Category Archives: History of Mathematics

How Renaissance Nürnberg became the Scientific Instrument Capital of Europe

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

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


Column Sundial by Christian Heyden Source: Museumslandschaft Hessen-Kassel

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


Paper and Wood Astrolabe Hartmann Source: MHS Oxford

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

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


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

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

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


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


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

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

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


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

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


The Imperial Regalia

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

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


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

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


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

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

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

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


The wirepulling mills of Nürnberg by Albrecht Dürer

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

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

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


The Cusanus Museum in Kue

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

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

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


The Erdapfel–the Behaim terrestial globe Germanische National Museum

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


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


In 1533 Apian’s Instrumentum Primi Mobilis 


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

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

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

WI12; WI33 WI3; WI2; WI30;

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


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












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

3 into 2 does go!

It would of course be totally unethical for me to review a book of which I am one of the authors. However, as my contribution is only six of two-hundred pages, of which three are illustrations, and the book is one that could/would/should interest some (many) of my readers, I’m going to be unethical and review it anyway.

Thinking 3D is an intellectual idea, it is a website, it is exhibitions and other events, it is a book but above all it is two people, whose idea it is: Daryl Green, who was Fellow Librarian of Magdalen College, Oxford and is now Special Collections Librarian of the University of Edinburg and Laura Moretti, who is Senior Lecturer in Art History at the University of St Andrews. The Thinking 3D idea is the historical investigation of the representation of the three-dimensional world on the two-dimensional page particular, but not exclusively, in print.

The Thinking 3D website explains in great detail what it is all about and contains a full description of the activities that have been carried out. For those quarantined there is a fairly large collection of essays on various topics from the project.

In 2019 Thinking 3D launched a major exhibition with The Bodleian Libraries Oxford as part of the commemorations of the 500th anniversary of Leonardo da Vinci’s death, Thinking 3D From Leonardo to the Present, which ran from March 2019 to February 2020 and which I have been told was quite exceptional.

thinking 3d002

As an extension and permanent record of that exhibition Bodleian Libraries published a book, Thinking 3D: Books, Images and Ideas from Leonardo to the Present[1], which appeared in autumn 2019. This is both a coffee table book but also, at the same time, a piece of serious academic literature.

thinking 3d001

The book opens with a long essay by Green and Moretti, The history of thinking 3D in forty books, which delivers exactly what the title says. This is an excellent survey of the topic and it is worth reading the book just for this. However, it does contain one historical error that I, in my alter ego of the HIST_SCI HULK, simply cannot ignore, at least not if I want to maintain my hard won reputation. Having introduced the topic of Copernicus’ De revolutionibus the authors write:

As mentioned above, the oft-published heliocentric diagram, and its theoretical propositions, are what launched this book into infamy (the book was immediately put on the Catholic Church’s Index of Prohibited Books [my emphasis]), but the execution of this relational illustration is simple and reductive.

De revolutionibus was published in 1543 but was first placed on the Index sixty-three years later in 1616 and more importantly, as I wrote very recently, not for the first time, it was placed on the Index until corrected. These corrections, which were fairly minimal, were carried out surprisingly quickly and the book became available to be studied by Catholics already in 1621.

Other than this I noticed no other errors in the highly informative introductory essay, which is followed by an essay from Matthew Landrus, Leonardo da Vinci, 500 years on, which examines Leonardo’s three-dimensional perception of the world and everything in it. It was for me an interesting addition to my previous readings on the Tuscan polymath.

The main body of the book is taken up by sixteen fairly short essays in four categories: Geometry, Astronomy, Architecture and Anatomy.

Geometry starts off with Ken Saito’s presentation of a ninth century manuscript of The Elements of Euclid, where he demonstrates very clearly that the author has no real consistent, methodology for presenting a 3D space on a 2D page.   This is followed by Renzo Baldasso’s essay on Luca Pacioli’s De divina proportione (1509). Here the three dimensional solids are presented perfectly by Pacioli’s friend, colleague and one time pupil Leonardo. We return to Euclid for Yelda Nasifoglu’s investigation of the English translation of The Elements by Henry Billingsley in 1570. This volume is totally fascinating as three-dimensional figures are present as pop-up figure like those that we all know from our children’s books. The geometry section closes with a book that I didn’t know, Max Brückner’s Vielecke und Vielflache (1900) presented by George Hart. This is a vast collection of photographs of paper models of three-dimensional figures, which I learnt also influenced M. C. Escher a master of the third dimension.

thinking 3d004

Luca Pacioli De divina proportione


Karl Galle, Renaissance Mathematicus friend and guest blogger, kicks of the astronomy section with Johannes Kepler’s wonderfully bizarre presentation of the planetary orbits embedded in the five regular Platonic solids from his Mysterium Cosmographicum (1596). Yours truly is up next with an account of Galileo’s Sidereus Nuncius (1610) and it’s famous washes of the Moon displaying three-dimension features. Also covered are the later pirate editions that screwed up those illustrations. Stephanie O’Rourke presents one of the most extraordinary nineteenth century astronomy books James Nasmyth’s and James Carpenter’s The Moon: Considered as a Planet, a World, and a Satellite(1874). This contains stunningly realistic photographic plates of the Moon’s surface but which are not actually real. The two Jameses constructed plaster models that they then lit and photographed to achieve the desired effect. We close the astronomy section with Thinking 3D’s co-chef, Daryl Green, taking on a survey of the surface of Mars with the United Stated Geological Survey, Geological Map of Mars (1978).

thinking 3d005

Johannes Kepler Mysterium Cosmographicum

Turning our attention to architecture, we travel back to the twelfth century, with Karl Kinsella as our guide, to Richard of St Victor’s In visionen Ezekielis; a wonderfully modern in its presentation but somewhat unique medieval architectural manuscript. The other half of the Thinking 3D team, Laura Moretti now takes us up to the sixteenth century and Sebastiano Serlio’s catalogue of the buildings of Rome (1544), which has an impossibly long Italian title that I’m not going to repeat here. We remain in the sixteenth century for Jacques Androuet du Cerceau’s Le premier [et second] volume des plus excellent bastiment de France (1576–9), where our guide is Frédérique Lemerle. Moving forward a century we close out the architecture section with Francesco Marcorin introducing us to Hans Vredeman de Vries’s absolutely stunning Perspective (1604–5).

thinking 3d008

Hans Vredeman de Vries Perspective

It would not be too difficult to guess that the anatomy section opens with one of the greatest medical books of all time, Andreas Vesalius’ De fabrica but not with the full version but the shorter (cheaper?) De humani corporis fabrica libroum epitome, like the full version published in 1543 in Basel. Our guide to Vesalius’ masterpiece is Mark Samos. Camilla Røstvik introduces us to William Hunter’s The Anatomy of the Human Gravid Uterus (1774), as she makes very clear a milestone in the study of women’s bodies with its revolutionary and controversial study of the pregnant body. For me this essay was a high point in a collection of truly excellent essays. We stay in the eighteenth century for Jacques Fabien Gautier D’Agoty’s Exposition anatomique des organes des sens (1775). Dániel Margócsy present a fascinating guide to the controversial work of this pioneer of colour printing. Anatomy, and the book as a whole, closes with Denis Pellerin’s essay on Arthur Thomson’s Anatomy of the Human Eye (1912). Thomson’s book was accompanied by a collection of stereoscopic images of the anatomy of the eye together with a stereoscope with which to view the 3D images thus created; a nineteenth century technology that was already dying out when Thomson published his work.

thinking 3d009

William Hunter The Anatomy of the Human Gravid Uterus

The book closes with a bibliography of five books for further reading for each essay, brief biography of each of the authors, a glossary of technical terms and a good general index. All sixteen of the essays are short, informative, light to read, easily accessible introductions to the volumes that they present and maintain a high academic quality throughout the entire book.

I said at the outset that this is also a coffee table book and that was not meant negatively. It measures 24X26 cm and is printed on environmentally friendly, high gloss paper. The typeface is attractive and light on the eyes and the illustrations are, as is to be expected for a book about the history of book illustration, spectacularly beautiful. The publishing team of the Bodleian Libraries are to be congratulated on an excellent publication. If you leave this on your coffee table then your visitors will soon be leafing though it admiring the pictures, whether they are interested in book history or not. I don’t usually mention the price of books that I review but at £35 this beautifully presented and wonderfully informative volume is very good value for money.

[1] Thinking 3D: Books, Images and Ideas from Leonardo to the Present, edited by Daryl Green and Laura Moretti, Bodleian Library, Oxford, 2019.


Filed under Book Reviews, Early Scientific Publishing, History of Astronomy, History of Mathematics, History of medicine

War, politics, religion and scientia

There is a strong tendency to view the history of science and the people who produced it in a sort of vacuum, outside of everyday society–Copernicus published this, Kepler published that, Newton synthesised it all… In fact the so-called scientific revolution took place in one of the most troubled times in European history, the age of the religious wars, the main one of which the Thirty Years War is thought to have been responsible directly and indirectly for the death of between one third and two thirds of the entire population of middle Europe. Far from being isolated from this turbulence the figures, who created modern science, were right in the middle of it and oft deeply involved and affected by it.

The idea for this blog post sort of crept into my brain as I was writing my review, two weeks ago, of two books about female spies during the English Revolution and Interregnum that is the 1640s to the 1660s. Isaac Newton was born during this period and grew up during it and, as I will now sketch, was personally involved in the political turbulence that followed on from it.

Born on Christmas Day in 1642 (os) shortly after the outbreak of the first of the three wars between the King and Parliament, Britain’s religious wars, he was just nine years old when Charles I was executed at the end of the second war.


Portrait of Newton by Godfrey Kneller, 1689 Source: Wikimedia Commons

Newton was too young to be personally involved in the wars but others whose work would be important to his own later developments were. The Keplerian astronomer William Gascoigne (1612-1644), who invented the telescope micrometer, an important development in the history of the telescope, died serving in the royalist forces at the battle of Marston Moor. The mathematician John Wallis (1616–1703), whose Arithmetica Infinitorum (1656) strongly influenced Newton’s own work on infinite series and calculus, worked as a code breaker for Cromwelland later for Charles II after the restoration.


John Wallis by Sir Godfrey Kneller

Newton first went up to university after the restoration but others of an earlier generation suffered loss of university position for being on the wrong side at the wrong time. John Wilkins (1614–1672), a parliamentarian and Cromwell’s brother-in-law, was appointed Master of Trinity College Cambridge, Newton’s college, in 1659 and removed from this position at the restoration. Wilkins’ Mathematical Magick (1648) had been a favourite of Newton’s in his youth.

Greenhill, John, c.1649-1676; John Wilkins (1614-1672), Warden (1648-1659)

Greenhill, John; John Wilkins (1614-1672), Warden (1648-1659); Wadham College, University of Oxford;

Newton’s political career began in 1689 following the so-called Glorious Revolution, when James II was chased out of Britain by William of Orange, his son-in-law, invited in by the parliament out of fear that James could reintroduce Catholicism into Britain. Newton sat in the House of Commons as MP for the University of Cambridge in the parliament of 1689, which passed the Bill of Rights, effectively a new constitution for England. Newton was not very active politically but he identified as a Whig, the party of his student Charles Montagu (1661–1715), who would go on to become one of the most powerful politicians of the age. It was Montagu, who had Newton appointed to lead the Royal Mint and it was also Montagu, who had Newton knighted in 1705in an attempt to get him re-elected to parliament.

In the standard version of story Newton represents the end of the scientific revolution and Copernicus (1473–1543) the beginning. Religion, politics and war all played a significant role in Copernicus’ life.


Copernicus, the “Torun portrait” (anonymous, c. 1580), kept in Toruń town hall, Poland.

Copernicus spent the majority of his life living in the autonomous prince-bishopric of Warmia, where as a canon of the cathedral he was effectively a member of the government. Warmia was a Catholic enclave under the protection of the Catholic Crown of Poland but as the same time was geographically part of Royal Prussia ruled over by Duke Albrecht of Prussia (1490–1568), who had converted to Lutheran Protestantism in 1552. Ironically he was converted by Andreas Osiander (1498–1552), who would go on the author the controversial ad lectorum in Copernicus’ De revolutionibus. Relations between Poland and Royal Prussia were strained at best and sometimes spilled over into armed conflict. Between 1519 and 1521 there was a war between Poland and Royal Prussia, which took place mostly in Warmia. The Prussians besieged Frombork burning down the town, but not the cathedral, forcing Copernicus to move to Allenstein (Olsztyn), where he was put in charge of organising the defences during a siege from January to February 1521.  Military commander in a religious war in not a role usually associated with Copernicus. It is an interesting historical conundrum that, during this time of religious strife, De revolutionibus, the book of a Catholic cathedral canon, was published by a Protestant printer in a strongly Protestant city-state, Nürnberg.

The leading figure of the scientific revolution most affected by the religious wars of the age must be Johannes Kepler. A Lutheran Protestant he studied and graduated at Tübingen, one of the leading Protestant universities. However, he was despatched by the university authorities to become the mathematics teacher at the Protestant school in Graz in Styria, a deeply Catholic area in Austria in 1594. He was also appointed district mathematicus.


Johannes Kepler Source: Wikimedia Commons

In 1598, Archduke Ferdinand, who became ruler of Styria in 1596, expelled all Protestant teachers and pastors from the province. Kepler was initially granted an exception because he had proved his worth as district mathematicus but in a second wave of expulsion, he too had to go. After failing to find employment elsewhere, he landed in Prague as an assistant to Tycho Brahe, the Imperial Mathematicus.


Tycho Brahe Source: Wikimedia Commons

Once again he, like Tycho, was a Protestant in a Catholic city serving a Catholic Emperor, Rudolf II. Here he soon inherited Tycho’s position as Imperial Mathematicus. However, Rudolf was tolerant, more interested in Kepler’s abilities as an astrologer than in his religious beliefs. Apart from a substantial problem in getting paid in the permanently broke imperial court, Kepler now enjoyed a fairly quiet live for the next twelve years, then everything turned pear shaped once more.

In 1612, Rudolf’s younger brother Archduke Matthias deposed him and although Kepler was allowed to keep his title of Imperial Mathematicus, and theoretically at least, his salary but he was forced to leave Prague and become district mathematicus in Linz. In Linz Kepler, who openly propagated ecumenical ideas towards other Protestant communities, most notably the Calvinists, ran into conflict with the local Lutheran pastor. The pastor demanded that Kepler sign the Formula of Concord, basically a commitment to Lutheran theology and a rejection of all other theologies. Kepler refused and was barred from Holy Communion, a severe blow for the deeply religious astronomer. He appealed to the authorities in Tübingen but they up held the ban.

In 1618 the Thirty Years War broke out and in 1620 Linz was occupied by the Catholic army of Duke Maximilian of Bavaria, which caused problems for Kepler as a Lutheran. At the same time he was fighting for the freedom of his mother, Katharina, who had been accused of witchcraft. Although he won the court case against his mother, she died shortly after regaining her freedom. In 1625, the Counterreformation reach Linz and the Protestants living there were once again persecuted. Once more Kepler was granted an exception because of his status as Imperial Mathematicus but his library was confiscated making it almost impossible for him to work, so he left Linz.

Strangely, after two years of homeless wandering Kepler moved to Sagen in Silesia in 1628, the home of Albrecht von Wallenstein the commander of the Catholic forces in the war and for whom Kepler had interpreted a horoscope much earlier in life. Kepler never found peace or stability again in his life and died in Ulm in 1630. Given the turbulence in his life and the various forced moves, which took years rather than weeks, it is fairly amazing that he managed to publish eighty-three books and pamphlets between 1596 and his death in 1630.

A younger colleague of Kepler’s who also suffered during the Thirty Years’ War was Wilhelm Schickard, who Kepler had got to know during his time in Württemberg defending his mother. Schickard would go on to produce the illustrations both Kepler’s Epitome Astronomiae Copernicanae and his Harmonice Mundi, as well as inventing a calculating machine to help Kepler with his astronomical calculations. In 1632 Württemberg was invaded by the Catholic army, who brought the plague with them, by 1635 Schickard, his wife and his four living children, his sister and her three daughters had all died of the plague.


Wilhelm Schickard, artist unknown Source: Wikimedia Commons

As I have pointed out on numerous occasions Galileo’s initial problems in 1615-16 had less to do with his scientific views than with his attempts to tell the theologians how to interpret the Bible, not an intelligent move at the height of the Counterreformation. Also in 1632 his problems were very definitely compounded by the fact that he was perceived to be on the Spanish side in the conflict between the Spanish and French Catholic authorities to influence, control the Pope, Urban VIII.


Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

I will just mention in passing that René Descartes served as a soldier in the first two years of the Thirty Year’s War, at first in the Protestant Dutch States Army under Maurice of Nassau and then under the Catholic Duke of Bavaria, Maximilian. In 1620 he took part in the Battle of the White Mountain near Prague, which marked the end of Elector Palatine Frederick V’s reign as King of Bohemia. During his time in the Netherlands Descartes trained as a military engineer, which was his introduction to the works of Simon Stevin and Isaac Beeckman.


René Descartes Portrait after Frans Hals Source: Wikimedia Commons

We have now gone full circle and are almost back to Isaac Newton. One interesting aspect of these troubled times is that although the problems caused by the wars, the religious disputes and the associated politics caused major problems in the lives of the astronomers and mathematicians, who were forced to live through them, and certainly affected their ability to carry on with their work, I can’t somehow imagine Copernicus working on De revolutionibus during the siege of Allenstein, the scholars themselves communicated quite happily across the religious divide.

Rheticus was treated as an honoured guest in Catholic Warmia although he was a professor at the University of Wittenberg, home to both Luther and Melanchthon. Copernicus himself was personal physician to both the Catholic Bishop of Frombork and the Protestant Duke of Royal Prussia. As we have seen, Kepler spent a large part of his life, although a devoted Protestant, serving high-ranking Catholic employers. The Jesuits, who knew Kepler from Prague, even invited him to take the chair for mathematics at the Catholic University of Bologna following the death of Giovanni Antonio Magini in 1617, assuring him that he did not need to convert. Although it was a very prestigious university Kepler, I think wisely, declined the invitation. The leading mathematicians of the time all communicated with each other, either directly or through intermediaries, irrespective of their religious beliefs. Athanasius Kircher, professor for mathematics and astronomy at the Jesuit Collegio Romano, collected astronomical data from Jesuits all over the world, which he then distributed to astronomers all over Europe, Catholic and Protestant, including for example the Lutheran Leibniz. Christiaan Huygens, a Dutch Calvinist, spent much of his life working as an honoured guest in Catholic Paris, where he met and influenced the Lutheran Leibniz.

When we consider the lives of scientists we should always bear in mind that they are first and foremost human beings, who live and work, like all other human beings, in the real world with all of its social, political and religious problems and that their lives are just as affected by those problems as everybody else.








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

The emergence of modern astronomy – a complex mosaic: Part XXX

As stated earlier the predominant medieval view of the cosmos was an uneasy bundle of Aristotle’s cosmology, Ptolemaic astronomy, Aristotelian terrestrial mechanics, which was not Aristotle’s but had evolved out of it, and Aristotle’s celestial mechanics, which we will look at in a moment. As also pointed out earlier this was not a static view but one that was constantly being challenged from various other models. In the early seventeenth century the central problem was, having demolished nearly all of Aristotle’s cosmology and shown Ptolemaic astronomy to be defective, without however yet having found a totally convincing successor, to now find replacements for the terrestrial and celestial mechanics. We have looked at the development of the foundations for a new terrestrial mechanics and it is now time to turn to the problem of a new celestial mechanics. The first question we need to answer is what did Aristotle’s celestial mechanics look like and why was it no longer viable?

The homocentric astronomy in which everything in the heavens revolve around a single central point, the earth, in spheres was created by the mathematician and astronomer Eudoxus of Cnidus (c. 390–c. 337 BCE), a contemporary and student of Plato (c. 428/27–348/47 BCE), who assigned a total of twenty-seven spheres to his system. Callippus (c. 370–c. 300 BCE) a student of Eudoxus added another seven spheres. Aristotle (384–322 BCE) took this model and added another twenty-two spheres. Whereas Eudoxus and Callippus both probably viewed this model as a purely mathematical construction to help determine planetary position, Aristotle seems to have viewed it as reality. To explain the movement of the planets Aristotle thought of his system being driven by friction. The outermost sphere, that of the fixed stars drove the outer most sphere of Saturn, which in turn drove the next sphere down in the system and so on all the way down to the Moon. According to Aristotle the outermost sphere was set in motion by the unmoved mover. This last aspect was what most appealed to the churchmen of the medieval universities, who identified the unmoved mover with the Christian God.


During the Middle Ages an aspect of vitalism was added to this model, with some believing that the planets had souls, which animated them. Another theory claimed that each planet had its own angel, who pushed it round its orbit. Not exactly my idea of heaven, pushing a planet around its orbit for all of eternity. Aristotelian cosmology said that the spheres were real and made of crystal. When, in the sixteenth century astronomers came to accept that comets were supralunar celestial phenomena, and not as Aristotle had thought sublunar meteorological ones, it effectively killed off Aristotle’s crystalline spheres, as a supralunar comet would crash right through them. If fact, the existence or non-existence of the crystalline spheres was a major cosmological debate in the sixteenth century. By the early seventeenth century almost nobody still believed in them.

An alternative theory that had its origins in the Middle Ages but, which was revived in the sixteenth century was that the heavens were fluid and the planets swam through them like a fish or flew threw them like a bird. This theory, of course, has again a strong element of vitalism. However, with the definitive collapse of the crystalline spheres it became quite popular and was subscribed to be some important and influential thinkers at the end of the sixteenth beginning of the seventeenth centuries, for example Roberto Bellarmino (1542–1621) the most important Jesuit theologian, who had lectured on astronomy at the University of Leuven in his younger days.


Robert Bellarmine artist unknown Source: Wikimedia Commons

It should come as no surprise that the first astronomer to suggest a halfway scientific explanation for the motion of the planets was Johannes Kepler. In fact he devoted quite a lot of space to his theories in his Astronomia nova (1609).


Astronomia Nova title page Source: Wikimedia Commons

That the periods between the equinoxes and the solstices were of unequal length had been known to astronomers since at least the time of Hipparchus in the second century BCE. This seemed to imply that the speed of either the Sun orbiting the Earth, in a geocentric model, or the Earth orbiting the Sun, in a heliocentric model, varied through out the year. Kepler calculated a table for his elliptical, heliocentric model of the distances of the Sun from the Earth and deduced from this that the Earth moved fastest when it was closest to the Sun and slowest when it was furthest away. From this he deduced or rather speculated that the Sun controlled the motion of the Earth and by analogy of all the planets. The thirty-third chapter of Astronomia nova is headed, The power that moves the planets resides in the body of the sun.

His next question is, of course, what is this power and how does it operate? He found his answer in William Gilbert’s (1544–1603) De Magnete, which had been published in 1600.


William Gilbert Source: Wikimedia Commons

Kepler speculated that the Sun was in fact a magnet, as Gilbert had demonstrated the Earth to be, and that it rotated on its axis in the same way that Gilbert believed, falsely, that a freely suspended terrella (a globe shaped magnet) did. Gilbert had used this false belief to explain the Earth’s diurnal rotation.


It should be pointed out that Kepler was hypothesising a diurnal rotation for the Sun in 1609 that is a couple of years before Galileo had demonstrated the Sun’s rotation in his dispute over the nature of sunspots with Christoph Scheiner (c. 1574–1650). He then argues that there is power that goes out from the rotating Sun that drives the planets around there orbits. This power diminishes with its distance from the Sun, which explains why the speed of the planetary orbits also diminishes the further the respective planets are from the Sun. In different sections of the Astronomia nova Kepler argues both for and against this power being magnetic in nature. It should also be noted that although Kepler is moving in the right direction with his convoluted and at times opaque ideas on planetary motion there is still an element of vitalism present in his thoughts.

Kepler conceived the relationship between his planetary motive force and distance as a simple inverse ratio but it inspired the idea of an inverse squared force. The French mathematician and astronomer Ismaël Boulliau (1605–1694) was a convinced Keplerian and played a central roll in spreading Kepler’s ideas throughout Europe.


Ismaël Boulliau portrait by Pieter van Schuppen Source: Wikimedia Commons

His most important and influential work was his Astronomia philolaica (1645). In this work Boulliau hypothesised by analogy to Kepler’s own law on the propagation of light that if a force existed going out from the Sun driving the planets then it would decrease in inverse squared ratio and not a simple one as hypothesised by Kepler. Interestingly Boulliau himself did not believe that such a motive force for the planet existed.


Another mathematician and astronomer, who looked for a scientific explanation of planetary motion was the Italian, Giovanni Alfonso Borelli (1608–1697) a student of Benedetto Castelli (1578–1643) and thus a second-generation student of Galileo.


Giovanni Alfonso Borelli Source: Wikimedia Commons

Borelli developed a force-based theory of planetary motion in his Theoricae Mediceorum Planatarum ex Causius Physicis Deductae (Theory [of the motion] of the Medicean planets [i.e. moons of Jupiter] deduced from physical causes) published in 1666. He hypothesised three forces that acted on a planet. Firstly a natural attraction of the planet towards the sun, secondly a force emanating from the rotating Sun that swept the planet sideway and kept it in its orbit and thirdly the same force emanating from the sun pushed the planet outwards balancing the inwards attraction.

The ideas of both Kepler and Borelli laid the foundations for a celestial mechanics that would eventually in the work of Isaac Newton, who knew of both theories, produced a purely force-based mathematical explanation of planetary motion.







Filed under History of Astronomy, History of Mathematics, History of Physics, History of science, Mediaeval Science

It’s all a question of angles.

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


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

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


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

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


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

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


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

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


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

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


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

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

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


Abū al-Wafā Source: Wikimedia Commons

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


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

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

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

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

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


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

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


Source: Wikimedia Commons

In the meantime Bartholomäus Pitiscus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuous in 1595.


Source: Wikimedia Commons

This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ Trigonometria were calculated to five or six places, where as those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as Magnus canon doctrinae triangulorum in 1607. He published a second further improved version under the title Thesaurus mathematicus in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.

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


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


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

Mathematics at the Meridian

Historically Greenwich was a village, home to a royal palace from the fifteenth to the seventeenth centuries, that lay to the southeast of the city of London on the banks of the river Thames, about six miles from Charing Cross. Since the beginning of the twentieth century it has been part of London. With the Cutty Sark, a late nineteenth century clipper built for the Chinese tea trade, the Queen’s House, a seventeenth-century royal residence designed and built by Inigo Jones for Anne of Denmark, wife of James I & VI, and now an art gallery, the National Maritime Museum, Christopher Wren’s Royal Observatory building and of course the Zero Meridian line Greenwich is a much visited, international tourist attraction.

Naturally, given that it is/was the home of the Royal Observatory, the Zero Meridian, the Greenwich Royal Hospital School, the Royal Naval College (of both of which more later), and most recently Greenwich University, Greenwich has been the site of a lot mathematical activity over the last four hundred plus years and now a collection of essays has been published outlining in detail that history: Mathematics at the Meridian: The History of Mathematics at Greenwich[1]


This collection of essays gives a fairly comprehensive description of the mathematical activity that took place at the various Greenwich institutions. As a result it also function as an institutional history, an often-neglected aspect of the histories of science and mathematics with their concentration on big names and significant theories and theorems. Institutions play an important role in the histories of mathematic and science and should receive much more attention than they usually do.

The first four essays in the collection cover the history of the Royal Observatory from its foundation down to when it was finally closed down in 1998 following several moves from its original home in Greenwich. They also contain biographies of all the Astronomers Royal and how they interpreted their role as the nation’s official state astronomer.

This is followed by an essay on the mathematical education at the Greenwich Royal Hospital School. The Greenwich Royal Hospital was established at the end of the seventeenth century as an institution for aged and injured seamen. The institution included a school for the sons of deceased or disabled sailors. The teaching was centred round seamanship and so included mathematics, astronomy and navigation.

When the Greenwich Royal Hospital closed at the end of the nineteenth century the buildings were occupied by the Royal Naval College, which was moved from Portsmouth to Greenwich. The next three chapters deal with the Royal Naval College and two of the significant mathematicians, who had been employed there as teachers and their contributions to mathematics.


Another institute that was originally housed at Greenwich was The Nautical Almanac office, founded in 1832. The chapter dealing with this institute concentrates on the life and work of Leslie John Comrie (1893–1950), who modernised the production of mathematical tables introducing mechanisation to the process.

Today, apart from the Observatory itself and the Meridian line, the biggest attraction in Greenwich is the National Maritime Museum, one of the world’s leading science museums and there is a chapter dedicated to the scientific instruments on display there.


Also today, the buildings that once housed the Greenwich Royal Hospital and then the Royal Naval College now house the University of Greenwich and the last substantial chapter of the book brings the reader up to the present outlining the mathematics that has been and is being taught there.

The book closes with a two-page afterword, The Mathematical Tourist in Greenwich.

Each essay in the book is written by an expert on the topic and they are all well researched and maintain a high standard throughout the entire book. The essays covers a wide and diverse range of topics and will most probably not all appeal equally to all readers. Some might be more interested in the history of the Royal Observatory, whilst the chapters on the mathematical education at the Greenwich Royal Hospital School and on its successor the Royal Naval College should definitely be of interest to the readers of Margaret Schotte’s Sailing School, which I reviewed in an earlier post.

Being the hopelessly non-specialist that I am, I read, enjoyed and learnt something from all of the essays. If I had to select the four chapters that most stimulated me I would chose the opening chapter on the foundation and early history of the Royal Observatory, the chapter on George Biddel Airy and his dispute with Charles Babbage over the financing of the Difference Engine, which I blogged about in December, the chapter on Leslie John Comrie, as I’ve always had a bit of a thing about mathematical tables and finally, one could say of course, the chapter on the scientific instruments in the National Maritime Museum.

The book is nicely illustrated with, what appears to have become the standard for modern academic books, grey in grey prints. There are extensive endnotes for each chapter, which include all of the bibliographical references, there being no general bibliography, which I view as the books only defect. There is however a good, comprehensive general index.

I can thoroughly recommend this book for anybody interested in any of the diverse topic covered however, despite what at first glance, might appear as a somewhat specialised book, I can also recommend it for the more general reader interested in the histories of mathematics, astronomy and navigation or those perhaps interested in the cultural history of one of London’s most fascinating district. After all mathematics, astronomy and navigation are all parts of human culture.

[1] Mathematics at the Meridian: The History of Mathematics at Greenwich, eds. Raymond Flood, Tony Mann, Mary Croarken, CRC Press, Taylor & Francis Group, Bacon Raton, London, New York, 2020.


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

The emergence of modern astronomy – a complex mosaic: Part XXVIII

One of the central problems in the transition from the traditional geocentric astronomy/cosmology to a heliocentric one was that the system that the Early Modern astronomers inherited from their medieval predecessors was not just an uneasy amalgam of Aristotelian cosmology and Ptolemaic astronomy but it also included Aristotle’s (384–322 BCE) theories of terrestrial and celestial motion all tied together in a complete package. Aristotle’s theory of motion was part of his more general theory of change and differentiated between natural motion and unnatural or violent motion.

The celestial realm in Aristotle’s cosmology was immutable, unchanging, and the only form of motion was natural motion that consisted of uniform, circular motion; a theory that he inherited from Plato (c. 425 – c.347 BCE), who in turn had adopted it from Empedocles (c. 494–c. 434 BCE).

His theory of terrestrial motion had both natural and unnatural motion. Natural motion was perpendicular to the Earth’s surface, i.e. when something falls to the ground. Aristotle explained this as a form of attraction, the falling object returning to its natural place. Aristotle also claimed that the elapsed time of a falling body was inversely proportional to its weight. That is, the heavier an object the faster it falls. All other forms of motion were unnatural. Aristotle believed that things only moved when something moved them, people pushing things, draught animals pulling things. As soon as the pushing or pulling ceased so did the motion.  This produced a major problem in Aristotle’s theory when it came to projectiles. According to his theory when a stone left the throwers hand or the arrow the bowstring they should automatically fall to the ground but of course they don’t. Aristotle explained this apparent contradiction away by saying that the projectile parted the air through which it travelled, which moved round behind the projectile and pushed it further. It didn’t need a philosopher to note the weakness of this more than somewhat ad hoc theory.

If one took away Aristotle’s cosmology and Ptolemaeus’ astronomy from the complete package then one also had to supply new theories of celestial and terrestrial motion to replace those of Aristotle. This constituted a large part of the development of the new physics that took place during the so-called scientific revolution. In what follows we will trace the development of a new theory, or better-said theories, of terrestrial motion that actually began in late antiquity and proceeded all the way up to Isaac Newton’s (1642–1726) masterpiece Principia Mathematica in 1687.

The first person to challenge Aristotle’s theories of terrestrial motion was John Philoponus (c. 490–c. 570 CE). He rejected Aristotle’s theory of projectile motion and introduced the theory of impetus to replace it. In the impetus theory the projector imparts impetus to the projected object, which is used up during its flight and when the impetus is exhausted the projectile falls to the ground. As we will see this theory was passed on down to the seventeenth century. Philoponus also rejected Aristotle’s quantitative theory of falling bodies by apparently carrying out the simple experiment usually attributed erroneously to Galileo, dropping two objects of different weights simultaneously from the same height:

but this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small. …

Philoponus also removed Aristotle’s distinction between celestial and terrestrial motion in that he attributed impetus to the motion of the planets. However, it was mainly his terrestrial theory of impetus that was picked up by his successors.

In the Islamic Empire, Ibn Sina (c. 980–1037), known in Latin as Avicenne, and Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī (c. 1080–1164) modified the theory of impetus in the eleventh century.


Avicenne Portrait (1271) Source: Wikimedia Commons

Nur ad-Din al-Bitruji (died c. 1204) elaborated it at the end of the twelfth century. Like Philoponus, al-Bitruji thought that impetus played a role in the motion of the planets.


Brought into European thought during the scientific Renaissance of the twelfth and thirteenth centuries by the translators it was developed by Jean Buridan  (c. 1301–c. 1360), who gave it the name impetus in the fourteenth century:

When a mover sets a body in motion he implants into it a certain impetus, that is, a certain force enabling a body to move in the direction in which the mover starts it, be it upwards, downwards, sidewards, or in a circle. The implanted impetus increases in the same ratio as the velocity. It is because of this impetus that a stone moves on after the thrower has ceased moving it. But because of the resistance of the air (and also because of the gravity of the stone) which strives to move it in the opposite direction to the motion caused by the impetus, the latter will weaken all the time. Therefore the motion of the stone will be gradually slower, and finally the impetus is so diminished or destroyed that the gravity of the stone prevails and moves the stone towards its natural place. In my opinion one can accept this explanation because the other explanations prove to be false whereas all phenomena agree with this one.


Jean Buridan Source

The impetus theory was now a part of medieval Aristotelian natural philosophy, which as Edward Grant pointed out was not Aristotle’s natural philosophy.

In the sixteenth century the self taught Italian mathematician Niccolò Fontana (c. 1500–1557), better known by his nickname, Tartaglia, who is best known for his dispute with Cardanoover the general solution of the cubic equation.


Niccolò Fontana Tartaglia Source: Wikimedia Commons

published the first mathematical analysis of ballistics his, Nova scientia (1537).


His theory of projectile trajectories was wrong because he was still using the impetus theory.


However, he was the first to demonstrate that an angle of 45° for a canon gives the widest range.


His book was massively influential in the sixteenth century and it also influenced Galileo, who owned a heavily annotated copy of the book.

We have traced the path of the impetus theory from its inception by John Philoponus up to the second half of the sixteenth century. Unlike the impetus theory Philoponus’ criticism of Aristotle’s theory of falling bodies was not taken up directly by his successors. However, in the High Middle Ages Aristotelian scholars in Europe did begin to challenge and question exactly those theories and we shall be looking at that development in the next section.







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