the movement, higher is the force. [This statement seems wrong, but what Leonardo means is possibly that the same torque applied to a is ten times more efficient (in raising the right-end weight) than when applied to b].

Moreover, a similar mechanism can be seen (see the lower image) in the famous encyclopedia Theatrum Machinarium of Jacob Leupold.

A page from Theatrum Machinarum of Jacob Leupold
A page from Theatrum Machinarium of Jacob Leupold

My correspondent Mr. Silvio Hénin from Milan (the author of historic investigations of the machines of other Italian inventors, like Luigi Torchi and Tito Gonnella), also thinks that Guatelli was almost certainly wrong. Mr. Hénin was so kind to give his expert opinion regarding this matter:
Leonardo was possibly studying the properties of gear trains in comparison with systems of levers; both can multiply forces (torques), but only gears can produce a continuous movement. In the other direction, the gear train can multiply rotation speed. On the same page, in fact, a compound pulley system is shown, which has the same force-multiplying properties as a gear train, a demonstration of what Leonardo was examining.
I can only add some points:
1. Leonardo’s drawing does not show any numbering on the gear wheels (mandatory for a calculator).
2. No way to set the operands is shown (which is mandatory).
3. No way (e.g. ratchet) to stop the wheels in precise discrete positions (which is mandatory) is shown.
4. Two weights are shown at the two ends (useless in a calculator).
5. The use of 13 decimal figures for calculations in the XV century is quite nonsense.

Guatelli worked for IBM until 1961 when he left and opened a workshop in New York. Interestingly, Guatelli built models not only of Leonardo da Vinci’s calculating device, but also of numerous other calculators; for example, the Pascaline by Blaise Pascal, the Stepped Reckoner by Gottfried Leibniz, the Differential Engine of Charles Babbage, the adding machine of Giovanni Poleni, the tabulating machine of Herman Hollerith, and the Millionaire direct multiplier of Steiger and Egli.

René Grillet de Roven

A great man is always willing to be little.
Ralph Waldo Emerson

The book Curiositez mathematiques de l'invention du Sr Grillet horlogeur a Paris of Grillet
The book Curiositez mathematiques de l’invention du Sr Grillet horlogeur a Paris

In 1673 the Parisian mechanic and watchmaker of King Louis XIV, René Grillet de Roven, published the small (49 pages) book Curiositez mathematiques de l’invention du Sr Grillet horlogeur a Paris. The book (see its title page nearby) describes several different inventions, but most of it is devoted to his Nouvelle machine d’arithmétique. However, the first edition of Grillet’s book (it was an edition of Jean-Baptiste Coignart, printer-bookseller of the King, and was reprinted in 1678) had a miserable fate, as it was seized by the Intendant of Police in Paris, because in his book Grillet described also a mercury barometer, for which he was accused of plagiarism from another Parisian instrument-maker, Hubin.

Five years later, in 1678, the earliest scientific journal published in Europe—the French Le Journal des Sçavans, published a short description (3 pages text and 1-page sketch) of the arithmetical machine of Grillet (see the image below).

Pages 161 and 162 from Journal des Sçavans, describing the machine of Grillet

Pages 163 and 164 from Journal des Sçavans, describing the machine of Grillet

Page 165 from Journal des Sçavans, describing the machine of Grillet
Pages 161, 162, 163, and 164 from Journal des Sçavans (1678), describing the machine of Grillet

Grillet obviously was so obsessed with keeping his design of the machine a secret, that he does very little to enlighten its mode of operation. After telling his readers that the idea for the machine originated from the rulers of Napier, mentioning that Pascal had invented an admirable machine for doing arithmetic and that Petit had given us a cylindre artihmetique (the French physicist Pierre Petit created in the early 1650s a kind of Napier’s bones engraved on a cylinder), Grillet stated that his device combines the wheels of Pascal with the cylinder of Petit, in order to provide a wonderful machine, which would perform all the arithmetic operations.

The sketch in the book depicts a box with 24 sets of wheels (3 rows by 8 wheels) on the lid. Each wheel consists of several concentric circles, while the bottom of the box contains a set of Napier’s bones, engraved on cylinders, reminiscent of those, described more than 20 years ago by Gaspard Shot. The construction of the machine is rather simple, with no connection between the group of wheels on the lid, which means, that tens carry cannot be performed automatically.

To perform an addition or subtraction on the lid dials, the operator would set up the first number on the upper line of wheels, the second number on the middle line of wheels, and then to perform the operation mentally, setting down the digits of the answer on the lower line of wheels. The wheels would have found their main use in adding up the partial products, generated by the set of the cylindrical Napier’s bones, located in the lower part of the machine.

The calculating machine of René Grillet de Roven (© CNAM, Paris)
The Nouvelle machine d’arithmétique of René Grillet de Roven (© CNAM, Paris)

Perhaps the most advanced aspect of Grillet’s machine was its size, as it was small enough to be carried in a (quite big) pocket (however, it was bigger, than the adding machine of Morland). Its dimensions are 14.5×32.5×5 cm, weight is 990 g. Materials used are wood, cardboard, paper, brass, and glass.

The reason for Grillet’s striving to keep the mode of operation and internal structure of his device a secret can be understood by the fact that he tried to make money from his machine, exposing it for a fee at fairs, and charging a silver coin to see it operate. It is known, that in 1673 the machine was exposed in Paris, at the Cloistre S. Jean de Latran and Quai de l’Horloge, №49. In 1681 the machine was exposed in Amsterdam, Netherlands. It seems Grillet tried to produce his machine in series and to sell it, but obviously without great market success :-), as only three examples of the device survived to the present day, two in the collection of Musée des Arts et Métiers in Paris, and one in private hands (on the image below you can see the 32.5cm x 14.5cm walnut cased mechanical calculator of Grillet, sold by Christie’s in 2013 for 119000 USD).

The Nouvelle machine d'arithmétique of René Grillet de Roven
The Nouvelle machine d’arithmétique of René Grillet de Roven

Biography of René Grillet de Roven

Not much is known about Grillet’s personal life. As it is clear from his name (spelled also René Grilliet), he originates from the town of Roven (Rouen), in northwestern France, the capital city of Normandy, the same town, where in the early 1640s Blaise Pascal created the famous Pascaline.

René Grillet’s parents were perhaps Jean Grillet (1605-1675) and Marie-Rosse Grillet. Jean Grillet was a king’s enameler (émailleur ordinaire de la Reyne), known as an author of a book—La Beauté des plus belles dames de la cour, les actions héroyques des plus vaillans hommes dece temps… et plusieurs autres pièces sur divers sujets gaillards et sérieux (Paris, 1648). Jean Grillet was interested in glassware, clocks, instrument making (he invented a thermometer, which was donated to his patron Monseigneur le comte de Montéclair) and all these sorts of curiosities, so obviously René inherited all these interests.

René Grillet became a well-known at the time french mechanic, instrument maker, and watchmaker, and he used to work for His Royal Highness King Louis XIV. Besides the calculating machine, which is of particular interest to us, he is known as a maker of several other devices—a hygrometer (anemometer); graphometers; drawing instrument set; protractor, sector, and square; set square, with plumb-bob, etc.

Let’s examine the curious history of one of Grillet’s inventions, the double mercury barometer. In early 1673 Hubin (he was an English enameller, who was established in Paris in the early 1670s as Emailleur ordinaire du Roy, and made thermometers, hygrometers, hydraulic, and air machines), published a pamphlet, in which he accused Grillet of fraud for using the principle of his mercury thermometer, already demonstrated to the French Academy and displayed it on his shop’s window sill. This accusation probably explains why Grillet’s 1673 book Curiositez mathematiques de l’invention du Sr Grillet horlogeur a Paris was seized by the Intendant of Police in Paris. That’s not the whole story, however. When sometime later the famous Dutch mathematician, astronomer, physicist, and horologist Christiaan Huygens presented his newly invented barometer, Grillet complained contra Huygens, stating that he had invented the double barometer two years before Huygens and that the latter stole the idea from him. Grillet even stressed he had demonstrated his invention during a meeting of the French Academy, which member was Huygens. Huygens however stated that Grillet had in fact shown a barometer to the Academy, but it had nothing particular to it. It seems Hubin was the only honest man in all this confusing story because he mentioned that the idea of putting serpentine tubes on his instrument he borrowed from a professor of Chartres, Laurent Cassegrain, while Grillet and Huygens didn’t give the proper credit to their precursors, trying to get all the glory.

In 1681 Grillet traveled to Amsterdam, Netherlands, where he exposed his calculating machine and instruments. It seems he stayed in the Netherlands for several years, working not only as an instrument-maker but also probably perfecting his printing art there under the care of the skilled Dutch master printers.

Later in the 1680s Grillet probably tried to establish a calico-printing workshop in France, but after the decree of 1686 prohibiting calico-printing in France, he went to try his fortune in England.

In 1690 René Grillet is mentioned to live already in England, where he took a patent for painting and printing calicos, and a factory for this purpose was opened in the Old Deer Park at Richmond, near London. It was the first calico-printing factory in England, but Grillet made the mistake to employ mostly Frenchmen and Roman Catholics, which led him into trouble with the local society and English authorities.

Samuel Morland

Very few people can afford to be poor.
George Bernard Shaw

Portrait of Sir Samuel Morland, by Peter Lely
Portrait of Sir Samuel Morland, by Peter Lely

The English polymath Samuel Morland invented in the early 1660s a total of three calculating machines: one for multiplication and division, one for trigonometry, and one for addition and subtraction. Morland’s calculating machines were simple but became rather popular and London instrument-makers were still selling Morland’s calculators as late as 1710.

Samuel Morland entered government service in 1653 when he was chosen to accompany a British diplomatic mission to the court of Sweden’s Queen Christina. Christina of Sweden was a noted patron of the sciences, and it was most likely at her court that Morland first became acquainted with the calculating machine of Blaise Pascal, who in 1652 placed at Queen’s disposal an example. This is probably the most important reason, why Morland became interested in the construction of calculating devices.

In his book, The description and use of two arithmetick instruments (published in 1673 in London), Morland described two invented by him calculating devices, which are working without charging the memory, disturbing the mind, or exposing the operations to any uncertainty. This is the first book on a calculator, written in English, and the first separate work on the subject after Napier’s Rabdologiae. The book may also be considered the first comprehensive book in computer literature, as Blaise Pascal published nothing about his own machine, except one 18-page pamphlet in 1644.

The description and use of two arithmetick instruments of Morland
The description and use of two arithmetick instruments of Morland

The Machina Nova Cyclologica Pro Multiplicatione of Morland

It seems the first calculating machine of Morland, devised probably in 1664, was so so-called multiplication machine, based on the principle of Napier’s bones. The device was described in the above-mentioned book of Morland under the name Machina Nova Cyclologica Pro Multiplicatione (A New Multiplying Instrument).

The sketch of the multiplication machine of Morland from his book <em>The description and use of two arithmetick instruments</em> (© United States Library of Congress)
The sketch of the multiplication machine of Morland from his book The description and use of two arithmetick instruments (© United States Library of Congress)

Morland ordered manufacturing of the device to the famous English mechanics Henri Sutton (London instrument-maker, active mainly between 1650 and 1661) and Samuel Knibb (1625-1674), and the particular machine, shown here, was donated by Morland himself to Grand Duke Cosimo III de’ Medici in 1679. The dedication to the Grand Duke contains an obvious error: it gives 1666 as the year of invention and 1664 as the year of manufacturing. Let’s examine the construction of the multiplication machine.

The multiplication machine of Morland (© Museo Galileo, Florence)
The multiplication machine of Morland, inside the case (© Museo Galileo, Florence)

In fact, the multiplication machine of Morland simplifies only the intermediate products, using the principle of Napier’s rods. There is no automatic or mechanical carry mechanism provided.

The device is made of silver, gilt, and silvered brass, wood, and crystal. The dimensions are 18 x 55.5 cm.

The digits from the ten Napier’s rods are inscribed across the periphery of 10 thin metal disks in such a manner, that units and tens of the rods are placed on the opposite ends of the circle (see the lower images for a closer view of the mechanism). There are two rows of axes in the device, the upper axes are fixed, while the lower can be rotated. In the row of windows, placed between the two axes can be set numbers, thus it serves as a mechanical memory. To perform a multiplication, the appropriate disk must be removed from the upper fixed axis and to be mounted to the lower working axis. Each of the lower axes is attached to a small pinion in the body of the machine, and this pinion is engaged with a toothed strip. This strip can be moved in the horizontal direction by means of the key, which can be seen in the lower right part of the lower figure, and its movement is marked by an arrow, which can be moved across a scale.

The multiplication machine of Morland , without the case (© Museo Galileo, Florence)
The multiplication machine of Morland, without the case (© Museo Galileo, Florence)

When the appropriate disks are set (according to the digits of the multiplicand), the lower part of the machine is covered by a lid, which has windows. The key must be rotated by the operator until the arrow comes to the digit of the multiplier on the scale, and during this rotation, the toothed strip will move and rotate the pinions, engaged with it. Thus in the lower row of windows can be seen the product. If the factors are multi-digital, then these actions must be repeated until all digits will be used.

For example, to multiply 23 by 7, the user would first take the discs for 2 and 3, place them on the central posts and close the door so that in the window the number 23 appears (3 in the left-most window and 2 in the left side of the second pair of windows). Then the user would turn the key until the pin on the slider scale pointed to 7. Each time the key is turned the discs are rotated once, which advances the display of the multiplication table for the selected numbers (2 and 3) by one. The windows are constructed so that a number on the leftmost edge of one disc appears next to the number on the rightmost edge of the next disc. The final answer must be obtained by adding the adjacent numbers in the windows, either with pen and paper or, as the inventor suggested, with the help of his instrument for addition. So, to finish the example, after discs 2 and 3 have been rotated 7 times, the numbers in the display window would read: 1 4 2 1. The final result is found by adding the adjacent digits to give 161.

The Machina Cyclologica Trigonometrica of Morland

Most probably in 1664, Samuel Morland invented a device, that can be used for trigonometric calculations, which he called the Machina Cyclologica Trigonometrica. The device was built in 1670 by John Marke (Maker of mathematical and astronomical instruments active in London between 1665 and 1679. Marke worked with Henry Sutton and succeeded him in 1665.) Materials used: silvered brass and silver, dimensions: 330×275 mm. The instrument is housed in an ebony case with a lid and engraved plate.

Trigonometric machine of Morland
Machina Cyclologica Trigonometrica of Morland (© Museo Galileo, Florence)

The trigonometric calculator of Morland allowed the operator to perform trigonometry by drawing out a problem and measuring the solution as with drawing instruments but without the need for pen and paper. It was a set of three rulers set into a divided circle that could be moved about using dials to form a triangle of any shape.

The device was used for determining the value of a trigonometric function (sine and cosine) of a known angle or, vice versa, for finding the value of an angle when its function is known. The instrument consists of a rectangular box holding a disk with a toothed circumference. At the center of the disk are mounted compasses with a fixed arm and a mobile arm rotating with the disk itself. Below are two smaller disks. The left-hand one shows the angle values. Its index is connected to the toothed circumference of the large disk and completes one revolution for every 30° traveled by the mobile compass arm. The right-hand disk shows the linear sine values. Its index is connected to the mobile horizontal rod on the upper half of the box and completes one revolution for every 25 units traveled by the rod on the side channels. The numbering of the horizontal rod and vertical channels basically represents the sine grid or reduction quadrant.

The machine for addition and subtraction of Samuel Morland

The adding machine of Samuel Morland (presented in the book as A new and most useful instrument for addition and subtraction of pounds, shillings, pence, and farthings…) was devised probably in 1665-1666, as four examples of the device survive to our time. The largest models can add up to one million, in either decimals or pounds, shillings, and pence.

The device (materials used are silver and brass), presented a true pocket calculator, measuring only 4 by 3 inches and less than a quarter of an inch thick (122 x 71 x 8 mm), weight 230 g. On page 12 of the book, Morland advertises that the instruments may be ordered from Humphry Adamson, living at Jonas Moore’s house in the Tower (Sir Jonas Moore was an English mathematician, surveyor, ordnance officer, and patron of astronomy, who warmly recommended the machines of Morland). Humphr(e)y Adamson was a very skillful English clock- and instrument-maker.

Morland's Adding Machine
Morland’s Adding Machine (© Science Museum Group Collection)

On the lid of the device are mounted 8 pairs of graduated dials. The scales of the dials are inscribed on the ring around them. The lower three scales are divided into 4, 12, and 20 parts and are inscribed and used for calculations in the English currency units of the 17th century—guinea (which was equal to 20 shillings), shilling (which was equal to 12 pennies), and penny (which was equal to 4 farthings). The upper five big dials have decimal scales and are inscribed with words unit, tens, hundreds, thousands, tenth.

Across the periphery of each big dial are made openings, according to the scale of the dial—4, 10, 12, or 20. In these openings can be put a stylus, and the dial can be rotated. During this rotation, in a little window in the upper part of each scale can be seen the appropriate number. Below each window is mounted a stop-pin, which is used for limiting the rotation during the adding operation. Over each big dial is mounted a smaller one, which serves as a counter to the revolutions of the big dial. For that purpose is used one-toothed gearing—the lower dial has one tooth, while the upper dial has ten teeth, so making a full revolution of the lower dial has as a result 1/10 revolution of the upper one.

Morland's Adding Machine, close view
Morland’s Adding Machine, close view (© Science Museum Group Collection)

The adding operation is performed by rotating the appropriate dials in the clockwise direction, pushing the stylus into the opening against the appropriate number, and turning the dial until the stylus will be stopped by the stop-pin. The subtraction can be done by rotating dials in the counter-clockwise direction, pushing the stylus in the opening below the window, and rotating the dial until it moves below the appropriate number.

The machine doesn’t have a tens carry mechanism, and this made it useless for practical needs. On 16 April 1668 Morland first printed short descriptions of his two adding devices in the London Gazette—As would now be expected, the adding instrument made its way into the pockets of modern young gentlemen (at a price of £3 10s).

Despite the excellent workmanship of the arithmetic devices of Morland, they were not very useful for practical needs, moreover, some of his contemporaries were not so fascinated by their usefulness also. Samuel Pepys (formerly Morland’s tutee at Cambridge from 1650, who did not think much of Morland) wrote in his 1667-8 years diary, that the machine of Morland is very pretty, but not very useful, while the famous scientist Robert Hooke, wrote in his diary for 31 January 1673: Saw Sir S. Morland’s Arithmetic engine Very Silly. The machines of Morland were, however, appreciated by King Charles II and Cosimo III de’ Medici, Grand Duke of Tuscany, which was more important for Morland as a typical courtier–inventor.

Portrait of Sir Samuel Morland, line engraving by P. Lombart from portrait by P. Lely

Biography of Samuel Morland

Samuel Morland was born in 1625 at Sulhamstead Bannister, near Reading in Berkshire, England. He was the son of Rev. Thomas Morland, a rector of the local parish church.

Morland entered Winchester School in Hampshire in 1638, and in May 1644, as a sizar (i.e. paying no fees but instead doing basic menial labor for the college between his studies) at Magdalene College, Cambridge. In 1649 he was elected a fellow of the society, and stayed there until 1653, but took no degree. While in College, Morland devoted much time to the study of mathematics, but he also became an accomplished Latinist and was proficient in Greek, Hebrew, and French (then the language of culture and diplomacy).

For the latter half of his time at university, Morland was a noted supporter of the Parliamentarian Party, and, from 1647 onward, he took part in student politics. In November 1653, he was included in the retinue of the English lawyer, writer, and parliamentarian Bulstrode Whitelocke, on an embassy to the Queen of Sweden for the purpose of concluding a military alliance. Whitelocke describes him as a very civil man and an excellent scholar; modest and respectful; perfect in the Latin tongue; an ingenious mechanist.

Morland as young
Samuel Morland a young

Upon his return from Sweden in July 1654, Morland was appointed as an assistant to John Thurloe, the secretary of the Lord Protector Oliver Cromwell. Later Cromwell appointed him as “commissioner extraordinary for the distribution of the collected moneys” and one of the clerks of the signet. In May 1655, Morland was sent by Cromwell to the King of France and to the Duke of Savoy to remonstrate with him on cruelties inflicted by him upon the sect of Waldenses or Vaudois, which had strongly excited the English public. Morland remained, for some time, in Geneva as an English resident, and his actions were highly appreciated after his return to England in December 1655.

Upon his return to England, Morland became intimately associated with the Government of the Commonwealth and he became a witness to several not-so-legitimate actions of his magnates, e.g. of Dr. Hewitt’s being “trepanned to death” in June 1658 by Thurloe and his agents. The most remarkable intrigue, however, which came to his knowledge was the so-called Sir Richard Willis’s Plot from the beginning of 1659. Its object was to induce King Charles II and his brother to effect a landing on the Sussex Coast, under the pretense of meeting their adherents and putting them both to death the moment they disembarked. This plot is said to have formed the subject of a conference between Cromwell, Thurloe, and Willis at Thurloe’s office, and the conversation was overheard by Morland, who pretended to be asleep at his desk. From this time, Morland endeavored to promote the Restoration and warned the King about the plot. In justifying to himself the abandonment of his former principles and associates, he observes that avarice could not be his object, as he was, at this time, living in greater plenty than he ever did after the Restoration, having a house well furnished, an establishment of servants, a coach, & co, and £1,000 a year to support all this, with several hundred pounds of ready money, and a beautiful young woman to his wife for a companion (in 1657, Morland married Susanne, daughter of Daniel de Milleville, Baron of Boissay in Normandy).

Sir Samuel Morland, painting from 1660-1661, artist Samuel Cooper
Sir Samuel Morland, painting from 1660-1661, artist Samuel Cooper

Morland officially joined the King’s party in May 1660, when in departed to Breda, bringing with him letters and notes of importance. The King welcomed him graciously and publicly acknowledged the services he had rendered for some years past, making him liberal promises of “future preferment” but these were, for the most part, unfulfilled. On 18 July 1660, Morland was made a baronet, being described as of Sulhamstead Bannister, although it does not appear very clear whether he was in possession of any considerable property in the parish. He was also made a Gentleman of the Privy Chamber; but this appointment, he says, was rather expensive than profitable, as he was obliged to spend £450 in two days on attending the ceremonies accompanying the coronation. He, indeed, obtained a pension of £500 from the post office, but his embarrassments obliged him to sell it, and, returning to his mathematical studies, he endeavored, by various experiments and the construction of machines, to earn a livelihood. But the King’s gratitude went no further. Morland was extravagant, sold his pension, wasted the proceeds in France, and ended as a beggar.

In 1666, Morland obtained, in conjunction with Richard Wigmore, Robert Lindsey, and Thomas Culpeper, a probably remunerative patent “for making metal fire-hearths”. From correspondence between Morland and Dr. Pell, it appears that about this same time (1666), the former had intended to publish a work “On the Quadrature of Curvilinear Spaces” and had actually proceeded to print part of it, but was happily persuaded, by Pell, to lay it aside. In carrying out his experiments in hydrostatics and hydraulics, Morland encountered many difficulties as a consequence of their expense. In December 1672, the King granted to him the sum of £260 to defray the charges of about five hundred looking-glasses “to be by him provided and set up in Olive wood frames for our special use and service,” as well as an annuity of £300 “in consideration of his keeping and maintaining in constant repair a certain private printing press… which by our Especial Order and Appointment he hath lately erected and set up”.

In 1681, Morland was appointed “magister mechanicorum” to the King, who, in recognition of his ingenuity, presented him with a medallion portrait of himself, set in diamonds, together with a medal as “an honorable badge of his signal loyalty”. In October 1684, the King advanced him £200 and, a year later, Morland received a similar sum by way of “bounty”.  According to his own account, his mechanical experiments pleased the King’s fancy; but when he had spent £500 or £1,000 upon them, he received sometimes only half, and sometimes only a third, of the cost.

In 1682, Charles II sent him to France “about the King’s waterworks,” but there also he seems to have lost more than he gained. On his return, King James II restored to him his pensions, which had been, for some reason, withdrawn and, likewise, granted him part of the arrears, but Morland was never repaid the expenses of the engine which he had constructed for bringing water from Blackmore Park, near Winkfield, to the top of Windsor Castle. In 1686, Morland was corresponding with Pepys about the new naval gun-carriages. In 1687, his pension was paid down to Lady Day 1689.

In 1689, Morland addressed a long letter to Archbishop Tenison, giving an account of his life, and concluding with a declaration that his only wish was to retire and spend his life “in Christian solitude”; and he begs the primate’s “helping hand to have his condition truly represented to his Majesty.” Tenison probably did something for him, as there is a letter of thanks for “favours and acts of charity,” dated 5 March 1695. The errors of his life were probably considerable, as he speaks of having been, at one time, excommunicated; but some of his writings show that he was a sincere penitent, particularly ‘The Urim of Conscience’ (1695), written, as the title says, “in blindness and retirement.”

Anne Fielding, Lady Morland
Anne Fielding, Lady Morland, third wife of Sir Samuel Morland

Morland married four times but was definitely not happy in his family life. In 1657 he married the Huguenot Susanne de Milleville, the beautiful daughter of Daniel de Milleville, seigneur de Boissay, Commissaire du roi au syndic provincial de Normandie, and Catherine de la Wespierre, his wife. They had three children (one son, Samuel), but Susanne died in 1668, when she returned with the children to France, mortally ill with dropsy (In his Sep. 1667 diary Samuel Pepys mentioned that they went to King’s Play House and …Here we saw Madam Morland, who is grown mighty fat but is very comely.)

Morland married secondly, on 26 October 1670, in Westminster Abbey, Carola Harsnett (1651-1674), daughter of Sir Roger Harsnett, knight. They had two children, but Carola died on 10 October 1674, aged 22. Morland married thirdly, on 10 November 1676, in Westminster Abbey, Anne Fielding (1661-1680), third daughter of George Fielding of Solihull, Warwickshire, by Mary, second daughter of Sir Thomas Shirley, knight, of Wiston, Sussex (Anne died on 20 February 1680, aged 19, leaving no issue). Lastly, Morland married, at the Knightsbridge Chapel, Middlesex, on 1 February 1687, Mary Ayliffe, a woman of low origin and infamous character, from whom he obtained a divorce for adultery on 16 July the following year.

Morland lost his sight about three years before his death. He died on 30 December 1695 and was buried in the Hammersmith Chapel on 6 January 1696. He must have been in an extremely weak condition, as he was unable to sign his will. By it, he disinherited his only son, Samuel (died Nov. 1716), who was the second and last baronet of the family, as the reason for this appears from a passage in a letter “I have been, in my youthful days, very undutiful to my parents, for which God has given me a son, altogether void of filial respect or natural affection”.

Charles Cotterell

You can easily judge the character of a man by how he treats those who can do nothing for him.
Malcolm Forbes

Cotterell Portrait

In 1667, Sir Charles Cotterell, an English courtier and translator, suggested a calculating instrument, called Arithmetical Compendium (Instrument for Arithmetick). It was a combination of Napier’s Bones with a bead-type abacus, used to avoid writing down the partial products of multiplication using the rods. In fact, Cotterell’s idea was popularized several years later, circa 1670, when Robert Jole (Jole was a scientific instrument maker, brother in the Clockmakers’ Company, who was active in London between 1664 and 1704), produced his version of the device.

Cotterell most probably knew and was influenced by Samuel Morland, who devised in the 1660s a total of three calculating machines, which were presented to King Charles II and the general audience.

Arithmetical Compendium of Cotterell
Arithmetical Compendium of Charles Cotterell (© National Museums Scotland)

The calculating instrument of Cotterell has a boxwood case (size: 184 mm x 59 mm x 19 mm, weight: 0.13 kg), and was made of brass (binding pieces), glass, iron, engraved paper, and vellum. There is a separate silvered brass tool, for pushing beads and lifting out rods.

Arithmetical Compendium of Cotterell
Arithmetical Compendium of Charles Cotterell in its boxwood case (© National Museums Scotland)

The user of Cotterell’s device would slide a window up and down to expose different parts of the times table, reading off the multiplicand on the edge of the instrument frame.

The box is inscribed with the text The Index of ye Quotient The Index of ye Multiplier Divisor Multiplic and The Fore Rule. The separate brass tool is inscribed with the text Robert Jole att ye Crowne Nere ye new Exchainge fecit.

Biography of Charles Cotterell

Charles Cotterell, portrait from 1660
Charles Cotterell, portrait from 1660 by Sir Peter Lely

Charles Cotterell (or Cottrell) was born on 7 April 1615 in Wilsford, Lincolnshire, England, and was baptized there two days later. He was the only son (there were four daughters) of Sir Clement Cotterell (1585–1631) and Anne Alleyne (d. 1660, an heiress and daughter of Henry Alleyne of Wilsford), who married in 1606.

Sir Clement Cotterell was appointed as muster-master of Buckinghamshire in 1616 and groom-porter to King James I in 1619, and was knighted in 1620. He was a Vice-Admiral of Lincolnshire from 1620 to 1631. In 1621, he was elected Member of Parliament for Grantham. In 1624 he was elected Member of Parliament again.

Charles Cotterell attended Queens’ College, Cambridge, from 1629 until 1632, but did not take any degree. In June 1632 he began touring Europe with aristocratic friends. Shortly after returning to England in 1634 he was engaged to take the Earl of Pembroke’s sons Charles and Philip Herbert on their travels. Though the tour was marred by the death of Charles, Cotterell entered Pembroke’s service on his return in 1636. Cotterell served the Earl of Pembroke under courtly and military service in the Cavalier army until knighted in Oxford in 1645.

In the early 1640s Cotterell served in the privy chamber troop against the Scots, and his real opportunity for advancement came with the death in July 1641 of the King’s master of the ceremonies, Sir John Finet. Sir Balthasar Gerbier had the reversion to Finet’s place and Cotterell replaced him as assistant master on 30 July. In December 1643 he was promoted to the rank of major and, having fought at Edgehill, at both battles of Newbury, and at Alresford, was knighted in Oxford on 21 March 1645.

The Cotterell Family, painted by Dutch artist Jan Mytens in 1657
The Cotterell Family, painted by Dutch artist Jan Mytens (Anne Cotterell obviously is missing, because she was left at nurse in England). The painting is said to be made in 1658, but this is very unlikely, because Cotterell’s wife Frances died in 1657, while their eldest daughter Frances died in December 1653.

In Oxford, Cotterell collaborated with William Aylesbury in translating Davila’s Storia delle guerre civile (published in 1647) from Italian. at the request of the king. In March 1649, after the king’s execution, Cotterell (along with his wife and elder daughter Frances (or Frank), born in 1646, as their infant daughter, Anne, born in 1648, remained in England), accompanied Aylesbury and the Duke of Buckingham into exile in Antwerp.

By 1652 Cotterell had moved to The Hague as steward to Elizabeth Stuart, Queen of Bohemia, while completing a translation of La Calprenède’s Cassandre. After resigning his stewardship in September 1655, Cotterell was appointed an adviser to the Duke of Gloucester, under whom he fought in three campaigns in Flanders.

In the summer of 1642, Charles Cotterell married Frances West (1621–1657), daughter of Edward West of Marsworth, Buckinghamshire, and half-sister of the architect Roger Pratt. They had three daughters and two sons: Frances (Frank), their eldest daughter, born in 1846, died on 31 December 1653; Anne Dormer (Cottrell) (1648-1695); Clement (1650-1672), their eldest son, was killed in a naval engagement against the Dutch at the Battle of Southwold Bay, in May 1672; Elizabeth Trumbull (Cottrell) (1652-1704); Charles Lodowick (1654–1710), followed his father as Master of the Ceremonies.

Sir Charles Cotterell by Robert Williams
Sir Charles Cotterell, by Robert Williams (© National Portrait Gallery, London)

On 29 May 1660, Cotterell returned with the royal party to London, with his children but without his wife Frances, who died around 1657, and was sworn Master of the Ceremonies on 5 June. The pursuit of a young widow, Anne Owen, led Cotterell to form a friendship with Katherine Phillips, whose husband was the Member of Parliament for Cardigan. Phillips was a young poet known in her salon as Orinda. Cotterell became a major figure in Orinda’s literary circle, and later took Phillips’s seat in Parliament from 1663 until 1678.

Sir Charles Cotterell resigned his seat as Master of the Ceremonies on 27 December 1686, in favor of his son Charles Lodowick. In 1693 he published a translation The Spiritual Year, a Spanish devotional tract.

Cotterell seems to have been well regarded by those who knew him. For example, the diarist and naval administrator Samuel Pepys described him as “ingenious”. The Italian historian and satirist Gregorio Leti lavished praise on him, believing he represented everything that any court in the world could seek in a master of ceremonies, describing him as “of kind disposition, soft and gentle, assiduous in his visits, of wise counsel, exemplary lifestyle, and the best conversation”.

After a sickness of some ten days, Charles Cotterell died at 6 p.m. on 7 June 1701 at his home in St Martin’s Lane, Westminster.

Salomon de Caus

Three Rules of Work: Out of clutter find simplicity; From discord find harmony; In the middle of difficulty lies opportunity.
Albert Einstein

Salomon de Caus

Jean Salomon de Caus was a French Huguenot engineer and scientist, who spent all his life moving across Europe. He worked as a hydraulic engineer and architect under Louis XIII from 1623 till his death in Paris in 1626. de Caus also was in service of the Prince of Wales and designed gardens in England, that of Somerset House among them. Salomon de Caus also designed the Hortus Palatinus, or Garden of the Palatinate, in Heidelberg, Germany (from 1614 till 1620 he was an engineer to the Elector Palatine, Frederick, at Heidelberg).

The book Les Raisons des Forces Mouvantes avec diverses machines tant utiles que plaisantes of de Caus
The book Les Raisons des Forces Mouvantes avec diverses machines tant utiles que plaisantes of de Caus

In 1615, de Caus published the book Les Raisons des Forces Mouvantes avec diverses machines tant utiles que plaisantes (The reasons for moving forces with various machines as useful as pleasant). It was an engineering treatise, which incorporated mechanical fountains, musical instruments, and other automata. His interest in these mechanical devices may have been influenced by the recovery of ancient texts by Heron of Alexandria around this time.

A water-driven organ, designed by de Caus for Hortus Palatinus
A water-driven organ, designed by de Caus for Hortus Palatinus

In his book, de Caus described an organ (see the upper image) in which a pegged cylinder, turned by a water wheel, activated levers that triggered bellows to force air through pipes.

A notable piece of work of de Caus was an automaton that had singing birds (see the picture below), directly influenced by a design by Heron. The birds flutter and chirp while an owl turns slowly toward them. When the intimidating owl faces the birds, they fall silent, but as he turns away, they resume their ruckus.

A water-driven singing bird automaton, designed by de Caus
A water-driven singing bird automaton, designed by de Caus
Another automaton of de Caus, a water-driven swan
Another automaton of de Caus, a water-driven swan

De Caus’s treatise also contains meticulous accounts of the mechanisms of hydraulic grottoes (small caves or caverns). In one (see the image below), Galatea rides astride a big seashell drawn by two dolphins. Behind her, Cyclops has put his club aside to play on a flageolet, while sheep gambol about. The mechanism is made entirely of wood, driven by two waterwheels, that are put in motion by jets of water from two pipes that emerge from a common reservoir. The pipes have valves that open and close alternately by means of a system of counterpoises so that the wheelwork turns one way and then the other as Galatea and her dolphins move back and forth across the scene. A third water wheel, through a train of gear wheels, drives a pinned barrel that is in turn connected with the keys of the flageolet.

Another automaton of Salomon de Caus, a hydraulic grotto
Another automaton of Salomon de Caus, a hydraulic grotto

Biography of Salomon de Caus

Salomon de Caus
Jean Salomon de Caus was born in 1576 in Dieppe, Normandy, into a Protestant family. The whole family moved to England, where Salomon obtained his education, studying painting, ancient languages, and mathematics. He was interested in engineering and architecture and studied ancient curiosities, such as the statue of Memnon and the Archite pigeons.

From 1595 till 1598 de Caus visited Italy, where he observed the garden of Bernardo Buontalenti in Pratolino in Florence with the mechanical arts and the Villa d’Este at Tivoli, near Rome. Inspired by Buontalenti he designed several gardens and related structures fountains, grottos, and machinery.

From 1600 till 1608 de Caus was an engineer (from 1605 “ingéniaire à la fontaine artificielle et en toutes autres choses que luy seront commandées”) at the court of the Austrian Archduke Albert VII of Habsburg, governor of the Netherlands, and his wife Isabella in Brussels.

During 1608 and 1609 de Caus was in London to teach drawing to Prince Henry, Prince of Wales, and to Princess Elizabeth, then continued his service at the court of King Charles I. Together with the architect Constantino de’ Servi he built water features and a picture gallery in Richmond Palace. De Caus built also gardens in Greenwich Park, Wilton House near Salisbury in Wiltshire, Hatfield House in Salisbury, and Somerset House (London), as well as some work in Richmond park, Gorhamburry, Camden House in Kensington.

In 1614, through the intervention of Princess Elizabeth Stuart, who had married Prince Elector Palatine, he answered the call to Heidelberg, to serve as the architect and engineer of Elector Friedrich V. De Caus was the architect of the part built under Frederick V at the Castle of Heidelberg. He was also the architect of Elisabethentors and the designer of Hortus Palatinus. This work occupied De Caus for some years and was not completed when, but he published in 1620 his complete designs in a work entitled Hortus Palatinus a Friderico Rege Boemiæ Electore Palatino Heidelbergæ exstructus.

He left Heidelberg for Rouen, then to Paris, in 1620, because of the war, to start in the service of Louis XIII, who employed him as “Engineer et architecte du Roy” (engineer and architect to the king) and was among others responsible for the sanitation and fountains.

Salomon’s brother, Isaac de Caus (1590–1648), also an engineer and landscape architect, is the creator of Hortus Penbrochianus (Wilton House) in 1615. Isaac also created several automata, similar to these of his brother, such as the water-powered machinery that counterfeited singing birds in Dieppe in 1617. In 1644 Isaac de Caus published a book, Nouvelle invention de level l’eau plus hault que sa source avec quelques machines mouvantes par le moyen de l’eau, et un discours de la conduite d’icelle, with print designs, which seem to have been taken directly from Salomon.

On 16 April 1606, Salomon de Caus married Esther Picart. On 24 February 1607, was born their son Guillaume.

Salomon de Caus died on 28 February 1626, in Paris, and was buried in cimetière de la Trinité.

William Pratt

The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.
G. K. Chesterton

The title page of Arithmeticall Jewell of William Pratt from 1617
The title page of Arithmeticall Jewell of William Pratt

A simple calculating device with the fancy name Arithmetical Jewell was designed by William Pratt in the middle 1610s, and described in the book of the same name (The Arithmeticall Jewell: or the use of a small Table Whereby is speedily wrought, as well all Arithmetical workes in whole Numbers, as all fractional operations, without fraction or reduction. Invented by William Pratt. Published by his Maiesties privilege, granted to the inventor, under the Great Seale of England), published in 1617 in London.

William Pratt, in association with John Harpur and Jeremy Drury, received a patent (privilege) on 27 March 1616, for the sole making of a table for casting accounts. The patent was for making a device “by which all questions arithmetical may be resolved without the use of pen or compters [counters]”. On 4 April 1616, the three men obtained a privilege for printing a book explaining an “instrument or table for cyfering and casting of accomptes”. Soon thereafter, the partners fell out, and published competing manuals of instruction: Harpur entered his in the Stationers’ Register on 8 March 1617, and Pratt on 21 June 1617.

The Arithmetical Jewell of Pratt is an instrument with a flat grid of semi-circular, rotating brass wedges, devised to facilitate addition, subtraction, multiplication, division, and the extraction of roots.

William Pratt was a mathematical practitioner and a member of the active circle of London’s mathematical teachers, close to the famous Gresham College, where the logarithms of Napier were popularized at that time.

The drawing of Arithmeticall Jewell of William Pratt from 1617
The drawing of Arithmetical Jewell of William Pratt from his book

Pratt’s device was nothing more than a rudimentary mechanical reconfiguration of the conventional reckoning technique: a portable, fancier, and gentlemanly adaptation of the century-old techniques of calculation like plume (manual calculations) and jetons of the abacus. With the exception here that you did not need paper to inscribe, for instance, the carry-over numbers of an addition; one could instead, using a small metallic stylus, “inscribe” them on the instrument’s appropriate sectors of brass. The reckoning method, nonetheless, was precisely the same as the plume and jetons.

The Arithmetical Jewell comprises two ivory-faced wooden tablets, with dimensions 122 mm x 65 mm x 5 mm each, put in tooled leather binding, 5″ x 3″, with a brass stylus 5 inches long. The weight of the device is only 0.14 kg. One tablet (below in the picture) has 14 columns, each with small brass parallel sectors, made from brass (copper, zinc alloy). The other has seven pairs of columns for laying out astronomical fractions to the base 60. Numbers are put in by moving the flags to reveal dots. Sums are then worked out with a pen and paper.

The Arithmeticall Jewell of William Pratt (© National Museums Scotland)
The Arithmetical Jewell of William Pratt (© National Museums Scotland)

There is a later account for Arithmetical Jewell by the English antiquary, natural philosopher, and writer John Aubrey (1626–1697):
Dr Pell told me, that one Jeremiah Grinken [a mathematical instrument maker] frequented Mr Gunters Lectures at Gresham College: He used an Instrument called a Mathematicall Jewell, by which he did speedily performe all Operations in Arithmeticke, without writing any figures, by little sectors of brasse [or some semi-circles] that did turn every one of them upon a Center. The Doctor has the booke… he told me, he thought his name is [William] Pratt.

Arithmetical Jewel in Science Museum Group Collection
The Arithmetical Jewell of William Pratt (© Science Museum Group Collection,