Category Archives: History of Navigation

Renaissance science – XXVII

Early on in this series I mentioned that a lot of the scientific developments that took place during the Renaissance were the result of practical developments entering the excessively theoretical world of the university disciplines. This was very much the case in the mathematical sciences, where the standard English expression for the Renaissance mathematicus is mathematical practitioner. In this practical world, areas that we would now regard as separate disciples were intertwined is a complex that the mathematical practitioners viewed as one discipline with various aspects, this involved astronomy, cartography, navigation, trigonometry, as well as instrument and globe making. I have already dealt with trigonometry, cartography and astronomy and will here turn my attention to navigation, which very much involved the other areas in that list.

The so-called Age of Discovery or Age of Exploration, that is when Europeans started crossing the oceans and discovering other lands and other cultures, coincides roughly with the Renaissance and this was, of course the main driving force behind the developments in navigation during this period. Before we look at those developments, I want to devote a couple of lines to the terms Age of Discovery and Age of Exploration. Both of them imply some sort of European superiority, “you didn’t exist until we discovered you” or “your lands were unknown until we explored them.” The populations of non-European countries and continents were not sitting around waiting for their lands and cultures to be discovered by the Europeans. In fact, that discovery very often turned out to be highly negative for the discovered. The explorers and discoverers were not the fearless, visionary heroes that we tend to get presented with in our schools, but ruthless, often brutal chancers, who were out to make a profit at whatever cost.  This being the case the more modern Contact Period, whilst blandly neutral, is preferred to describe this period of world history.

As far as can be determined, with the notable exception of the Vikings, sailing in the Atlantic was restricted to coastal sailing before the Late Middle Ages. Coastal sailing included things such as crossing the English Channel, which, archaeological evidence suggests, was done on a regular basis since at least the Neolithic if not even earlier. I’m not going to even try to deal with the discussions about how the Vikings possibly navigated. Of course, in other areas of the world, crossing large stretches of open water had become common place, whilst the European seamen still clung to their coast lines. Most notable are the island peoples of the Pacific, who were undertaking long journeys across the ocean already in the first millennium BCE. Arab and Chinese seamen were also sailing direct routes across the Indian Ocean, rather than hugging the coastline, during the medieval period. It should be noted that European exploited the navigation skills developed by these other cultures as they began to take up contact with the other part of the world. Vasco da Gamma (c. 1460–1524) used unidentified local navigators to guide his ships the first time he crossed the Indian Ocean from Africa to India. On his first voyage of exploration of the Pacific Ocean from 1768 to 1771, James Cook (1728–1779) used the services of the of the Polynesian navigator, Tupaia (c. 1725–1770), who even drew a chart, in cooperation with Cook, Joseph Banks, and several of Cooks officer, of his knowledge of the Pacific Ocean. 

Tupaia’s map, c. 1769 Source: Wikimedia Commons

There were two major developments in European navigation during the High Middle Ages, the use of the magnetic compass and the advent of the Portolan chart. The Chinese began to use the magnetic properties of loadstone, the mineral magnetite, for divination sometime in the second century BCE. Out of this they developed the compass needle over several centuries. It should be noted that for the Chinese, the compass points South and not North. The earliest Chinese mention of the use of a compass for navigation on land by the military is before 1044 CE and in maritime navigation in 1117 CE.

Diagram of a Ming Dynasty (1368–1644) mariner’s compass Source: Wikimedia Commons

Alexander Neckam (1157–1219) reported the use of the compass for maritime navigation in the English Channel in his manuscripts De untensilibus and De naturis rerum, written between 1187 and 1202.

The sailors, moreover, as they sail over the sea, when in cloudy whether they can no longer profit by the light of the sun, or when the world is wrapped up in the darkness of the shades of night, and they are ignorant to what point of the compass their ship’s course is directed, they touch the magnet with a needle, which (the needle) is whirled round in a circle until, when its motion ceases, its point looks direct to the north.

This and other references to the compass suggest that it use was well known in Europe by this time.

A drawing of a compass in a mid 14th-century copy of Epistola de magnete of Peter Peregrinus. Source: Wikimedia Commons

The earliest reference to maritime navigation with a compass in the Muslim world in in the Persian text Jawāmi ul-Hikāyāt wa Lawāmi’ ul-Riwāyāt (Collections of Stories and Illustrations of Histories) written by Sadīd ud-Dīn Muhammad Ibn Muhammad ‘Aufī Bukhārī (1171-1242) in 1232. There is still no certainty as to whether there was a knowledge transfer from China to Europe, either direct or via the Islamic Empire, or independent multiple discovery. Magnetism and the magnetic compass went through a four-hundred-year period of investigation and discovery until William Gilbert (1544–1603) published his De magnete in 1600. 

De Magnete, title page of 1628 edition Source: Wikimedia Commons

The earliest compasses used for navigation were in the form of a magnetic needle floating in a bowl of water. These were later replaced with dry mounted magnetic needles. The first discovery was the fact that the compass needle doesn’t actually point at the North Pole, the difference is called magnetic variation or magnetic declination. The Chinese knew of magnetic declination in the seventh century. In Europe the discovery is attributed to Georg Hartmann (1489–1564), who describes it in an unpublished letter to Duke Albrecht of Prussia. However, Georg von Peuerbach (1423–1461) had already built a portable sundial on which the declination for Vienna is marked on the compass.

NIMA Magnetic Variation Map 2000 Source: Wikimedia Commons

There followed the discovery that magnetic declination varies from place to place. Later in the seventeenth century it was also discovered that declination also varies over time. We now know that the Earth’s magnetic pole wanders, but it was first Gilbert, who suggested that the Earth is a large magnet with poles. The next discovery was magnetic dip or magnetic inclination. This describes the fact that a compass needle does not sit parallel to the ground but points up or down following the lines of magnetic field. The discovery of magnetic inclination is also attributed to Georg Hartmann. The sixteenth century English, seaman Robert Norman rediscovered it and described how to measure it in his The Newe Attractive (1581) His work heavily influenced Gilbert. 

Illustration of magnetic dip from Norman’s book, The Newe Attractive Source: Wikimedia Commons

The Portolan chart, the earliest known sea chart, emerged in the Mediterranean in the late thirteenth century, not long after the compass, with which it is closely associated, appeared in Europe. The oldest surviving Portolan, the Carta Pisana is a map of the Mediterranean, the Black Sea and part of the Atlantic coast.

Source: Wikimedia Commons

The origins of the Portolan chart remain something of a mystery, as they are very sophisticated artifacts that appear to display no historical evolution. A Portolan has a very accurate presentation of the coastlines with the locations of the major harbours and town on the coast. Otherwise, they have no details further inland, indicating that they were designed for use in coastal sailing. A distinctive feature of Portolans is their wind roses or compass roses located at various points on the charts. These are points with lines radiating outwards in the sixteen headings, on later charts thirty-two, of the mariner’s compass.

Central wind rose on the Carta Pisana

Portolan charts have no latitude or longitude lines and are on the so-called plane chart projection, which treats the area being mapped as flat, ignoring the curvature of the Earth. This is alright for comparatively small areas, such as the Mediterranean, but leads to serious distortion, when applied to larger areas.

During the Contact Period, Portolan charts were extended to include the west coast of Africa, as the Portuguese explorers worked their way down it. Later, the first charts of the Americas were also drawn in the same way. Portolan style charts remained popular down to the eighteenth century.

Portolan chart of Central America c. 1585-1595 Source:

A central problem with Portolan charts over larger areas is that on a globe constant compass bearings are not straight lines. The solution to the problem was found by the Portuguese cosmographer Pedro Nunes (1502–1578) and published in his Tratado em defensam da carta de marear (Treatise Defending the Sea Chart), (1537).

Image of Portuguese mathematician Pedro Nunes in Panorama magazine (1843); Lisbon, Portugal. Source: Wikimedia Commons

The line is a spiral known as a loxodrome or rhumb lines. Nunes problem was that he didn’t know how to reproduce his loxodromes on a flat map.

Image of a loxodrome, or rhumb line, spiraling towards the North Pole Source: Wikimedia Commons

The solution to the problem was provided by the map maker Gerard Mercator (1512–1594), when he developed the so-called Mercator projection, which he published as a world map, Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata (New and more complete representation of the terrestrial globe properly adapted for use in navigation) in 1569.

Source: Wikimedia Commons
The 1569 Mercator world map Source: Wikimedia Commons.

On the Mercator projection lines of constant compass bearing, loxodromes, are straight lines. This however comes at a price. In order to achieve the required navigational advantage, the lines of latitude on the map get further apart as one moves away from the centre of projection. This leads to an area distortion that increases the further north or south on goes from the equator. This means that Greenland, slightly more than two million square kilometres, appear lager than Africa, over thirty million square kilometres.

Mercator did not publish an explanation of the mathematics used to produce his projection, so initially others could reproduce it. In the late sixteenth century three English mathematicians John Dee (1527–c. 1608), Thomas Harriot (c. 1560–1621), and Edward Wright (1561–1615) all individually worked out the mathematics of the Mercator projection. Although Dee and Harriot both used this knowledge and taught it to others in their respective functions as mathematical advisors to the Muscovy Trading Company and Sir Walter Raleigh, only Wright published the solution in his Certaine Errors in Navigation, arising either of the Ordinarie Erroneous Making or Vsing of the Sea Chart, Compasse, Crosse Staffe, and Tables of Declination of the Sunne, and Fixed Starres Detected and Corrected. (The Voyage of the Right Ho. George Earle of Cumberl. to the Azores, &c.) published in London in 1599. A second edition with a different, even longer, title was published in the same year. Further editions were published in 1610 and 1657. 

Source: Wikimedia Commons
Wright explained the Mercator projection with the analogy of a sphere being inflated like a bladder inside a hollow cylinder. The sphere is expanded uniformly, so that the meridians lengthen in the same proportion as the parallels, until each point of the expanding spherical surface comes into contact with the inside of the cylinder. This process preserves the local shape and angles of features on the surface of the original globe, at the expense of parts of the globe with different latitudes becoming expanded by different amounts. The cylinder is then opened out into a two-dimensional rectangle. The projection is a boon to navigators as rhumb lines are depicted as straight lines. Source: Wikimedia Commons

His mathematical solution for the Mercator projection had been published previously with his permission and acknowledgement by Thomas Blundeville (c. 1522–c. 1606) in his Exercises (1594) and by William Barlow (died 1625) in his The Navigator’s Supply (1597). However, Jodocus Hondius (1563–1612) published maps using Wright’s work without acknowledgement in Amsterdam in 1597, which provoked Wright to publish his Certaine Errors. Despite its availability, the uptake on the Mercator projection was actually very slow and it didn’t really come into widespread use until the eighteenth century.

Wright’s “Chart of the World on Mercator’s Projection” (c. 1599), otherwise known as the Wright–Molyneux map because it was based on the globe of Emery Molyneux (died 1598) Source: Wikimedia Commons

Following the cartographical trail, we have over sprung a lot of developments in navigation to which we will return in the next episode. 

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Filed under History of Cartography, History of Mathematics, History of Navigation, Renaissance Science

The black sheep of the Provence-Paris group

I continue my sketches of the seventeenth century group pf mathematicians and astronomers associated with Nicolas-Claude Fabri de Peiresc (1580-1637) in Provence and Marin Mersenne (1588–1648) in Paris with Jean-Baptiste Morin (1583–1656), who was born in Villefranche-sur-Saône in the east of France.

Jean-Baptiste Morin Source: Wikimedia Commons

He seems to have come from an affluent family and at the age of sixteen he began his studies at the University of Aix-en-Provence. Here he resided in the house of the Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix and Peiresc’s observing partner. For the last two years of his time in Aix, the young Pierre Gassendi, also lived in Gaultier de la Valette’s house and the two became good friends and observing partners.

In 1611, Morin moved to the University of Avignon, where he studied medicine graduating MD in 1613. For the next eight years, until 1621, he was in the service of Claude Dormy (c.1562–1626) the Bishop of Boulogne, in Paris, who paid for him to travel extensively in Germany, Hungary and Transylvania to study the metal mining industry. As well as serving Dormy as physician, he almost certainly acted as his astrologer, this was still in the period when astro-medicine or iatromathematics was the mainstream medical theory.

The tomb of Claude Dormy Source

From 1621 to 1629 he served Philip IV, King of Spain, and Duke of Luxembourg, also probably as astrologer. 

In 1630, he was indirectly asked by Marie de’ Medici, the Queen Mother, to cast a horoscope for her son, Louis XIII, who was seriously ill and whose doctor had predicted, on his own astrological reading, that he would die. Morin’s astrological analysis said that Louis would be severely ill but would survive. Luckily for Morin, his prediction proved accurate, and Marie de’ Medici used her influence to have him appointed professor for mathematics at the Collège Royal in Paris, a position he held until his death in 1656.

Marie de Médici portrait by Frans Pourbus, the Younger Source: Wikimedia Commons

In Paris, Morin he took up his friendship with Gassendi from their mutual student days and even continued to make astronomical observations with him in the 1630s, at the same time becoming a member of the group around Mersenne. However, in my title I have labelled Morin the black sheep of the Provence-Paris group and if we turn to his scholarly activities, it is very clear why. Whereas Peiresc, Mersenne, Boulliau, and Gassendi were all to one degree or another supporters of the new scientific developments in the early seventeenth century, coming to reject Aristotelean philosophy and geocentric astronomy in favour of a heliocentric world view, Morin stayed staunchly conservative in his philosophy and his cosmology.

Already in 1624, Morin wrote and published a defence of Aristotle, and he remained an Aristotelian all of his life. He rejected heliocentricity and insisted that the Earth lies at the centre of the cosmos and does not move. Whereas the others in the group supported the ideas of Galileo and also tried to defend Galileo against the Catholic Church, Morin launched an open attack on Galileo and his ideas in 1630, continuing to attack him even after his trial and house arrest. In 1638, he also publicly attacked René Descartes and his philosophy, not critically like Gassendi, but across-the-board, without real justification. He famously wrote that he knew that Descartes philosophy was no good just by looking at him when they first met. This claim is typical of Morin’s character, he could, without prejudice, be best described as a belligerent malcontent. Over the years he managed to alienate himself from almost the entire Parisian scholarly community. 

It would seem legitimate to ask, if Morin was so pig-headed and completely out of step with the developments and advances in science that were going on around him, and in which his friends were actively engaged, why bother with him at all? Morin distinguished himself in two areas, one scientific the other pseudo-scientific and it is to these that we now turn.

The scientific area in which made a mark was the determination of longitude. With European seamen venturing out into the deep sea for the first time, beginning at the end of the fifteenth century, navigation took on a new importance. If you are out in the middle of one of the Earth’s oceans, then being able to determine your exact position is an important necessity. Determining one’s latitude is a comparatively easy task. You need to determine local time, the position of the Sun, during the day, or the Pole Star, during the night and then make a comparatively easy trigonometrical calculation. Longitude is a much more difficult problem that relies on some method of determining time differences between one’s given position and some other fixed position. If one is one hour time difference west of Greenwich, say, then one is fifteen degrees of longitude west of Greenwich. 

Finding a solution to this problem became an urgent task for all of the European sea going nations, including France, and several of them were offering substantial financial rewards for a usable solution. In 1634, Morin suggested a solution using the Moon as a clock. The method, called the lunar distance method or simply lunars, was not new and had already suggested by the Nürnberger mathematicus, Johannes Werner (1468–1522) in his Latin translation of Ptolemaeus’ GeographiaIn Hoc Opere Haec Continentur Nova Translatio Primi Libri Geographicae Cl Ptolomaei, published in Nürnberg in 1514 and then discussed by Peter Apian (1495–1552) in his Cosmographicus liber, published in Landshut in 1524.

The lunar distance method relies on determining the position of the Moon relative to a given set of reference stars, a unique constellation for every part of the Moon’s orbit. Then using a set of tables to determine the timing of a given constellation for a given fixed point. Having determined one’s local time, it is then possible to calculate the time difference and thus the longitude. Because it is pulled hither and thither by both the Sun and the Earth the Moon’s orbit is extremely erratic and not the smooth ellipse suggested by Kepler’s three laws of planetary motion. This led to the realisation that compiling the tables to the necessary accuracy was beyond the capabilities of those sixteenth century astronomers and their comparatively primitive instruments, hence the method had not been realised. Another method that was under discussion was taking time with you in the form of an accurate clock, as first proposed by Gemma Frisius (1508–1555), Morin did not think much of this idea:

“I do not know if the Devil will succeed in making a longitude timekeeper but it is folly for man to try.”

Morin was well aware of the difficulties involved and suggested a comprehensive plan to overcome them. Eager to win the offered reward money, Morin put his proposal to Cardinal Richelieu (1585–1642), Chief Minister and most powerful man in France. Morin suggested improved astronomical instruments fitted out with vernier scales, a recent invention, and telescopic sights, also comparatively new, along with improvements in spherical trigonometry. He also suggested the construction of a national observatory, with the specific assignment of collected more accurate lunar data. Richelieu put Morin’s proposition to an expert commission consisting of Étienne Pascal (1588–1651), the father of Blaise, Pierre Hérigone (1580–1643), a Parisian mathematics teacher, and Claude Mydorge (1585–1647), optical physicist and geometer. This commission rejected Morin’s proposal as still not practical, resulting in a five year long dispute between Morin and the commission. It would be another century before Tobias Mayer (1723–1762) made the lunar distance method viable, basically following Morin’s plan.

Although his proposal was rejected, Morin did receive 2000 livre for his suggestion from Richelieu’s successor, Cardinal Mazarin (1602–1661) in 1645. Mazarin’s successor Jean-Baptiste Colbert (1619–1683) set up both the Académie des sciences in 1666 and the Paris Observatory in 1667, to work on the problem. This led, eventually to Charles II setting up the Royal Observatory in Greenwich, in 1675 for the same purpose.

Today, Morin is actually best known as an astrologer. The practice of astrology was still acceptable for mathematicians and astronomers during Morin’s lifetime, although it went into steep decline in the decades following his death. Although an avid astronomer, Peiresc appears to have had no interest in astrology. This is most obvious in his observation notes on the great comet of 1618. Comets were a central theme for astrologers, but Peiresc offers no astrological interpretation of the comet at all. Both Mersenne and Gassendi accepted the scientific status of astrology and make brief references to it in their published works, but neither of them appears to have practiced astrology. Boulliau also appear to have accepted astrology, as amongst his published translations of scientific texts from antiquity we can find Marcus Manilius’ Astronomicom (1655), an astrological poem written about 30 CE, and Ptolemaeus’ De judicandi jacultate (1667). Like Mersenne and Gassendi he appears not to have practiced astrology.

According to Morin’s own account, he initially thought little of astrology, but around the age of thirty he changed his mind and then spent ten years studying it in depth.

Jean-Baptiste Morin’s with chart as cast by himself

He then spent thirty years writing a total of twenty-six volumes on astrology that were published posthumously as one volume of 850 pages in Den Hague in 1661, as Astrologia Galllica (French Astrology). Like Regiomontanus, Tycho Brahe, and Kepler before him, he saw astrology as in need of reformation and himself as its anointed reformer. 

Source: Wikimedia Commons

The first eight volumes of Astrologia Galllica hardly deal with astrology at all but lay down Morin’s philosophical and religious views on which he bases his astrology. The remaining eighteen volumes then deal with the various topics of astrology one by one. Central to his work is the concept of directio in interpreting horoscopes. This is a method of determining the times of major events in a subject’s life that are indicated in their birth horoscope. Originally, to be found in Ptolemaeus’ Tetrabiblos, it became very popular during the Renaissance. The standard text was Regiomontanus’ Tabulae Directionum, originally written in 1467, and large numbers of manuscripts can still be found in libraries and archives. It was published in print by Erhard Ratdolt in Augsburg in 1490 and went through eleven editions, the last being published in 1626. Aware of Kepler’s rejection of both the signs of the zodiac and the system of houses, Morin defends both of them.

Coming, as it did at a time when astrology was in decline as an accepted academic discipline, Morin’s Astrologia Galllica had very little impact in the seventeenth century, but surprisingly, in English translation, it enjoys a lot of popularity amongst modern astrologers.

Morin was cantankerous and belligerent, which cost him most of his contacts with the contemporary scholars in Paris and his adherence to Aristotelian philosophy and a geocentric world view put him out of step with the rest of the Provence-Paris group, but he certainly didn’t suffer from a lack of belief in his own abilities as he tells us in this autobiographical quote:

“… I am excessively inclined to consider myself superior to others on account of my intellectual endowments and scientific attainments, and it is very difficult for me to struggle against this tendency, except when the realization of my sins troubles me, and I see myself a vile man and worthy of contempt. Because of all this my name has become famous throughout the world.”

 

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Filed under History of Astrology, History of Astronomy, History of medicine, History of Navigation

OHMS or everything you wanted to know about the history of trigonometry and didn’t know who to ask

When I was a kid, letters from government departments came in buff, manila envelopes with OHMS printed on the front is large, black, capital letters. This acronym stood for, On Her Majesty’s Service and earlier during Liz’s father’s reign (and no I’m not that old, although I was just born in his reign), On His Majesty’s Service, implying that civil servants worked directly for the monarch.  This was, of course, the origin of the title of Ian Fleming’s eleventh James Bond novel, On Her Majesty’s Secret Service

When I started learning trigonometry at school this acronym took on a whole new meaning as a mnemonic for the sine relation in right angle triangles, Opposite over Hypotenuse Means Sine. Recently it occurred to me that we had no mnemonic for the other trigonometric relations. Now in those days or even later when the trigonometry I was taught got more complex, I wasn’t aware of the fact that this mathematical discipline had a history. Now, a long year historian of mathematics, I am very much aware of the fact that trigonometry has a very complex, more than two-thousand-year history, winding its way from ancient Greece over India, the Islamic Empire and Early Modern Europe down to the present day. 

The Canadian historian of mathematics, Glen van Brummelen has dedicated a large part of his life to researching, writing up and publishing that history of trigonometry. The results of his labours have appeared in three volumes, over the years, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013 and most recently The Doctrine of TrianglesA History of Modern Trigonometry, Princeton University Press, Princeton and Oxford, 2021. He describes himself as the “best trigonometry historian, and the worst trigonometry historian”, as he is the only one[1]

A review of these three volumes could be written in one sentence, if you are interested in the history of trigonometry, then these three masterful volumes are essential. One really doesn’t need to say more, but in what follows I will give a brief sketch of each of the books. 

The Mathematics of the Heavens and the Earth: The Early History of Trigonometry delivers exactly what it says on the cover. The book opens with a brief but detailed introduction to the basics of spherical astronomy, because for a large part of the period covered, what we have is not the history of plane trigonometry, that’s the stuff we all learnt at school, but spherical trigonometry, that is the geometry of triangles on the surface of a sphere, which was developed precisely to do spherical astronomy. 

A friendly warning for potential readers this is not popular history but real, hardcore history of mathematics with lots of real mathematical examples worked through in detail. However, given the way Van Brummelen structures his narrative, it is possible to skip the worked examples and still get a strong impression of the historical evolution of the discipline. This is possible because Van Brummelen gives a threefold description of every topic that he elucidates. First comes a narrative, fairly non-technical, description of the topic he is discussing. This is followed by an English translation of a worked example from the historical text under discussion, followed in turn by a technical explication of the text in question in modern terminology. Van Brummelen’s narrative style is clear and straightforward meaning that the non-expert reader can get good understanding of the points being made, without necessarily wading through the intricacies of the piece of mathematics under discussion. 

The book precedes chronologically. The first chapter, Precursors, starts by defining what trigonometry is and also what it isn’t. Having dealt with the definitions, Van Brummelen moves onto the history proper dealing with things that preceded the invention of trigonometry, which are closely related but are not trigonometry. 

Moving on to Alexandrian Greece, Van Brummelen takes the reader through the beginnings of trigonometry starting with Hipparchus, who produced the first chord table linking angles to chords and arcs of circles, Moving on through Theodosius of Bithynia and Menelaus of Alexandria and the emergence of spherical trigonometry. He then arrives at Ptolemy his astronomy and geography. Ptolemy gets the longest section of the book, which given that everything that follows in some way flows from his work in logical. Here we also get two defining features of the book. The problem of calculating trigonometrical tables and what each astronomer or mathematician contributed to this problem and the trigonometrical formulas that each of them developed to facilitate calculations. 

From Greece we move to India and the halving of Hipparchus’ and Ptolemy’s chords to produce the sine function and later the cosine that we still use today. Van Brummelen takes his reader step for step and mathematician for mathematician through the developments of trigonometry in India. 

The Islamic astronomers took over the baton from the Indians and continued the developments both in astronomy and geography. It was Islamic mathematicians, who developed the plane trigonometry that we know today rather than the spherical trigonometry. As with much other mathematics and science, trigonometry came into medieval Europe through the translation movement out of Arabic into Latin. Van Brummelen traces the development in medieval Europe down to the first Viennese School of mathematics, John of Gmunden, Peuerbach, and Regiomontanus. This volume closes with Johannes Werner and Copernicus, with a promise of a second volume. 

In the book itself, the brief sketch above is filled out in incredible detail covering all aspects of the evolution of the discipline, the problems, the advances, the stumbling stones and the mathematicians and astronomers, who discovered each problem, solved, or failed to solve them. To call Van Brummelen comprehensive would almost be an understatement. Having finished this first volume, I eagerly awaited the promised second volume, but something else came along instead.

Having made clear in his first book that the emphasis is very much on spherical trigonometry rather than plane trigonometry, in his second book Van Brummelen sets out to explain to the modern reader what exactly spherical trigonometry is, as it ceased to be part of the curriculum sometime in the modern period. What we have in Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry is a spherical trigonometry textbook written from a historical perspective. The whole volume is written in a much lighter and more accessible tone than The Mathematics of the Heavens and the Earth. After a preface elucidating the purpose of the book there follow two chapters, Heavenly Mathematics and Exploring the Sphere, which lay out and explain the basics in clear and easy to follow steps.

Next up, we have the historical part of the book with one chapter each on The Ancient Approach and The Medieval Approach. These chapters could be used as an aid to help understand the relevant sections of the authors first book. But fear not the reader must not don his medieval personality to find their way around the complexities of spherical trigonometry because following this historical guide we are led into the modern textbook.

The bulk of the book consists of five chapters, each of which deals in a modern style with an aspect of spherical trigonometry: Right Angle Triangles, Oblique Triangles, Areas, Angles and Polyhedra, Stereographic Projection, and finally Navigation by the Stars. The chapter on stereographic projection is particularly interesting for those involved with astrolabes and/or cartography. 

The book closes with three useful appendices. The first is on Ptolemy’s determination of the position of sun. The second is a bibliography of textbooks on or including spherical trigonometry with the very helpful indication, which of them are available on Google Books. The final appendix is a chapter by chapter annotated list of further reading on each topic. 

If you wish to up your Renaissance astrology game and use the method of directions to determine your date of death, which require spherical trigonometry to convert from one celestial coordinate system to another, then this is definitely the book for you. It is of course also a brilliant introduction for anybody, who wishes to learn the ins and outs of spherical trigonometry. 

I bought Van Brummelen’s first book when it was published, in 2009, and read it with great enthusiasm, but experienced a sort of coitus interruptus, when in stopped in the middle of the Renaissance, the period that interested me most. I was consoled by the author’s declaration that a second volume would follow, which I looked forward to with great expectations. Over the years those expectations dimmed, and I began to fear that the promised second volume would never appear, so I was overjoyed when the publication of The Doctrine of Triangles was announced this year and immediately placed an advanced order. I was not disappointed. 

The modern history of trigonometry continues where the early history left off, tracing the developments of trigonometry in Europe from Regiomontanus down to Clavius and Gunter in the early seventeenth century. There then follows a major change of tack, as Van Brummelen delves into the origins of logarithms.

Today in the age of the computer and the pocket calculator, logarithmic tables are virtually unknown, a forgotten relic of times past. I, however, grew up using my trusty four figure log tables to facilitate calculations in maths, physics, and chemistry. Now, school kids only know logarithms as functions in analysis. One thing that many, who had the pleasure of using log tables, don’t know is that Napier’s first tables were of the logarithms of trigonometrical factions in order to turn the difficult multiplications and divisions of sines, cosines et al in spherical trigonometry into much simpler additions and subtractions and therefore Van Brummelen’s detailed presentation of the topic.

Moving on, in his third chapter, Van Brummelen now turns to the transition of trigonometry as a calculation aid in spherical and plane triangles to trigonometrical functions in calculus. There where they exist in school mathematics today. Starting in the period before Leibniz and Newton, he takes us all the way through to Leonard Euler in the middle of the eighteenth century. 

The book now undergoes a truly major change of tack, as Van Brummelen introduces a comparative study of the history of trigonometry in Chinese mathematics. In this section he deals with the Indian and Islamic introduction of trigonometry into China and its impact. How the Chinese dealt with triangles before they came into contact with trigonometry and then the Jesuit introductions of both trigonometry and logarithms into China and to what extent this influenced Chinese geometry of the triangle. A fascinating study and an enrichment of his already excellent book.

The final section of the book deals with a potpourri of developments in trigonometry in Europe post Euler. To quote Van Brummelen, “A collection of short stories is thus more appropriate here than a continuous narrative.” The second volume of Van Brummelen’s history is just as detailed and comprehensive as the first. 

All three of the books display the same high level of academic rigour and excellence. The two history volumes have copious footnotes, very extensive bibliographies, and equally extensive indexes. The books are all richly illustrated with many first-class explanatory diagrams and greyscale prints of historical title pages and other elements of the books that Van Brummelen describes. All in all, in his three volumes Van Brummelen delivers a pinnacle in the history of mathematics that sets standards for all other historians of the discipline. He really does live up to his claim to be “the best historian of trigonometry” and not just because he’s the only one.

Coda: If the potential reader feels intimidated by the prospect of the eight hundred and sixty plus pages of the three volumes described here, they could find a gentle entry to the topic in Trigonometry: A Very Short Introduction (OUP, 2020), which is also authored by Van Brummelen, a sort of Van Brummelen light or Van Brummelen’s greatest hits.

In this he covers a wide range of trigonometrical topics putting them into their historical context. But beware, reading the Very Short Introduction could well lead to further consumption of Van Brummelen’s excellent work. 


[1] This is not strictly true as Van Brummelen has at least two predecessors both of who he quotes in his works. The German historian Anton von Braunmühl, who wrote several articles and a two volume Vorlesung über Geschichte der Trigonometrie (Leipzig, 1900/1903) and the American Sister Mary Claudia Zeller, The Development of Trigonometry from Regiomontanus to Pitiscus (Ann Arbor 1944)

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Filed under History of Astronomy, History of Cartography, History of Islamic Science, History of Mathematics, History of Navigation

The amateur, astronomical, antiquarian aristocrat from Aix

In a recent blog post about the Minim friar, Marin Mersenne (1588–1648), I mentioned that when Mersenne arrived in Paris in 1619 he was introduced to the intellectual elite of the city by Nicolas-Claude Fabri de Peiresc (1580-1637). In another recent post on the Republic of Letters I also mentioned that Peiresc was probably, the periods most prolific correspondent, with more than ten thousand surviving letters. So, who was this champion letter writer and what role did he play in the European scientific community in the first third of the seventeenth century?

Nicolas-Claude Fabri de Peiresc by Louis Finson Source: Wikimedia Commons

Nicolas-Claude Fabri was born, into a family of lawyers and politicians, in the town Belgentier near Toulon on 1 December in 1580, where his parents had fled to from their hometown of Aix-en-Provence to escape the plagues. He was educated at Aix-en-Provence, Avignon, and the Jesuit College at Tournon. Having completed his schooling, he set off to Padua in Italy, nominally to study law, but he devoted the three years, 1600–1602, to a wide-ranging, encyclopaedic study of the history of the world and everything in it. 

In this he was aided in that he became a protégé of Gian Vincenzo Pinelli (1535–1601) a humanist scholar and book collector, his library numbered about 8,500 printed works, with all-embracing intellectual interests, whose main areas were botany, optics, and mathematical instruments.

Gian Vincènzo Pinelli Source: Rijksmuseum via Wikimedia Commons

Pinelli introduced Fabri to many leading scholars including Marcus Welser (1558–1614), Paolo Sarpi (1552–1623) and indirectly Joseph Scaliger (1540–1609). Pinelli also introduced him to another of his protégés, Galileo Galilei (1564–1642). One should always remember that although he was thirty-eight years old in 1602, Galileo was a virtually unknown professor of mathematics in Padua. When Pinelli died, Fabri was living in his house and became involved in sorting his papers.

In 1602, Fabri returned to Aix-en-Provence and completed his law degree, graduating in 1604. In the same year he assumed the name Peiresc, it came from a domain in the Alpes-de-Haute-Provence, which he had inherited from his father. He never actually visited Peiresc, now spelt Peyresq.

Village of Peyresq Source: Wikimedia Commons

Following graduation Peiresc travelled to the Netherlands and England via Paris, where he made the acquaintance of other notable scholars, including actually meeting Scaliger and also meeting the English antiquarian and historian William Camden (1551–1623).

Returning to Provence, in 1607, he took over his uncle’s position as conseiller to the Parliament of Provence under his patron Guillaume du Vair (1556–1621), cleric, lawyer, humanist scholar and president of the parliament.

Guillaume-du-Vair Source: Wikimedia Commons

In 1615 he returned to Paris with du Vair as his secretary, as du Vair was appointed keeper of the seals during the regency of Marie de’ Medici (1575–1642). Peiresc continued to make new contacts with leading figures from the world of scholarship, and the arts, including Peter Paul Rubens (1577–1640).

Peter Paul Rubens self-portrait 1623

Peiresc acted as a go between in the negotiations between Reubens and the French court in the commissioning of his Marie de’ Medici Cycle. Just one of Peiresc’s many acts of patronage in the fine arts.

Marie de’ Medici Cycle in the Richelieu wing of the Louvre Source: Wikimedia Commons

In 1621 de Vair died and in 1623 Peiresc returned to Provence, where he continued to serve in the parliament until his death in 1637.

Peiresc was an active scholar and patron over a wide range of intellectual activities, corresponding with a vast spectrum of Europe’s intellectual elite, but we are interested here in his activities as an astronomer. Having developed an interest for astronomical instruments during his time as Pinelli’s protégé, Peiresc’s astronomical activities were sparked by news of Galileo’s telescopic discoveries, which reached him before he got a chance to read the Sidereus Nuncius. He rectified this lack of direct knowledge by ordering a copy from Venice and borrowing one from a friend until his own copy arrived.

Source: Wikimedia Commons

He immediately began trying to construct a telescope to confirm or refute Galileo’s claims, in particular the discovery of the first four moons of Jupiter. At this point in his life Peiresc was still a geocentrist, later he became a convinced heliocentrist. We know very little about where and how he acquired his lenses, but we do know that he had various failures before he finally succeeded in observing the moons of Jupiter for himself, in November 1610. In this he was beaten to the punch by his fellow Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix. At this point it is not clear whether the two were competing or cooperating, as they would then later do with Gaultier de la Valette becoming a member of Peiresc’s Provencal astronomical observation group. Shortly thereafter, Peiresc became the first astronomer to make telescopic observations of the Orion Nebular and Gaultier de la Valette the second. This is rather strange as the Orion Nebular is visible to the naked eye. However, apparently none of the telescopic astronomy pioneers had turned their telescopes to it before Peiresc.

In one of the most detailed astronomical images ever produced, NASA/ESA’s Hubble Space Telescope captured an unprecedented look at the Orion Nebula. … This extensive study took 105 Hubble orbits to complete. All imaging instruments aboard the telescope were used simultaneously to study Orion. Source: Wikimedia Commons

Peiresc, like Galileo, realised that the moons of Jupiter could be used as a clock to determine longitude and began an observation programme of the moons, viewing them every single day that the weather conditions permitted, well into 1612. Having compiled tables of his observations he sent one of his own protégés Jean Lombard, about whom little is known, equipped with suitable instruments on a tour of the Mediterranean. Lombard observed the satellites at Marseille in November 1611 and then proceeded to Malta, Cyprus and to Tripoli observing as he went, until May 1612. Meanwhile, Peiresc made parallel observation in Aix and Paris, he hoped by comparing the time differences in the two sets of observations to be able to accurately determine the longitude differences. Unfortunately, the observations proved to be not accurate enough for the purpose and the world would have to wait for Giovanni Domenico Cassini (1625–1712) to become the first to successfully utilise this method of determining longitude. Peiresc’s own observation were, however, the longest continuous series of observations of the Jupiter moons made in the seventeenth century and displayed a high level of accuracy even when compared with this of Galileo.

I mentioned, above, Peiresc’s Provencal astronomical observing group. Peiresc employed/sponsored young astronomers to help him with his observation programmes, supplying them with instruments and instructions on how to use them. This group included such notable, future astronomers, as Jean-Baptiste Morin (1583–1556),

Jean-Baptiste Morin Source: Wikimedia Commons

Ismaël Boulliau (1605–1694),

Ismaël Boulliau Source: Wikimedia Commons

and Pierre Gassendi (1592–1655). Peiresc’s patronage extended well beyond this. Gassendi had held the chair of philosophy at the University of Aix-en-Provence since 1617 but in 1623 the university was taken over by the Jesuits and Gassendi was replaced by a Jesuit and became unemployed.

Portrait of Pierre Gassendi by Louis-Édouard Rioult Source: Wikimedia Commons

From then until he again found regular employment in 1634, Peiresc provided him with a home base in his own house and financed his travels and research. Similarly, Peiresc, having introduced Mersenne to Parisian intellectual circles in 1619, continued to support him financially, Mersenne as a Minim friar had no income, supplying him with instruments and financing his publications. 

Marin Mersenne Source: Wikimedia Commons

Patronage played a central role in Peiresc’s next venture into astronomy and another attempt to solve the longitude problem. There has been much talk in recent decades about so-called citizen science, in which members of the public are invited to participate in widespread scientific activities. Annual counts of the birds in one’s garden is a simple example of this. Citizen science is mostly presented as a modern phenomenon, but there are examples from the nineteenth century. Peiresc had already launched a variation on citizen science in the seventeenth century.

In order to determine longitude Peiresc further developed a method that had been in use since antiquity. Two astronomers situated in different location would observe a lunar or solar eclipse and then by comparing their observations they could determine the local time difference between their observations and thus the longitude difference between the locations. By the seventeenth century predicting eclipses had become a fairly accurate science and Peiresc thought that if he could organise and coordinated a world spanning network of observers to accurately observe and record an eclipse, he could then calculate a world spanning network of longitude measurements. The idea was good in theory but failed in practice.

Most of Peiresc’s team of observers were amateurs–missionaries, diplomats, traders, travellers–whom he supplied with astronomical instruments and written instructions on how to use them, even paying travelling expenses, where necessary. Peiresc organised mass observations for lunar eclipses in 1628, 1634, and 1635 and a solar eclipse in 1633. Unfortunately, many of his observers proved to be incompetent and the results of their observations were too inaccurate to be usable. One positive result was that Peiresc was able to correct the value for the length of the Mediterranean. Before one is too hard on Peiresc’s amateur observers, one should remember that in the middle of the eighteenth century the world’s professional astronomical community basically failed in their attempt to use the transits of Venus to determine the astronomical unit, despite being equipped with much better instruments and telescopes.

Although, Peiresc’s various astronomical activities and their results were known throughout Europe by word of mouth through his various colleagues and his correspondence, he never published any of his work. Quite why, is not really known although there are speculations.

Peiresc was a high ranking and highly influential Catholic and he applied that influence in attempts to change the Church’s treatment of astronomers he saw as being persecuted. He interceded on behalf Tommaso Campanella (1568–1639), actively supporting him when he fled to France in 1634.

Tommaso Campanella portrait by Francesco Cozza Source: Wikimedia Commons

More famously he personally interceded with the Church on behalf of Galileo, without any great success.

Nicolas-Claude Fabri de Peiresc’s career is, like that of his friend Mersenne, a good illustration that the evolution of science is a product of widespread cooperation of a community of practitioners and not the result of the genial discoveries of a handful of big names, as it is unfortunately too often presented. Morin, Boulliau, Gassendi and Mersenne, who all made serious contributions to the evolution of science in the seventeenth century, did so with the encouragement, guidance, and very active support of Peiresc.

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Filed under History of Astronomy, History of Navigation, History of science, Renaissance Science, The Paris Provencal Connection

The alchemist, who became a cosmographer

As an Englishman brought up on tales, myths and legends of Francis Drake, Walter Raleigh, Admiral Lord Nelson, the invincible Royal Navy and Britannia rules the waves, I tend not to think about the fact that Britain was not always a great seafaring nation. As an island there were, of course, always fisher boats going about their business in the coastal waters and archaeology has shown us that people have been crossing the strip of water between Britain and the continent, as long as the island has been populated. However, British sailors only really began to set out onto the oceans for distant lands in competition to their Iberian brethren during the Early Modern Period. Before the start of these maritime endeavours there was a political movement in England to get those in power to take up the challenge and compete with the Spanish and the Portuguese in acquiring foreign colonies, gold, silver and exotic spices. One, today virtually unknown, man, whose writings played a not insignificant role in this political movement was the alchemist Ricard Eden[1] (c. 1520–1576).

Richard Eden[2] was born into an East Anglian family of cloth merchants and clerics, the son of George Eden a cloth merchant. He studied at Christ’s College Cambridge (1534–1537) and then Queen’s College, where he graduated BA in 1538 and MA in 1544. He studied under Sir Thomas Smith (1533–1577) a leading classicist of the period, who was also politically active and a major supporter of colonialism, which possibly influenced Eden’s own later involvement in the topic.

Sir_Thomas_Smith,_ob._1577_(c._early_19th_century)

A c. 19th-century line engraving of Sir Thomas Smith. Source: Wikimedia Commons

Through Smith, Eden was introduced to John Cheke (1514–1557), Roger Ascham (c. 1515–1568) and William Cecil (1520–1598), all of whom were excellent classicists and statesmen. Cecil would go on under Elizabeth I to become the most powerful man in England. From the beginning Eden moved in the highest intellectual and political circles.

After leaving Cambridge Eden was appointed first to a position in the Treasury and then distiller of waters to the royal household, already indicating an interest in and a level of skill in alchemy. Eden probably acquired his interest in alchemy from his influential Cambridge friends, who were all eager advocates of the art. However, he lost the post, probably given to someone else by Somerset following Henry VIII’s death in 1547 and so was searching for a new employer or patron.

Through a chance meeting he became acquainted with the rich landowner Richard Whalley, who shared his interest in alchemy. Whalley provided him with a house for his family and an income, so that he could devote himself to both medicinal and transmutational alchemy. His activities as an alchemist are not of interest here but one aspect of his work for Whalley is relevant, as it marked the beginning of his career as a translator.

Whalley was obviously also interested in mining for metal ores, because he commissioned Eden to translate the whole of Biringuccio’s Pirotechnia into English. Although he denied processing any knowledge of metal ores, Eden accepted the commission and by 1552 he had completed twenty-two chapters, that is to the end of Book 2. Unfortunately, he lent the manuscript to somebody, who failed to return it and so the project was never finished. In fact, there was no English translation of the Pirotechnia before the twentieth century. Later he produced a new faithful translation of the first three chapters dealing with gold, silver and copper ores, only omitting Biringuccio’s attacks on alchemy, for inclusion, as we shall see, in one of his later works.

800px-de_la_pirotechnia_1540_title_page_aq1_1

Title page, De la pirotechnia, 1540, Source: Science History Museum via Wikipedia Commons

In 1552, Eden fell out with Whalley and became a secretary to William Cecil. It is probable the Cecil employed him, as part of his scheme to launch a British challenge to the Iberian dominance in global trade. In his new position Eden now produced a translation of part of Book 5 of Sebastian Münster’s Cosmographia under the title A Treatyse of the New India in 1553. As I explained in an earlier blog post Münster’s Cosmographia was highly influential and one of the biggest selling books of the sixteenth century.

treatyseofnewein00mnst

This first cosmographical publication was followed in 1555 by his The Decades of the newe worlde or west India, containing the nauigations and conquests of the Spanyardes… This was a compendium of various translations including those three chapters of Biringuccio, probably figuring that most explorers of the Americas were there to find precious metals. The main parts of this compendium were taken from Pietro Martire d’Anghiera’s De orbe novo decades and Gonzalo Fernández de Oviedo y Valdés’ Natural hystoria de las Indias.

edens-decades-of-the-c_13_a_8_title_page

Source: The British Library

Pietro Martire d’Anghiera (1457–1526) was an Italian historian in the service of Spain, who wrote the first accounts of the explorations of Central and South America in a series of letters and reports, which were published together in Latin. His De orbe novo (1530) describes the first contacts between Europeans and Native Americans.

Deorbonovo_(1)

Source: Wikimedia Commons

Gonzalo Fernández de Oviedo y Valdés (1478–1557) was a Spanish colonist, who arrived in the West Indies a few years after Columbus. His Natural hystoria de las Indias (1526) was the first text to introduce Europeans to the hammock, the pineapple and tobacco.

Oviedo_MS_pineapple

MS page from Oviedo’s La Natural hystoria de las Indias. Written before 1535, this MS page is the earliest known representation of a pineapple Source: Wikimedia Commons

Important as these writings were as propaganda to further an English involvement in the new exploration movement in competition to the Iberian explorers, it was probably Eden’s next translation that was the most important.

As Margaret Schotte has excellently documented in her Sailing School (Johns Hopkins University Press, 2019) this new age of deep-sea exploration and discovery led the authorities in Spain and Portugal to the realisation that an active education and training of navigators was necessary. In 1552 the Spanish Casa de la Contratación established a formal school of navigation with a cátedra de cosmografia (chair of cosmography). This move to a formal instruction in navigation, of course, needed textbooks, which had not existed before. Martín Cortés de Albacar (1510–1582), who had been teaching navigation in Cádiz since 1530, published his Breve compendio de la sphere y de la arte de navegar in Seville in 1551.

2880px-Retrato_de_Martín_Cortes._De_Bujaraloz

Retrato de Martín Cortés, ilustración del Breve compendio de la sphera y de la arte de navegar, Sevilla, 1556. Biblioteca Nacional de España via Wikimedia Commons

In 1558, an English sea captain from Dover, Stephen Borough (1525–1584), who was an early Artic explorer, visited Seville and was admitted to the Casa de la Contratación as an honoured guest, where he learnt all about the latest instruments and the instruction for on going navigators. On his return to England, he took with him a copy of Cortés’ Breve compendio, which he had translated into English by Richard Eden, as The Arte of Navigation in 1561. This was the first English manual of navigation and was immensely popular going through at least six editions in the sixteenth century.

RE84TP

In 1562, Eden became a companion to Jean de Ferrières, Vidame of Chartres, a Huguenot aristocrat, who raised a Protestant army in England to fight in the French religious wars. Eden, who was acknowledged as an excellent linguist, stayed with de Ferrières until 1573 travelling extensively throughout France and Germany. Following the St. Batholomew’s Day massacre, which began in the night of 23–24 August 1572, Eden together with de Ferrières party fled from France arriving in England on 7 September 1573. At de Ferrières request, Elizabeth I admitted Eden to the Poor Knights of Windsor, a charitable organisation for retired soldiers, where he remained until his death in 1576.

After his return to England Eden translated the Dutch musician and astrologer, Jean Taisnier’s Opusculum perpetua memoria dignissimum, de natura magnetis et ejus effectibus, Item de motu continuio, which was a plagiarism of Petrus Peregrinus de Maricourt’s (fl. 1269) Epistola de magnete and a treatise on the fall of bodies by Giambattista Benedetti (1530–1590) into English.

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This was published posthumously together with his Arte of Navigation in 1579. His final translation was of Ludovico de Varthema’s (c. 1470–1517) Intinerario a semi-fictional account of his travels in the east. This was published by Richard Willes in The History of Travayle an enlarged version of his Decades of the newe worlde in 1577.

Eden’s translations and publications played a significant role in the intellectual life of England in the sixteenth century and were republished by Richard Hakluyt (1553–1616) in his The Principal Navigations, Voiages, Traffiques and Discoueries of the English Nation (1589, 1598, 1600), another publication intended as propaganda to promote English colonies in America.

001-3-principal-navigations

Unlike Sebastian Münster or Richard Hakluyt, Eden has been largely forgotten but he made important and significant contributions to the history of cosmography and deserves to be better known.

[1] I want to thank Jenny Rampling, whose book The Experimental Fire, which I reviewed here, made me aware of Richard Eden, although, I have to admit, he turns up, managing to slip by unnoticed in other books that I own and have read.

[2] The biographical details on Eden are mostly taken from the ODNB article. I would like to thank the three wonderful people, who provided me with a pdf of this article literally within seconds of me asking on Twitter

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Filed under Early Scientific Publishing, History of Cartography, History of Navigation

Renaissance Science – V

According to the title, this series is supposed to be about Renaissance science but as we saw in the last episode the Renaissance started off as anything but scientific, so what exactly is Renaissance science, does it even exist, and does it actually have anything to do with the language and linguistics movement that kicked of the period that is now known as the Renaissance? I will start with the second of these questions and return later to the other two.

The history of science in its present form is actually a very young discipline, which really only came to fruition in the twentieth century. There are of course early elements of the discipline scattered around the past but the structured academic discipline as we know it only really began in the decades between the two world wars and came to maturity following the second world war. The early discipline was of course very euro-centric, and a major element was the so-called scientific revolution, which was initially seen as a single historical block. Maria Boas Hall (1919–2009) was, as far as I know, the first to divide that block into two parts, a sort of proto scientific revolution, her The Scientific Renaissance 1450–1630 (published, 1962), followed by the full scientific revolution. She was followed in this bifurcation by Peter Dear in his book Revolutionizing the Sciences: European Knowledge in Transition 1500–1700 (originally 2001, 3rd ed. 2019), who sees two phases, 1500-1600 and 1600-1700. These two books established, I think correctly, the idea of a separate Scientific Renaissance, which preceded the Scientific Revolution.

So, what is the nature of this Renaissance science, how did it differ from the existing medieval science and what changed and when going forward into the so-called scientific revolution? There is quite a lot to unpack here and the first thing we need to do is to stop talking about science and instead talk about knowledge, the more correct translation of the Latin term, scientia used in this period. Also, within the scope of scientia, what we might regard as the areas of hard science, which Aristotle called physics, meaning the study of nature, should more appropriately be referred to as natural philosophy. However, medieval natural philosophy was a very restricted area, it included cosmology but did not for example include astronomy, which was a mathematical discipline. Aristotle rejected mathematics as scientia, because its objects were not real. The mathematical disciplines, such as astronomy and optics, were not regarded as belonging to natural philosophy but were given a sort of halfway status. Natural philosophy also didn’t include any of what we would now call the life sciences.

Knowledge in the European medieval context was divided into two completely distinct areas, which didn’t intersect in anyway. On the one side there was the knowledge propagated by the medieval universities, which, as I explained in an earlier post, was almost totally theoretical book knowledge, with almost no practical aspects to it at all. This knowledge was not static, as it is often falsely presented, but evolved over time. However, this evolution was also a theoretical process. The knowledge progressed through debate and the application of argumentation and logic, not through the acquisition of new empirical facts.

The other area of knowledge was artisanal knowledge, that is the knowledge of the maker, the craftsman. This knowledge was empirical and practical, consisting of directions or instruction on how to complete a given task, how to achieve a given aim or fulfil a given assignment. It might, for example, be how to make bricks out of clay, or how to build a stone arch that would be stable and not collapse under load. This knowledge covered a vast range of activities and had been accumulated from a very wide range of sources over virtually the whole of human existence. This knowledge was, traditional, rarely written down but was usually passed on by word of mouth and direct training from master to apprentice, often from father to son over many generations. This knowledge was in general not viewed as knowledge by scholars within the university system.

Starting around fourteen hundred a process of what we would today call crossover began between these two previously distinct and separate areas of knowledge. Scholars began to write learned works about specific areas of artisanal knowledge, a classic example being Georgius Agricola’s De re metallica, published posthumously in 1556, and craftsmen began to write books explaining and elucidating their forms of knowledge, for example the goldsmith Lorenzo Ghiberti’s I commentarii, which remained unfinished in manuscript and unpublished at the time of his death in 1455. It should be noted that before the Renaissance the people we now call artists were regarded as craftsmen. Crossover is here perhaps the wrong term, as people didn’t just cross the boundary in both directions but the boundary itself began to dissolve producing 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.

What provoked this move towards practical, empirical knowledge during the Renaissance? There are two major areas of development driving this shift in emphasis, as to what constitutes knowledge. The first is general social, political, economical and cultural developments. The rapid increase in long distant trade produced a demand for new methods of navigation and cartography. Changes in concepts of land ownership also drove developments in cartography and the closely associated surveying. Developments in warfare again drove developments in cartography but also in gunnery, a new discipline, and military tactics in general. The invention of gunpowder and with-it military gunnery drove developments in metallurgy, as did other areas where the use of metals increased, for example in the wider use of metal coinage. The greater demand for metals in turn drove the development of mining. Greater wealth in society in general and the perceived need for rulers to display their power through ostentatious display increased the demand for architecture and fine art. The introduction of gunpowder and gunnery also drove the development of architecture because of the need for better defences. These are just some examples of the growing demand for artisanal knowledge within an increasingly urban culture financed by long distance trade.

But what of the movement that gave the Renaissance its name, which we saw was initially language and linguistic based movement, how did this play a role in this move towards the elevation of the status of empirical and practical knowledge if at all? This is in fact our second area of development. Those early Renaissance scholars, who searched for Latin literature texts and orations in the monastic libraries also unearthed Greek and Latin texts on science, technology, mathematics and medicine and in the general renewal of the culture of antiquity also translated and made these texts available, often arguing for their purity in comparison to the texts from the same authors that had come into Europe through the filter of translation into Arabic and then back into Latin. Example of texts that became available for the first time are Vitruvius’ work on architecture De architectura and Ptolemaeus’ Geographia. The latter had been known to the Islamic cartographers but had not been translated into Latin from Arabic during the twelfth century translation movement. As well as bringing new original Greek and Latin manuscripts into circulation the Renaissance scholars introduced a strong empirical element through their philological work. This work was based on an empirical analysis of various copies of a given work as well as an investigation of the plausibility of a given word, phrase or sentence, which didn’t appear to make sense. Beyond this in some areas the Renaissance scholars, as we shall see in more detail later, began to try and understand what the scholars were referring to in specific instances. For example, which plants was Dioscorides referring to in his De meteria medica? The answer to such questions required real empirical research.

The Renaissance opened up a whole new world of practical, empirical knowledge alongside the theoretical book knowledge of the medieval university. The last question is how did this differ from the knowledge of the following period and when did this transition take place?

The emphasis on this Renaissance empirical knowledge was very much on the practical. How can we use it, where and how can it be applied? During the seventeenth century the emphasis changed to one of devising theoretical explanations for all of the freshly won empirical knowledge from the previous two hundred years. The transition is from how do we use or apply it, to how do we explain it. It is impossible to set a firm date for this transition as it was by its very nature a gradual one, so both Boas Hall and Dear are in a certain sense correct with their respective 1630 and 1600. The transition had definitely already begun by 1600 and probably wasn’t finished, yet by 1630. In my case I follow Francis Yates in choosing the end of the Thirty Year’s War in 1648, as I think the transition had been completed by then at the latest.

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Filed under History of Cartography, History of Navigation, History of science, Renaissance Science

Illuminating medieval science

 

There is a widespread popular vision of the Middle ages, as some sort of black hole of filth, disease, ignorance, brutality, witchcraft and blind devotion to religion. This fairly-tale version of history is actively propagated by authors of popular medieval novels, the film industry and television, it sells well. Within this fantasy the term medieval science is simply an oxymoron, a contradiction in itself, how could there possible be science in a culture of illiterate, dung smeared peasants, fanatical prelates waiting for the apocalypse and haggard, devil worshipping crones muttering curses to their black cats?

Whilst the picture I have just drawn is a deliberate caricature this negative view of the Middle Ages and medieval science is unfortunately not confined to the entertainment industry. We have the following quote from Israeli historian Yuval Harari from his bestselling Sapiens: A Brief History of Humankind (2014), which I demolished in an earlier post.

In 1500, few cities had more than 100,000 inhabitants. Most buildings were constructed of mud, wood and straw; a three-story building was a skyscraper. The streets were rutted dirt tracks, dusty in summer and muddy in winter, plied by pedestrians, horses, goats, chickens and a few carts. The most common urban noises were human and animal voices, along with the occasional hammer and saw. At sunset, the cityscape went black, with only an occasional candle or torch flickering in the gloom.

On medieval science we have the even more ignorant point of view from American polymath and TV star Carl Sagan from his mega selling television series Cosmos, who to quote the Cambridge History of Medieval Science:

In his 1980 book by the same name, a timeline of astronomy from Greek antiquity to the present left between the fifth and the late fifteenth centuries a familiar thousand-year blank labelled as a “poignant lost opportunity for mankind.” 

Of course, the very existence of the Cambridge History of Medieval Science puts a lie to Sagan’s poignant lost opportunity, as do a whole library full of monographs and articles by such eminent historians of science as Edward Grant, John Murdoch, Michael Shank, David Lindberg, Alistair Crombie and many others.

However, these historians write mainly for academics and not for the general public, what is needed is books on medieval science written specifically for the educated layman; there are already a few such books on the market, and they have now been joined by Seb Falk’s truly excellent The Light Ages: The Surprising Story of Medieval Science.[1]  

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How does one go about writing a semi-popular history of medieval science? Falk does so by telling the life story of John of Westwyk an obscure fourteenth century Benedictine monk from Hertfordshire, who was an astronomer and instrument maker. However, John of Westwyk really is obscure and we have very few details of his life, so how does Falk tell his life story. The clue, and this is Falk’s masterstroke, is context. We get an elaborate, detailed account of the context and circumstances of John’s life and thereby a very broad introduction to all aspects of fourteenth century European life and its science.

We follow John from the agricultural village of Westwyk to the Abbey of St Albans, where he spent the early part of his life as a monk. We accompany some of his fellow monks to study at the University of Oxford, whether John studied with them is not known.

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Gloucester College was the Benedictine College at Oxford where the monks of St Albans studied

We trudge all the way up to Tynemouth on the wild North Sea coast of Northumbria, the site of daughter cell of the great St Alban’s Abbey, main seat of Benedictines in England. We follow John when he takes up the cross and goes on a crusade. Throughout all of his wanderings we meet up with the science of the period, John himself was an astronomer and instrument maker.

Falk is a great narrator and his descriptive passages, whilst historically accurate and correct,[2] read like a well written novel pulling the reader along through the world of the fourteenth century. However, Falk is also a teacher and when he introduces a new scientific instrument or set of astronomical tables, he doesn’t just simply describe them, he teachers the reader in detail how to construct, read, use them. His great skill is just at the point when you think your brain is going to bail out, through mathematical overload, he changes back to a wonderfully lyrical description of a landscape or a building. The balance between the two aspects of the book is as near perfect as possible. It entertains, informs and educates in equal measures on a very high level.

Along the way we learn about medieval astronomy, astrology, mathematics, medicine, cartography, time keeping, instrument making and more. The book is particularly rich on the time keeping and the instruments, as the Abbott of St Albans during John’s time was Richard of Wallingford one of England’s great medieval scientists, who was responsible for the design and construction of one of the greatest medieval church clocks and with his Albion (the all in one) one of the most sophisticated astronomical instruments of all time. Falk’ introduction to and description of both in first class.

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The book is elegantly present with an attractive typeface and is well illustrated with grey in grey prints and a selection of colour ones. There are extensive, informative endnotes and a good index. If somebody reads this book as an introduction to medieval science there is a strong chance that their next question will be, what do I read next. Falk gives a detailed answer to this question. There is an extensive section at the end of the book entitled Further Reading, which gives a section by section detailed annotated reading list for each aspect of the book.

Seb Falk has written a brilliant introduction to the history of medieval science. This book is an instant classic and future generations of schoolkids, students and interested laypeople when talking about medieval science will simply refer to the Falk as a standard introduction to the topic. If you are interested in the history of medieval science or the history of science in general, acquire a copy of Seb Falk’s masterpiece, I guarantee you won’t regret it.

[1] American edition: Seb Falk, The Light Ages: The Surprising Story of Medieval Science, W. W. Norton & Co., New York % London, 2020

British Edition: Seb Falk, The Light Ages: A Medieval Journey of Discover, Allen Lane, London, 2020

[2] Disclosure: I had the pleasure and privilege of reading the whole first draft of the book in manuscript to check it for errors, that is historical errors not grammatical or orthographical ones, although I did point those out when I stumbled over them.

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Filed under History of Astrology, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Mediaeval Science, Myths of Science

A scientific Dutchman

For many decades the popular narrative version of the scientific revolution started in Poland/Germany with Copernicus moving on through Tycho in Denmark, Kepler in Germany/Austria, Galileo et al in Northern Italy, Descartes, Pascal, Mersenne etc., in France and then Newton and his supporters and opponents in London. The Netherlands simply didn’t get a look in except for Christiaan Huygens, who was treated as a sort of honorary Frenchman. As I’ve tried to show over the years the Netherlands and its scholars–Gemma Frisius, Simon Stephen, Isaac Beeckman, the Snels, and the cartographers–actually played a central role in the evolution of the sciences during the Early Modern Period. In more recent years efforts have been made to increase the historical coverage of the contributions made in the Netherlands, a prominent example being Harold J Cook’s Matters of Exchange: Commerce, Medicine and Science in the Dutch Golden Age.[1]

A very strange anomaly in the #histSTM coverage concerns Christiaan Huygens, who without doubt belongs to the seventeenth century scientific elite. Whereas my bookcase has an entire row of Newton biographies, and another row of Galileo biographies and in both cases there are others that I’ve read but don’t own. The Kepler collection is somewhat smaller but it is still a collection. I have no idea how many Descartes biographies exist but it is quite a large number. But for Christiaan Huygens there is almost nothing available in English. The only biography I’m aware of is the English translation of Cornelis Dirk Andriesse’s scientific biography of Christiaan Huygens, The Man Behind the Principle.[2] I read this several years ago and must admit I found it somewhat lacking. This being the case, great expectation have been raised by the announcement of a new Huygens biography by Hugh Aldersey-Williams, Dutch Light: Christiaan Huygens and the Making of Science in Europe.[3]

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So does Aldersey-Williams fulfil those expectations? Does he deliver the goods? Yes and no, on the whole he has researched and written what is mostly an excellent biography of the Netherland’s greatest scientist[4] of the Early Modern Period but it is in my opinion marred by sloppy history of science fact checking that probably won’t be noticed by the average reader but being the notorious #histSTM pedant that I am I simply can’t and won’t ignore.[5]

My regular readers will known that I describe myself as a narrative contextual historian of science and I personally believe that if we are to understand how science has evolved historical then we have to tell that story with its complete context. This being the case I’m very happy to report that Aldersey-Williams is very much a narrative contextual historian, who tells the complete story of Christiaan Huygens life within its wider context and not just offering up a list of his scientific achievements. In fact what the reader gets for his money is not just a biography of Christiaan but also a biography of his entire family with some members being given more space than other. In particular it is a full biography of Christiaan and his father Constantijn, who played a significant and central role in shaping Christiaan’s life.

The book opens by setting the scientific scene in the early seventeenth-century Netherlands. We get introduced to those scientists, who laid the scientific foundations on which Christiaan would later build. In particular we get introduced to Simon Steven, who shaped the very practice orientated science and technology of the Early Modern Netherlands. We also meet other important and influential figures such as Hans Lipperhey, Isaac Beeckman, Willebrord Snel, Cornelius Drebbel and others.

There now follows what might be termed a book within a book as Aldersey-Williams delivers up a very comprehensive biography of Constantijn Huygens diplomat, poet, composer, art lover and patron and all round lover of knowledge. Constantijn was interested in and fascinated by almost everything both scientific and technological. His interest was never superficial but was both theoretical and practical. For example he was not only interested in the newly invented instruments, the telescope and the microscope, but he also took instruction in how to grind lenses and that from the best in the business. Likewise his love for art extended beyond buying paintings and patronising artists, such as Rembrandt, but to developing his own skills in drawing and painting. Here Aldersey-Williams introduces us to the Dutch term ‘kenner’ (which is the same in German), which refers to someone such Constantijn Huygens, whose knowledge of a subject is both theoretical and practical. Constantijn Huygens married Suzanna von Baerle for love and they had five children over ten years, four sons and a daughter, Christiaan was the second oldest, and Suzanna died giving birth to their daughter, also named Suzanna.

Constantijn Huygens brought up his children himself educating them in his own polymathic diversity with the help of tutors. When older the boys spent brief periods at various universities but were largely home educated. We now follow the young Christiaan and his older brother, also Constantijn, through their formative young years. The two oldest boys remained close and much of Christiaan’s astronomical work was carried out in tandem with his older brother. We follow Christiaan’s early mathematical work and his introduction into the intellectual circles of Europe, especially France and England, through his father’s widespread network of acquaintances. From the beginning Christiaan was set up to become either a diplomat, like his father, grandfather and brothers, or a scientist and it is the latter course that he followed.

Aldersey-Williams devotes an entire chapter to Christiaan’s telescopic observations of Saturn, with a telescope that he and Constantijn the younger constructed and his reputation making discovery of Titan the largest of Saturn’s moons, and the first discovered, and his determination that the strange shapes first observed by Galileo around Saturn were in fact rings. These astronomical discoveries established him as one of Europe’s leading astronomers. The following chapter deals with Huygens’ invention of the pendulum clock and his excursions into the then comparatively new probability theory.

Saturn and the pendulum clock established the still comparatively young Huygens as a leading light in European science in the second half of the seventeenth century and Aldersey-Williams now takes us through ups and downs of the rest of Christiaan’s life. His contact with and election to the Royal Society in London, as its first foreign member. His appointment by Jean-Baptist Colbert, the French First Minister of State, as a founding member of the Académie des sciences with a fairy generous royal pension from Louis XIV. His sixteen years in Paris, until the death of Colbert, during which he was generally acknowledged as Europe’s leading natural philosopher. His initial dispute over light with the young and comparatively unknown Newton and his tutorship of the equally young and unknown Leibniz. His fall from grace following Colbert’s death and his reluctant return to the Netherlands. The last lonely decade of his life in the Netherlands and his desire for a return to the scientific bustle of London or Paris. His partial rapprochement with Newton following the publication of the Principia. Closing with the posthumous publication of his works on gravity and optics. This narrative is interwoven with episodes from the lives of Constantijn the father and Constantijn his elder brother, in particular the convoluted politics of the Netherlands and England created by William of Orange, whose secretary was Constantijn, the younger, taking the English throne together with his wife Mary Stewart. Christiaan’s other siblings also make occasional appearances in letters and in person.

Aldersey-Williams has written a monumental biography of two generations of the Huygens family, who played major roles in the culture, politics and science of seventeenth century Europe. With a light, excellent narrative style the book is a pleasure to read. It is illustrated with 37 small grey in grey prints and 35 colour plates, which I can’t comment on, as my review proof copy doesn’t contain them. There are informative footnotes scattered through out the text and the, by me hated, hanging endnotes referring to the sources of direct quotes in the text. Here I had the experience more than once of looking up what I took to be a direct quote only to discover that it was not listed. There is an extensive bibliography of both primary and secondary sources and I assume an extensive index given the number of blank pages in my proof copy. There were several times when I was reading when I had wished that the index were actually there.

On the whole I would be tempted to give this book a glowing recommendation were it not for a series of specific history of science errors that simple shouldn’t be there and some general tendencies that I will now detail.

Near the beginning Aldersey-Williams tells us that ‘Stevin’s recommendation to use decimals in arithmetical calculations in place of vulgar fractions which could have any denominator [was] surely the sand-yacht of accountancy … Thirty years later, the Scottish mathematician John Napier streamlined Stevin’s notation by introducing the familiar comma or point to separate off the fractional part…” As is all too often the case no mention is made of the fact that Chinese and Arabic mathematicians had been using decimal fractions literally centuries before Stevin came up with the concept. In my opinion we must get away from this Eurocentric presentation of the history of science. Also the Jesuit mathematician Christoph Clavius introduced the decimal point less than ten years after Stevin’s introduction of decimal fractions, well ahead of Napier, as was its use by Pitiscus in 1608, the probable source of Napier’s use.

We also get told when discussing the Dutch vocabulary that Stevin created for science that, “Chemistry becomes scheikunde, the art of separation, an acknowledgement of the beginnings of a shift towards an analytical science, and a useful alternative to chemie that severs the etymological connections with disreputable alchemy.” This displays a complete lack of knowledge of alchemy in which virtually all the analytical methods used in chemistry were developed. The art of separation is a perfectly good term from the alchemy that existed when Stevin was creating his Dutch scientific vocabulary. Throughout his book Aldersey-Williams makes disparaging remarks about both alchemy and astrology, neither of which was practiced by any of the Huygens family, which make very clear that he doesn’t actually know very much about either discipline or the role that they played in the evolution of western science, astrology right down to the time of Huygens and Newton and alchemy well into the eighteenth century. For example, the phlogiston theory one of the most productive chemical theories in the eighteenth century had deep roots in alchemy.

Aldersey-Williams account of the origins of the telescope is a bit mangled but acceptable except for the following: “By the following spring, spyglasses were on sale in Paris, from where one was taken to Galileo in Padua. He tweaked the design, claimed the invention as his own, and made dozens of prototypes, passing on his rejects so that very soon even more people were made aware of this instrument capable of bringing the distant close.”

Firstly Galileo claimed that he devised the principle of the telescope and constructed his own purely on verbal descriptions without having actually seen one but purely on his knowledge of optics. He never claimed the invention as his own and the following sentence is pure rubbish. Galileo and his instrument maker produced rather limited numbers of comparatively high quality telescopes that he then presented as gifts to prominent political and Church figures.

Next up we have Willebrord Snel’s use of triangulation. Aldersey-Williams tells us, “ This was the first practical survey of a significant area of land, and it soon inspired similar exercises in England, Italy and France.” It wasn’t. Mercator had previously surveyed the Duchy of Lorraine and Tycho Brahe his island of Hven before Snel began his surveying in the Netherlands. This is however not the worst, Aldersey-Williams tells us correctly that Snel’s survey stretched from Alkmaar to Bergen-op-Zoom “nearly 150 kilometres to the south along approximately the same meridian.” Then comes some incredible rubbish, “By comparing the apparent height of his survey poles observed at distance with their known height, he was able to estimate the size of the Earth!”

What Snel actually did, was having first accurately determined the length of a stretch of his meridian using triangulation, the purpose of his survey and not cartography, he determined astronomically the latitude of the end points. Having calculated the difference in latitudes it is then a fairly simple exercise to determine the length of one degree of latitude, although for a truly accurate determination one has to adjust for the curvature of the Earth.

Next up with have the obligatory Leonard reference. Why do pop history of science books always have a, usually erroneous, Leonardo reference? Here we are concerned with the camera obscura, Aldersey-Williams writes: “…Leonardo da Vinci gave one of the first accurate descriptions of such a design.” Ibn al-Haytham gave accurate descriptions of the camera obscura and its use as a scientific instrument about four hundred and fifty years before Leonardo was born in a book that was translated into Latin two hundred and fifty years before Leonardo’s birth. Add to this the fact that Leonardo’s description of the camera obscura was first published late in the eighteenth century and mentioning Leonardo in this context becomes a historical irrelevance. The first published European illustration of a camera obscura was Gemma Frisius in 1545.

When discussing Descartes, a friend of Constantijn senior and that principle natural philosophical influence on Christiaan we get a classic history of mathematics failure. Aldersey-Williams tells us, “His best known innovation, of what are now called Cartesian coordinates…” Whilst Descartes did indeed cofound, with Pierre Fermat, modern algebraic analytical geometry, Cartesian coordinates were first introduced by Frans van Schooten junior, who of course features strongly in the book as Christiaan’s mathematics teacher.

Along the same lines as the inaccurate camera obscura information we have the following gem, “When applied to a bisected circle (a special case of the ellipse), this yielded a new value, accurate to nine decimal places, of the mathematical constant π, which had not been improved since Archimedes” [my emphasis] There is a whole history of the improvements in the calculation of π between Archimedes and Huygens but there is one specific example that is, within the context of this book, extremely embarrassing.

Early on when dealing with Simon Stevin, Aldersey-Williams mentions that Stevin set up a school for engineering, at the request of Maurits of Nassau, at the University of Leiden in 1600. The first professor of mathematics at this institution was Ludolph van Ceulen (1540–1610), who also taught fencing, a fact that I find fascinating. Ludolph van Ceulen is famous in the history of mathematics for the fact that his greatest mathematical achievement, the Ludophine number, is inscribed on his tombstone, the accurate calculation of π to thirty-five decimal places, 3.14159265358979323846264338327950288…

Next up we have Christiaan’s correction of Descartes laws of collision. Here Aldersey-Williams writes something that is totally baffling, “The work [his new theory of collision] only appeared in a paper in the French Journal des Sçavans in 1669, a few years after Newton’s laws of motion [my emphasis]…” Newton’s laws of motion were first published in his Principia in 1687!

Having had the obligatory Leonardo reference we now have the obligatory erroneous Galileo mathematics and the laws of nature reference, “Galileo was the first to fully understand that mathematics could be used to describe certain laws of nature…” I’ve written so much on this that I’ll just say here, no he wasn’t! You can read about Robert Grosseteste’s statement of the role of mathematics in laws of nature already in the thirteenth century, here.

Writing about Christiaan’s solution of the puzzle of Saturn’s rings, Aldersey-Williams say, “Many theories had been advanced in the few years since telescopes had revealed the planet’s strange truth.” The almost five decades between Galileo’s first observation of the rings and Christiaan’s solution of the riddle is I think more than a few years.

Moving on Aldersey-Williams tells us that, “For many however, there remained powerful reasons to reject Huygens’ discovery. First of all, it challenged the accepted idea inherited from Greek philosophers that the solar system consisted exclusively of perfect spherical bodies occupying ideal circular orbits to one another.” You would have been hard put to it to find a serious astronomer ín 1660, who still ascribed to this Aristotelian cosmology.

The next historical glitch concerns, once again, Galileo. We read, “He dedicated the work [Systema Saturnium] to Prince Leopoldo de’ Medici, who was patron of the Accademia del Cimento in Florence, who had supported the work of Huygens’ most illustrious forebear, Galileo.” Ignoring the sycophantic description of Galileo, one should perhaps point out that the Accademia del Cimento was founded in 1657 that is fifteen years after Galileo’s death and so did not support his work. It was in fact founded by a group of Galileo’s disciples and was dedicated to continuing to work in his style, not quite the same thing.

Galileo crops up again, “the real power of Huygens’ interpretation was its ability to explain those times when Saturn’s ‘handles’ simply disappeared from view, as they had done in 1642, finally defeating the aged Galileo’s attempts to understand the planet…” In 1642, the year of his death, Galileo had been completely blind for four years and had actually given up his interest in astronomy several years earlier.

Moving on to Christiaan’s invention of the pendulum clock and the problem of determining longitude Aldersey-Williams tells us, “The Alkmaar surveyor Adriaan Metius, brother of the telescope pioneer Jacob, had proposed as long ago as 1614 that some sort of seagoing clock might provide the solution to this perennial problem of navigators…” I feel honour bound to point out that Adriaan Metius was slightly more than simply a surveyor, he was professor for mathematics at the University of Franeker. However the real problem here is that the clock solution to the problem of longitude was first proposed by Gemma Frisius in an appendix added in 1530, to his highly popular and widely read editions of Peter Apian’s Cosmographia. The book was the biggest selling and most widely read textbook on practical mathematics throughout the sixteenth and well into the seventeenth century so Huygens would probably have known of Frisius’ priority.

Having dealt with the factual #histSTM errors I will now turn to more general criticisms. On several occasions Aldersey-Williams, whilst acknowledging problems with using the concept in the seventeenth century, tries to present Huygens as the first ‘professional scientist’. Unfortunately, I personally can’t see that Huygens was in anyway more or less of a professional scientist than Tycho, Kepler or Galileo, for example, or quite a long list of others I could name. He also wants to sell him as the ‘first ever’ state’s scientist following his appointment to the Académie des sciences and the accompanying state pension from the king. Once again the term is equally applicable to Tycho first in Denmark and then, if you consider the Holy Roman Empire a state, again in Prague as Imperial Mathematicus, a post that Kepler inherited. Galileo was state ‘scientist’ under the de’ Medici in the Republic of Florence. One could even argue that Nicolas Kratzer was a state scientist when he was appointed to the English court under Henry VIII. There are other examples.

Aldersey-Williams’ next attempt to define Huygens’ status as a scientist left me somewhat speechless, “Yet it is surely enough that Huygens be remembered for what he was, a mere problem solver indeed: pragmatic, eclectic and synthetic and ready to settle for the most probable rather than hold out for the absolutely certain – in other words. What we expect a scientist to be today.” My ten years as a history and philosophy of science student want to scream, “Is that what we really expect?” I’m not even going to go there, as I would need a new blog post even longer than this one.

Aldersey-Williams also tries to present Huygens as some sort of new trans European savant of a type that had not previously existed. Signifying cooperation across borders, beliefs and politics. This is of course rubbish. The sort of trans European cooperation that Huygens was involved in was just as prevalent at the beginning of the seventeenth century in the era of Tycho, Kepler, Galileo, et al. Even then it was not new it was also very strong during the Renaissance with natural philosophers and mathematici corresponding, cooperating, visiting each other, and teaching at universities through out the whole of Europe. Even in the Renaissance, science in Europe knew no borders. It’s the origin of the concept, The Republic of Letters. I suspect my history of medieval science friend would say the same about their period.

In the partial rapprochement between Huygens and Newton following the Publication of the latter’s Principia leads Aldersey-Williams to claim that a new general level of reasonable discussion had entered scientific debate towards the end of the seventeenth century. Scientists, above all Newton, were still going at each other hammer and tongs in the eighteenth century, so it was all just a pipe dream.

Aldersey-Williams sees Huygens lack of public profile, as a result of being in Newton’s shadow like Hooke and others. He suggests that popular perception only allows for one scientific genius in a generation citing Galileo’s ascendance over Kepler, who he correctly sees as the more important, as another example. In this, I agree with him, however he tries too hard to put Huygens on the same level as Newton as a scientist, as if scientific achievement were a pissing contest. I think we should consider a much wider range of scientists when viewing the history of science but I also seriously think that no matter how great his contributions Huygens can’t really match up with Newton. Although his Horologium oscillatorium sive de motu pendularium was a very important contribution to the debate on force and motion, it can’t be compared to Newton’s Principia. Even if Huygens did propagate a wave theory of light his Traité de la lumière is not on a level with Newton’s Opticks. He does have his Systema saturniumbut as far as telescopes are concerned Newton’s reflector was a more important contribution than any of Huygens refractor telescopes. Most significant, Newton made massive contributions to the development of mathematics, Huygens almost nothing.

Talking of Newton, in his discussion of Huygens rather heterodox religious views Aldersey-Williams discussing unorthodox religious views of other leading scientists makes the following comment, “Newton was an antitrinitarian, for which he was considered a heretic in his lifetime, as well as being interested in occultism and alchemy.” Newton was not considered a heretic in his lifetime because he kept his antitrinitarian views to himself. Alchemy yes, but occultism, Newton?

I do have one final general criticism of Aldersey-Williams’ book. My impression was that the passages on fine art, poetry and music, all very important aspects of the life of the Huygens family, are dealt with in much greater depth and detail than the science, which I found more than somewhat peculiar in a book with the subtitle, The Making of Science in Europe. I’m not suggesting that the fine art, poetry and music coverage should be less but that the science content should have been brought up to the same level.

Despite the long list of negative comments in my review I think this is basically a very good book that could in fact have been an excellent book with some changes. Summa summarum it is a flawed masterpiece. It is an absolute must read for anybody interested in the life of Christiaan Huygens or his father Constantijn or the whole Huygens clan. It is also an important read for those interested in Dutch culture and politics in the seventeenth century and for all those interested in the history of European science in the same period. It would be desirable if more works with the wide-ranging scope and vision of Aldersey-Williams volume were written but please without the #histSTM errors.

[1] Harold J Cook, Matters of Exchange: Commerce, Medicine and Science in the Dutch Golden Age, Yale University Press, New Haven & London, 2007

[2] Cornelis Dirk Andriesse, The Man Behind the Principle, scientific biography of Christiaan Huygens, translated from Dutch by Sally Miedem, CUP, Cambridge, 2005

[3] Hugh Aldersey-Williams, Dutch Light: Christiaan Huygens and the Making of Science in Europe, Picador, London, 2020.

[4] Aldersey-Williams admits that the use of the term scientist is anachronistic but uses it for simplicity’s sake and I shall do likewise here.

[5] I have after all a reputation to uphold

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Filed under Book Reviews, History of Astronomy, History of Mathematics, History of Navigation, History of Optics, History of Physics, History of science, Newton

How Renaissance Nürnberg became the Scientific Instrument Capital of Europe

This is a writen version of the lecture that I was due to hold at the Science and the City conference in London on 7 April 2020. The conference has for obvious reasons been cancelled and will now take place on the Internet. You can view the revised conference program here.

The title of my piece is, of course, somewhat hyperbolic, as far as I know nobody has ever done a statistical analysis of the manufacture of and trade in scientific instruments in the sixteenth century. However, it is certain that in the period 1450-1550 Nürnberg was one of the leading European centres both for the manufacture of and the trade in scientific instruments. Instruments made in Nürnberg in this period can be found in every major collection of historical instruments, ranging from luxury items, usually made for rich patrons, like the column sundial by Christian Heyden (1526–1576) from Hessen-Kassel

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Column Sundial by Christian Heyden Source: Museumslandschaft Hessen-Kassel

to cheap everyday instruments like this rare (rare because they seldom survive) paper astrolabe by Georg Hartman (1489–1564) from the MHS in Oxford.

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Paper and Wood Astrolabe Hartmann Source: HSM Oxford

I shall be looking at the reasons why and how Nürnberg became such a major centre for scientific instruments around 1500, which surprisingly have very little to do with science and a lot to do with geography, politics and economics.

Like many medieval settlements Nürnberg began simply as a fortification of a prominent rock outcrop overlooking an important crossroads. The first historical mention of that fortification is 1050 CE and there is circumstantial evidence that it was not more than twenty or thirty years old. It seems to have been built in order to set something against the growing power of the Prince Bishopric of Bamberg to the north. As is normal a settlement developed on the downhill slopes from the fortification of people supplying services to it.

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A fairly accurate depiction of Nürnberg from the Nuremberg Chronicle from 1493. The castles (by then 3) at the top with the city spreading down the hill. Large parts of the inner city still look like this today

Initially the inhabitants were under the authority of the owner of the fortification a Burggraf or castellan. With time as the settlement grew the inhabitants began to struggle for independence to govern themselves.

In 1200 the inhabitants received a town charter and in 1219 Friedrich II granted the town of Nürnberg a charter as a Free Imperial City. This meant that Nürnberg was an independent city-state, which only owed allegiance to the king or emperor. The charter also stated that because Nürnberg did not possess a navigable river or any natural resources it was granted special tax privileges and customs unions with a number of southern German town and cities. Nürnberg became a trading city. This is where the geography comes into play, remember that important crossroads. If we look at the map below, Nürnberg is the comparatively small red patch in the middle of the Holy Roman Empire at the beginning of the sixteenth century. If your draw a line from Paris to Prague, both big important medieval cities, and a second line from the border with Denmark in Northern Germany down to Venice, Nürnberg sits where the lines cross almost literally in the centre of Europe. Nürnberg also sits in the middle of what was known in the Middle Ages as the Golden Road, the road that connected Prague and Frankfurt, two important imperial cities.

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You can also very clearly see Nürnberg’s central position in Europe on Erhard Etzlaub’s  (c. 1460–c. 1531) pilgrimage map of Europe created for the Holy Year of 1500. Nürnberg, Etzlaub’s hometown, is the yellow patch in the middle. Careful, south is at the top.

1024px-Rompilger-Karte_(Erhard_Etzlaub)

Over the following decades and centuries the merchant traders of Nürnberg systematically expanded their activities forming more and more customs unions, with the support of various German Emperors, with towns, cities and regions throughout the whole of Europe north of Italy. Nürnberg which traded extensively with the North Italian cities, bringing spices, silk and other eastern wares, up from the Italian trading cities to distribute throughout Europe, had an agreement not to trade with the Mediterranean states in exchange for the Italians not trading north of their northern border.

As Nürnberg grew and became more prosperous, so its political status and position within the German Empire changed and developed. In the beginning, in 1219, the Emperor appointed a civil servant (Schultheis), who was the legal authority in the city and its judge, especially in capital cases. The earliest mention of a town council is 1256 but it can be assumed it started forming earlier. In 1356 the Emperor, Karl IV, issued the Golden Bull at the Imperial Diet in Nürnberg. This was effectively a constitution for the Holy Roman Empire that regulated how the Emperor was to be elected and, who was to be appointed as the Seven Prince-electors, three archbishops and four secular rulers. It also stipulated that the first Imperial Diet of a newly elected Emperor was to be held in Nürnberg. This stipulation reflects Nürnberg’s status in the middle of the fourteenth century.

The event is celebrated by the mechanical clock ordered by the town council to be constructed for the Frauenkirche, on the market place in 1506 on the 150th anniversary of the Golden Bull, which at twelve noon displays the seven Prince-electors circling the Emperor.

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Mechanical clock on the Frauenkirche overlooking the market place in Nürnberg. Ordered by the city council in 1506 to celebrate the 150th anniversary of the issuing of the Golden Bull at the Imperial Diet in 1356

Over time the city council had taken more and more power from the Schultheis and in 1385 they formally bought the office, integrating it into the councils authority, for 8,000 gulden, a small fortune. In 1424 Emperor, Sigismund appointed Nürnberg the permanent residence of the Reichskleinodien (the Imperial Regalia–crown, orb, sceptre, etc.).

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The Imperial Regalia

This raised Nürnberg in the Imperial hierarchy on a level with Frankfurt, where the Emperor was elected, and Aachen, where he was crowned. In 1427, the Hohenzollern family, current holders of the Burggraf title, sold the castle, which was actually a ruin at that time having been burnt to the ground by the Bavarian army, to the town council for 120,000 gulden, a very large fortune. From this point onwards Nürnberg, in the style of Venice, called itself a republic up to 1806 when it was integrated into Bavaria.

In 1500 Nürnberg was the second biggest city in Germany, after Köln, with a population of approximately 40,000, about half of which lived inside the impressive city walls and the other half in the territory surrounding the city, which belonged to it.

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Map of the city-state of Nürnberg by Abraham Ortelius 1590. the city itself is to the left just under the middle of the map. Large parts of the forest still exists and I live on the northern edge of it, Dormitz is a neighbouring village to the one where I live.

Small in comparison to the major Italian cities of the period but even today Germany is much more decentralised with its population more evenly distributed than other European countries. It was also one of the richest cities in the whole of Europe.

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Nürnberg, Plan by Paul Pfinzing, 1594 Castles in the top left hand corner

Nürnberg’s wealth was based on two factors, trading, in 1500 at least 27 major trade routes ran through Nürnberg, which had over 90 customs unions with cities and regions throughout Europe, and secondly the manufacture of trading goods. It is now time to turn to this second branch of Nürnberg’s wealth but before doing so it is important to note that whereas in other trading centres in Europe individual traders competed with each other, Nürnberg function like a single giant corporation, with the city council as the board of directors, the merchant traders cooperating with each other on all levels for the general good of the city.

In 1363 Nürnberg had more than 1200 trades and crafts masters working in the city. About 14% worked in the food industry, bakers, butchers, etc. About 16% in the textile industry and another 27% working leather. Those working in wood or the building branch make up another 14% but the largest segment with 353 masters consisted of those working in metal, including 16 gold and silver smiths. By 1500 it is estimated that Nürnberg had between 2,000 and 3,000 trades and crafts master that is between 10 and 15 per cent of those living in the city with the metal workers still the biggest segment. The metal workers of Nürnberg produced literally anything that could be made of metal from sewing needles and nails to suits of armour. Nürnberg’s reputation as a producer rested on the quality of its metal wares, which they sold all over Europe and beyond. According to the Venetian accounts books, Nürnberg metal wares were the leading export goods to the orient. To give an idea of the scale of production at the beginning of the 16th century the knife makers and the sword blade makers (two separate crafts) had a potential production capacity of 80,000 blades a week. The Nürnberger armourers filled an order for armour for 5,000 soldiers for the Holy Roman Emperor, Karl V (1500–1558).

The Nürnberger craftsmen did not only produce goods made of metal but the merchant traders, full blood capitalists, bought into and bought up the metal ore mining industry–iron, copper, zinc, gold and silver–of Middle Europe, and beyond, (in the 16th century they even owned copper mines in Cuba) both to trade in ore and to smelt ore and trade in metal as well as to ensure adequate supplies for the home production. The council invested heavily in the industry, for example, providing funds for the research and development of the world’s first mechanical wire-pulling mill, which entered production in 1368.

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The wirepulling mills of Nürnberg by Albrecht Dürer

Wire was required in large quantities to make chainmail amongst other things. Around 1500 Nürnberg had monopolies in the production of copper ore, and in the trade with steel and iron.  Scientific instruments are also largely made of metal so the Nürnberger gold, silver and copper smiths, and toolmakers also began to manufacture them for the export trade. There was large scale production of compasses, sundials (in particular portable sundials), astronomical quadrants, horary quadrants, torquetum, and astrolabes as well as metal drawing and measuring instruments such as dividers, compasses etc.

The city corporation of Nürnberg had a couple of peculiarities in terms of its governance and the city council that exercised that governance. Firstly the city council was made up exclusively of members of the so-called Patrizier. These were 43 families, who were regarded as founding families of the city all of them were merchant traders. There was a larger body that elected the council but they only gave the nod to a list of the members of the council that was presented to them. Secondly Nürnberg had no trades and crafts guilds, the trades and crafts were controlled by the city council. There was a tight control on what could be produced and an equally tight quality control on everything produced to ensure the high quality of goods that were traded. What would have motivated the council to enter the scientific instrument market, was there a demand here to be filled?

It is difficult to establish why the Nürnberg city corporation entered the scientific instrument market before 1400 but by the middle of the 15th century they were established in that market. In 1444 the Catholic philosopher, theologian and astronomer Nicolaus Cusanus (1401–1464) bought a copper celestial globe, a torquetum and an astrolabe at the Imperial Diet in Nürnberg. These instruments are still preserved in the Cusanus museum in his birthplace, Kues on the Mosel.

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The Cusanus Museum in Kue

In fact the demand for scientific instrument rose sharply in the 15th & 16th centuries for the following reasons. In 1406 Jacopo d’Angelo produced the first Latin translation of Ptolemy’s Geographia in Florence, reintroducing mathematical cartography into Renaissance Europe. One can trace the spread of the ‘new’ cartography from Florence up through Austria and into Southern Germany during the 15th century. In the early 16th century Nürnberg was a major centre for cartography and the production of both terrestrial and celestial globes. One historian of cartography refers to a Viennese-Nürnberger school of mathematical cartography in this period. The availability of the Geographia was also one trigger of a 15th century renaissance in astronomy one sign of which was the so-called 1st Viennese School of Mathematics, Georg von Peuerbach (1423–1461) and Regiomontanus (1436–176), in the middle of the century. Regiomontanus moved to Nürnberg in 1471, following a decade wandering around Europe, to carry out his reform of astronomy, according to his own account, because Nürnberg made the best astronomical instruments and had the best communications network. The latter a product of the city’s trading activities. When in Nürnberg, Regiomontanus set up the world’s first scientific publishing house, the production of which was curtailed by his early death.

Another source for the rise in demand for instruments was the rise in interest in astrology. Dedicated chairs for mathematics, which were actually chairs for astrology, were established in the humanist universities of Northern Italy and Krakow in Poland early in the 15th century and then around 1470 in Ingolstadt. There were close connections between Nürnberg and the Universities of Ingolstadt and Vienna. A number of important early 16th century astrologers lived and worked in Nürnberg.

The second half of the 15th century saw the start of the so-called age of exploration with ships venturing out of the Iberian peninsular into the Atlantic and down the coast of Africa, a process that peaked with Columbus’ first voyage to America in 1492 and Vasco da Gama’s first voyage to India (1497–199). Martin Behaim(1459–1507), son of a Nürnberger cloth trading family and creator of the oldest surviving terrestrial globe, sat on the Portuguese board of navigation, probably, according to David Waters, to attract traders from Nürnberg to invest in the Portuguese voyages of exploration.  This massively increased the demand for navigational instruments.

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The Erdapfel–the Behaim terrestial globe Germanische National Museum

Changes in the conduct of wars and in the ownership of land led to a demand for better, more accurate maps and the more accurate determination of boundaries. Both requiring surveying and the instruments needed for surveying. In 1524 Peter Apian (1495–1552) a product of the 2nd Viennese school of mathematics published his Cosmographia in Ingolstadt, a textbook for astronomy, astrology, cartography and surveying.

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The Cosmographia went through more than 30 expanded, updated editions, but all of which, apart from the first, were edited and published by Gemma Frisius (1508–1555) in Louvain. In 1533 in the third edition Gemma Frisius added an appendix Libellus de locorum describendum ratione, the first complete description of triangulation, the central method of cartography and surveying down to the present, which, of course in dependent on scientific instruments.

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In 1533 Apian’s Instrumentum Primi Mobilis 

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was published in Nürnberg by Johannes Petreius (c. 1497–1550) the leading scientific publisher in Europe, who would go on ten years later to publish, Copernicus’ De revolutionibus, which was a high point in the astronomical revival.

All of this constitutes a clear indication of the steep rise in the demand for scientific instruments in the hundred years between 1450 and 1550; a demand that the metal workers of Nürnberg were more than happy to fill. In the period between Regiomontanus and the middle of the 16th century Nürnberg also became a home for some of the leading mathematici of the period, mathematicians, astronomers, astrologers, cartographers, instrument makers and globe makers almost certainly, like Regiomontanus, at least partially attracted to the city by the quality and availability of the scientific instruments.  Some of them are well known to historians of Renaissance science, Erhard Etzlaub, Johannes Werner, Johannes Stabius (not a resident but a frequent visitor), Georg Hartmann, Johannes Neudörffer and Johannes Schöner.**

There is no doubt that around 1500, Nürnberg was one of the major producers and exporters of scientific instruments and I hope that I have shown above, in what is little more than a sketch of a fairly complex process, that this owed very little to science but much to the general geo-political and economic developments of the first 500 years of the city’s existence.

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One of the most beautiful sets on instruments manufactured in Nürnberg late 16th century. Designed by Johannes Pretorius (1537–1616), professor for astronomy at the Nürnberger University of Altdorf and manufactured by the goldsmith Hans Epischofer (c. 1530–1585) Germanische National Museum

**for an extensive list of those working in astronomy, mathematics, instrument making in Nürnberg (542 entries) see the history section of the Astronomie in Nürnberg website, created by Dr Hans Gaab.

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Filed under Early Scientific Publishing, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, History of Technology, Renaissance Science

It’s all a question of angles.

Thomas Paine (1736–1809) was an eighteenth-century political radical famous, or perhaps that should be infamous, for two political pamphlets, Common Sense (1776) and Rights of Man (1791) (he also wrote many others) and for being hounded out of England for his political views and taking part in both the French and American Revolutions.

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Thomas Paine portrait of Laurent Dabos c. 1792 Source: Wikimedia Commons

So I was more than somewhat surprised when Michael Brooks, author of the excellent The Quantum Astrologer’s Handbook, posted the following excerpt from Paine’s The Age of Reason, praising trigonometry as the soul of science:

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My first reaction to this beautiful quote was that he could be describing this blog, as the activities he names, astronomy, navigation, geometry, land surveying make up the core of the writings on here. This is not surprising as Ivor Grattan-Guinness in his single volume survey of the history of maths, The Rainbow of Mathematics: A History of the Mathematical Sciences, called the period from 1540 to 1660 (which is basically the second half of the European Renaissance) The Age of Trigonometry. This being the case I thought it might be time for a sketch of the history of trigonometry.

Trigonometry is the branch of mathematics that studies the relationships between the side lengths and the angles of triangles. Possibly the oldest trigonometrical function, although not regarded as part of the trigonometrical cannon till much later, was the tangent. The relationship between a gnomon (a fancy word for a stick stuck upright in the ground or anything similar) and the shadow it casts defines the angle of inclination of the sun in the heavens. This knowledge existed in all ancient cultures with a certain level of mathematical development and is reflected in the shadow box found on the reverse of many astrolabes.

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

Trigonometry as we know it begins with ancient Greek astronomers, in order to determine the relative distance between celestial objects. These distances were determined by the angle subtended between the two objects as observed from the earth. As the heavens were thought to be a sphere this was spherical trigonometry[1], as opposed to the trigonometry that we all learnt at school that is plane trigonometry. The earliest known trigonometrical tables were said to have been constructed by Hipparchus of Nicaea (c. 190–c. 120 BCE) and the angles were defined by chords of circles. Hipparchus’ table of chords no longer exist but those of Ptolemaeus (fl. 150 CE) in his Mathēmatikē Syntaxis (Almagest) still do.

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

With Greek astronomy, trigonometry moved from Greece to India, where the Hindu mathematicians halved the Greek chords and thus created the sine and also defined the cosine. The first recoded uses of theses function can be found in the Surya Siddhanta (late 4th or early 5th century CE) an astronomical text and the Aryabhatiya of Aryabhata (476–550 CE).

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Statue depicting Aryabhata on the grounds of IUCAA, Pune (although there is no historical record of his appearance). Source: Wikimedia Commons

Medieval Islam in its general acquisition of mathematical knowledge took over trigonometry from both Greek and Indian sources and it was here that trigonometry in the modern sense first took shape.  Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850), famous for having introduced algebra into Europe, produced accurate sine and cosine tables and the first table of tangents.

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Statue of al-Khwarizmi in front of the Faculty of Mathematics of Amirkabir University of Technology in Tehran Source: Wikimedia Commons

In 830 CE Ahmad ibn ‘Abdallah Habash Hasib Marwazi (766–died after 869) produced the first table of cotangents. Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī (c. 858–929) discovered the secant and cosecant and produced the first cosecant tables.

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (940–998) was the first mathematician to use all six trigonometrical functions.

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Abū al-Wafā Source: Wikimedia Commons

Islamic mathematicians extended the use of trigonometry from astronomy to cartography and surveying. Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (1201–1274) is regarded as the first mathematician to present trigonometry as a mathematical discipline and not just a sub-discipline of astronomy.

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Iranian stamp for the 700th anniversary of Nasir al-Din Tusi’s death Source: Wikimedia Commons

Trigonometry came into Europe along with astronomy and mathematics as part the translation movement during the 11th and 12th centuries. Levi ben Gershon (1288–1344), a French Jewish mathematician/astronomer produced a trigonometrical text On Sines, Chords and Arcs in 1342. Trigonometry first really took off in Renaissance Europe with the translation of Ptolemaeus’ Geōgraphikḕ Hyphḗgēsis (Geographia) into Latin by Jacopo d’Angelo (before 1360–c. 1410) in 1406, which triggered a renaissance in cartography and astronomy.

The so-called first Viennese School of Mathematics made substantial contributions to the development of trigonometry in the sixteenth century. John of Gmunden (c. 1380–1442) produced a Tractatus de sinibus, chodis et arcubus. His successor, Georg von Peuerbach (1423–1461), published an abridgement of Gmunden’s work, Tractatus super propositiones Ptolemaei de sinibus et chordis together with a sine table produced by his pupil Regiomontanus (1436–1476) in 1541. He also calculated a monumental table of sines. Regiomontanus produced the first complete European account of all six trigonometrical functions as a separate mathematical discipline with his De Triangulis omnimodis (On Triangles) in 1464. To what extent his work borrowed from Arabic sources is the subject of discussion. Although Regiomontanus set up the first scientific publishing house in Nürnberg in 1471 he died before he could print De Triangulis. It was first edited by Johannes Schöner (1477–1547) and printed and published by Johannes Petreius (1497–1550) in Nürnberg in 1533.

At the request of Cardinal Bessarion, Peuerbach began the Epitoma in Almagestum Ptolomei in 1461 but died before he could complete it. It was completed by Regiomontanus and is a condensed and modernised version of Ptolemaeus’ Almagest. Peuerbach and Regiomontanus replaced the table of chords with trigonometrical tables and modernised many of the proofs with trigonometry. The Epitoma was published in Venice in 1496 and became the standard textbook for Ptolemaic geocentric astronomy throughout Europe for the next hundred years, spreading knowledge of trigonometry and its uses.

In 1533 in the third edition of the Apian/Frisius Cosmographia, Gemma Frisius (1508–1555) published as an appendix the first account of triangulationin his Libellus de locorum describendum ratione. This laid the trigonometry-based methodology of both surveying and cartography, which still exists today. Even GPS is based on triangulation.

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With the beginnings of deep-sea exploration in the fifteenth century first in Portugal and then in Spain the need for trigonometry in navigation started. Over the next centuries that need grew for determining latitude, for charting ships courses and for creating sea charts. This led to a rise in teaching trigonometry to seamen, as excellently described by Margaret Schotte in her Sailing School: Navigating Science and Skill, 1550–1800.

One of those students, who learnt their astronomy from the Epitoma was Nicolaus Copernicus (1473–1543). Modelled on the Almagest or more accurately the Epitoma, Copernicus’ De revolutionibus, published by Petreius in Nürnberg in 1543, also contained trigonometrical tables. WhenGeorg Joachim Rheticus (1514–1574) took Copernicus’ manuscript to Nürnberg to be printed, he also took the trigonometrical section home to Wittenberg, where he extended and improved it and published it under the title De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published. He would dedicate a large part of his future life to the science of trigonometry. In 1551 he published Canon doctrinae triangvlorvm in Leipzig. He then worked on what was intended to be the definitive work on trigonometry his Opus palatinum de triangulis, which he failed to finish before his death. It was completed by his student Valentin Otho (c. 1548–1603) and published in Neustadt an der Haardt in 1596.

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Source: Wikimedia Commons

In the meantime Bartholomäus Pitiscus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuous 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, where as those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. 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. 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.

We have come a long way from ancient Greece in the second century BCE to Germany at the turn of the seventeenth century CE by way of Early Medieval India and the Medieval Islamic Empire. During the seventeenth century the trigonometrical relationships, which I have up till now somewhat anachronistically referred to as functions became functions in the true meaning of the term and through analytical geometry received graphical presentations completely divorced from the triangle. However, I’m not going to follow these developments here. The above is merely a superficial sketch that does not cover the problems involved in actually calculating trigonometrical tables or the discovery and development of the various relationships between the trigonometrical functions such as the sine and cosine laws. For a detailed description of these developments from the beginnings up to Pitiscus I highly recommend Glen van Brummelen’s The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009.

 

[1] For a wonderful detailed description of spherical trigonometry and its history see Glen van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013

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Filed under History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Mediaeval Science, Renaissance Science