Category Archives: Renaissance Science

The Seaman’s Secrets

Regular readers of my series of posts on English mathematical practitioners in the late sixteenth and early seventeenth centuries might have noticed the name John Davis popping up from time to time. Unlike most of the other mathematical practitioners featured here in the early modern history of cartography, navigation, and scientific instrument design, who were basically mathematicians who never or seldom went to sea, John Davis (c. 1550–1605) was a mariner and explorer, who was also a mathematician, who wrote an important and widely read book on the principles of navigation, which included the description of an important new instrument that he had designed.  

Miniature engraved portrait of navigator John Davis (c. 1550-1605), detail from the title page of Samuel Purchas’s Hakluytus Posthumus or Purchas his Pilgrimes (1624) Source: Wikimedia Commons

John Davis was born and grew up in Sandridge Barton, the manor farm on the Sandrige Estate of Stoke Gabriel in Devon, a small village on the river Dart about six kilometres up-river from Dartmouth, which was an important port in the early modern period, so it seems that Davis was destined to go to sea. Amongst his neighbours, on the Sandrige Estate, were the five sons of the Gilbert-Raleigh family, Humphrey, John and Adrian Gilbert and their half brothers Carew and Walter Raleigh. Both Humphrey Gilbert (c. 1539–1583) and Walter Raleigh (c. 1552–1618) were important Elizabethan explorers and Carew Raleigh (c. 1550–c. 1625) was a naval commander. Adrian Gilbert (c. 1541–1628), who became an MP was an intimate friend of the young John Davis as was Walter Raleigh. Little is known of his childhood and youth, but we do know that he early became a friend and pupil of the leading Elizabethan mathematical practitioner, John Dee (1527–c. 1608). Davis’ friendship with the Gilbert-Raleigh brothers and John Dee would prove helpful in his first major exploration endeavour, the search for the Northwest Passage. 

Stoke Gabriel Source
The Dartmouth Town Council blue plaque erected in memory of Davis Source: Wikimedia Commons

Throughout the Middle Ages, Europe had imported good, in particular spices, from Asia via a complex, largely overland route that ended in Northern Italy, from whence the city of Nürnberg distributed them all over Europe.  As the European began to venture out onto the high seas in the fifteenth century, the question arose, whether it was possible to reach Asia directly by sea? The Portuguese began to edge their way down the West African coast and in 1487/88 Bartolomeu Dias (c. 1450–1500) succeeded in rounding the southern end of Africa.

Source: Wikimedia Commons

Between 1479 and 1499 Vasco da Gama (c. 1460–1524) succeeded, with the help of an Arabic pilot, in crossing the Indian Ocean and bringing back a cargo of spices from India to Portugal. This established an initial Portuguese dominance over the oceanic route sailing eastwards to Asia, which with time they extended to the so-called Spice Islands. 

Vasco da Gama Source: Wikimedia Commons

As every school kid knows, Christopher Columbus (1451–1506) believed that there was open water between the west coast of Europe and east coast of China, and that he could reach Asia faster and easier sailing west across the ocean rather the east around Africa. In 1492, he put his theory to the test and, having vastly underestimated the distance involved, just as he was running out of food, his small fleet fortuitously ran into the Americas, although they weren’t called that yet. In 1519, the Spanish seaman Ferdinand Magellan (1480–1521) proved it was possible to get past the southern tip of America and into the Pacific Ocean. The last remnants of his very battered fleet returning to Spain, without Magellan, who was killed on the way, in 1522, becoming the first people to circumnavigate the globe. 

In 1577, Francis Drake (c. 1540­–1596) set out to attack the Spanish on the west coast of the America, decided to return via the Pacific Ocean arriving back in England in 1580, becoming the second to circumnavigate the globe, and the first commander to survive the journey. Between 1586 and 1588, Thomas Cavendish (1560-­1592), a protégé of Walter Raileigh, became the third man to circumnavigate the globe, on what was the first planned voyage to do so. 

Thomas Cavendish An engraving from Henry Holland’s Herōologia Anglica (1620). Source: Wikimedia Commons

The successful circumnavigations via the southern tip of the Americas led to speculation whether it was possible to reach the Pacific Ocean by rounding the northern end of the Americas. These speculations led to the search for the so-called Northwest Passage, an endeavour in which English mariners would dominate. 

Already in 1497, Henry VII sent the Italian mariner, John Cabot (C. 1450–c. 1500) to attempt to find the Northwest Passage. He is thought to have landed once somewhere on the coast of what is now Canada before returning to Bristol. In 1508, Cabot’s son Sebastian (c. 1474–1557) followed his father in trying to find the Northwest Passage. He is thought to have sailed as far north as Hudson Bay. In 1524, the Portuguese mariner, Estêvão Gomes (c. 1483–1538), who had mutinied on the Magellan circumnavigation, bringing his ship back to Spain in 1521, was commissioned by the Spanish Crown to seek a northern route through the Americas, reaching Nova Scotia before returning to Spain. 

In 1551, the Muscovy Trading Company was founded in London with the specific intention of finding a Northeast Passage to China by sailing around the northern coast of Russia. A project for which they were granted exclusive rights by the English Crown. The Muscovy Company employed John Dee to teach cartography and navigation to its ships’ officers. They failed in their endeavour to find the Northeast Passage but did establish successful trading deals with Russia.

According to Charlotte Fell Smith, this portrait was painted when Dee was 67. It belonged to his grandson Rowland Dee and later to Elias Ashmole, who left it to Oxford University. Source: Wikimedia Commons

In the 1560s Humphrey Gilbert wrote a detailed treatise supporting the idea of a government supported endeavour to search for the Northwest Passage. In 1574, the privateer Martin Frobisher (c. 1535–1594) petitioned the Privy Council for permission and financial support for an expedition to find the Northwest Passage. They referred him to the Muscovy Company, who eventually agreed to licence his voyage. Altogether Frobisher undertook three attempts, in 1596 with three ships, in 1597 with a much larger fleet and finally in 1578 with a total of fifteen ships. Although he explored much of the coast and islands of Northern Canada the undertaking was basically an expensive flop. On the second expedition Frobisher’s master was Christopher Hall. Frobisher and Hall were coached by Dr John Dee in geometry and cosmography in order to improve their use of the instruments for navigation in their voyage. 

Full-length life-size oil painting portrait of English explorer Martin Frobisher commissioned by the Company of Cathay to commemorate his 1576 Northwest Passage voyage and promote the planned follow-up expedition of 1577 painted by Cornelius Ketel Source:Wikimedia Commons

In 1583, Humphrey Gilbert launched an attempt, based on letters patent, that he had acquired from the crown in 1578, to establish an English colony in North America. His half-brother Walter Raleigh sailed with him but had to turn back due to lack of food on his ship. Having taken possession of Newfoundland by force, he then left again without establishing a colony due to lack of supplies. The return journey was a disaster with the loss of the biggest vessel with most of there stores and Gilbert died of blood poisoning, having stepped on a nail.

Portrait Sir Humphrey Gilbert artist unknown Source: Wikimedia Commons

The only halfway positive outcome was that Walter Raleigh received a royal charter based on Gilbert’s letters patent and would in turn go on to found, with Thomas Harriot (c. 1560–1621), as his cartography and navigation advisor, the first English colony in North America on Roanoke Island in 1584. Only halfway positive because the Roanoke colony was also a failure. 

It was against this background of one hundred years of failure, from John Cabot to Martin Frobisher, to find a northwest passage that John Davis became involved in the launching of yet another expedition to find one, initiated by his childhood friend Adrian Gilbert and John Dee. Gilbert and Dee, appealed to Sir Francis Walsingham (1573–1590) Secretary of State for funding in 1583. Whilst Walsingham favoured the idea politically, no money from the state was forthcoming. Instead, the planned expedition was financed privately by the London merchant, William Sanderson (c.1548–1638). 

Sanderson was trained by Thomas Allen, an assistant to the Muscovy Company, who supplied the Queen’s Navy with hemp, rope, flax, and tallow, which he imported from the Baltic countries. As a young man, Sanderson travelled with Allen throughout the Baltic, France, Germany, and the Netherlands. According to his son, he became wealthy when he inherited the family estates following the death of his elder brother. In either 1584 or 1585 he married Margaret Snedall, daughter of Hugh Snedall, Commander of the Queen’s Navy Royal, and Mary Raleigh sister to Walter Raleigh. Sanderson would go on to become Walter Raleigh’s financial manager.

Here we have once again a merchant financing exploration in the early stumbling phase of the British Empire, a concept that was first floated by John Dee and was propagated by the various members of the Gilbert-Raleigh clan. As we saw in an earlier post, it was the merchants Thomas Smith and John Wolstenholme, who later founded the East India Trading Company, who financed the mathematical lectures of Thomas Hood (1556–1620). Above, we saw that the Muscovy Trading Company financed Frobisher’s efforts to find the Northwest Passage. The founding of the British Empire was driven by trade, and it remained a trading empire throughout its existence. Trade in spices, gold, opium, tea, slaves and other commodities drove and financed the existence of the Empire.

Davis led three expeditions in search of the Northwest Passage in 1585, 1586, and 1587. He failed to find the passage but carried out explorations and surveys of much territory between Greenland and Northern Canada liberally spraying the map with the names of Sanderson, Raleigh, and Gilbert. On these voyages Davis proved his skill as a navigator and marine commander, his logbooks being a model for future mariners and although the expeditions failed in their main aim, they can certainly be counted as successful. 

Map showing Davis’s northern voyages. From A life of John Davis, the navigator by Clements R. Markham, (1889) Source: Wikimedia Commons

In 1588 he is thought to have commanded the Black Dog against the Spanish Armada. In 1589 he was part of the Earl of Cumberland’s voyage to the Azores and became acquainted with Edmund Wright.

George Clifford, 3rd Earl of Cumberland after Nicholas Hilliard  Source: Wikimedia Commons

In 1591, he was part of Thomas Cavendish’s voyage to attempt to find the Northwest Passage from the western end in the Pacific. The voyage was a disaster, Cavendish losing most of his crew in a battle with the Portuguese and setting sail for home. Davis carried on to the Straits of Magellan but was driven back by bad weather, also turning for home. He too lost most of his crew on the return journey but is purported in 1592 to be the first English man to discover the Falkland Islands, a claim that is disputed.

Davis sailed as master with Walter Raleigh on his voyages to Cádiz and the Azores in 1596 and 1597. He sailed as pilot with a Dutch expedition to the East Indies between 1598 and 1600. From 1601 to 1603 he was pilot-major on the first English East India Company voyage led by Sir James Lancaster (c. 1554–1618), a privateer and trader.

James Lancaster in 1596 artist unknown Source: Wikimedia Commons
Lancaster’s Ship the Red Dragon

Although a success, the voyage led to a dispute between Davis and Lancaster, the later accusing the pilot of having supplied false information on details of trading. Annoyed, Davis sailed in 1604 once again to the East Indies as pilot to Sir Edward Michelbourne (c. 1562–1609) an interloper who had been granted a charter by James I & VI despite the East India Company’s crown monopoly on trade with the East. On this voyage he was killed off Singapore by a Japanese pirate whose ship he had seized.  Thus, ending the eventful life of one of Elizabethan England’s greatest navigators.

All the above is merely an introduction to the real content of this post, Davis’ book on navigation and his contribution to the development of navigation instruments. However, this introduction should serve to show two things. Firstly, that when Davis wrote about navigation and hydrography, he did so as a highly experienced mariner and secondly just how incestuous the exploration and navigation activities in late sixteenth century England were. 

In 1594, Davis published his guide to navigation for seamen, which could with some justification be called Navigation for Dummies. It was the first book on navigation actually written by a professional navigator. To give it its correct title:

THE SEAMAN’S SECRETS; Deuided into 2, partes, wherein is taught the three kindes of Sayling, Horizontall, Peradoxal, and sayling vpon a great Circle: also an Horizontall Tyde Table for the easie finding of the ebbing and flowing of the Tydes, with a Regiment newly calculated for the finding of the Declination of the Sunne, and many other necessary rules and Instruments, not heretofore set foorth by any. 

 Newly published by Iohn Dauis of Sandrudge, neere Dartmouth, in the County of Deuon. Gent.

Imprinted at London by Thomas Dawson, dwelling at the three Cranes in the Vinetree, and are these to be solde. 1595

David Waters write, “his work gives in the briefest compass the clearest picture of the art of navigation at this time.”[1]

Davis defines his three kinds of sailing thus:

Horizontal [plane] Navigation manifesteth all the varieties [changes] of the ship’s motion within the Horizontal plain superfices [on a plane chart], where every line [meridian] is supposed parallel.

This was the traditional and most common form of navigation at the time Davis wrote his book and he devotes the whole of the first part of the book to it.

Paradoxal Navigation demonstrateth [on circumpolar charts] the true motion of the ship upon any corse assigned … neither circular nor strait, but concurred or winding … therefore called paradoxal, because it is beyond opinion that such lines should be described by plain horizontal motion.

What Davis is defining here is rhumb line or Mercator sailing.

Great circle navigation he considered as the ‘chiefest of all the three kinds of sayling’, and defined it as one ‘in whom all the others are contained … continuing a corse by the shortest distance between places not limited to any one corse.’

He lists the instruments necessary for a skilful seaman:

A sea compass, a cross staff, a quadrant, an Astrolaby, a chart, an instrument magnetical for finding the variations of the compass, an Horizontal plain sphere, a globe and a Paradoxal compass.

He then qualifies the list:

But the sea Compass, Chart and Cross Staff are instruments sufficient for the Seaman’s use … for the Cross Staffe, Compass and the chart are so necessarily joined together as that the one say not well be without the other … for as the Chart sheweth the courses, so doth the compasse direct the same, and the cross-staffe by every particular observed latitude doth informe the truth of such course, and also give the certaine distance that the ship hath sayled upon the same. 

Davis describes the technique of plane (horizontal) sailing as–’the god observation of latitude, careful reckoning of the mean course steered (corrected for variation), and careful estimation of the distance run’. Of these ‘the pilot has only his height [latitude] in certain.’[2]

Davis gives clear definition of special terms such as course and traverses and delivers an example of how he wrote up his ship’s journal. His was the first book published to give such things.

Source: Waters’ The Art of Navigation
Source: Waters’ The Art of Navigation

He gave much space to how to calculate the tides, including the use of ‘An Horizontal Tyde-Table,’ an instrument for calculating tide times. 

Davis goes into a lot of details on how to calibrate the cross-staff, he paid particular attention to the problem of parallax produced by placing the end of the cross-staff in the wrong position on the face. This is interesting given his development of the back-staff.

In order to determine one’s latitude, it was necessary to determine the altitude of the sun at noon. This was usually done using a cross-staff, also known as a Jacob’s staff, but could also be done with a quadrant or a mariner’s astrolabe.

Source: Waters’ The Art of Navigation

The cross-staff suffered from a couple of problems. As well as the eye parallax problem, already discussed, the user had to hold the staff so that the lower tip of the traverse rested on the horizon, whilst the upper tip was on the sun, then the angle of altitude could be read off on the calibrated scale on the staff.  There were different sized traverses for different latitudes and there were scales on the staff for each traverse, a topic that Davis delt with in great detail. It was difficult for the user to view both tips at the same time. Added to this the user was basically staring directly into the sun. 

The cross-staff Wikimedia Commons

To get round these problems Davis invented the backstaff. At the end of the staff was a horizon vane through which the user viewed the horizon with his back to the sun. An arc, ewith a shadow vane, was attached to the staff which could slide back and forth until its shadow fell on the horizon vane the angle of altitude could be read off on the calibrated staff. This staff did not suffer from the eye parallax problem, the user only had to observe the horizon and not the sun at the same time, and the user did not have to look directly into the sun.

Source: Waters’ The Art of Navigation
Figure 1 – A simple precursor to the Davis Quadrant after an illustration in his book, Seaman’s Secrets. The arc was limited to measuring angles to 45°. Source: Wikimedia Commons

Davis’ original back staff could only measure a maximum angle of altitude of 45°, which was OK as long as he was sailing in the north but was too small when he started sailing further south, so he developed a more advanced model that could measure angles up to 90°.

Source: Waters’ The Art of Navigation
Figure 2 – The second Davis Quadrant after an illustration in his book, Seaman’s Secrets. The arc above is replaced with an arc below and a shadow-casting transom above. This instrument can now measure up to 90°. Source: Wikimedia Commons

This evolved over time into the so-called Davis quadrant. 

Source: Waters’ The Art of Navigation
Figure 3 – The Davis Quadrant as it evolved by the mid-17th century. The upper transom has been replaced with a 60° arc. Source: Wikimedia Commons
Late 17th-century engraving of Davis holding his double quadrant Source: Wikimedia Commons

Better than the cross-staff for measuring the sun’s altitude, the back-staff became the instrument of choice, particularly for English mariners for more than a century, but it was not perfect. Unlike the cross-staff, it could not be used at night to determine latitude by measuring stellar altitudes, also its use was limited by overcast weather when the sun was not strong enough to cast a shadow. To help with the latter problem, John Flamsteed replaced the shadow vane with a lens that focused the sunlight on the horizon vane instead of a shadow. The weak sunlight focused by the lens could be better seen that the faint shadow. The backstaff with lens evolved into the Hadley quadrant, which in turn evolved into the sextant still in use today. 

Davis also gives an extensive description of how to navigate using a terrestrial globe. This was very innovative because mass produced printed globes were a fairly recent invention, Johannes Schöner (1477–1547) produced the first serial printed terrestrial globe in 1515, and were not easy to come by. It was Davis, who persuaded his own patron, William Sanderson, to finance Emery Molyneux’s creation of the first printed terrestrial and celestial globes in England in 1592.

Source: Waters’ The Art of Navigation

Davis emphasised that the terrestrial globe was particularly good for instruction in navigation because all three forms of sailing–plane, rhumb line, great circle–could be demonstrated on it.

In his original list of instruments for the seaman, Davis included the Paradoxal compass but he doesn’t actually explain anywhere what this instrument is. John Dee, who remember was John Davis’ teacher, also mentions the Paradoxal compass in his writings without explanation. There is talk of how he created a Paradoxal chart for Humphrey Gilbert for his fatal 1583 expedition. It turns out that the Paradoxal compass and Paradoxal chart are one and the same and that it is an azimuthal equidistant circumpolar chart, with the north pole at its centre and the lines of latitude at 10° interval as concentric circles. The azimuthal equidistant projection goes back at least to al-Bīrūnī (973–after 1050) in the eleventh century.

An azimuthal projection showing the Arctic Ocean and the North Pole. The map also shows the 75th parallel north and 60th parallel north. Source: Wikimedia Commons Davis Paradoxal compass would have covered a similar area.

In his book on plane sailing, Davis discusses the drawbacks of the plane chart or equirectangular projection, which assumes that the world is flat and on which both lines of longitude and latitude are straight equidistant parallel line which cross at right angles, which according to Ptolemaeus was invented by Marinus of Tyre (c. 70–130 CE) in about 100 CE. A plane chart is OK for comparatively small areas, the Mediterranean for example, and Davis praises its usefulness for coastal regions. However, it distorts badly the further you move away from its standard parallel.

Equirectangular projection with Tissot’s indicatrix of deformation and with the standard parallels lying on the equator Source: Wikimedia Commons

As a result, it is useless for exploration in the far north and hence the use of the Paradoxal compass. The use of such circumpolar maps became standard for polar exploration in the following centuries.

Straight forward, clear and direct The Seaman’s Secret was very popular and went through several new editions in the decades following Davis’ death. A year after it was published Davis published a second book, his The World’s Hydrographical Description or to give it its full title:




Wherein is proved not only by Aucthoritie of Writers, but also by late experience of Travellers and Reasons of Substantial Probabilitie, that the Worlde in all
his Zones, Clymats, and places, is habitable
and inhabited, and the Seas likewise universally navigable without any
naturall anoyance to hinder
the same,

Whereby appears that from England there is a short and 

speedie passage into the South Seas, to China, 

Molucca, Philippina, and India, by Northerly


To the Renowne, Honour, and Benifit of Her Majesties State and 


Published by

In the Countie of Devon, Gentleman. ANNO 1595, May 27.

Imprinted at London

Dwelling at the Three Cranes in the Vinetree, and there to be sold.


 The ‘by Northerly Navigation’ reveals that it is in fact a long plea for a return to exploration to find the Northwest Passage. 

With his The Seaman’s Secrets based on his own extensive experience as an active navigator and his invention of the backstaff, John Davis made a substantial contribution to the development of mathematical navigation in the Early Modern Period.

[1] David Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, New Heaven, 1958, p.201 

[2] All the above is distilled from Water’s page 202.

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

Renaissance garbage ­– V

This is the fifth in a series of discussion of selected parts of Paul Strathern’s The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). For more general details on both the author and his book see the first post in this series.

Today, I’m looking at Strathern’s chapter on Vesalius. It goes without saying that Strathern evokes the mythical religious taboo on dissection of the human body. 

The dissection of human bodies had been a religious taboo in the western world since well before the birth of Christ. This taboo extended through all Abrahamic religions – i.e. Judaism, Christianity and Islam – as well as most of the heterodox sects and cults which pervaded the Mediterranean region and the rest of Europe. 

Whilst there was a prohibition on the dissection of human cadavers from about 150 BCE in Ancient Rome that affected the work of Galen (129–c. 216 CE), who as Strathern reports carried out most of his anatomical work on animals, which of course led to anatomical errors; there was no taboo in the European Middle Ages or the Renaissance, when Leonardo, yes, he gets discussed too, and Vesalius carried out their anatomical studies. Human dissection had already begun in the fourteenth century with the work of Mondino de Luzzi (C. 1270–1326) and by 1400 regular anatomical dissections were an established part of the curriculum in most medical schools. Strathern then delivers up another of his errors on authority:

Despite such mistakes, Galen’s ‘authority’ on medical matters reigned supreme throughout the medieval era, alongside that of Aristotle. Not until the Renaissance would his errors come to light. 

Galen was just one of several medical authors whose texts were used on the medieval universities and despite challenges continued to be used throughout the Renaissance. In fact, there was during the Renaissance a strong neo-Galenic movement that challenged Vesalius. 

It’s almost unavoidable that Strathern, like everybody else, includes Leonardo, he writes:

A century prior to Vesalius, Leonardo da Vinci’s obsessive curiosity led him to carry out dissections of human cadavers, which he recorded in his notebooks. By now the prohibition of such activities had become somewhat more relaxed, though they remained frowned upon. 

As already noted, there was no prohibitions and Leonardo, like all the apprentices of Andrea del Verrochio (1435–1488), who insisted that his apprentices gain a thorough grounding in anatomy, would have attended dissections as an apprentice.

Leonardo carried out systematics anatomical investigations together with Marcantonio della Torre (1481–1511), lecturer on anatomy at the universities of Pavia and Padua, between 1510 and 1511. Vesalius’ principle anatomical work, his De fabrica, of which more later, in 1543. According to my arithmetic that is a span of thirty-one years and not a century. Maybe Strathern uses a different number system?

We get no account of Leonardo’s systematic work with Andrea del Verrochio this would spoil the image that Strathern creates of the chaotic nature of Leonardo’s work and notes. We do, however get a lengthy anecdote about his dissection of a man who claimed to be one hundred years old. Then Strathern drops the following gem:

But why this diversion? What possible relevance does such work by Leonardo have to the northern Renaissance? In fact, none. And that is the point. By recounting these pioneering anatomical experiments – unique in their breadth, depth and explication – we gain an insight into the immense difficulties involved in human dissection during this period. We can also witness the birth of a new, forbidden science coming into being. Or apparently so. For this infant body of learning would not survive its premature birth – stillborn before it could draw breath – largely through the procrastination of Leonardo himself. 

As already stated, we do not have “the birth of a new, forbidden science”, dissection of human cadavers was routine by the time that Leonardo was active. Also, if he poses the question, “What possible relevance does such work by Leonardo have to the northern Renaissance?” then he must also pose it for Vesalius, who although he came from the Netherlands did all his anatomical work in Padua and was very much an integral part of the Italian Renaissance. 

We then get a brief description of the fate of Leonardo’s papers and drawings which closes with the repeat of the arithmetical error:

Working a century later, Vesalius would remain unaware of Leonardo’s pioneering work, which remained lost to history. 

Leonardo lived from 1452 to 1519, Vesalius was born in 1514 and carried out his anatomical work in Padua beginning in 1537, publishing the De fabrica in 1543. No matter how I try I can’t make a century out of these dates! The century is merely a lead into a piece of pseudo historical pathos:

Such a lacuna leads one to speculate on how much more, of genuine worth, was lost during this period. Such discovery and progress had little place in the medieval era. The Renaissance would have to find its own way of accommodating and preserving the innovations it produced. All we know are the successes which eluded loss or destruction: Copernicus’s revolutionary work published by his friends and gifted to him on his deathbed; Paracelsus’s haphazard discoveries, and superstitious lapses, disseminated by means of Gutenberg’s invention, which was itself wrested from the hands of its creator. In this aspect, more than most, no history can be any more than an incomplete account. Fortunately for history, Vesalius would do his utmost to gainsay this fact, his work being both painstaking and thorough from the outset. And his motives throughout his long and arduous task would be single-mindedly focused on public recognition and public reward. 

In the history of science many new ideas, theories and discoveries have been lost, forgotten, or suffered a delayed reception. The reasons are numerous but, and it’s a very big but, they were not somehow actively repressed during the Middle Ages as Strathern would have us believe.  I have already pointed out in my review of the chapter on Copernicus, his work was not published by his friends in an act of deception as Strathern claims, but he sent his book to Petreius in Nürnberg to be published himself. Paracelsus’ work was mostly posthumously printed and published by his disciples, a case of late reception and in Gutenberg’s case, if you build up your business with large sums of borrowed cash and then get into a dispute with your financier, then you tend to lose your business. However, the invention of printing with moveable type was in no way effected by Gutenberg’s financial problems. Strathern’s examples don’t illustrate his argument at all. 

We now arrive at Vesalius and the usual brief biography. Strathern tries to paint his father as somehow inferior and suffering from an inferiority complex. Claiming that it was his mother who raised him and set him on his way entering him, like Mercator and Gemma Frisius before him, in a school of the Brotherhood of the Common life, in his case in Brussels. It was actually his father who entered him in the school. Like Mercator and Gemma Frisius, Vesalius now entered the University of Leuven, and Strathern displays his total ignorance of the medieval university system:

Surprisingly, Vesalius did not register at the school of medicine, but instead chose to study the arts and humanities, which included learning Latin and Greek, at which he thrived. (The books in his grandfather’s library would mostly have been written in Latin, and Vesalius almost certainly had extended his schoolboy Latin by reading these works.) After young Vesalius graduated with a good arts degree in 1532, he was accepted to study at the prestigious University of Paris, where he entered the school of medicine. Only now did he begin the formal study of this subject. 

On the medieval university the undergraduate course of study, in the lower faculty, was nominally the seven liberal arts. First when a student had graduated from the lower faculty, was he permitted to matriculate in one of the higher faculties to study, theology, law, or medicine. So, it was not a surprise that “Vesalius did not register at the school of medicine.”

We get some more misinformation from Strathern:

Despite its reputation, the University of Paris remained firmly committed to the teachings of Aristotle, and its school of medicine was still dominated by the 1,300-year-old ideas of Galen. Lectures in practical anatomy were a comparatively rare novelty, having only recently received limited Church dispensation [my emphasis].

Lectures in practical anatomy were a standard part of the medical curriculum in Paris, there being no Church restriction on them as I’ve already explained above. Strathern contradicts himself by explaining that anatomical lectures, with public dissections, were a standard part of the curriculum, although he correctly observes that the student were only allowed to observe but not to dissect themselves. He notes correctly that the one professor of anatomy, Jacobus Sylvius (1478–1555) was a strict adherent of Galen but that another, Johann Winter von Andernach (1505–1574) was more open minded and even allowed students to participate in dissections. Winter became Vesalius’ mentor and even employed Vesalius as an assistant in the preparation of his four volume Institutiones anatomicae (Paris, 1536) for the press, praising him in the preface; it would become a standard work, of which Vesalius published a second updated edition in 1539. It should, however, be pointed out that Winter was one of those who triggered to renaissance in Galenic anatomy when he produced and published a Latin translation of Galen’s newly discovered and most important De Anatomicis Administrationibus (On Anatomical Procedures) 9 vols. Paris in 1531, which Strathern doesn’t mention at all.

Strathern delivers up some waffle about what happened next when Vesalius was forced by war to leave Paris and return to the Netherlands, where he re-entered the University of Leuven to complete his medicine degree. Here he wrote his doctoral thesis and Strathern once again displays his ignorance:

At the same time, Vesalius began composing his graduation thesis. Interestingly, he chose for his subject the tenth-century Persian physician and alchemist known in the west as Rhazes. (In the Arabic world his full name was Abu Bakr Muhammad ibn Zakariya al-Razi.) The important fact about Rhazes was that he not only based his science upon the experiments he conducted himself, but he also wrote these out in detail, step by step. This meant that they could be precisely repeated by other scientists. Here, reliance upon the word of a universal and unchanging ‘authority’ was skilfully circumvented. 

This important lesson would soon begin to permeate the world of science in both the northern and the Italian Renaissance. The days when scientists – from mathematicians to alchemists – kept their discoveries secret in order to gain advantage over their rivals were coming to an end. Science was entering the public domain. Experimenters would publish their work in books, and their results could be verified (or shown to be faulty) by their peers. 

Vesalius’ doctoral thesis was actually Paraphrasis in nonum librum Rhazae medici Arabis clarissimi ad regem Almansorem, de affectuum singularum corporis partium curatione, a commentary on the ninth book of Rhazes.

Strathern tries to make it seem as if Vesalius’ thesis was in somehow exceptional in its choice of topic and in some way ground-breaking, whereas it was perfectly normal.  The book of Rhazes referred to here is his The Virtuous Life (al-Hawi), a nine-volume posthumous collection of his medical notebooks, which was translated into Latin in the late thirteenth century and was a standard textbook in the medical faculties of the European medieval universities, so there was nothing exception about Vesalius writing his doctoral thesis on part of it. Strathern continues:

Vesalius’s reputation as a talent of great promise seems to have spread far and wide, almost certainly aided by Andernach’s description of him in Institutiones Anatomicae. Immediately upon his graduation from Leuven, Vesalius received an invitation to become a professor of anatomy and surgery at the University of Padua, one of the finest centres of scientific research in Italy. 

This proving that Vesalius was very much part of the Italian Renaissance and not the Northern Renaissance! Now Strathern starts off on a path where he will begin to mix fact with fiction:

More importantly for Vesalius, Padua was just twenty miles from Venice, the commercial and cultural capital of the region, and it was here that he met the German-born artist Jan von Calcar, who had served his apprenticeship under Titian. Calcar’s particular talents were his ability to imitate the works of others and his supreme skill with woodcuts. 

In 1538 Vesalius collaborated with Calcar on the production of his first anatomical text, Tabulae Anatomicae Sex (Six Anatomical Charts.) Three of these charts were produced by Calcar, taken from a full-scale skeleton of the human body which Vesalius had put together. The other three made use of charts which Vesalius himself had drawn for lectures to his students. 

Although the story is well document, Strathern can’t get the facts right. The Tabulae Anatomicae Sex were originally six large, woodcut, wall posters that Vesalius had created for his lecture theatre. He discovered that students were copying them, so he decided to make a professional printed edition of them. Of the printed edition the first three were entirely his own work but for the second set of three he employed Jan van Calcar. Strathern notes correctly that here Vesalius corrects some of Galen’s anatomical errors but repeats some others.

Tabulae Anatomicae Sex the first three illustrations are by Vesalius the three skeletons by Calcar Source: Welcome Collection

Strathern now delivers up a classic historical myth in a footnote:

From now on, the more Vesalius continued with his investigations of the human body, the bolder he became. By this stage he had reached an agreement with the Paduan authorities, who allowed him to dissect the regular supply of cadavers of prisoners executed on the gallows. Vesalius’s retelling of how he carried out his researches paints a vivid, if lurid, picture. He described how he ‘would keep in my bedroom for several weeks bodies from graves or given me after public executions’. How did his neighbours put up with the appalling stench? To say nothing of their suspicions that he might be indulging in necromancy or demonology? The answer is that they may well have been unable to distinguish the stench from the general pervasive malodorousness.* 

* During this era the waterways of Padua, like the canals of Venice and its nearby lagoon, emitted powerful smells, especially in the summer. This was hardly helped by the customary lack of bathing and personal hygiene which pervaded all classes throughout Europe. [my emphasis] Indeed, such habits accounted for the constant use of sweet- smelling nosegays in genteel society. These consisted of flowers or herbs intended to mask the sense of smell. It is said that in Venice a certain type of nosegay evolved which went further, using citrus oil or extracts of resin intended to numb the olfactory sense altogether, rather than simply distracting it. 

The sentence that I have emphasised is, unfortunately, a widespread myth and total piffle! Europeans in the Renaissance bathed regularly and took great care of their personal hygiene. Nobody claiming to be an academic historian, as Strathern does, should be repeating this garbage in 2023! On the subject of demonology Strathern drops the following gem:

As for the suspicion that Vesalius might have been involved in occult practices – presumably he remained under the protection and good name of the university. This was still an era when a large majority of the population believed in demonology, witchcraft and the like – general superstition was rife. Here was one area where the power of the Church and its insistence on orthodoxy was beneficial: in its suppression of heretical practices and beliefs, it undoubtedly reduced the credulity [my emphasis] which led to the outbreaks of mass hysteria that were prevalent during this period. 

It was the Church with its insistence on the real existence of the Devil, demons, black magic, witches, and all the rest that was the main driving force fuelling the credulity.

It is now that Strathern begins mixing fact with fiction or maybe fantasy.

Vesalius now began assembling, together with Calcar, the large, precisely delineated drawings that would become the body of the master- piece which assured his lasting place in medical history. Apart from Leonardo’s, previous books containing anatomical illustrations had tended to be schematic, or cartoon-like, mostly drawn by their medical author – whose talent would often be amateurish at best. By contrast, Vesalius’s De Humani Corporis Fabrica (The Apparatus of the Human Body) would not only be comprehensive and encyclopedic in its knowledge, but its precise illustrations would also be works of art as much as science. Calcar’s large exact drawings, made under Vesalius’s painstaking direction, would in their own distinctly different style be a match for the as-yet-unseen drawings of Leonardo.* Meanwhile Vesalius’s text would set medicine free from the stranglehold of Galen. 

Vesalius Fabrica fronticepiece Source: Wikimedia Commons

That Calcar was the artist, who created the illustration in De fabrica is an unsubstantiated claim made by Giorgio Vasari (1511–1574) in his Le Vite de’ più eccellenti pittori, scultori, ed architettori (Lives of the Most Excellent Painters, Sculptors, and Architects), 1st edition 1550, 2nd expanded edition 1558, a book not exactly renowned for its historical accuracy. There is no mention in the De fabrica, who the artist actually was. In a footnote in her The Scientific Renaissance 1450–1630 (ppb. Dover, 1994) Marie Boas Hall writes about the illustrations:

These are attributed to Jan Stephen van Calcar (1499–c. 1550) by the sixteenth-century art historian, Vasari. Modern students have doubted this, because the figures are as superior to those of the Tabulae Sex as the text of the Fabrica is to that of the earlier work–though it is possible that the artist had learned as rapidly as the author. In place of Jan Stephen van Calcar, the only candidate is an unknown, also a member of Titian’s studio. It seems difficult to believe that so spirited a draughtsman as the artist who drew the pictures for the Fabrica should be otherwise unknown; though it is odd that Vesalius, who had given Jan Stephen credit for his work on the Tabulae Sex, did not mention the name of the artist of the Fabrica.

It has also been speculated that is unlikely that a single artist created all 273 illustrations in such a short period of time. So, Jan van Calcar as the author of the medical illustrations in De fabrica is anything but an established fact but this doesn’t stop Strathern writing the following in a footnote to the paragraph quoted above: 

* Such artistry did not come cheap. Indeed, Vesalius was unable to pay Calcar, and in lieu of a fee he signed over to the artist any future profits the Fabrica would make. 

Either Strathern is making things up or he is quoting a source, which he doesn’t name, that is making things up without checking on the accuracy of the claim made. It doesn’t stop here. Later in his lengthy description of the book itself he writes:

Alas, Vesalius’s perfectionism would result in an increasing number of quarrels with Calcar. Breaks in their collaboration now followed, and Vesalius began drawing a number of the anatomical illustrations in the Fabrica himself. 

A couple of paragraphs further on:

By now, it appears, Calcar had quit the project altogether. We must imagine him storming off in some indignation at Vesalius’s tenacious insistence upon the minutest detail. (This was woodcut, remember, not drawing; erasure was no simple matter with a gouged wooden surface.) 

Here also Strathern appears to not know the difference between the artist and the woodblock cutter. Calcar or whoever was the artist, would draw the images onto the surface of the woodblock, but the actual cutting would be done by a professional woodblock cutter and not the artist. 

I’m not going to do a blow-by-blow analysis of Strathern’s long account of De fabrica, as this review is already over long, but just mention a couple of salient points. To start with Strathern makes no mention of the fact that just as Copernicus modelled De revolutionibus on the Epytoma…in Almagestum Ptolomei of Peuerbach and Regiomontanus, so Vesalius modelled his De fabrica on Galen’s De Anatomicis Administrationibus (On Anatomical Procedures), which as I mentioned above was first translated into Latin and published by his mentor Johann Winter von Andernach.

At one point Strathern tells us, “As the work continues, the illustrations become less precise and their interpretation less exact.” The final chapter of De fabrica, Book VII, deals with the brain and Strathern writes, apparently contradicting himself, “Ensuing books of the Fabrica would prove similarly perceptive – especially Vesalius’s investigations of the human brain.” Do these images appear imprecise to you? 

Fabrica Book VII Source: Wikimedia Commons
Fabrica Book VII Source: Wikimedia Commons

Strathern also writes, And the illustration of the pregnant uterus containing a foetus is undeniably medieval in its crudity…” Vesalius has no illustration of the pregnant uterus containing a foetus;maybe he was confusing it with the illustration of the placenta with its attached foetus?

Fabric Book VII Source: Wikimedia Commons

Having completed his tour of De fabrica, Strathern now jumps the shark!

When he had completed the manuscript of Fabrica, he sent the text and illustrations north to Basel in Switzerland. This was the home of Johannes Oporinus, who sixteen years previously had worked as Paracelsus’s long-suffering assistant. Oporinus had now succeeded Paracelsus as a rather more orthodox professor of medicine in Basel. He also happened to come from a family renowned for their printing and engraving skills, and combined his medical knowledge with an expert understanding of the entire printing process. This was the only man in Europe whom Vesalius could trust with the production of his masterwork. 

Johannes Oporinus (1507–1568) had very little to do with Paracelsus, he was merely for a brief period in 1527 his famulus.  The son of a painter he studied law and Hebrew at Basel University, whilst working as a proofer in the print workshop of Johann Froben (c. 1460–1527). He also worked as a schoolteacher for Latin. From 1538 to 1542, he was professor for Greek at the University of Basel, resigning to devote himself fulltime to his own print workshop. 

Portrait of Johannes Oporinus by Hans Bock Source: Wikimedia Commons

Strathern closes his chapter on Vesalius with a long-winded account of further biography as Imperial physician to Charles V and later Philip II. Strathern of course cannot resist including the unsubstantiated anecdote that in Spain Vesalius started to carry out an autopsy on a corpse only to discover that the man wasn’t actually dead. There are numerous cases of this happening throughout history, and it even still occasionally occurs today, but whether it actually happened to Vesalius is, as I said, unsubstantiated.

In his chapter on Vesalius, as usual Strathern delivers up a collection of inaccuracies, myths, and in the case of the relationship between Vesalius and Calcar some pure fantasy. Once more I am forced to ask how did this book ever get published?

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Filed under Book Reviews, History of medicine, Renaissance Science

Renaissance garbage ­– IV

This is the fourthin a series of discussion of selected parts of Paul Strathern’s The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). For more general details on both the author and his book see the first post in this series.

Strathern introduces us to today’s subjects thus:

We now come to two figures who used ingenious mathematical techniques to unravel their own versions of the truth. These were Gerardus Mercator and François Viète, both of whom lived exciting lives (though not always pleasantly so), and whose works would play a part in transforming the world in which we live. 

Although Mercator’s biography is well documented Strathern still manages to screw up his facts. He tells us that his father was from Gangelt and was therefore German. Gangelt was at this time in the Duchy of Jülich and the inhabitants spoke a dialect of what would become Dutch. I do wish people would look more deeply at nationality, ethnicity etc in history, just because somewhere is German or whatever today doesn’t mean it was in the sixteenth century. Then he tells us:

During Mercator’s youth, two historic events took place which would change Europe forever. Mercator was just five when Luther instigated what would become the Reformation, and he was ten years old when the survivors of Magellan’s three-year expedition to circumnavigate the globe arrived back in Seville. By this time young Mercator’s father had died, and his uncle had taken on the role of his guardian. 

The Frans Hogenberg portrait of 1574, showing Mercator pointing at the North magnetic pole Source: Wikimedia Commons

Mercator was actually fifteen when his father died, and his uncle placed him in the school of the Brethren of the Common Life in ‘s-Hertogenbosch. Here Strathern drops a paragraph that brought tears to my eyes the first time I read it, not really believing what I had just read. The second time through I started weeping and the third time I just wanted to burn the whole thing down.  

Even so, the main curriculum was still based on the traditional scholastic trivium of grammar, logic and rhetoric, all of which were of course taught in Latin. However, in a gesture towards the renaissance of classical knowledge, the curriculum had been extended to include Ptolemy and his Geography. The Ancient Greek polymath had written this work in Alexandria around AD 150. The fact that it was written in Ancient Greek meant that it had remained unknown to Europe during the medieval era, as scholars only knew Latin. It was not translated until 1406, when its appearance created a great stir. Meanwhile Ptolemy’s geocentric cosmology, which Aristotle had passed on, would not be refuted by Copernicus until 1543, when Mercator was in his thirties. But much of Ptolemy’s Geography, especially his map of the world – consisting of a chart which stretched from the Atlantic coast in the west to Sinae (China) in the east – had come as a revelation to the young Mercator. 

Ptolemy’s Mathēmatikē Syntaxis (Almagest) and his Tetrabiblos were available and widely read in Latin in the medieval period, both of them having been translated directly from the original Greek in the twelfth century, but apparently his Geōgraphikḕ Hyphḗgēsis (Geographia) was not as, “written in Ancient Greek meant that it had remained unknown to Europe during the medieval era, as scholars only knew Latin.” This is of course total bullshit. There was no Latin translation of the Geographia in Europe in the Middle Ages because there was neither a Greek nor an Arabic manuscript of the work known before a Greek manuscript was discovered in Constantinople in the early fifteenth century and translated by Jacobus Angelus in 1406.

Meanwhile Ptolemy’s geocentric cosmology, which Aristotle had passed on [my emphasis], would not be refuted by Copernicus until 1543, when Mercator was in his thirties.

Please savour this gem of a sentence, you will probably search high and low to find its equivalent in stupidity in a supposedly serious, ‘academic’ publication. Strathern claims to be an academic author. Aristotle (384–322 BCE) passed on the geocentric cosmology of Ptolemy, written c. 150 CE! 

Having imbibed Ptolemaic geography at school, Mercator now goes off to university:

In 1530, at the age of eighteen, Mercator travelled to the similarly prestigious University of Leuven. Here he passed the entry matriculation, where his name appears in the Latin form he had adopted at school followed by the classification pauperes ex castro (poor students of the castle). This indicated that he was given lodgings in one of the communal dormitories set aside for unprivileged students in the castle by the fish market. Rich students lived separately in their own rooms in a more salubrious quarter of the city. 

Pauperes does in fact mean that he was a poor student but ex castro refers to the college he was in Castle College (Dutch: De Burcht or Het Kasteel, Latin: Paedagogium Castri) the oldest of the Leuven colleges, founded in 1431. All the students, rich or poor, lived in the same college building, although the quality of their rooms varied.

Strathern now slips in a reference to Vesalius:

Despite such domestic segregation, all students mingled freely, attending the same lectures, and it was here that Mercator formed a friendship with one of his more privileged contemporaries, named Andreas Vesalius, of whom we will hear more later. Suffice to say that Vesalius would become one of the great luminaries of the northern Renaissance, on a par with Mercator himself, with whom he retained a lifelong friendship. 

Although they almost certainly knew each other, I know of no special friendship between Mercator and Vesalius. However, there was one between Vesalius and Gemma Frisius, about whom more soon, they even, infamously, stole part of a corpse on a gibbet together.

Having graduated MA in 1532, Mercator took himself off to Antwerp for two years, rather than progressing on to one of the higher faculties to study, theology, law, or medicine. During these two years, he took up contact with Franciscus Monachus, (c. 1490 – 1565), a Minorite friar at the monastery in Mechelen, who had earlier taught geography at the University of Leuven. Strathern introduces Monachus thus:

As we have seen, in 1494 Pope Alexander VI had brokered the Treaty of Tordesillas, which aimed to avert a dangerous clash between the two Catholic countries most involved in exploration – namely, Portugal and Spain. The pope had drawn a line north–south through the middle of the Atlantic Ocean: all land discovered to the west of this line (i.e. the New World) would belong to Spain, while all land discovered to the east of it (Africa and Asia) would belong to Portugal. Illustrating this ruling, as well as making allowances for consequent discoveries, Monachus drew two circular maps. One depicted the western hemisphere of the Americas, and the other outlined the eastern hemisphere: Africa, India and the lands to the east, which he named Alta India (in effect ‘Beyond India’). In the light of Magellan’s circumnavigation, the next obvious step was to create a model of the world in the form of a globe. 

His globe, which did not survive, came first, and was constructed with the engraver Gaspard van der Heyden(c. 1496 – c. 1549). The two hemispherical maps are in an open letter describing the globe to his patron, Jean II Carondelet (1469–1545), Archbishop of Palermo, entitled De Orbis Situ ac descriptione ad Reverendiss. D. archiepiscopum Panormitanum, Francisci, Monachi ordinis Franciscani, epistola sane qua luculenta. (A very exquisite letter from Francis, a monk of the Franciscan order, to the most reverend Archbishop of Palermo, touching the site and description of the globe) in 1524. 

De Orbis Situ title page Source: Wikimedia Commons
De Orbis Situ maps showing Portuguese and Spanish hemispheres Sourec: Wikimedia Commons

Strathern now launches into a brief history of terrestrial globe-making, of which I will only give extracts that mostly need correcting:

Monachus was not the first to do this. Indeed, in line with the rebirth of classical knowledge, it was known that the Ancient Greek philosopher Crates of Mallus (now south-east Turkey) had produced a globe as early as the second century BC. 

Nothing to criticise here but Strathern then goes into a discussion in which he states:

This illustrated Crates’s belief that the world consisted of five distinct climactic zones. 

The climate zones or climata are, of course, standard Greek cosmography and predate Crates. First hypothesised by Parmenides and then modified by Aristotle. We move on:

A rather more accurate representation appeared during the Arab Golden Age, when in 1267 the Persian astronomer Jamal al-Din travelled to Beijing and created a terrestrial globe for Kublai Khan. 

Jamal al-Din didn’t create a terrestrial globe for Kublai Khan in Beijing. When he travelled to Beijing, to become head of the Islamic Astronomical Bureau he took seven astronomical instruments of Islamic type with him, namely an armillary sphere, a parallactic ruler, an instrument for determining the time of the equinoxes, a mural quadrant, a celestial and a terrestrial globe, and an astrolabe with him. 

Just prior to the geographical revolution which had taken place during Mercator’s childhood, the German navigator, merchant and map-maker Martin Behaim constructed the Erdapfel (earth apple), the earliest-known surviving globe, which followed the prevailing ideas held by Columbus, omitting any large land mass between western Europe and China. 

Martin Behaim was not a navigator.

This appeared in 1492, and over the coming years it inspired a number of more accurate globes. One, constructed out of two glued-together lower halves of an ostrich egg, was among the first to include the New World. Another, cast in copper, imitated medieval maps which illustrated undiscovered regions with dragons, monsters or mythical beasts. It also labelled the unknown region to the south of China Hic sunt dracones (Here be dragons), which would become a popular appellation covering unknown regions in later maps.

Both the provenance of the Ostrich Egg Globe and its supposed date (1604) are, to say the least, disputed and I would not include it in any serious account of the history of globes.

The Ostrich Egg globe Source: Wikimedia Commons

The copper globe, that it is very similar to, is the Lenox Globe (1610) and its undiscovered regions are not illustrated with dragons, monsters, or mythical beasts. It is in fact only one of two maps known to bear the legend HC SVNT DRACONES (Latin: hic sunt dracones means here are dragons), the other is the Ostrich Egg Globe.

The Lenox Globe Source: Wikimedia Commons
Northern hemisphere of the Lenox Globe can you see the dragons, monsters or mythical beasts? Source: Wikimedia Commons
Southern hemisphere of the Lenox Globe can you see the dragons, monsters or mythical beasts? Source: Wikimedia Commons
The Lenox Globe, by B.F. De Costa 1879. I can see a couple of sea creatures but no dragons, monsters or mythical beasts Source: Wikimedia commons
Close-up of the text ‘Hic Sunt Dracones’ Source: Wikimedia Commons

The handful of globes that Strathern has mentioned in his brief survey are all so-called manuscript globes i.e., they are handmade unique examples. Strathern makes no mention whatsoever of the most important development in globe history, a very significant one for Mercator, the advent of the printed globe. The earliest known printed globe, of which only sets of gores exist, was the small globe printed of the Waldseemüller world map that gave America its name.

Waldseemüller globe gores 1507

This globe was relatively insignificant is the history of the globe, the major breakthrough came with the work of the Nürnberger mathematicus, Johannes Schöner (1477–1547). Schöner went into serial production of a terrestrial globe in 1515 and a matching celestial globe in 1517.

Schöner terrestrial globe 1515, Historisches Museum Frankfurt via Wikimedia Commons Not the original stand

In the 1530s he produced a new updated pair of globes. We will return to Schöner and the influence of his globes on Mercator.

But first back to Strathern: 

However, the most significant feature of these globes for Mercator was that, unlike with previous medieval maps, their geographical features were drawn or painted upon solid round surfaces. A map on a globe represented the actual size and shape of its geographical features, whereas a continuous map on a flat rectangular chart was bound to distort shapes, stretching them the further they were from the Equator [my emphasis]. The understanding of this fundamental distinction would be the making of Mercator. 

That all flat maps distort was well-known to Ptolemy, who in his Geographia explicitly states that a globe is the best representation of the world. To transfer the map from the globe to a flat map one needs a projection, Ptolemy describes three different ones, and each projection, of which there are numerous, distorts differently. Strathern seems to imply here that there is only one map projection and the distortion that he describes here is that of the Mercator projection!

But first of all he [Mercator] would have to understand the complexities of maps and globes.

These he learned from a curious character by the name of Gemma Frisius…


Although only four years older than Mercator, at this stage he may well have taught Mercator mathematics. 

Gemma Frisius 17th C woodcut E. de Boulonois Source: Wikimedia Commons

Why is Gemma Frisius (1508–1555) a curious character? Strathern gives no explanation for this statement. There is also no ‘may well’ about it, when Mercator returned to Leuven in 1534 after his two-year time-out, he spent two years studying geography, mathematics, and astronomy under Gemma Frisius’ guidance. He also in this period learnt the basics of instrument and globe making from Frisius. Strathern now delivers up a very garbled and historically highly inaccurate account of how Frisius and Mercator became globe makers.

Around 1530, when Frisius was in his early twenties, a local goldsmith called Gaspar van der Heyden produced ‘an ingenious all-in-one terrestrial/celestial globe’. This incorporated a geographical map of the world, on which were also inscribed the main stars of the heavens. Such was the complexity of this muddled enterprise that it required a three-part booklet to explain how to understand it. The task of writing this was assigned to Frisius, and its title gives an indication of the difficulties involved: On the Principles of Astronomy and Cosmography, with Instruction for the Use of Globes and Information on the World and on Islands and Other Places Recently Discovered

What actually happened is somewhat more complex. Schöner had become a highly successful globe maker and his globes were being sold over all in Europe. However, there was a greater demand than he could supply. 

Jean II Carondelet, the Archbishop of Palermo, who as we saw above was Franciscus Monachus’ patron and dedicatee of his De Orbis Situ, commissioned the Antwerp printer/publisher Roeland Bollaert, who had printed the De Orbis Situ, to reprint Schöner’s Appendices in opusculum Globi Astriferi, in 1527, and the engraver Gaspard van der Heyden was commissioned to engrave the celestial globe to accompany it. In 1529, Gemma Frisius edited an improved second edition of Peter Apian’s Cosmographia, which was printed and published by Roeland Bollaert. Gemma Frisius, who had earlier studied under Monarchus, began to work together with Gaspar van der Heyden, and it was Gemma Frisius who created the ‘ingenious all-in-one terrestrial/celestial globe’, which van der Heyden engraved. Gemma wrote the accompanying booklet Gemma Phrysius de Principiis Astronomiae & Cosmographiae deque usu globi ab eodem editi (1530), which was published by the Antwerp publisher Johannes Graheus. It is probably that Roeland Bollaert had died in the meantime. In this book Gemma Frisius acknowledges his debt to Johannes Schöner. Monarchus had also acknowledged his debt to both Schöner and Peter Apian in his De Orbis Situ. Gemma Frisius and van der Heyden later produced a new pair of globes, 1536, terrestrial and 1537, celestial, and this time Mercator was employed to add the cartouches in italic script to the globes, his introduction to globe making. 

Strathern now tells us about Gemma Frisius’ book and its influence on Mercator:

Within this cornucopia of often extraneous knowledge were to be found the sound principles which Frisius would later pass on to Mercator. Most importantly, these involved such vital cartographic elements as the principles of longitude and latitude, which form a network covering the surface of the globe. The lines of longitude are drawn down the surface of the globe at regular intervals from the North Pole to the South Pole.* As long as a ‘meridian’ or middle point (line zero) is established, it is possible to record how far one’s position lies east or west of this line from pole to pole. By this time, navigators were beginning to carry shipboard clocks. As a rough-and-ready method for discovering how far east (or west) they had travelled from their home port, they could measure the time discrepancy between noon on the shipboard clock (i.e. noon at their home port) and noon at their current location (the sun’s zenith). 

* Both of these were of course theoretical concepts at the time, conjectured from the fact that a globe must have a top (northernmost point) and a bottom (southernmost point.) It would be some five centuries before the existence of the actual poles was confirmed by discovery. 

The lines of latitude are drawn around the globe, beginning at its widest girth (the Equator), and then ascending in regular diminishing circles towards the North Pole, and also descending at regular intervals to the South Pole. In order to establish their longitude, navigators had learned to measure the precise location above the horizon of stars in the sky. This also could be compared to their location when at the home port. Such measurements were taken with an astrolabe (literally ‘star taker’), the forerunner of the sextant. 

Reading these atrocious paragraphs, I asked myself why do I bother? Why don’t I just throw the whole thing in the next trash can and walk quietly away? However, being a glutton for punishment, I persevere. But where to begin? I will start with the origins of the longitude and latitude system, at the same time dealing with the mind bogglingly stupid starred footnote. 

Most people don’t realise but the longitude and latitude system of cartographical location was first developed in astronomy to map the skies. In the northern hemisphere, if you look up into the night sky, the heavens appear to form a sphere around the Earth and there are stars that every night circle the same point in the heavens, that point is the astronomical north pole. In fact, as we now know it’s the Earth that turns not those circumpolar stars, but for our mapping purpose that is irrelevant. The astronomical or celestial north pole is of course directly above the terrestrial north pole, on a straight line perpendicular to the plane of the equator. You can observe the same phenomenon in the southern hemisphere, defining the south celestial and terrestrial poles, but as the European astronomers could not see the heavens further south than the Tropic of Capricorn, that doesn’t need to concern us at the moment. Note the north and south poles are not theoretical concepts but real points on both the celestial and terrestrial spheres. The lines of longitude are the theoretical great circles around the celestial sphere passing through the north and south poles. The annual path of the Sun defines the Equator and the Tropics of Cancer and Capricorn, the principal lines of latitude. The Poles, the Equator, and the two Tropics are the principal features on the armillary sphere, the earliest three-dimensional model of the celestial sphere created by astronomers, sometime around the third century BCE. 

Diagram of an armillary sphere with the Poles, the Equator, and the two Tropics Source: Wikimedia Commons

At some point somebody had the clever idea of shrinking this handy mapping network down from the celestial sphere on to the terrestrial sphere, the Earth. The first cartographer to use longitude and latitude for terrestrial maps was probably Eratosthenes (C. 276–c. 195 BCE). His prime meridian (line of longitude) passed throughAlexandria and Rhodes, while his parallels (lines of latitude) were not regularly spaced, but passed through known locations, often at the expense of being straight lines. (Duane W. Roller, Eratosthenes Geography, Princeton University Press, 2010 pp. 25–26). Hipparchus (c. 190–c. 120 BCE) was already using the same system that we use today. Ptolemy, of course, used the longitude and latitude system in his Geographia, in fact a large part of the book consists of tables of longitude and latitude from hundreds of places from which it is possible to reconstruct maps. If as Strathern claims, Mercator studied the Geographia at school then he didn’t need Gemma Frisius to explain longitude and latitude to him.

Strathern’s “By this time, navigators were beginning to carry shipboard clocks. As a rough-and-ready method for discovering how far east (or west) they had travelled from their home port, they could measure the time discrepancy between noon on the shipboard clock (i.e. noon at their home port) and noon at their current location (the sun’s zenith) can only be described as a historical cluster fuck! Dave Sobel’s Longitude (Walker & Company, 1995), for all its errors, and it has many, which tells the story of how John Harrison (1693–1776) produced the first marine chronometer, that is a clock accurate and reliable enough under testing condition to enable the determination of longitude, his H4 in 1761, was almost certainly the biggest popular history of science best-seller ever! Apparently, Strathern has never heard of it! 

The whole is much, much worse when you know that the first person to hypothesise the determination of longitude using an accurate mechanical clock was Gemma Frisius and he did so in Chapter nine of his On the Principles of Astronomy and Cosmography, the only one of his publications that Strathern mentions: 

… it is with the help of these clocks and the following methods that longitude is found. … observe exactly the time at the place from which we are making our journey. … When we have completed a journey … wait until the hand of our clock exactly touches the point of an hour and, at the same moment by means of an astrolabe… find out the time of the place we now find ourselves. … In this way I would be able to find the longitude of places, even if I was dragged off unawares across a thousand miles.

Gemma Frisius was, however, of the difficulties that the construction of such a clock would involve:

… it must be a very finely made clock which does not vary with change of air.

More than a hundred years later the French astronomer Jean-Baptiste Morin (1583–1656), who propagated the lunars method of determining longitude wrote:

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

Strathern is not much better on latitude, The lines of latitude are drawn around the globe, beginning at its widest girth (the Equator), and then ascending in regular diminishing circles towards the North Pole, and also descending at regular intervals to the South Pole. In order to establish their longitude [sic, I assume that should read latitude!] navigators had learned to measure the precise location above the horizon of stars in the sky. This also could be compared to their location when at the home port. Such measurements were taken with an astrolabe (literally ‘star taker’), the forerunner of the sextant. Latitude is determined by measuring either the height of the Sun, during the day, or the Pole Star, at night. That’s why in marine slang the daytime measurement is called “shooting” the sun. As David King is fond of repeating, the astrolabe was never used for navigation. It is possible that fifteenth century navigators used a mariner’s astrolabe, but more likely that they used a quadrant or a Jacob’s staff. Frisius’ lifetime is too early for the backstaff, which was first described by John Davis (c.1550­–1605) in his The Seaman’s Secrets in 1594. It is the backstaff that was the forerunner of the sextant not the astrolabe. 

A Jacob’s staff, from John Sellers’ Practical Navigation (1672) Source: Wikimedia Commons

The errors continue:

In order to prepare the maps for incorporation on the globe, they first had to be copied to a uniform scale so that they could be aligned with other maps. All this required a sophisticated understanding of the maps involved, and required the use of geometry, trigonometry and especially triangulation. 

This last method enabled the map-makers to calculate the precise location of a distant geographical feature – such as a mountain, town or river mouth – using the known location of two other features. The modern version of this method was invented by Frisius in 1533, and worked as follows. First a line of known length was drawn between two features (Brussels and Antwerp in Frisius’s early experiment). Then the surveyor would draw a line from each end of the known line directly towards the unlocated feature (Middelberg, in Frisius’s case) and measure the angles between these lines and the ends of the known line. This gave him a triangle with a base of known length, and two base angles. From these it was a simple matter of geometry to ‘triangulate’ the distances to and position of the unlocated feature. 

Gemma Frisius was indeed the first to describe triangulation in the third edition of Apian’s Cosmographia in 1533 but Strathern’s account of how it works is arse backwards. Triangulation is a trigonometrical method of surveying, which is then used to draw maps. First you have to accurately measure your baseline on the ground, in Gemma Frisius’ example between Brussels and Antwerp. Then from the two endpoints the angles of observation of a third point, Middelburg in Gemma’s example, are measured enabling the completion of the triangle on the drawing board and thus the determination of the distances between the endpoints of the baseline and the third point using trigonometry. Gemma Frisius’ example is purely theoretical as you can’t actually see Middelburg from either Brussels or Antwerp. 

Gemma Frisius triangulation example from 3rd edition Apian/Frisius Cosmographia Source: Wikimedia Commons

Strathern devotes some time to Mercator’s biography, his setting up as an independent cartographer and instrument maker and his marriage, then delivers the next piece of history of cartography ignorance:

A year later, in 1538, he produced his first etched map of the world, Orbis Imago. This map is highly ingenious in its representation of the globe on a flat surface. The map is in two parts, which join at a tangent. The first part views the world from above the North Pole, the second from above the South Pole. But instead of showing two semicircles, each view is a rounded heart-shape with an indentation curving in towards the pole. This tearing-apart of the semicircle enabled Mercator to represent the land masses without the distorted exaggeration which would have occurred if the maps had stretched to contain two semicircles. A cut-out of these two-dimensional shapes can be twisted and folded into a semblance of a three-dimensional globe, and there is no doubt that Mercator had something similar in mind. When presented in this form, a flat map of the world did not distort the land masses; however, it also did not provide an accurate picture of the distances between various geographical features so was of little use to mariners. 

Mercator Orbis Imago Source: Wikimedia Commons

Mercator’s Orbis Imago is a double cordiform (heart shaped) polar projection and Strathern seems to think that Mercator invented it, he didn’t. The cordiform projection is also known as the Stab-Werner projection named after Johannes Stabius (1540–1522), who invented it and Johannes Werner (1468–1522), who first published/publicised it, in his partial translation of Ptolemy’s Geographia (1514). The two mathematici were friends, who knew each other from their mutual time at the University of Ingolstadt.  Both Peter Apian in 1530 and the French mathematicus Oronce Fine (1494–1555) in 1531 produced single cordiform projection world maps, of which Mercator was almost certainly aware as the sixteenth century, European, cartography scene was strongly networked.

Peter Apian cordiform world map 1530 Source: British Library
Oronce Fine cordiform world map 1534 Source: Wikimedia Commons

More importantly in 1532 Oronce Fine also produced a double cordiform polar projection world map and Mercator’s Orbis Imago is fairly obviously merely an improved version of Fine’s map.

Oronce Fine double cordiform (heart shaped) polar projection Source: Wikimedia Commons

The Stab-Werner projection is Equal-area i.e., area measure is conserved everywhere and Equidistant i.e., all distances from one (or two) points are correct. It was never intended for use by mariners.

Apart from its geometric ingenuity, Mercator’s Orbis Imago has two other features of note. The view over the South Pole includes a large-scale representation of Antarctica, which he named Terra Australis Incognita (Unknown Southern Land). According to historical records, neither Australia nor Antarctica had yet been discovered by Europeans; however, the existence of such a land mass had long been a theoretical supposition – considered a necessary counterbalance to the land masses of the northern hemisphere. 

The Terra Australis Incognita first appeared in the sixteenth century on the globes of Johannes Schöner and it has been shown that Oronce Fine took the details for his maps from Schöner’s work and that Mercator took his from Fine. 

Mercator’s map also included the word ‘America’ as a name for the large land mass to the west of Europe. 

The German map-maker Martin Waldseemüller had been the first to use the name ‘America’ on a map, in 1507. This labelled a large island, straddling the Equator, which he had named after Amerigo Vespucci, the Florentine explorer whose voyages had provided extensive mapping of the south-east coast of this territory which Vespucci first named the New World. 

However, in later maps new evidence had led Waldseemüller to take a more tentative view of Vespucci’s claims, and he replaced ‘America’ with the inscription ‘Terra Incognita’, suggesting that the Terra de Cuba discovered by Columbus was in fact an eastern part of Asia. Mercator’s labelling of America, as well as his clear outlining of the northern and southern parts of this landmass, confirmed once and for all this name. 

Both Johannes Schöner and Peter Apian had followed Martin Waldseemüller in his misnaming of the newly discovered continent America after Vespucci, and both continued to do so after Waldseemüller has realised his error and withdrawn the name, which had nothing to do with his believing that Cuba was an eastern part of Asia, so the name had become fixed long before Mercator got involved. However, it was in fact Mercator who first applied the name America to North America. 

There now follows a long biographical section that I won’t comment on; I’m only here for the history of cartography. We now arrive at the ominous 1569 world map, and what is probably the worst account of the Mercator projection that I have ever read.

From now on Mercator buried himself in his work. His ambition was no less than to produce a complete map of the world which could be used by navigators. 

Throughout history, large-scale maps had usually been centred upon a known location. For instance, Ptolemy’s map was centred on the Mediterranean. Later maps, such as the large round medieval Mappa Mundi,* had Jerusalem as their centre, with the known world radiating outwards from this central holy point. Mercator decided that his map would have no centre. Instead it would be projected onto a grid of longitude and latitude lines – which would become known as Mercator’s projection. On a globe these lines are curved, but on Mercator’s flat surface they were rectilinear straight lines. This inevitably stretched the scale of the map the further it moved from the Equator. For instance, on Mercator’s map the Scandinavian peninsula appeared to be three times the size of the Indian subcontinent, whereas in fact India is one and a half times larger than Scandinavia. But this would in no way hamper navigation, which relied upon location established by lines of latitude and longitude. A ship could sail across an ocean following a constant compass bearing. This may have appeared curved on Mercator’s flat map, but owing to the bulge of the globe it did in fact represent the most direct route. 

Ptolemy’s world map in not centred on the Mediterranean; the Mediterranean lies in the top half of the map on the lefthand side.

A printed map from the 15th century depicting Ptolemy’s description of the Ecumene by Johannes Schnitzer (1482). Source: Wikimedia Commons Note the longitude and latitude grid

The starred footnote to “the large round medieval Mappa Mundi,” “*This remains on public display at Hereford Cathedral in England,” seems very strongly to imply that Strathern thinks there was only ever one large round medieval Mappa Mundi, which is of course total rubbish. 

Mercator’s infamous 1596 map is centred on the Atlantic Ocean setting a standard for European world maps that would lead to the cartographers being accused of politically portraying the world from a Eurocentric standpoint.

Mercator world map 1569 Source: Wikimedia Commons

Projecting a map on to a grid of longitude and latitude lines is not the Mercator projection. The printed Ptolemaic world maps of the late fifteenth century are projected on to a grid of longitude and latitude lines (see above), as are the world maps of John Ruysch (1507), Martin Waldseemüller (1507), Francesco Rosseli (1508), Dürer-Stabius (1515), Peter Apian (1530) and Oronce Fine (1536) all on various map projections.

On a globe these lines are curved, but on Mercator’s flat surface they were rectilinear straight lines. This inevitably stretched the scale of the map the further it moved from the Equator.

This is rubbish! In order to have a map on which a loxodrome or rhomb line is a straight-line Mercator systematically widened the distance between the lines of latitude towards the north and south poles, according to a set mathematical formular, which he didn’t reveal. Strathern makes no mention of Pedro Nunes (1520–1578), who first determined the rhumb line as the course of constant bearing on a globe was a spiral, the basis of Mercator’s work. Mercator had drawn rhumb line spirals on his globe from 1541. 

A ship could sail across an ocean following a constant compass bearing. This may have appeared curved on Mercator’s flat map, but owing to the bulge of the globe it did in fact represent the most direct route. 

Once again Strathern is spouting rubbish.  As already stated above on the Mercator projection a course of constant compass bearing, the rhumb line, is a straight-line, the whole point of the projection, and it does not represent the most direct route. The most direct route is the arc of the great circle of the globe that passes through the point of departure and the destination. However, to sail such a course means having to constantly change the compass bearing, so although longer the course of constant compass bearing is easier to navigate.  

A rhumb line (blue) compared to a great-circle arc (red) between Lisbon, Portugal and Havana, Cuba. Top: orthographic projection. Bottom: Mercator projection. Jacob Run Source. Wikimedia Commons

Yet what happened when a ship travelled beyond the edge of the map? If a ship set sail from China, heading east across the Pacific Ocean, it would soon reach the limit. But if the navigator rolled the map into a cylinder, with the eastern edge of the map attached to the western edge, the solution to this problem was obvious. The navigator could simply continue from the eastern border of the map across the Pacific to the west coast of America. In this he would also be aided by corresponding map references on lines of longitude and latitude. 

This is simply cringe worthy. If someone was sailing from China across the Pacific to the west coast of America, they would use a chart of the Pacific for the voyage. 

From now on navigators would adopt Mercator’s projection, both for continental and for local charts. The entire world had become ‘orientated’. Originally this word meant ‘aligned to the east’; on Mercator’s projection the world was aligned north, south, east and west, by means of longitude and latitude. And any point on this flattened globe could be pinpointed, as if on a graph, by reading off its precise position in numbers along the lines of longitude and latitude. Dangerous shoals, rocks, river mouths, cities and towns, mountains, borders and even entire countries could be mapped and ‘orientated’. Mercator completed his task in 1569, and to this day Mercator’s projection is how we envisage the world when it is mapped onto a flat surface. 

Nobody adopted the Mercator projection in 1569 because Mercator did not explain how to construct it. It first came into use at the end of the century when Edward Wright (1581–1626) revealed the mathematics of the Mercator projection in his Certaine Errors in Navigation (1599). Even then the take up of the Mercator projection for marine charts was a slow process only really becoming general in the early eighteenth century. Strathern still seems to be under the illusion that the cartographical longitude and latitude grid somehow originated with Mercator, whereas by the time Mercator created his 1569 world map it had been in use for about eighteen centuries. The Mercator projection is only one of numerous ways that we envisage the world when it is mapped onto a flat surface and there is in fact a major debate which projection should be used. The use of alongitude and latitude grid does not necessarily imply that a map has to have north at the top.

But Mercator’s task was not complete. For the next twenty-six years he painstakingly created more than a hundred maps, all scaled according to his projection. During the final years of his life he started binding these together with the intention of making them into a book. For the front cover he planned to have an engraving of the Ancient Greek Titan named Atlas, kneeling, with the world balanced on his shoulders. Hence the name which would come to be attached to such compilations of maps. 

 Mercator did not start binding his maps together with the intention of making them into a book during the final years of his life. His Atlas was part of a major complex publishing project, beginning in 1564, when he began compiling his Chronologia, which was first published in 1569:

The first element was the Chronologia, a list of all significant events since the beginning of the world compiled from his literal reading of the Bible and no less than 123 other authors of genealogies and histories of every empire that had ever existed. (Wikipedia)

The Chronologia developed into an even wider project, the Cosmographia, a description of the whole Universe. Mercator’s outline was (1) the creation of the world; (2) the description of the heavens (astronomy and astrology); (3) the description of the earth comprising modern geography, the geography of Ptolemy and the geography of the ancients; (4) genealogy and history of the states; and (5) chronology. Of these the chronology had already been accomplished, the account of the creation and the modern maps would appear in the atlas of 1595, his edition of Ptolemy appeared in 1578 but the ancient geography and the description of the heavens never appeared. (Wikipedia)

The maps, that would eventually appear posthumously in his Atlas, were not drawn using the Mercator projection, which is totally unsuitable for normal regional maps. The Atlas was not named after the Titan, who carried the world on his shoulders, but after a mythical king of Mauretania credited with creating the first globe, who Mercator described in the preface to his 1589 map collection, “Italiae, Sclavoniae, Grecia”, thus “I have set this man Atlas, so notable for his erudition, humaneness, and wisdom as a model for my imitation.” The name Atlas was first used on the 1595 posthumous map collection Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura, published by his son Rumold Mercator (1541–199). King Atlas was first replaced on the cover by the Titan Atlas in later edition in the seventeenth century.

Cover of Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura (facsimile)
Showing King Atlas of Mauretania

Having royally screwed up the life and work of Mercator, Strathern now turns to the French jurist and mathematician François Viète (1540–1603).

François Viète (1540–1603). Source: Wikimedia Commons

Having started Viète’s biography Strathern delivers this gem:

In 1564, Viète’s mathematical skills led to him entering the service of the Parthenay family, so that he could act as tutor to the twelve-year-old mathematical prodigy Catherine de Parthenay. Together they wrote a number of treatises on astronomy and trigonometry. In these, Viète used decimal notation several decades before this was introduced to the northern Renaissance by the Dutch mathematician Simon Stevin.

Decimal notation had been in use for a couple of centuries before Viète came along, what he tried to introduce without success was the use of decimal fractions. 

After lots more biographical detail covering Viète’s political involvements, we get the following.

Viète had become involved in a dispute over the new calendar with the Jesuit monk Christopher Clavius, who had been charged with overseeing its compilation. Such was the subtlety of Viète’s mathematical argument that it was not until more than twenty years later (after his death) that a flaw was discovered in Viète’s calculations. 

The Jesuits are an apostolic and not a monastic order, so Christoph Clavius is not a monk. Clavius was not charged with compiling the Gregorian Calendar, but with explicating and defending it after it had been introduced. Viète attacked both the new calendar and Clavius in a series of pamphlets in 1600, in particular the calculation of the lunar cycle. He gave a new timetable, which Clavius refuted, after Viète’s death, in his Explicatio in 1603. I don’t know but in my world from 1600 to 1603 is not twenty years.

What is more surprising is that, during the course of his hectic royal employment, he managed to produce a body of transformative mathematics. In this, Viète attempted to give algebra a foundation as rigid as that of the geometry of Euclid, whose theorems were built upon a number of self-evident axioms. 

Viète did try to give algebra a new foundation but the analogy with Euclid’s Elements is badly chosen. The Elements, with its axiomatic approach, is the epitome of the synthetic proof methodology in mathematics. What Viète started was on the way to setting up algebra as the epitome of the analytical proof methodology; in fact, it was Viète, who replaced the term algebra with the term analysis. 

At the same time he advocated the viewing of geometry in a more algebraic fashion. Instead of the necessarily inexact measurement with a ruler of lines, curves and figures drawn on paper, these were to be reduced to algebraic formulas, thus enabling them to be calculated in algebraic fashion, giving precise numerical answers. 

This is a misrepresentation of what Viète actually did. He revived the geometric algebra that can be found in Euclid’s Elements. Here problems and theorems that we would present algebraically are handled as geometrical constructions. This is the reason why in our terminology x2 is referred to as x squared and an equation with x2 is a quadratic equation. For Euclid x is literally the side of a square or quadrate and x2 is its area. Similarly, x3 is the volume of a cube of side length x, hence the terms x cubed and cubic equation. Viète took this route because he wanted to demonstrate that the variables in an algebraic expression could represent geometrical objects, such as a line segment, and not just numbers. He didn’t develop these thoughts very far.

As we have seen, in the previous century Regiomontanus had attempted a similar standardization of algebra – but this had not become widely accepted. 

As I have explained in an earlier post Regiomontanus did not attempt a similar standardization of algebra.

Now Viète would attempt his own fundamental transformation of algebra. This branch of mathematics still largely consisted of a number of algorithms: rules of thumb to be followed in order to find the answer to a calculation. These had been set down in prose form – as indeed had all algebraic formulas. For instance: ‘In order to obtain the cubic power, multiply the unknown by its quadratic power.’ In modern notation, this can be simply put: 

y2 = y

Unfortunately Viète was hampered by the lack of an agreed symbol for ‘equals’ (=), as well as agreed symbols for ‘multiplication’ (x) and ‘division’ (÷) – which had also hampered acceptance of Regiomontanus’s notation. However, although Viète’s attempt to rationalize algebraic notation failed to gain widespread acceptance, it made many realize that such reform was long overdue. 

The transformation of algebra from rhetorical algebra, in which everything is expressed purely in words, to symbolic algebra, in which symbols are used to express almost everything, had been taking place step for step for a couple of centuries, in the form of syncopated algebra which uses a mixture of words, abbreviations, and symbols in its expressions, before Viète made his contribution As is mostly the case in the evolution of science this was not a smooth linear progress but often a case of two steps forward and one step back. With his In artem analyticem isagoge (Introduction to the art of analysis) in 1591, Viète made a significant and important contribution to that progress. His major contribution was the introduction of letters, vowels, such as A, for variables and consonants, such as Z, for parameters in algebraic expressions. Strathern is correct is saying that Viète lacked symbols for some operators. Interestingly our equals sign, =, had been in use in Northern Italy for some time and had famously been introduced into Northern Europe by Robert Recorde (c. 1512–1558) in his Whetstone of Witte in 1557. 

Viète In artem analyticem isagoge Source: MacTutor

Viète actually managed a ‘one step back’ in his Isagoge. In an earlier step in syncopated algebra quadrate had been abbreviated to q and cube to c, so A2 was written Aq and A3 as Ac. A later development was to drop the abbreviation and write A2 as AA and A3 as AAA, an important step towards our use of superscripts to indicate the multiplicity of a variable. Viète reverted to using the abbreviations q and c. His Isagoge found quite a high level of acceptance; Regiomontanus’ notation, however, found no acceptance because it never existed!

More ambitiously, Viète pressed ahead with his attempt to unite algebra and geometry, though here too any general answer eluded him. But Viète’s efforts were not to be in vain. The very fact that he had attempted such innovations would reinforce the movement of maths in the direction of its modern incarnation, where solutions to both these problems would be found. 


It would be the following century when Descartes managed to solve such problems, with the introduction of Cartesian coordinates: two lines at right angles, one representing the x-axis and the other the y-axis. Here the answers to an algebraic formula could be transformed into a line on a graph; likewise geometric lines could be seen as algebraic formulas. 

Here we are talking about the creation of analytical geometry, which was developed independently, but contemporaneously, by both Pierre Fermat (1607–1665) and René Descartes (1596–1650). Fermat, who was according to his own account influenced by Viète, circulated his Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum, based on work from 1629, in manuscript from 1636, although it was first published, posthumously in 1679. It was less influential in analytical geometry, Descartes having garnered the laurels, but had an important influence on the development of calculus, as acknowledged by Newton.

Fermat Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum, Source: Royal Astronomical Society

Descartes, for whom no influence by Viète has been determined, famously published his La Géométrie, as an appendix to his Discours de la méthode in 1637.

Descartes La Géométrie Source: Wikimedia Commons

The real impact of the work coming with the publication of the second, expanded, Latin edition by Frans van Schooten Jr (1615–1660) in 1649. You can search La Géométrie, as much as you want but you won’t find any trace of an orthogonal, Cartesian coordinate system, as used today. This was first introduced by Frans van Schooten in the Latin edition. 

Sorce MAA

The resemblance between these coordinates and the lines of latitude and longitude which Mercator drew on his maps is indicative. It was in this way that Mercator, and to a certain extent Viète, enabled the northern Renaissance to lay the foundations for our present world view. It was they who sought to devise a coordinated representation of our modern physical world in geography, and pointed the way to our modern theoretical world of multidimensional mathematics. 

Well at least Strathern recognises that a longitude and latitude grid as used by Mercator is an orthogonal, coordinate system but as is fairly clear from this final paragraph he definitely suffers from the illusion that Mercator invented the orthogonal longitude and latitude grid, which is simply historical hogwash.

If someone was intending to write an essay about Gerard Mercator, one might think that they would first acquaint themselves with an extensive knowledge of cartography and its history, in which Mercator played a highly significant role. Paul Strathern apparently didn’t feel this was necessary and obviously didn’t bother, the result is a steaming heap of bovine manure masquerading as history. 


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

A cosmologist screws up the history of cosmology

I got pulled into a Twitter exchange with a self-proclaimed science writer called Spencer (@Unpop_Science), who was ranting about “abysmal ecological education in Western society. His rant contained the tweet:

Ecology challenges the Abrahamic religions of the West, which mythologize human supremacy over the vast, complex Universe our species inhabits. It’s no coincidence that Western history’s most persecuted scientists are Galileo & Darwin — whose theories challenged human supremacy.

One of my Twitter followers, not surprisingly, asked me about Darwin being persecuted!?

Darwin was of course not persecuted but I couldn’t resist tweeting the following:

Excuse me for asking, but in what way did Galileo’s theories challenge human supremacy?

Mr ScienceWriter: They disproved the myth that the Universe revolves around us.

Me: Go away and learn some history of cosmology, that was never the belief of people before the acceptance of heliocentricity. Also, Galileo proved absolutely nothing in this context.

You are repeating a myth created in the late 18th earth 19th centuries by people such as Goethe and Kant that in the geometric cosmos humanity was in the exalted position, at the centre. For the real picture read this

And I linked to this quote on my blog:



Since, however, almost everyone has been of the conviction that the earth is immobile since it is a heavy body, the dregs, as it were, of the universe and for this reason situated in the middle or the lowest region of the heaven

Otto von Guericke; The New (So-Called) Magdeburg Experiments of Otto von Guericke, trans. with pref. by Margaret Glover Foley Ames. Kluwer Academic Publishers, Dordrecht/Boston/London, 1994, pp 15 – 16. (my emphasis)

Mr ScienceWriter: That’s it? Your own blog with one quote saying Earth is “the lowest region of the heaven”? What an amazingly lazy attempt to revise history.

Me: Not revising history that was the standard view in the Early Modern Period. You are the person who very obviously doesn’t know any history.

After his last tweet above Mr ScienceWriter linked to the webpage A Brief History, which is part of the website What is the Ultimate Fate of the Universe? Created by Molly Read of the Observational Cosmology Group at the University of Wisconsin-Madison.

Let us examine what a cosmologist, not a historian, has to tell us about the history of 2000 years of cosmology in a couple of hundred words. But before I do let us re-examine how we got here. I, for my sins, a professional historian of science, who specialises in the history of the mathematical sciences in the Early Modern Period, which of course includes the histories of astronomy and cosmology presented Mr ScienceWriter with a referenced quote from a leading seventeenth-century scholar that explains in plain terms that the people in this period didn’t think that the universe revolves around them but rather thought they lived in the garbage pit. Mr ScienceWriter, apparently an ecologist sneered because the quote was presented on my blog and cancelled me with, “What an amazingly lazy attempt to revise history.” As a counter he links to a webpage (is a webpage somehow superior to a blog?), where a cosmologist, not a historian, presents an incredibly brief history of two thousand years of cosmology without any sources, references and this is supposed to represent the “real” history! 

Since early times, man has been fascinated with discovering the origins of the cosmos. Similarly, man has often been influenced by his creationist ideas: that some divine power created the universe and everything in it. For example, the Ancient Greeks developed some of the earliest recorded theories of the origin of the universe. Unfortunately, many of these Greek philosophers and astronomers placed the Earth in the center of their models of the universe. They thought, if the heavens are divine, and the gods created man, well then certainly the universe must be geocentric, meaning the Earth is the center of the universe. 

The argument presented here as to why all early models of the cosmos are geocentric is so bizarre it hurts. I don’t know how oft I have repeated this over the years, but we live in a geocentric world. A couple of simple thought experiments. Imagine standing on a plane with a distant horizon in every direction. There can be mountains, hills, forests, or even an open stretch of water on the horizon in any particular direction. Now imagine rotating on your own axis through 360 degrees, stopping, then rotating back through 360 degrees. Everything you perceive whilst you turn has you at the centre, it’s a fact of life. Your perception is “youcentric” by definition. 

Now we turn to cosmology. Go out at night somewhere in the middle of the country, where there is no light pollution and lay down on you back in the middle of a meadow and observe the night sky. If you lie there long enough, you will see the night sky revolving over your head. If you lie there for many nights, you will see a steady change in the night sky. Do this for many years and you will recognise repeating patterns in the changes in the night sky. This is how astronomy and cosmology were both born and because the observer was per definition is at the centre of the observations the picture of the perceptible cosmos that the observer develops is by necessity geocentric. It is virtually impossible for anybody to perceive the cosmos as anything else. This is what makes thinkers like Aristarchus of Samos or Copernicus so unique in human history. It’s thinking outside of the box with a vengeance. 

One should also not forget that whereas everything observable in the heavens appears to move that platform from which our observer is making their observations, the Earth, shows no signs of moving in anyway whatsoever. Providing empirical proof that the Earth moves proved incredibly difficult. Such a proof that the Earth orbits the Sun was first delivered by Bradley’s discovery of stella aberration in 1725 (published 1728) that’s 182 years after the publication of Copernicus’ De revolutionibus! We had to wait until 1851, that’s 308 years after the publication of De revolutionibus, before we had an empirical proof of diurnal rotation!

Ancient societies were obsessed with the idea that God must have placed humans at the center of the cosmos (a way of referring to the universe).

Those pioneering astronomers were far from thinking that they were supreme, as Mr ScienceWriter would have us believe. In almost all of those early cultures that developed astronomy, when they discovered the planets, they thought they were gods. We still use the Roman gods’ names for the planets today. Those humans considered themselves to be at the mercy of the capriciousness of those, anything but friendly, gods, who controlled their fate, having dumped them on the hostile Earth, while they, the gods, reigned supreme in the heavens. 

An astronomer named Eudoxus created the first model of a geocentric universe around 380 B.C. Eudoxus designed his model of the universe as a series of cosmic spheres containing the stars, the sun, and the moon all built around the Earth at its center. Unfortunately, as the Greeks continued to explore the motion of the sun, the moon, and the other planets, it became increasingly apparent that their geocentric models could not accurately nor easily predict the motion of the other planets.

Eudoxus (c. 400–c. 350 BCE) did not create the first model of a geocentric universe. He did not even create the first Greek, model of a geocentric universe. Maybe our cosmologist meant to say that Eudoxus created the first mathematical (speak geometrical) model of a geocentric universe but even that wouldn’t be true. At best we can say that Eudoxus created the first Greek, mathematical, model of a geocentric universe, there were earlier Babylonian mathematical models, which were however based on algebraic algorithms, not geometry. The Greek’s were aware of this having acquired much of their astronomical data from the Babylonians. 

The various Greek geocentric models might have been complicated but they were surprisingly accurate in their ability to track the movement of the celestial bodies. Our cosmologist now delivers a standard explanation of the phenomenon of retrograde planetary motion, which closes with a snarky comment.

Take the apparent motion of Mars from an observer on the Earth, for example. As the Earth and Mars orbit around the sun, Mars appears to advance forwards, and then stop and start moving backwards, and then stop and change direction once again to start moving forwards (shown in the picture at left). You can see in the picture that this phenomenon is easily explained by a heliocentric universe (“heliocentric” meaning the sun is the center of the universe), but imagine being an ancient Greek and trying to understand why Mars would follow such an unusual orbit (when, according to them, it was supposed to have a circular orbit) if the Earth was the center of the universe!

Although the apparent motion of the planets contradicted the philosophical axioms of Ancient Greek cosmology, the astronomers did succeed in producing ingenious, functional models of that described fairly accurately that errant planetary behaviour. Our cosmologist nowhere explains the ingenious method that Eudoxus developed to solve this apparent contradiction, he system of nested homocentric spheres. She links to a diagram of part of Eudoxus’ system without any explanation of what it is and how it works.

After Aristotle developed a more intricate geocentric model (which was later refined by Ptolemy), general cosmology clung to these misconstrued ideas for the next 2,000 years. 

We now spring direct to Aristotle, what no Callipus (c. 370­–c. 300 BCE)? If you are going to talk about the nested homocentric spheres model developed by Eudoxus, then you should mention the improvement made by Callipus before moving on to the improvements made by Aristotle (382–322 BCE). Having completely ignored Apollonius (c. 240–c. 190 BCE) and Hipparchus (c. 190–c. 120 BCE), our cosmologist now jumps the shark stating, “which was later refined by Ptolemy,” the Eudoxian/Aristotelian system of homocentric spheres is completely different to the deferent/epicycle models developed by Ptolemy (c. 100–c. 170 CE). The two systems where contradictory and their supporters competed with each other over the next fourteen centuries.

Eudoxus’s model of planetary motion. Each of his homocentric spheres is represented here as a ring which rotates on the axis shown. The outermost (yellow) sphere rotates once per day; the second (blue) describes the planet’s motion through the zodiac; the third (green) and fourth (red) together move the planet along a figure-eight curve (or hippopede) to explain retrograde motion. Source: Wikimedia Commons
The epicycles of the planets in orbit around Earth (Earth at the centre). The path-line is the combined motion of the planet’s orbit (deferent) around Earth and within the orbit itself (epicycle). Source: Wikipedia Commons

Our cosmologist completely ignores the discussions, challenges, criticisms, and improvements made to the geocentric astronomical models, first in late antiquity, then by Islamic astronomer, and finally by medieval European astronomers, during those fourteen centuries before Copernicus emerged with his heliocentric model. She also completely ignores the geoheliocentric model first discussed by Martianus Capella (fl. c. 410), in which Venus and Mercury orbited the Sun which in turn orbited the Earth with the other planets,

Naboth’s representation of Martianus Capella’s geo-heliocentric astronomical model (1573) Source: Wikimedia Commons

as well as the geocentric model with diurnal rotation first proposed by Heraclides Ponticus (c. 390–c. 310 BCE). Both of these models were well-known and much discussed throughout the Middle Ages and are important developments in the direction of a heliocentric system. Calling those models misconstrued ideas displays an incredible ignorance on the part of our cosmologist. They are rational attempts to model the empirical evidence available to the senses at the time.   

We then get the classic piece of bullshit ignorance:

Even when Nicholas Copernicus, introduced the notion of a heliocentric universe, many contemporary societies greatly influenced by religious beliefs refused to accept it. 

That a professional academic from a large, public American university is writing crap like this in the twenty-first century beggars belief. Professional astronomers in the second half of the sixteenth century and the first half of the seventeenth didn’t accept a heliocentric model of the cosmos because there was no empirical evidence to support a moving Earth theory. As already noted above the necessary empirical evidence was first delivered 182 years, orbit around the Sun, and 308 years, diurnal rotation, after the publication of Copernicus’ De revolutionibus! Around the end of the first decade of the seventeenth century there were at least nine models of the cosmos vying for attention–Ptolemaic geocentric with or without diurnal rotation, Aristotelian homocentric, Capellan with or without diurnal rotation, Tychonic with or without diurnal rotation, Copernican heliocentric, Keplerian heliocentric and yes, the Copernican and Keplerian systems were regarded as rivals. 

Someone might ask why all the details, isn’t a simplified version of the history as presented by here by our cosmologist enough? The answer is a categorical no. The highly simplified and factually inaccurate version presented here is a falsification and corruption of the actually historical evolution of astronomy and cosmology over a period of more than two thousand years, which leads to statements such as “the Ptolemaic model of the cosmos ruled uncontested for 1400 years until Nicholas Copernicus challenged it.” Over the years I have stumbled across many variants of this historically incorrect statement. In reality from well before Eudoxus developed his geometrical model of the cosmos, cosmology and astronomy evolved in the constant ebb and flow of an intensive and complex philosophical and scientific debate. As I have explained in the past, Copernicus’ contribution was made during a period when that debate was, for various reasons, raging particularly strongly. 

Turning back to the original debate on Twitter that led to all of this, I will give a hopefully brief account of the real perception of humanities place and status in the cosmos at the time Galileo made his contributions to the ongoing debate. The primary model of the cosmos in Europe in the High Middle Ages and going on into the Renaissance, that is the one Galileo would have been acquainted with when he first became interested in astronomy, was a mixture of Aristotelian cosmology, Ptolemaic astronomy, and Catholic theology. There were variations and alternatives, as I have outlined above, but they don’t need to concern us here. The philosophical picture was determined by the Aristotelian cosmology, and Catholic theology, the Ptolemaic astronomy being regarded as purely an instrumental device to determine astronomical data for astrology, computus, that is calculating the movable church feast days, ect. 

Aristotle’s cosmology divided the cosmos into two, everything above the moons orbit, the supralunar sphere, and everything under the moons orbit, the sublunar sphere. Everything in the supralunar sphere, that is the heavens, consisting of the so called fifth element, quintessence or aether, was perfect, unchanging, eternal, and incorruptible. Everything in the sublunar sphere, that is the Earth and its atmosphere, was made of the four elements–earth, water, air, fire–was subject to change, decay, and corruption. There were initially seven heavens, corresponding to the orbits of the seven planets–Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn–a concept that predated Christianity. Each of these orbits was encased in a celestial sphere. Outside of the orbit of Saturn was the eighth sphere of the fixed stars. Over time a nineth and tenth spheres were added for mechanical reason. All of these spheres or heavens rotate, the nineth or tenth, depending on model, model the Primum Mobile providing the driving force for the whole construction.  In the Catholic theology, outside of the tenth sphere was a sphere that did not rotate, the Empyrean, the abode of God.  

Geocentric celestial spheres; Peter Apian’s Cosmographia (Antwerp, 1539) Source: Wikimedia Commons

Here it is obvious that the central point is the Empyrean, the abode of God, and in importance one either ascend into the heavens to God, or descend to the corrupt and sinful Earth, or even deeper into the pits of hell beneath the Earth.  It is very clear in this concept there is no human supremacy but as Otto von Guericke put it so succinctly, the Earth is the dregs… or the lowest region of the cosmos. Cicero in his Dream of Scipio describes an ascent through the celestial spheres, “compared to which the Earth and “Scipio now noticed that the stars were globes which easily outstripped to earth in size. Indeed the earth now appeared so small in comparison that the Roman Empire, which was hardly more than a point on that tiny surface, excited his contempt.” Macrobius in his commentary to Cicero’s work strengthened this image and spread it through the Middle Ages.

The Universe, the Earth in the centre, surrounded by the seven planets within the zodiacal signs. Image from a 12th-century manuscript of Macrobius’ Commentarii in Somnium Scipionis via Wikimedia Commons

In the Paradiso, in Dante’s Divine Comedy the image of ascending away from the corrupt Earth into the perfect Empyrean is the central theme of his narrative.

Illustration for Paradiso (of The Divine Comedy) by Gustave Doré Source: Wikimedia Commons

Galileo was well acquainted with the Divine Comedy; his very first public lecture was a reconstruction of the dimensions of hell in the Inferno.

A final example comes from the work of the highly influential medieval French philosopher, astronomer, physicist, and theologian Nicole Oresme (c.1320­–1382). In his Le livre du Ciel et du Monde, a translation of and commentary on Aristotle’s De caelo produced for his patron, Charles V, the illustration of the spheres is in the conventional order but they are curved concave upwards, centred on God, rather than concave downwards, centred on the Earth.

A page from Oresme’s Livre du ciel et du monde, 1377, showing the celestial spheres Source: Wikimedia Commons

In my initial Twitter exchange with Mr ScienceWriter, I asked him “in what way did Galileo’s theories challenge human supremacy?” He answered, “They disproved the myth that the Universe revolves around us.” I later then asked him what exactly Galileo proved, a question that he failed to answer. It is obvious that he suffers from the, unfortunately, very widespread illusion that Galileo somehow proved the validity of the heliocentric system. Regular readers of this blog will know that he did nothing of the sort. As a historian of science, I find it sad that somebody, who claims to be scientifically knowledgeable, spread falsehoods and myths to his 11,6K followers on Twitter and when challenged persists that they are right and their critics are wrong, instead of checking their claims.  I also find it depressing that a large American university posts on their official website an article on the history of astronomy and cosmology written by a non-historian, who very clearly doesn’t know what they are talking about. 


Filed under History of Astronomy, History of cosmology, Myths of Science, Renaissance Science

Renaissance garbage ­– III

This is the third in a series of discussion of selected parts of Paul Strathern’s The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). For more general details on both the author and his book see the first post in this series.

Today’s subject is Nicolaus Copernicus and I have to admit that, based on the two chapters that I had already read, I expected the Copernicus chapter to be more of a train wreck than it is. That is not to say that it isn’t a train wreck but there are less derailed carriages, decapitated corpses, and severed limbs than I expected.

One interesting aspect is that Strathern here reveals a couple of the sources that he used to write this chapter and they don’t cast a particularly good light on the level of his research for the book. The two sources that the names are Arthur Koestler’s The SleepwalkersA History of Man’s Changing Vision of the Universe, and Jack Repcheck’s Copernicus’ Secret. He describes Recheck as Copernicus’ biographer. I am not knocking Koestler’s The Sleepwalkers; it is a beautifully written book and above all it is one of the books from my youth that inspired me to become a historian of science. But let’s face it, the book was not free of errors when it was written and that was sixty years ago, the research has moved on a bit since then. Repcheck’s book is more recent, 2007, it is a pop biography strong on the personal, but weak on the science, the “secret” of the title, which forms the central part of the book is the fact that Nicky was apparently banging his housekeeper. A common practice amongst “celibate” Catholic clerics, which reliable sources tell me is still practiced in Catholic villages in Germany today. Strathern, of course, writes about the episode and manages to produce a wonderful howler in doing so.

Strathern manages to stumble several times in the opening paragraphs to the chapter. He opens with:

Perhaps the greatest renaissance, in its literal form of the rebirth of ancient learning, took place in the field of astronomy. During the third century BC, the Ancient Greek Aristarchus of Samos was a mathematician in the greatest centre of learning in the ancient world – namely Alexandria, in northern Egypt. His only surviving work is On the Sizes and Distances of the Sun and Moon, which assumes a heliocentric world view. And his contemporary, the supreme mathematician Archimedes, elaborated on Aristarchus’s ideas as they appeared in his lost works: ‘His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit.’ Aristarchus proposed that the stars were merely other suns, which despite their fixed motions did in fact move relative to each other and the earth. Unfortunately, the telescope had yet to be invented, which meant that he was unable to prove or demonstrate in any way his seemingly non-intuitive hypotheses. 

That Aristarchus worked in Alexandria is speculative and by no means an established fact. On the Sizes and Distances of the Sun and Moon, assumes a geocentric world view and not a heliocentric one. I will return to Aristarchus, heliocentricity, and rebirth. Strathern follows this with:

Nearly four centuries later, the Ancient Roman astronomer Claudius Ptolemy, who also worked in Alexandria, wrote The Almagest, which proposed an alternative astronomical picture that appeared to accord more closely with human observation. This was a geocentric model, placing the earth at the centre of the universe. 

Does one really have to point out that the geocentric model of the universe predated Aristarchus’ musing on a heliocentric model and that Ptolemy was merely codifying the work of numerous other astronomers in his Mathēmatikē Syntaxis (to give it its correct title, Almagest was a title it acquired centuries later in Arabic translation). Following a description of the Mathēmatikē Syntaxis,

Ptolemy’s geocentric system was in time accepted by the Roman Catholic Church, and thus became part of theological doctrine. This meant that throughout the Middle Ages the Ptolemaic system could not be contradicted. Anyone who did so was committing heresy. This fact is central to the life and conduct of Nicolaus Copernicus, the man who is credited with resurrecting the ideas of Aristarchus. 

The Catholic Church accepted Aristotle’s geocentric cosmology. Ptolemy’s geocentric astronomy (not the same thing at all, in fact the two systems were contradictory in numerous points) was viewed merely as a mathematical system used by astronomers for their calculations. The Ptolemaic system was often contradicted during the medieval period, by Aristotelians, who thought that his deferent/epicycle models for the planetary orbits contradicted Aristotle’s homocentric cosmology and by adherents of the geo-heliocentric model of Martianus Capella (fl. c. 410–420), which was very popular in the Middle Ages. To contradict the Ptolemaic system was not in anyway heretical. 

Here we return to Strathern’s opening statement, “Perhaps the greatest renaissance, in its literal form of the rebirth of ancient learning, took place in the field of astronomy” here given flesh, “Nicolaus Copernicus, the man who is credited with resurrecting the ideas of Aristarchus.” There is absolutely no evidence that Copernicus resurrected the ideas of Aristarchus. As far as can be determined, Copernicus conceived the concept of a heliocentric cosmos without any knowledge of Aristarchus and his theory. Later when writing De revolutionibus he played the game of seeking out and quoting predecessors. He was justifying his own concept, after the fact, by saying look I’m not the only one who had these ideas. 

There is a puzzle about Copernicus’ knowledge of Aristarchus and his heliocentric theory. He originally included the following in the manuscript of De revolutionibus but deleted it in the printed version. 

Philolaus believed in the earth’s motion for these and similar reasons. This is plausible because Aristarchus of Samos too held the same view according to some people, who were not motivated by the argumentation put forward by Aristotle and rejected by him. 

Commentators have argued that Copernicus could not have known about the Archimedes text as it was first published the year after his death. However, there is a plausible possibility how he could have known about it. The Archimedes was published by the Nürnberger Thomas Venatorius (1488–1551). Rheticus became friends with Venatorius when he was in Nürnberg before he travelled to Frombork to visit Copernicus and ask him about De revolutionibus.  Rheticus took six books containing related material with him as a present for Copernicus, as well as astronomical observations made earlier in Nürnberg by Bernhard Walther (c. 1430–1504). He could well have also brought information on the Archimedes text. Why Copernicus originally included this knowledge and then deleted it is not known.

Strathern continues:

The importance of Copernicus’s feat is all but impossible to exaggerate. Nearly three centuries later, the German poet and polymath Goethe would write in an oft-quoted passage: 

Of all discoveries and opinions, none may have exerted a greater effect on the human spirit than the doctrine of Copernicus. The world had scarcely become known as round and complete in itself when it was asked to waive the tremendous privilege of being the centre of the universe. 

Unfortunately, Goethe is here contributing to a couple of eighteenth-century myths. The European world had known the world to be round since at least the fourth century BCE and far from the Earth being the privileged centre of the universe in the geocentric model, it was to quote Otto von Guericke, the dregs: (details here)

Since, however, almost everyone has been of the conviction that the earth is immobile since it is a heavy body, the dregs, as it were, of the universe and for this reason situated in the middle or the lowest region of the heaven… [my emphasis]

Strathern now starts a fairly standard biography of Copernicus but runs into problems here:

When Nicolaus was just ten years old, his father died. All the children were then taken into the care of their maternal uncle Lucas Watzenrode the Younger, an intellectual who moved in humanist circles. Besides being a wealthy man, Watzenrode was also a canon – a member of the clergy who took ‘first orders’, including the vow of chastity. This post was often, but not always, held prior to taking full ‘higher orders’ and becoming an ordained priest. Watzenrode had ambitions: he was a canon of Frombork Cathedral, and within six years he would become Prince-Bishop of Warmia, a district of north-eastern Poland abutting the Baltic Sea. 

A canon is a member of the chapter of a cathedral, a college of clerics formed to advise a bishop. Put differently, the cathedral chapter was effectively the ruling council of the bishopric. In the case of Warmia the cathedral chapter were the government of the Prince Bishopric, which was not a district of north-eastern Poland but an automatous, self-governing region under the protection of the Polish Crown.

Turning to Copernicus’ education Strathern bizarrely writes the following: 

Nicolaus was enrolled to study canon law,* with the aim of following his uncle’s path to an important clerical post. 

* This was the body of ecclesiastical law which in general governed ordained members of the Catholic Church. Those outside the Church – the lay population – were governed by what is known as civil or common law. This system of dual jurisdiction prevailed throughout the medieval era, but generally fell into abeyance following the Reformation. 

I say bizarrely, because he appeared to be totally unaware of the existence of canon law in his chapter on Cusanus.

We then get a standard idiot-level historical claim:

Quite apart from the ever-evolving Renaissance, this was a time of world-changing discovery. In 1493, Columbus returned from his first transatlantic voyage to what he thought was Cathay (China). Ironically, it was this mistake which meant most to the natural philosophers (scientists) of the time. To them it proved, once and for all, that the world was round. 

Firstly, nobody in Europe doubted that the world was round. Secondly, Columbus sailing across the Atlantic to the Americas and then back to Europe in no way whatsoever proves that the world was round. Sailing about 7000 km west and then 7000km back east proves nothing and anybody, who thinks it does, needs their head examined.

Strathern now turns to Copernicus’ studies at the University of Kraków and his time with Albert Brudzewski (c. 1445–c. 1497) the leading scholar of astronomy there. Here Strathern drops the bomb that I already revealed in the last post about Nicholas of Cusa and Regiomontanus:

Brudzewski had read Regiomontanus and shared his belief that the geocentric Ptolemaic model had its flaws. He had also read the work of the Austrian Georg von Peuerbach, who had lived during the earlier years of the century (1423–61). Peuerbach had been taught by Regiomontanus [my emphasis] and had collaborated with him, using instruments which he invented to measure the passage of the stars in the heavens. In 1454 Peuerbach completed his Theoricae Novae Planetarium (New Theories of the Planets), which presented a more simplified form of Ptolemy’s system. 

I’ve already dealt with the “Peuerbach had been taught by Regiomontanus” last time, Peuerbach was of course Regiomontanus’ teacher, but I still shudder every time I read this sentence. Peuerbach’s Theoricae Novae Planetarium, became the first ever printed mathematical science book, it was printed and published by Regiomontanus in Nürnberg in 1473. It is not a more simplified form of Ptolemy’s system, it’s a cosmology of his system. It shows how the epicycle/deferent orbits of the planets fitted into the crystalline spheres. For a long time, it was thought that the work was original to Peuerbach, but in the 1960s an Arabic manuscript of Ptolemy’s Planetary Hypotheses (Ὑποθέσεις τῶν πλανωμένων, lit. Hypotheses of the Planets), previously unknown, was found and it was realised that the Peuerbach was just a modernised version of Ptolemy’s work. Strathern doesn’t mention that Brudzewski wrote Commentariolum super Theoricas novas planetarum Georgii Purbachii […] per Albertum de Brudzewo — a commentary on Georg von Peuerbach’s text. He also makes no mention of The Epitoma in Almagestum Ptolemae co-authored by Peuerbach, first six books, and Regiomontanus, remaining seven books, a manuscript of which was taken to Kraków by Marcin Bylica (c.1433–1493) from Budapest. Copernicus first became aware of this important text in Kraków, which would later become the model for his own De revolutionibus. The two books Peuerbach’s Theoricae Novae Planetarium and the Peuerbach/Regiomontanus Epitoma in Almagestum Ptolemae were the primary texts from which Copernicus learnt his astronomy.


In 1495 Copernicus left the University of Kraków without a degree and returned to stay with his uncle Lucas Watzenrode who had now become Prince-Bishop of Warmia. Watzenrode intended to make his nephew a canon, a sinecure which would have supported him during his ensuing studies, but this appointment was held up over a dispute concerning another candidate. So Watzenrode decided to send Copernicus to study in Italy, with the aim of furthering his career in the Church. Two years later, while he was away in Italy, Copernicus would be appointed a canon by proxy, thus guaranteeing him an income. 

We don’t know why the appointment of Copernicus as a canon was initially not granted in 1495 but there is no talk of a dispute concerning another candidate. More likely this blatant act of nepotism was viewed negatively by some, especially given Copernicus age and obvious lack of any formal qualifications. 

Although Copernicus was still supposed to be studying canon law, at the University of Bologna he so impressed the authorities with his astronomical knowledge that he was able to become an assistant to the renowned astronomer Domenico Maria Novara da Ferrara. Together, Copernicus and Novara observed a lunar occultation of Aldebaran. 

Copernicus didn’t have to impress any “authorities with his astronomical knowledge” to take up extracurricular studies of astronomy with Domenico Maria Novara da Ferrara. He just had to go and ask the man.

The year 1500 saw huge celebrations in Rome, marking one and a half millennia since the birth of Christ. Copernicus was present in the Holy City, as his uncle had arranged for him to undertake an apprenticeship at the Curia. He took this opportunity to deliver a number of lectures in Rome, casting doubt on the mathematical calculations of Ptolemaic astronomy. 

Copernicus’ supposed astronomical lectures in Rome in 1500 are only mentioned posthumously by Rheticus and there is considerable doubt concerning their existence and/or contents. 

Later, Copernicus would study at the universities of Padua and Ferrara, between times making further visits back to his uncle in Warmia. During one of these visits, Uncle Lucas instructed Copernicus to broaden his studies by learning medicine at Padua, which was a renowned centre of medical studies at the time. In 1503, Copernicus also completed a doctorate in canon law. All this gives an indication of the depth and breadth of Copernicus’s learning. It was also during this time that Leonardo da Vinci’s wide variety of pursuits made him the epitome of what came to be known as a Renaissance man. One of the great distinctions of this period was the ever-expanding breadth of knowledge of those who contributed to its discoveries. Ideas from one field were likely to inspire breakthroughs in other fields. The greatest advances in Renaissance thought, literature, ideas, science and the arts all took place in the field they referred to as the humanities. Nowadays the humanities are contrasted with the sciences, but in the Renaissance the humanities included the sciences. In accord with its name, this was the study of humanity in all its manifestations, and anything to do with it – in contrast to religious studies. 

Difficult to know where to start with this word salad. Copernicus had a degree in canon law in which he never appeared to be very interested and had studied medicine without a degree, he had a driving passion for astronomy is this an indication of depth and breadth? Medicine was a standard further study for astronomer/mathematicians during the period because of the dominance of astro-medicine. Why the comparison with Leonardo? Two completely different animals. But it’s the waffle about the humanities that I find more than somewhat bizarre. 

Let’s look what the etymological dictionary has to say on the subject:

1702; plural of humanity (n.), which had been used in English from late 15c. in a sense “class of studies concerned with human culture” (opposed variously and at different times to divinity or sciences). Latin literae humaniores, the “more human studies” (literally “letters”) are fondly believed to have been so called because they were those branches of literature (ancient classics, rhetoric, poetry) which tended to humanize or refine by their influence, but the distinction was rather of secular topics as opposed to divine ones (literae divinae).


From the late Middle Ages, the singular word humanity served to distinguish classical studies from natural sciences on one side and sacred studies (divinity) on the other side. … The term’s modern career is not well charted. But by the eighteenth century humanity in its academic sense seems to have fallen out of widespread use, except in Scottish universities (where it meant the study of Latin). Its revival as a plural in the course of the following century apparently arose from a need for a label for the multiple new ‘liberal studies’ or ‘culture studies’ entering university curricula. [James Turner, “Philology,” 2014]

We are of course dealing with Renaissance Humanism here, which as I wrote in my blog post on the topic:

They [Renaissance Humanists] referred to their own activities as studia humanitatis, from the Latin humanitas meaning education befitting a cultivated man. Once again, the origin of the modern words: humanism, humanist, and the name, the humanities. These student of humanitas devoted themselves to searching out manuscripts in monastic libraries in Latin but also in Greek that fulfilled their concept of such an education, history, music, art, literature and poetry predominating.

The relationship between the original Renaissance Humanists and science was complex and to say the least fraught, and their concept of the humanities did not include the sciences, whereas the basic, medieval, scholastic university education did.  At least nominally, the undergraduate degree at the scholastic university was based on the seven liberal arts, which included the quadrivium–arithmetic, geometry, music, astronomy. 

By the time Copernicus returned home to Warnia he was thirty years old. Apart from brief visits to Kraków, Thorn and Gdańsk he would remain in Warmia for the rest of his life, living as a canon of Frombork Cathedral. He would characterize this spot as ‘the remotest corner of the world’. Despite Copernicus’s great learning, he was not an ambitious man in any way. According to his biographer Jack Repcheck, ‘He was a retiring hermit like scholar who wanted nothing more than to be left alone.’ His uncle’s intention that he should one day succeed him as Prince-Bishop of Warmia was politely declined, and he lived out the rest of his days as a lowly canon, fulfilling just the minimum of duties for which he was being paid, and occasionally being called into service as his uncle’s physician. 

Copernicus returned to Warmia in 1503 and from then till at least 1510 and probably till 1512 he lived in the prince-bishop’s castle at Heilsberg (Lidzbark) serving as his uncle’s personal physician and secretary. In this capacity he took part in all of his uncle’s administrative and diplomatic activities, travelling throughout Warmia, Royal Prussia, and Poland. Only around the time of his uncle’s death did he take up residence at Frombork Cathedral. Also, here he took on many administrative activities in his function as a canon, which was expected of him. He continued to work as a physician treating the various prince-bishops, who succeeded his uncle, other residents and even on occasion Duke Albrecht of Royal Prussia. Between 1516 and 1521 Copernicus was resident in Allenstein (Olsztyn), the chapters economic and administrative centre. He was there during the siege of Allenstein in 1521 and was responsible for organising the towns defences. There is much more but I’m not writing a biography of Copernicus. However, a large part of his life in Warmia as a canon of the cathedral chapter was anything but “fulfilling just the minimum of duties for which he was being paid”! 

Strathern now begins to quote Koestler on Copernicus and encapsulates Koestler’s theory of The Sleepwalkers thus, “The scientists of this new age were blundering forward, towards they knew not what.” He then produces another of his extraordinarily wrong assessment of the Middle Ages:

To a greater or lesser extent, this is true of all great ages of discovery. Each new addition contributes its part in the construction of a world which was previously inconceivable. In the case of the Renaissance, this was very much so. Although the Ancient Greeks and Romans had known much of what was rediscovered during the Renaissance, the medieval world had persisted for centuries in a kind of Aristotelian dream. No matter that many of Aristotle’s findings did not in fact accord with reality: his words and his ideas, together with all the ideas which had accrued to his vision of the world (such as the Ptolemaic view of the cosmos), were simply not permitted to be questioned, any more than it was permissible to question one’s faith in God. 

I could give Strathern a long list of books on the history of medieval science that would explain to him why it was not only permitted to question many of Aristotle’s finding but was actually done with great vigour by many scholars, starting at the latest with John Philoponus in the sixth century CE. As I like to point out, “the Ptolemaic view of the cosmos” and Aristotle’s cosmology actually contradicted each other in several key points and both of them were subjected to widespread criticism during the medieval period. 

After more quotes from Koestler, Strathern writes: 

Newton famously characterized his great discoveries by claiming that he had only made them ‘by standing on the shoulders of giants’. In the case of Copernicus, we have seen how he stood on the shoulders of Aristarchus, Nicholas of Cusa, Peuerbach and Regiomontanus (to name but a few).

As already pointed out Copernicus did not stand on the shoulders of Aristarchus and almost certainly first became aware of Aristarchus’ theories long after he had formulated his own heliocentric hypothesis. Nicholas of Cusa doesn’t appear anywhere in De revolutionibus and to quote Edward Rosen from his notes to the English translation of Copernicus’ magnum opus, “it cannot be shown that Copernicus was acquainted with the writings of Nicholas of Cusa,” so no shoulders there. Copernicus did learn much of his astronomy from the readings of Peuerbach and Regiomontanus and De revolutionibus is modelled on the Peuerbach-Regiomontanus Epitoma in Almagestum Ptolema, so we’ll give him that one. However, although Peuerbach and Regiomontanus contributed substantially to Copernicus’ astronomical education, it’s unlikely that anything either of them wrote contributed to his “discovery”. Interestingly, we don’t actually know what motivated Copernicus to adopt a heliocentric model; there are lots of theories put out by historians of astronomy, but they are all speculative. In terms of Strathern’s insistence that Copernicus dared to reject Aristotle, in fact, the only motivation that Copernicus gives is the desire to remove Ptolemy’s equant point because it contradicted Aristotle’s cosmological axioms.

In 1517, Luther’s theses burst upon Europe, creating turmoil throughout western Christendom. Meanwhile Copernicus continued painstakingly setting down his ‘larger work’, making the requisite mathematical calculations based upon astronomical observations he had made in Italy, and further observations which he now continued to make despite the cloudy northern nights of Warmia. 

There are only sixty known observations made by Copernicus over a period of the fifty years that he was interested in astronomy, so not in any way a significant observational astronomer.

The breadth of knowledge and wisdom possessed by the scholarly canon soon began to spread further afield. In 1519, King Sigismund I of Poland became distressed at the sheer economic muddle which had overtaken his kingdom. Trade was clogged, and no one seemed to know what to do about it. For a start there were no less than three currencies in circulation: that of Royal Prussia, the Polish currency, and the currency minted by the Teutonic Order of Knights. In an attempt to retain the Order’s wealth and status, the last of these three currencies had been systematically debased. (Similar practices prevailed amongst the other currencies, though on a lesser scale.) Meanwhile, citizens had begun hoarding more valuable coins for their higher true-metal content, and only using debased or ‘clipped’ coins.

Copernicus wrote a paper for King Sigismund I, suggesting a reform of the currency situation, with the maintenance of a single uniform currency. 

Copernicus’ initial work on theories of currency was conducted together with and on request of the Royal Prussians and not Sigismund of Poland. It is true that both the Prussians and the Poles read his finished document but neither of them acted upon it. Strathern goes off on a long spiel that Copernicus anticipated Gresham’s Law. 

Almost at the end of the chapter, Strathern finally reaches De revolutionibus again emphasising the observations and claiming that one of them confirmed his heliocentric hypothesis. It didn’t! He then makes a bizarre statement that I can’t explain, can any of my readers? Personally, I think it’s garbage. 

Copernicus’s heliocentric system also explained certain anomalies. As the earth orbited on its circular path around the sun, it tilted slightly on its axis. This accounted for why certain eclipses took place a few minutes before or after their predicted time. 

We then get the big Copernicus myth:

As he grew older, Copernicus began to attract a number of visiting intellectuals, keen to hear his latest ideas. One of these was the twenty- five-year-old Austrian-born Lutheran scholar Georg Joachim Rheticus, who would remain with the ageing Copernicus as his assistant. 

Rheticus, as well as others, kept urging Copernicus to publish his work, but Copernicus demurred. Being a canon of the Church he was loath to promulgate ideas which might be mistaken for heresy. [my emphasis]

This myth has been rejected by historians of astronomy for decades. Various high Church official, well aware of Copernicus’ heliocentric hypothesis, were urging him to publish. For example, Cardinal Nikolaus von Schönberg (1472–1537), Archbishop of Capua wrote to Copernicus in 1536:

Some years ago word reached me concerning your proficiency, of which everybody constantly spoke. At that time I began to have a very high regard for you… For I had learned that you had not merely mastered the discoveries of the ancient astronomers uncommonly well but had also formulated a new cosmology. In it you maintain that the earth moves; that the sun occupies the lowest, and thus the central, place in the universe… Therefore with the utmost earnestness I entreat you, most learned sir, unless I inconvenience you, to communicate this discovery of yours to scholars, and at the earliest possible moment to send me your writings on the sphere of the universe together with the tables and whatever else you have that is relevant to this subject …

He even offered to cover the costs of having a fine copy of Copernicus’ manuscript made.

It is fairly certain that Copernicus was reluctant to publish because he couldn’t prove the truth of his heliocentric hypothesis, which he had claimed he would do in the Commentariolus, his first early outline of his hypothesis. Not wishing—as he confessed—to risk the scorn “to which he would expose himself on account of the novelty and incomprehensibility of his theses.”

We then get the story of Dantiscus banishing Copernicus’s housekeeper, Anna Schilling, from Frombork in spring 1539. Here Strathern once again manages to produce some true garbage:

Indeed, Copernicus’s indiscretion might have been overlooked while his uncle Lucas was prince-bishop, but this was not the case with some of his successors. Prince-Bishop Johannes Dantiscus was particularly upset when in 1537 he succeeded to the bishopric, and wrote to his predecessor [my emphasis] Tiedemann Giese, whom he knew was a friend of Copernicus, remonstrating about his canon’s behaviour: ‘In his old age… he is still said to let his mistress in frequently in secret assignations. Your Reverence would perform a great act of piety if you warned the fellow privately and in the friendliest terms to stop this disgraceful behaviour.’ And not only was Copernicus living with his mistress, he was also entertaining in his house Lutherans such as Rheticus. 

Prince-Bishop [my emphasis] Giese, who had influence at the royal court and was senior to Prince-Bishop[my emphasis] Dantiscus, proved a true friend. 

Tiedemann Giese (1480–1550), who was Copernicus’ best friend, was Bishop of Kulm (Chełmno). He was not at that time a prince-bishop and was in the Church hierarchy junior to Danticus, whose successor he was and not his predecessor. Dantiscus, who was a highly educated and knowledgeable man, would have had no problems with Rheticus’ presence in Warmia. Rheticus was a protégé of Philip Melanchthon. Dantiscus knew Melanchthon personally and respected him highly, only regretting that he was a Protestant, his protégé Rheticus would have been a welcome guest. Dantiscus was a fan of Copernicus’ astronomical work and even invited Gemma Frisius, who he knew personally, to come to Frombork to work with Copernicus. 

Strathern continues with more nonsense: 

Not only did he protect Copernicus from censure, but in league with Rheticus he persuaded Copernicus to part with a copy of De Revolutionibus Orbium Coelestium, allegedly so that they could study it in greater detail. Between the two of them, Giese and Rheticus then contrived to have Copernicus’s work published in Nuremberg in 1543. 

I have no idea where Strathern dredged up this little piece of fantasy. The was no conspiracy, no skulduggery. Buoyed by the positive reception of Rheticus’ Narratio Prima, published in Danzig in 1540 and again in Basel in 1541, and convinced by the quality of the science books published by Johannes Petreius, brought by Rheticus as a gift to Frombork, Copernicus gave a manuscript of his De revolutionibus to Rheticus to take to Nürnberg to be published by Petreius.

The rubbish continues:

They intended Copernicus’s work to be judged on its own merit; and wished, as far as possible, to avoid any controversy. Unfortunately, Rheticus was called away from supervising the publication in Nuremberg, and this task was taken over by a Lutheran theologian called Andreas Osiander, who had ideas of his own. Osiander was a reformer of considerable influence; indeed, some years previously he had been instrumental in Nuremberg becoming a Protestant city. Purely on his own initiative Osiander inserted an anonymous preface of his own into De Revolutionibus Orbium Coelestium, giving the impression that the preface had in fact been penned by Copernicus himself. In doing so, Osiander managed to undermine the two most important principles which Copernicus wished to uphold: the avoidance of controversy, and the certain truth of his system. Not only did Osiander make a show of how offensive Copernicus’s system was to long-accepted orthodoxy; at the same time he stated that the author believed its contents to be a mere hypothesis. At a stroke, he left Copernicus open to a charge of heresy yet simultaneously implied that he did not believe in the truth of what he was saying. 

I know this is nit picking but Osiander wrote an ad lectorum, address to the reader, not a preface and it was obviously inserted in the book with the full knowledge and approval of Petreius the publisher. This is very obvious from the very rude answer that Petreius gave the Nürnberg City Council, when they questioned him following the receipt of a letter from Tiedemann Giese complaining about the inclusion of the ad lectorum. He basically said, I’ll include in my books what I want. To anybody, who actually reads the ad lectorum, it is obvious from how it is phrased that it was not written by the author of the book. 

The rest of Strathern’s comment is a total inversion of the effect of the ad lectorum. Far from leaving Copernicus open to a charge of heresy; by indicating that the book could also be simply read as a mathematical hypothesis, Osiander actually defused the possibility of such an accusation. 

One last piece of Strathern confusion:

Fortunately, Copernicus would never see these words. By now he was seventy and lay on his deathbed, having suffered a stroke. However, according to legend, the final printed pages of Copernicus’s masterpiece were delivered to him in Warmia, where he still lived in an isolated tower outside Frombork, whereupon he is said to have woken briefly from his coma. 

Copernicus acquired a house outside of the cathedral walls in 1512 in which he was still living when he died. In 1514 he purchased the north-western tower within the walls of the Frombork stronghold, which it is assumed that he used for his astronomical observations. It would appear that Strathern has a black belt in mangling facts. 


Filed under History of Astronomy, Renaissance Science

Correcting navigational errors, the Wright way

Today I’m continuing my occasional series on the English mathematical practitioners of the Early Modern Period. In the post in this series about Edmund Gunter (1581–1626) I quoted the historian of navigation David Waters as follows: 

Gunter’s De Sectore & Radio must rank with Eden’s translation of Cortes’s Arte de Navegar and Wright’s Certain Errors as one of the three most important English books ever published for the improvement of navigation.[1]

I have already written about Richard Eden (c. 1520–1576) back in 2021, so today I am turning my attention to the third of Water’s trio of navigation improvers, Edward Wright (1561–1615).

As is fairly obvious from the David Waters quote, Edward Wright is one of the most important figures in the history of, not just English but European, navigation during the Early Modern Period. However, as is, unfortunately, all to often the case with mathematical practitioners from this period, we have very little biographical detail about his life and can only fill the gaps with speculation. 

The younger son of Henry and Margaret Wright, he was baptised in the village of Garveston in Norfolk on 8 October 1561. His father, a man of “mediocrisfortunae” (modest means), was already deceased, when his elder brother Thomas entered Gonville and Caius College, Cambridge as a pensioner in 1574. Edward was probably educated by John Hayward at Hardingham school, like his elder brother, and also entered Gonville and Caius College, as a sizar, a student who earns part of his fees by working as a servant for other students, in December 1576. Unfortunately, Thomas died early in 1579. Edward graduated BA in the academic year 1580-81 and MA in 1584. He became a fellow of Gonville and Caius in 1587 and resigned his fellowship in 1596, having married Ursula Warren (died 1625) 8 August 1595. Oxbridge fellows were not permitted to marry. They had a son Thomas Wright (1596–1616), who was admitted sizar at Gonville and Caius in 1612.

Wright’s career in Cambridge parallels that of another significant mathematical practitioner born in the same year, Henry Briggs (1561–1630). Briggs went up to St John’s College in 1577, graduated BA in 1581 or 1582 and MA in 1585. He was awarded a fellowship in 1588. The two became friends and interacted over the years up till Wright’s death.

Henry Briggs

Another acquaintance of Wright’s, who he possibly got to know at Cambridge, was the aristocrat Robert Devereux (1565–1601), who graduated MA at Trinity College Cambridge in 1581, and who had succeeded to the title of Earl of Essex in in 1576 at the death of his father. Devereux, a soldier, was incredibly well connected in Elizabethan society becoming a favourite at Elizabeth’s Court and so would initially have been a good contact for the commoner Wright. However, he was still a close friend of Wright’s when he rebelled against Elizabeth at the end of the century, which could have proved dangerous for the mathematical practitioner, but apparently didn’t.

Melancholy youth representing the Earl of Essex, c.1588, miniature by Nicholas Hilliard Source: Wikimedia Commons

Another soldier and a good friend of Devereux’s, who also became a friend of Wright’s at Cambridge, was the astronomer and astrologer, Sir Christopher Heydon (1561–1623), who graduated BA in 1589 at Peterhouse Cambridge. 

Sir Christopher Hendon’s birth horoscope

What we don’t know is who taught Edward Wright mathematics at Cambridge and how, why, and when he became deeply interested in navigation and cartography, which he very obviously did, whilst still at the university. The interest in sea voyages and all things navigational associated with them was very strong in England in the latter part of the sixteenth century, with England beginning to flex its deep-sea muscles and challenge the Spanish Portuguese duopoly on marine exploration and discovery, particularly following the defeat of the Spanish Armada in 1588. This could well have been Wright’s motivation as a mathematical practitioner to follow the lead of other practitioners such as John Dee (1527–1608/9) and Thomas Harriot (c. 1560–1621) and specialise in navigation. 

That Wright had taken up the study of navigation and already acquired a substantial reputation is indicated by the Royal Mandate, issued by Elizabeth in 1589, instructing Gonville and Caius College to grant Wright leave of absence to carry out navigational studies on a raiding expedition to the Azores under the command of Sir George Clifford, 3rd Earl of Cumberland (1588–1605).

George Clifford, 3rd Earl of Cumberland after Nicholas Hilliard  Source: Wikimedia Commons

Cumberland was sailing as what is known as a privateer, which means piracy licensed by the Crown in exchange for a share of the profits. Sailing to the Azores, on the way Cumberland seized French Catholic league and Flemish vessels. In the Azores he attacked both the islands and various Portuguese and Spanish vessels making rich killings. Up till now, the expedition was a success, but the return journey was pretty much a disaster. Hit by storms many of the crew died of hunger and thirst on the return journey and the English ship the Margaret was shipwrecked off the coast of Cornwall. All the while Wright was carrying out his navigational studies. On the voyage he was accompanied by Richard Hues (1553–1632) a cartographical and navigational pupil of Thomas Harriot and one of the Wizard Earl’s mathematici. He also became acquainted with the navigator and explorer John Davis (c. 1550–1605). 

Miniature engraved portrait of navigator John Davis (c. 1550-1605), detail from the title page of Samuel Purchas’s Hakluytus Posthumus or Purchas his Pilgrimes (1624) Source: Wikimedia Commons

Wright recorded his experiences of the Azores’ voyage in his most important publication: 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.), London: Printed … by Valentine Sims. 

Edward Wright’s map “for sailing to the Isles of Azores” (c. 1595), the first to be prepared according to his projection Source: Wikimedia Commons

Another version of the work published in the same year was entitled: Wright, Edward (1599), Errors in nauigation 1 Error of two, or three whole points of the compas, and more somtimes, by reason of making the sea-chart after the accustomed maner … 2 Error of one whole point, and more many times, by neglecting the variation of the compasse. 3 Error of a degree and more sometimes, in the vse of the crosse staffe … 4 Error of 11. or 12. minures in the declination of the sunne, as it is set foorth in the regiments most commonly vsed among mariners: and consequently error of halfe a degree in the place of the sunne. 5 Error of halfe a degree, yea an whole degree and more many times in the declinations of the principall fixed starres, set forth to be obserued by mariners at sea. Detected and corrected by often and diligent obseruation. Whereto is adioyned, the right H. the Earle of Cumberland his voyage to the Azores in the yeere 1589. wherin were taken 19. Spanish and Leaguers ships, together with the towne and platforme of Fayal, London: Printed … [by Valentine Simmes and W. White] for Ed. Agas.

Before we turn to the navigational errors that Wright illuminated in his book, it also contains another piece of interesting information. Wright states that he sailed with Cumberland under the name Edward Carelesse. When he introduces himself in the book, he also states that he sailed with Sir Francis Drake, as Captain of the Hope, on his West Indian voyage of 1585-86, which evacuated Sir Walter Raleigh’s Virginia colony and brought the survivors back to England. Wright would have had the opportunity to make the acquaintance of Thomas Harriot, who was one of the rescued colonists. Capt. Walter Bigges and Lt. Crofts’ book A Summarie and True Discourse of Sir Frances Drakes West Indian Voyage (1589) confirms that Edward Carelesse was commander of the Hope. This voyage would fit into the gap between Wright’s MA, 1585 and the start of his fellowship in 1587. 

The principal navigational error that Wright’s book addresses, and the reason why it is so important, is the problem of sailing the shortest route between two places on a sea voyage. In the early phase of European deep-sea exploration, mariners adopted the process of latitude sailing. Mariners could not determine longitude but could determine latitude fairly easily. Knowing the latitude of their destination they would sail either north or south until they reached that latitude and then sail directly east or west until they reached their desired destination. This was by no means the most direct route but prevented getting lost in the middle of the ocean. 

The actual shortest route is a great circle, that is a circumference of the globe passing through both the point of departure and the destinations. However, it is very difficult to sail a great circle using a compass as you have to keep adjusting your compass bearing. Although not as short, far more practical for mariners would be a course that is a constant compass bearing, such a course is known as a rhumb line, rhumb, or loxodrome:

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path of constant bearing as measured relative to true north. (Wikipedia)

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

The first to analyse the mathematics of rhumb lines, which takes the form of a spiral on the surface of a sphere, was the Portuguese mathematical practitioner Pedro Nunes (1502–1578) in his Tratado em defensam da carta de marear (Treatise Defending the Sea Chart), (1537).

Pedro Nunes, 1843 print Source: Wikimedia Commons

Nunes determined that a course of constant bearing would be a rhumb, but he did not solve the problem of how to construct a marine chart on which a rhumb line would be a straight-line enabling navigators to simply read off the required compass bearing from the chart. This problem was first solved by the Flemish globe maker and cartographer, Gerard Mercator (1512–1594), who was friends with Nunes, with the publication of his world map of 1569, which introduced for the first time what is now known as the Mercator projection on which a course of constant compass bearing is a straight line.  

Gerhard Mercator, Kupferstich von Frans Hogenberg, 1574 Source: Wikimedia Commons

Mercator explained in simple terms how he had achieved this, “We have progressively increased the degrees of latitude towards each pole in proportion to the lengthenings of the parallels with reference to the equator” but gave no exact mathematical prescription how to produce such a chart. 

Mercator World Map 1569 Source: Wikimedia Commons

Both John Dee (1527–1609?), who personally knew Pedro Nunes and studied cartography under Mercator’s teacher Gemma Frisius (1508–155) and Thomas Harriot (c. 1560–1621) solved the problem of how to mathematically construct the Mercator projection. However, although both of them taught cartography and navigation, Dee to the captains of the Muscovy Trading Company and Harriot to Walter Raleigh’s captains, neither of them made their solution public. Enter Edward Wright.

Wright took up the problem of the marine chart and rhumb lines where Pedro Nunes had stopped, openly acknowledging his debt to Nunes in the preface to his Certaine Errors in Navigation:

Cover of Wright’s Certaine Errors Source: Wikimedia Commons

Yet it may be, I shall be blamed by some, as being to busie a fault-finder myself. For when they shall, see their Charts and other instruments controlled which so long time have gone for current, some of them perhappes will scarcely with pacience endure it. But they may be pacified, if not by reason of the good that ensueth hereupon, yet towards me at the least because the errors I poynt at in the chart, have beene heretofore poynted out by others, especially by Petrus Nonius, out of whom most part of the first Chapter of the Treatise following is almost worde for worde translated;

He goes on to solve the problem of constructing the Mercator projection:

By help of this planisphaere with the meridians, rumbes, and parallels thus described therein, the rumbs may much more easily & truly be drawn in the globe then by these mechanical wayes which Petrus Nonius [Pedro Nunes] teacheth cap. 26 lib. 2 de obser. Reg. et Instr. Geom..

The problem that Wright solved is that as one proceeds north or south from the equator the circles of latitude get progressively smaller but when one unwraps the globe on the surface of a cylinder in the Mercator projection all the lines of latitude need to be the same length so that they cross all lines of latitude at right angles. 

The Mercator projection shows rhumbs as straight lines. A rhumb is a course of constant bearing. Bearing is the compass direction of movement. Source: Wikimedia Commons

Wright’s principle was very simple: to increase the distance apart of the parallels of latitude to match the exaggeration arising from the assumption that they were equally long. Since the lengths of the parallels varied according to a factor cos λ, the correction factor was sec λ at any point. In order to plot the parallels on the new charts, Wright had effectively to perform the integration’ sec λdλ. This was done numerically—in his own words, “by perpetual addition of the Secantes answerable to the latitudes of each point or parallel into the summe compounded of all the former secantes. . . .,” (P. J. Wallace, Dictionary of Scientific Biography)

To save others having to repeat the protracted and tedious numerical iterations that he had carried out, Wright published a table of the necessary correcting factors for the distance between the lines of latitude. In the first edition of the book this table was only six pages long and contained the correction factors for every 10 minutes of latitude. In the second edition of the book, Certaine Errors in Navigation, Detected and Corrected with Many Additions that were not in the Former Edition…, published in London in 1610, the table had grown to 23 pages with factors for every minute of latitude. 

Cover of Wright’s Certaine Errors second edition 1610 Source

The emergence of both Wright’s book and his method of constructing the Mercator projection into the public sphere is rather complex. He obviously wrote the major part of the manuscript of the book when he returned to Cambridge in 1598 but there are sections of the book based on observation made in London between 1594 and 1597. Wright’s development of the Mercator projection was first published, with his consent, in Thomas Blundevile’s His Exercises containing six Treatises in 1594, the first publication in English on plane trigonometry, he wrote:

Blundevile’s Exercises 1613 edition Source: Wikimedia Commons

[the new (Mercator) arrangement, which had been constructed] “by what rule I knowe not, unless it be by such a table, as my friende M.Wright of Caius College in Cambridge at my request sent me (I thanke him) not long since for that purpopse which table with his consent. I have here plainlie set down together with the use of thereof as followeth”. The table of meridiional parts was given at degree intervals.

The Dutch engraver and cartographer Jodocus Hondius (1563–1612), who in his time in London had obviously had access to Wright’s work, published his “Christian Knight World Map” based on Wright’s calculations without acknowledgement in 1597.

The so-called “Christian Knight map”, published by Flemish map-maker Jodocus Hondius in Amsterdam in 1597. To produce the map, Hondius made use of Edward Wright’s mathematical methods without acknowledgement. Source: Wikimedia Commons

Although he wrote a letter of apology to Wright, Wright condemned him for it in the preface to Certaine Errors:

“But the way how this [Mercator projection] should be done, I learned neither of Mercator, nor of any man els. And in that point I wish I had beene as wise as he in keeping it more charily to myself”

Hondius was by no means the only one to publish Wright’s method before he himself did so. William Barlow (1544–1625) included in his The Navigator’s Supply (1597) a demonstration of Wright’s projection “obtained of a friend of mone of like professioin unto myself”.

In 1598–1600 Richard Hakluyt published his Principle Navigations which contains two world charts on the new projection, that of 1600 a revision of the first. Although not attributed to Wright it is clear that they are his work. 

The title page of the first edition of Hakluyt’s The Principall Navigations, Voiages, and Discoveries of the English Nation (1589) Source: Wikimedia Commons
Wright’s “Chart of the World on Mercator’s Projection” (c. 1599), otherwise known as the Wright–Molyneux map published in Hakluyt’s The Principall Navigations Source: Wikimedia Commons

Earlier, the navigator Abraham Kendall had borrowed a draft of Wright’s manuscript and unknown to Wright made a copy of it. He took part in Drake’s expedition to the West Indies in 1595 and died at sea in 1596. The copy was found in his possessions and believing it to be his work it was brought to London to be published. Cumberland showed the manuscript to Wright, who, of course recognised it as his own work.

Wright first publicly staked his claim to his work when he finally published the first edition of Certaine Errors in 1599. A claim that he reinforced with the publication of the second, expanded edition in 1610. However, it should not be assumed that mariners all immediately began to use Mercator projection sea charts for navigating. The acceptance of the Mercator marine chart was a slow process taking several decades. As well the method of producing the Mercator projection, Certaine Errors also includes other useful information on the practice of navigation such as a correction of errors arising from the eccentricity of the eye when making observations using the cross-staff, tables of declinations, and stellar and solar observations that he had made together with Christopher Haydon. The work also includes a translation of Compendio de la Arte de Navegar (Compendium of the Art of Navigation, 1581, 2nd ed., 1588) by the Spanish cosmographer Rodrigo Zamorano (1542–1620).

It is not clear how Wright lived after he had resigned from his fellowship. There are suggestions that he took up the position of Mathematicall Lecturer to the Citie of London when Thomas Hood resigned from the post after only four years in 1592. However, there is no evidence to support this plausible suggestion. Wright’s friend, Henry Briggs, was appointed the first Gresham professor of geometry in 1596, a position to hold public lectures also in London, which may have made the earlier lectureship superfluous. However, Wright was definitely employed by Thomas Smith and John Wolstenholme, who had sponsored Hood’s lectureship, as a lecturer in navigation for the East India Company at £50 per annum, probably from 1612 but definitely from 1614.  Before his employment by the East India Company, he had been mathematical tutor to Prince Henry (1594–1612), the eldest son of King James I/IV, from about 1608, to whom he dedicated the second edition of Certaine Errors

In the 1590s Wright was one of the investigators whose work contributed to William Gilbert’s De Magnete(1600) for which he wrote the opening address on the author and according to one source contributed Chapter XII of Book IV, Of Finding the Amount of Variation


In this context he also wrote, Description and Use of the Two Instruments for Seamen to find out the Latitude … First Invented by Dr. Gilbert, published in Blundeville, Thomas; Briggs, Henry; Wright, Edward (1602),The Theoriques of the Seuen Planets… a work on the dip circle. 

He also authored The Description and Vse of the Sphære. Deuided into Three Principal Partes: whereof the First Intreateth especially of the Circles of the Vppermost Moueable Sphære, and of the Manifould Vses of euery one of them Seuerally: the Second Sheweth the Plentifull Vse of the Vppermost Sphære, and of the Circles therof Ioyntly: the Third Conteyneth the Description of the Orbes whereof the Sphæres of the Sunne and Moone haue beene supposed to be Made, with their Motions and Vses. By Edward Wright. The Contents of each Part are more particularly Set Downe in the Table first published in London in 1613 with a second edition in 1627. This could be viewed as a general introduction to the armillary sphere, but was actually written was a textbook for Prince Henry. A year later he published A Short Treatise of Dialling Shewing, the Making of All Sorts of Sun-dials, Horizontal, Erect, Direct, Declining, Inclining, Reclining; vpon any Flat or Plaine Superficies, howsoeuer Placed, with Ruler and Compasse onely, without any Arithmeticall Calculationprobably also written for the Prince.

As well as the translation of Zamorano’s Compendio de la Arte de Navegar included in his Certaine Errors, he translated Simon Stevin’s The Hauen-finding Art, or The VVay to Find any Hauen or Place at Sea, by the Latitude and Variation. Lately Published in the Dutch, French, and Latine Tongues, by Commandement of the Right Honourable Count Mauritz of Nassau, Lord High Admiral of the Vnited Prouinces of the Low Countries, Enioyning all Seamen that Take Charge of Ships vnder his Iurisdiction, to Make Diligent Obseruation, in all their Voyages, according to the Directions Prescribed herein: and now Translated into English, for the Common Benefite of the Seamen of England, a text on determining longitude using magnetic variation.

Full text here

In 1605, he also edited Robert Norman’s translation out of Dutch of The Safegarde of Saylers, or Great Rutter. Contayning the Courses, Dystances, Deapths, Soundings, Flouds and Ebbes, with the Marks for the Entring of Sundry Harboroughs both of England, Fraunce, Spaine, Ireland. Flaunders, and the Soundes of Denmarke, with other Necessarie Rules of Common Nauigation. 

His most important work of translation was certainly that of John Napier’s Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms, 1614) from the original Latin into English: A Description of the Admirable Table of Logarithmes: With a Declaration of the … Use thereof. Invented and Published in Latin by … L. John Nepair … and Translated into English by … Edward Wright. With an Addition of an Instrumentall Table to Finde the Part Proportionall, Invented by the Translator, and Described in the Ende of the Booke by Henry Brigs, etc approved by Napier and first published posthumously by Henry Briggs in 1616 and then again in 1618. 


The development of mathematical navigation, cartography, and surveying in the Early Modern Period, in which Wright along with others played a central role, was by nature predominantly trigonometrical. Napier’s invention of logarithms made the complex trigonometrical calculations much easier to manage. This was something that Napier himself was acutely aware of and the majority of tables in his work were, in fact, logarithms of trigonometrical functions. By translating Napier’s work into English, Wright made it accessible to those mariners confronted with trigonometrical navigational problems, who couldn’t read Latin. The introduction contains the following poem:

            The toylesome rules of due proportion

            Done here by addition and subtraction,

            By tripartition and tripartition,

            The square and cubicke roots extraction:

                        And so, all questions geometricall,

                        But with most ease triangles-sphericall.

            The use in great in all true measuring

            of lands, plots, buildings, and fortification,

So in astronomy and dialling,

Geography and Navigation.

             In these and like, yong students soon may gaine

             The skilfull too, may save cost, time, & paine.

Wright was also acknowledged as a skilled designer of scientific instruments, but like his friend Edmund Gunter (1561–1626), he didn’t make them himself. He is known to have designed instruments for the astronomer/astrologer Sir Christopher Haydon and to have made astronomical observations with him in London in the early 1590s. We don’t know Wright’s attitude to astrology, but that of his two Cambridge friends was diametrically opposed. Haydon was the author of the strongest defence of astrology written in English in the early seventeenth century, his A Defence of Judiciall Astrologie (1603), whereas Henry Briggs was one of the few mathematical practitioners of the period, who completely rejected it, unlike John Napier who as an ardent supporter. 

Wright’s work in navigation was highly influential on both sides of the North Sea. 

His work formed the basis of the navigation treatise Tiphys Batavus (1624) of the Netherlander mathematical practitioner Willebrord Snel (1580­–1626),

Source: Wikimedia Commons

as well as the Primum mobile astronomice, sciographice, geometrice & hydrographice nova methodo explicatum of his student Adriaan Adriaanszoon Metius (1571–1635) son of the mathematical practitioner Adriaan Anthoniszoon (1541–1620) and brother of Jacob Metius (1572–1628) one of the inventors of the telescope. Wright’s friend Edmund Gunter included a scale for the meridional parts of the Mercator chart on his sector.


Wright gets positively acknowledge, both in The Navigator (1642) by Charles Saltonstall (1607–1665) and in the Navigation by the Mariners Plain Scale New Plain’d (1659) by John Collins (1625–1683).

In England Wright’s work was also taken up by Richard Norwood (1590? –1675), the surveyor of Bermuda, who using Wright’s methods determined one degree of a meridian to be 367,196 feet (111,921 metres), surprisingly accurate, publishing the result in his The Seaman’s Practice, 1637. However, in his Norwood’s Epitomy, being the Application of the Doctrine of Triangles, 1645, he gives a clear sign that the Mercator chart still hasn’t been totally accepted 46 years after Wright first published the solution of how to construct it. 

Although the ground of the Projection of the ordinary Sea-Chart being false, (as supposing the Earth and Sea to be plain Superficies [sufaces]) and so the conclusions thence derived must also for the most part erroneous; yet because it is most easy, and much used, and the errors in small distances not so evident, we will not wholly neglect it. 

He actually devotes as much space, in this work, which continued to be published throughout the century in various editions, to plain sailing as he does to Mercator sailing. Interestingly in the section on Mercator sailing, he doesn’t, following Wright, just give a table of meridional parts but explains how to use trigonometry to calculate them.

Now that which he [Edward Wright] hath shewed to performe by the Chart it selfe [the table of meridional parts], we will shew to work by the Doctrine of plaine Triangles, using the helpe of the Table of Logarithme Tangents

Although its impact was drawn out over several decades it is impossible to over emphasise Wright’s contribution to the histories of cartography and navigation by his publication of the mathematical means of constructing a Mercator chart.

[1] David Walters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, 1958 p. 359


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

Renaissance garbage ­– II

This is the second in a series of discussion of selected parts of Paul Strathern’s The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). For more general details on both the author and his book see the first post in this series.

Today, I turn my attention to his chapter on the fifteenth century, German philosopher and theologian, Nicholas of Cusa, a large part of which is, as we shall see, actually devoted to another fifteenth century German scholar. Right in his opening paragraph to this chapter, Strathern lets a historical bomb of major dimensions explode, he writes:

As the world of art in northern Europe began its drastic transformation, shedding the stylistic formalism and religious subject- matter of medieval art, so the northern intellectual world underwent a similar revolution. The origins of the humanistic way of thought and its empirical attitude to learning were not the sole preserve of the Italian Renaissance. Indeed, the ‘father of humanism’ is generally recognized as a German, born in 1401 in the Electorate of Trier: Nicholas of Cusa [my emphasis].

Regular readers will already know that here at the Renaissance Mathematicus we react extremely allergically to the phrase ‘father of anything’, but to title Nicholas of Cusa ‘father of humanism’ is really breath takingly stupid, further to also claim that this is ‘generally recognised’ pushes the claim into the mindboggling. As I explained in the fourth part of my series on Renaissance science, Renaissance Humanism, which originated in Northern Italy, did so almost a century before Nicholas of Cusa was born, so if he was, as Strathern claims, the ‘father of humanism’ then his birth in 1401 must have been a reincarnation. 

Nicholas of Cusa, by Master of the Life of the Virgin Source: Wikimedia Commons

Strathern writes that Nicholas was “the son of ‘a prosperous boatman and ferryman’.” A boatman is, according to the dictionary, “a man who takes people or goods somewhere in a small boat, or who has small boats that you can rent for a period of time.” The Stanford Encyclopedia of Philosophy says his father was “a prosperous merchant who became one of the landed gentry in Trier” and German Wikipedia say he was “als Schiffer ein wohlhabender Kaufmann,” that is, “as shipper a wealthy merchant”. Once again according to the dictionary “a shipper is a company that arranges for goods to be taken somewhere by ship.” There appears to be a major disparity concerning the real profession of the father and Strathern’s version.  

Nicholas was a precocious student. In his early teens he entered the University of Heidelberg, the oldest in Germany, which had been established in the middle of the previous century. Here he studied law, before transferring to the University of Padua near Venice. He graduated in 1423, but instead of becoming a lawyer he took up minor holy orders. 

Early teens was a perfectly normal age to enter university in the fifteenth century, so not necessarily precocious. Strathern seems not to be aware that law was divided into civil law and canon law, that is church law, at medieval universities and Nicholas obtained a doctorate in canon law in Padua in 1423, so it was perfectly normal for him to take minor holy orders on graduating. 

From the outset, Nicholas was an imaginative polymath, his mind fecund with novel ideas on all manner of subjects. Under normal circumstances such ideas would have been controversial, and might even have put his life in mortal danger (almost 150 years later, the Italian philosopher Giordano Bruno would be burned at the stake for expressing similar ideas). However, it seems that the sheer brilliance of Nicholas of Cusa’s mind won him friends in high places. 

Here we have an oblique reference to Nicholas’ cosmological speculation, which did include, like Bruno, the idea that the stars were other suns and there might be other inhabited planets orbiting around them; speculations also shared by Nicole Oresme in the previous century. However, unlike Bruno, Nicholas did not travel around Europe pissing off everybody who was anybody, and also did not deny the divinity of Christ or the Virgin Birth, so his life was never in danger, because of his cosmological speculations. 

In 1450 Nicholas completed a work in the form of dialogues between a layman and a priest. This was entitled Idiota de Mente (literally translated as ‘An Idiot Speaks His Mind’). Surprisingly, it is the ‘idiot’ who puts forward Nicholas’s bold proposals, which contrast sharply with the orthodox Aristotelian views proposed by the priest. It should be borne in mind that during this period Aristotle was regarded as the highest authority on intellectual matters: his word was seen as little less than Holy Writ. 

In medieval Latin Idiota means layman so the title actually translates as A Layman Speaks his Mind. Strathern actually says this in a footnote, so I really don’t understand his next sentence. It is time for my favourite Edward Grant saying, medieval “Aristotelian philosophy is not Aristotle’s philosophy” and in fact it was constantly changing and evolving. Scholars were constantly discussing, criticising, and modifying Aristotle’s thoughts throughout the Middle Ages, so no, it was not Holy Writ.

Next up we have a rather thin presentation of Nicholas’ theological philosophy and his use therein of mathematics and measurement, which is not particular accurate, but I can’t be bothered to unravel it. However, Strathern makes the following claim: 

Despite the abstract flavour of Nicholas’s mathematical pronouncements, his motives were entirely practical. Delineating discrete parts of the world by measurement was what led to knowledge, which was essentially a practical matter. Such thoughts opened the way to an entirely different method of learning. 

He then states, “In order to understand the magnitude of Nicholas’s mode of thought it is necessary for the moment to take the wider view,” and proceeds to give a very thin and not very accurate account of the supposed decline of China. Re-enter Nicholas:

Ironically, it was the very opposite to the process which was taking place in Europe. And it was Nicholas of Cusa who was giving voice to this new direction. Mathematical measurement should be applied to the world. Architecture, commerce, shipbuilding, the very nature of tools and machines – all would undergo major developments during the Renaissance era as a result of this new attitude towards the practical world. 

I wrote a whole series of blog posts about those developments in practical knowledge in the Renaissance and the theological-philosophical ramblings of Nicholas of Cusa did not in any way play a significant role in them. 

As examples of Nicholas’ practical applications of mathematics and measurement he gives the following from his attempts to square the circle (of which more later):

However, in the course of his attempts by pure geometry to solve this problem he managed to calculate the value of π as 3.1423, a figure of greater accuracy than any before – including that calculated by Archimedes, who in fact only worked out its limits of between 223/71 and 22/7 (3.14084 and 3.14285). 

I love the “only” by Archimedes’ process of calculating Pi. It is one of the puzzles of the history of maths, as to why Archimedes stopped where he did and didn’t carry out the next iteration(s)of his calculation, which would, naturally, have given him value for Pi much more accurate than that of Nicholas. Some have suggested that there was a second, now missing, book where he completed his calculations.

In fact, Nicholas’ value is no more accurate than the value used by Ptolemaeus in the second century CE and less accurate than the value calculated by the Indian mathematician Āryabhaṭīya in the sixth century. Closer to Nicholas’s time in the fourteenth century the Indian mathematician Mādhava of Sangamagrāma calculated a value for Pi accurate to eleven decimal places and in 1425, the Persian mathematician Jamshīd al-Kāshī calculated Pi accurate to sixteen decimal places. 

Next up we have Nicholas as calendar reformer:

Nicholas also argued that there was a need to calculate a new calendar, as the seasons were gradually falling out of synchronization with the dates and the months (it would be almost 150 years before his suggestion was taken up by Pope Gregory XIII). 

The recognition of the need to reform the Julian calendar, to bring it back into line with the solar year, goes back at least to the Venerable Bede in the eighth century CE. Notable mathematicians, who made reform suggestions earlier that Nicholas, include Johannes de Sacrobosco (c. 1195–c. 1256), Roger Bacon (c. 12220–c. 1290) and Johannes de Muris (c. 1290–1344). When Gregory XII finally put that reform into practice, he was not taking up the suggestion of Nicholas of Cusa.

Strathern’s next claim completely blew my mind and sent me down a major rabbit hole:

Perhaps Nicholas’s most important invention was a new type of spectacle lens. Previously, lenses had been ground to a convex shape. This was an easier process, and it enabled the viewer to achieve long-sightedness. Nicholas tried the opposite method, grinding a lens into a concave shape, and found that it enabled the viewer to achieve near-sightedness. This brought about a revolution. Old men with failing sight could continue reading, learning, making suggestions, discoveries, inventions. It is little exaggeration to say that intellectual life almost doubled over the coming century as a result of Cusa’s innovation. 

Now, the history of optics, including the history of spectacles, has been a special area of interest of mine for at least thirty years and I have a rather large literature collection on the subject, as a result, but I have never ever come across the claim that Nicholas of Cusa invented the concave spectacle lens, indeed a major development in the history of optics. I was, as I said above, mind blown. I first of all googled Cusanus and spectacles and to my amazement came up with hundreds or even thousands of websites making exactly this claim that Nicholas of Cuse invented the concave spectacle lens in 1450/1. Mostly there was just one simple sentence with no explanation, no source, no history, nothing! Still not convinced I dug deeper and consulted Vincent Ilardi an expert on the history of spectacles and found the answer to this conundrum. 

More certain in this respect, on the other hand, is the often-cited quotation from Cardinal Nicholas of Cusa’s De beryllo (On the Beryl) as the first mention of concave lenses for the correction of myopia. In this treatise, written over a five-year period and completed in in 1485, Nicholas treated the beryl metaphorically but also as a practical magnifying device:

The beryl is a clear, bright, and transparent stone, to which is given a concave as well as a convex form, and by looking through it, one attains what was previously invisible. If the intellectual beryl, which possesses both the maximum and the minimum in the same way, is adapted to the intellectual eyes, the indivisible principle of all things is attained.

Shorn of its convolution, for which Nicholas had a special aptitude, this passage seems to indicate that the beryl used in its concave shape aided distant vision (“the maximum”) whereas the convex shaped one brought short distance images into focus (“the minimum”). And in another passage from his Compendium, completed in 1463, he again cited beryl as lenses to aid vision in a celebration of human creativeness and inventiveness to remedy the deficiencies of nature and master the environment at a level for superior than the capabilities of the animals. 


For man alone discovers how to supplement weakness of light with a burning candle, so that he can see, how to aid deficient vision with lenses [berylli], and how to correct errors concerning vision with the perspectival art.


The above quotations seem to indicate that Nicholas was familiar with spectacles fitted both with concave and convex lenses just a few years before we have unequivocal proof of the former’s availability in quantity.[1]

It is very clear that Nicholas is in no way claiming to have invented the concave spectacle lens, but is merely describing the fact that they exist. It would be an interesting exercise to try and discover who first misinterpreted this passage in this way. As an interesting side note, the use of beryl to make lenses, because of the poor quality of the available glass, led to the fact that spectacles are called Brillen in German. Of course, as Ilardi says, concave lenses aided distant vision and did not as Starter writes enable, old men with failing sight to continue reading, that task had already been covered by the convex spectacle lens. Personally, I think that a historian when confronted by this claim should weigh up the probability that a cardinal and high-ranking Church diplomat ground lenses in his spare time, possible but highly improbable.

A further revolution was instigated when Nicholas turned his attention to a study of the heavens. Despite the fact that the telescope had yet to be invented, his observations enabled him to reach some highly original conclusions. While several of the Ancient Greeks had speculated on such matters, drawing their own similar conclusions, Nicholas was perhaps the first to put these together into a truly universal structure. 

Nicholas’ thoughts on cosmology were based on speculation not observations and although interesting had almost no impact on the actual astronomy/cosmology debate in the Renaissance. He was also by no means “the first to put these together into a truly universal structure.”

However, none of this accounts for the sheer originality of his thinking. Besides the subjects already mentioned, Nicholas made original contributions in fields ranging from biology to medicine. By applying his belief in rigorous measurement to the field of medicine, he would introduce the practice of taking precise pulse rates to use as an indication of a patient’s health. Previously, physicians had been in the habit of taking a patient’s pulse and using their own estimation of its rate to infer the state of their health. Nicholas of Cusa introduced an exact method, weighing the quantity of water which had run from a water clock during one hundred pulse beats. 

As far as I can see, measuring the pulse using a water clock is the only original contributions in fields ranging from biology to medicine that he made. How original it was is debateable:

Pulse rate was first measured by ancient Greek physicians and scientists. The first person to measure the heartbeat was Herophilos of Alexandria, Egypt (c. 335–280 BC) who designed a water clock to time the pulse. (Wikipedia)

In the middle of a lot of stuff about Nicholas’ role as a Church diplomat we get:

Nicholas’s scientific work would go on to influence thinkers of the calibre of the German philosopher-mathematician Gottfried Leibniz, a leading philosopher of the Enlightenment who lived two centuries later. 

It is interesting to note that Nicholas of Cusa is regarded as one of the great Renaissance thinkers and although he was very widely read, his influence on others was actually minimal. Whether or not he influenced Leibniz is actually an open question.

For whatever reason, Strathern now turns to a completely different Renaissance thinker:

The work and thought of Nicholas of Cusa is indicative of the wide-ranging re-examination of the human condition which was beginning to take place, especially amongst thinkers of the northern Renaissance. Another leading German scientific thinker from this period, who would become a friend of Nicholas of Cusa, was Regiomontanus, who was born Johannes Müller in rural Bavaria, southern Germany, in 1436. 

Regiomontanus holding up an astrolabe , see below, Woodcut from the 1493 Nuremberg Chronicle vis Wikimedia Commons

We get a long spiel about scholars adopting Latin names during the Renaissance and the use of Latin in general during the medieval period, but nowhere does he mention that Johannes Müller never actually used the name Regiomontanus, which was first coined by Philip Melanchthon in 1535, that is almost seventy years after his death. 

“[Regiomontanus] would become a friend of Nicholas of Cusa”, really‽ I can find no references whatsoever to this ‘friendship’. There is no correspondence between the two of them, no record of their having ever met. Although, a meeting would have been possible as Regiomontanus lived and worked in Italy during the last three years of Nicholas’ life (1461–64), and even lived in Rome, where Nicholas was living, for some of this time. 

Regiomontanus’ view of Nicholas of Cusa can best be taken from his analysis of Nicholas’s attempts to square the circle.  Nicholas wrote four texts on the topic–De circuli quadratura , 1450, Quadratura circuli 1450, Dialogus de circuli quadratura  1457 and De caesarea circuli quadratura , 1457–all of which he sent to Georg von Peuerbach in Vienna. Regiomontanus wrote a series of notes analysing these texts during his time in Vienna and his conclusion was far from flattering, “Cusanus makes a laughable figure as a geometer; he has, through vanity, increased the claptrap in the world.” Regiomontanus’ very negative analysis of Nicholas of Cusa geometry was first published by Johannes Schöner as an appendix to Regiomontanus’ De triangulis omnimodis in 1533.

Nicholas of Cusa was a good friend of Regiomontanus’ teacher Georg von Peuerbach (1423–1461). Georg von Peuerbach travelled through Italy between graduating BA in 1448 and when he returned to Vienna to graduate MA in 1443.  In Italy he became acquainted with the astronomers Giovanni Bianchini (1410–after 1469), Paolo dal Pozzo Toscanelli (1397–1482), and Nicholas of Cusa. In fact, he lived with Nicholas in his apartment in Rome for a time. Later Georg von Peuerbach and Nicholas corresponded with each other. During his travels in Italy Regiomontanus met Toscanelli and Bianchini and also corresponded with both of them but for Nicholas we have no record of any personal contact whatsoever. As we have seen Regiomontanus heavily criticised Nicholas’ mathematics, but this only became public long after both of them were dead.

Strathern tells us:

Regiomontanus was sent to the University of Leipzig in 1437 [my emphasis], at the age of eleven. Five years later he was studying at the University of Vienna, where he took a master’s degree and began lecturing in optics and classical literature at the age of twenty-one. 

Note there is here no mention of Georg von Peuerbach, in fact, in the whole section about Regiomontanus Georg von Peuerbach gets no mention whatsoever. This is quite incredible! Writing about Regiomontanus without mentioning Georg von Peuerbach is like writing about Robin the Boy Wonder without mentioning Batman! Peuerbach was Regiomontanus’ principal and most influential teacher in Vienna and after Regiomontanus graduated MA, the worked closely together as a team, reforming, and modernising astronomy up till Peuerbach’s death in 1461. Their joint endeavours played a massive role in the history of European astronomy. 

But be warned gentle readers there is far worse to come. If we go and search for the good Georg von Peuerbach, reported missing here, we find the following horror in the chapter on Copernicus:

He had also read the work of the Austrian Georg von Peuerbach, who had lived during the earlier years of the century (1423–61). Peuerbach had been taught by Regiomontanus [my emphasis] and had collaborated with him, using instruments which he invented to measure the passage of the stars in the heavens. 

I don’t know whether to laugh or cry or simply to don rubber gloves, pick up the offending tome, and dump it in the garbage disposal.

You might also note that in 1437, Regiomontanus was one year old not eleven!

While Regiomontanus was teaching at the University of Vienna, the city was visited by the Greek scholar Bessarion, who would play a significant role in Regiomontanus’s subsequent career. As such, it is worth examining Bessarion’s unusual background. 

This is followed by a reasonable brief synopsis of Bessarion’s life prior to his visit to Vienna but no explanation of why he was there or what he did respective Peuerbach and Regiomontanus whilst he was there. This is important in order to understand future developments. Bessarion came to Vienna in 1460 as papal legate to negotiate with the Holy Roman Emperor Frederick III. He also sought out Georg von Peuerbach, who was acknowledged as one of the leading astronomer/mathematicians in Europe, for a special commission. Earlier Bessarion had commissioned another Greek scholar, Georg of Trebizond (1395–1472) to produce a new translation Ptolemy’s Mathēmatikē Syntaxis or as it is better known the Almagest from the original Greek into Latin, providing him with a Greek manuscriptGeorg of Trebizond made a mess of the translation and Bessarion asked Georg von Peuerbach to do a new translation. Georg von Peuerbach couldn’t read Greek, but he knew the Almagest inside out and offered instead to produce an improved, modernised Epitome of it instead. Bessarion accepted the offer and Georg von Peuerbach set to work. Bessarion then asked Georg von Peuerbach if he would become part of his familia (household) and accompany him back to Italy. Georg von Peuerbach agreed on the condition that Regiomontanus could accompany them; Bessarion accepted the condition. Unfortunately, Georg von Peuerbach, only having completed six of the thirteen books of the Almagest, died in 1461, so it was only Regiomontanus, who accompanied Bessarion back to Italy as a member of his familia. A more detailed version is here.

Back to Strathern:

Under Bessarion’s guidance, many works of Ancient Greece – of which western Europe was ignorant – were translated into Latin. And it was in this way that Regiomontanus learned sufficient Greek for him to be accepted as a member of Bessarion’s entourage while he travelled through Italy. 

Most of those works were actually already known in Europe, either through poor quality translations from the Greek or translation from Arabic. This was not how Regiomontanus learnt Greek. He was part of Bessarion’s familia and Bessarion taught him Greek during their travels. 

During these years, Regiomontanus would complete a new translation of the second-century Greek Almagest by Ptolemy. 

Regiomontanus didn’t complete a translation of Ptolemaeus’ Almagest, he completed Georg von Peuerbach’s Epitome of the Almagest (Epytoma in almagesti Ptolemei), fulfilling a death bed promise to Georg von Peuerbach to do so. To quote Michael H Shank

The Epitome is neither a translation (an oft repeated error) nor a commentary but a detailed sometimes updated, overview of the Almagest. Swerdlow once called it “the finest textbook of Ptolemaic astronomy ever written.

Epytoma in almagesti Ptolemei. frontispiece Source: Wikimedia Commons

Strathern continues:

This is the work in which Ptolemy describes the movements of the sun, the moon, the planets and the stars around the earth, which was deemed to be the centre of the universe. For many centuries, such geocentric teaching had been accepted by the early Christians as Holy Writ, and as such its authority lay beyond question. 

Strathern is perpetuating a popular myth. Geocentric cosmology and the Ptolemaic version of it were very often questioned and subjected to criticism throughout the medieval period, both by Islamic and European astronomers and philosophers, as I have documented in numerous blog posts. In fact, Copernicus’ heliocentric model appeared during an intense period of criticism of the accepted astronomy, which began around 1400. Strathern himself in this chapter details Nicholas of Cusa’s unorthodox cosmological speculations!

Strathern now delivers the standard speculation that Regiomontanus was moving towards a heliocentric view of the cosmos based on an over interpretation of a couple of quotes but then tells us:

Some suspect that Regiomontanus must surely have thought through the obvious implications of these remarks, i.e. that the earth moves around the sun. But there is no evidence for this. On the contrary, despite his suspicions as to the accuracy of Ptolemy’s universe, Regiomontanus seems to have continued to use geocentric astronomical mathematics, as well as accepting the authority of Aristotle’s pronouncement that ‘comets were dry exhalations of Earth that caught fire high in the atmosphere or similar exhalations of the planets and stars’. This reliance on ‘authority’ was certainly the case when he made observations of the comet which remained visible for two months during early 1472. He calculated this comet’s distance from the earth as 8,200 miles, and its coma (the diameter at its head) as 81 miles. According to the contemporary astronomer David A. J. Seargent: ‘These values, of course, fail by orders of magnitude, but he is to be commended for this attempt at determining the physical dimensions of the comet.’* 

In the footnote indicated by the *. Strathern writes:

* This comet is visible on earth at intervals ranging from seventy-four to seventy-nine years. Its first certain observation was recorded in a Chinese chronicle dating from 240 bc. When it was observed by the English astronomer Edmond Halley in 1705 it was named after him. The justification for this is that Halley was the first to realize that it was the same comet as had appeared at 74–79-year intervals since time immemorial. Even so, Regiomontanus deserves more than a little credit for his observation of the comet, for in the words of the twentieth-century American science writer Isaac Asimov: ‘This was the first time that comets were made the object of scientific study, instead of serving mainly to stir up superstitious terror.’ 

There is quite a lot to unpack in these two paragraphs, but we can start with the very simple fact that the Great Comet of 1472 was not Comet Halley! The most important point of Regiomontanus’ comet observations is that he tried to determine its distance from the Earth using parallax, this was an important development in the history of astronomy despite his highly inaccurate results. He wrote a book De Cometae, outlining how to determine the parallax of a moving object that was published in Nürnberg in 1531 and played an important role in the attempts to determine the nature of comets in the sixteenth century. 

Comet of 1472 Woodcut from the 1493 Nuremberg Chronicle via Wikimedia Commons

Regiomontanus was not the first to make comets “the object of scientific study” that honour goes to Paolo dal Pozzo Toscanelli, who began treating comets as celestial objects and trying to track their path through the heavens beginning with the comet of 1433, and continuing with the comets of 1449-50, Halley’s comet of 1456, the comet of May, 1457, of June-July-August, 1457, and that of 1472.  He did not publish his observations, but he almost certainly showed them to Georg von Peuerbach when they met. Georg von Peuerbach went back to Vienna in thee 1440s he applied Toscanelli’s methods of comet observation to Comet Halley in 1456 together with his then twenty-year-old student Regiomontanus, as did Toscanelli in Italy. 

Paolo dal Pozzo Toscanelli Source: Wikimedia Commons

Following on to the comet disaster Strathern writes: 

However, Regiomontanus would make two contributions of lasting importance. In his work on rules and methods applicable to arithmetic and algebra, Algorithmus Demonstratus, he reintroduced the symbolic algebraic notation used by the third-century Greek mathematician Diophantus of Alexandria. He also added certain improvements of his own. Basically, this is the algebra we use today, where unknown quantities are manipulated in symbolic form, such as ax by c. Here and are variable unknowns, and ab, and are constants. 

My first reaction was basically, “Yer wot!” I am, for my sins, supposed to be something of a Regiomontanus expert and I have never heard of a book titled Algorithmus Demonstratus and I know for a fact that Regiomontanus did not introduce or reintroduce symbolic algebra, so it was rabbit hole time again.

During his travels in Italy and Hungary, Regiomontanus collected a large number of mathematical, astrological, and astronomical manuscripts, a number of which he intended to print and publish when he settled down in Nürnberg; of which more later. Unfortunately, he died before he could print more than a handful and it turns out that the Algorithmus Demonstratus was one of those manuscripts, which was then edited by Johannes Schöner and published by Johannes Petreius in Nürnberg in 1535. Although it has been falsely attributed to both Regiomontanus, and to the thirteenth century mathematician Jordanus de Nemore, it is not actually known who the author was. Although it has some very primitive attempts to introduce letters for numbers It is in no way an (re)introduction of symbolic algebra as you can judge for yourself here.

As Frank J. Swetz, an expert for Early Modern arithmetic, writes on the MAA website:

On page 10 of the Algorithmus, we find crude attempts to employ symbolic notation. For example, the third paragraph down notes that digit a multiplied by digit b will result in articulum c. An example is given in the margin: 5 x 4 = 20; also articulum times articulum b gives [the product n, 50 x 40 = 2000].

Source: MAA see link above

It is obvious that Strathern literally doesn’t know what he’s talking about and has never even bothered to take a look at the book he is describing.

The garbage continues:

Regiomontanus also made considerable advances in trigonometry, although it has since been discovered that at least part of this was plagiarized from the twelfth-century Arab writer Jabir ibn Aflah. On top of this, Regiomontanus drew up books of trigonometric tables: these lists provided ready answers in the calculation of angles and lengths of sides of right-angle triangles. 

Strathern is here referencing Regiomontanus’ De triangulis omnimodis (On Triangles of All Kinds) edited by Johannes Schöner and printed and published posthumously by Johannes Petreius in Nürnberg in 1533. This is the book he should have featured and not the spurious  Algorithmus Demonstratus. The accusation that he had plagiarised Jābir ibn Aflah was already made in the sixteenth century by the Italian polymath Gerolamo Cardano (1501–1576), whose books were also printed and published by Petreius in Nürnberg. For its role in the history of trigonometry I quote Glen van Brummelen (In his own words, he is the “best trigonometry historian, and the worst trigonometry historian” (as he is the only one)):

[…] what separates the De triangulis from its predecessors is–as the title say–its universal coverage of all cases of triangles, plane or spherical, and its demonstrations from first principles of the most important theorems. It is remarkable in the way that Euclid’s Elements is: not because its results were new, but its structure codified the subject for the future. Although not published until 1533, the De triangulis was to be the foundation of trigonometrical work for centuries, and was a source of inspiration for Copernicus, Rheticus, and Brahe, among many others.[2]

Van Brummelen follows this with a section on possible sources, which Regiomontanus might have used:

There are several possible Arabic sources that Regiomontanus might have used for the De triangulis.


Rather, as the absence of the tangent function in the De triangulis suggests, Regiomontanus’s debt seems to lie mostly in the tradition of the Toledan Tables and Jābir ibn Aflah, whose writings were still being published after Regiomontanus’s death. Several Arabic antecedents have been suggested for particular theorems in De triangulis, but the smoking gun of transmission awaits discovery.[3]

Title page of De triangulis omnimodis  Source MAA

De triangulis does not include the tangent function because Regiomontanus had already dealt with that in his earlier Tabula directionem, which was written in 1467 but first published by Erhard Ratdolt in Augsburg in 1490. This book was a Renaissance bestseller and went through eleven edition the last appearing at the beginning of the seventeenth century. 

This is followed by another piece of misinformation from Strathern:

And it is in these tables that Regiomontanus popularized yet another notational advance. Instead of fractions, which could become increasingly complex, he started using decimal point notation, which was much easier to manipulate. A simple example: the sum of 1/8 + 1/5 is much easier to calculate when these numbers are written as 0.125 + 0.2. The answer in fractional form is 13/40, but in decimal form it is simply 0.325. Furthermore, the decimal answer is much more amenable to further addition, multiplication and so forth with other numbers in decimal form. 

Regiomontanus did not use decimal point notation, to quote Wikipedia, which paraphrases E. J Dijksterhuis, Simon StevinScience in the Netherlands around 1600, 1970 (Dutch original, 1943):

Simon Stevin in his book describing decimal representation of fractions (De Thiende), cites the trigonometric tables of Regiomontanus as suggestive of positional notation.  

Decimal positional notation had existed in Arabic mathematics since the tenth century and there is a complex history of its use over the centuries. Stevin is credited with having introduced it in European mathematics in1585, although, as stated, he credits Regiomontanus as a predecessor, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.[4]   However, Stevin did not use a decimal point, this innovation is often falsely attributed to John Napier in his Mirifici logarithmorum canonis constructio written before 1614, but first published posthumously in 1620. However, Christoph Clavius had already used the decimal point in the goniometric tables of his astrolabium text in 1593.  

After leaving Rome, Regiomontanus travelled around Europe, continuing to compile his tables and frequently constructing ingenious objects for his hosts. In Hungary, for King Matthias I, he created a handheld astrolabe. Such devices were first made by the Ancient Greeks in around 200 BC. They contain many moving parts, which mirror the movements of the planets and the stars. Astrolabes can be put to a variety of uses, including astronomy, navigation, the calculation of tides, and the determining of horoscopes for astrologers. 

Strathern seems to be under the impression that Regiomontanus spent his four years as a member of Bessarion’s familia living in Rome, whereas in fact he spent most of his time travelling around Italy visiting libraries and archives to search out manuscripts which he copied both for himself and Bessarion. He left Italy in 1465 and for the next two years we don’t know where he was. In 1467 he was on the court of János Vitéz the Archbishop of Esztergom in Hungary, about 45 kilometres northwest of Budapest, working as his librarian. It was Vitéz, who commissioned him to write his Tabula directionem. In 1468 he moved to the court of the King, Matthias Corvinus (1443–1490), again as librarian, where he stayed until 1471, when he moved to Nürnberg. 

Strathern’s few sentences on the astrolabe are amongst to worst that I have ever read on the instrument. I shall forgive him the, “Such devices were first made by the Ancient Greeks in around 200 BC,” as variation on this myth can be found everywhere, including on Wikipedia, usually crediting the invention of the astrolabe to either Hipparchus or Apollonius. I shall take the opportunity to correct this myth.

We don’t actually know where or when the astrolabe first put in an appearance. The earliest mention of the stereographic projection of the celestial sphere that is at the heart of an astrolabe was the Planisphaerium of Ptolemy written in the second century CE. This text only survived as an Arabic translation. The earliest known description of the astrolabe and how to use it was attributed to Theon of Alexandria (c. 335–c. 405 CE), it hasn’t survived but is mentioned in the Suda, a tenth century Byzantine encyclopaedia of the ancient Mediterranean world, as well as Arabic sources. The extant treatises on the astrolabe of John Philoponus (c. 490–c. 570) and of Severus Sebokht (575–667) both draw on Theon’s work. The development of the instrument is attributed to Islamic astronomers; the oldest surviving astrolabe is a tenth century Arabic instrument.

An astrolabe usually only has two moving parts, the rete, a cut out star map with the ecliptic and, in the northern hemisphere, the tropic of cancer, that rotates on the front side over the stereographic projection of the celestial sphere. On the back of the astrolabe is an alidade, a sighting device. Some astrolabes also have a rotating rule on the front to make taking readings easier.

Arabic Astrolabe 1208 Above: Cut out Rete with eccentric ecliptic and stars as points of thorns. Below right: front of astrolabe with rete Left: rear of astrolabe with alidade Source: Wikimedia commons

Regiomontanus wrote a text on the construction and use of the astrolabe, whilst he was in Vienna. He is thought to have constructed several instruments of which the most famous is one he made for and dedicated to Bessarion in 1462. The instrument he made for Corvinus has not survived.

Regiomontanus astrolabe from 1462 dedicated to Bessarion

We move on:

In Nuremberg, Regiomontanus established a novel type of printing press, the first of its kind devoted entirely to the printing of scientific and mathematical works. 

I’m not sure how to interpret this sentence. Does Strathern mean that Regiomontanus’ printing press, meaning printing house, was novel, because “it was the first of its kind devoted entirely to the printing of scientific and mathematical works,” which is true. Or does he mean that Regiomontanus had created a novel mechanical printing press, which is not true.

He also oversaw the building of the earliest astronomical observatory, in Germany. 

This is simply not true, there was no observatory. Regiomontanus and his partner Bernhard Walter made their astronomical observations with portable instruments out in the street. 

Finally returning to Rome, he constructed a portable sundial for Pope Paul II. Later he also seems to have re-established contact with his friend and mentor, Cardinal Bessarion, who was in Rome in 1471 for the conclave to elect a new pope after the death of Paul II. 

Regiomontanus remained in Nürnberg from 1471 to 1475, when he was called to Rome to assist in a calendar reform. He died there in 1476 probably in an epidemic.

To call this capital of Strathern’s shoddy would be akin to praising it. It creates the impression that he gathered together a pile of out-of-date references and debunked myths, threw them up in the air and then sent the ones that landed on his desk to the publishers. 

[1] Vincent Ilardi, Renaissance Vision from Spectacles to Telescopes, American Philosophical Society, 2007, pp. 80-81

[2] Glen van Brummelen, The Mathematics of the Heavens and the EarthThe Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009 pp. 260-261

[3] Van Brummelen p. 261

[4] Dijksterhuis, Stevin. 


Filed under Book Reviews, Renaissance Science, Uncategorized

Renaissance garbage ­– I

In case you hadn’t noticed there is a four-weekly cycle of blog posts here at the Renaissance Mathematicus. Week one is a new series post, week two a fairly random #histSTM post, week three the next new series post, and week four a book review. Sometimes I throw in a random HISTSCI_HULK post just because. Today should be the next series post but I posted the last episode of my Renaissance Science series two weeks ago. You might be pleased or dismayed to learn that I have a new series lined up, at least in my head, but I want to take a little breathing space before I start in on it. To bridge the gap, I shall be posting a series of posts, which are sort of related to the Renaissance Science series, on a book by Paul Strathern, The Other RenaissanceFrom Copernicus to Shakespeare, (Atlantic Books, 2023). The Other Renaissance is, of course, what is normally referred to as the Northern Renaissance.

I must admit that I had never heard of Paul Strathern, but a friend, who shall remain nameless, received a review copy of this book, and thought it was so bad that he decided not to review it at all. My historical garbage antenna went up and I made further inquiries, whereupon said friend was kind enough to send me a pdf of said book. And yes folks, it is truly a stinker. I don’t intend to review the whole thing, but I thought I would post a series of HISTSCI_HULK typical put downs of selected elements of the text.

However, before we get down to the nitty-gritty a few words about our author. According to his Wikipedia page, Paul Strathern (born 1940!!) is a Scots Irish writer and academic, and an incredibly prolific writer he is too. To date, he has published five novels, fifteen “academic” books, nineteen titles in his Philosophers in 90 Minutesseries, eleven titles in his Great Writers in 90 Minutes series, twelve titles in his The Big Idea: Scientists Who Changed the World series, and finally three travel guides. I wonder what he does in his spare time? Also, if his The Big Idea: Scientists Who Changed the World are as bad as his The Other RenaissanceFrom Copernicus to Shakespeare, then I’m glad that the HISTSCI_HULK hasn’t read any of them.

As I will be only commenting on selected bits of the book here is the full contents list:

Map viii–ix Timeline of Significant Events during the Northern Renaissance x-xi 

Prologue: Lifting the Lid 1 

1 Gutenberg 13 

2 Jan van Eyck 21 

3 Nicholas of Cusa 33 

4 Francis I and the French Renaissance 47 

5 A New Literature: Rabelais 63 

6 Martin Luther and the Protestant Reformation 73 

7 The Rise of England 89 

8 The Rise and Rise of the Fuggers 105 

9 Copernicus 125 

10 Erasmus 141 

11 Dürer 157

12 Straddling Two Ages: Paracelsus and Bruegel the Elder 175 

13 Versions of the True: Mercator and Viète 193 

14 Vesalius 211 

15 Catherine de’ Medici 229 

16 Montaigne 249 

17 Elizabethan England 265 

18 Brahe and Kepler 281 

19 Europe Expands 301 

Conclusion: A Last Legacy 317 

And the timeline: (The formatting is a bit weird, result of my cut and paste)


1415 Jan Hus, founder of the Hussites, burned at the stake
1432 Van Eyck completes the Ghent Altarpiece
1460s Regiomontanus oversees the building of the first observatory in Europe (Rubbish!)
1464 Death of Nicholas of Cusa
1474 Technique of painting in oils spreads from the Netherlands to Italy
1494 Signing of the Treaty of Tordesillas: Pope Alexander VI draws a line down the Atlantic Ocean, dividing the globe between Spain and Portugal
1497 John Cabot sails from Bristol and reaches North America 

1509 Erasmus writes In Praise of Folly
1512 Torrigiano is commissioned by Henry VIII to create a Renaissance tomb for Henry VII

1514  Dürer produces Melencolia I 

1515  Francis I ascends to the throne of France

1517 Martin Luther nails his Ninety-Five Theses to the door of Wittenberg Castle Church
1519 The death of Leonardo da Vinci in France
1519 Charles V is crowned Holy Roman Emperor
1525 Death of Jacob Fugger the Rich
1529 The Colloquy of Marburg fails to unite Protestants 

1533 Henry VIII breaks with Rome 

1536 John Calvin arrives in Geneva; Calvinist missionaries soon begin to spread over northern Europe 

1536 French explorer Jacques Cartier brings Chief Donnacona from the New World to see Francis I 

1541 Death of Paracelsus
1543 Copernicus is shown the first published edition of his De Revolutionibus Orbium Coelestium, describing the solar system, while on his deathbed
1543 Vesalius publishes De Fabrica, describing human anatomy
1547 The completion of the Château de Chambord in the Loire Valley 

1547 The death of Francis I of France
1553 The death of Rabelais
1553 English explorer Richard Chancellor visits the court of Ivan the Terrible
1558 The death of Holy Roman Emperor Charles V
1569 Mercator publishes his Atlas, containing his cylindrical projection map of the world
1572 The St Bartholomew’s Day massacre of Huguenots in France 

1588  The Spanish Armada fails to invade Elizabethan England 

1589  The death of Catherine de’ Medici, mother of French kings and ruler of France 

1596 Johannes Kepler publishes his laws describing the elliptical orbit of plants (Love this typo!)

1600  The founding of the East India Company in London 

1601  Tycho Brahe dies in Prague 

1608 Invention of the perspicillium in Holland, which inspires Galileo to create the telescope (Rubbish!)
1616 The death of Shakespeare
1642 Cardinal Richelieu dies in France
1648 The Peace of Westphalia ends the Thirty Years’ War 

I decided to start this week with Strathern’s chapter on Albrecht Dürer, as I have a local connection to the man. 

After a conventional biographical introduction from birth to marriage, Strathern tells us:

Dürer did not travel on his own. He is thought to have been accompanied by an ebullient rugged-faced companion named Willibald Pirckheimer, [my emphasis] whose appearance belied his acute intelligence and thirst for learning. Pirckheimer came from a distinguished family in Nuremberg, was a year older than Dürer, and was filled with patrician self-confidence. He was studying law at Padua, and during the course of their friendship Pirckheimer would fill the huge gaps in Dürer’s education, introducing him to the humanist ideas he had picked up at university and amongst his father’s intellectual circle in Nuremberg. 

When I read that Dürer, on his first journey to Italy, was accompanied by Willibald Pirckheimer, I did a double take. Having read quite a lot about Dürer and Pirckheimer and I’ve never come across any such claim. So back to the literature. My first stop was German Wikipedia, where to my surprise I read the following:

In der Folgezeit bis 1500 schuf er eine Serie von kleinen Landschaftsaquarellen mit Nürnberger Motiven bzw. mit Motiven von Stationen seiner ersten Italienreise, die er in der ersten Hälfte des Oktobers 1494, bereits drei Monate nach seiner Hochzeit, antrat. Diese Reise verstärkte sein Interesse an der Kunst des Quattrocento. Im Mai 1495 kehrte er zurück nach Nürnberg.

Von der jüngeren Forschung wird angezweifelt, dass Dürer im Rahmen dieser Reise jemals die Grenzen des deutschen Sprachgebiets überschritt, und die Indizien, die gegen einen Aufenthalt in Venedig sprechen, häufen sich: Dürer selbst erwähnte in seiner Familienchronik 1494/95 keine Reise nach Venedig. Die italienischen Züge in seinen Werken ab 1497 interpretieren manche als direkten Einfluss des paduanischen Malers Andrea Mantegna, der 1494/95 zwar nicht in Padua war, dessen Werke Dürer aber dort gesehen haben könnte. Beweisbar ist nur, dass Dürer in Innsbruck, Trient und Arco beim Gardasee war. Von Orten südlich von Arco gibt es bei Dürers Aquarellen keine Spur, also auch nicht von Venedig. Auch die Route spricht gegen die Venedig-Theorie: Für Dürer hätte es näher gelegen, den für Nürnberger (Kaufleute) üblichen Weg nach Venedig zu nehmen, der über Cortina und Treviso verlief und „Via Norimbergi“ genannt wurde. Die Bilder aus seiner späteren, nachweisbar venezianischen Zeit ab 1505 haben deutlich stärker venezianische Charakteristika.

For those who don’t read German, it basically says that recent research doubts that Dürer ever left the German language area in 1494 and thus was never in Italy on this journey. This was new to me as I have always read about and accepted that Dürer made two journeys to Italy, the first in 1494. Happily, in 2021, the National Gallery in London put on a major expedition Dürer’s JourneysTravels of a Renaissance Artist for which there is an amazing book[1], which thanks to my very generous stepmother I own a copy. Turning to this wonderful tome I discovered that it is really so that historians now believe that Dürer did not reach Italy in 1494. Apparently, the whole story of the first Italian journey is based on two very short ambiguous quotes and the rest has been built up over the years based on reading the tealeaves in Dürer’s work. 

I actually began to question these two paragraphs of Strathern because of his claim that Pirckheimer had accompanied Dürer on this journey. No journey so no Pirckheimer but there is more. Strathern correctly states that Pirckheimer was studying in law in Padua, which he did for seven years from 1488, first returning to Nürnberg on 1495. This was when Willibald and Albrecht first met!

Later Strathern turns to the “second” Italian journey, the one that really did take place and dishing up a myth that has been long debunked, he tells us:

Between 1507 and 1509 Dürer paid a second visit to Italy, passing beyond Venice to Padua and maybe even Mantua. He certainly visited Bologna, for it was here that he met Luca Pacioli, the friar mathematician and friend of Leonardo da Vinci. It was Pacioli who had taught Leonardo mathematics, and it seems that Dürer too studied with him. Dürer’s meticulous and exact art inclined him to mathematics, and it would play an increasing role in both his painting and his other intellectual interests. Pacioli is known to have taught Dürer linear perspective, which was by now widely developed amongst Italian Renaissance artists. But Pacioli probably taught Dürer much more than this useful mathematical–artistic device, for Dürer would continue to study mathematics over the coming years. 

It is simply not known from whom Dürer learnt the basics of, the then still comparatively new, linear perspective. The question has a certain historical importance, as he is credited with having introduced linear perspective into Northern European art. I have no idea who first introduced the theory that he learnt it from Luca Pacioli during his time in Bologna but that is absolutely no evidence to support it. The theory was finally totally debunked, when somebody pointed out that when Dürer was in Bologna, Pacioli was in Milano! Maybe Pacioli taught him telepathically? As for the implication that Pacioli also taught Dürer mathematics, we know from fairly solid evidence that Dürer didn’t need to go to Italy for his maths lessons, he got them at home in Nürnberg from Johannes Werner. (1468–1522)

Source: Wikimedia Commons

Next up we get a strange twist in the Dürer timeline from Strathern:

By the time Dürer returned home from his second visit to Italy, he was known by reputation throughout Europe. In 1512, the Holy Roman Emperor Maximilian I became one of his patrons. Despite this, Dürer found that he was making insufficient income from his paintings, and even abandoned this art form for several years in favour of making woodcuts and engravings – which could be reproduced and thus sold many times over. He may not have been the best painter in Europe, but his engravings were unsurpassed. 

Dürer served his apprenticeship in the studio of Michael Wolgemut between 1486 and 1490. Wolgemut specialised in producing woodblock prints as book illustrations. For example, his studio produced the illustrations for the famous  Liber Chronicarum, better known as the Nuremberg Chronicle in English and Die Schedel’sche Weltchronik in German. There are even speculations by art historians as to whether the young apprentice was responsible for some of those illustrations. From the very beginning when he set up his own workshop in 1495, Dürer specialised in woodblock printing. He also developed his skill in engraving, almost certainly learnt in his original apprenticeship under his father, a goldsmith. To illustrate a couple of quotes from Wikipedia:

Arguably his best works in the first years of the workshop were his woodcut prints, mostly religious, but including secular scenes such as The Men’s Bath House (ca. 1496). These were larger and more finely cut than the great majority of German woodcuts hitherto, and far more complex and balanced in composition.

His series of sixteen designs for the Apocalypse] is dated 1498, as is his engraving of St Michael Fighting the Dragon. He made the first seven scenes of the Great Passion in the same year, and a little later, a series of eleven on the Holy Family and saints. The Seven Sorrows Polyptych, commissioned by Frederick III of Saxony in 1496, was executed by Dürer and his assistants c. 1500. In 1502, Dürer’s father died. Around 1503–1505 Dürer produced the first 17 of a set illustrating the Life of the Virgin which he did not finish for some years. Neither these nor the Great Passion were published as sets until several years later, but prints were sold individually in considerable numbers.

In 1496 he executed the Prodigal Son, which the Italian Renaissance art historian Giorgio Vasari singled out for praise some decades later, noting its Germanic quality. He was soon producing some spectacular and original images, notably Nemesis (1502), The Sea Monster (1498), and Saint Eustace (c. 1501), with a highly detailed landscape background and animals.

Prints are highly portable and these works made Dürer famous throughout the main artistic centres of Europe within a very few years.

As you can see this is not post 1512 but we have just reached 1505 and Dürer is a highly prolific and famous producer of fine art prints. In fact, rather than being a painter, who turned to fine art printing for financial reasons, as Strathern would have us believe, Dürer was a highly successful fine art printer, who painted on the side.

We get the standard discussions of The Rhinoceros, Dürer’s most famous print, and Melencolia I, his most enigmatic and most interpreted print. I have only one question about Strathern’s waffle here, he writes about Melencolia I:

Though mathematics, especially geometry (Plato’s favourite), underlies much of the scene. 

There is no other reference to Plato anywhere in his convoluted discussion of Melencolia I, so why shove him in here? Earlier he makes the equally strange comment:

Set into the wall above the angel’s head is a four-by-four magic square, indisputable evidence of Dürer’s continuing mathematical interest. 

There was nothing to say that Dürer had ever stopped being interested in mathematics.

Having dealt with The Rhinoceros and before Melencolia I, Strathern enlightens us with the following paragraph:

As we have seen, the year 1500 marked one and a half millennia since the birth of Christ, and there was a widespread belief around this time that it heralded the Second Coming of Christ, which is mentioned in the Bible: ‘This same Jesus, which is taken up from you into heaven, shall so come in like manner as ye have seen him go into heaven.’ Such an event would precede the Last Judgement, after which our souls would be despatched to Purgatory, Hell or Heaven.

Various dates were considered by various people to signify the second coming but I personally have never come across a reference to 1500 as one of them.

Strathern also turns his spotlight on the Ehrenpforte Maximilians I, known in English as The Triumphal Arch or the Arch of Maximillian I, he writes:

Dürer created a number of works for his most important patron, Maximilian I. Amongst these is a large, highly complex woodcut of a triumphal arch, which measures almost ten feet by ten feet. Dürer spent over two years – on and off – busying himself with this work, which includes 195 separate woodcuts printed on 36 sheets of paper. The intention was that it should be hung in princely palaces and city halls throughout the Holy Roman Empire. Indeed, Maximilian I made a habit of giving away copies of this work with this intention. 

The work itself is a suitably grandiose hotchpotch of styles – resembling, if anything, an example of Indian architecture rather than any classical triumphal arch (such as Marble Arch in London, or the Washington Square Arch in New York). It stands more as a monument to Dürer’s indefatigable technical expertise than any aesthetic achievement. Such a work made him rich, allowing him independence – even if it contributed nothing to his artistic attainment and was otherwise a complete waste of his time. [my emphasis]

Ehrenpforte Maximilians I, Source: Wikimedia Commons

As is, unfortunately, all to common Strathern attributes this work to Dürer alone but it was the work of a group of people, quoting, yet again, Wikipedia:

The design program and explanations were devised by Johannes Stabius, the architectural design by the master builder and court-painter Jörg Kölderer and the woodcutting itself by Hieronymous Andreae, with Dürer as designer-in-chief. […] the flanking round towers are attributed to Albrecht Altdorfer.

The closing clause from Strathern, that I have emphasised, displays, in my opinion, his ignorance of the professional life of an artist and in particular that of his subject Albrecht Dürer. Dürer ran a highly professional, commercial fine art print studio, with which he not only earned the money on which he and his family lived but also the money with which he paid his employees. The Ehrenpforte was anything but a complete waste of time, as the commission raised the status of his studio and did in fact contribute to his artistic attainment, as it displayed his mastership in woodblock printing to the world. 

Here we have the name of the mathematician, Johannes Stabius 1450–1522), who was the Imperial Court historian, was the director of the project, as he would employ Dürer on two further commissions, neither of which Strathern considers worth mentioning, despite his continued references to Dürer’s interest in mathematics.

Johannes Stabius portrait by Albrecht Dürer Source: Wikimedia Commons

The first was the Stabius-Dürer World Map

Stabius-Dürer World Map published in 1515, a perspective representation of the earth as a globe. Source: Wikimedia Commons

and the second, and historically much more important the Stabius-Dürer-Heinfogel planispheres of the southern and northern hemispheres, the first European, printed celestial maps, which I wrote about here.

The Dürer Star Maps in a hand coloured edition Source: Ian Ridpath Star Tales

 We get accounts of Dürer’s journey to the Netherlands to get his Imperial pension renewed following the death of Maximillian, including an account of his portrait of Erasmus and Strathern’s rather bizarre interpretation of its Greek inscription. 

Almost at the end of his chapter, Strathern turns to Dürer the mathematician:

With Dürer’s eyesight fading, he devoted less of his energies to his art. Instead he concentrated on writing treatises on such subjects as ‘human proportions’ and ‘fortifications’. However, his most important work was his Four Books on Measurement. These contain the wealth of mathematical knowledge he accumulated during his life – including the geometrical construction of shadows in prints (projective geometry), as well as several ideas by the Tuscan artist Piero della Francesca which had not yet been published. (These Dürer had almost certainly learned from Luca Pacioli.) Very little of this vast compendium of work is original, but it was written in the vernacular German rather than in Latin. This established Dürer as the first figure of the northern Renaissance to outline in German Euclidean geometry and demonstrate the construction of the five Platonic solids, other Archimedean semi-regular truncated solids, and a number of constructed figures which are thought to have been of his own invention. His treatises were the first printed north of the Alps to view art in a scientific fashion, exposing the mathematical bones upon which much artistic flesh is based. 

I’m sure Dürer would be delighted to know that his Four Books on Measurement was his most important work, whereas his ‘human proportions’ was just a treatise on “such subjects”! 

About the time of his “second” journey to Italy, Dürer became obsessed with the idea that the secret of beauty lies in the mathematical theory of proportions. He began working on his Vier Bücher von menschlicher Proportion (Four Books on Human Proportion) in 1512 and the four books, written at different time over the years, deal with various aspects of exactly that, human proportion. An appendix to the book explains Dürer’s theories on ideal beauty.

Title page of Vier Bücher von menschlicher Proportion Source: Wikimedia Commons

The book was written for apprentice artists and in the middle of the 1520s, Dürer realised that the geometry of the book was too advanced for the intended readers, so he sat down and wrote his Four Books on Measurement (Underweysung der Messung mit dem Zirckel und Richtscheyt or Instructions for Measuring with Compass and Ruler), (which I wrote about here) an introductory textbook on geometry for apprentice artists. It is the book on human proportions that is his most important work, the Four Books on Measurement merely developed the mathematical tools needed to understand it. 

Underweysung der Messung mit dem Zirkel und Richtscheyt Title Page

I have no idea which not yet published ideas from Piero della Francesca Dürer’s book supposedly contains. It goes without saying that Dürer didn’t learn anything from Luca Pacioli, whom he never met and with whom, as far as we know, he didn’t correspond. However, he might have accessed della Francesca work via Pacioli, who had plagiarised it in his Divina proportione, published in 1509, which was possibly owned by one of the Nürnberg mathematicians, Werner or Stabius, but that’s just speculation. 

Far from being a vast compendium of work, Underweysung der Messung is 27cm X 18cm and probably less than 200 pages long, it’s not paginated so I had to guess. Dürer’s Underweysung der Messung is actually the very first mathematics book printed in German and like most textbooks it is of course derivative. However, it does contain one important geometrical innovation. Dürer introduced the geometry net, which is the two-dimensional figure that arises when you open a three-dimensional figure along edges.

Underweysung der Messung geometrical net

In one sense Underweysung der Messung did become more important than Vier Bücher von menschlicher Proportion both were into Latin and Underweysung der Messung was translated into several different European languages. Vier Bücher von menschlicher Proportion appealed to a very limited readership but Underweysung der Messung became a very widely read geometry textbook throughout Europe for most of the next hundred years. 

Strathern has obviously not bothered to do serious research for his book but has just thrown it together from the first sources that crossed his path without bothering to check whether they were factually correct or not. As we will see in later chapters this sloppy approach is not confined to Dürer but is characteristic of the whole book. 

[1] Susan Foister and Peter van den Brink eds., Dürer’s JourneysTravels of a Renaissance Artist, National Gallery, London, distributed by Yale University Press, 2021. 


Filed under Book Reviews, Renaissance Science, Uncategorized

Renaissance Science – LII

Another Renaissance Mathematicus series comes to an end a little more than two years after it began with the questions Renaissance Science? Which Renaissance? and eight weeks later, is there any such thing as Renaissance science and if so, what is it? Having established that we were talking about the Humanist Renaissance, which began in Northern Italy in the fourteenth century, as a literary movement, and expanded into other areas in the fifteenth and sixteenth centuries. I took the middle of the seventeenth century as its culmination. However, already in that early episode I ended thus:

A important closing comment is that there is actually a very high level of continuity rather than disruption from the High Middle Ages through the Renaissance and one can regard the Renaissance both as a phase of the Middle Ages but also of the Early Modern Period; all historical periodisations are of course artificial and also to some extent arbitrary.

Raphael’s School of Athens an idealisation of the Humanist Renaissance by one of it’s greatest artists

The School of Athens is a fresco by Raphael. The fresco was painted between 1509 and 1511 as a part of Raphael’s commission to decorate the rooms now known as the Stanze di Raffaello , in the Apostolic Palace in the Vatican. It depicts a congregation of philosophers, mathematicians, and scientists from Ancient Greece, including Plato, Aristotle, Pythagoras, Archimedes, and Heraclitus. The Italian artists Leonardo da Vinci and Michelangelo are also featured in the painting, shown as Plato and Heraclitus respectively.

The painting notably features accurate perspective projection, a defining characteristic of the Renaissance era. Raphael learned perspective from Leonardo, whose role as Plato is central in the painting. The themes of the painting, such as the rebirth of Ancient Greek philosophy and culture in Europe (along with Raphael’s work) were inspired by Leonardo’s individual pursuits in theatre, engineering, optics, geometry, physiology, anatomy, history, architecture and art.

Description and picture both taken from Wikipedia

I then posed the all-important question, is there such a thing as Renaissance science, and if so, what is it? If I wished to write a series of episodes about it, then I should first establish what it is I’m writing about. To give a brief summary of that episode I stated that in my defined period of Renaissance science, from c. 1400 to c. 1650, a crossover took place between academic book knowledge and the empirical and practical knowledge of the artisan, areas that had previously been separated from each other. This crossover was driven by external forces drawn from political, social, cultural, and economic developments. Added to this was the literary Humanist drive to recover the knowledge of classical antiquity, which didn’t restrict itself to works of literature but revived interest in many half-forgotten scientific texts. These two developments blended together to produce a new wave of empirical, practice orientated knowledge, which when theorised in a further evolution, which began in the sixteenth century following into the seventeenth led to the so-called scientific revolution. 

Having delineated an area that, I was happy to label Renaissance science, I then over the next forty-five episodes tried to expand on this theoretical definition by displaying how these changes developed in various areas of knowledge production. In the last two episodes I gave first a brief outline of the philosophical revivals that contributed to the downfall of the all-embracing Aristotelian philosophy of the medieval scholastics and then explained why I believe that the much-quoted Francis Bacon actually presented a summation of Renaissance science rather than pointing to the future and modern science as he is all too oft presented

Having defined a general development in the sciences that took place in the historical period I defined as the scientific Renaissance i.e., a period defined by the scientific development that took place during it, we now have to apply the same caution that we applied above to the time period itself. Expressed very simply, there is no point in time were people stopped doing medieval science and started doing Renaissance science. Equally there is no point in time were people stopped doing Renaissance science and started doing modern science. These developments were both gradual and included much takeover of thoughts and practices from one era into the other. A key concept here is continuity, although we must be careful not to evoke a Whig historical concept of progress. I will now look back at a couple of the topics we have discussed over the last two years and point out the threads of continuity that they contain.

I will start with the botanical gardens, in themselves part of the much wider complex of natural history, materia medica, medical education, and botany. As I pointed out the concept of the specialist, medicinal herb garden had already existed in antiquity, and had also been fostered in the medieval monasteries, whose gardens also served as a role model for the university botanical gardens that emerged in the Renaissance. Here, there was a change of emphasis, as well as serving as a practical resource for medicinal herbs, to save having to scavenge them from nature, the university botanical gardens served as a centre for teaching and research. Also, botanical gardens did not cease to exist with the advent of modern science but are still going strong, to quote Wikipedia:

Worldwide, there are now about 1800 botanical gardens and arboreta in about 150 countries (mostly in temperate regions) of which about 550 are in Europe (150 of which are in Russia), 200 in North America, and an increasing number in East Asia. These gardens attract about 300 million visitors a year.

The last sentence shows that botanical gardens now function as tourist attractions, the entrance fees helping to finance their upkeep. Although, the Renaissance botanical gardens also attracted a steady flow of visitors from all over Europe. The modern botanical gardens, many of which a government sponsored, are major scientific research centres and are networked worldwide to increase their effectivity, exchanging plants, seeds, and knowledge. This networking and scientific exchange was already developing during the Renaissance, albeit on a much smaller scale. 

As you can see there is a strong continuity in the concept and existence of botanical gardens from some point deep in antiquity down to the present day. However, that continuity is not a smooth curve but has suffered breaks and seen changes in the people operating them and the functions that they have fulfilled. 

I don’t intend to deal with all the topics that I have covered in the episodes of this series in the same way here, but I will bring one more example from a completely different area, mathematics and in particular algebra.

Although the word algebra is a comparatively modern coinage, stemming as it does from the title of a ninth-century book, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (The Compendious Book on Calculation by Completion and Balancing) by the Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850), the commonly used definition for elementary algebra is to quote Wikipedia:

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as “does an equation have a solution?”, “how many solutions does an equation have?”, “what can be said about the nature of the solutions?” are considered.

A definition that covers all of the algebra under discussion here.

Both the ancient Egyptians and Babylonians wrote about the solution of equations in their mathematical texts. The Babylonian’s even had a version of the general solution for quadratic equations as early as 1700 BCE. It is in a different form to that taught in schools today and of course only accepts positive solutions. Babylonian algebra grew out of the commercial arithmetic that they developed for their central, state-controlled economy.

We find our form of the general solution for quadratic equations with both positive and negative solutions in the Brāhma-sphuṭa-siddhānta (Correctly Established Doctrine of Brahma) of the Indian mathematician and astronomer Brahmagupta (c. 598­–c. 668) a text that did much to inform the work of al-Khwārizmī. 

Al-Khwārizmī’ s book was first translated into Latin by Robert of Chester in 1145 but initially had little impact in Europe. Algebra first became truly establish in Europe with the publication of the Liber Abbaci (The Book of Calculation) by Leonardo Pisano (c. 1170–c. 1245) in 1201. This established algebra as commercial arithmetic, which was then taught throughout Europe in so-called Abbacus schools to apprentice traders, in order to be able to calculate interest rates on loans, exchange rates of currencies when crossing borders, and profit shares in joint trading ventures, amongst other things. This had also been the primary use of algebra in Islamicate culture from whom Leonardo had directly taken his knowledge of the discipline. 

It was first in the sixteenth century, also within my defined timeframe for Renaissance science, that algebra first became recognised as a proper branch of mathematics during the disputes over the discovery of the general solutions of the cubic, and quartic equations. Some even refer to Cardano’s Ars Magna (Nürnberg 1543), a central text in those disputes, as the first modern mathematics book. Algebra only became truly establish as the core of analytical mathematics in the seventeenth century as part of the so-called scientific revolution. The sixteenth and seventeenth centuries also saw a gradual development from rhetorical algebra, written entirely in words, over syncopated algebra, with some symbolism, to our symbolic algebra.

Algebra, as the doctrine of the solution of equations, is of course a central element of the modern school mathematics curriculum. In German the general solution for quadratic equations is called the Mitternachtsformel (Midnight formula), because school children are expected to be able to rattle it off if woken up by their maths teacher at midnight. 

As I have outlined above algebra winds its way through history from at least 2000 BCE down to the present, changing in presentation, and function over the centuries. It is by no means a continuous evolution but continuity over time in its history is just as important, if not more so, that the developments in any given, artificially defined period such as the Humanist Renaissance.

What I hope I have made clear in this blog post is that, although historically useful, the concepts, of a time period that we call the Humanist Renaissance, and the developments in the sciences within that period, that I, following others, have chosen the define as Renaissance science, are artificial constructs and when we use them, we should be very much aware of the continuity that exists with the periods and the science within them, that exists both before and after our defined period. 


Filed under Renaissance Science

Renaissance Science – LI

One of the most ubiquitous figures in the history of science in the first half of the seventeenth century was Francis Bacon, 1st Viscount St Alban (1561–1626), jurist, and politician, who rose to become Lord High Chancellor of England.

Portrait of Francis Bacon by Paul van Somer 1617 Source: Wikimedia Commons

A prolific author of polemical text, he gets labelled the father of empiricism, the father of the scientific method, and even the father of modern science. Regular readers of this blog will know, without asking, that I reject all three labels. In fact, I go much further, rejecting the deification of Francis Bacon in the hierarchy of modern science. First, and it gets said far too little, Bacon was not a scientist and secondly, he didn’t even understand science or, if it comes to that the scientific method. In my opinion, Bacon is not the signpost to the future of science that his fans claim him to be, but someone, who looks back at the development in science that had taken place in the recent past and collated, idealised, and systemised them, whilst projecting them into an imaginary future.  

Bacon’s record on the leading scientific developments in the early seventeenth century is so abysmal that it is difficult to understand how anybody ever took him seriously as a philosopher of science. 

His attitude to the already advancing mathematisation of the sciences was to say the least retrograde or even reactionary. In his writings he, like Aristotle, says that pure mathematics has no place in natural philosophy, because its objects are not material. At one point he specifically rejects the developments that had been made in algebra, as it had not been well perfected. He, also like Aristotle, allows mixed mathematics, even acknowledging an increasing list of areas where this applies listing, perspective, music, astronomy, cosmography, architecture, engineering, and diverse others. This list encapsulates many of the developments during the Renaissance that we have examined in various episodes of this series. However, he only allows mathematics a measuring role in the, for him all important, empirical investigations, but not a determining or philosophical one. It should, however, be noted that in his own examples of empirical investigations there are no quantitative tables of measurement. His attitude to the role of mathematics is best illustrated by his rejection of Copernican astronomy. Bacon feared abstract reasoning not based upon experience, and rejected purely theoretical science such as Copernican astronomy, a purely mathematical model. A model for which there was no empirical, observational evidence. One must admit, a fairly reasonable argument at the time.

Bacon rejected another important milestone in the history of science, William Gilbert’s De Magnete, which had been published in 1600. This was a work solidly based on experience, experiments, and empirical observations, so one would have thought that it would be acceptable to Bacon, but this was not the case. He criticised Gilbert heavily because although based on a wealth of experiments, he had made a philosophy out of the loadstone, indulging in extravagant speculation.

Perhaps surprisingly, Bacon also raised serious doubts about both the telescope and the microscope, empirical research instruments. He found Galileo’s initial telescopic discoveries admirable, but then there was nothing more and he thus found the handful of discoveries suspicious because they were so few and had then petered out. As I’ve noted elsewhere there was indeed a lull after 1613 in telescopic discoveries, which lasted until astronomers adopted the astronomical telescope, which had a much greater magnification than the original Dutch or Galilean telescope. Bacon, who was not an optician or astronomer, had no real understanding of that which he was criticising. His doubts concerning the microscope can possibly be excused as he died much to early to see any real results of microscopic investigations, although I wonder if he attended any of Cornelis Drebble’s public demonstrations of his Keplerian microscopes in the early 1620s.

There are three major publications outlining Bacon’s views on education, which include his views on natural philosophy and his thoughts on how it should be practiced i.e., his much-praised methodology. The first of these is his The Advancement of Learning from 1605. This is a polemic advocating for a general state sponsored education. The emphasis in this polemic is very much on religion and civics. There is very little in this work that in anyway relates to the developing sciences of the period and his highly abstract discussion of natural philosophy is, for a man who supposedly dethroned Aristotle, highly Aristotelian. 

Title page Source: Wikimedia Commons

It is in fact first in his Novum Organum from 1620 that he seeks to dethrone Aristotle replacing, as the title states Aristotle’s Organon, his six books on logical analysis, which underly his physics, that is the description of nature, with Bacon’s own new empirical inductive logic, which is so often falsely claimed to be “the” modern scientific methodology.

Title page Source: Wikimedia Commons

There of course being no singular scientific method and also those who believe there is one describe something very different to Bacon’s model. Bacon rejects Aristotle’s top-down methodology, which starts with supposedly obvious first principle or axioms to which deductive logic is systematically applied until one arrives at empirically observed facts. He wishes to replace it with a bottom-up system, which starts with empirically observed facts and then uses inductive logic to arrive at general statements derived from those facts. 

Bacon’s system is very naïve and primitive and consists of creating lists of empirical observations. For a given phenomenon, the example Bacon uses is heat, he collects in a list all the empirical instances where heat occurs. He then complies a second list of all the instances where heat doesn’t occur. This is of course a major problem as, whilst not infinite, such a list would be impossibly long, so he makes some arbitrary decisions to reduce the list. He then compares the properties of the lists to eliminate any that appear in both lists. Finally in the parred down list of heat occurrences he removes those properties that are not in all instances, for example light, which is in fire but not in hot water. In the list that is left over the form (cause) of heat should naturally emerge. He explicitly warns against speculating too far from the acquired evidence. 

This is of course not how science works. Is it argued that Bacon plays an important role in the development of the scientific method because he suggests experimentation as a method to produce more empirical instances. Of course, Bacon is not the first to introduce experimentation into scientific research, alchemy, which Bacon disdained, had been using experimentation for centuries and experimental laboratories were a feature of Renaissance science. Bacon’s insistence on empirical observation and induction appears to me to be a very similar, but formalised, approach to that of the work of the Renaissance researchers, who developed the materia medica and botany. 

I think the best comment on Bacon’s approach was supposedly made by William Harvey, in his Brief Lives, John Aubrey tells us that Harvey:

“had been physitian to the Lord Chancellour Bacon, whom he esteemed much for his witt and style, but would not allow him to be a great Philosopher. Said he to me, ‘He writes Philosophy like a Lord Chancellour,’ speaking in derision, ‘I have cured him.'”

One of the most often referenced of Bacon’s texts in his utopia, The New Atlantis, the House of Salomon in which supposedly inspired the foundation of the Royal Society. It was never completed and first published posthumously.

Title page Source: Wikimedia Commons

In the modern English version that I own, it is forty-nine pages long and the first thirty-six pages tell the story of a ship blown of course arriving at the Island of Bensalem, apparently Bacon’s concept of an ideal society. I’m not going to describe the culture of Bensalem, which appears to me to be basically a form of theocracy but will briefly sketch his account of the House of Salomon. The official of the House of Salomon, who gives a verbal guided tour to the book’s narrator, a member of the ship’s crew, who is not more closely identified, just rattles of long lists of all the wonderful things that each section or division of the house contains. There is no real attempt to describe the science that produced these wonders or explain the methodology behind them. In general, large parts of this pean to the scientific achievements of the Bensalemites read like an idealised cross between the Renaissance botanical gardens of Northern Italy and the curiosity cabinets of the German aristocrats. In fact, elsewhere Bacon suggests that systemised curiosity cabinets could be used for his type of inductive scientific research. Some of them, such as that of Rudolf II, were already used for scientific research but not using Baconian methodology.

Returning to my original position, I contend that Bacon is not the father of modern science shining a methodological beacon into the future of scientific research but rather a man with very little real understanding of how science works, who held up a mirror, which reflected various aspects of the Renaissance science that had preceded him.


Filed under Early Scientific Publishing, Renaissance Science