Category Archives: Renaissance Science

The Bees of Ingolstadt

The tittle of this blog post is a play on the names of a father and son duo of influential sixteenth century Renaissance mathematici. The father was Peter Bienewitz born 16 April 1495 in Leisnig in Saxony just south of Leipzig. His father was a well off shoemaker and Peter was educated at the Latin school in Rochlitz and then from 1516 to 1519 at the University of Liepzig. It was here that he acquired the humanist name Apianus from Apis the Latin for a bee, a direct translation of the German Biene. From now on he became Petrus Apianus or simply Peter Apian.

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Apianus on a 16th-century engraving by Theodor de Bry Source: Wikimedia Commons

In 1519 he went south to the University of Vienna to study under Georg Tannstetter a leading cosmographer of the period.

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Georg Tannstetter Portrait ca. 1515, by Bernhard Strigel (1460 – 1528) Source: Wikimedia Commons

Tannstetter was a physician, mathematician astronomer and cartographer, who studied mathematics at the University of Ingolstadt under Andreas Stiborius and followed Conrad Celtis and Stiborius to Vienna in 1503 to teach at Celtis’ Collegium poetarum et mathematicorum. The relationship between teacher and student was a very close one. Tannstetter edited a map of Hungary that was later printed by Apian and the two of them produced the first printed edition of Witelo’s Perspectiva, which was printed and published by Petreius in Nürnberg in 1535. This was one of the books that Rheticus took with him to Frombork as a gift for Copernicus.

In 1520 Apian published a smaller updated version of the Waldseemüller/ Ringmann world map, which like the original from 1507 named the newly discovered fourth continent, America. Waldseemüller and Ringmann had realised their original error and on their 1513 Carte Marina dropped the name America, However, the use by Apian and by Johannes Schöner on his 1515 terrestrial globe meant that the name became established.

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Apian’s copy of the Waldseemüller world map, naming the new fourth continent America Source: Wikimedia Commons

Apian graduated BA in 1521 and moved first to Regensburg then Landshut. In 1524 he printed and published his Cosmographicus liber, a book covering the full spectrum of cosmography – astronomy, cartography, navigation, surveying etc. The book became a sixteenth century best seller going through 30 expanded editions in 14 languages but after the first edition all subsequent editions were written by Gemma Frisius.

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Title page of Apian’s Cosmpgraphia

In 1527 Apian was called to the University of Ingolstadt to set up a university printing shop and to become Lektor for mathematics. He maintained both positions until his death in 1552.

In 1528 he printed Tannstetter’s Tabula Hungariaethe earliest surviving printed map of Hungary. In the same year Apian dedicated his edition of Georg von Peuerbach’s New Planetary Theory to his “famous teacher and professor for mathematics” Tannstetter.

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Tabula Hungarie ad quatuor latera Source: Wikimedia Commons

One year earlier he published a book on commercial arithmetic, Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen(A new and well-founded instruction in all Merchants Reckoning in three books, understood with fine rules and exercises). It was the first European book to include (on the cover), what is know as Pascal’s triangle, which was known earlier to both Chinese and Muslim mathematicians.

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This is one of the volumes lying on the shelf in Holbein’s painting The Ambassadors. Like his Cosmographicusit was a bestseller.

In the 1530s Apian was one of a group of European astronomers, which included Schöner, Copernicus, Fracastoro and Pena, who closely observed the comets of that decade and began to question the Aristotelian theory that comets are sublunar meteorological phenomena. He was the first European to observe and publish that the comet’s tail always points away from the sun, a fact already known to Chinese astronomers. Fracastoro made the same observation, which led him and Pena to hypothesise that the comet’s tail was an optical phenomenon, sunlight focused through the lens like translucent body of the comet. These observations in the 1530s led to an increased interest in cometary observation and the determination in the 1570s by Mästlin, Tycho and others that comets are in fact supralunar objects.

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Diagram by Peter Apian from his book Astronomicum Caesareum (1540) demonstrating that a comet’s tail points away from the Sun. The comet he depicted was that of 1531, which we now know as Halley’s Comet. Image courtesy Royal Astronomical Society.

Through the Cosmographicus he became a favourite of Karl V, the Holy Roman Emperor, and Apian became the Emperor’s astronomy tutor. Karl granted him the right to display a coat of arms in 1535 and knighted him in 1541. In 1544 Karl even appointed him Hofpfalzgraf (Imperial Count Palatine), a high ranking court official.

Apian’s association with Karl led to his most spectacular printing project, one of the most complicated and most beautiful books published in the sixteenth century, his Astronomicum Caesareum (1540). This extraordinary book is a presentation of the then Standard Ptolemaic astronomy in the form of a series of highly complex and beautifully designed volvelles. A vovelle or wheel chart is a form of paper analogue computer. A series of rotating paper discs mounted on a central axis or pin that can be used to calculate various mathematical functions such as the orbital positions of planets.

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Astronomicum Caesareum title page

The Astronomicum Caesareumcontains two volvelles for each planet, one to calculate its longitude for a given time and one to calculate its latitude.

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Astronomicum Caesareum volvelle for longitude for Saturn

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Astronomicum Caesareum volvelle for the latitude for Saturn

There is also a calendar disc to determine the days of the week for a given year.

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Astronomicum Caesareum calendar volvelle

Finally there are vovelles to determine the lunar phases  as well as lunar and solar eclipse.

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Astronomicum Caesareum : Disc illustrating a total eclipse of the moon 6 Octobre 1530

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Astronomicum Caesareum solar eclisse volvelle

Johannes Kepler was very rude about the Astronomicum Caesareum, calling it a thing of string and paper. Some have interpreted this as meaning that it had little impact. However, I think the reverse is true. Kepler was trying to diminish the status of a serious rival to his endeavours to promote the heliocentric system. Owen Gingerich carried out a census of 111 of the approximately 130 surviving copies of the book and thinks that these represent almost the whole print run. This book is so spectacular and so expensive that the copies rarely got seriously damaged of thrown away.

Like other contemporary mathematici Apian designed sundials and astronomical instruments as well as marketing diverse volvelles for calculation purposes. Apian died in 1552 and was succeeded on his chair for mathematics by his son Philipp, the second of the bees from Ingolstadt.

Philipp Apian was born 14 September 1531, as the fourth of fourteen children (nine sons and five daughters) to Peter Apian and his wife Katharina Mesner.

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Philipp Apian painting by Hans Ulrich Alt Source: Wikimedia Commons

He started receiving tuition at the age of seven together with Prince Albrecht the future Duke of Bavaria, who would become his most important patron.

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Duke Albrecht V of Bavaria Hans Muelich Source: Wikimedia Commons

He entered the University of Ingolstadt at the age of fourteen and studied under his father until he was eighteen. He completed his studies in Burgundy, Paris and Bourges. In 1552 aged just 21 he inherited his fathers printing business and his chair for mathematics on the University of Ingolstadt. As well as teaching mathematics at the university, which he had started before his father died, Philipp studied medicine. He graduated in medicine several years later during a journey to Italy, where he visited the universities of Padua, Ferrara and Bolgna.

In 1554 his former childhood friend Albrecht, now Duke of Bavaria, commissioned him to produce a new map of Bavaria. During the summers of the next seven years he surveyed the land and spent the following two years drawing the map. The 5 metres by 6 metres map at the scale of 1:45,000, hand coloured by Bartel Refinger was hung in the library of the Bavarian palace.

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Philipp Apian’s map of Bavaria

In 1566 Jost Amman produced 24 woodblocks at the smaller scale of 1:144,000, which Apian printed in his own print shop. Editions of this smaller version of the map continued to be issued up to the nineteenth century.

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Overview of the 24 woodblock prints of Apian’s map of Bavaria

In 1576 he also produced a terrestrial globe for Albrecht. Map, woodblocks, woodblock prints and globe are all still extant.

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Apian’s terrestrial globe

In 1568 Phillip converted to Protestantism and in the following year was forced by the Jesuit, who controlled the University of Ingolstadt to resign his post. In the same year, he was appointed professor for mathematics at the Protestant University of Tübingen. In Tübingen his most famous pupil was Michael Mästlin, who succeeded him as professor for mathematics at the university and would become Johannes Kepler’s teacher. An irony of history is that Philipp was forced to resign in Tübingen in 1583 for refusing to sign the Formal of Concord, a commitment to Lutheran Protestantism against Calvinism. He continued to work as a cartographer until his death in 1589.

There is a genealogy of significant Southern German Renaissance mathematici: Andreas Stiborius (1464–1515) taught Georg Tannstetter (1482–1535), who taught Peter Apian (1495–1552), who taught Philipp Apian (1531–1589), who taught Michael Mästlin (1550–1631), who taught Johannes Kepler (1571–1630)

 

 

 

 

 

 

 

 

 

 

 

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

Tycho’s last bastion

In the history of science, scholars who end up on the wrong side of history tend to get either forgotten and/or vilified. What do I mean by ‘end up on the wrong side of history’? This refers to scholars who defend a theory that in the end turns out to be wrong against one that in the end turns out to be right. My very first history of science post on this blog was about just such a figure, Christoph Clavius, who gets mocked by many as the last Ptolemaic dinosaur in the astronomy/cosmology debate at the beginning of the seventeenth century. In fact there is much to praise about Clavius, as I tried to make clear in my post and he made many positive contributions to the evolution of the mathematical sciences. Another man, who ended up on the wrong side of history in the same period is the Danish astronomer, Christen Sørensen, better known, if at all, by the name Longomontanus, the Latinised toponym based on Lomborg, the Jutland village where he was born on 4 October 1562 the son of a poor labourer, who died when he was only eight years old.

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Longomontanus Source: Wikimedia Commons

Tycho Brahe backed the wrong astronomical theory in this period, a theory that is generally named after him although several people seem to have devised it independently of each other in the closing quarter of the sixteenth century. However, Tycho has not been forgotten because he delivered the new data with which Johannes Kepler created his elliptical model of the solar system. However, what people tend to ignore is that Tycho did not produce that data single-handedly, far from it.

The island of Hven, Tycho’s fiefdom, was a large-scale research institute with two observatories, an alchemy laboratory, a paper mill and a printing workshop.

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Map of Hven from the Blaeu Atlas 1663, based on maps drawn by Tycho Brahe in the previous century Source: Wikimedia Commons

This enterprise was staffed by a veritable army of servants, technicians and research assistant with Tycho as the managing director and head of research.

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Engraving of the mural quadrant from Brahe’s book Astronomiae instauratae mechanica (1598) Showing Tyco direction observations Source: Wikimedia Commons

Over the years the data that would prove so crucial to Kepler’s endeavours was collected, recorded and analysed by a long list of astronomical research assistants; by far and away the most important of those astronomical research assistants was Christen Sørensen called Longomontanus, who also inherited Tycho’s intellectual mantle and continued to defend his system into the seventeenth century until his death in 1647.

Christen Sørensen came from a very poor background so acquiring an education proved more than somewhat difficult. After the death of his father he was taken into care by an uncle who sent him to the village school in Lemvig. However, after three years his mother took him back to work on the farm; she only allowed him to study with the village pastor during the winter months. In 1577 he ran away to Viborg, where he studied at the cathedral school, supporting himself by working as a labourer. This arrangement meant that he only entered the university in Copenhagen in 1588, but with a good academic reputation. It was here at the university that he acquired his toponym, Longomontanus. In 1589 his professor recommended him to Tycho Brahe and he entered into service on the island of Hven.

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Tycho Brahe’s Uraniborg main building from the 1663 Blaeu’s Atlas Major Centre of operations Source: Wikimedia Commons

He was probably instructed in Tycho’s methods by Elias Olsen Morsing, who served Tycho from 1583 to 1590, and Peter Jacobsen Flemløse, who served from 1577-1588 but stayed in working contact for several years more and became a good friend of Longomontanus. Longomontanus proved to be an excellent observer and spent his first three years working on Tycho’s star catalogue.

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Stjerneborg Tycho Brahe’s second observatory on Hven: Johan Blaeu, Atlas Major, Amsterdam Source: Wikimedia Commons

Later he took on a wider range of responsibilities. In 1597, Tycho having clashed with the new king, the entire research institute prepared to leave Hven. Longomontanus was put in charge of the attempt to bring Tycho’s star catalogue up from 777 stars to 1,000. When Tycho left Copenhagen, destination unknown, Longomontanus asked for and received his discharge from Tycho’s service.

While Tycho wandered around Europe trying to find a new home for his observatory, Longomontanus also wandered around Europe attending various universities–Breslau, Leipzig and Rostock–and trying to find a new patron. He graduated MA in Rostock. During their respective wanderings, Tycho’s and Longomontanus’ paths crossed several times and the corresponded frequently, Tycho always urging Longomontanus to re-enter his service. In January 1600 Longomontanus finally succumbed and joined Tycho in his new quarters in Prague, where Johannes Kepler would soon join the party.

When Kepler became part of Tycho’s astronomical circus in Prague, Longomontanus the senior assistant was working on the reduction of the orbit of Mars. Tycho took him off this project putting him instead onto the orbit of the Moon and giving Mars to Kepler, a move that would prove history making. As should be well known, Kepler battled many years with the orbit of Mars finally determining that it was an ellipse thereby laying the foundation stone for his elliptical astronomy. The results of his battle were published in 1609, together with his first two laws of planetary motion, in his Astronomia nova.

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Portrait of Johannes Kepler. Source: Wikimedia Commons

Meanwhile, Longomontanus having finished Tycho’s lunar theory and corrected his solar theory took his final departure from Tycho’s service, with letters of recommendation, on 4 August 1600.  When Tycho died 24 October 1601 it was thus Kepler, who became his successor as Imperial Mathematicus and inherited his data, if only after a long dispute with Tycho’s relatives, and not Longomontanus, which Tycho would certainly have preferred.

Longomontanus again wandered around Northern Europe finally becoming rector of his alma mater the cathedral school in Viborg in 1603. In 1605, supported by the Royal Chancellor, Christian Friis, he became extraordinary professor for mathematics at the University of Copenhagen, moving on to become professor for Latin literature in the same year. In 1607 he became professor for mathematics, and in 1621 his chair was transformed into an extraordinary chair for astronomy a post he held until his death.

As a professor in Copenhagen he was a member of an influential group of Hven alumni: Cort Aslakssøn (Hven 159-93) professor for theology, Christian Hansen Riber (Hven 1586-90) professor for Greek, as well as Johannes Stephanius (Hven 1582-84) professor for dialectic and Gellius Sascerides (Hven 1585-86) professor for medicine.

Kepler and Longomontanus corresponded for a time in the first decade of the seventeenth century but the exchange between the convinced supporter of heliocentricity and Tycho’s most loyal lieutenant was not a friendly one as can be seen from the following exchange:

Longomontanus wrote to Kepler 6th May 1604:

These and perhaps all other things that were discovered and worked out by Tycho during his restoration of astronomy for our eternal benefit, you, my dear Kepler, although submerged in shit in the Augean stable of old, do not scruple to equal. And you promise your labor in cleansing them anew and even triumph, as if we should recognise you as Hercules reborn. But certainly no one does, and prefers you to such a man, unless when all of it has been cleaned away, he understands that you have substituted more appropriate things in the heaven and in the celestial appearances. For in this is victory for the astronomer to be seen, in this, triumph. On the other hand, I seriously doubt that such things can ever be presented by you. However, I am concerned lest this sordid insolence of yours defile the excellent opinion of all good and intelligent men about the late Tycho, and become offensive.

Kepler responded early in 1605:

The tone of your reference to my Augean stable sticks in my mind. I entreat you to avoid chicanery, which is wont to be used frequently with regard to unpopular things. So that you might see that I have in mind how the Augean stable provided me with the certain conviction that I have not discredited astronomy – although you can gather from the present letter – I will use it with the greatest possible justification. But it is to be used as an analogy, not for those things that you or Tycho were responsible for constructing – which either blinded by rage or perverted by malice you quite wrongfully attributed to me – but rather in the comparison of the ancient hypotheses with my oval path2. You discredit my oval path. I hold up to you the hundred-times-more-absurd spirals of the ancients (which Tycho imitated by not setting up anything new but letting the old things remain). If you are angry that I cannot eliminate the oval path, how much more ought you to be angry with the spirals, which I abolished. It is as though I have sinned with the oval I have left, even though to you all the rest of the ancients do not sin with so many spirals. This is like being punished for leaving behind one barrow full of shit although I have cleaned the rest of the Augean stables. Or in your sense, you repudiate my oval as one wagon of manure while you tolerate the spirals which are the whole stable, to the extent that my oval is one wagon. But it is unpleasant to tarry in rebutting this most manifest slander.

 Whereas, as already mentioned above, Kepler presented his heliocentric theory to the world in 1609, Longomontanus first honoured Tycho’s memory with his Astronomia Danica in 1622. Using Tycho’s data Longomontanus provided planetary models and planetary tables for Tycho’s geo-heliocentric system. Longomontanus, however, differed from Tycho in that he adopted the diurnal rotation of Helisaeus Roeslin, Nicolaus Raimarus and David Origanus.

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The Astronomia Danica saw two new editions in 1640 and 1663. For the five decades between 1620 and 1670 Kepler’s elliptical astronomy and the Tychonic geo-heliocentric system with diurnal rotation competed for supremacy in the European astronomical community with Kepler’s elliptical system finally triumphing.

 In 1625 Longomontanus suggested to the King, Christian IV, that he should build an observatory to replace Tycho’s Stjerneborg, which had been demolished in 1601. The observatory, the Rundetaarn (Round Tower), was conceived as part of the Trinitatis Complex: a university church, a library and the observatory. The foundation stone was laid on 7 July 1637 and the tower was finished in 1642. Longomontanus was appointed the first director of the observatory, after Leiden 1632 only the second national observatory in Europe.

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Copenhagen – Rundetårn Source: Wikimedia Commons

Both Kepler and Longomontanus, who lost their fathers early, started life as paupers Both of them worked they way up to become leading European astronomers. Kepler has entered the pantheon of scientific gods, whereas Longomontanus has largely been assigned to the dustbin of history. Although Longomontanus cannot be considered Kepler’s equal, I think he deserves better, even if he did back the wrong theory.

 

 

 

 

 

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Spicing up the evolution of the mathematical sciences

When we talk about the history of mathematics one thing that often gets forgotten is that from its beginnings right up to the latter part of the Early Modern Period almost all mathematics was developed to serve a particular practical function. For example, according to Greek legend geometry was first developed by the ancient Egyptians to measure (…metry) plots of land (geo…) following the annual Nile floods. Trade has always played a very central role in the development of mathematics, the weights and measures used to quantify the goods traded, the conversion rates of different currencies used by long distance traders, the calculation of final prices, taxes, surcharges etc. etc. A good historical example of this is the Islamic adoption of the Hindu place value decimal number system together with the associated arithmetic and algebra for use in trade, mirrored by the same adoption some time later by the Europeans through the trader Leonardo Pisano. In what follows I want to sketch the indirect impact that the spice trade had on the evolution of the mathematical sciences in Europe during the Renaissance.

The spice trade does not begin in the Renaissance and in fact had a long prehistory going back into antiquity. Both the ancient Egyptians and the Romans had extensive trade in spices from India and the Spice Islands, as indeed the ancient Chinese also did coming from the other direction.

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The spice trade from India attracted the attention of the Ptolemaic dynasty, and subsequently the Roman empire. Source: Wikimedia Commons

Throughout history spice meant a much wider range of edible, medicinal, ritual and cosmetic products than our current usage and this trade was high volume and financially very rewarding. The Romans brought spices from India across the Indian Ocean themselves but by the Middle Ages that trade was dominated by the Arabs who brought the spices to the east coast of Africa and to the lands at the eastern end of the Mediterranean, known as the Levant; a second trade route existed overland from China to the Levant, the much fabled Silk Road. The Republic of Venice dominated the transfer of spices from the Levant into Europe, shipping them along the Mediterranean.

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The economically important Silk Road (red) and spice trade routes (blue) blocked by the Ottoman Empire c. 1453 with the fall of the Byzantine Empire, spurring exploration motivated initially by the finding of a sea route around Africa and triggering the Age of Discovery. Source: Wikimedia Commons

Here I go local because it was Nürnberg, almost literally at the centre of Europe, whose traders collected the spices in Venice and distributed them throughout Europe. As Europe’s premier spice traders the Nürnberger Patrizier (from the Latin patrician), as they called themselves, grew very rich and looking for other investment possibilities bought up the metal ore mines in central Europe. In a short period of time they went from selling metal ore, to smelting the ore themselves and selling the metal, to working the metal and selling the finished products; each step producing more profit. They quite literally produced anything that could be made of metal from sewing needles to suits of armour. Scientific and mathematical instruments are also largely made of metal and so Nürnberg became Europe’s main centre for the manufacture of mathematical instruments in the Renaissance. The line from spice to mathematical instruments in Nürnberg is a straight one.

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Torquetum designed by Johannes Praetorius and made in Nürnberg

By the middle of the fifteenth century the Levant had become a part of the Ottoman Empire, which now effectively controlled the flow of spices into Europe and put the screws on the prices. The Europeans needed to find an alternative way to acquire the much-desired products of India and the Spice Islands, cutting out the middlemen. This need led to the so-called age of discovery, which might more appropriately be called the age of international sea trade. The most desirable and profitable trade goods being those spices.

The Portuguese set out navigating their way down the west coast of Africa and in 1488 Bartolomeu Dias succeeded in rounding the southern most tip of Africa and entering the Indian Ocean.

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Statue of Bartolomeu Dias at the High Commission of South Africa in London. Source: Wikimedia Commons

This showed that contrary to the Ptolemaic world maps the Indian Ocean was not an inland sea but that it could be entered from the south opening up a direct sea route to India and the Spice Islands.

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A printed map from the 15th century depicting Ptolemy’s description of the Ecumene, (1482, Johannes Schnitzer, engraver). Showing the Indian Ocean bordered by land from the south Source: Wikimedia Commons

In 1497 Vasco da Gama took that advantage of this new knowledge and sailed around the Cape, up the east coast of Africa and then crossing the Indian Ocean to Goa; the final part of the journey only being made possible with the assistance of an Arab navigator.

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The route followed in Vasco da Gama’s first voyage (1497–1499) Source: Wikimedia Commons

Famously, Christopher Columbus mistakenly believed that it would be simpler to sail west across, what he thought was, an open ocean to Japan and from there to the Spice Islands. So, as we all learn in school, he set out to do just that in 1492.

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Columbus sailed the ocean blue.

The distance was of course much greater than he had calculated and when, what is now called, America had not been in the way he and his crews would almost certainly have all died of hunger somewhere out on the open seas.

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Columbus’ voyage. Modern place names in black, Columbus’s place names in blue Source: Wikimedia Commons

The Portuguese would go on over the next two decades to conquer the Spice Islands setting up a period of extreme wealth for themselves. Meanwhile, the Spanish after the initial disappointment of realising that they had after all not reached Asia and the source of the spices began to exploit the gold and silver of South America, as well as the new, previously unknown spices, most famously chilli, that they found there. In the following centuries, eager also to cash in on the spice wealth, the English and French pushed out the Portuguese in India and the Dutch did the same in the Spice Islands themselves. The efforts to establish sea borne trading routes to Asia did not stop there. Much time, effort and money was expended by the Europeans in attempts to find the North West and North East Passages around the north of Canada and the north of Russia respectively; these efforts often failed spectacularly.

So, you might by now be asking, what does all this have to do with the evolution of the mathematical science as announced in the title? When those first Portuguese and Spanish expedition set out their knowledge of navigation and cartography was to say the least very rudimentary. These various attempts to reach Asia and the subsequent exploration of the Americas led to an increased effort to improve just those two areas of knowledge both of which are heavily based on mathematics. This had the knock on effect of attempts to improve astronomy on which both navigation and cartography depend. It is not chance or coincidence that the so-called age of discovery is also the period in which modern astronomy, navigation and cartography came of age. Long distance sea trading drove the developments in those mathematically based disciplines.

This is not something that happened overnight but there is a steady curve of improvement in this disciplines that can be observed over the two plus centuries that followed Dias’ first rounding of the Cape. New instruments to help determine latitude and later longitude such as mariners’ astrolabe (which is not really an astrolabe, around 1500) the backstaff (John Davis, 1594) and the Hadley quadrant (later sextant, 1731) were developed. The Gunter Scale or Gunter Rule, a straight edge with various logarithmic and trigonometrical scales, which together with a pair of compasses was used for cartographical calculations (Edmund Gunter, early seventeenth century). William Oughtred would go on to lay two Gunter Scales on each other and invent the slide rule, also used by navigators and cartographers to make calculations.

New surveying instruments such as the surveyor’s chain (also Edmund Gunter), the theodolite (Gregorius Reisch and Martin Waldseemüller independently of each other but both in 1512) and the plane table (various possible inventors, middle of the sixteenth century). Perhaps the most important development in both surveying and cartography being triangulation, first described in print by Gemma Frisius in 1533.

Cartography developed steadily throughout the sixteenth century with cartographers adding the new discoveries and new knowledge to their world maps (for example the legendary Waldseemüller world map naming America) and searching for new ways to project the three-dimensional earth globe onto two-dimensional maps. An early example being the Stabius-Werner cordiform projection used by Peter Apian, Oronce Fine and Mercator.

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Cordiform projection in a map of the world by Apianus 1524 which is one of the earliest maps that shows America Source: Wikimedia Commons

This development eventually leading to the Mercator-Wright projection, a projection specifically designed for marine navigators based on Pedro Nunes discovery that a path of constant bearing is not a great circle but a spiral, known as a loxodrome or rhumb line. Nunes is just one example of a mathematical practitioner, who was appointed to an official position to develop and teach new methods of navigation and cartography to mariners, others were John Dee and Thomas Harriot.

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Pedro Nunes was professor of mathematics at the University of Coimbra and Royal Cosmographer to the Portuguese Crown. Source: Wikimedia Commons

To outline all of the developments in astronomy, navigation and cartography that were driven by the demands the so-called age of discovery, itself triggered by the European demand for Asian spices would turn this blog post into a book but I will just mention one last thing. In his one volume history of mathematics, Ivor Grattan-Guinness calls this period the age of trigonometry. The period saw a strong development in the use of trigonometry because this is the mathematical discipline most necessary for astronomy, navigation and cartography. One could say a demand for spices led to a demand for geometrical angles.

 

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Sobel’s five books

 

Five Books is an Internet website that invites an expert to discuss in interview format five books that they recommend in a given discipline or academic area. Somebody recently drew my attention to a Five Books interview with pop science writer Dava Sobel asking my opinion of her chosen five books. Although I actually own all of the books that she recommends I have serious problems with her choices that start with the title of interview, The best books on The Early History of Astronomy recommended by Dava Sobel.

I remain a sceptic about a lot of the claims made by archeoastronomers concerning supposed astronomical alignments of various archaeological features but I am quite happy to admit that Stonehenge, for example, does have such an alignment, which would place early astronomy at least as early as the third millennium BCE. Maybe astronomy and not archaeoastronomy was meant it which case we would be in the second millennium BCE with the Babylonians. Perhaps Ms Sobel thinks astronomy doesn’t really start until we reach the ancient Greeks meaning about five hundred BCE. But wait, all five of her books are about astronomy in the sixteenth and seventeenth centuries CE! This is not by any definition the early history of astronomy. What is in fact meant is the early history of the Copernican heliocentric theory.

We now turn to the books themselves. I should point out before I start that I actually own and have read all five of the books that Sobel has chosen, so my criticisms are well informed.

First up we have Owen Gingerich’s The Book Nobody Read. This is not actually a book on the history of astronomy. During his years of research into the history of astronomy Gingerich carried out a census of the existing copies of the first and second editions of Copernicus’ De revolutionibus, which I also own. The Book Nobody Read is a collection of personal anecdotes about episodes involved in the creation of that census. Sobel also repeats a major error that Gingerich made in choosing his title.

Five Books: And that is why the 20th century author and journalist Arthur Koestler dismissed it as “the book that nobody read”, which is something that Owen Gingerich is at pains to correct with this book.

Sobel: Yes, he is referring to Koestler’s comment with his title. This was the insult hurled at Copernicus’s book because it is so long and mathematical.

During his census Gingerich recorded the annotations in all of the copies of De revolutionibus that he examined showing that people in the sixteenth and seventeenth centuries did indeed read the book. However, Koestler’s comment was not addressed at those original readers but at the wanna be historians in the nineteenth century during the Copernicus renaissance (Copernicus effectively disappeared out of the history of astronomy in the early seventeenth century and only returned with Kant’s “Copernican Turn” in the late eighteenth century leading to the concept of the Copernican revolution), who claimed that De revolutionibus was mathematically simpler than the prevailing geocentric model, as Koestler showed this was not the case prompting him to make his famous quip about “the book nobody read.”

Next up we have Robert Westman’s The Copernican Question. Now I’m a Westman fan, who has learnt much over the years reading almost every thing that he has written. However, The Copernican Question is a complex, highly disputed book that I would not recommend for somebody new to the subject.

Sobel’s third choice is Galileo’s Sidereus Nuncius, once again not a book that I would recommend for a beginner. To understand Sidereus Nuncius you really need to understand it in the context in which it was written. There are also several comments made by Sobel that are to say the least dubious.

Sobel: This is a thrilling book. It is the moment that astronomy became an observational science.

Astronomy has always been an observational science!

Sobel: Until Galileo’s time, the most that anyone could know about a planet was where it was.

You could also determine its orbit, its speed and its apparent relative distance from the earth.

Sobel: With his telescope Galileo was able to determine the composition of the moon.

Galileo could determine that the moon was not smooth but was mountainous like the earth, which is not quite the same as determining its composition. We had to wait for the Apollo Programme for that.

Five Books: How did he manage to get hold of the telescope?

Sobel: He had heard of such a thing being invented as a novelty and so he figured out how to build one. And although at first he considered it a military tool, which was passed to the navy in Italy to keep watch on the horizon for enemy ships, he very soon realised he could turn it skywards. So he made these amazing discoveries and published them.

The telescope was not invented as a novelty; its inventor, Lipperhey, offered it to the States General in the Dutch Republic as a military tool. There was of course no navy in Italy; in fact there was in that sense no Italy. Galileo offered his telescope to the Venetian Senate, in fact to be able to observe ships approaching the port earlier than with the naked eye, both for trade and military purposes.

Number four is Stillman Drake’s Galileo at Work. On the face of it an excellent choice but however one with a slight blemish, Drake is a straight up Galileo groupie, which makes his descriptions and judgements somewhat less than objective. Here once again we find a more than somewhat strange claim by Sobel

Five Books: And the church didn’t have an issue with what he was doing?

Sobel: Not at that point. The minute he started agreeing out loud with Copernicus and writing about it in Italian and not Latin then he became more controversial. The Sidereal Messenger is written in Latin but soon after that he switched to Italian and that is when it became an issue. His controversial views were investigated by the Roman Inquisition which concluded that his ideas could only be supported as a possibility and not an established fact, and he spent the rest of his life under house arrest.

Galileo’s choice of Italian as the language in which he wrote his Dialogo had little or nothing to do with his trial and eventual condemnation by the Inquisition.

Sobel’s final choice is more than somewhat bizarre, Arthur Koestler’s The Sleepwalkers.

Five Books: Lastly, you have chosen The Sleepwalkers by Arthur Koestler, which is an overview of that period, though he is not quite so complimentary about Copernicus and Galileo as the other authors you have chosen.

 Sobel: Arthur Koestler was a journalist with an interest in science. He really got fascinated by this subject. So this book traces the early history of astronomy because he too found it fascinating. Unfortunately, as you say, he didn’t like Copernicus, or Galileo for that matter. The only one he seems to really have liked was Kepler. So one reads his book sceptically. But it is a book that was widely read and it had a tremendous influence on people. Even though it came out in the 1950s you still meet people who will talk about that book. And for many it was the book that got them interested in astronomy. I read it years ago as well and it has stayed with me.

Now, Sleepwalkers is without doubt one of the five most influential books in my development as a historian of science and I still have my much thumbed copy bought when I was still comparatively young, but it is severely dated and I would certainly not recommend it today as an introductory text on the history of astronomy. Koestler’s book started out as the first full length English biography of Kepler and this is why Kepler takes the central position in his book. On Koestler’s treatment of Copernicus and Galileo we get the following:

Five Books: Why do you think he was so scathing of Copernicus and Galileo?

 Sobel: It is hard to say. He found Copernicus dull, and I admit that his book On the Revolution makes dull reading for a person who is not capable of understanding the maths. But Copernicus is far from dull.

Both Copernicus and Galileo acolytes detest Koestler’s book for his portrayals of their heroes. He didn’t find Copernicus dull he labels him “The Timid Canon “ because he thought that Copernicus lacked the courage of his convictions as far as his heliocentric theory was concerned. This is a hard but not unfair judgement of Copernicus’s behaviour. As far as Galileo is concerned, Koestler is one of the earliest authors to attack and demolish the Galileo hagiography, in particular with reference to his problems with the Church.

I wrote this blog post because one of my followers on Twitter asked my opinion of Sobel’s list. As I said at the beginning I own all of these five books and think all of them are in some sense good, however as a recommendation for somebody to learn about the early phase of heliocentricity in the Early Modern Period I find it a not particularly appropriate collection.

This of course immediately raises the question what I would recommend for this purpose. I hate this question. I have acquired my knowledge of the subject over the years by reading umpteen books and even more academic papers and filtering out the reliable facts and information from this vast collection of material. The moment I recommend a book I start to qualify my recommendation but you must also read this paper and chapter 10 in that book and you really need to look at… On the whole I would recommend people to start with John North’s Cosmos: An Illustrated History of Astronomy and Cosmology and if they want to discover more to proceed with North’s bibliographical recommendations.

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Filed under Book Reviews, History of Astronomy, Renaissance Science

Some good Copernican mythbusting

For those who haven’t already seen it Tim O’Neill, Renaissance Mathematicus friend and guest blogger, has posted a superb essay on his excellent blog, History for Atheists, on the myths surrounding the dissemination, publication and reception of Copernicus’ heliocentric theory, The Great Myths 6: Copernicus’ Deathbed Publication. Regular readers of the Renaissance Mathematicus won’t learn anything new but it is an excellent summary of the known historical facts and well worth a read. As with this blog the comments are also well worth reading.

The earliest mention of Copernicus’ theory – Matthew of Miechów’s 1514 catalogue

Go on over and give Tim a boost!

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400 Years of The Third Law–An overlooked and neglected revolution in astronomy

Four hundred years ago today Johannes Kepler rediscovered his most important contribution to the evolution of astronomy, his third law of planetary motion.

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Portrait of Johannes Kepler 1610 by unknown artist. Source: Wikimedia Commons

He had originally discovered it two months earlier on 8 March but due to a calculation error rejected it. On 15 May he found it again and this time recognised that it was correct. He immediately added it to his Harmonices Mundi:

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For when the true distances between the spheres were found, through the observations of Brahe, by continuous toil for a very long time, at last, at last, the genuine proportion of the periodic times to the proportion of the spheres –

Only at long last did she look back at him as she lay motionless,

But she look back and after a long time she came [Vergil, Eclogue I, 27 and 29.]

And if you want the exact moment in time, it was conceived mentally on the 8th of March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely exact that proportion between the periodic times of any two planets is precisely the sesquialterate[1] proportion of their mean distances, that is of the actual spheres, though with this in mind, that the arithmetic mean between the two diameters of the elliptical orbit is a little less than the longer diameter. Thus if one takes one third of the proportion from the period, for example, of the Earth, which is one year, and the same from the period of Saturn, thirty years, that is, the cube roots, and one double that proportion, by squaring the roots, he has in the resulting numbers the exactly correct proportion of the mean distances of the Earth and Saturn from the Sun.[2]

writing a few days later:

Now, because eighteen months ago the first dawn, three months ago the broad daylight, but a very few days ago the full sun of a most remarkable spectacle has risen, nothing holds me back. Indeed, I give myself up to a sacred frenzy.

He finished the book on 27 May although the printing would take a year.

In modern terminology:

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The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit: i.e. for two planets with P = orbital period and R = semi-major axis P12/P22=R13/R23

Kepler’s third law is probably the most important discovery on the way to the establishment of a heliocentric astronomy but its importance was initially overlooked and its implications were somehow neglected until Isaac Newton displayed its significance in his Principia Mathematica, published in 1687 sixty-eight years after the third law first appeared in print.

What the third law gives us is a direct mathematical relationship between the size of the orbits of the planets and their duration, which only works in a heliocentric system. In fact as we will see later it’s actually equivalent to the law of gravity. There is nothing comparable for either a full geocentric system or for a geo-heliocentric Tychonic or semi-Tychonic system. It should have hit the early seventeenth-century astronomical community like a bomb but it didn’t, which raises the question why it didn’t.

The main answer lies in Kepler’s own writings. Although he viewed its discovery as the crowning glory of his work on the Harmonices Mundi Kepler didn’t give it any prominence in that work. The Harmonices Mundi is a vast sprawling book explicating Kepler’s version of the Pythagorean theory of the harmony of the spheres in five books. After four introductory books covering plane geometry, music theory and astrology Kepler gets down to harmonic planetary theory in the fifth and final book. Book V, 109 pages in the English translations, contains lots of musical relationships between various aspects of the planetary orbits, with the third law presented as just one amongst the many with no particular emphasis. The third law was buried in what is now regarded as a load of unscientific dross. Or as Carola Baumgardt puts it, somewhat more positively,  in her Johannes Kepler life and letters (Philosophical Library, 1951, p. 124):

Kepler’s aspirations, however, go even much higher than those of modern scientific astronomy. As he tried to do in his “Mysterium Cosmographicum” he coupled in his “Harmonice Mundi” the precise mathematical results of his investigations with an enormous wealth of metaphysical, poetical, religious and even historical speculations. 

Although most of Kepler’s contemporaries would have viewed his theories with more sympathy than his modern critics the chances of anybody recognising the significance of the harmony law for heliocentric astronomical theory were fairly minimal.

The third law reappeared in 1620 in the second part of Kepler’s Epitome Astronomiae Copernicanae, a textbook of heliocentric astronomy written in the form of a question and answer dialogue between a student and a teacher.

How is the ratio of the periodic times, which you have assigned to the mobile bodies, related to the aforesaid ratio of the spheres wherein, those bodies are borne?

The ration of the times is not equal to the ratio of the spheres, but greater than it, and in the primary planets exactly the ratio of the 3/2th powers. That is to say, if you take the cube roots of the 30 years of Saturn and the 12 years of Jupiter and square them, the true ration of the spheres of Saturn and Jupiter will exist in those squares. This is the case even if you compare spheres that are not next to each other. For example, Saturn takes 30 years; the Earth takes one year. The cube root of 30 is approximately 3.11. But the cube root of 1 is 1. The squares of these roots are 9.672 and 1. Therefore the sphere of Saturn is to the sphere of the Earth as 9.672 is to 1,000. And a more accurate number will be produced, if you take the times more accurately.[3]

Here the third law is not buried in a heap of irrelevance but it is not emphasised in the way it should be. If Kepler had presented the third law as a table of the values of the orbit radiuses and the orbital times and their mathematical relationship, as below[4], or as a graph maybe people would have recognised its significance. However he never did and so it was a long time before the full impact of the third law was felt in astronomical community.

third law001

The real revelation of the significance of the third law came first with Newton’s Principia Mathematica. By the time Newton wrote his great work the empirical truth of Kepler’s third law had been accepted and Newton uses this to establish the empirical truth of the law of gravity.

In Book I of Principia, the mathematics and physics section, Newton first shows, in Proposition 11[5], that for a body revolving on an ellipse the law of the centripetal force tending towards a focus of the ellipse is inversely as the square of the distance: i.e. the law of gravity but Newton is not calling it that at this point. In Proposition 14[6] he then shows that, If several bodies revolve about a common center and the centripetal force is inversely as the square of the distance of places from the center, I say that the principal latera recta of the orbits are as the squares of the areas which bodies describe in the same time by radii drawn to the center. And Proposition 15[7]: Under the same supposition as in prop. 14, I say the square of the periodic times in ellipses are as the cubes of the major axes. Thus Newton shows that his law of gravity and Kepler’s third law are equivalent, although in this whole section where he deals mathematically with Kepler’s three laws of planetary motion he never once mentions Kepler by name.

Having established the equivalence, in Book III of The Principia: The System of the World Newton now uses the empirical proof of Kepler’s third law to establish the empirical truth of the law of gravity[8]. Phenomena 1: The circumjovial planets, by radii drawn to the center of Jupiter, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 2: The circumsaturnian planets, by radii drawn to the center of Saturn, describe areas proportional to the times, and their periodic times—the fixed stars being et rest—are as 3/2 powers of their distances from that center. Phenomena 3: The orbits of the five primary planets—Mercury, Venus, Mars, Jupiter, and Saturn—encircle the sun. Phenomena 4: The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun—the fixed stars being at rest—are as the 3/2 powers of their mean distances from the sun. “This proportion, which was found by Kepler, is accepted by everyone.”

Proposition 1: The forces by which the circumjovial planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the center of Jupiter and are inversely as the squares of the distances of their places from that center. “The same is to be understood for the planets that are Saturn’s companions.” As proof he references the respective phenomena from Book I. Proposition 2: The forces by which the primary planets are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the sun and are inversely as the squares of the distances of their places from its center. As proof he references the respective phenomenon from Book I:

One of the ironies of the history of astronomy is that the general acceptance of a heliocentric system by the time Newton wrote his Principia was largely as a consequence of Kepler’s Tabulae Rudolphinae the accuracy of which convinced people of the correctness of Kepler’s heliocentric system and not the much more important third taw of planetary motion.

[1] Sesquialterate means one and a half times or 3/2

[2] The Harmony of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aiton, A.M. Duncan & J.V. Field, Memoirs of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, Volume 209, 1997 pp. 411-412

[3] Johannes Kepler, Epitome of Copernican Astronomy & Harmonies of the World, Translated by Charles Glenn Wallis, Prometheus Books, New York, 1995 p. 48

[4] Table taken from C.M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy, CUP, Cambridge etc., 2004 p. 198

[5] Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy, A New Translation by I: Bernard Cohen and Anne Whitman assisted by Julia Budenz, Preceded by A Guide to Newton’s Principia, by I. Bernard Cohen, University of California Press, Berkley, Los Angeles, London, 1999 p. 462

[6] Newton, Principia, 1999 p. 467

[7] Newton, Principia, 1999 p. 468

[8] Newton, Principia, 1999 pp. 797–802

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Filed under History of Astronomy, History of science, Newton, Renaissance Science, Uncategorized

Who cares about facts? – Make up your own, it’s much more fun!

Math Horizons is a magazine published by Taylor & Francis for the Mathematical Association of America aimed at undergraduates interested in mathematics: It publishes expository articles about “beautiful mathematics” as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews. (Description taken from Wikipedia, which attributes it to the Math Horizons instructions for authors January 3 2009). Apparently, however, authors are not expected to adhere to historical facts, they can, it seems, make up any old crap.

The latest edition of Math Horizons (Volume 25, Issue 3, February 2018) contains an article by a Stephen Luecking entitled Albrecht Dürer’s Celestial Geometry. As I am currently, for other reasons, refreshing my knowledge of Albrecht the mathematician I thought, oh that looks interesting I must read that. I wish I hadn’t.

Luecking’s sub-title seems innocent enough: Renaissance artist Albrecht Dürer designed a specialty compass for astronomical drawings, but when you read the article you discover that Luecking says an awful lot more and most of it is hogwash. What does he have to say?

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Albrecht Dürer Self-Portrait 1500 Source: Wikimedia Commons

Albrecht Dürer (1471–1528), noted Renaissance printer and painter, twice left his native Germany for sojourns to Italy, once from 1494 to 1495 and again from 1505 to 1507. During those years his wide-ranging intellect absorbed the culture and thinking of noted artists and mathematicians. Perhaps the most important
 outcome of these journeys was his
introduction to scientific methods. 
His embrace of these methods
 went on to condition his thinking 
for the rest of his life. 


So far so good. However what Dürer absorbed on those journeys to Italy was not scientific methods but linear perspective, the mathematical method, developed in Northern Italy in the fifteenth century, to enable artists to represent three dimensional reality realistically in a two dimensional picture. Dürer played a significant role in distributing these mathematical techniques in Europe north of the Alps. His obsession with mathematics in art led to him developing the theory that the secret of beauty lay in mathematical proportion to which de devoted a large part of the rest of his life. He published the results of his endeavours in his four-volume book on human proportions, Vier Bücher von Menschlicher Proportion, in the year of his death, 1528.

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Title page of Vier Bücher von menschlicher Proportion showing the monogram signature of artist Source: Wikimedia Commons

If Dürer wanted to learn scientific methods, by which, as we will see Luecking means astronomy, he could and probably did learn them at home in Nürnberg. Dürer was part of the humanist circle of Willibald Pirckheimer, he close friend and patron.

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Engraving of Willibald Pirckheimer at 53 by Albrecht Dürer, 1524. We live by the spirit. The rest belongs to death. Source: Wikimedia Commons

Franconian houses are built around a courtyard; Dürer was born in the rear building of the Pirckheimer house on the market square in Nürnberg. Although his parents bought their own house a few years later Albrecht and Willibald remained close friends and possibly even lovers all of their lives. Pirckheimer was a big supporter of the mathematical sciences—astronomy, mathematics, cartography and astrology—and his circle included, amongst others, Johannes Stabius, Johannes Werner, Erhard Etzlaub, Georg Hartmann, Konrad Heinfogel and Johannes Schöner all of whom were either astronomers, mathematicians, cartographers, instrument makers or globe makers some of them all five and all of them friends of Dürer.

Next up Luecking tells us:

One notable
consequence was Dürer’s abandonment of astrological subject
matter—a big seller for a printer
and publisher such as himself—in favor of astronomy.

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Albrecht Dürer Syphilis 1496 Syphilis was believed to have an astrological cause Source: Wikimedia Commons

Luecking offers no evidence or references for this claim, so I could offer none in saying that it is total rubbish, which it is. However I will give one example that shows that Albrecht Dürer was still interested in astrology in 1517. Lorenz Beheim (1457–1521) was a humanist, astrologer, physician and alchemist, who was a canon of the foundation of the St Stephan Church in Bamberg, he was a close friend of both Pirckheimer and Dürer and corresponded regularly with Pirckheimer. In a letter from 8 December 1517 he informed Pirckheimer that Johannes Schöner was coming to Nürnberg with printed celestial globes that could be used for astrology, which if his wished could be acquired by him and Albrecht Dürer. He would not have passed on the information if he thought that they wouldn’t be interested. Beheim also cast horoscopes for both Pirckheimer and Dürer.

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Gores for Johannes Schöner’s Celestial Globe 1517  Source: Hans Gaab, Die Sterne Über Nürnberg: Albrecht Dürer und seine Himmelskarten von 1515, Nürnberger Astronomische Gesellschaft, Michael Imhof Verlag, 2015 p. 115

 

Next up Luecking starts, as he means to go on, with pure poppycock. All of the above Nürnberger mathematician, who all played significant roles in Dürer’s life, were of course practicing astrologers.

Astronomy was not to be a casual interest. Just before his second trip to Italy, Dürer published De scientia motus orbis, a cosmological treatise by the Persian Jewish astronomer Masha’Allah ibn Atharī (ca. 740–815 CE). Since Masha’Allah wrote the treatise for laymen and included ample illustrations, it was a good choice for introducing Europeans to Arabic astronomy.

The claim that Dürer published Masha’Allah’s De scientia motus orbis is so mind bogglingly wrong anybody with any knowledge of the subject would immediately stop reading the article, as it is obviously a complete waste of time and effort. The book was actually edited and published by Johannes Stabius and printed by Weissenburger in Nürnberg in 1504.

The woodcut illustrations came from the workshop of Albrecht Dürer, but probably not from Dürer himself. There were traditionally attributed to Hans Süß von Kulmbach (1480–1522), one of Dürer’s assistants, who went on to become a successful painter in his own right, but modern research has shown that Süß didn’t move to Nürnberg until 1505, a year after the book was published.

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Hans Süß portrait  Source: Wikimedia Commons

Although Luecking wants Masha’Allah to be an astronomer he was in fact a very famous astrologer, who amongst other things cast the horoscope for the founding of Bagdad. De scientia motus orbis is indeed a book on Aristotelian cosmology and physics but it includes his theory that there are ten heavenly spheres not eight as claimed by Aristotle. His extra heavenly spheres play a significant role in his astrological theories. It is very common practice for astrologers, starting with Ptolemaeus, to publish their astronomy and astrology in separate books but they are seen as complimentary volumes. From their beginnings in ancient Babylon down to the middle of the seventeenth century astronomy and astrology were always seen as two sides of the same coin.

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Title page De scientia motus orbis Although this woodcut is usually titled The Astronomer I personally think the figure looks more like an astrologer Source: Wikimedia Commons

In 1509 Dürer purchased the entire library of Regiomontanus (1436–1476 CE) from the estate of Nuremberg businessman Bernhard Walther. Regiomontanus was Europe’s leading astronomer,
a noted mathematician, and a designer of astronomical instruments. Walther had sponsored Regiomontanus’s residency in Nuremberg between 1471 and 1475. Part of Walther’s largesse was to provide a print shop from which Regiomontanus published the world’s first scientific texts ever printed.

Regiomontanus was of course first and foremost an astrologer and most of those first scientific texts that he published in Nürnberg were astrological texts. Walther did not sponsor Regiomontanus’ residency in Nürnberg but was his colleague and student in his endeavours in the city. An analysis of Walther’s astronomical observation activities in Nürnberg after Regiomontanus’ death show that he too was an astrologer rather than an astronomer. When Regiomontanus came to Nürnberg he brought a very large number of manuscripts with him, intending to edit and publish them. When he died these passed into Walther’s possession, who added new books and manuscripts to the collection. The story of what happened to this scientific treasure when Walther died in 1504 is long and very complicated. In fact Dürer bought not “the entire library” but a mere ten manuscripts not when he bought Walther’s house, the famous Albrecht Dürer House, in 1509 but first in 1522.

In 1515, Dürer and Austrian cartographer and mathematician Johannes Stabius produced the first map of the world portraying the earth as a sphere.

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Johannes Statius portrait by Albrecht Dürer Source: Wikimedia Commons

The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The earliest know printed world map portraying the earth as a sphere is a woodcut in a Buchlein über die Kunst Corsmographia, (Booklet about the Art of Cosmographia) published in Nürnberg in about 1490. There are others that predate the Stabius-Dürer map most notably on the title page of Waldseemüller’s Die Welt Kugel (The Earth Sphere) published in Straßburg in 1509.

There are no surviving copies of the Stabius-Dürer world map from the sixteenth century so we don’t actually know what it was produced for. The woodblocks survived and were rediscovered in the 18th century.

It is however dedicated to both the Emperor Maximilian, Stabius’s employer who granted the printing licence, and Cardinal Matthäus Lang, so it might well have been commissioned by the latter. Lang commissioned the account of Magellan’s circumnavigation on which Schöner based his world map of that circumnavigation.

Afterward, Stabius proposed continuing their collaboration by publishing a star map—the first such map published in Europe. Their work relied heavily on data assembled by Regiomontanus, plus refinements from Walther.

It will probably not surprise you to discover that this was not “the first such map published in Europe. It’s the first printed one but there are earlier manuscript ones, two of which from 1435 in Vienna and 1503 in Nürnberg probably served as models for the Stabius–Dürer–Heinfogel one. Their work did not rely “heavily on data assembled by Regiomontanus, plus refinements from Walther” but was based on Ptolemaeus’ star catalogue from the Almagest. There is a historical problem in that there was not printed copy of that star catalogue available at the time so they probably work from one or more manuscripts and we don’t know which one(s). The star map contains the same dedications to Maximilian and Lang as the world map so one again might have been a commission from Lang, Stabius acting as the commissioning agent. Stabius and Lang studied together at the University of Ingolstadt.

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Stabs-Dürer-Heinfogel Star Map Northern Hemisphere Source: Ian Ridpath’s Star Tales

For more details on the star maps go here

The star map required imprinting the three- dimensional dome of the heavens onto a two- dimensional surface without extreme distortions, a task that fell to Stabius. He used a stereographic projection. In this method, rays originate at the pole in the opposite hemisphere, pass through a given point in the hemisphere, and yield a point on a circular surface.

You will note that I have included the name of Konrad Heinfogel to the producers of the map and it was actually he, and not Stabius, who was responsible for the projection of the map and the location of the individual stars. In fact in this project Johannes Stabius as commissioning agent was project leader, Konrad Heinfogel was the astronomical expert and Albrecht Dürer was the graphic artist hired to draw the illustration. Does one really have to point out that in the sixteenth century star maps were as much, if not more, for astrologers than for astronomers.

Luecking now goes off on an excurse about the history of stereographic projection, which ends with the following paragraph.

As the son of a goldsmith, Dürer’s exposure to stereographic projection would have been by way of the many astrolabes being fabricated in Nuremburg, then Europe’s major center for instrument makers. As the 16th century moved on, the market grew for such scientific objects as astrology slipped into astronomy. Handcrafted brass instruments, however, were affordable only to the wealthy, whereas printed items like the Dürer-Stabius maps reached a wider market.

Nürnberg was indeed the major European centre for the manufacture of scientific instruments during Dürer’s lifetime but scientific instrument makers and goldsmiths are two distinct professional groups, so Luecking’s argument falls rather flat, although of course Dürer would have well acquainted with the astrolabes made by his mathematical friends. Astrolabes are of course both astrological and astronomical instruments and astrology did not slip into astronomy during the 16th century. In fact the 16th century is regarded by historians as the golden age of astrology.

There now follows another excurse on the epicycle-deferent model of planetary orbits as a lead up to the articles thrilling conclusion.

In his 1525 book Die Messerung (On Measurement), Dürer presents an instrument of his own design used to draw these and other more general curves. This compass for drawing circles upon circles consisted of four telescoping arms and calibrated dials. An arm attached to the first dial could rotate in a full circle, a second arm fixed to another dial mounted on the end of this first arm could rotate around the end of the first arm, and so on.

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Dürer’s four arm compass

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Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

The title of Dürer’ 1525 book is actually Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Straightedge). It is a basic introduction to geometry and its applications, which Dürer wrote when he realised that his Vier Bücher von Menschlicher Proportion was too advanced for the artist apprentices that he thought should read it. The idea was first read and digest the Underweysung then read the Vier Bücher von Menschlicher Proportion.

Luecking tells us that:

As a trained metalsmith, Dürer possessed the expertise to craft this complex tool. Precision calibration and adjustable arms allowed its user to plot an endless number of curves by setting the length of each telescoping arm and determining the rate at which the arms turned. This, in effect, constituted manual programming by setting the parameters of each curve plotted.

As a teenager Dürer did indeed serve an apprenticeship under his father as a goldsmith, but immediately on completing that apprenticeship he undertook a second apprenticeship as a painter with Michael Wolgemut from 1486 to 1490 and dedicated his life to painting and fine art printing. Luecking has already correctly stated that Nürnberg was the major European centre for scientific instrument making and Dürer almost certainly got one of those instrument makers to produce his multi-armed compass. Luecking describes the use to which Dürer put this instrument in drawing complex geometrical curves. He then goes on to claim that Dürer might actually have constructed it to draw the looping planetary orbits produced by the epicycle-deferent model. There is absolutely no evidence for this in the Underweysung and Luecking’s speculation is simple pulled out of thin air.

To summarise for those at the back who haven’t been paying attention. Dürer did not absorb scientific methods in Italy. He did not abandon astrology for astronomy. He didn’t publish Masha’Allah’s De scientia motus orbis, Johannes Stabius did. Dürer only bought ten of Regiomontanus’ manuscripts and not his entire library. The Stabius-Dürer world map was not “the first map of the world portraying the earth as a sphere”. The Stabius–Dürer–Heinfogel star charts were the first star-charts printed in Europe but by no means the first ones published. Star charts are as much astrological, as they are astronomical. Astrology did not slip into astronomy in the 16th century, which was rather the golden age of astrology. There is absolutely no evidence that Dürer’s multi-arm compass, as illustrated in his geometry book the Underweysung, was ever conceived for drawing the looping orbits of epicycle-deferent planetary models, let alone used for this purpose.

It comes as no surprise that Stephen Luecking is not a historian of mathematics or art for that matter. He is the aged (83), retired chairman of the art department of DePaul University in Chicago.

Whenever I come across an article as terrible as this one published by a leading scientific publisher in a journal from a major mathematical organisation such as the MAA I cringe. I ask myself if the commissioning editor even bothered to read the article; it was certainly not put out to peer review, as any knowledgeable Dürer expert would have projected it in an elegant geometrical curve into his trashcan. Above all I worry about the innocent undergraduates who are subjected to this absolute crap.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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