Category Archives: History of Mathematics

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.

Leave a comment

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

Counting the people

There is a widespread misconception amongst people, who are not particularly good at mathematics that mathematicians can do mathematics, by which I mean that a mathematicians can do the whole range of subdisciplines that are collected together under the term mathematics. Nothing could be further from the truth. It is claimed that there is a fundamental divide between mathematicians who think in diagrams, i.e., at the entry level geometry, and those who think in symbols, i.e., algebra and analysis. Not that I’m a mathematician but I certainly think in symbols, not diagrams. At a higher level, most mathematicians have a special discipline where they are at home and excel and other disciplines that that they have difficulties even comprehending. When I studied mathematics at university the introductory courses were allotted in rotation to the professors. My introductory analysis lectures were held by a really sweet guy, who was a world-renowned finite group theory specialist, where world renowned means that the fifty people in the world in his field all knew him. One day he came into the lecture hall and said, “today we should start with step functions, but I’ve never understood them, so we’ll do something else instead.” 

Although I studied mathematics at university up to about BSc level, worked in a research project into mathematical logic for a number of years, and have tutored school leaving/university entrance level (A-level, Abitur, Baccalauréat etc.) mathematics for the last twenty years, I do not consider myself a mathematician. Having said that, I definitely have my strengths and weaknesses in different mathematical disciplines. I could always do basic calculus without even think about it, in fact it was my love of calculus that led me into the history of mathematics, when I discovered that Newton and Leibniz had both invented/discovered (choose your preferred term) calculus independently of each other. I later discovered that this wasn’t true but that’s another story. An entry level discipline which I could never get my head around was probability theory and statistics and I used to groan inwardly when I had to teach them to one of my private pupils. 

When I studied mathematics at school before the second ice age, I did A-level maths, probability theory and statistics were not part of the curriculum. However, teaching Abitur in Germany over the last four years probability theory and statistics were very much part of the curriculum. Today, there is a widespread and very dynamic discussion in many lands about changing, extending, and improving the mathematics courses taught in schools at all levels, in order to combat a perceived mathematical illiteracy. See, for example, the Tory government’s call for maths to eighteen for school kids in the UK. One prominent argument in these discussions is for the reduction or removal of much of the tradition diet of algebra, geometry, and calculus and replacing it with a much-expanded emphasis on probability and statistics because these are the areas of mathematics that people need to understand and even use in everyday life. 

It is in fact true that we stumble across probability theory and statistics on a daily basis in the media, on the news, in advertising etc. often misused and mostly misunderstood by the people reading it. Probability turns up in the weather forecast, there’s a X% chance of rain, in sport betting, which has become a vast industry, in other forms of gambling, but also in areas of science and medicine. What is the probability of blah, blah blah… Here the quoted probabilities are derived from statistical analysis. Today, it is perfectly normal for all aspects of human existence to be analysed statistically. From trivial things like what percentage of the population is left-handed, to serious topics such as what is the probability of someone developing a particular type of cancer. We have statical analyses of opinion polls, election results, school exam results and …

We are so inundated with statistical information, oft presented as a probability, that we simple accept it without really thinking about it. However, where do these two areas of mathematics come from, when were they developed and why? Although there are earlier simple examples of the calculation of probabilities, both probability theory and statistics first emerged in the Early Modern Period. Not surprisingly, probability theory first emerged in the calculation of odds in gambling. The first major work on probability theory was written by one of my favourite Renaissance polymaths, Gerolamo Cardano (1501–1576), who was a passionate, and at time professional, gambler. His book, Liber de ludo aleae (Book on Games of Chance), written in the 1560s but first published in 1663, also includes advice on how to cheat. In their correspondence in 1654, Pierre Fermat (1607–1655) and Blaise Pascal (1623–1662) discussed various aspects of probability theory after being asked how the pot should be divided in an interrupted game. Christiaan Huygens (1629­–1695) came across this correspondence in Paris and wrote the most coherent and at that time, most advanced book on probability theory his De Ratiociniis in Ludo Aleae (On reasoning in games of chance). Originally written in Dutch, it was translated into Latin and published by Frans van Schooten Jr. in 1657. The mathematics of probability was firmly established in the early eighteenth century by Jacob Bernoulli (1655–1705) with his posthumously published Ars Conjectandi (The Art of Conjecturing) in 1713, which covers combinatorics and probability, and Abraham De Moivre (1667–1754) with his The doctrine of chances: or, a method for calculating the probabilities of events in play in 1718 with an expanded second edition in 1738, and a further expanded edition published posthumously in 1756. 

Statistics, however, developed from the start through a desire to count people, a development that had a long and complex prehistory before it began to become formalised on a very simple level in the second half of the seventeenth century. That formalisation took place in the work of John Graunt (1620–1674), Edmond Halley (1656–1741), John Arbuthnot (1667–1735), Gregory King (1648–1712), Charles Davenant (1656–1714), and William Petty (1623–1687), who gave the counting of people the early name of Political Arithmetick

Today, we use the term demography derived from the Ancient Greek demos meaning people, society and graphía meaning writing, drawing, description, and meaning the statistical study of populations. The American historian Ted McCormick, who teaches at Univesité Concordia in Montreal earlier wrote William Petty and the Ambitions of Political Arithmetic (OUP, 2009), using the manuscripts of Sir William Petty (1623-1687) to show how a mixture of alchemical and natural-philosophical ideas were brought bear governing colonial populations in Ireland and the Atlantic, as well as confessional and labouring populations in Britain (taken from his university webpage), which I haven’t read but I’ve now added to the infinite reading list. He has now followed up with Human EmpireMobility and Demographic Thought in the British Atlantic World1500–1800,[1] which details the gradual development of demography in Britain, Ireland and the American Colonies in the Early Modern Period.

Before I write more about McCormick’s book in any detail, a couple of important notes about it. Although this book relates the historical developments over a couple of centuries that led to the first low level uses of statistics by the scholars I named above, it is in no way whatsoever a history of mathematics text. In fact, actual numbers are strikingly absent from McCormick’s narrative. Rather it examines the social, political, environmental, cultural, philosophical, and economic circumstances that led authorities and individual to consider it necessary to enumerate elements of the population. Secondly, having said this, it should be fairly obvious from my general description that this is also in no way a popular book, but rather a deeply and intensively researched academic book. 

The word demography was first coined in the nineteenth century, but societies have been indulging in demographic thought at least since the emergence of the earliest civilisations. McCormick’s book might well be regarded as an extended case study into the structure and content of such thought in Britain and its colonies over four centuries. McCormick himself illustrates the ancient origins of the discipline, in his introduction, with references to the Bible. He also touched briefly on Aristotle’s thoughts on the topic, as these were of course relevant to those engaged in debates in the Early Modern and Modern periods. He writes the following:

Aristotle thus presented population not just as a measurable number of inhabitants but also, more saliently, as the living material of the city-state. Its size would be constrained by the territory that it occupied. It should be limited, too, by the counterpoised imperatives of magnitude and order in the context of polity conceived as an organic unit – a body politic – with a constitution. More significant than absolute size was the relative proportions of the body’s parts: The balance between citizens, slaves and foreigners, [my emphasis]and between soldiers, husbandmen and artisans (page, 29). 

Reading these lines and in particular the clause I have emphasised I was instantly reminded of current political and cultural debates that are currently raging in many countries about exactly that balance, (ignoring the slaves of course) or as many see the lack of what they see as the correct balance–too many foreigners, migrants, asylum seekers… take your pick. This was the first time that such parallels between the historical debates that McCormick outlines in great detail and the actual political debates of our times, but it was by no means the last time. Again and again, I found myself thinking this is all too familiar. My feelings were confirmed when in the closing pages of his book McCormick remarked:

To put it another way: what might the early modern history of demographic governance tell us about the persistence or reemergence of concerns about the mobility, mixture and mutability of populations in the nineteenth, twentieth and twenty-first centuries) Instances of such concerns are not only numerous but also fundamental to received interpretations of modern history (page 249)

The main text of the book covers four phases or periods of debate about perceived demographic topics. The first concerning the fifteenth century reminded me of my days as a field archaeologist. I spent several seasons, both Easter and summer, working on the major excavation of a deserted medieval village, or DMV, in the north of England. DMVs, shrunken medieval villages, SMVs and expanded medieval villages, otherwise known as towns, are the result of a major shift in population distribution during the High Middle Ages largely brought about by the enclosures. That is turning agricultural land, which peasants farmed to make a subsistence living, into pasture for sheep grazing, thereby forcing the ploughmen to abandon their villages and try to seek employment elsewhere. 

The socio-political debate that McCormick covers, about this depopulation, concerns the reformers, who wished to restore the honest ploughman to his rightful place in society. At this time the talk, however, is not of populations, but of the much more imprecise multitudes. In the second section of the book, we still have to do with multitudes but here, in the Elizabethan era, it is not a debate about a positive part of the body politic, the humble ploughman, but concerns about negative sections of that body, vagrants and the poor. Once again, the main participants in the public debate are reformers offering and discussing potential solutions what they see as the proliferation of undesirable elements in society. 

This second phase moves out of England into Ireland where the English had problems with various aspects of various parts of the island’s population.

In these sections and also in the ones to follow, McCormick outlines the perceived problems with selected parts of the population, the multitudes, and then presents the solutions proposed by the various reformers. He lets the participants in the various debates present their polemic themselves, in lengthy direct quotes all delivered up in the original English of the period, with its own vocabulary, orthography, and grammar. I must admit that I found it difficult reading some of these passages that appeared almost to be in a foreign language and not the English with which I grew up. However, it pays to persevere because it gives a much clearer picture of what the participants were aiming for than a simple modern English paraphrase. 

In the third section McCormick brings the political philosophers into play and the discussions on over population. Here we see the emergence of colonialism as a potential solution, as to what to do with surplus multitudes. Already practiced with little success in Ireland, we see the beginnings of the establishment of colonies in North America. The seventeenth century sees the move in the discussion from the multitudes of the earlier reformers to the perception of population and the slow move towards encapsulation through mathematics in the form of statistics driven by the writings of such as Francis Bacon and the Hartlib Circle, a precursor to the Royal Society. The latter offering up various projects for empire, colonialism, and population. 

Enter William Petty. Petty, an associate of the Hartlib Circle and a founding member of the Royal Society, no longer simply delivered polemics on population and population reform but in his survey of Ireland, on behalf of Cromwell, to organise the distribution of land to Cromwell’s soldiers, as payment for their services and also to dilute the troublesome Irish population, Petty was instrumental in putting a plan into action. McCormick covers Petty’s survey and its background in great detail. Out of his work in Ireland he developed his economic theories, marking him as a pioneer in economic science, and also his Political Arithmetick. It’s worth quoting McCormick here on Petty’s innovation:

“Political arithmetic” was coined sometime around 1670. It has appeared ever since as the invention of a new, scientific and, above all, quantitative age. Its inventor, Petty, has often seemed precocious in his focus on “number, weight and measure,” his imaginative exploitation of demographic and economic figures running well ahead of the empirical data at his disposal over a century before the census. While John Graunt’s “shop arithmetique” revealed a world of relationships hidden in the rows and columns of London’s weekly bills of mortality and in the patchy parish registers of baptisms, marriages and burials, Petty promised nothing less than a new “Instrument of Government” for the Stuart kingdoms and the growing colonial empire, predicted on the collection and analysis of vast amounts of information. Most of this was numerical. 

This introduction is followed by an in-depth analysis of what exactly Petty’s innovative creation was, what it meant and how it influenced future developments. 

The final section of McCormick’s main narrative follows how the rhetoric about population and demography evolved throughout the eighteenth century both in Britain and in its colonies in America. In Britain the reformist mode of thought is still dominant. In America the problem of increasing the colonial population to take over the land is predominant with Benjamin Franklin taking a lead in the debate. On the one side his acknowledgement that the land was occupied by Indians when the settlers first arrived but, on the other, that it would only be settled when the colonial population had grown enough shows a level of casual racism that I found deeply disturbing. There is also an extreme and dismissive level of racism shown by Franklin and others towards the negro slave population. 

McCormick closes his stimulating and fascinating narrative in the conclusion to his book with a discussion of Thomas Malthus’ 1798 Essay on the Principle of Population, a work that famous influenced by Charles Darwin and Alfred Russel Wallace in the formulation of their theories of evolution by natural selection. Darwin and Wallace do not put in an appearance here as McCormick is concerned with Malthus’ influence on the development of demography, which he presents as revolutionary:

From the perspective of the Henrician humanists, Elizabethan pamphleteers, Jacobian colonial promotors, Interregnum projectors, Restoration political arithmeticians and Enlightenment-era physicians, philanthropists and philosophers who have populated this book, T. R. Malthus’s 1798 Essay on the Principle of Populationappears scarcely less than the French Revolution, to mark the end of an age. 

Carrying on, McCormick runs revue over the polemics, reforms, schemes. and projects that he has described in the previous chapters and concludes:

Into this baroque edifice slammed the wrecking ball of Malthusian principle.

He then follows up with an analysis of that Malthusian wrecking ball. McCormick’s book closes with some very open and honest general thoughts on the limitations of his own research.

Fans of footnotes will love McCormick, the book has literally tons of them listing vast amounts of sources, that are included in a twenty-two-page bibliography, the whole closing out with an excellent index. This book does not have illustrations.

This is not an easy read. It is a book packed with intensive historical information and evidence with an equally intensive analysis. However, if you have any interest in the topics covered this is a must read.

Having started out with my personal relationship to probability theory and a very brief sketch of its early history, when the philosopher of science Ian Hacking died whilst I was writing this, I immediately got his excellent The Emergence of ProbabilityA Philosophical Study of Early Ideas about Probability, Induction and Statistical inference (CUP, 1975), a title that can be found in McCormick’s extensive bibliography, out of the university library. A book that I first consulted more than thirty years hence and is now my current bed time reading. 

[1] Ted McCormick, Human EmpireMobility and Demographic Thought in the British Atlantic World1500–1800, CUP, Cambridge, 2022


Filed under Book Reviews, History of Mathematics

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

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 Science – XLIX

The mathematisation of science is considered to be one of the principle defining characteristics of the so-called scientific revolution in the seventeenth century. Knowledge presentation on the European, medieval universities was predominantly Aristotelian in nature and Aristotle was dismissive of mathematics. He argued that the objects of mathematics were not real and therefore mathematics could not produce knowledge (episteme/scientia). He made an exception for the so-called mixed disciplines: astronomy, geometrical optics, and statics. These were, however, merely functionally descriptive, and not knowledge. So, mathematical astronomy described how to determine the positions of celestial bodies at a given point in time, but it was non-mathematical cosmology that described the true nature of those celestial bodies. Knowledge production and knowledge acquisition was, for Aristotle and those who adopted his philosophy, non-mathematical.

With just a relatively superficial examination, it is very clear that the new knowledge delivered up in astronomy, physics etc in the seventeenth century was very mathematical, just consider the title of Newton’s magnum opus, Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy), something major had changed and two central questions for the historian of science are what and why? 

When I first started learning the history and philosophy of science several decades ago, there was a standard pat answer to this brace of questions. It was stated that there had been a change in philosophical systems underlying knowledge acquisition, Aristotelian philosophy and been replaced by a mathematical neo-Platonic philosophy; Plato had, according to the legend, famously the dictum, May no one ignorant of Geometry enter here Inscribed above the entrance to his school, The Academy. The belief that the mathematisation of science was driven by a Platonic renaissance was probably strengthened by the fact that the title page of Copernicus’ De Revolutionibus, the book that supposedly signalled the start of the scientific revolution, carried the same dictum. In fact, there is no evidence in Greek literature from the entire time that The Academy was open that the dictum existed. It was first mentioned by John Philoponus (c. 490­–c. 570) after Justinian had ordered The Academy closed in 529. De Revolutionibus is also not in anyway Platonic. 

To be fair to the proposers of the Plato replaced Aristotle thesis, Plato’s philosophy was definitively more mathematical than Aristotle’s and there was a neo-Platonic revival during the Renaissance, but it was more the esoteric and mystical Plato rather than the mathematical Plato, as I’ve already outlined in an earlier episode in this series.

So, what did drive the mathematisation? As already explained in explained in earlier episodes there were major expansions and developments in astronomy, cartography, surveying, and navigation starting in the fifteenth century during the Renaissance. All four disciplines demanded an intensive use of geometry and especially trigonometry. This can be seen in the publication of the first printed edition of Euclid by Erhard Ratdolt (1442–1528) in 1482, which was followed by significant printed translations in the vernacular throughout Europe.

A page with marginalia from the first printed edition of Euclid’s Elements, printed by Erhard Ratdolt in 1482
Folger Shakespeare Library Digital Image Collection
Source: Wikimedia Commons

In trigonometry, Johannes Petreius (c. 1497–1550) published Regiomontanus’ De triangulis omnimodis (On Triangles of All Kinds), edited by Johannes Schöner, in 1533. This was the first almost complete account of trigonometry published in Europe, the only thing that was missing was the tangent, but Regiomontanus had included the tangent in his earlier Tabula directionum, written in 1467 but first published in print in 1490. Regiomontanus’ trigonometry was followed by several important volumes on the topic during the sixteenth century. 


These areas of mathematical development were however for the Aristotelian academics at the universities not scientia and the mathematical practitioners, who did the mathematics were not considered to be academics but mere craftsmen. However, the widening reliance on mathematics in what had become important political areas of Renaissance society did much to raise the general status of mathematics.

Another area where a mathematical subdiscipline was on the advance was algebra, the basis for the analytical mathematics that would become so important in the seventeenth century. Already introduced in the twelfth century, with the translation of al-Khwarizmi’s text on the Hindu-Arabic number system into Latin, Algoritmi de numero Indorum along with the book that gave algebra its name, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah. It initially had minimal impact. However, reintroduced in the next century by Leonardo Pisano, with his Liber Abbaci, and although the acceptance was slow, only beginning to accelerate with the introduction of double entry bookkeeping at the end of the thirteenth century and beginning of the fourteenth. By the sixteenth century many abbaco schools (scuole d’abbaco or botteghe d’abbco) had been established throughout Europe teaching the Hindu-Arabic number system and algebra to apprentices using Libri d’abbaco, (abbacus books). In fact, the first ever printed mathematics book was an abbacus book, the so-called Treviso Arithmetic or Arte dell’Abbaco written in vernacular Venetian and published in Treviso in 1478. 

Once again, however, what was being taught here was not an academic discipline, which could generate scientia, but commercial arithmetic, very useful for an increasing commercial Europe dependent on extensive trade but not for the acquisition of academic knowledge. This began to change slowly in the sixteenth century beginning with the Summa de arithmetica, geometria, proportioni et proportionalita of Luca Pacioli (c. 1447–1517) published in 1496, still a book of practical mathematics and so not academic, but one which contained the false claim that there could not be a general solution of the cubic equation. This led on to Scipione del Ferro (1465–1526) discovering such a solution, Tartaglia (c. 1499–1557) rediscovering it and Gerolamo Cardano (1501–1576) seducing Tartaglia into revealing his solution and then publishing it in his Ars Magna. All of which I have outlined in more detail here. Cardano’s Ars magna, published in Nürnberg in 1545 by Johannes Petreius, has been called the first modern mathematics book, a term I don’t particularly like, but it did bring algebra into the world of academia, although it still wasn’t considered to be knowledge producing.


So, how did the change in status of mathematics on the universities come about and who was responsible for it if it wasn’t Plato? The change was brought about by Italian, humanist scholars in the sixteenth century and the responsibility lies not with a philosopher but with a mathematician, Archimedes. 

Bronze statue of Archimedes in Syracuse Source: Wikimedia Commons

Archimedes of Syracuse (c.287–c. 212 BCE) mathematician, physicist, engineer, and inventor is one of the most well-known figures in the entire history of science. Truly brilliant in a range of fields of study and immersed in a cloud of myths and legends. He is famous for the machines he invented and a legend for alleged machines he constructed to defend his hometown of Syracuse against the Romans. It is these war machines and the myth of his death at the hands of a Roman soldier that dominate the accounts of his life all written posthumously in antiquity. His mathematical work, which is what interest us here, remained largely unknown in antiquity. There only began to become known in the Early Middle Ages but, although translated into Arabic by Thābit ibn Qurra (836–901) and from there into Latin by Gerard of Cremona (c. 1114–1187) and again directly from Geek into Latin by William of Moerbeke (c. 1215–1286) and once more by Iacobus Cremonnensis (c. 1400–c. 1454), his work received very little attention in the Middle Ages.

Beginning, already in the fifteenth century, Renaissance humanist began to seriously re-evaluate the leading Greek mathematicians, in particular Euclid and Archimedes. Euclidian geometry was playing a much greater role in the evolving optics, in particular linear perspective, than it had ever played on the medieval universities. This led, as already noted above, to the publication of the first printed edition of The Elements, by Erhard Ratdolt, in 1482. Interestingly, the manuscript that Ratdolt used for edition was one that Regiomontanus (1436–1476) had brought with him to Nürnberg, where he established the world’s first scientific published endeavour, intending to publish it himself, as he announced in his published catalogue of intended publications. Unfortunately, he died before he could print most of this extensive catalogue of scientific and mathematical texts. 

This catalogue also included a manuscript of the works of Archimedes in the Latin translation of Iacobus Cremonnensis, which also fell foul of the Franconian mathematician’s early death. This manuscript would eventually be published in Basel, together with a Greek original brought from Rome by Willibald Pirckheimer (1470–1530), in a bilingual edition of the works edited by the Nürnberger theologian, humanist, and mathematician, Thomas Venatorius (1488–1551), in 1544. 

Venatorius’ edition of the works of Archimedes Source

The Italian astrologer, astronomer, and mathematician, Luca Gaurico (1475–1558), had published Archimedes’ works On the Parabola and On the Circle in the Latin translation by William of Moerbeke in 1503. Niccolò Fontana Tartaglia, who had published an Italian translation of The Elements in 1543, also published On the ParabolaOn the CircleCentres of Gravity, and On Floating Bodies in the Moerbeke translation in 1543. Later he would publish translation into Italian of these works, some of which appeared posthumously. Unlike, other later, Italian mathematicians, Tartaglia did not incorporate much of Archimedes’ work into his own highly influential, original Nova Scientia (1537), a mathematical work that did deliver, as the title says, scientia or knowledge. It is difficult to say how much Tartaglia was influenced by Archimedes in his approach to physics. 

Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi per Nicolaum Tartaleam … (facsimile) Source

We have already seen in the episode on hydrostatics how Archimedes work On Floating Bodies, informed and influenced the work of both Tartaglia’s one time student, Giambattista Benedetti (1530–1590), and the engineer, Simon Stevin (1448–1620) in the Netherlands in their work on hydrostatics and on the laws of fall. As I have also outlined in great detail, Archimedes work on statics had a major influence on the Italian mathematicians of the so-called Urbino School. Federico Commandino (1509 – 1575), Guidobaldo dal Monte (1545 – 1607), and Bernardino Baldi (1553–1617), the first two also producing and publishing new improved translations of various of Archimedes work. Once again Simon Stevin also produced a major, Archimedes inspired work on statics. 

These mathematicians had, directly inspired, and heavily influenced by the work of Archimedes, now produced, in several fields, work that was indisputably scientia or knowledge by the use of mathematics and thus instigated the turn from Aristotelian philosophical knowledge to the mathematisation of knowledge production and this movement began to spread in the seventeenth century.

The Urbino School, in particular dal Monte, who was his patron, influenced Galileo, who openly declared that he had in his natural philosophy replaced Aristotle with Archimedes. Galileo’s pupils Vincenzo Viviani (1622–1703) and Evangelista Torricelli (1608–1647) followed his lead on this. Stevin’s work, written in Dutch was translated into Latin by Willebrord Snel (1580–1626) and heavily influenced the French natural philosophers such as, Marin Mersenne (1588–1648), who together with Pierre Gassendi (1592–1655) led the informal academic society, the Academia Parisiensis, a weekly gathering from 1633 onwards, which included the most important French, English and Dutch natural philosophers of the period. This group was of course also influenced by the work of Galileo. 

It was not the thoughts of a philosopher, Plato, that pushed Aristotelian philosophy from its throne on the medieval university as the purveyor of factual knowledge of the real world and replaced it with a mathematics-based system, but the work of a mathematician, Archimedes. As the century progressed Euclid and Archimedes would in turn be replaced by the algebra-based analytical mathematics that would eventually develop into calculus, although also here the method of exhaustion first developed by Eudoxus of Cnidus (c. 408–c. 355 BCE), but popularised by Archimedes was the basis of the integral calculus half of the new mathematics.


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

Starting off Fibonacci year…

The HistSci-Hulk woke up briefly from his winter slumbers to cast a bleary eye over a piece by Katie Steckles on the web site SciLogs celebrating, what some are calling the Fibonacci New Year, because it starts with 1/1/23 the first four digits of the so-called Fibonacci sequence. It doesn’t really because the sequence correctly begins with a zero and Fibonacci began it with 1, 2 not 1, 1, 2.

“I bet she doesn’t point out that old Leo wasn’t the first to elucidate this sequence,” the grumpy beast muttered as rubbed the sleep out of his eyes. 

Surprisingly he was wrong, as Steckles does actually point out that the sequence first appeared historically in Indian grammatical studies of Sanskrit prosody. However, much to the annoyance of out grumpy friend, her second paragraph is loaded historical with historical errors.

Leonardo Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician born circa 1170 CE, who – like many historical mathematicians – is primarily remembered for one book he wrote. 

His name was never Leonardo Fibonacci. He was Leonardo Pisano, which translates as Leonardo of Pisa. The name Fibonacci, which unfortunately has become universal, was created in the nineteenth century by a French historian. 

In Fibonacci’s case, the book was called “Liber Abaci” – literally, “The Book of the Abacus” – although it was actually presented as an alternative to the then-common use of abaci for calculation. 

Liber Abaci, which on the title page is actually spelt Liber Abbaci, although Leonardo uses both spellings in the text, has absolutely nothing to do with the abacus or counting board. Abbaci comes from the then Italian term for calculate or reckon and the correct translation of the title is Book of Calculations.

Published in 1202, this was the first European work covering Indian and Arabian mathematics, and introduced the idea of Hindu-Arabic numerals – the standard digits 0-9 with a decimal system we use today – to Europe for the first time.

This was not the first European work covering Indian and Arabian mathematics, that honour probably goes to the Latin translations of al-Khwarizmi’s On the Calculation with Hindu Numerals, (Arabic original c. 825), Latin Algoritmi de numero Indorum, al-Kindi’s On the Use of the Hindu Numerals (Arabic original c. 830), and al-Khwarizmi’s The Compendious Book on Calculation by Completion and Balancing (Arabic original c. 820) all of which were available in Europe earlier than the Liber Abbaci.

The Liber Abbaci was the first presentation of the Hindu-Arabic number system written by a European author, and had a greater impact than the Latin translations of the Arabic works, because it was presented as commercial arithmetic, leading to the abbacus schools and abbacus books to teach commercial arithmetic to apprentice traders in a Europe, when trading was increasing exponentially.

A minor but important point is that the digits for 0-9 introduced by Leonardo looked very different to the ones we know today.

Evolution of the digits 0-9 Source: Wikimedia Commons

Another minor point is that the digits for 0-9 had already been introduced in the tenth century by Gerbert d’Aurillac (c. 946–1003) but not the number system. He used them to label the counters on his abacus.

A more detailed post on the history of the Hindu-Arabic number system can be found here


Filed under History of Mathematics

Renaissance Science – XLVIII

Using the simplest and widest definition as to what constitutes a scientific instrument, it is literally impossible to say who first created, devised, used a scientific instrument or when and where they did it. My conjecture would be that the first scientific instrument was some sort of measuring device, a rod, or a cord to standardise a unit of measurement, almost certainly taken from the human body: a forearm, the length of a stride or pace, maybe a foot, a unit that we still use today. It is obviously that all the early great civilisation, Indus valley, Yellow River, Yangtze River, Fertile Crescent and so on, definitely used measuring devices, possibly observational devices, instruments to measure or lay out angles, simple compasses to construct circles, all of them probably as much to do with architecture and surveying, as with anything we might now label science.

This is the Royal cubit rod of Amenemope – a 3320-year-old measuring rod which revealed that Egyptians used units of measurement taken from the human body. The basic unit was the cubit – the length from the elbow to the tip of the middle finger, about 45cm. Source: British Museum

Did the early astronomers in China, India, Babylon use some sorts of instruments to help them make their observations? We know that later people used sighting tubes, like a telescope without the lenses, to improve the quality of their observations, did those first astronomers already use something similar. Simple answer, we don’t really know, we can only speculate. We do know that Indian astronomers used a quadrant in their observation of solar eclipses around 1000 BCE. 

Turning to the Ancient Greeks we initially have a similar lack of knowledge. The first truly major Greek astronomer Hipparkhos (c. 190–c. 120 BCE) (Latin Hipparchus) definitely used astronomical instruments but we have no direct account of his having done so. Our minimal information of his instruments comes from later astronomers, such as Ptolemaios (c. 100–c. 170 CE). Ptolemaios tells us in his Mathēmatikē Syntaxis aka Almagest that Hipparkhos made observations with an equatorial ring.

The easiest way to understand the use of an equatorial ring is to imagine a ring placed vertically in the east-west plane at the Earth’s equator. At the time of the equinoxes, the Sun will rise precisely in the east, move across the zenith, and set precisely in the west. Throughout the day, the bottom half of the ring will be in the shadow cast by the top half of the ring. On other days of the year, the Sun passes to the north or south of the ring, and will illuminate the bottom half. For latitudes away from the equator, the ring merely needs to be placed at the correct angle in the equatorial plane. At the Earth’s poles, the ring would be horizontal. Source: Wikipedia

At another point in the book Ptolemaios talks of making observations with an armillary sphere and compares his observations with those of Hipparkhos, leading some to think that Hipparkhos also used an armillary sphere. Toomer in his translation of the Almagest say there is no foundation for this speculation and that Hipparkhos probably used a dioptra. [1]

Ptolemaios mentions four astronomical instruments in his book, all of which are for measuring angles: 

1) A double ring device and

Toomer p. 61

2) a quadrant both used to determine the inclination of the ecliptic.

Toomer p. 62

3) The armillary sphere, which he confusingly calls an astrolabe, used to determine sun-moon configurations. 

Toomer p. 218

4) His parallactic rulers, used to determine the moon’s parallax, which was called a triquetrum in the Middle Ages. 

Toomer p. 245

Ptolemaios almost certainly also used a dioptra a simple predecessor to the theodolite used for measuring angles both in astronomy and in surveying. As I outlined in the post on surveying, ancient cultures were also using instruments to carry out land measuring.

Graphic reconstruction of the dioptra, by Venturi, in 1814. (An incorrect interpretation of Heron’s description) Source: Wikimedia Commons

Around the same time as the armillary sphere began to emerge in ancient Greece it also began to emerge in China, with the earliest single ring device probably being used in the first century BCE. By the second century CE the complete armillary sphere had evolved ring by ring. When the armillary sphere first evolved in India is not known, but it was in full used by the time of Āryabhata in the fifth century CE.

Armillary sphere at Beijing Ancient Observatory, replica of an original from the Ming Dynasty

A parallel development to the armillary sphere was the celestial globe, a globe of the heavens marked with the constellations. In Greece celestial globes predate Ptolemaios but none of the early ones have survived.  In his Almagest, Ptolemaios gives instruction on how to produce celestial globes. Chinese celestial globes also developed around the time of their armillary spheres but, once again, none of the early ones have survived. As with everything else astronomical, the earliest surveying evidence for celestial globes in India is much later than Greece or China.

The Farnese Atlas holding a celestial globe is the oldest known surviving celestial globe dating from the second century CE Source: Wikimedia Commons

In late antiquity the astrolabe emerged, its origins are still not really clear. Ptolemaios published a text on the planisphere, the stereographic projection used to create the climata in an astrolabe and still used by astronomers for star charts today. The earliest references to the astrolabe itself are from Theon of Alexandria (c. 335–c. 414 CE). All earlier claims to existence or usage of astrolabes are speculative. No astrolabes from antiquity are known to have survived. The earliest surviving astrolabe is an Islamic instrument dated AH 315 (927-28 CE).

North African, 10th century AD, Planispheric Astrolabe Khalili Collection via Wikimedia Commons

Late Antiquity and the Early Middle Ages saw a steady decline in the mathematical sciences and with it a decline in the production and use of most scientific instruments in Europe until the disappeared almost completely. 

When the rapidly expanding Arabic Empire began filing their thirst for knowledge across a wide range of subjects by absorbing it from Greek, Indian and Chinese sources, as well as the mathematical disciplines they also took on board the scientific instruments. They developed and perfected the astrolabe, producing hundreds of both beautiful and practical multifunctional instruments. 

As well large-scale astronomical quadrants they produced four different types of handheld instruments. In the ninth century, the sine or sinical quadrant for measuring celestial angles and for doing trigonometrical calculations was developed by Muḥammad ibn Mūsā al-Khwārizmī. In the fourteenth century, the universal (shakkāzīya) quadrant used for solving astronomical problems for any latitude. Like astrolabes, quadrants are latitude dependent and unlike astrolabes do not have exchangeable climata. Origin unknown, but the oldest known example is from 1300, is the horary quadrant, which enables the uses to determine the time using the sun. An equal hours horary quadrant is latitude dependent, but an unequal hours one can be used anywhere, but its use entails calculations. Again, origin unknown, is the astrolabe quadrant, basically a reduced astrolabe in quadrant form. There are extant examples from twelfth century Egypt and fourteenth century Syria.

Horary quadrant for a latitude of about 51.5° as depicted in an instructional text of 1744: To find the Hour of the Day: Lay the thread just upon the Day of the Month, then hold it till you slip the small Bead or Pin-head [along the thread] to rest on one of the 12 o’Clock Lines; then let the Sun shine from the Sight G to the other at D, the Plummet hanging at liberty, the Bead will rest on the Hour of the Day. Source: Wikimedia Commons
Astrolabic quadrant, made of brass; made for latitude 33 degrees 30 minutes (i.e. Damascus); inscription on the front saying that the quadrant was made for the ‘muwaqqit’ (literally: the timekeeper) of the Great Umayyad Mosque of Damascus. AH 734 (1333-1334 CE) British Museum

Islamicate astronomers began making celestial globes in the tenth century and it is thought that al-Sufi’s Book of the Constellations was a major source for this development. However, the oldest surviving Islamic celestial globe made by Ibrahim Ibn Saîd al-Sahlì in Valencia in the eleventh century show no awareness of the forty-eight Greek constellations of al-Sufi’s book.

Islamicate mathematical scholars developed and used many scientific instruments and when the developments in the mathematical sciences that they had made began to filter into Europe during the twelfth century scientific renaissance those instruments also began to become known in Europe. For example, the earliest astrolabes to appear in Europe were on the Iberian Peninsula, whilst it was still under Islamic occupation.  

Canterbury Astrolabe Quadrant 1388 Source Wikimedia Commons
Astrolabe of Jean Fusoris, made in Paris, 1400 Source: Wikimedia Commons

The medieval period in Europe saw a gradual increase in the use of scientific instruments, both imported and locally manufactured, but the use was still comparatively low level. There was some innovation, for example the French Jewish scholar, Levi ben Geshon (1288–1344), published the first description of the cross staff or Jacob’s staff, used in astronomy, surveying, and navigation, in his Book of the Wars of the Lord (originally in Hebrew but also translated into Latin). 

…of a staff of 4.5 feet (1.4 m) long and about one inch (2.5 cm) wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars

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

Also, the magnetic compass came into use in Europe in the twelfth century, first mentioned by Alexander Neckam (1157–1217) in his De naturis rerum at the end of the century.

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

Petrus Pereginus (fl. 1269) gave detailed descriptions of both the floating compass and the dry compass in his Epistola de magnete

However, it was first in the Renaissance that a widespread and thriving culture of scientific instrument design, manufacture, and usage really developed. The steep increase in scientific instrument culture was driving by the substantial parallel developments in astronomy, navigation, surveying, and cartography that began around fourteen hundred that I have already outlined in previous episodes of this series. Renaissance scientific instrument culture is too large a topic to cover in detail in one blog post, so I’ll only do a sketch of some major points and themes with several links to other earlier related posts.

Already, the first Viennese School of Mathematics, which was heavily involved in the development of both astronomy and cartography was also a source of scientific instrument design and manufacture.Johannes von Gmunden (c. 1380–1442) had a notable collection of instruments including an Albion, a multipurpose instrument conceived by Richard of Wallingford (1292–1336).

Albion front side Source: Seb Falk’s Twitter feed
Albion rear Source: Seb Falk’s Twitter feed

Georg von Peuerbach (1423–1461) produced several instruments most notably the earliest portable sundial marked for magnetic declination.

Folding sundial by Georg von Peuerbach

His pupil Regiomontanus (1436–1476) wrote a tract on the construction and use of the astrolabe and there is an extant instrument from 1462 dedicated to Cardinal Bessarion and signed IOHANNES, which is assumed to have been made by him. At least eleven other Regiomontanus style astrolabes from the fifteenth century are known.

Regiomontanus style astrolabe Source: Wikimedia Commons

Elements of his design were adopted by both Johannes Stöffler (1452–1531), the first professor of astronomy at the University of Tübingen, and by the Nürnberger mathematicus Georg Hartmann (1489–1564).

Stöffler also made celestial globes and an astronomical clock.

Celestial Globe, Johannes Stöffler, 1493; Landesmuseum Württemberg Source: Wikimedia Commons

Mechanical astronomical clocks began to emerge in Europe in the fourteenth century, but it would not be until the end of the sixteenth century that mechanical clocks became accurate enough to be used as scientific instruments. The earliest clockmaker, who reached this level of accuracy being the Swiss instrument maker, Jost Bürgi (1552–1632)

Bürgi made numerous highly elaborate and very decorative mechanical clocks, mechanised globes and mechanised armillary spheres that were more collectors items for rich patrons rather than practical instruments.

Bürgi Quartz Clock 1622-27
Source: Swiss Physical Society

This illustrates another driving force behind the Renaissance scientific instrument culture. The Renaissance mathematicus rated fairly low in the academical hierarchy, actually viewed as a craftsman rather than an academic. This made finding paid work difficult and they were dependent of rich patrons amongst the European aristocracy. It became a standard method of winning the favour of a patron to design a new instrument, usually a modification of an existing one, making an elaborate example of it and presenting it to the potential patron. The birth of the curiosity cabinets, which often also included collections of high-end instruments was also a driving force behind the trend. Many leading instrument makers produced elaborate, high-class instruments for such collections. Imperial courts in Vienna, Prague, and Budapest employed court instrument makers. For example, Erasmus Habermel (c. 1538–1606) was an incredibly prolific instrument maker, who became instrument maker to Rudolf II. A probable relative Josua Habermel (fl. 1570) worked as an instrument maker in southern Germany, eventually moving to Prague, where he probably worked in the workshop of Erasmus.

 1594 armillary sphere by Erasmus Habermel of Prague.

Whereas from Theon onwards, astrolabes were unique, individual, instruments, very often beautiful ornaments as well as functioning instruments, Georg Hartmann was the first instrument maker go into serial production of astrolabes. Also, Hartmann, although he didn’t invent them, was a major producer of printed paper instruments. These could be cut out and mounted on wood to produce cheap, functional instruments for those who couldn’t afford the expensive metal ones. 

Hartmann astrolabe front
Hartmann astrolabe rear
Paper and Wood Astrolabe Hartmann Source: HSM Oxford

Hartmann lived and worked in Nürnberg, which as I have sketched in an earlier post, was for more than a century the scientific instrument capital of Europe with a massive produce of instruments of all sorts.

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

As well as astrolabes and his paper instruments Hartmann also produced printed globes, none of which have survived. Another Nürnberger mathematicus, Johannes Schöner (1477–1547) launched the printed pairs of terrestrial and celestial globes onto the market.

Celestial Globe by Johannes Schöner c. 1534 Source

His innovation was copied by Gemma Frisius (1508–1555), whose student Gerard Mercator (1512–1594) took up globe making on a large scale, launching the seventeenth century Dutch globe making industry. 

Gemma Frisius set up a workshop producing a range of scientific instruments together with his nephew (?) Gualterus Arsenius (c. 1530–c. 1580).  

Astronomical ring dial Gualterus Arsenius Source

In France, Oronce Fine (1494–1555), a rough contemporary, who was appointed professor at the Collège Royal, was also influenced by Schöner in his cartography and like the Nürnberger was a major instrument maker. In Italy, Egnatio Danti (1536–1586) the leading cosmographer was also the leading instrument maker. 

Egnation Danti, Astrolabe, ca. 1568, brass and wood. Florence, Museo di Storia della Scienza Source: Fiorani The Marvel of Maps p. 49

A major change during the Renaissance was the emergence, for the first time in Early Modern Europe, of large-scale astronomical observatories, Wilhelm IV (1532–1592) in Hessen-Kassel beginning in about 1560 and Tycho Brahe (1546–1601) on the Island of Hven beginning in 1575. Both men commissioned new instruments, many of which were substantially improved in comparison with their predecessors from antiquity.

Sternwarte im Astronomisch-Physikalischen Kabinett, Foto: MHK, Arno Hensmanns Reconstruction of Wilhelm’s observatory
Tycho Brahe, Armillary Sphere, 1581 Source
Tycho Brahe quadrant

Their lead was followed by others, the first Vatican observatory was established in the Gregorian Tower in 1580.

View on the Tower of Winds (Gregorian tower) in Vatican City (with the dome of Saint Peter’s Basilica in the background). Source: Wikimedia Commons

In the early seventeenth century, Leiden University in Holland established the first European university observatory and Christian Longomontanus (1562–1647), who had been Tycho’s chief assistant, established a university observatory in Copenhagen 

Drawing of Leiden Observatory in 1670, seen on top of the university building. Source: Wikimedia Commons
Copenhagen University Observatory Source: Wikimedia Commons

As in all things mathematical England lagged behind the continent but partial filled the deficit by importing instrument makers from the continent, the German Nicolas Kratzer (c. 1487–1550) and the Netherlander Thomas Gemini (c. 1510–1562). The first home grown instrument maker was Humfrey Cole (c. 1530­–1591). By the end of the sixteenth century, led by John Dee (1527–c. 1608), who studied in Louven with Frisius and Mercator, and Leonard Digges (c. 1515–c. 1559), a new generation of English instrument makers began to dominate the home market. These include Leonard’s son Thomas Digges (c. 1546–1595), William Bourne (c. 1535–1582), John Blagrave (d. 1611), Thomas Blundeville (c. 1522–c. 1606), Edward Wright (1561–1615), Emery Molyneux (d. 1598), Thomas Hood (1556–1620), Edmund Gunter (1581–1626) Benjamin Cole (1695–1766), William Oughtred (1574–1660), and others.

The Renaissance also saw a large amount of innovation in scientific instruments. The Greek and Chinese armillary spheres were large observational instruments, but the Renaissance armillary sphere was a table top instrument conceived to teach the basic of astronomy.

Armillary Sphere by Carlo Plato, Rome, 1588 Museum of the History of Science

In navigation the Renaissance saw the invention various variations of the backstaff, to determine solar altitudes.

Davis quadrant (backstaff), made in 1765 by Johannes Van Keulen. On display at the Musée national de la Marine in Paris. Source: Wikimedia Commons

Also new for the same purpose was the mariner’s astrolabe.

Mariner’s Astrolabe c. 1600 Source: Wikimedia Commons

Edmund Gunter (1581–1626) invented the Gunter scale or rule a multiple scale (logarithmic, trigonometrical) used to solve navigation calculation just using dividers.

Gunter scale front
Gunter scale back Source

William Oughtred (1574–1660) combined two Gunter scales to produce the slide rule.

New in surveying were the surveyor’s chain,

A Gunter chain photographed at Campus Martius Museum. Source: Wikimedia Commons

the plane table,

Surveying with plane table and surveyor’s chain

the theodolite

Theodolite 1590 Source:

and the circumferentor.

18th century circumferentor

All of which were of course also used in cartography. Another Renaissance innovation was sets of drawing instruments for the cartographical, navigational etc draughtsmen.

Drawing instruments Bartholomew Newsum, London c. 1570 Source

The biggest innovation in instruments in the Renaissance, and within its context one of the biggest instrument innovation in history, were of course the telescope and the microscope, the first scientific instruments that not only aided observations but increased human perception enabling researchers to perceive things that were previously hidden from sight. Here is a blog post over the complex story of the origins of the telescope and one over the unclear origins of the microscope.

The Renaissance can be viewed as the period when instrumental science began to come of age. 

[1] The information on Ptolemaios’ instruments and the diagrams are taken from Ptolemy’s Almagest, translated and annotated by G. J. Toomer, Princeton Paperbacks, 1998


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

God made all things by measure, number and weight[1]

God made all things by measure, number and weight[1]

The first history of science, history of mathematics book I ever read was Lancelot Hogben’s Man Must Measure: The Wonderful World of Mathematics, when I was about six years old.

It almost certainly belonged to my older brother, who was six years older than I. This didn’t matter, everybody in our house had books and everybody could and did read everybody’s books. We were a household of readers. I got my first library card at three; there were weekly family excursions to the village library. But I digress.

It is seldom, when people discuss the history of mathematics for them to think about how or where it all begins. It begins with questions like how much? How many? How big? How small? How long? How short? How far? How near? All of these questions imply counting, comparison, and measurement. The need to quantify, to measure lies at the beginning of all systems of mathematics. The histories of mathematics, science, and technology all have a strong stream of mensuration, i.e., the act or process of measuring, running through them. Basically, without measurement they wouldn’t exist. 

Throughout history measuring and measurement have also played a significant role in politics, often leading to political disputes. In modern history there have been at least three well known cases. The original introduction of the metric system during the French revolution, the battle of the systems, metric contra imperialism, during the nineteenth century, and most recently the bizarre wish of the supporters of Brexit to reintroduce the imperial system into the UK in their desire to distance themselves as far as possible from the evil EU. 

It was with some anticipation that I greeted the news that James Vincent had written and published Beyond MeasureThe Hidden History of Measurement.[2] Vincent’s book is not actually a history of measurement on a nuts and bolts level i.e., systems of measurement, units of measurement and so on, but what I would call a social history of the uses of measurement. This is not a negative judgement; some parts of the book are excellent exactly because it is about the use and abuse of methods of measurement rather than the systems of measurement themselves.

Although roughly chronological, the book is not a systematic treatment of the use of measurement from the first group of hunter gatherers, who tried to work out an equitable method of dividing the spoils down to the recent redefinition of the kilogram in the metric system. The latter being apparently the episode that stimulated Vincent into writing his book. Such a volume would have to be encyclopaedic in scope, but is rather an episodic examination of various passages in the history of mensuration. 

The first episode or chapter takes a rather sweeping look at what the author sees as the origins of measurement in the early civilisations of Egypt and Babylon. Whilst OK in and of itself, what about other cultures, civilisations, such as China or India just to mention the most obvious. This emphasises something that was already clear from the introduction this is the usual predominantly Eurocentric take on history. 

The second chapter moves into the realm of politics and the role that measurement has always played in social order, with examples from all over the historical landscape. Measurement as a tool of political control. This demonstrates one of the strengths of Vincent’s socio-political approach. Particularly, his detailed analysis of how farmers, millers, and tax collectors all used different tricks to their advantage when measuring grain and the regulation that as a result were introduced is fascinating.

Vincent is, however, a journalist and not a historian and is working from secondary sources and in the introduction, we get the first of a series of really bad takes on the history of science that show Vincent relying on myths and clichés rather than doing proper research. He delivers up the following mess:

Consider, for example, the unlikely patron saint of patient measurement that is the sixteenth-century Danish nobleman Tycho Brahe. By most accounts Brahe was an eccentric, possessed of a huge fortune (his uncle Jørgen Brahe was one of the wealthiest men in the country), a metal nose (he lost the original in a duel), and a pet elk (which allegedly died after drinking too much beer and falling down the stairs of one of his castles). After witnessing the appearance of a new star in the night sky in 1572, one of the handful of supernovae ever seen in our galaxy, Brahe devoted himself to astronomy.

Tycho’s astronomical work was financed with his apanage from the Danish Crown, as a member of the aristocratical oligarchy that ruled Denmark. His uncle Jørgen, Vice-Admiral of the Danish navy, was not wealthier than Tycho’s father or his independently wealthy mother. Tycho had been actively interested in astronomy since 1560 and a serious astronomer since 1563, not first after observing the 1572 supernova.

After describing Tycho’s observational activities, Vincent writes:

It was the data collected here that would allow Brahe’s apprentice, the visionary German astronomer Johannes Kepler, to derive the first mathematical laws of planetary motion which correctly described the elliptical orbits of the planets…

I don’t know why people can’t get Kepler’s status in Prague right. He was not Tycho’s apprentice. He was thirty years old, a university graduate, who had studied under Michael Mästlin one of the leading astronomers in Europe. He was the author of a complex book on mathematical astronomy, which is why Tycho wanted to employ him. He was Tycho’s colleague, who succeeded him in his office as Imperial Mathematicus. 

It might seem that I’m nit picking but if Vincent can’t get simple history of science facts right that he could look up on Wikipedia, then why should the reader place any faith in the rest of what he writes?

The third chapter launches its way into the so-called scientific revolution under the title, The Proper Subject of Measurement. Here Vincent selectively presents the Middle Ages in the worst possible anti-science light, although he does give a nod to the Oxford Calculatores but of course criticises them for being purely theoretical and not experimental. In Vincent’s version they have no predecessors, Philoponus or the Arabic scholars, and no successors, the Paris physicists. He then moves into the Renaissance in a section titled Measuring art, music, and time. First, we get a brief section on the introduction of linear perspective. Here Vincent, first, quoting Alberti, tells us:

I set this up between the eye and the object to be represented, so that the visual pyramid passes through the loose weave of the veil.

The ‘visual pyramid’ described by Alberti refers to medieval theories of optics. Prior to the thirteenth century, Western thinkers believed that vision worked via ‘extramission,’ with the eye emitting rays that interacted with the world like a ‘visual finger reaching out to palpated things’ (a mechanism captured by the Shakespearean imperative to ‘see feelingly’). Thanks largely to the work of the eleventh-century Arabic scholar Ibn al-Haytham, known in the West as Alhacen, this was succeeded by an ‘intromisionist’ explanation, which reverses the causality so that it is the eye that receives impressions from reality. It’s believed that these theories informed the work of artists like Alberti, encouraging the geometrical techniques of the perspective grids and creating a new incentive to divide the world into spatially abstract units.

Here, once again, we have Vincent perpetuating myths because he hasn’t done his homework. The visual pyramid is, of course, from Euclid and like the work of the other Greek promoters of geometrical optics was indeed based on an extramission theory of vision. As I have pointed out on numerous occasions the Greeks actually had both extramission and intromission theories of vision, as well as mixed models. Al-Haytham’s great achievement was not the introduction of an intromission theory, but was in showing that an intromission theory was compatible with the geometrical optics, inclusive visual pyramid, of Euclid et al. The geometrical optics of Alberti and other perspective theorists is pure Euclidian and does not reference al-Haytham. In fact, Alberti explicitly states that it is irrelevant whether the user of his system of linear perspective believes in an extramission or an intromission theory of vision. 

Linear perspective is followed by a two page romp through the medieval invention of musical notation before turning to the invention of the mechanical clock. Here, Vincent makes the standard error of over emphasising the influence of the mechanical clock in the early centuries after its invention and introduction. 

Without mentioning Thomas Kuhn, we now get a Kuhnian explanation of the so-called astronomical revolution, which is wonderfully or should that be horrifyingly anachronistic:

This model [the Aristotelian geocentric one] sustained its authority for centuries, but close observation of the night skies using increasingly accurate telescopes [my emphasis] revealed discrepancies. These were changes that belied its immutable status and movements that didn’t fit the predictions of a simple geocentric universe. A lot of work was done to make the older models account for such eccentricities, but as they accrued mathematical like sticky notes, [apparently sticky notes are the 21st century version of Kuhnian ‘circles upon circles’] doubts about their veracity became unavoidable. 

Where to begin with what can only be described as a clusterfuck. The attempts to reform the Aristotelian-Ptolemaic geocentric model began at the latest with the first Viennese School of Mathematics in the middle of the fifteenth century, about one hundred and fifty years before the invention of the telescope. Those reform attempts began not because of any planetary problems with the model but because the data that it delivered was inaccurate. Major contributions to the development of a heliocentric model such as the work of Copernicus and Tycho Brahe, as well as Kepler’s first two laws of planetary motion also all predate the invention of the telescope. Kepler’s third law is also derived from pre-telescope data. The implication that the geocentric model collapsed under the weight of ad hoc explanation (the sticky notes) was Kuhn’s explanation for his infamous paradigm change and is quite simply wrong. I wrote 52 blog posts explaining what really happened, I’m not going to repeat myself here.

We now get the usual Galileo hagiography for example Vincent tells us: 

It was Galileo who truly mathematised motion following the early attempts of the Oxford Calculators, using practical experiments to demonstrate flaws in Aristotelian wisdom.

Vincent ignores the fact that Aristotle’s concepts of motion had been thrown overboard long before and completely ignores the work of sixteenth century mathematicians, such as Tartaglia and Benedetti. 

He then writes:

In one famous experiment he dropped cannonballs and musket balls from the Leaning Tower of Pisa (an exercise that likely never took place in the way Galileo claims [my emphasis]) and showed that, contra to Aristotle, objects accelerate at a uniform rate, not proportionally to their mass.

Galileo never claimed to have dropped anything from the Leaning Tower, somebody else said that he had and if it ‘never took place’, why fucking mention it?

Now the telescope:

From 1609, Galileo’s work moved to a new plane itself. Using home-made telescopes he’d constructed solely by reading descriptions of the device…

The myth, created by himself, that Galileo had never seen a telescope before he constructed one has been effectively debunked by Mario Biagioli. This is followed by the usually one man circus claims about the telescopic discoveries, completely ignoring the other early telescope observers. Copernicus and Kepler now each get a couple of lines before we head off to Isaac Newton. Vincent tells us that Newton devised the three laws of motion and the universal law of gravitation. He didn’t he took them from others and combined them to create his synthesis.

The fourth chapter of the book is concerned with the invention of the thermometer and the problems of creating accurate temperature scales. This chapter is OK, however, Vincent is a journalist and not a historian and relies on secondary sources written by historians. There is nothing wrong with this, it’s how I write my blog posts. In this chapter his source is the excellent work of Hasok Chang, which I’ve read myself and if any reader in really interested in this topic, I recommend that they read Chang rather than Chang filtered by Vincent. Once again, we have some very sloppy pieces of history of science, Vincent writes: 

Writing in 1693, the English astronomer Edmond Halley, discoverer of the eponymous comet…”

Just for the record, Halley was much more than just an astronomer, he could for example have been featured along with Graunt in chapter seven, see below. It is wrong to credit Halley with the discovery of Comet Halley. The discoverer is the first person to observe a comet and record that observation. Comet Halley had been observed and recorded many times throughout history and Halley’s achievement was to recognise that those observations were all of one and the same comet.

 A few pages further on Vincent writes: 

Unlike caloric, phlogiston had mass, but Lavoisier disproved this theory, in part by showing how some substances gain weight when burned. (This would eventually lead to the discovery of oxygen as the key element in combustion.) [my emphasis]

I can hear both Carl Scheele and Joseph Priestley turning in their graves. Both of them discovered oxygen, independently of each other; Scheele discovered it first bur Priestly published first, and both were very much aware of its role in combustion and all of this well before Lavoisier became involved. 

Chapter five is dedicated to the introduction of the metric system in France correctly giving more attention to the political aspects than the numerical ones. Once again, an excellent chapter.

Chapter six which deals primarily with land surveying had a grandiose title, A Grid Laid Across the World, but is in fact largely limited to the US. Having said that it is a very good and informative chapter, which explains how it came about that the majority of US towns and properties are laid out of a unified rectangular grid system. Most importantly it explains how the land grant systems with its mathematical surveying was utilised to deprive the indigenous population of their traditional territories. The chapter closes with a brief more general look at how mathematical surveying and mapping played a significant role in imperialist expansion, with a very trenchant quote from map historian, Matthew Edney, “The empire exists because it can be mapped; the meaning of empire is inscribed into each map.”

Unfortunately, this chapter also contains some more sloppy history of science, Vincent tells us:

In such a world, measurement of the land was of the utmost importance. As a result, sixteenth-century England gave rise to one of the most widely used measuring tools in the world: the surveyor’s chain, or Gunter’s chain, named after its inventor the seventeenth-century English priest and mathematician Edmund Gunter. 

Sixteenth or seventeenth century? Which copy editor missed that one? It’s actually a bit of a problem because Gunter’s life starts in the one century and ends in the other, 1581–1626. However, we can safely say that he produced his chain in the seventeenth century. Vincent makes the classic error of attributing the invention of the surveyors’ chain to Gunter, to quote myself from my blog post on Renaissance surveying:

In English the surveyor’s chain is usually referred to as Gunter’s chain after the English practical mathematician Edmund Gunter (1581–1626) and he is also often referred to erroneously as the inventor of the surveyor’s chain but there are references to the use of the surveyor’s chain in 1579, before Gunter was born. 

Even worse he writes:

Political theorist Hannah Arendt described the work of surveying and mapping that began with the colonisation of America as one of three great events that ‘stand at the threshold of the modern age and determine its character’ (the other two being the Reformation of the Catholic Church and the cosmological revolution begun by Galileo) [my emphasis]

I don’t know whether to attribute this arrant nonsense to Arendt or to Vincent. Whether he is quoting her or made this up himself he should know better, it’s complete bullshit. Whatever Galileo contributed to the ‘cosmological revolution,’ and it’s much, much less than is often claimed, he did not in anyway begin it. Never heard of Copernicus, Tycho, Kepler, Mr Vincent? Oh yes, you talk about them in chapter three!

Chapter seven turns to population statistics starting with the Royal Society and John Graunt’s Natural and Political Observations Made Upon the Bills of Mortality. Having dealt quite extensively with Graunt, with a nod to William Petty, but completely ignoring the work of Caspar Neumann and Edmond Halley, Vincent now gives a brief account of the origins of the new statistics. Strangely attributing this entirely to the astronomers, completely ignoring the work on probability in games of chance by Cardano, Fermat, Pascal, and Christian Huygens. He briefly mentions the work of Abraham de Moivre but ignores the equally important, if not more important work of Jacob Bernoulli. He now gives an extensive analysis of Quetelet’s application of statistics to the social sciences. Quetelet, being an astronomer, is Vincent’s reason d’être for claiming that it was astronomers, who initial developed statistics and not the gamblers. Quetelet’s the man who gave us the ubiquitous body mass index. The chapter then closes with a good section on the abuses of statistics in the social sciences, first in Galton’s eugenics and secondly in the misuse of IQ tests by Henry Goddard. All in all, one of the good essays in the book

Continuing the somewhat erratic course from theme to theme, the eighth chapter addresses what Vincent calls The Battle of the Standards: Metric vs Imperial and metrology’s culture war. A very thin chapter, more of a sketch that an in-depth analysis, which gives as much space to the post Brexit anti-metric loonies, as to the major debates of the nineteenth century. This is mainly so that Vincent can tell the tale of his excursion with said loonies to deface street signs as an act of rebellion. 

In the ninth chapter, Vincent turns his attention to replacement of arbitrary definitions of units of measurement with definitions based on constants of nature, with an emphasis on the recent new definition of the kilogram. At various point in the book, Vincent steps out from his role of playing historian and presents himself in the first person as an investigative journalist, a device that I personally found irritating. In this chapter this is most pronounced. He opens with, “On a damp but cheerful Friday in November 2018, I travelled to the outskirts of Paris to witness the overthrow of a king.” He carries on in the same overblown style finally revealing that he, as a journalist was attending the conference officially launching the redefining of the kilogram, going on to explain in equally overblown terms how the kilogram was originally defined. The purple prose continues with the introduction of another attendee, his acquaintance, the German physicist, Stephan Schlamminger:

Schlamminger is something of a genius loci of metrology: an animating spirit full of cheer and knowledge, as comfortable in the weights and measures as a fire in a heath. He is also a key player in the American team that helped create the kilogram’s new definition. I’d spoken to him before, but always delighted in his enthusiasm and generosity. ‘James, James, James,’ he says in a rapid-fire German accent as he beckoned me to join his group. ‘Welcome to the party.’

We then get a long, overblown speech by Schlamminger about the history of the definitions in the metric system ending with an explanation, as to why the kilogram must be redefined.

This is followed by a long discourse over Charles Sanders Peirce and his attempts to define the metre using the speed of light, which failed. Vincent claims that Peirce was the first to attempt to attempt to define units of measurement using constants of nature, a claim that I find dubious, but it might be right. This leads on to Michelson and Morley defining the metre using the wavelength of sodium light, a definition that in modified form is still used today. The chapter closes with a long, very technical, and rather opaque explanation of the new definition of the kilogram based on Planck’s constant, h

The final chapter of Vincent’s book is a sociological or anthropological mixed basket of wares under the title The Managed Life: Measurements place in modern society in our understanding of ourselves, which is far too short to in anyway fulfil its grandiose title.

The book closes with an epilogue that left me simply baffled. He tells a personal story about how he came to listen to Beethoven’s Ninth Symphony only when he had a personal success in his life and through this came to ruin his enjoyment of the piece. Despite his explanation I fail to see what the fuck this has to do with measurement.

The book has a rather small, random collection of colour prints, related to various bits of the text, in the middle. There are extensive endnotes relating bits of the text to there bibliographical sources, but no separate bibliography, and an extensive index.

I came away feeling that there is a good book contained in Vincent’s tome, struggling to get out. However, there is somehow too much in the way for it to emerge. Some of the individual essays are excellent and I particularly liked his strong emphasis on some of the negative results of applying systems of measurement. People reading this review might think that I, as a historian of science, have placed too much emphasis on his truly shoddy treatment of that discipline; ‘the cosmological revolution begun by Galileo,’ I ask you? However, as I have already stated if we can’t trust his research in this area, how much can we trust the rest of his work?

[1] Wisdom of Solomon 11:20

[2] James Vincent, Beyond MeasureThe Hidden History of Measurement, Faber & Faber, London, 2022


Filed under Book Reviews, History of Astronomy, History of Mathematics, History of science


Due to the impact of Isaac Newton and the mathematicians grouped around him, people often have a false impression of the role that England played in the history of the mathematical sciences during the Early Modern Period. As I have noted in the past, during the late medieval period and on down into the seventeenth century, England in fact lagged seriously behind continental Europe in the development of the mathematical sciences both on an institutional level, principally universities, and in terms of individual mathematical practitioners outside of the universities. Leading mathematical practitioners, working in England in the early sixteenth century, such as Thomas Gemini (1510–1562) and Nicolas Kratzer (1486/7–1550) were in fact immigrants, from the Netherlands and Germany respectively.

In the second half of the century the demand for mathematical practitioners in the fields of astrology, astronomy, navigation, cartography, surveying, and matters military was continually growing and England began to produce some home grown talent and take the mathematical disciplines more seriously, although the two universities, Oxford and Cambridge still remained aloof relying on enthusiastic informal teachers, such as Thomas Allen (1542–1632) rather than instituting proper chairs for the study and teaching of mathematics.

Outside of the universities ardent fans of the mathematical disciplines began to establish the so-called English school of mathematics, writing books in English, giving tuition, creating instruments, and carrying out mathematical tasks. Leading this group were the Welsh man, Robert Recorde (c. 1512–1558), who I shall return to in a later post, John Dee (1527–c. 1608), who I have dealt with in several post in the past, one of which outlines the English School, other important early members being, Dee’s friend Leonard Digges, and his son Thomas Digges (c. 1446–1595), who both deserve posts of their own, and Thomas Hood (1556–1620) the first officially appointed lecturer for mathematics in England.  I shall return to give all these worthy gentlemen, and others, the attention they deserve but today I shall outline the life and mathematical career of John Blagrave (d. 1611) a member of the landed gentry, who gained a strong reputation as a mathematical practitioner and in particular as a designer of mathematical instruments, the antiquary Anthony à Wood (1632–1695), author of Athenae Oxonienses. An Exact History of All the Writers and Bishops, who Have Had Their Education in the … University of Oxford from the Year 1500 to the End of the Year 1690, described him as “the flower of mathematicians of his age.”

John Blagrave was the second son of another John Blagrave of Bullmarsh, a district of Reading, and his wife Anne, the daughter of Sir Anthony Hungerford of Down-Ampney, an English soldier, sheriff, and courtier during the reign of Henry VIII, John junior was born into wealth in the town of Reading in Berkshire probably sometime in the 1560s. He was educated at Reading School, an old established grammar school, before going up to St John’s College Oxford, where he apparently acquired his love of mathematics. This raises the question as to whether he was another student, who benefitted from the tutoring skills of Thomas Allen (1542–1632). He left the university without graduating, not unusually for the sons of aristocrats and the gentry. He settled down in Southcot Lodge in Reading, an estate that he had inherited from his father and devoted himself to his mathematical studies and the design of mathematical instruments. He also worked as a surveyor and was amongst the first to draw estate maps to scale.

Harpsden a small parish near Henley-on-Thames Survey by John Blagrave 1589 Source

There are five known surviving works by Blagrave and one map, as opposed to a survey, of which the earliest his, The mathematical ievvel, from1585, which lends its name to the title of this post, is the most famous. The full title of this work is really quite extraordinary:


Shewing the making, and most excellent vse of a singuler Instrument So called: in that it performeth with wonderfull dexteritie, whatsoever is to be done, either by Quadrant, Ship, Circle, Cylinder, Ring, Dyall, Horoscope, Astrolabe, Sphere, Globe, or any such like heretofore deuised: yea or by most Tables commonly extant: and that generally to all places from Pole to Pole. 

The vse of which Ievvel, is so aboundant and ample, that it leadeth any man practising thereon, the direct pathway (from the first steppe to the last) through the whole Artes of Astronomy, Cosmography, Geography, Topography, Nauigation, Longitudes of Regions, Dyalling, Sphericall triangles, Setting figures, and briefely of whatsoeuer concerneth the Globe or Sphere: with great and incredible speede, plainenesse, facillitie, and pleasure:

The most part newly founde out by the Author, Compiled and published for the furtherance, aswell of Gentlemen and others desirous or Speculariue knowledge, and priuate practise: as also for the furnishing of such worthy mindes, Nauigators,and traueylers,that pretend long voyages or new discoueries: By John Blagave of Reading Gentleman and well willer to the Mathematickes; Who hath cut all the prints or pictures of the whole worke with his owne hands. 1585•

Dig the spelling!
Title Page Source Note the title page illustration is an  armillary sphere and not the Mathematical Jewel

Blagrave’s Mathematical Jewel is in fact a universal astrolabe, and by no means the first but probably the most extensively described. The astrolabe is indeed a multifunctional instrument, al-Sufi (903–983) describes over a thousand different uses for it, and Chaucer (c. 1340s–1400) in what is considered to be the first English language description of the astrolabe and its function, a pamphlet written for a child, describes at least forty different functions. However, the normal astrolabe has one drawback, the flat plates, called tympans of climata, that sit in the mater and are engraved with the stereographic projection of a portion of the celestial sphere are limited in their use to a fairly narrow band of latitude, meaning that if one wishes to use it at a different latitude you need a different climata. Most astrolabes have a set of plates each engraved on both side for a different band of latitude. This problem led to the invention of the universal astrolabe.

Full-page figure of the rete of Blagrave’s Jewel (Peterborough A.8.13) For more illustration from The Mathematical Jewel go here

The earliest known universal astrolabes are attributed to Abū Isḥāq Ibrāhīm ibn Yaḥyā al-Naqqāsh al-Zarqālī al-Tujibi (1029-1100), known simply as al-Zarqālī and in Latin as Arzachel, an Arabic astronomer, astrologer, and instrument maker from Al-Andalus, and another contemporary Arabic astronomer, instrument maker from Al-Andalus, Alī ibn Khalaf: Abū al‐Ḥasan ibn Aḥmar al‐Ṣaydalānī or simply Alī ibn Khalaf, about whom very little is known. In the Biographical Encyclopedia of Astronomers (Springer Reference, 2007, pp. 34-35) Roser Puig has this to say about the two Andalusian instrument makers: 

ʿAlī ibn Khalaf is the author of a treatise on the use of the lámina universal (universal plate) preserved only in a Spanish translation included in the Libros del Saber de Astronomía (III, 11–132), compiled by the Spanish King Alfonso X. To our knowledge, the Arabic original is lost. ʿAlī ibn Khalaf is also credited with the construction of a universal instrument called al‐asṭurlāb al‐maʾmūnī in the year 1071, dedicated to al‐Maʾmūn, ruler of Toledo. 

The universal plate and the ṣafīḥa (the plate) of Zarqalī (devised in 1048) are the first “universal instruments” (i.e., for all latitudes) developed in Andalus. Both are based on the stereographic meridian projection of each hemisphere, superimposing the projection of a half of the celestial sphere from the vernal point (and turning it) on to the projection of the other half from the autumnal point. However, their specific characteristics make them different instruments.

Al-Zarqālī’s universal astrolabe was known as the Azafea in Arabic and as the Saphaea in Europe.

A copy of al-Zarqālī’s astrolabe Source: Wikimedia Commons

Much closer to Blagrave’s time, Gemma Frisius (1508–1555) wrote about a universal astrolabe, published as the Medici ac Mathematici de astrolabio catholico liber quo latissime patientis instrumenti multiplex usus explicatur, in 1556. Better known than Frisius’ universal instrument was that of his one-time Spanish, student Juan de Rojas y Samiento (fl. 1540-1550) published in his Commentariorum in Astrolabium libri sex in 1551.


Although he never really left his home town of Reading and his work was in English, Blagrave, like the other members of the English School of Mathematics, was well aware of the developments in continental Europe and he quotes the work of leading European mathematical practitioners in his Mathematical Jewel, such as the Tübingen professor of mathematics, Johannes Stöffler (1452–1531), who wrote a highly influential volume on the construction of astrolabes, his Elucidatio fabricae ususque astrolabii originally published in 1513, which went through 16 editions up to 1620

or the works of Gemma Frisius, who was possibly the most influential mathematical practitioner of the sixteenth century. Blagrave’s Mathematical Jewel was based on Gemma Frisius astrolabio catholico.

Blagrave’s Mathematical Jewel was obviously popular because Joseph Moxon (1627–1691), England first specialist mathematical publisher, cartographer, instrument, and globe maker republished it under the title:

The catholique planisphaer which Mr. Blagrave calleth the mathematical jewel briefly and plainly discribed in five books : the first shewing the making of the instrument, the rest shewing the manifold vse of it, 1. for representing several projections of the sphere, 2. for resolving all problemes of the sphere, astronomical, astrological, and geographical, 4. for making all sorts of dials both without doors and within upon any walls, cielings, or floores, be they never so irregular, where-so-ever the direct or reflected beams of the sun may come : all which are to be done by this instrument with wonderous ease and delight : a treatise very usefull for marriners and for all ingenious men who love the arts mathematical / by John Palmer … ; hereunto is added a brief description of the cros-staf and a catalogue of eclipses observed by the same I.P.

Engraved frontispiece to John Palmer (ed.), ‘The Catholique Planispaer, which Mr Blagrave calleth the Mathematical Jewel’ (London, Joseph Moxon, 1658); woman, wearing necklace, bracelet, jewels in her hair, and a veil, and seated at a table, on which are a design of a mathematical sphere, a compass, and an open book; top left, portrait of John Blagrave, wearing a ruff; top right, portrait of John Palmer; top centre, an angel with trumpets.
Engraving David Loggan Source: British Museum

John Palmer (1612-1679), who was apparently rector of Ecton and archdeacon of Northampton, is variously described as the author or the editor of the volume, which was first published in 1658 and went through sixteen editions up to 1973.

Following The Mathematical Jewel, Blagrave published four further books on scientific instruments that we know of: 

Baculum Familliare, Catholicon sive Generale. A Booke of the making and use of a Staffe, newly invented by the Author, called the Familiar Staffe (London, 1590)

Astrolabium uranicum generale, a necessary and pleasaunt solace and recreation for navigators … compyled by John Blagrave (London, 1596)

An apollogie confirmation explanation and addition to the Vranicall astrolabe (London, 1597)

None of these survive in large numbers.

Blagrave also manufactured sundials and his fourth instrument book is about this: 

The art of dyalling in two parts (London, 1609)


Here there are considerably more surviving copies and even a modern reprint by Theatrum Orbis Terrarum Ltd., Da Capo Press, Amsterdam, New York, 1968.

People who don’t think about it tend to regard books on dialling, that is the mathematics of the construction and installation of sundials, as somehow odd. However, in this day and age, when almost everybody walks around with a mobile phone in their pocket with a highly accurate digital clock, we tend to forget that, for most of human history, time was not so instantly accessible. In the Early Modern period, mechanical clocks were few and far between and mostly unreliable. For time, people relied on sundials, which were common and widespread. From the invention of printing with movable type around 1450 up to about 1700, books on dialling constituted the largest genre of mathematical books printed and published. Designing and constructing sundials was a central part of the profession of mathematical practitioners. 

As well as the books there is one extant map:

Noua orbis terrarum descriptio opti[c]e proiecta secundu[m]q[ue] peritissimos Anglie geographos multis ni [sic] locis castigatissima et preceteris ipsiq[ue] globo nauigationi faciliter applcanda [sic] per Ioannem Blagrauum gen[er]osum Readingensem mathesibus beneuolentem Beniamin Wright Anglus Londinensis cµlator anno Domini 1596 

This is described as:

Two engraved maps, the first terrestrial, the second celestial (“Astrolabium uranicum generale …”). Evidently intended to illustrate Blagrave’s book “Astrolabium uranicum generale” but are not found in any copy of the latter.
The original is in the Bodleian Library.

When he died in 1611, Blagrave was buried in the St Laurence Church in Reading with a suitably mathematical monument. 

Blagrave is depicted surrounded by allegorical mathematical figures, with five women each holding the five platonic solids and Blagrave (in the center) depicted holding a globe and a quadrant.
The monument was the work of the sculptor Gerard Christmas (1576–1634), who later in life was appointed carver to the navy. It is not known who produced the drawing of the monument. 
Modern reconstruction of the armillary sphere from the cover of The Mathematical Jewel created by David Harber a descendent of John Blagrave

Blagrave was a minor, but not insignificant, participant in the mathematical community in England in the late sixteenth century. His work displays the typical Renaissance active interest in the practical mathematical disciplines, astronomy, navigation, surveying, and dialling. He seems to have enjoyed a good reputation and his Mathematical Jewel appears to have found a wide readership.  


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

The Wizard Earl’s mathematici 

In my recent post on the Oxford mathematician and astrologer Thomas Allen, I mentioned his association with Henry Percy, 9th Earl of Northumberland, who because of his strong interest in the sciences was known as the Wizard Earl.

HENRY PERCY, 9TH EARL OF NORTHUMBERLAND (1564-1632) by Sir Anthony Van Dyck (1599-1641). The ‘Wizard Earl’ was painted posthumously as a philosopher, hung in Square Room at Petworth. This is NT owned. via Wikimedia Commons

As already explained there Percy actively supported four mathematici, or to use the English term mathematical practitioners, Thomas Harriot (c. 1560–1621), Robert Hues (1553–1632), Walter Warner (1563–1643), and Nathaniel Torporley (1564–1632). Today, I’m going to take a closer look at them.

Thomas Harriot is, of course, the most well-known of the four; I have already written a post about him in the past, so I will only brief account of the salient point here.

Portrait often claimed to be Thomas Harriot (1602), which hangs in Oriel College, Oxford. Source: Wikimedia Commons

He graduatied from Oxford in 1580 and entered the service of Sir Walter Raleigh (1552–1618) in 1583. At Raleigh’s instigation he set up a school to teach Raleigh’s marine captains the newest methods of navigation and cartography, writing a manual on mathematical navigation, which contained the correct mathematical method for the construction of the Mercator projection. This manual was never published but we can assume he used it in his teaching. He was also directly involved in Raleigh’s voyages to establish the colony of Roanoke Island.

Sir Walter Ralegh in 1588 artist unknown. Source: Wikimedia Commons

In 1590, he left Raleigh’s service and became a pensioner of Henry Percy, with a very generous pension, the title to some land in the North of England, and a house on Percy’s estate, Syon House, in Middlesex.[1] Here, Harriot lived out his years as a research scientist with no obligations.

Syon House Attributed to Robert Griffier

After Harriot, the most significant of the Wizard Earl’s mathematici was Robert Hues. Like Harriot, Hues attended St Mary’s Hall in Oxford, graduating a couple of years ahead of him in 1578. Being interested in geography and mathematics, he was one of those who studied navigation under Harriot in the school set up by Raleigh, having been introduced to Raleigh by Richard Hakluyt (1553–1616), another student of Thomas Allen and a big promoter of English colonisation of North America.  

Hakluyt depicted in stained glass in the west window of the south transept of Bristol Cathedral – Charles Eamer Kempe, c. 1905. Source: Wikimedia Commons

Hues went on to become an experienced mariner. During a trip to Newfoundland, he came to doubt the published values for magnetic declination, the difference between magnetic north and true north, which varies from place to place.

In 1586, he joined with Thomas Cavendish (1560–1592), a privateer and another graduate of the Harriot school of navigation, who set out to raid Spanish shipping and undertake a circumnavigation of the globe, leaving Plymouth with three ships on 21 July. After the usual collection of adventures, they returned to Plymouth with just one ship on 9 September 1588, as the third ever ship to complete the circumnavigation after Magellan and Drake. Like Drake, Cavendish was knighted by Queen Elizabeth for his endeavours.

Thomas Cavendish An engraving from Henry Holland’s Herōologia Anglica (1620). Animum fortuna sequatur is Latin for “May fortune follow courage.” Source: Wikimedia Commons

Hues undertook astronomical observations throughout the journey and determined the latitudes of the places they visited. In 1589, he served with the mathematicus Edward Wright (1561–1615), who like Harriot worked out the correct mathematical method for the construction of the Mercator projection, but unlike Harriot published it in his Certaine Errors in Navigation in 1599.

Source: Wikimedia Commons

In August 1591, he set out once again with Cavendish on another attempted circumnavigation, also accompanied by the navigator John Davis (c. 1550–1605), another associate of Raleigh’s, known for his attempts to discover the North-West passage and his discovery of the Falkland Islands.

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

Cavendish died on route in 1592 and Hues returned to England with Davis in 1683. On this voyage Hues continued his astronomical observations in the South Atlantic and made determinations of compass declinations at various latitudes and the equator. 

Back in England, Hues published the results of his astronomical and navigational research in his Tractatus de globis et eorum usu (Treatise on Globes and Their Use, 1594), which was dedicated to Raleigh.

The book was a guide to the use of the terrestrial and celestial globes that Emery Molyneux (died 1598) had published in 1592 or 1593.

Molyneux CEltial Globe Middle Temple Library
A terrestrial globe by Emery Molyneux (d.1598-1599) is dated 1592 and is the earliest such English globe in existence. It is weighted with sand and made from layers of paper with a surface coat of plaster engraved with elaborate cartouches, fanciful sea-monsters and other nautical decoration by the Fleming Jodocus Hondius (1563-1611). There is a wooden horizon circle and brass meridian rings.

Molyneux belong to the same circle of mariners and mathematici, counting Hues, Wright, Cavendish, Davis, Raleigh, and Francis Drake (c. 1540–1596) amongst his acquaintances. In fact, he took part in Drake’s circumnavigation 1577–1580. These were the first globes made in England apparently at the suggestion of John Davis to his patron the wealthy London merchant William Sanderson (?1548–1638), who financed the construction of Molyneux’s globes to the tune of £1,000. Sanderson had sponsored Davis’ voyages and for a time was Raleigh’s financial manager. He named his first three sons Raleigh, Cavendish, and Drake.

Molyneux’s terrestrial globe was his own work incorporating information from his mariner friends and with the assistance of Edward Wright in plotting the coast lines. The circumnavigations of Drake and Cavendish were marked on the globe in red and blue line respectively. His celestial globe was a copy of the 1571 globe of Gerard Mercator (1512–1594), which itself was based on the 1537 globe of Gemma Frisius (1508–1555), on which Mercator had served his apprenticeship as globe maker. Molyneux’s globes were engraved by Jodocus Hondius (1563–1612), who lived in London between 1584 and 1593, and who would upon his return to the Netherlands would found one of the two biggest cartographical publishing houses of the seventeenth century.

Hues’ Tractatus de globis et eorum usu was one of four publications on the use of the globes. Molyneux wrote one himself, The Globes Celestial and Terrestrial Set Forth in Plano, published by Sanderson in 1592, of which none have survived. The London public lecturer on mathematics Thomas Hood published his The Vse of Both the Globes, Celestiall and Terrestriall in 1592, and finally Thomas Blundeville (c. 1522–c. 1606) in his Exercises containing six treatises including Cosmography, Astronomy, Geography and Navigation in 1594.

Hues’ Tractatus de globis has five sections the first of which deals with a basic description of and use of Molyneux’s globes. The second is concerned with matters celestial, plants, stars, and constellations. The third describes the lands, and seas displayed on the terrestrial globe, the circumference of the earth and degrees of a great circle. Part four contains the meat of the book and explains how mariners can use the globes to determine the sun’s position, latitude, course and distance, amplitudes and azimuths, and time and declination. The final section is a treatise, inspired by Harriot’s work on rhumb lines, on the use of the nautical triangle for dead reckoning. Difference of latitude and departure (or longitude) are two legs of a right triangle, the distance travelled is the hypotenuse, and the angle between difference of latitude and distance is the course. If any two elements are known, the other two can be determined by plotting or calculation using trigonometry.

The book was a success going through numerous editions in various languages. The original in Latin in 1593, Dutch in 1597, an enlarged and corrected Latin edition in 1611, Dutch again in 1613, enlarged once again in Latin in 1617, French in 1618, another Dutch edition in 1622, Latin again in 1627, English in 1638, Latin in 1659, another English edition also in 1659, and finally the third enlarged Latin edition reprinted in 1663. There were others.

The title page of Robert Hues (1634) Tractatvs de Globis Coelesti et Terrestri eorvmqve vsv in the collection of the Biblioteca Nacional de Portugal via Wikimedia Commons

Hues continued his acquaintance with Raleigh in the 1590s and was one of the executors of Raleigh’s will. He became a servant of Thomas Grey, 15th Baron Gray de Wilton (died 1614) and when Grey was imprisoned in the Tower of London for his involvement in a Catholic plot against James I & VI in 1604, Hues was granted permission to visit and even to stay with him in the Tower. From 1605 to 1621, Northumberland was also incarcerated in the Tower because of his family’s involvement in the Gunpowder Plot. Following Grey’s death Hues transferred his Tower visits to Northumberland, who paid him a yearly pension of £40 until his death in 1632.

He withdrew to Oxford University and tutored Henry Percy’s oldest son Algernon, the future 10th Earl of Northumberland, in mathematics when he matriculated at Christ’s Church in 1617.

Algernon Percy, 10th Earl of Northumberland, as Lord High Admiral of England, by Anthony van Dyck. Source: Wikimedia Commons

In 1622-23 he would also tutor the younger son Henry.

Oil painting on canvas, Henry Percy, Baron Percy of Alnwick (1605-1659) by Anthony Van Dyck Source: Wikimedia Commons

During this period, he probably visited both Petworth and Syon, Northumberland’s southern estates. He in known to have had discussion with Walter Warner on reflection. He remained in Oxford discussing mathematics with like minded fellows until his death.

Compared to the nautical adventures of Harriot and Hues, both Warner and Torporley led quiet lives. Walter Warner was born in Leicestershire and educated at Merton College Oxford graduating BA in 1579, the year between Hues and Harriot. According to John Aubrey in his Brief Lives, Warner was born with only one hand. It is almost certain that Hues, Warner, and Harriot met each other attending the mathematics lectures of Thomas Allen at Oxford. Originally a protégé of Robert Dudley, 1st Earl of Leicester, (1532–1588), he entered Northumberland’s household as a gentleman servitor in 1590 and became a pensioner in 1617. Although a servant, Warner dined with the family and was treated as a companion by the Earl. In Syon house, he was responsible for purchasing the Earl’s books, Northumberland had one of the largest libraries in England, and scientific instruments. He accompanied the Earl on his military mission to the Netherlands in 1600-01, acting as his confidential courier.       

Like Harriot, Warner was a true polymath, researching and writing on a very wide range of topics–logic, psychology, animal locomotion, atomism, time and space, the nature of heat and light, bullion and exchange, hydrostatics, chemistry, and the circulation of the blood, which he claimed to have discovered before William Harvey. However, like Harriot he published almost nothing, although, like Harriot, he was well-known in scholarly circles. Some of his work on optics was published posthumously by Marin Mersenne (1588–1648) in his Universæ geometriæ (1646).

Source: Google Books

It seems that following Harriot’s death Warner left Syon house, living in Charing Cross and at Cranbourne Lodge in Windsor the home of Sir Thomas Aylesbury, 1st Baronet (!576–1657), who had also been a student of Thomas Allen, and who had served both as Surveyor of the Navy and Master of the Mint. Aylesbury became Warner’s patron.

This painting by William Dobson probably represents Sir Thomas Aylesbury, 1st Baronet. 
Source: Wikimedia Commons

Aylesbury had inherited Harriot’s papers and encouraged Warner in the work of editing them for publication (of which more later), together with the young mathematician John Pell (1611–1685), asking Northumberland for financial assistance in the endeavour.

Northumberland died in 1632 and Algernon Percy the 10th Earl discontinued Warner’s pension. In 1635, Warner tried to win the patronage of Sir Charles Cavendish and his brother William Cavendish, enthusiastic supporters of the new scientific developments, in particular Keplerian astronomy. Charles Cavendish’s wife was the notorious female philosopher, Margaret Cavendish. Warner sent Cavendish a tract on the construction of telescopes and lenses for which he was rewarded with £20. However, Thomas Hobbes, another member of the Cavendish circle, managed to get Warner expelled from Cavendish’s patronage. Despite Aylesbury’s support Warner died in poverty. 

Nathaniel Torporley was born in Shropshire of unknow parentage and educated at Shrewsbury Grammar Scholl before matriculating at Christ Church Oxford in 1581. He graduated BA in 1584 and then travelled to France where he served as amanuensis to the French mathematician François Viète (1540–1603).

François Viète Source: Wikimedia Commons

He is thought to have supplied Harriot with a copy of Viète’s Isagoge, making Harriot the first English mathematician to have read it.


Torporley returned to Oxford in 1587 or 1588 and graduated MA from Brasenose College in 1591. 

He entered holy orders and was appointed rector of Salwarpe in Worcestershire, a living he retained until 1622. From 1611 he was also rector of Liddington in Wiltshire. His interest in mathematics, astronomy and astrology attracted the attention of Northumberland and he probably received a pension from him but there is only evidence of one payment in 1627. He was investigated in 1605, shortly before the Gunpowder Plot for having cast a nativity of the king. At some point he published a pamphlet, under the name Poulterey, attacking Viète. In 1632, he died at Sion College, on London Wall and in a will written in the year of his death he left all of his books, papers, and scientific instrument to the Sion College library.

Although his papers in the Sion College library contain several unpublished mathematical texts, still extant today, he only published one book his Diclides Coelometricae; seu Valuae Astronomicae universales, omnia artis totius munera Psephophoretica in sat modicis Finibus Duarum Tabularum methodo Nova, generali et facillima continentes, (containing a preface, Directionis accuratae consummata Doctrina, Astrologis hactenus plurimum desiderata and the Tabula praemissilis ad Declinationes et coeli meditations) in London in 1602.


This is a book on how to calculate astrological directions, a method for determining the time of major incidents in the life of a subject including their point of death, which was a very popular astrological method in the Renaissance. This requires spherical trigonometry, and the book is interesting for containing new simplified methods of solving right spherical triangles of any sort, methods that are normally attributed to John Napier (1550–1617) in a later publication. The book is, however, extremely cryptic and obscure, and almost unreadable. Despite this the surviving copies would suggest that it was widely distributed in Europe.

Our three mathematici came together as executors of Harriot’s will. Hues was charged with pricing Harriot’s books and other items for sale to the Bodleian Library. Hues and Torporley were charged with assisting Warner with the publication of Harriot’s mathematical manuscripts, a task that the three of them managed to bungle. In the end they only managed to publish one single book, Harriot’s algebra Artis Analyticae Praxis in 1631 and this text they castrated.


Harriot’s manuscript was the most advanced text on the topic written at the time and included full solutions of algebraic equations including negative and complex solutions. Either Warner et al did not understand Harriot’s work or they got cold feet in the face of his revolutionary new methods, whichever, they removed all of the innovative parts of the book making it basically irrelevant and depriving Harriot of the glory that was due to him.

For myself the main lesson to be learned from taking a closer look at the lives of this group of mathematici is that it shows that those interested in mathematics, astronomy, cartography, and navigation in England the late sixteenth and early seventeenth centuries were intricately linked in a complex network of relationships, which contains hubs one of which was initially Harriot and Raleigh and then later Harriot and Northumberland. 

[1] For those who don’t know, Middlesex was a small English county bordering London, in the South-West corner of Essex, squeezed between Hertfordshire to the north and Surry in the South, which now no longer exists having been largely absorbed into Greater London. 


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