Category Archives: Renaissance Science

Renaissance Science – XIV

In the previous episode we saw how the Renaissance rediscovery of Vitruvius’ De architectura influenced the development of architecture during the Renaissance and dissolved the boundary between the intellectual theoreticians and the practical artisans. However, as stated there Vitruvius was not just an architect, but was also an engineer and his Book X deals quite extensively with machines both civil and military. This had a massive influence on a new type of artisan the Renaissance artist-engineer and it is to these that we now turn our attention. 

Artist-engineers were very much a Northern Italian Renaissance phenomenon, but even earlier artists had been categorised as craftsmen or artisans and not as artists as we would understand the term. The occupation of artist-engineer was very much influenced by the popularity of Vitruvius’ De architectura. The most well-known Renaissance artist engineer is, of course, Leonardo da Vinci (1452–1519), but he was by no means unique, as he is often presented in popular accounts, but he stood at the end of a line of other artist-engineers, who are known to have influenced him. Here I will deal principally with those artisan artist-engineers, who dissolved the boundary between practice and theory by witing and circulating treatises on their work.

At the beginning of the line were the Florentine rival, goldsmiths Lorenzo Ghiberti (1378–1455) and Filippo Brunelleschi (1377–1446). In 1401 there was a competition to design the first set of new doors for the Florence Baptistery. Ghiberti and Brunelleschi were two of the seven artists on the short list. Ghiberti won the commission and set up a major engineering workshop to carry out the work. 

It took Ghiberti twenty-one years to complete the first set of doors featuring twenty New Testament Bible scenes, the four evangelists and four of the Church Fathers, but once finished they established his reputation, as a great Renaissance artist. In 1425 he was awarded a second commission for another set of doors, these featuring ten Old Testament scenes in realistic perspective presentation took another twenty-seven years. The second set of doors included portraits of both Ghiberti and his father Bartolomeo Ghiberti. 

Ghiberti self portrait from his second set of doors (modern copy Source: Wikimedia Commons

We don’t need to go into any great detail here about the doors or the other commissions that Ghiberti’s workshop finished.

Ghiberti’s second set of doors, known as the Gates of Paradise (modern copy) Source: Wikimedia Commons

What is much more relevant to our theme is his activities as an author. Although he was the artisan son of an artisan father, Ghiberti crossed the medieval boundary between theory and practice with his Commentarii, a thesis on the history of art, written in Italian. He drew on various sources from antiquity including the first century BCE illustrated Greek text on machines by Athenaeus Mechanicus and Pliny’s Naturalis Historia, a text much discussed by the Renaissance Humanists, but his major source was Vitruvius’ De architectura. Ghiberti died without finishing his Commentarii and it was never published. However, many important Renaissance artist, such as Donatello and Paolo Uccello, served their apprenticeships in his workshop, so his influence on future generations was very large.

One probable graduate of Ghiberti’s workshop was Antonio Averlino (c. 1400–c. 1469) known as Filarete, a sculptor and architect. 



Filarete, Self-portrait medal, obverse, c. 1460, bronze. London, V & A

 Between 1461 and 1464, he wrote a vernacular volume on architecture in twenty-five books, his illustrated Trattato di Architettura, which circulated widely in manuscript. Central to his theory of architecture was the Vitruvian ideal of practice combined with theory. The most significant part of his book was his design for Sforzinda an ideal city named after his patron Francesco Sforza (1401–1466). This was the first of several ideal cities, which became a feature of the Renaissance. It is thought that his inspiration came from the works of Plato and his knowledge of this came from his friend at the Sforza court, the humanist scholar and philologist Francesco da Tolentino (1398–1481) known as Filelfo. Once again, we have, as in the last episode, a cooperation across the old boundaries between a scholar and an artisan.

Filarete Sforzinda

Filippo Brunelleschi poses a different problem. Like Ghiberti trained as a goldsmith, he went on to become the epitome of a Renaissance Vitruvian architect. However, there is no direct evidence that connects him with De architectura or its author. There is no direct evidence that connects him with anything except for the products of his life’s work, most notably the dome of the Santa Maria del Fiori cathedral in Florence. He is also renowned as the inventor or discoverer of the mathematical principles of linear perspective, as explained in episode seven of this series. This links him indirectly to Vitruvius, as some authors insist that he only rediscovered linear perspective, quoting Book 7 of De architectura, where Vitruvius describes the use of some form of perspective on the ancient Greek theatre flats. 

Filippo Brunelleschi in an anonymous portrait of the 2nd half of the 15th century (Louvre, Paris) Source: Wikimedia Commons

More importantly, Brunelleschi, as an architect, not only designed and supervised the construction of the buildings that he was commissioned to build but also devised and constructed the machines that he needed on his building sites to facilitate those constructions. For his work on the Santa Maria dome, for example he designed a crane to lift the building materials up to the top of the cathedral.

Brunelleschi’s revolving crane

A drawing of that crane can be found in Leonardo’s manuscripts. He was also granted a patent by the ruling council of Florence for the design of a ship to transport heavy loads of stone on rivers and canals.

Reproduction of Brunelleschi’s patent boat Source: Wikimedia Commons

Brunelleschi was also like, Vitruvius, a successful hydraulic engineer. It is hard to believe that he wasn’t influenced by De architectura.

There is no doubt about the Vitruvian influence of our next artist-engineer, Mariano di Jacopo (1382–c. 1453) known as Taccola (the jackdaw), who, as I explained in an earlier post on that Renaissance iconic figure, included a Vitruvian Man in his drawings. Taccola, who is known to have worked as a sculptor, superintendent of roads and hydraulic engineer, was from Sienna. He met and talked with Brunelleschi, one of the few people known to have done so. 

Taccola produced two annotated manuscripts the four books of De ingeneis, written between 1419 and 1433, and De machnis issued in 1449, which was partially an improved version of his De ingeneis.


ResearchGate
Jacopo Mariano Taccola, De ingeneis, Book I. Codex Latinus 197,..

Both manuscripts contain numerous illustrations of machines for hydraulic engineering, milling (and mills were one of the most important types of machines in medieval and Renaissance culture), construction and military machinery, all topics covered by Vitruvius.

First European depiction of a piston pump by Taccola, c.1450 Source: Wikimedia Commons

His manuscripts also some of Brunelleschi’s construction machines. Taccola is in one sense a transitional figure as his representations, of three-dimensional machines, often use medieval drawing conventions rather than Brunelleschi’s recently discovered linear perspective. 

Taccola’s works were never printed but copies of his manuscripts are known to have circulated widely during his lifetime and to have been highly influential. After his death his influence waned as his work was superceded by the more advance work of Francesco di Giorgio Martini and Leonardo da Vinci both of whom were heavily influenced by Taccola.

Francesco di Giorgio Martini (1439–1501) was, like Taccola, from Siena and was an architect, engineer, painter, sculptor, and writer.

His Vitruvian influence is very obvious in his work, as also the influence of Taccola. Francesco worked for much of his life on an Italian translation of Vitruvius’ De architectura, which he never published. Like Filarete he wrote an architectural treatise Trattato di archtettura, ingegneria e arte militare, worked on over decades and finished sometime after 1482. Many of his machines are taken from Taccola’s manuscripts. As can be seen from the title, it continues the Vitruvian tradition. Like Filarete’s volume it contains a design for an ideal town. Probably inspired by Sulpizio’s first printed edition of De architectura and Alberti’s De re aedificatoria, he produced a new edition of his own book known as Trattato II. 

Edificij et machine, Martini, Francesco di Giorgio, 1439-1501, brown ink and wash, ca. 1475-ca. 1480, The volume comprises 103 drawings by Francesco di Giorgio Martini and his assistants, featuring machines and devices for lifting columns and other heavy weights, schemes for transporting water, and mechanisms for milling and moving boats. There are also a few drawings showing how people could walk or float on water standing on inflatable containers and using an oar to propel themselves. PUBLICATIONxINxGERxSUIxAUTxONLY Copyright: LCD2_180906_23583

Both Taccola and Francesco are known to have influenced the most famous of the Renaissance artist-engineers, Leonardo da Vinci. As well as the obvious direct influence of Vitruvius, many of the machines illustrated in Leonardo’s manuscripts are taken from the work of Brunelleschi, Taccola and Francesco di Giorgio. As an apprentice, Leonardo had worked on the final phase of Brunelleschi’s dome for the Santa Maria Cathedral, and he took the opportunity to study Brunelleschi’s building site machines and scaffolding. He owned copies of the manuscripts of both Taccola and Francesco, the latter of which he annotated heavily. Leonardo, as is well known, wrote reams of annotated manuscripts on his machines but never published any of them.

Watter wheel, just one of Leonardo’s hundreds of drawings of machines Source

All of the artist-engineers that I have briefly sketched here are examples of artisans who crossed over or better dissolved the boundaries between theoretical and practical knowledge. They are also, so to speak, the stars of a much larger and widespread group of Renaissance artist-engineers, whose influence spread throughout the Renaissance, changing and elevating the status of the skilled artisan.  

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Filed under Book History, History of Technology, Renaissance Science

They also serve…

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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Renaissance Science – XII

There is a popular misconception that the emergence of modern science during the Renaissance, or proto-scientific revolution as we defined it in episode V of this series, and the scientific revolution proper includes a parallel rejection of the so-called occult sciences. Nothing could be further from the truth. This period sees a massive revival of all sorts of occult studies, covering a wide spectrum, we will look at this in more details in further episodes, but today I wish to deal with astrology. It is generally acknowledged that the period we know as the Renaissance was the golden age of astrology in Europe. There are multiple reasons for this rise of interest in and practice astrology in the period from roughly fourteen hundred and the middle of the seventeenth century.

As already explained in the previous episode, one reason for the rise in the status of the mathematical sciences during the Renaissance was the rise of astrological-medicine, or iatromathematics, within school medicine, something that we will look at in more detail when discussing Renaissance medicine. This rise in iatromathematics was, naturally, also a driving force in the increasing acceptance of astrology, but it was by no means the only one. This brings us to the important fact, that whereas most people on hearing the term astrology automatically think of natal astrology (also known as genethliacal astrology), that is birth horoscopes, but this is only one branch of the discipline and often in a given context not the most important one.

As well as natal astrology and iatromathematics there are also mundane astrology, electional astrology, horary astrology, locational astrology also called astrogeography, and meteorological astrology, each of which played a significant role in the world of astrology in the Renaissance. 

Mundane astrology is the application of astrology to world affairs and world events rather than to individuals and is generally acknowledged as the oldest form of astrology.

Electional astrology is the attempt to determine the most auspicious time to stage an event or undertake a venture, or even to show that no time would be auspicious for a given event of venture. The range of events or ventures can and did include, starting a war, or staging a battle, but also peaceful activities such as launching a diplomatic mission, simply going on a journey, or planning the date for an important, i.e., political, wedding.  

Horary astrology attempts to answer questions, interrogations, posed to the astrologer by casting a horoscope at the time that question is received and understood by the astrologer. The range of possible questions is entirely open, but few would waste the time of the astrologer or incur the costs that they might levy with trivial questions.

Locational astrology assumes that geographical locations play a specific role in astrological interpretation. For example, although time and latitude are the principle initial condition for casting a horoscope, two babies born at exactly the same time on the same day but in differing locations would have differing horoscopes, even if born at the same latitude, because of the influence of the geographical location.

Meteorological astrology, or astrometeorology, is the belief that the weather is caused by the position and motion of celestial objects, and it is therefor possible to predict or forecast the weather through astrological means.  

There are also special procedures such as lots of fortune and prorogation to determine special or important events in a subjects life, too detailed for this general survey. 

Mundane, natal, electional, horary and locational astrology are all grouped together under the term judicial astrology. Iatromathematics and astrometeorology are referred to as natural astrology. Those who objected to or rejected astrology, including at times the Catholic Church, usually rejected judicial astrology but accepted natural astrology as a branch of knowledge.

Western astrology has its origins in the omen astrology of the Babylonians, which was originally purely mundane astrology. Individual horoscope astrology emerged in Babylon around the sixth century BCE, and it was this that the ancient Greeks adopted and developed further. This is basically the astrology that was still in use in Renaissance Europe. After some reluctance the Romans adopted the Greek astrology and in the second century CE Ptolemaeus produced the most comprehensive text on the philosophy and practice of astrology, his Tetrabiblos, also known in Greek as Apotelesmatiká (Ἀποτελεσματικά) “Effects”, and in Latin as Quadripartitum. It should, however, be noted that this is by no means the only astrology text from antiquity. 

With the general collapse of learning in Europe in the Early Middle Ages from the fifth century onwards, astrology disappeared along with other scholarly disciplines. It was first revived by the Arabic, Islamic culture via the Persians in the eighth century. Arabic scholars developed and expanded the Greek astrology. Astrological texts were amongst the earliest ones translated into Arabic during the big translation movement in the eighth and ninth centuries. The same was true when European scholars began translating Arabic texts into Latin in the twelfth century. They translated both Greek and Arabic texts on astrology.

The Church could have rejected Greek astrology in the High Middle Ages as it was deterministic and as such contradicted the theological principle of free will, which is fundamental to Church doctrine. However, Albertus Magnus and Thomas Aquinas, who made Aristotelian philosophy acceptable to the Church also did the same for astrology reinterpreting it as contingent rather than determinist. By the thirteenth century all the forms of astrology had become established in Europe.

So, astrology in its various forms were well established in Europe in the High Middle Ages. This raises the question, why did it flourish and bloom during the Renaissance? As already stated above it was not just the rise of iatromathematics although this was a contributary factor.

One factor was the rise of the court astrologer, as a member of the retinue serving the ruler at court. Several Roman emperors had employed court astrologers, but the practice re-entered Europe in the Middle Ages via the Islamic culture. The Abbasid Caliphs, who started the major translation movement of Greek knowledge into Arabic, adopted the practice of employing a court astrologer from the Persians. In the Middle Ages, one of the first European potentates to adopt the practice was the Hohenstaufen Holy Roman Emperor, Frederick II (1194–1250), whose court was on the island of Sicily an exchange hub between North African Arabic-Islamic and European cultures. Frederick was a scholar, who not only traded goods with his Islamic neighbours but also knowledge. Following the Abbasid example, he installed an astrologer in his court. Both the prominent astrologers Michael Scot (1175–c. 1232) and Guido Bonatti (c. 1210–c. 1300) served in this function. The fashion spread and by the fifteenth century almost all rulers in Europe employed a court astrologer, either as a direct employee at court or when employed elsewhere on a consultant basis. The role of the court astrologer was that of a political advisor and whilst casting birth horoscopes, their main activities were in electional and horary astrology. Many notable mathematicians and astronomers served as court astrologers including Johannes Regiomontanus (1436–1476), Georg von Peuerbach (1423–1461), Peter Apian (1495–1552), Tycho Brahe (1546–1601), Michael Mästlin (1550–1631), and Johannes Kepler (1571–1630).

The upper echelons were thus firmly anchored in an astrological culture but what of the masses? Here, an important factor was the invention of movable type printing. This, of course, meant that the major Greek and Arabic astrological volumes became available in printed form. Ptolemaeus’ Tetrabiblos, translated from Arabic into Latin in the twelfth century, was first printed and published in Venice by Erhard Ratdolt (1442–1528) in 1484. However, much more important for the dissemination and popularisation of astrology were the astrological ephemera that began to appear from the very beginning of the age of print–wall calendars, prognostica, writing calendars and almanacs. The wall calendars, and Guttenberg printed a wall calendar to help finance the printing of his Bible, and writing calendars were a product of the iatromathematics, whereas the prognostica and almanacs dealt with astrometeorology and mundane astrology. These ephemera were comparatively cheap and were produced in print runs that often ran into the tens of thousands, making them very profitable for printer-publishers. Often containing editorial sections, the prognostica and almanacs came in a way to fulfil the function of the tabloid press today. For most households the annual almanac was the only print item that the purchased, apart perhaps from a Bible. 

But what of the Humanist Renaissance, did its basic philosophy or principles play a role in the rise of astrology? The answer is yes, very much so. Although the Tetrabiblos was translated into Latin comparatively early, the majority of important astrological texts in the Middle Ages were Arabic ones and these also found their way early into print editions. This circumstance kicked off a back to Greek purity–remove the Arabic influence debate amongst Renaissance astrologers. The humanists insisted that the only permissible astrological methods were those found in the Tetrabiblos and anything else was Arabic corruption. This meant they wanted to eliminate elections and interrogations, which Ptolemaeus does not deal with. Ironical both practices came into Arabic astrology via Persian astrology from Greek astrology that was older than Ptolemaeus’ work.

We don’t need to discuss the details of this debate but leading scholars, and the astrologers were leading mathematicians, astronomers and physicians were exchanging theoretical broadsides in print over decades. This, of course, raised the public perception and awareness of astrology and contributed to the Renaissance rise in astrology.

The Renaissance surge in astrology held well into the seventeenth century. With the notable exception of Copernicus, who apparently had little interest in astrology, all of the astronomers, who contributed to the so-called astronomical revolution including Tycho, Kepler and Galileo were practicing astrologers. Later in the seventeenth century, astrology went into decline but we don’t need to address that here.

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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.

Astrolabium_Masha'allah_Public_Library_Brugge_Ms._522.tif

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

Renaissance Science – VII

In the last post we looked at the European re-invention of moveable-type and the advent of the printed book, which played a highly significant role in the history of science in general and in Renaissance science in particular. I also emphasised the various print technologies developed for reproducing images, because they played a very important role in various areas of the sciences during the Renaissance, as we shall see in later posts in this series. Parallel to these technological developments there were two major developments in the arts, which would have a very major impact on the illustration in Renaissance science publications, the (re?)-discovery of linear perspective and the development of naturalism.

Linear perspective is the geometrical method required to reproduce three-dimensional objects realistically on a two-dimensional surface; the discovery or invention of linear perspective is usually attributed to the Renaissance artist-engineer and architect, Filippo Brunelleschi (1377–1446), about whom more below, but already in the Renaissance it was often referred to as a re-discovery. This Renaissance re-discovery trope was very much in line with the general Renaissance concept of a rebirth of classical knowledge. Here the belief that linear perspective was a re-discovery is based on the concept of skenographia in ancient Greek theatre, which consists of using painted flat panels on a stage to give the illusion of depth. This is mentioned in Aristotle’s Poetics (c. 335 BCE) a general work on drama. More importantly, from a Renaissance perspective, it is briefly described in Vitruvius’ De Architectura libri dicem (Ten Books on Architecture) from the first century BCE. Once again, as we shall see later, Vitruvius’ De Architectura played a central role in Renaissance thought. In his Book 7 On Finishing, Vitruvius wrote in the preface:

In Athens, when Aechylus was producing tragedies, Agathachus was the first to work for the theatre and wrote a treatise about it. Learning from this, Democritus and Anaxagoras wrote on the same subject, namely how the extension of rays from a certain established centre point ought to correspond in a natural ration to the eyes’ line of sight, so that they could represent the appearance of buildings in scene painting, no longer by some uncertain method, but precisely, both the surfaces that were depicted frontally, and those that seemed either to be receding or projecting[1].

Of course, ancient Greek theatre flats no longer exist, but some Greek and many more Roman wall paintings have survived, which very obviously display some degree of perspective. However, closer analysis of these paintings has shown that while they are in fact constructed on some sort of perspective scheme it is not the linear perspective that was developed in the Renaissance.

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Villa of P. Fannius Synistor Cubiculum M alcove Panel with temple at east end of the alcove, the north end of the east wall Middle of the first century B.C. Boscoreale (Pompeii), Italy Source:

Although linear perspective was not strictly a re-discovery, it also didn’t emerge at the beginning of the fifteenth century out of thin air. Already, more than a century earlier the so-called proto-Renaissance artists, in particular Giotto (1267–1337), were producing paintings that displayed depth based on a mathematical model, when not quite that of linear perspective and not consistent.

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‘Jesus Before the Caïf’, by Giotto (1305). The ceiling rafters show the Giotto’s introduction of convergent perspective. B. Detailed analysis, however, reveals that the ceiling has an inconsistent vanishing point and that the Caïf’s dais is in parallel perspective, with no vanishing point. Source

At the beginning of the fifteenth century, the Renaissance sculptor Lorenzo Ghiberti (1378–1455) used linear perspective in the panels of the second set of bronze doors he was commissioned to produce for the Florence Baptistry, dubbed the Gates of Paradise by Michelangelo.

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A panel of Adam and Eve in Ghiberti’s “Gate’s of Paradise”. Photo by Thermos.Source: Wikimedia Commons

As already stated, Brunelleschi is credited with having invented linear perspective according to his biographer Antonio di Tuccio Manetti (1423–1497), he compared the reality of his painting using linear perspective of the Florence Baptistery with the building itself using mirrors.

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Filippo Brunelleschi in an anonymous portrait of the 2nd half of the 15th century (Louvre, Paris) via Wikimedia Commons

According to Manetti, he used a grid or set of crosshairs to copy the exact scene square by square and produced a reverse image. The results were compositions with accurate perspective, as seen through a mirror. To compare the accuracy of his image with the real object, he made a small hole in his painting, and had an observer look through the back of his painting to observe the scene. A mirror was then raised, reflecting Brunelleschi’s composition, and the observer saw the striking similarity between the reality and painting. Both panels have since been lost. (Wikipedia)

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Brunelleschi left no written account of how he constructed his painting and the first written account we have of the geometry of linear perspective is from another Renaissance humanist artist and architect, Leon Battista Alberti (1404–1472) in his book On painting, published in Tuscan dialect as Della Pittura in 1436/6 and in Latin as De pictura first in 1450, although the Latin edition was also written in 1435. The book contains a comparatively simple account of the geometrical rudiments of linear perspective.

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Presumed self-portrait of Leon Battista Alberti Source: Wikimedia Commons

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Figure from the 1804 edition of Della pittura showing the vanishing point Source: Wikimedia Commons

A much fuller written account of the mathematics of linear perspective was produced in manuscript by the painter Piero della Francesca (c. 1415–1492), De Prospectiva pingendi (On the Perspective of painting), around 1470-80.

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An icosahedron in perspective from De Prospectiva pingendi Source: Wikimedia Commons

He never published this work, but his ideas on perspective were incorporated in his book Divina proportione by the mathematician Luca Pacioli (c. 1447–1517), written around 1498 but first published in 1509. Pacioli’s book also plagiarised another manuscript of della Francesca’s on perspective, his De quinque corporibus regularibus (The Five Regular Solids).

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Piero della Francesca by Giorgio Vasari Source: Wikimedia Commons

Mathematicians and artists continued over the centuries to write books describing and investigating the geometrical principles of linear perspective the most notable of, which during the Renaissance was Albrecht Dürer’s Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Ruler) published in 1525, which contains the first account of two point perspective. Dürer is credited with introducing linear perspective into the Northern Renaissance.

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Dürer, draughtsman Making a Perspective Drawing of a Reclining Woman

Naturalism is, as its name would suggest, the development in art to depict things naturally i.e., as we see them with our own eyes. Linear perspective is actually one aspect of naturalism. In her The Body of the Artisan, Pamala H. Smith writes the following:

It is difficult to know where to begin a discussion of naturalism (which can encompass the striving for “verisimilitude,” “illusionism,” “realism,” and the “imitation of nature”) in the early modern period, for the secondary literature in art history alone is vast. David Summers has defined naturalism as the attempt to make the elements of the artwork (in his account primarily painting) coincide with the elements of the optical experience[2]. (Her endnote: Summers, The Judgement of Sense, p. 3)

Smith also quotes in this context Alberti, “[He] put it in about 1435, making a picture that was an “open window” through which the world was seen.[3]” There is no neat timeline of events for Naturalism, as I have recreated above for linear perspective. Smith gives as her first historical example of Naturalism the so-called Carrara Herbal produced in Padua around 1400, with till then unknown, for this type of literature, unprecedented naturalism in its illustrations.[4]

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Violet plant – Carrara Herbal (c.1400), f.94 – BL Egerton MS 2020.jpg Source: Wikimedia Commons

As we will see in a later blog post it was in natural history, in particular in botany, that naturalism made a major impact in printed scientific illustrations.

Although, they still hadn’t really adopted the techniques of linear perspective it was the artists of the Northern Renaissance, rather than their Southern brethren, who first extensively adopted Naturalism, most notably Jan van Eyck (before 1390 – 1441). An attribute of the Naturalism of these painters was the use of mirrors in their paintings to symbolise the reflection of nature or reality.

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Jan van Eyck Detail with mirror and signature; Arnolfini Portrait, 1434 Source: Wikimedia Commons

Once again, we meet here Albrecht Dürer, who is justifiably renowned for his lifelike reproduction of various aspects of nature in his artwork.

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Albrecht Dürer Young Hare, (1502), Source: Wikimedia Commons

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Albrecht Dürer Great Piece of Turf, 1503 Source: Wikimedia commons

It is important to note here that although this picture looks very realistic, when first viewed, it is actually an example of illusion or hyperrealism. There are none of the old or withered plants that such a scene in nature would inevitably have. Also none of the plants obscure other plants with their shadows, as they would in reality. What Dürer delivers up here is an idealised naturalism, almost a contradiction in terms. This conflict between real naturalism and the demands of clear to interpret illustrations would play a significant role in the illustrations of Renaissance books on natural history.

However, as we shall see in later posts both linear perspective and Naturalism made a massive impact on the scientific and technological book illustrations that were produced during the Renaissance.

[1] Vitruvius, Ten Books on Architecture, Eds. Ingrid D. Rowland & Thomas Noble Howe, CUP, 1999 p. 86

[2] Pamala H. Smith, The Body of the Artisan: Art and Experience in the Scientific Revolution, University of Chicago Press, 2004 p. 9

[3] Smith, p. 33

[4] Smith p. 33

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Filed under Book History, History of science, Renaissance Science