Category Archives: History of Mathematics

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In 1610, Galileo published his Sidereus Nuncius, the first publication to make known the new astronomical discoveries made with the recently invented telescope.

Source: Wikimedia Commons

Although, one should also emphasise that although Galileo was the first to publish, he was not the first to use the telescope as an astronomical instrument, and during that early phase of telescopic astronomy, roughly 1609-1613, several others independently made the same discoveries. There was, as to be expected, a lot of scepticism within the astronomical community concerning the claimed discoveries. The telescopes available at the time were generally of miserable quality and Galileo’s discoveries proved difficult to replicate. It was the Jesuit mathematicians and astronomers in the mathematics department at the Collegio Romano, who would, after initial difficulties, provide the scientific confirmation that Galileo desperately needed. The man, who led the endeavours to confirm or refute Galileo’s claims was the acting head of the mathematics faculty Christoph Grienberger (the professor, Christoph Clavius, was old and infirm). Grienberger is one of those historical figures, who fades into the background because they made no major discoveries or wrote no important books, but he deserves to be better known, and so this brief sketch of the man and his contributions.

As is all too oft the case with Jesuit scholars in the Early Modern Period, we know almost nothing about Grienberger before he joined the Jesuit Order. There are no know portraits of him. The problems start with his name variously given as Bamberga, Bamberger, Banbergiera, Gamberger, Ghambergier, Granberger, Panberger and a total of nineteen variations, history has settled on Grienberger. He was born 2 July 1561 in Hall a small town in the Tyrol in the west of Austria. That’s all we know till he entered the Jesuit Order in 1580. He studied rhetoric and philosophy in Prague from 1583 to 1584. From 1587 he was a mathematics teacher at the Jesuit university in Olmütz. He began his theology studies, standard for a Jesuit, in Vienna in 1589, also teaching mathematics. His earliest surviving letter to Christoph Clavius, who he had never met but who he describes as his teacher, he had studied the mathematical sciences using Clavius’ books, is dated from 1590. In 1591 he moved to the Collegio Romano, where he became Clavius’ deputy. 

In 1595, Clavius went to Naples, the purpose of his journey is not clear, but he was away from Rome for somewhat more than a year. During his absence Grienberger took over direction of the mathematics department at the Collegio Romana. From the correspondence between the two mathematicians, during this period, it becomes very obvious that Grienberger does not enjoy being in the limelight. He complains to Clavius about having to give a commencement speech and also about having to give private tuition to the sons of aristocrats. Upon Clavius’ return he fades once more into the background, only emerging again with the commotion caused by the publication of Galileo’s Sidereus Nuncius.

Rumours of Galileo’s discoveries were already making the rounds before publication and, in fact on the day the Sidereus Nuncius appeared, the wealthy German, Humanist Markus Welser (1558–1614) from Augsburg wrote to Clavius asking him his opinion on Galileo’s claims.

Markus Welser artist unknown Source: Wikimedia Commons

We know from letters that the Jesuit mathematicians in the Collegio Romano already had a simple telescope and were making astronomical observations before the publication of the Sidereus Nuncius. They immediately took up the challenge of confirming or refuting Galileo’s discoveries. However, their telescope was not powerful enough to detect the four newly discovered moons of Jupiter. Grienberger was away in Sicily attending to problems at the Jesuit college there, so it was left to Giovanni Paolo Lembo (c. 1570–1618) to try and construct a telescope good enough to complete the task. We know that Lembo was skilled in this direction because between 1615 and 1617 he taught lens grinding and telescope construction to the Jesuits being trained as missionaries to East Asia at the University of Coimbra. 

Lembo’s initial attempts to construct a suitable instrument failed and it was only after Grienberger returned from Sicily that the two of them were able to make progress. At this point Galileo was corresponding with Clavius and urging the Jesuit astronomer on provide the confirmation of his discoveries that he so desperately needed, the general scepsis was very high, but he was not prepared to divulge any details on how to construct good quality telescopes. Eventually, Grienberger and Lembo succeeded in constructing a telescope with which they could observe the moons of Jupiter but only under very good observational conditions. They first observed three of the moons on 14 November 1610 and all four on 16 November. 

Clavius wrote to the merchant and mathematician Antonio Santini (1577–1662) in Venice, who had been to first to confirm the existence of the Jupiter moons in 1610, with a telescope that he constructed himself, detailing observation from 22, 23, 26, and 27 November but stating that they were still not certain as to the nature of the moons. Santini relayed this information to Galileo. On 17 December, Clavius wrote to Galileo:

…and so we have seen [the Medici Stars] here in Rome many times. At the end of the letter I will put some observations, from which it follows very clearly that they are not fixed but wandering stars, because they change position with respect to each other and Jupiter.

Much of what we know about the efforts of the Jesuit astronomers under the leadership of Grienberger to build an adequate telescope to confirm Galileo’s discoveries come from a letter that Grienberger wrote to Galileo in February 1611. One interesting aspect of Grienberger’s letter is that the Jesuit astronomers had also been observing Venus and there is good evidence that they discovered the phases of Venus independently at least contemporaneously if not earlier than Galileo. This was proof that Venus, and by analogy probably also Mercury, orbit the Sun and not the Earth. This was the death nell for a pure Ptolemaic geocentric system and the acceptance at a minimum of a Capellan system where the two inner planets orbit the Sun, which orbits the Earth, if not a full blown Tychonic system or even a heliocentric one. This was in 1611 troubling for the conservative leadership of the Jesuit Order, but would eventually lead to them adopting a Tychonic system at the beginning of the 1620s. 

Clavius died 6 February 1612 and Grienberger became his official successor as the professor for mathematics at the Collegio Romano, a position he retained until 1633, when he was succeeded in turn by Athanasius Kircher (1602–1680). The was a series of Rules of Modesty in Ignatius of Loyola’s rules for the Jesuit Order and individual Jesuits were expected to self-abnegate. The most extreme aspect of this was that many scientific works were published anonymously as a product of the Order and not the individual. Different Jesuit scholars reacted differently to this principle. On the one hand, Christoph Scheiner (1573–1650), Galileo’s rival in the sunspot dispute and author of the Rosa Ursina sive Sol(1626–1630) presented himself as a great astronomer, which did not endear him to his fellow Jesuits.

Christoph Scheiner artist unknown Source: Wikimedia Commons

On the other hand, Grienberger put his name on almost none of his own work preferring it to remain anonymous. There is only a star catalogue and a set of trigonometrical tables that bear his name.

However, as head of the mathematics department at the Collegio Romano he was responsible for controlling and editing all of the publications in the mathematical disciplines that went out from the Jesuit Order and it is know that he made substantial improvements to the works that he edited both in the theoretical parts and in the design of instruments. A good example is the heliotropic telescope, which enables the observer to track the movement of the Sun whilst observing sunspots, illustrated in Scheiner’s Rosa Ursina.

Heliotropic telescope on the left. On the right Scheiner’s acknowledgement that Grienberger was the inventor

This instrument is known to have been designed and constructed by Grienberger, who, however, explicitly declined Scheiner’s offer to add a text under his own name describing its operation. Grienberger also devised a system of gearing theoretical capable of lifting the Earth

Reconstruction of Grienberger’S Earth lifting gearing

Grienberger, admired Galileo and took his side, if only in the background, in Galileo’s dispute with the Aristotelians over floating bodies. He was, however, disappointed by Galileo’s unprovoked and vicious attacks on the Jesuit astronomer Orazio Grassi on the nature of comets and explicitly said that it had cost Galileo the support of the Jesuits in his later troubles. He also clearly stated that if Galileo had been content to propose heliocentricity as a hypothesis, its actual scientific status at the time, he could have avoided his confrontation with the Church.

Élie Diodati (1576–1661) the Calvinist, Genevan lawyer and friend of Galileo, who played a central role in the publication of the Discorsi, quoted Grienberger in a letter to Galileo from 25 July 1634, as having said, “If Galileo had recognised the need to maintain the favour of the Fathers of this College, then he would live gloriously in the world, and none of his misfortune would have occurred, and he could have written about any subject, as he thought fit, I say even about the movement of the Earth…”

Several popular secondary sources claim the Grienberger supported the Copernican system. However, there is only hearsay evidence for this claim and not actual proof. He might have but we will never know. 

Grienberger made no major discoveries and propagated no influential new theories, which would launch him into the forefront of the big names, big events style of the history of science. However, he played a pivotal role in the very necessary confirmation of Galileo’s telescopic discoveries. He also successfully helmed the mathematical department of the Collegio Romano for twenty years, which produced many excellent mathematicians and astronomers, who in turn went out to all corners of the world to teach others their disciplines. By the time Athanasius Kircher inherited Grienberger’s post there was a world-wide network of Jesuit astronomers, communicating data on important celestial events. One such was Johann Adam Schall von Bell (1591–1666), who studied under Grienberger and went on to lead the Jesuit mission in China.

Johann Adam Schall von Bell Source: Wikimedia Commons

Science is a collective endeavour and figures such as Grienberger, who serve inconspicuously in the background are as important to the progress of that endeavour as the shrill public figures, such as Galileo, hogging the limelight in the foreground. 

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Filed under History of Astronomy, History of Mathematics, Renaissance Science

Renaissance Science – XIII

As already explained in the fourth episode of this series, the Humanist Renaissance was characterised by a reference for classical literature, mostly Roman, recovered from original Latin manuscripts and not filtered and distorted through repeated translations on their way from Latin into Arabic and back into Latin. It was also a movement that praised a return to classical Latin, away from the, as they saw it, barbaric medieval Latin. In the fifth episode we also saw that, what I am calling, Renaissance science was characterised by a break down of the division that had existed between theoretical book knowledge as taught on the medieval universities and the empirical, practical knowledge of the artisans. As also pointed there this was not so much a breaking down of boundaries or a crossover between the two fields of knowledge as a meld between the two types of knowledge that would over the next two and a half centuries lead to the modern concept of knowledge or science.

One rediscovered classical Latin text that very much filled the first criterium, which at the same time played a major role in the second was De architectura libri decem (Ten Books on Architecture) by the Roman architect and civil and military engineer Marcus Vitruvius Pollio (c.80-70–died after 15 BCE), who is usually referred to simply as Vitruvius and there are doubts about the other two names that are ascribed to him. 

From the start we run into problems about the standard story that the manuscript was rediscovered by the Tuscan, humanist scholar Poggio Bracciolini (1380–1459) in the library of Saint Gall Abbey in 1416, as related by Leon Battista Alberti (1404–1472) in his own architecture treatise De re aedificatoria (1452), which was modelled on Vitruvius’ tome. In reality, De architectura had never been lost during the Middle Ages; there are about ninety surviving medieval manuscripts of the book.

Manuscript of Vitruvius; parchment dating from about 1390 Source: Wikimedia Commons

The oldest was made during the Carolingian Renaissance in the early nineth century. Alcuin of York was consulted on the technical terms in the text. During the Middle Ages many leading scholars including Hermann of Reichenau (1013–1054), a central figure of the Ottonian Renaissance, and both Albertus Magnus (c. 1200– 1280) and Thomas Aquinas (1225–1274), who laid the foundations of medieval Aristotelian philosophy, read the text, and commented on it. 

However, although well-known it had little impact on architecture in the medieval period. The great medieval cathedrals and castle were built by master masons, whose knowledge was practical artisanal knowledge passed on by word of mouth from master to apprentice. This changed with Poggio’ rediscovery of Vitruvius’ work and the concept of the theoretical and practical architect began to emerge.

Before we turn to the impact of De architectura in the Renaissance we first need to look at the book and its author. Very little is known about Vitruvius, as already stated above, the other names attributed to him are based on speculation, most of what we do know is pieced together from the book itself. Vitruvius was a military engineer under Gaius Julius Caesar (100–44 BCE) and apparently received a pension from Octavian (63 BCE–14 CE), the later Caesar Augustus, to whom the book is dedicated. The book was written around twenty BCE. Vitruvius wrote it because he believed in making knowledge public and available to all, unlike those artisans, who kept their knowledge secret.

The ten books are organised as follows:

  1. Town planning, architecture or civil engineering in general and the qualification required by an architect or civil engineer
  2. Building materials
  3. Temples and the orders of architecture
  4. As book 3
  5. Civil buildings
  6. Domestic buildings
  7. Pavements and decorative plasterwork
  8. Water supplies and aqueducts
  9. The scientific side of architecture – geometry, measurement, astronomy, sundials
  10. Machines, use and construction – siege engines, water mills, drainage machines, technology, hoisting, pneumatics

In terms of its reception and influence during the Renaissance the most important aspect is Vitruvius’ insistence that architecture requires both ratiocinatio and fabrica, that is reasoning or theory, and practice or construction. This Vitruvian philosophy of architecture took architecture out of the exclusive control of the master mason and into the hands of the theoretical scholars in union with the artisans. This move was also motivated by the humanist drive to study archaeologically the Roman remains in Rome the Eternal City. Vitruvius provided a guide to understanding the Roman architecture, which would become the model for the construction of new buildings. 

But for it to become influential Vitruvius’s text first had to become widely available. The first printed Latin edition was edited by the humanist scholar Fra. Giovanni Sulpizio da Veroli (fl. c. 1470–1490) and published in 1486 with a second edition in 1495 or 1496.

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The first printed edition had no illustrations. Fra. Giovanni Giocondo da Verona (c. 1433–1515) produced the first edition with woodcut illustrations, published in Venice in 1511. A second improved edition was published in Florence in 1521. 

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In order for De architectura to reach artisans it needed to be translated into the vernacular, as most of them couldn’t read Latin. This process began in Italy and during the sixteenth century spread throughout Europe. The process started already before De architectura appeared in print. As mentioned above Alberti’s De re aedificatoria (On the Art of Buildings), not a translation of De architectura but a book strongly modelled on it appeared in Latin in print in 1452.

Source: Wikimedia Commons

The first Italian edition appeared in 1486 A second Italian edition, by the humanist mathematician Cosimo Bartoli (1503-1572), which became the standard edition, appeared in 1550. Alberti was very prominent in Renaissance culture and very widely read. His influence can be measured by the fact that a collective bilingual, English/Italian, edition of his works on architecture, painting and sculpture was published as late as 1726. 

The first Italian edition of De architectura with new illustration and added commentary by Cesare Cesariano (1475-1543) was published at Como in 1521.

1521 Italian edition title page Source
1521 Italian edition

A plagiarised version was published in Venice in 1524. The first French edition, translated by Jean Martin (died 1553), which is said to contain many errors, was published in Paris in 1547.

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The first German edition was translated by Walther Hermann Ryff (c. 1500–1548). As far as can be determined, it appears the Ryff was an apothecary but work mostly as what today would probably be described as a hack. He published as editor, translator, adapter, and compiler a large number of books, around 40, over a wide range of topics, although the majority were in some sense medical, and was seemingly very successful. He was often accused of plagiarism. The physician and botanist, Leonhart Fuchs (1501–1566) described him as an “extremely brazen, careless, fraudulent author.” Apart from his medical works, Ryff obviously had a strong interest in architecture. He edited and published a Latin edition of De architectura in Strasbourg in 1543. This was followed by a commentary on De architectura in German, Der furnembsten, notwendigsten, der gantzen Architectur angehörigen Mathematischen vnd Mechanischen künst, eygentlicher bericht, vnd vast klare, verstendliche vnterrichtung, zu rechtem verstandt der lehr Vitruuij, in drey furneme Bücher abgetheilet (The most distinguished, necessary, mathematical and mechanical arts belonging to the entire architecture, actual report and clear, understandable instruction of the teachings of Vitruvius shared in three distinguished books), published by Johannes Petreius, the leading European scientific publisher of the period, in Nürnberg in 1547. For obvious reasons this is usually simply referred to as Architektur. This was obviously a product of the German translation of De architectura, which Petreius had commissioned Ryff to produce and, which he published in Nürnberg in 1548 under the title, Vitruvius Teutsch. Nemlichen des aller namhafftigisten vñ hocherfahrnesten römischen Architecti vnd kunstreichen Werck zehn Bücher von der Architectur und künstlichem Bawen… (Vitruvius in German…).

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We return now to Italy and the story of the stone mason, Andrea di Pietro della Gondola, born in Padua in 1508. Having served his apprenticeship, he worked as a stone mason until he was thirty years old. In 1538–39, he was employed to rebuild the villa of the humanist poet and scholar, Gian Giorgio Trissino (1478–1550) to rebuild his villa in Cricoli.

Gian Giorgio Trissino, portrayed in 1510 by Vincenzo Catena Source: Wikimedia Commons
Villa Trissino Source: Wikimedia Commons

Trissino ran a small private learned academy for young gentlemen in his renovated villa and apparently, having taken a shine to the young stone mason invited him to become a member. Andrea accepted the offer and Trissino renamed him Palladio.

Portrait of Palladio by Alessandro Maganza Source: Wikimedia Commons

The two became friends and colleagues, and Trissino, who was deeply interested in classical architecture and Vitruvius took the newly christened Palladio with him on trips to Rome to study the Roman ruins. Palladio became an architect in 1540 and became a specialist for designing and building neo-classical, Palladian, villas. 

Villa Barbaro begun 1557 Source: Wikimedia Commons

Trissino died in 1550 but Palladio acquired a new patron, Daniele Barbaro (1514–1570), a member of one of the most prominent and influential aristocratical families of Venice.

Daniele Barbaro by Paolo Veronese (the book in the painting is Barbaro’s translation of De architectura)

Daniele Barbaro studied philosophy, mathematics, and optics at the University of Padua. He was a diplomat and architect, who like Trissino, before him, accompanied Palladio on expeditions to study Roman architecture. In 1556, Barbaro published a new Italian translation of De architectura with an extended commentary, Dieci libri dell’architettura di M. Vitruvio.

I dieci libri dell’architettura di M. Vitruvio tradutti et commentati da monsignor Barbaro eletto patriarca d’aquileggia 1556 Images by Palladio Source

In 1567, he, simultaneously published, a revised Italian and a Latin edition entitled M. Vitruvii de architectura. The illustrations for Barbaro’s editions were provided by Palladio. Barbaro provided the best, to date, explanations of much of the technical terminology in De architectura, also acknowledging Palladio’s theoretical contributions to the work.

Palladio had become one of the most important and influential architects in the whole of Europe, designing many villas, palaces, and churches. He also became an influential author publishing L’Antichida di Roma (The Antiquities of Rome) in 1554,

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and I quattro libri dell’architettura (The Four Books of Architecture) in 1570, which was heavily influenced by Vitruvius. His books were translated into many different languages and went through many editions right down into the eighteenth and nineteenth centuries. His work inspired leading architects in France and Germany.

Title page from 1642 edition Source: Wikimedia Commons

Up till now we have said nothing about England, which as usual lagged behind the continent in things mathematical, although in the second half of the sixteenth century both Leonard Digges and John Dee, of the so-called English school of mathematics, counted architecture under the mathematical disciplines. In 1563 John Shute (died 1563) included Vitruvian elements in his The First and Chief Grounds of Architecture.

John Shute The First and Chief Grounds of Architecture.

Inigo Jones (1573–1652) was born into a Welsh speaking family in Smithfield, London. There is minimal evidence that he was an apprentice joiner but at some point, before 1603 he acquired a rich patron, who impressed by his sketches, sent him to Italy to study drawing in Italy.

Inigo Jones by Anthony van Dyck

In a second visit to Italy in 1606 he came under the influence of Sir Henry Wotton (1568–1639) the English ambassador to Venice.

Henry Wotton artist unknown Source: Wikimedia Commons

Wotton was interested in astronomy, and it was he, who sent two copies of Galileo’s Sidereus Nuncius (1610) to London on the day it was published. Wotton convinced Jones to learn Italian and introduced him to Palladio’s I quattro libri dell’architettura. Jones’ copy of the book has marginalia that references Wotton. In 1624, Wotton published The Elements of Architecture a loose translation of De architectura into English. The first proper translation appeared only in 1771. 

19th century reprint Source

Inigo Jones introduced the Vitruvian–Palladian architecture into England and became the most influential architect in the country, becoming Surveyor of the King’s Works.

The Queen’s House in Greenwich designed and built by Inigo Jones Source: Wikimedia Commons

His career was ended with the outbreak of the English Civil War in 1642. England’s most famous architect Christopher Wren (1632–1723), a mathematician and astronomer turned architect stood in a line with Vitruvius, Palladio, and Jones. It is very clear that the humanist rediscovery and promotion of De architectura had a massive influence on the development of architecture in Europe in the sixteenth and seventeenth centuries, in the process dissolving the boundaries between the theoretical intellectuals and the practical artisans. 

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Filed under History of Mathematics, History of Technology, Renaissance Science

The deviser of the King’s horologes

There can’t be many Renaissance mathematici, whose existence was ennobled by a personal portrait by the master of the Renaissance portraits, Hans Holbein the younger. In fact, I only know of one, the German mathematicus, Nicolas Kratzer.

Nicolas Kratzer Portrait by Hans Holbein the younger

One might be excused for thinking that having received this singular honour that Kratzer had, in his lifetime, achieved something truly spectacular in the world of the Renaissance mathematical disciplines; however, almost the opposite is true. Kratzer appears to have produced nothing of any significance, was merely the designer and maker of sundials, and an elementary maths teacher, who was only portrayed by Holbein, because for a time they shared the same employers and were apparently mates. 

So, who was Kratzer and how did he and Holbein become mates? Here we find a common problem with minor scientific figures in the Renaissance, there are no biographies, no handy archives giving extensive details of his life. All we have are a few, often vague, sometimes contradictory, traces in the proverbial sands of time from which historians have attempted to reconstruct at least a bare outline of his existence. 

Kratzer was born in 1487 in Munich, the son of a saw-smith and it is probably that he learnt his metal working skills, as an instrument maker, from his father. He matriculated at the University of Köln 18 November 1506 and graduated BA 14 June 1509. He moved onto the University of Wittenberg, famous as the university of Martin Luther. However, this was before the Reformation and Wittenberg, a young university first founded in 1502, was then still Catholic. We now lose track of Kratzer, who is presumed to have then worked as an instrument maker. Sometime in the next years, probably in 1517, he copied some astronomical manuscripts at the Carthusian monastery of Maurbach, near Vienna. 

In January 1517, Pieter Gillis (1486–1533) wrote to his erstwhile teacher Erasmus (1466–1536) that the skilled mathematician Kratzer was on his way with astrolabes and spheres, and a Greek book.

HOLBEIN, Hans the Younger (b. 1497, Augsburg, d. 1543, London) Portrait of Erasmus of Rotterdam 1523 Wood, 76 x 51 cm National Gallery, London

This firmly places Kratzer in the circle of humanist scholars, most famously Erasmus and Thomas More (1478–1535) author of Utopia, who founded the English Renaissance on the court of Henry VIII (1491–1547). Holbein was also a member of this circle. Erasmus and Holbein had earlier both worked for the printer/publisher collective of Petri-Froben-Amerbach in Basel. Erasmus as a copyeditor and Holbein as an illustrator. Holbein produced the illustrations for Erasmus’ In Praise of Folly (written 1509, published by Froben 1511)

Holbein’s witty marginal drawing of Folly (1515), in the first edition, a copy owned by Erasmus himself

Kratzer entered England either at the end of 1517 or the beginning of 1518. His first identifiable employment was in the household of Thomas More as maths teacher for a tutorial group that included More’s children. It can be assumed that it was here that he got to know Holbein, who was also employed by More. 

Thomas More Portrait by Hans Holbein 1527

For his portraits, Holbein produced very accurate complete sketches on paper first, which he then transferred geometrically to his prepared wooden panels to paint them. Around 1527, Holbein painted a group portrait of the More family that is no longer extant, but the sketch is. The figures in the sketch are identified in writing and the handwriting has been identified as Kratzer’s. 

Like Holbein, Kratzer moved from More’s household to the court of Henry VIII, where he listed in the court accounts as the king’s astronomer with an income of £5 a quarter in 1529 and 1531. It is not very clear when he entered the King’s service but in 1520 Cuthbert Tunstall (1474–1559), later Prince-Bishop of Durham, wrote in a letter:

Met at Antwerp with [Nicolas Kratzer], an Almayn [German], devisor of the King’s horologes, who said the King had given him leave to be absent for a time.

Both Tunstall and Kratzer were probably in Antwerp for the coronation of Charles V (1500–1558) as King of Germany, which took place in Aachen. There are hints that Kratzer was there to negotiate with members of the German court on Henry’s behalf. Albrecht Dürer (1471–1528) was also in the Netherlands; he was hoping that Charles would continue the pension granted to him by Maximilian I, who had died in 1519. Dürer and Kratzer met in the house of Erasmus and Kratzer was present as Dürer sketched a portrait of Erasmus. He also drew a silver point portrait of Kratzer, which no longer exists. 

 

Dürer sketch of Erasmus 1520
Dürer engraved portrait of Erasmus based on 1520 sketch finished in 1526. Erasmus reportedly didn’t like the portrait

Back in England Kratzer spent some time lecturing on mathematical topics at Oxford University during the 1520s. Here once again the reports are confused and contradictory. Some sources say he was there at the behest of the King, others that he was there in the service of Cardinal Wolsey. There are later claims that Kratzer was appointed a fellow of Corpus Christi College, but the college records do not confirm this. However, it is from the Oxford records that we know of Kratzer’s studies in Köln and Wittenberg, as he was incepted in Oxford as BA and MA, on the strength of his qualifications from the German institutions, in the spring of 1523. 

During his time in Oxford, he is known to have erected two standing sundials in the college grounds, one of which bore an anti-Lutheran inscription.

Drawing of Kratzer’s sundial made for the garden of Corpus Christi College Oxford

Neither dial exists any longer and the only dial of his still there is a portable brass dial in the Oxford History of Science Museum, which is engraved with a cardinal’s hat on both side, which suggests it was made for Wolsey.

Kratzer polyhedral sundial presumably made for Cardinal Wolsey Museum for the History of Science Oxford

On 24 October 1524 Kratzer wrote the following to Dürer in Nürnberg

Dear Master Albert, I pray you to draw for me a model of the instrument that you saw at Herr Pirckheimer’s by which distances can be measured, and of which you spoke to me at Andarf [Antwerp], or that you will ask Herr Pirckheimer to send me a description of the said instrument… Also I desire to know what you ask for copies of all your prints, and if there is anything new at Nuremberg in my craft. I hear that our Hans, the astronomer, is dead. I wish you to write and tell me what he has left behind him, and about Stabius, what has become of his instruments and his blocks. Greet in my name Herr Pirckheimer. I hope shortly to make a map of England which is a great country, and was not known to Ptolemy; Herr Pirckheimer will be glad to see it. All who have written of it hitherto have only seen a small part of England, no more… I beg of you to send me the likeness of Stabius, fashioned to represent St. Kolman, and cut in wood…

Herr Pirckheimer is Willibald Pirckheimer (1470–1530), who was a lawyer, soldier, politician, and Renaissance humanist, who produced a new translation of Ptolemaeus’ Geographia from Greek into Latin.

Engraved portrait of Willibald Pirckheimer Dürer 1524

He was Dürer’s life-long friend, (they were born in the same house), patron and probably lover.  He was at the centre of the so-called Pirckheimer circle, a group of mostly mathematical humanists that included “Hans the astronomer, who was Johannes Werner (1468–1522), mathematician, astronomer, astrologer, geographer,

Johannes Werner artist unknown

and cartographer and Johannes “Stabius” (c.1468–1522) mathematician, astronomer, astrologer, and cartographer.

Johannes Stabius portrait by Dürer

Werner was almost certainly Dürer’s maths teacher and Stabius worked together with Dürer on various projects including his star maps. The two are perhaps best known for the Werner-Stabius heart shaped map projection. 

Dürer replied to Kratzer 5 December 1524 saying that Pirckheimer was having the required instrument made for Kratzer and that the papers and instruments of Werner and Stabius had been dispersed.

Here it should be noted that Dürer, in his maths bookUnderweysung der Messung mit dem Zirkel und Richtscheyt (Instruction in Measurement with Compass and Straightedge), published the first printed instructions in German on how to construct and orientate sundials. The drawing of one sundial in the book bears a very strong resemblance to the polyhedral sundial that Kratzer made for Cardinal Wolsey and presumably Kratzer was the original source of this illustration. 

Dürer drawing of a sundial

Kratzer is certainly the source of the mathematical instruments displayed on the top shelf of Holbein’s most famous painting the Ambassadors, as several of them are also to be seen in Holbein’s portrait of Kratzer.

in’s The AmbassadorsHolbe

Renaissance Mathematicus friend and guest blogger, Karl Galle, recently made me aware of a possible/probable indirect connection between Kratzer and Nicolas Copernicus (1473–1543). Georg Joachim Rheticus (1514–1574) relates that Copernicus’ best friend Tiedemann Giese (1480–1550) possessed his own astronomical instruments including a portable sundial sent to him from England. This was almost certainly sent to him by his brother Georg Giese (1497–1562) a prominent Hanseatic merchant trader, who lived in the Steelyard, the Hansa League depot in London, during the 1520s and 30s.

London’s Steelyard

He was one of a number of Hanseatic merchants, whose portraits were painted by Holbein, so it is more than likely that the sundial was one made by Kratzer. 

Georg Giese portrait by Hans Holbein 1532

Sometime after 1530, Kratzer fades into the background with only occasional references to his activities. After 1550, even these ceased, so it is assumed that he had died around this time. In the first half of the sixteenth century England lagged behind mainland Europe in the mathematical disciplines including instrument making, so it is a natural assumption that Kratzer with his continental knowledge was a welcome guest in the Renaissance humanist circles of the English court, as was his younger contemporary, the Flemish engraver and instrument maker, Thomas Gemini (1510–1562). Lacking homegrown skilled instrument makers, the English welcomed foreign talent and Kratzer was one who benefited from this. 

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Renaissance Science – XI

The Renaissance saw not only the introduction of new branches of mathematics, as I have outlined in the last three episodes in this series, but also over time major changes in the teaching of mathematics both inside and outside of the universities. 

The undergraduate or arts faculty of the medieval university was nominally based on the so-called seven liberal arts, a concept that supposedly went back to the Pythagoreans. This consisted of the trivium – grammar, logic, and rhetoric – and the quadrivium – arithmetic, geometry, music, and astronomy – whereby the quadrivium was the mathematical disciplines. However, one needs to take a closer look at what the quadrivium actually entailed. The arithmetic was very low level, as was the music, actually in terms of mathematics the theory of proportions. Astronomy was almost entirely non-technical being based on John of Sacrobosco’s (c. 1195–c. 1256) Tractatus de Sphera (c. 1230). Because Sacrobosco’s Sphera was very basic it was complemented with a Theorica planetarum, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Only geometry had a serious mathematical core, being based on the first six books of The Elements of Euclid

I said above, nominally, because in reality on most universities the quadrivium only had a niche existence. Maths lectures were often only offered on holidays, when normal lectures were not held. Also, the mathematical disciplines were not examination subjects. If a student didn’t have the necessary course credit for a mathematical discipline, they could often acquire it simply by paying the requisite tuition fees. Put another way, the mathematical disciplines were not taken particularly seriously in the early phase of the European universities. There were some exceptions to this, but they were that, exceptions. 

Through out much of the Middle Ages there were no chairs for mathematics and so no professors. Very occasionally a special professor for mathematics would be appointed such as the chair created by Francois I at the Collège Royal in the 16th century for Oronce Fine (1494–1555) initially there were only chairs and professorships for the higher faculties, theology, law, and medicine. On the arts faculty the disciplines were taught by the postgraduate masters. The MA was a teaching licence. If somebody was particularly talented in a given discipline, they would be appointed to teach it, but otherwise the masters were appointed each year by drawing lots. To get the lot for mathematics was the equivalent of getting the short straw. This changed during the Renaissance, and we will return to when and why below but before we do, we need to first look at mathematics outside of the university. 

During the medieval period preceding the Renaissance, trades people who had to do calculations used an abacus or counting board and almost certainly a master taught his apprentice, often his own son, how to use one. This first began to change during the so-called commercial revolution during which long distance trade increased significantly, banks were founded for the first time, double entry bookkeeping was introduced, and both the decimal place value number system and algebra were introduced to aid business and traded calculations. As I said earlier this led to the creation of the so-called abbacus, or in English reckoning schools with their abbacus or reckoning books.

The reckoning schools and books not only taught the new arithmetic and algebra but also elementary geometry and catered not only for the apprentice tradesmen but also for apprentice artists, engineers, and builder-architects.  It is fairly certain, for example, that Albrecht Dürer, who would later go on to write an important maths textbook for apprentice artists, acquired his first knowledge of mathematics in a reckoning school. This was a fairly radical development in the formal teaching of mathematics at an elementary level, as the Latin schools, which prepared youths for a university education, taught no mathematics at all. 

The first major change in the mathematic curriculum on the European universities was driven by astrology, or more precisely by astrological medicine or iatromathematics, as it was then called. As part of the humanist Renaissance, astro-medicine became the dominant form of medicine followed on the Renaissance universities; a development we will deal with later. In the early fifteenth century, in order to facilitate this change in the medical curriculum the humanist universities of Northern Italy and also the University of Cracow introduced chairs and professorships for mathematics, whose principal function was to teach astrology to medical students. Before they could practice astro-medicine the students had to learn how to cast a horoscope, which meant first acquiring the necessary mathematical and astronomical skills to do so. This was still the principal function of professors of mathematics in the early seventeenth century and Galileo, would have been expected to teach such courses both at Pisa and Padua.

©Photo. R.M.N. / R.-G. OjŽda Source: Wikimedia Commons

As with other aspects of the humanist Renaissance this practice spread to northwards to the rest of Europe. The first chair for mathematics at a German university was established at the University of Ingolstadt, also to teach medical student astrology. Here interestingly, Conrad Celtis, know in Germany as the Arch Humanist, when he was appointed to teach poetics subverted the professors of mathematics slightly to include mathematical cartography in their remit. He took two of those professors, Johannes Stabius and Andreas Stiborius, when he moved to Vienna and set up his Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, this helped to advance the study and practice of mathematical cartography on the university.

Astrology also played a central role in the next major development in the status and teaching of mathematics on school and universities. Philipp Melanchthon (1497–1560) was a child prodigy. Having completed his master’s at the University of Heidelberg in 1512 but denied his degree because of his age, he transferred to the University of Tübingen, where he became enamoured with astrology under the influence of Johannes Stöffler (1452–1531), the recently appointed first professor of mathematics, a product of the mathematics department at Ingolstadt.

Contemporary Author’s Portrait Stöfflers from his 1534 published Commentary on the Sphaera of the Pseudo-Proklos (actually Geminos) Source: Wikimedia Commons

Melanchthon was appointed professor of Greek at Wittenberg in 1518, aged just twenty-one. Here he became Luther’s strongest supporter and was responsible for setting up the Lutheran Protestant education system during the early years of the reformation. Because of his passion for astrology, he established chairs for mathematics in all Protestant schools and university. Several of Melanchthon’s professors played important rolls in the emergence of the heliocentric astronomy.

The Lutheran Protestants thus adopted a full mathematical curriculum early in the sixteenth century, the Catholic education system had to wait until the end of the century for the same development. Founded in 1540, the Society of Jesus (the Jesuits) in their early years set up an education system to supply Catholics with the necessary arguments to combat the arguments of the Protestants. Initially this strongly Thomist education system did not include mathematics. Christoph Clavius (1538–1612), who joined the Jesuits in 1555, was a passionate mathematician, although it is not exactly clear where he acquired his mathematical education or from whom. By 1561 he was enrolled in the Collegio Romano, where he began teaching mathematics in 1563 and was appointed professor of mathematics in 1567. Clavius created an extensive and comprehensive mathematical curriculum that he wanted included in the Jesuit educational programme. Initially, this was rejected by conservative elements in the Society, but Clavius fought his corner and by the end of the century he had succeeded in making mathematics a central element in Jesuit education. He personally taught the first generation of teachers and wrote excellent modern textbooks for all the mathematical disciplines, including the new algebra. By 1626 there were 444 Jesuit colleges and 56 seminaries in Europe all of which taught mathematics in a modern form at a high level. Many leading Catholic mathematicians of the seventeenth century such as Descartes, Gassendi, and Cassini were products of this Jesuit education network.

Christoph Clavius Source: Wikimedia Commons

By the beginning of the seventeenth century mathematics had become an established high-level subject in both Protestant and Catholic educational institutions throughout the European mainland, the one exception which lagged well behind the rest of Europe was England. 

Well aware that the mathematical education in England was abysmal, a group of influential figures created a public lectureship for mathematics in London at the end of the seventeenth century. These lectures intended for soldiers, artisans and sailors were held from 1588 to 1592 by Thomas Hood (1556–1620), who also published books on practical mathematics in the same period. Other English practical mathematicians such as Robert Recorde, Leonard and Thomas Digges, Thomas Harriot and John Dee also gave private tuition and published books aimed at those such as cartographers and navigators, who needed mathematics. 

In 1597, Gresham College was set up in London using money bequeathed by Sir Thomas Gresham (c. 1519–1597) to provide public lectures in both Latin and English in seven subjects, including geometry and astronomy. The professorships in these two mathematical disciplines have been occupied by many notable mathematical scholars over the centuries.

Gresham College engraving George Vertue 1740 Source: Wikimedia Commons

The two English universities, Oxford, and Cambridge, still lagged behind their continental colleagues, as far as the mathematical sciences were concerned. The first chairs at Oxford University for astronomy and geometry were the result of a private initiative. Henry Savile (1549–1622), an Oxford scholar, like many others in this period, travelled on the continent in order to acquire a mathematical education, primarily at the North German Universities, where several prominent Scottish mathematicians also acquired their mathematical education.

Henry Savile Source: Wikimedia Commons

In 1619, he founded and endowed the Savilian Chairs for Astronomy and Geometry at Oxford. Many leading English mathematical scholars occupied these chairs throughout the seventeenth century, several of whom had previously been Gresham professors. 

Cambridge University held out until 1663, when Henry Lucas founded and endowed the Lucasian Chair for Mathematics, with Isaac Barrow (1630–1677) as its first incumbent, and Isaac Newton (1642–1626) as his successor. Despite this, John Arbuthnot (1667–1735) could write in an essay from 1705 that there was not a single grammar school in England where mathematics was taught.

John Arbuthnot, by Godfrey Kneller Source: Wikimedia Commons

In the High Middle Ages the mathematical disciplines were treated as niche subjects on the medieval university. Throughout the Renaissance period this changed and with it the status and importance of mathematics. This change was also driven by the need for mathematics in the practical disciplines of cartography, navigation, surveying, astrology, and the emerging new astronomy; we will deal with these developments in future episodes. However, by the end of the Renaissance, mathematics had gained the high academic status that it still enjoys today.

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Renaissance Science – X

Almost certainly the most important development in mathematics during the Renaissance was the introduction of algebra to the European academic canon. For part of this appropriation process it followed the same path as the base ten place value number system and the Hindu-Arabic numerals, with which it was in the European context intimately intertwined. However, there are enough differences to justify a separate post. 

In the High Middle Ages formal mathematics was totally dominated by Euclidian geometry, an inheritance from ancient Greece. In the Early Modern Period it would become slowly supplanted by algebra leading eventually to the creation of both analytical geometry and the calculus, the mathematics at the heart of the scientific revolution. However, in antiquity algebra predated Euclid’s geometry. Some form of algebra existed in China, India, Egypt, and Babylon before Euclid’s masterpiece was created. 

The various states that occupied the so-called fertile crescent were states with all the power in the hands of a central government. Resources we collected in central warehouses and then distributed to the population. This required a numerate administration, who were responsible for recording, dividing up and accounting for those resources. To do this they first developed a sophisticated base sixty place value number system and then the arithmetic and algebra to manipulate that number system. 

The Babylonian mathematical clerks developed their algebra to quite a high level. They could solve linear equation, including indeterminant ones. They had the general solution to the general quadratic equations but only considered positive solutions, as they had no concept of negative numbers. They could also solve various cubic equations but did no appear to have the general solution. Babylonian astronomers applied algebra to their work analysing centuries of observational data of the planetary positions and of solar and lunar eclipses. Their analysis led to accurate algebraic algorithms for predicting the positions of the planets. They also produced algebraic algorithms to predict lunar and solar eclipses. The algorithm for lunar eclipses was very accurate and reliable. For technical reasons the algorithm for solar eclipses could only predict when a solar eclipse might take place but not if it actually would take place. This correctly predicted the actual solar eclipses but produced more false predictions and no means in advance of deciding which was which.

Although the ancient Greeks inherited their astronomy and astrology from the Babylonians, they rejected the Babylonian algebraic approach to the mathematical problems substituting geometrical models for the Babylonian algebraic algorithms. This insistence on solving algebraic problems with geometry, not exclusive to the Greeks, led to much of our terminology for equations. X2 is the area of a square with a side length of X and so second order equations became quadratic, that is square, equations. The same argument leads to third order equations being named cubic equations.  

Indian mathematicians, like the Babylonians, also developed a strong arithmetical/algebraic tradition, having like the Babylonians a place value number system. Of interest, there are algebraic problems, in the form of mathematic riddles, that turn up in very similar forms in several ancient cultures, India, Babylon and Egypt, which suggest some form of knowledge transfer in the past, but none has been found to date. Jens Høyrup hypothesised an earlier common source on which they all drew, rather than a direct knowledge transfer.

As with the Hindu-Arabic numerals and the rules for their use it was Brahmagupta (c. 598–c. 668), who in his Brāhmasphuṭasiddhānta provided the most developed presentation of the Indian algebra. He gave solutions to linear equations and the first presentation of the general solution to the quadratic equation in the form that it is still taught today with both positive and negative solutions. In general, his algebra was more advanced than the Babylonian. 

As we saw in the last post the Brāhmasphuṭasiddhānta was translated into Arabic in about 770, where it became established. Once again it was a text from Muhammad ibn Musa al-Khwārizmī (c. 780–c. 850), his al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (the compendium on calculation by restoring and balancing”), which was translated into Latin by Robert of Chester in 1145. As also noted in the previous post al- Khwārizmī’s name gave us the word algorithm, although its meaning has changed down the years. The Arabic word al-Jabr, which in everyday language means “reunion of broken parts” becomes the European word algebra. In Spain, which was the major interface between Islamic and Christian culture, an algebrista was a bonesetter, a use that spread throughout Europe.

It should be noted that there were other Arabic algebra texts that were far more advanced mathematically than al-Khwārizmī’s al-Kitāb that were in the Middle Ages never translated into Latin.

As with the introduction of the Hindu-Arabic numerals Robert of Chester’s translation initially had comparatively little impact on the world of formal mathematics. Once again as with the Hindu-Arabic numerals, it was Leonardo of Pisa’s Liber Abbaci (1202, 2nd edition 1227), which also drew heavily on al-Khwārizmī, that established algebra, initially in Northern Italy as part of the commercial arithmetic taught in the abacus schools. Again, as with the Hindu-Arabic numerals the introduction of double entry bookkeeping along with other aspects of the commercial revolution accelerated the spread of the use of algebra. An acceleration increased by the publication of Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita in 1494.

Although algebra started life very much as a form of practical mathematics, throughout its history, its practitioners had developed it beyond their initial or basic needs. The clerks in Babylon and in the Islamic period doing mathematics for its own sake, pushing the boundaries of the discipline. The same took place in Europe during the Renaissance.

One such was the Frenchman Nicolas Chuquet (c. 1450– c. 1495), whose Triparty en la science des nombres was definitely more of an algebra book rather than a reckoning book.

Le Triparty en la Science des Nombres par Maistre Nicolas Chuquet Parisien – an extract from Chuquet’s original 1484 manuscript Source: Wikimedia Commons

It was never published in his lifetime but was heavily plagiarised. In Germany, in the sixteenth century a movement known as the Cossists developed, who wrote and published Coss books. These are algebra textbooks named after their use of the word Coss, derived from the Italian ‘cosa’ meaning thing, itself a translation of the Latin ‘res’, as a universal term for the unknown in an algebraic problem. The Cossists were generally reckoning masters, but their Coss books are different to their reckoning books. 

The earliest Coss author was Christoff Rudolff (c.1500–before 1543), who published his Behend und hübsch Rechnung durch die kunstreichen regeln Algebre, so gemeinicklich die Coß genennt werden (Deft and nifty reckoning with the artful rules of Algebra, commonly called the Coss) in Straßburg in1525.

Christoff Rudolff Source: Wikimedia Commons

An extended improved edition under the title, Die Coss Christoffs Rudollfs, was published by Michael Stiffel (c. 1487–1567) in Königsberg in 1553. The last edition of Die Coss was published in Amsterdam in 1615. In the eighteenth century, Leonard Euler (1707–1783) used Rudollfs Coss as his algebra textbook. Michael Stiffel had published his own Coss, Arithmetica Integra, in Nürnberg in 1544.

Michael Stifel Source: Wikimedia Commons
Michael Stifel’s Arithmetica Integra (1544), p. 225. Source: Wikimedia Commons

Robert Recorde’s The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng the extraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers, published in London in 1557 is, as its title clearly states, in the Coss tradition.

Robert Recorde Source: Wikimedia Commons

Also, in the same tradition was the L’arithmétique by the Netherlander Simon Stevin (1548–1620) published in 1585. 

Simon Stevin Source: Wikimedia Commons

The Coss authors were all outside the university system and the algebra remained outside too. The process that led to the acceptance of algebra in the university system was a slightly different parallel one. To some extent it begins with Pacioli’s claim in his Summa that, unlike the quadratic equation, there was no general solution to the cubic equation. However, at the beginning of the sixteenth century Scipione del Ferro (1465–1526), the professor of mathematics at the University of Bologna found a general solution to one form of the cubic equation. He died without revealing his discovery, which was then inherited by one of his students, Antonio Maria Fior. It was common in this period for mathematicians to challenge each other to public problem-solving competitions and Fior challenged Niccolo Fontana (1500–1557), known as Tartaglia, a leading mathematical exponent, letting it be known that he had a solution to the cubic equation. Tartaglia realised he was on a hiding to nothing and set about studying the problem and came up with a more general solution. On the day of the competition Tartaglia won hands down and achieved overnight fame. Polymath Gerolamo Cardano (1501–1576) seduced Tartaglia into revealing his solution in exchanged for introductions into higher social circles. The condition was however that Cardano was not allowed to publish the solution before Tartaglia had done so. Cardano, however, travelled to Bologna and discovered del Ferro’s solution. In the meantime, Cardano’s student Ludovico Ferrari (1522-1565) had discovered the general solution to the bi-quadratic or quartic equation.

Gerolamo Cardano Source: Wikimedia Commons

Having expanded del Ferro’s solution to a general one for cubic equations, Cardano combined it with Ferrari’s general solution of the quartic and published them both in his major algebra book Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra), published by Johannes Petreius (c. 1497–1550) in Nürnberg in 1545.

Source: Wikimedia Commons

He attributed the solution of the cubic to del Ferro much to the annoyance of Tartaglia, who still hadn’t published. Petreius had previously published Cardano’s abbacus book Practica arithmetice et mensurandi singularis in 1538.

Source: Research Gate

Cardano, a physician, astrologer, philosopher, and mathematician, was a leading intellectual figure in Europe and Petreius was the leading scientific publisher, so the Ars Magna had a major impact helping to establish algebra as an accepted mathematical discipline. Although it’s a claim that I view sceptically, this impact in reflected in the general claim that Ars Magna was the first modern mathematics book.

Cardano had not only accepted positive and negative solutions to all equations but had also to a limited extent worked with complex numbers in the solution of equations as long as they cancelled out. Another Italian mathematician Rafael Bombelli (1526–1572) fully developed the use of complex numbers in his L’Algebra from 1572, an algebra textbook. 

Source: Wikimedia Commons

The last significant Renaissance algebra book published before the seventeenth century was probably the most significant, the In artem analyticem isagoge (1591) by the French mathematician François Viète (1540–1603). Viète pulled together, systemised, and provided a foundation for much of what had gone before. Most important he produced an algebra that was to a large extent symbolic in its presentation, which introduces a topic that I haven’t dealt with yet. 

François Viète Source: Wikimedia Commons
In artem analyticem isagoge 2nd ed.

Historically there were three presentational forms of algebra. Rhetorical algebra in which everything is written out in full sentences with words and not symbols or numerals, for example X+1 = 2 becomes “the thing plus one equals two”. Babylonian and al-Khwārizmī’s algebra were both rhetorical algebras. The second form is syncopated algebra in which some of the words that repeated occur are reduced to abbreviations. This was the case with Brahmagupta’s Brāhmasphuṭasiddhānta. It was also the case with Diophantus’ Arithmetica (difficult to date but probably 3rd century CE). Diophantus’ Arithmetica is something that only became known again in Europe during the Renaissance. If Diophantus’ Arithmetica was originally syncopated is not known, as the oldest known manuscript dates from the 13th century CE. Viète drew heavily for his Isagoge from Diophantus. The third form is symbolic algebra in which both variables and operations are expressed with symbols. Some of the Arabic algebraists developed symbolic algebra but this was not transferred to Europe during the translation movement.

Most of the 16th century algebras are a mixture of rhetorical and syncopated algebra with occasional symbols for operations. The German university lecturer, Johannes Widmann (c. 1460–after 1498) introduced the symbols for plus and minus in his Mercantile Arithmetic oder Behende und hüpsche Rechenung auff allen Kauffmanschafft an abbacus book published in 1489. 

Mercantile Arithmetic (1489) Source: Wikimedia Commons

Famously, Robert Recorde introduced the equals sign in his Whetstone of witte, although he didn’t invent it, as it had already been in use earlier in Italy.

The equals sign in the Whetstone of witte Source: Wikimedia Commons

Viète systematically used symbols for variables and constants, using vowels for the former and consonants for the latter. He, however, did not use symbols for operations. Although his system did not last long it had a massive influence on the symbolisation of algebra. 

The introduction of symbols for operations was a rather haphazard process that had begun in the fifteenth century and wasn’t really completed until the eighteenth century. A significant publication was William Oughtred’s Clavis Mathematicae (1631), which was almost entirely symbolic, although most of the symbols he invented did not survive in the long run.

Source: Wikimedia Commons

The Clavis Mathematicae went through numerous Latin editions during the seventeenth century and was used widely as a textbook both in Britain and on the continent. Many leading seventeenth century mathematicians, including John Wallis, Christopher Wren, Seth Ward, Isaac Newton, and Edmond Halley, learnt their algebra from it. The first English edition was produced in 1647 and Halley produced a new English translation in 1694.

Title page Clavis Mathematicae 5th ed 1698
Ed John Wallis Source: Wikimedia Commons

Thomas Harriot (c. 1560–1621) and René Descartes (1596–1650) both made substantial contributions to the modern form of symbolic algebra. 

 In the seventeenth century both Pierre de Fermat (1601–1665) and Descartes combined algebra with geometry to create analytical geometry. Both of them, also contributed substantially to the conversion of calculus, which in its origins was geometrical, into an analytical algebraic discipline. 

The acceptance and establishment of algebra in the first half of the 17th century was not uncontested and was certainly not a case of the modern scientists accepting and the last generation of traditionalists rejecting.  Just to give one interesting example, Galileo, who is oft celebrated as the ‘father of modern science’, rejected it, whereas Christoph Clavius, the last of the great Ptolemaic astronomers not only accepted it but also wrote a textbook based on Viète’s algebra, which was then taught to and by all Jesuit mathematicians, and this although he was involved in a dispute with Viète about the calendar reform. 

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Renaissance Science – IX

The part of mathematics that we most use in our lives is numbers, the building blocks of arithmetic. Today, we mostly use the Hindu-Arabic numerals and the associated place value decimal system, but this was not always the case. In fact, although this number system first entered Europe during the 12th century translation movement, it didn’t become truly established until well into the Renaissance.

First, we will briefly track the Hindu-Arabic place value decimal system from its origins till its advent in Europe. The system emerged in India sometime late in the sixth century CE. Āryabhaṭa (476–550) a leading mathematician and astronomer doesn’t mention them in his Aryasiddhanta. The earliest known source being in the Āryabhaṭīyabhāṣya of Bhāskara I (c. 600–c. 680) another leading astronomer mathematician. The full system, as we know it today, was described in the Brāhmasphuṭasiddhānta of Brahmagupta (c. 598–c. 668 n. Chr.). The only difference is that he allows division by zero, which as we all learnt in the school is not on.

The Brāhmasphuṭasiddhānta was translated into Arabic in about 770 by Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (d. 777), Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (d. c. 800) and Yaʿqūb ibn Ṭāriq (d. c. 796). The first two are father and son. Having teams doing scientific translations in the middle ages was actually very common. I won’t go into detail, but it should be noted that it took several hundred years for this system to replace the existing number systems in Arabic culture, many mathematicians preferring their own systems, which they considered superior.

The system first came into Europe in the 12th century through the translation of a work by Muhammad ibn Musa al-Khwārizmī (c. 780–c. 850) by an unknown translator. No Arabic manuscript of this work is known to exist, and it is only known by its Latin title Algoritmi de Numero Indorum, where Algoritmi is a corruption of al-Khwārizmī.

This translation only had a very limited impact. The new number system was adopted by the scholars at the universities as part of computus in order to calculate the date of Easter and the other moveable Church feasts. Leading scholars such as Sacrobosco wrote textbooks to teach the new discipline, which was Algorimus, another corruption of al-Khwārizmī. The other mostly university-based scholars, who used mathematics extensively, the astronomers, continued to use a sexagesimal i.e., base sixty, number system that they had inherited from both the Greek and the Arabic astronomers. This system would stay in use by astronomers down to Copernicus’ De revolutionibus (1543) and beyond.

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This is the opening page of a 1490 manuscript copy of Johannes de Sacro Bosco’s Tractatus de Arte Numerandi, also referred to as his Algorismus Source:

What about the world outside of the universities? In the outside world the new number system was simply ignored. Which raises the question why? People generally believe that the base ten place value number system is vastly superior to the Roman numeral system that existed in Europe in the Middle ages, so why didn’t the people immediately adopt it? After all you can’t do arithmetic with Roman numerals. The thing is people didn’t do arithmetic with Roman numerals, although it would have been possible using different algorithm to the ones we use for the decimal place-value system. People did the calculations using either finger reckoning

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of counting boards, also known as reckoning boards or abacuses. They only used the roman numerals to record the results.

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Counting Board

In the hands of a skilled operator the counting board is a powerful instrument. It can be used very simply for addition and subtraction and using the halving and doubling algorithms, almost as simply for multiplication and division. A skilled operator can even extract roots using a counting board. The counting board also offers the possibility in a business deal for the reckoning masters of both parties to observe and control the calculations on the counting board.

The widespread use of counting boards over many centuries is still reflected in modern word usage. The serving surface in a shop is called a counter because it was originally the counting board on which the shop owner did their calculations. The English finance ministry is called the Exchequer after a special kind of counting board on which they did they calculations in the past. Nobody pays much attention to the strange term bankrupt, which also has its origins in the use of counting boards. The original medieval banks in Northern Italy were simply tables, Italian banca, on the marketplace, on which a printed cloth counting board was spread out. If the bankers were caught cheating their customers, then the authorities came and symbolically destroyed their table, in Italian, banca rotta, broken table.

Things first began to change slowly with the second introduction of Hindu-Arabic numerals by Leonardo from Pisa (c. 1175–c. 1250) in his Liber Abbaci (1202, 2nd edition 1227).

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Leonardo from Pisa Liber Abbaci

This was basically a book on commercial arithmetic, following its Arabic origins. The Arabic/Islamic culture used different number systems for different tasks and used the Hindu-Arabic numerals and the decimal place-value system extensively in commercial arithmetic, in general account keeping, to calculate rates of interest, shares in business deals and the division of inheritance according to the complex Islamic inheritance laws. Leonardo’s father was a customs officer in North Africa, and it was here that Leonard learnt of the Hindu-Arabic numerals and the decimal place-value system from Arab traders in its usage as commercial arithmetic.

This new introduction saw the gradual spread in Norther Italy of Scuole or Botteghe D’abbaco (reckoning schools) lead by a Maestri D’abbaco (reckoning master), who taught this new commercial arithmetic to apprentice traders from Abbaco Libro (reckoning books), which he usually wrote himself. Many leading Renaissance mathematici, including Peter Apian (1495–1552, Niccolò Fontana Tartaglia (c. 1500–1557), Gerolamo Cardano (1501–1576), Gemma Frisius (1508–1555) and Robert Recorde (c. 1512–1558), wrote a published abbacus books. The very first printed mathematics book the Arte dell’Abbaco also known as the Treviso Arithmetic (1478) was , as the title clearly states, an abacus book.

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Un maestro d’abaco. Filippo Calandri, De arimetricha opusculum, Firenze 1491

This practice began to accelerate with the introduction of double entry bookkeeping. This was part of the more general so-called commercial revolution, which included the founding of the first banks and the introduction of bills of exchange to eliminate the necessity of traders carrying large amounts of gold or silver. Developments in Europe that lead to the Renaissance. The earliest known example of double entry bookkeeping is the Messari Report of the Republic of Genoa, 1340. The earliest account of double entry bookkeeping is the Libro dell’arte di mercatura by Benedetto Cotrugli (1416–1469), which circulated in manuscript but was never printed. The first printed account was in the highly successful Summa de arithmetica, geometria, proportioni et proportionalita of Fra. Luca Bartolemeo Pacioli (c.1447–1517) published in 1494, which contain the twenty-seven-page introduction to double entry bookkeeping, Particularis de computis et scripturis.

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Particularis de computis et scripturis, about double-entry bookkeeping.

Beginning with the Southern German trading centres of Augsburg, Regensburg and Nürnberg, which all traded substantially with the Northern Italian commercial centres, the new commercial arithmetic and double entry bookkeeping began to expand throughout Europe. This saw the fairly rapid establishment of reckoning schools and the printing of reckoning books throughout the continent. We can see the partial establishment of the Hindu-Arabic numerals some four hundred years after their first introduction, although they were used principally for recording, the reckoning continuing to be done on a counting board, in many cases down to the eighteenth century.

Already in the fifteenth century we can see the glimmer of the base ten system moving into other mathematical areas. Peuerbach and Regiomontanus started using circles with radii of 10,000 or 100,000, suggesting base ten, to calculate their trigonometrical tables instead of radii of 60,000, base sixty. The use of such large radii was to eliminate the need for fractional values.

By the end of the sixteenth century, the base ten positional value number system with Hindu-Arabic numerals had become well established across the whole spectrum of number use, throughout Europe. The Indian decimal system had no fractions and decimal fractions were first introduced into the Hindu-Arabic numerals by Abu’l Hasan Ahmad ibn Ibrahim Al-Uqlidisi in his Kitab al-Fusul fi al-Hisab al-Hindi around 952 and then again independently by Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (c. 1380–1429) in his Key to Arithmetic (1427). They first emerged in Europe in 1585 in Simon Stevin’s De Thiende also published in French as La Disme. The decimal point or comma was first used in Europe by Christoph Clavius (1538–1612) in the goniometric tables for his astrolabe in 1593. Its use became widespread through its adoption by John Napier in his Mirifici Logarithmorum Canonis Descriptio (1614).

However, at the end of the seventeenth century we still find both John Evelyn (1620–1706) and John Arbuthnot (1667–1735) discussing the transition from Roman to Hindu-Arabic numerals in their writings; the former somewhat wistfully, the later thankfully.

In the eighteenth century, Pierre-Simon Laplace reputedly said:

‘It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’

A very positive judgement, with hindsight, of the base ten place value number system with Hindu-Arabic numerals but one that was obviously not shared in the Early Modern period when the system was initially on offer in Europe.

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Renaissance Science – VIII

In the last two episodes we have looked at developments in printing and art that, as we will see later played an important role in the evolving Renaissance sciences. Today, we begin to look at another set of developments that were also important to various areas of the newly emerging practical sciences, those in mathematics. It is a standard cliché that mathematisation played a central roll in the scientific revolution but contrary to popular opinion the massive increase in the use of mathematics in the sciences didn’t begin in the seventeenth century and certainly not as the myth has it, with Galileo.

Medieval science was by no means completely devoid of mathematics despite the fact that it was predominantly Aristotelian, and Aristotle thought that mathematics was not scientia, that is, it did not deliver knowledge of the natural world. However, the mathematical sciences, most prominent astronomy and optics, had a fairly low status within medieval university culture.

One mathematical discipline that only really became re-established in Europe during the Renaissance was trigonometry. This might at first seem strange, as trigonometry had its birth in Greek spherical astronomy, a subject that was taught in the medieval university from the beginning as part of the quadrivium. However, the astronomy taught at the university was purely descriptive if not in fact even prescriptive. It consisted of very low-level descriptions of the geocentric cosmos based largely on John of Sacrobosco’s (c. 1195–c. 1256) Tractatus de Sphera (c. 1230). Sacrobosco taught at the university of Paris and also wrote a widely used Algorismus, De Arte Numerandi. Because Sacrobosco’s Sphera was very basic it was complimented with a Theorica planetarum, by an unknown medieval author, which dealt with elementary planetary theory and a basic introduction to the cosmos. Mathematical astronomy requiring trigonometry was not hardy taught and rarely practiced.

Both within and outside of the universities practical astronomy and astrology was largely conducted with the astrolabe, which is itself an analogue computing device and require no knowledge of trigonometry to operate.

Before we turn to the re-emergence of trigonometry in the medieval period and its re-establishment during the Renaissance, it pays to briefly retrace its path from its origins in ancient Greek astronomy to medieval Europe.

The earliest known use of trigonometry was in the astronomical work of Hipparchus, who reputedly had a table of chords in his astronomical work. This was spherical trigonometry, which uses the chords defining the arcs of circles to measure angles. Hipparchus’ work was lost and the earliest actual table of trigonometrical chords that we know of is in Ptolemaeus’ Mathēmatikē Syntaxis or Almagest, as it is usually called today.

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The chord of an angle subtends the arc of the angle. Source: Wikimedia Commons

When Greek astronomy was appropriated in India, the Indian astronomers replaced Ptolemaeus’ chords with half chords thus creating the trigonometrical ratios now known to us, as the sine and the cosine.

It should be noted that the tangent and cotangent were also known in various ancient cultures. Because they were most often associated with the shadow cast by a gnomon (an upright pole or post used to track the course of the sun) they were most often known as the shadow functions but were not considered part of trigonometry, an astronomical discipline. So-called shadow boxes consisting of the tangent and cotangent used for determine heights and depths are often found on the backs of astrolabes.

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Shadow box in the middle of a drawing of the reverse of Astrolabium Masha’Allah Public Library Bruges [nl] Ms. 522. Basically the tangent and cotangent functions when combined with the alidade

  Islamic astronomers inherited their astronomy from both ancient Greece and India and chose to use the Indian trigonometrical half chord ratios rather than the Ptolemaic full cords. Various mathematicians and astronomers made improvements in the discipline both in better ways of calculating trigonometrical tables and producing new trigonometrical theorems. An important development was the integration of the tangent, cotangent, secant and cosecant into a unified trigonometry. This was first achieved by al-Battãnī (c.858–929) in his Exhaustive Treatise on Shadows, which as its title implies was a book on gnonomics (sundials) and not astronomy. The first to do so for astronomy was Abū al-Wafā (940–998) in his Almagest.

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Image of Abū al-Wafā Source: Wikimedia Commons

It was this improved, advanced Arabic trigonometry that began to seep slowly into medieval Europe in the twelfth century during the translation movement, mostly through Spain. It’s reception in Europe was very slow.

The first medieval astronomers to seriously tackle trigonometry were the French Jewish astronomer, Levi ben Gershon (1288–1344), the English Abbot of St Albans, Richard of Wallingford (1292–1336) and the French monk, John of Murs (c. 1290–c. 1355) and a few others.

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Richard of Wallingford Source: Wikimedia Commons

However, although these works had some impact it was not particularly widespread or deep and it would have to wait for the Renaissance and the first Viennese School of mathematics, Johannes von Gmunden (c. 1380­–1442), Georg von Peuerbach (1423–1461) and, all of whom were Renaissance humanist scholars, for trigonometry to truly establish itself in medieval Europe and even then, with some delay.

Johannes von Gmunden was instrumental in establishing the study of mathematics and astronomy at the University of Vienna, including trigonometry. His work in trigonometry was not especially original but displayed a working knowledge of the work of Levi ben Gershon, Richard of Wallingford, John of Murs as well as John of Lignères (died c. 1350) and Campanus of Novara (c. 200–1296). His Tractatus de sinibus, chordis et arcubus is most important for its probable influence on his successor Georg von Peuerbach.

Peuerbach produced an abridgement of Gmunden’s Tractatus and he also calculated a new sine table. This was not yet comparable with the sine table produced by Ulugh Beg (1394–1449) in Samarkand around the same time but set new standards for Europe at the time. It was Peuerbach’s student Johannes Regiomontanus, who made the biggest breakthrough in trigonometry in Europe with his De triangulis omnimodis (On triangles of every kind) in 1464. However, both Peuerbach’s sine table and Regiomontanus’ De triangulis omnimodis would have to wait until the next century before they were published. Regiomontanus’ On triangles did not include tangents, but he rectified this omission in his Tabulae Directionum. This is a guide to calculating Directions, a form of astrological prediction, which he wrote at the request for his patron, Archbishop Vitéz. This still exist in many manuscript copies, indicating its popularity. It was published posthumously in 1490 by Erhard Ratdolt and went through numerous editions, the last of which appeared in the early seventeenth century.

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A 1584 edition of Regiomontanus’Tabulae Directionum Source

Peuerbach and Regiomontanus also produced their abridgement of Ptolemaeus’ Almagest, the Epitoma in Almagestum Ptolemae, published in 1496 in Venice by Johannes Hamman. This was an updated, modernised version of Ptolemaeus’ magnum opus and they also replaced his chord tables with modern sine tables. A typical Renaissance humanist project, initialled by Cardinal Basilios Bessarion (1403–1472), who was a major driving force in the Humanist Renaissance, who we will meet again later. The Epitoma became a standard astronomy textbook for the next century and was used extensively by Copernicus amongst others.

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Title page Epitoma in Almagestum Ptolemae Source: Wikimedia Commons

Regiomontanus’ De triangulis omnimodis was edited by Johannes Schöner and finally published in Nürnberg in 1533 by Johannes Petreius, together with Peuerbach’s sine table, becoming a standard reference work for much of the next century. This was the first work published, in the European context, that treated trigonometry as an independent mathematical discipline and not just an aide to astronomy.

Copernicus (1473–1543,) naturally included modern trigonometrical tables in his De revolutionibus. When Georg Joachim Rheticus (1514–1574) travelled to Frombork in 1539 to visit Copernicus, one of the books he took with him as a present for Copernicus was Petreius’ edition of De triangulis omnimodis. Together they used the Regiomontanus text to improve the tables in De revolutionibus. When Rheticus took Copernicus’ manuscript to Nürnberg to be published, he took the trigonometrical section to Wittenberg and published it separately as De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published.

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Rheticus’ action was the start of a career in trigonometry. Nine years later he published his Canon doctrinae triangvlorvmin in Leipzig. This was the first European publication to include all of the six standard trigonometrical ratios six hundred years after Islamic mathematics reached the same stage of development. Rheticus now dedicated his life to producing what would become the definitive work on trigonometrical tables his Opus palatinum de triangulis, however he died before he could complete and publish this work. It was finally completed by his student Valentin Otto (c. 1548–1603) and published in Neustadt and der Haardt in 1596.

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Source: Wikimedia Commons

In the meantime, Bartholomäus Piticus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solution triangulorum tractatus brevis et perspicuous, one year earlier, in 1595.

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Source:. Wikimedia Commons

This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ Trigonometria were calculated to five or six places, whereas those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. In comparison Peuerbach’s sine tables from the middle of the fifteenth century were only accurate to three places of decimals. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as Magnus canon doctrinae triangulorum in 1607.

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He published a second further improved version under the title Thesaurus mathematicus in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.

In the seventeenth century a major change in trigonometry took place. Whereas throughout the Renaissance it had been handled as a branch of practical mathematics, used to solve spherical and plane triangles in astronomy, cartography, surveying and navigation, the various trigonometrical ratios now became mathematical functions in their own right, a branch of purely theoretical mathematics. This transition mirroring the general development in the sciences that occurred between the Renaissance and the scientific revolution, from practical to theoretical science.

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Filed under History of Astronomy, History of Islamic Science, History of Mathematics, History of science, Renaissance Science

The man who printed the world of plants

Abraham Ortelius (1527–1598) is justifiably famous for having produced the world’s first modern atlas, that is a bound, printed, uniform collection of maps, his Theatrum Orbis Terrarum. Ortelius was a wealthy businessman and paid for the publication of his Theatrum out of his own pocket, but he was not a printer and had to employ others to print it for him.

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Abraham Ortelius by Peter Paul Rubens , Museum Plantin-Moretus via Wikimedia Commons

A man who printed, not the first 1570 editions, but the important expanded 1579 Latin edition, with its bibliography (Catalogus Auctorum), index (Index Tabularum), the maps with text on the back, followed by a register of place names in ancient times (Nomenclator), and who also played a major role in marketing the book, was Ortelius’ friend and colleague the Antwerp publisher, printer and bookseller Christophe Plantin (c. 1520–1589).

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Plantin also published Ortelius’ Synonymia geographica (1578), his critical treatment of ancient geography, later republished in expanded form as Thesaurus geographicus (1587) and expanded once again in 1596, in which Ortelius first present his theory of continental drift.

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Plantin’s was the leading publishing house in Europe in the second half of the sixteenth century, which over a period of 34 years issued 2,450 titles. Although much of Plantin’s work was of religious nature, as indeed most European publishers of the period, he also published many important academic works.

Before we look in more detail at Plantin’s life and work, we need to look at an aspect of his relationship with Ortelius, something which played an important role in both his private and business life. Both Christophe Plantin and Abraham Ortelius were members of a relatively small religious cult or sect the Famillia Caritatis (English: Family of Love), Dutch Huis der Leifde (English: House of Love), whose members were also known as Familists.

This secret sect was similar in many aspects to the Anabaptists and was founded and led by the prosperous merchant from Münster, Hendrik Niclaes (c. 1501–c. 1580). Niclaes was charged with heresy and imprisoned at the age of twenty-seven. About 1530 he moved to Amsterdam where his was once again imprisoned, this time on a charge of complicity in the Münster Rebellion of 1534–35. Around 1539 he felt himself called to found his Famillia Caritatis and in 1540 he moved to Emden, where he lived for the next twenty years and prospered as a businessman. He travelled much throughout the Netherlands, England and other countries combining his commercial and missionary activities. He is thought to have died around 1580 in Cologne where he was living at the time.

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Niclaes wrote vast numbers of pamphlets and books outlining his religious views and I will only give a very brief outline of the main points here. Familists were basically quietists like the Quakers, who reject force and the carrying of weapons. Their ideal was a quite life of study, spiritualist piety, contemplation, withdrawn from the turmoil of the world around them. The sect was apocalyptic and believed in a rapidly approaching end of the world. Hendrik Niclaes saw his mission in instructing mankind in the principal dogma of love and charity. He believed he had been sent by God and signed all his published writings H. N. a Hillige Nature (Holy Creature). The apocalyptic element of their belief meant that adherents could live the life of honest, law abiding citizens even as members of religious communities because all religions and authorities would be irrelevant come the end of times. Niclaes managed to convert a surprisingly large group of successful and wealthy merchants and seems to have appealed to an intellectual cliental as well. Apart from Ortelius and Plantin, the great Dutch philologist, humanist and philosopher Justus Lipsius (1574–1606) was a member, as was Charles de l’Escluse (1526–1609), better known as Carolus Clusius, physician and the leading botanist in Europe in the second half of the sixteenth century. The humanist Andreas Masius (1514–1573) an early syriacist (one who studies Syriac, an Aramaic language) was a member, as was Benito Arias Monato (1527–1598) a Spanish orientalist. Emanuel van Meteren (1535–1612) a Flemish historian and nephew of Ortelius was probably also Familist. The noted Flemish miniature painter and illustrator, Joris Hoefnagel (1542–1601), was a member as was his father a successful diamond dealer. Last but by no means least Pieter Bruegel the Elder (c. 1525– 1569) was also a Familist. As we shall see the Family of Love and its members played a significant role in Plantin’s life and work.

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Christophe Plantin by Peter Paul Rubens Museum Platin-Moretus  via Wikimedia Commons Antwerp in the time of Plantin was a major centre for artists and engravers and Peter Paul Rubins was the Plantin house portrait painter.

Christophe Plantin was born in Saint-Avertin near Tours in France around 1520. He was apprenticed to Robert II Macé in Caen, Normandy from whom he learnt bookbinding and printing. In Caen he met and married Jeanne Rivière (c. 1521–1596) in around 1545.

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Jeanne Rivière School of Rubens Museum Plantin-Moretus via Wikimedia Commons

They had five daughters, who survived Plantin and a son who died in infancy. Initially, they set up business in Paris but shortly before 1550 they moved to the city of Antwerp in the Spanish Netherlands, then one of Europe’s most important commercial centres. Plantin became a burgher of the city and a member of the Guild of St Luke, the guild of painter, sculptors, engravers and printers. He initially set up as a bookbinder and leather worker but in 1555 he set up his printing office, which was most probably initially financed by the Family of Love. There is some disagreement amongst the historians of the Family as to how much of Niclaes output of illegal religious writings Plantin printed. But there is agreement that he probably printed Niclaes’ major work, De Spiegel der Gerechtigheid (Mirror of Justice, around 1556). If not the house printer for the Family of Love, Plantin was certainly one of their printers.

The earliest book known to have been printed by Plantin was La Institutione di una fanciulla nata nobilmente, by Giovanni Michele Bruto, with a French translation in 1555, By 1570 the publishing house had grown to become the largest in Europe, printing and publishing a wide range of books, noted for their quality and in particular the high quality of their engravings. Ironically, in 1562 his presses and goods were impounded because his workmen had printed a heretical, not Familist, pamphlet. At the time Plantin was away on a business trip in Paris and he remained there for eighteen months until his name was cleared. When he returned to Antwerp local rich, Calvinist merchants helped him to re-establish his printing office. In 1567, he moved his business into a house in Hoogstraat, which he named De Gulden Passer (The Golden Compasses). He adopted a printer’s mark, which appeared on the title page of all his future publications, a pair of compasses encircled by his moto, Labore et Constantia (By Labour and Constancy).

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Christophe Plantin’s printers mark, Source: Wikimedia Commons

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Engraving of Plantin with his printing mark after Goltzius Source: Wikimedia Commons

Encouraged by King Philip II of Spain, Plantin produced his most famous publication the Biblia Polyglotta (The Polyglot Bible), for which Benito Arias Monato (1527–1598) came to Antwerp from Spain, as one of the editors. With parallel texts in Latin, Greek, Syriac, Aramaic and Hebrew the production took four years (1568–1572). The French type designer Claude Garamond (c. 1510–1561) cut the punches for the different type faces required for each of the languages. The project was incredibly expensive and Plantin had to mortgage his business to cover the production costs. The Bible was not a financial success, but it brought it desired reward when Philip appointed Plantin Architypographus Regii, with the exclusive privilege to print all Roman Catholic liturgical books for Philip’s empire.

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THE BIBLIA SACRA POLYGLOTTA, CHRISOPHER PLANTIN’S MASTERPIECE. IMAGE Chetham’s Library

In 1576, the Spanish troops burned and plundered Antwerp and Plantin was forced to pay a large bribe to protect his business. In the same year he established a branch of his printing office in Paris, which was managed by his daughter Magdalena (1557–1599) and her husband Gilles Beys (1540–1595). In 1578, Plantin was appointed official printer to the States General of the Netherlands. 1583, Antwerp now in decline, Plantin went to Leiden to establish a new branch of his business, leaving the house of The Golden Compasses under the management of his son-in-law, Jan Moretus (1543–1610), who had married his daughter Martine (1550–16126). Plantin was house publisher to Justus Lipsius, the most important Dutch humanist after Erasmus nearly all of whose books he printed and published. Lipsius even had his own office in the printing works, where he could work and also correct the proofs of his books. In Leiden when the university was looking for a printer Lipsius recommended Plantin, who was duly appointed official university printer. In 1585, he returned to Antwerp, leaving his business in Leiden in the hands of another son-in-law, Franciscus Raphelengius (1539–1597), who had married Margaretha Plantin (1547–1594). Plantin continued to work in Antwerp until his death in 1589.

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Source: Museum Plantin-Moretus

After this very long introduction to the life and work of Christophe Plantin, we want to take a look at his activities as a printer/publisher of science. As we saw in the introduction he was closely associated with Abraham Ortelius, in fact their relationship began before Ortelius wrote his Theatrum. One of Ortelius’ business activities was that he worked as a map colourer, printed maps were still coloured by hand, and Plantin was one of the printers that he worked for. In cartography Plantin also published Lodovico Guicciardini’s (1521–1589) Descrittione di Lodovico Guicciardini patritio fiorentino di tutti i Paesi Bassi altrimenti detti Germania inferiore (Description of the Low Countries) (1567),

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Source: Wikimedia Commons

which included maps of the various Netherlands’ cities.

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Engraved and colored map of the city of Antwerp Source: Wikimedia Commons

Plantin contributed, however, to the printing and publication of books in other branches of the sciences.

Plantin’s biggest contribution to the history of science was in botany.  A combination of the invention of printing with movable type, the development of both printing with woodcut and engraving, as well as the invention of linear perspective and the development of naturalism in art led to production spectacular plant books and herbals in the Early Modern Period. By the second half of the sixteenth century the Netherlands had become a major centre for such publications. The big three botanical authors in the Netherlands were Carolus Clusius (1526–1609), Rembert Dodoens (1517–1585) and Matthaeus Loblius (1538–1616), who were all at one time clients of Plantin.

Matthaeus Loblius was a physician and botanist, who worked extensively in both England and the Netherlands.

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Matthias de Lobel (Lobelius),by Francis Delaramprint, 1615 Source: Wikimedia Commons

His Stirpium aduersaria noua… (A new notebook of plants) was originally published in London in 1571, but a much-extended edition, Plantarum seu stirpium historia…, with 1, 486 engravings in two volumes was printed and published by Plantin in 1576. In 1581 Plantin also published his Dutch herbal, Kruydtboek oft beschrÿuinghe van allerleye ghewassen….

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Source: Wikimedia Commons

There is also an anonymous Stirpium seu Plantarum Icones (images of plants) published by Plantin in 1581, with a second edition in 1591, that has been attributed to Loblius but is now thought to have been together by Plantin himself from his extensive stock of plant engravings.

Carolus Clusius also a physician and botanist was the leading scientific horticulturist of the period, who stood in contact with other botanist literally all over the worlds, exchanging information, seeds, dried plants and even living ones.

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Portrait of Carolus Clusius painted in 1585 Attributed to Jacob de Monte – Hoogleraren Universiteit Leiden via Wikimedia Commons

His first publication, not however by Plantin, was a translation into French of Dodoens’ herbal of which more in a minute. This was followed by a Latin translation from the Portuguese of Garcia de Orta’s Colóquios dos simples e Drogas da India, Aromatum et simplicium aliquot medicamentorum apud Indios nascentium historia (1567) and a Latin translation from Spanish of Nicolás Monardes’  Historia medicinal delas cosas que se traen de nuestras Indias Occidentales que sirven al uso de la medicina, , De simplicibus medicamentis ex occidentali India delatis quorum in medicina usus est (1574), with a second edition (1579), both published by Plantin.His own  Rariorum alioquot stirpium per Hispanias observatarum historia: libris duobus expressas (1576) and Rariorum aliquot stirpium, per Pannoniam, Austriam, & vicinas quasdam provincias observatarum historia, quatuor libris expressa … (1583) followed from Plantin’s presses. His Rariorum plantarum historia: quae accesserint, proxima pagina docebit (1601) was published by Plantin’s son-in-law Jan Moretus, who inherited the Antwerp printing house.

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Our third physician-botanist, Rembert Dodoens, his first publication with Plantin was his Historia frumentorum, leguminum, palustrium et aquatilium herbarum acceorum, quae eo pertinent (1566) followed by the second Latin edition of his  Purgantium aliarumque eo facientium, tam et radicum, convolvulorum ac deletariarum herbarum historiae libri IIII…. Accessit appendix variarum et quidem rarissimarum nonnullarum stirpium, ac florum quorumdam peregrinorum elegantissimorumque icones omnino novas nec antea editas, singulorumque breves descriptiones continens… (1576) as well as other medical books.

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Rembert Dodoens Theodor de Bry – University of Mannheim via Wikimedia Commons

His most well known and important work was his herbal originally published in Dutch, his Cruydeboeck, translated into French by Clusius as already stated above.

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Title page of Cruydt-Boeck,1618 edition Source: Wikimedia Commons

Plantin published an extensively revised Latin edition Stirpium historiae pemptades sex sive libri XXXs in 1593.

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This was largely plagiarised together with work from Loblius and Clusius by John Gerrard (c. 1545–1612)

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John Gerard Source: Wikimedia Commons

in his English herbal, Great Herball Or Generall Historie of Plantes (1597), which despite being full of errors became a standard reference work in English.

The Herball, or, Generall historie of plantes / by John Gerarde

Platin also published a successful edition of Juan Valverde de Amusco’s Historia de la composicion del cuerpo humano (1568), which had been first published in Rome in 1556. This was to a large extent a plagiarism of Vesalius’ De humani corporis fabrica (1543).

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Another area where Platin made a publishing impact was with the works of the highly influential Dutch engineer, mathematician and physicist Simon Stevin (1548-1620). The Plantin printing office published almost 90% of Stevin’s work, eleven books altogether, including his introduction into Europe of decimal fractions De Thiende (1585),

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Source: Wikimedia Commons

his important physics book De Beghinselen der Weeghconst (The Principles of Statics, lit. The Principles of the Art of Weighing) (1586),

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Source: Wikimedia Commons

his Beghinselen des Waterwichts (Principles of hydrodynamics) (1586) and his book on navigation De Havenvinding (1599).

Following his death, the families of his sons-in-law continued the work of his various printing offices, Christophe Beys (1575–1647), the son of Magdalena and Gilles, continued the Paris branch of the business until he lost his status as a sworn printer in 1601. The family of Franciscus Raphelengius continued printing in Leiden for another two generations, until 1619. When Lipsius retired from the University of Leiden in 1590, Joseph Justus Scaliger (1540-1609) was invited to follow him at the university. He initially declined the offer but, in the end, when offered a position without obligations he accepted and moved to Leiden in 1593. It appears that the quality of the publications of the Plantin publishing office in Leiden helped him to make his decision.  In 1685, a great-granddaughter of the last printer in the Raphelengius family married Jordaen Luchtmans (1652 –1708), who had founded the Brill publishing company in 1683.

The original publishing house in Antwerp survived the longest. Beginning with Jan it passed through the hands of twelve generations of the Moretus family down to Eduardus Josephus Hyacinthus Moretus (1804–1880), who printed the last book in 1866 before he sold the printing office to the City of Antwerp in 1876. Today the building with all of the companies records and equipment is the Museum Plantin-Moretus, the world’s most spectacular museum of printing.

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2-021 Museum Plantin Moretus

There is one last fascinating fact thrown up by this monument to printing history. Lodewijk Elzevir (c. 1540–1617), who founded the House of Elzevir in Leiden in 1583, which published both Galileo’s Discorsi e dimostrazioni matematiche intorno a due nuove scienze in 1638 and Descartes’ Discours de la Méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences in 1637, worked for Plantin as a bookbinder in the 1560s.

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Nikolaes Heinsius the Elder, Poemata (Elzevier 1653), Druckermarke Source: Wikimedia Commons

The House of Elzevir ceased publishing in 1712 and is not connected to Elsevier the modern publishing company, which was founded in 1880 and merely borrowed the name of their famous predecessor.

The Platntin-Moretus publishing house played a significant role in the intellectual history of Europe over many decades.

 

 

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Reading Euclid

This is an addendum to yesterday review of Reading Mathematics in Early Modern Europe. As I noted there the book was an outcome of two workshops held, as part of the research project Reading Euclid that ran from 2016 to 2018. The project, which was based at Oxford University was led by Benjamin Wardhaugh, Yelda Nasifoglu (@YeldaNasif) and Philip Beeley.

The research project has its own website and Twitter account @ReadingEuclid. As well as Benjamin Wardhaugh’s The Book of Wonders: The Many Lives of Euclid’s Elements, which I reviewed here:

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And Reading Mathematics in Early Modern EuropeStudies in the Production, Collection, and Use of Mathematical Books, which I reviewed yesterday.

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There is also a third online publication Euclid in print, 1482–1703: A catalogue of the editions of the Elements and other Euclidian Works, which is open access and can be downloaded as a pdf for free.

All of this is essential reading for anybody interested in the history of the most often published mathematics textbook of all times.

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There’s more to reading than just looking at the words

When I first became interested in the history of mathematics, now literally a lifetime ago, it was dominated by a big events, big names approach to the discipline. It was also largely presentist, only interested in those aspects of the history that are still relevant in the present. As well as this, it was internalist history only interested in results and not really interested in any aspects of the context in which those results were created. This began to change as some historians began to research the external circumstances in which the mathematics itself was created and also the context, which was often different to the context in which the mathematics is used today. This led to the internalist-externalist debate in which the generation of strictly internalist historians questioned the sense of doing external history with many of them rejecting the approach completely.

As I have said on several occasions, in the 1980s, I served my own apprenticeship, as a mature student, as a historian of science in a major research project into the external history of formal or mathematical logic. As far as I know it was the first such research project in this area. In the intervening years things have evolved substantially and every aspect of the history of mathematics is open to the historian. During my lifetime the history of the book has undergone a similar trajectory, moving from the big names, big events modus to a much more open and diverse approach.

The two streams converged some time back and there are now interesting approaches to examining in depth mathematical publications in the contexts of their genesis, their continuing history and their use over the years. I recently reviewed a fascinating volume in this genre, Benjamin Wardhaugh’s The Book of Wonder: The Many Lives of Euclid’s Elements. Wardhaugh was a central figure in the Oxford-based Reading Euclid research project (2016–2018) and I now have a second volume that has grown out of two workshops, which took place within that project, Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books[1]. As the subtitle implies this is a wide-ranging and stimulating collection of papers covering many different aspects of how writers, researchers, and readers dealt with the mathematical written word in the Early Modern Period.

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In general, the academic standard of all the papers presented here is at the highest level.  The authors of the individual papers are all very obviously experts on the themes that they write about and display a high-level of knowledge on them. However, all of the papers are well written, easily accessible and easy to understand for the non-expert. The book opens with a ten-page introduction that explains what is being presented here is clear, simple terms for those new to the field of study, which, I suspect, will probably the majority of the readers.

The first paper deals with Euclid, which is not surprising given the origin of the volume. Vincenzo De Risi takes use through the discussion in the 16th and 17th centuries by mathematical readers of the Elements of Book 1, Proposition 1 and whether Euclid makes a hidden assumption in his construction. Risi points out that this discussion is normally attributed to Pasch and Hilbert in the 19th century but that the Early Modern mathematicians were very much on the ball three hundred years earlier.

We stay with Euclid and his Elements in the second paper by Robert Goulding, who takes us through Henry Savile’s attempts to understand and maybe improve on the Euclidean theory of proportions. Savile, best known for giving his name and his money to establish the first chairs for mathematics and astronomy at the University of Oxford, is an important figure in Early Modern mathematics, who largely gets ignored in the big names, big events history of the subject, but quite rightly turns up a couple of times here. Goulding guides the reader skilfully through Savile’s struggles with the Euclidean theory, an interesting insight into the thought processes of an undeniably, brilliant polymath.

In the third paper, Yelda Nasifoglu stays with Euclid and geometry but takes the reader into a completely different aspect of reading, namely how did Early Modern mathematicians read, that is interpret and present geometrical drawings? Thereby, she demonstrates very clearly how this process changed over time, with the readings of the diagrams evolving and changing with successive generations.

We stick with the reading of a diagram, but leave Euclid, with the fourth paper from Renée Raphael, who goes through the various reactions of readers to a problematic diagram that Tycho Brahe used to argue that the comet of 1577 was supralunar. It is interesting and very informative, how Tycho’s opponents and supporters used different reading strategies to justify their standpoints on the question. It illuminates very clearly that one brings a preformed opinion to a given text when reading, there is no tabula rasa.

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We change direction completely with Mordechai Feingold, who takes us through the reading of mathematics in the English collegiate-humanist universities. This is a far from trivial topic, as the Early Modern humanist scholars were, at least superficially, not really interested in the mathematical sciences. Feingold elucidates the ambivalent attitude of the humanists to mathematical topics in detail. This paper was of particular interest to me, as I am currently trying to deepen and expand my knowledge of Renaissance science.

Richard Oosterhoff, in his paper, takes us into the mathematical world of the relatively obscure Oxford fellow and tutor Brian Twyne (1581–1644). Twyne’s manuscript mathematical notes, complied from various sources open a window on the actual level and style of mathematics’ teaching at the university in the Early Modern Period, which is somewhat removed from what one might have expected.

Librarian William Poole takes us back to Henry Savile. As well as giving his name and his money to the Savilian mathematical chairs, Savile also donated his library of books and manuscripts to be used by the Savilian professors in their work. Poole takes us on a highly informative tour of that library from its foundations by Savile and on through the usage, additions and occasional subtractions made by the Savilian professors down to the end of the 17th century.

Philip Beeley reintroduced me to a recently acquired 17th century mathematical friend, Edward Bernard and his doomed attempt to produce and publish an annotated, Greek/Latin, definitive editions of the Elements. I first became aware of Bernard in Wardhaugh’s The Book of Wonder. Whereas Wardhaugh, in his account, concentrated on the extraordinary one off, trilingual, annotated, Euclid (Greek, Latin, Arabic) that Bernard put together to aid his research and which is currently housed in the Bodleian, Beeley examines Bernard’s increasing desperate attempts to find sponsors to promote the subscription scheme that is intended to finance his planned volume. This is discussed within the context of the problems involved in the late 17th and early 18th century in getting publishers to finance serious academic publications at all. The paper closes with an account of the history behind the editing and publishing of David Gregory’s Euclid, which also failed to find financial backers and was in the end paid for by the university.

Following highbrow publications, Wardhaugh’s own contribution to this volume goes down market to the world of Georgian mathematical textbooks and their readers annotations. Wardhaugh devotes a large part of his paper to the methodology he uses to sort and categorise the annotations in the 366 copies of the books that he examined. He acknowledges that any conclusions that he draws from his investigations are tentative, but his paper definitely indicates a direction for more research of this type.

Boris Jardine takes us back to the 16th century and the Pantometria co-authored by father and son Leonard and Thomas Digges. This was a popular book of practical mathematics in its time and well into the 17th century. Jardine examines how such a practical mathematics text was read and then utilised by its readers.

Kevin Tracey closes out the volume with a final contribution on lowbrow mathematical literature and its readers with an examination of John Seller’s A Pocket Book, a compendium of a wide range of elementary mathematical topics written for the layman. Following a brief description of Seller’s career as an instrument maker, cartographer and mathematical book author, Tracey examines marginalia in copies of the book and shows that it was also actually used by university undergraduates.

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The book is nicely presented and in the relevant papers illustrated with the now ubiquitous grey in grey prints. Each paper has its own collection of detailed, informative, largely bibliographical endnotes. The books referenced in those endnotes are collected in an extensive bibliography at the end of the book and there is also a comprehensive index.

As a whole, this volume meets the highest standards for an academic publication, whilst remaining very accessible for the general reader. This book should definitely be read by all those interested in the history of mathematics in the Early Modern Period and in fact by anybody interested in the history of mathematics. It is also a book for those interested in the history of the book and in the comparatively new discipline, the history of reading. I would go further and recommend it for general historians of the Early Modern Period, as well as interested non experts.

[1] Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books, eds. Philip Beeley, Yelda Nasifoglu and Benjamin Wardhaugh, Material Readings in Early Modern Culture, Routledge, New York and London, 2021

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