Category Archives: History of Mathematics

The Arch-Humanist

The name Conrad Celtis is not one that you’ll find in most standard books on the history of mathematics, which is not surprising as he was a Renaissance humanist scholar best known in his lifetime as a poet. However, Celtis played an important role in the history of mathematics and is a good example of the fact that if you really wish to study the evolution of the mathematical sciences it is necessary to leave the narrow confines of the mathematics books.

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507 Source: Wikimedia Commons

Conrad Celtis: Gedächtnisbild von Hans Burgkmair dem Älteren, 1507
Source: Wikimedia Commons

Born Konrad Bickel or Pyckell, (Conrad Celtis was his humanist pseudonym) the son of a winemaker, in Franconian Wipfield am Main near Schweinfurt on 1 February 1459, he obtained his BA at the University of Cologne in 1497. Unsatisfied with the quality of tuition in Cologne he undertook the first of many study journeys, which typified his life, to Buda in 1482, where he came into contact with the humanist circle on the court of Matthias Corvinus, the earlier patron of Regiomontanus. 1484 he continued his studies at the University of Heidelberg specialising in poetics and rhetoric, learning Greek and Hebrew and humanism as a student of Rudolf Agricola, a leading Dutch early humanist scholar. Celtis obtained his MA in 1485. 1486 found him underway in Italy, where he continued his humanist studies at the leading Italian universities and in conversation with many leading humanist scholars. Returning to Germany he taught poetics at the universities of Erfurt, Rostock and Leipzig and on 18 April 1487 he was crowned Poet Laureate by Emperor Friedrich III in Nürnberg during the Reichstag. In Nürnberg he became part of the circle of humanists that produced the Nürnberger Chronicle to which he contributed the section on the history and geography of Nürnberg. It is here that we see the central occupation of Celtis’ life that brought him into contact with the Renaissance mathematical sciences.

During his time in Italy he suffered under the jibes of his Italian colleges who said that whilst Italy had perfect humanist credentials being the inheritors of the ancient Roman culture, Germany was historically a land of uncultured barbarians. This spurred Celtis on to prove them wrong. He set himself the task of researching and writing a history of Germany to show that its culture was the equal of Italy’s. Celtis’ concept of history, like that of his Renaissance contemporaries, was more a mixture of our history and geography the two disciplines being regarded as two sides of the same coin. Geography being based on Ptolemaeus’ Geographia (Geographike Hyphegesis), which of course meant cartography, a branch of the mathematical sciences.

Continuing his travels in 1489 Celtis matriculated at the University of Kraków specifically to study the mathematical sciences for which Kraków had an excellent reputation. A couple of years later Nicolaus Copernicus would learn the fundamentals of mathematics and astronomy there. Wandering back to Germany via Prague and Nürnberg Celtis was appointed professor of poetics and rhetoric at the University of Ingolstadt in 1491/92. Ingolstadt was the first German university to have a dedicated chair for mathematics, established around 1470 to teach medical students astrology and the necessary mathematics and astronomy to cast a horoscope. When Celtis came to Ingolstadt there were the professor of mathematics was Andreas Stiborius (born Stöberl 1464–1515) who was followed by his best student Johannes Stabius (born Stöberer before 1468­–1522) both of whom Celtis convinced to support him in his cartographic endeavours.

In 1497 Celtis received a call to the University of Vienna where he established a Collegium poetarum et mathematicorum, that is a college for poetry and mathematics, with Stiborius, whom he had brought with him from Ingolstadt, as the professor for mathematics. In 1502 he also brought Stabius, who had succeeded Stiborius as professor in Ingolstadt, and his star student Georg Tanstetter to Vienna. Stiborius, Stabius and Tanstetter became what is known, to historians of mathematics, as the Second Viennese School of Mathematics, the First Viennese School being Johannes von Gmunden, Peuerbach and Regiomontanus, in the middle of the fifteenth century. Under these three Vienna became a major European centre for the mathematical sciences producing many important mathematicians the most notable being Peter Apian.

Although not a mathematician himself Conrad Celtis, the humanist poet, was the driving force behind one of the most important German language centres for Renaissance mathematics and as such earns a place in the history of mathematics. A dedicated humanist, wherever he went on his travels Celtis would establish humanist societies to propagate humanist studies and it was this activity that earned him the German title of Der Erzhumanist, in English the Arch Humanist. Celtis died in 1508 but his Collegium poetarum et mathematicorum survived him by twenty-two years, closing first in 1530

 

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

Christmas Trilogy 2015 Part 3: Roll out the barrel.

The village master taught his little school

The village all declared how much he knew,

‘Twas certain he could write, and cipher too;

Lands he could measure, times and tides presage,

And e’en the story ran that he could gauge

Oliver Goldsmith – The Deserted Village

As I have commented on a number of occasions in the past, although most people only know Johannes Kepler, if they have heard of him at all, as the creator of his eponymous three laws of planetary motion in fact he published more than eighty books and pamphlets in his life covering a very wide range of scientific and mathematical subjects. One of those publications, which often brings a smile to the faces of those not aware of its mathematical significance, is his Nova stereometria doliorum vinariorum (which translates as The New Art of Measuring the Contents of Wine Barrels) published in 1615. A whole book devoted to determining the volume of wine barrels! Surely not a suitable subject for a man who determined the laws of the cosmos and helped lay the foundations of modern optics, had the good Johannes taken to drink in the face of his personal problems?

Title page of Kepler's 1615 Nova stereometria doliorum vinariorum (image used by permission of the Carnegie Mellon University Libraries)

Title page of Kepler’s 1615 Nova stereometria doliorum vinariorum (image used by permission of the Carnegie Mellon University Libraries)

Because he is now regarded as one of the earliest ‘modern’ mathematicians people tend to forget that Kepler lived not in the age of the mathematician but in that of the mathematical practitioner. This means that as district mathematician in Graz, and later in Linz, Kepler would have been expected to carry out a large range of practical mathematical tasks including surveying, cartography, dialling (that is the design and construction of sundials), writing astrological prognostica, almanacs and calendars and gauging amongst others. We know that Kepler carried out a lot of these tasks but as far as I know he was never employed as a gauger, that is a man responsible for measuring and/or calculating the volume of barrels and their contents.

Nowadays with the wooden barrel degraded to the role of garden ornament in the forecourts of kitschy country pubs it is hard for people to imagine that for more than half a millennium the art of gauging and the profession of the gauger were a widespread and important part of the political and business life of Europe. Wooden barrels first made their appearance during the iron age, that is sometime during the first millennium BCE, iron making it possible to make tools with which craftsmen could work and shape the hard woods used to make barrels. It seems that we owe the invention of the barrel to the Celtic peoples of Northern Europe, who were making wooden barrels at least as early as five hundred BCE, although wooden buckets go back much earlier, with the earliest known one being from Egypt, 2690 BCE. The early wooden buckets were carved from single blocks of wood unlike barrels that are made from staves assembled and held together with hoops of saplings, rope or iron.

Source: Wood, Whiskey and Wine: A History of Barrels by Henry H. Work

Source: Wood, Whiskey and Wine: A History of Barrels by Henry H. Work

The ancient Greeks and Romans used large clay vessels called amphora to transport goods, in particular liquids such a wine and oil.

Roman Amphorae Source: Wikimedia Commons

Roman Amphorae
Source: Wikimedia Commons

However by about two hundred to three hundred CE the Romans, to whom we owe our written knowledge (supported by archaeological finds) of the Celtic origins of barrel making, were transporting wine in barrels. Wooden barrels appear to be a uniquely European invention appearing first in other parts of the world when introduced by Europeans.

By the Middle Ages wooden barrels had become ubiquitous throughout Europe used for transporting and storing a bewildering range of both dry and wet goods including books and corpses, the latter conserved in alcohol. With the vast increase in trade, both national and international, came the problem of taxes and custom duties on borders or at town gates. Wine, beer and spirits were taxed according to volume and the tax officials were faced with the problem of determining the volumes of the diverse barrels that poured daily across borders or through town gates, enter the gauger and the gauging rod.

Gauger with gauging rod Source:

Gauger with gauging rod
Source:

The simplest method of determining the volume of liquid contained in a barrel would be to pour out contents into a measuring vessel. This was of course not a viable choice for tax or customs official, so something else had to be done. Because of its shape determining the volume of a barrel-shaped container is not a simple geometrical exercise like that of determining the volume of a cylinder, sphere or cube so the mathematicians had to find another way. The solution was a gauging rod. This is a rod marked with a scale that was inserted diagonally into the barrel through the bung hole and by reading off the number on the scale the gauger could then calculate a good approximation of the volume of fluid in the barrel and then calculate the tax or custom’s duty due. From some time in the High Middle ages through to the nineteenth century gaugers and their gauging rods and gauging slide rules were a standard part of the European trade landscape.

A gauging slide rule Source

A gauging slide rule
Source

The mathematical literature on the art of gauging, particularly from the Early Modern Period is vast. As a small side note Antonie van Leeuwenhoek, the famous seventeenth-century microscopist, also worked for a time as gauger for the City of Delft.

A Cooper Jan Luyken Source

A Cooper Jan Luyken
Source

However after this brief excursion into the history of barrels and barrel gauging it is time to turn attention back to Kepler and his Nova stereometria doliorum vinarioru. In 1613, now living in Linz, Kepler purchased some barrels to lay in a supply of wine for his family. The wine dealer filled the casks and proceeded to measure the volume they contained using a gauging rod. Kepler being a notoriously exacting mathematician was horrified by the inaccuracy of this method of measurement and set about immediately to see if he produce a better mathematical method of determining the volume of barrels. Returning to the Eudoxian/Archimedian method of exhaustion that he had utilized to determine his second law of planetary motion he presented the volume of the barrel as the sum of a potentially infinite sum of a series of slices through the barrels. In modern terminology he used integral calculus to determine the volume. Never content to do half a job Kepler extended his mathematical investigations to determining the volumes of a wide range of three-dimensional containers and his efforts developed into a substantial book. Because he lacked the necessary notions of limits and convergence when summing infinite series, Kepler’s efforts lack mathematical rigour, as had his determination of his second law, a fact that Kepler was more than aware of. However, as with his second law he was prepared to sacrifice rigour for a practical functioning solution and to leave it to prosperity posterity to clean up the mess.

Having devoted so much time and effort to the task Kepler decided to publish his studies and immediately ran into new problems. There was at the time no printer/publisher in Linz so Kepler was forced to send his manuscript to Markus Welser, rich trader and science patron from Augsburg, who initiated the sunspot dispute between Galileo and Christoph Scheiner, to get his book published there. Unfortunately none of the printer/publishers in Augsburg were prepared to take on the risk of publishing the book and when Welser died in 1614 Kepler had to retrieve his manuscript and make other arrangements. In 1615 he fetched the printer Johannes Plank from Erfurt to Linz and paid him to print the book at his own cost. Unfortunately it proved to be anything but a best seller leaving Kepler with a loss on his efforts. In order to make his new discoveries available to a wider audience Kepler edited a very much simplified German edition in the same year under the title Ausszug auss der Vralten Messkunst Archimedis (Excerpts from the ancient art of mensuration by Archimedes). This book is important in the history of mathematics for provided the first German translations of numerous Greek and Latin mathematical terms. Plank remained in Linz and became Kepler’s house publisher during his time there.

Ausszug auss der Vralten Messkunst Archimedis title page Source

Ausszug auss der Vralten Messkunst Archimedis title page
Source

Although not one of his most successful works Kepler’s Nova stereometria doliorum is historically important for two different reasons. It was the first book to present a systematic study of the volumes of barrels based on geometrical principles and it also plays an important role in the history of infinitesimal calculus.

 

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Filed under History of Computing, History of Mathematics

Hans Holbein and the Nürnberg–Ingolstadt–Vienna Renaissance mathematical nexus.

There is a strong tendency, particularly in the popular history of science, to write about or present scientists as individuals. This leads to a serious distortion of the way that science develops and in particular propagates the lone genius myth. In reality science has always been a collective endeavour with its practitioners interacting in many different ways and on many different levels. In the Renaissance, when travelling from one end of Europe to the other would take weeks and letters often even longer, one might be excused for thinking that such cooperation was very low level but in fact the opposite was the truth, with scholars in the mathematical sciences exchanging information and ideas throughout Europe. A particularly strong mathematical nexus existed in the Southern German speaking area between the cities of Nürnberg, Ingolstadt and Vienna in the century between 1450 and 1550. Interestingly two of the paintings of the Northern Renaissance artist Hans Holbein the Younger open a door into this nexus.

Holbein (c. 1497–1543) was born in Augsburg the son of the painter and draughtsman Hans Holbein the Elder. As a young artist he lived and worked for a time in Basel where he became acquainted with Erasmus and worked for the printer publisher Johann Froben amongst others. Between 1526 and 1528 he spent some time in England in the household of Thomas More and it is here that he painted the second portrait I shall be discussing. The next four years find him living in Basel again before he returned to England in 1532 where he became associated with the court of Henry VIII, More having fallen out of favour. It was at the court that he painted, what is probably his most well know portrait, The Ambassadors in 1533.

Hans Holbein The Ambassadors Source: Wikimedia Commons

Hans Holbein The Ambassadors
Source: Wikimedia Commons

The painting shows two courtiers, usually identified as the French Ambassador Jean de Dinteville and Georges de Selve, Bishop of Lavaur standing on either side of a set of shelves laden with various books and instruments. There is much discussion was to what the instruments are supposed to represent but it is certain that, whatever else they stand for, they represent the quadrivium, arithmetic, geometry music and astronomy, the four mathematical sciences taught at European medieval universities. There are two globes, on the lower shelf a terrestrial and on the upper a celestial one. The celestial globe has been positively identified, as a Schöner globe and the terrestrial globe also displays characteristics of Schöner’s handwork.

Terrestrial Globe The Ambassadors Source Wikimedia Commons

Terrestrial Globe The Ambassadors
Source Wikimedia Commons

Celestial Globe The Ambassadors Source Wikimedia Commons

Celestial Globe The Ambassadors
Source Wikimedia Commons

Johannes Schöner (1477–1547) was professor for mathematics at the Egidienöberschule in Nürnberg, the addressee of Rheticus’ Narratio Prima, the founder of the tradition of printed globe pairs, an editor of mathematical texts for publication (especially for Johannes Petreius the sixteenth centuries most important scientific publisher) and one of the most influential astrologers in Europe. Schöner is a central and highly influential figure in Renaissance mathematics.

On the left hand side of the lower shelf is a copy of Peter Apian’s Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen (published in Ingolstadt in 1527) held open by a ruler. This is a popular book of commercial arithmetic, written in German, typical of the period. Peter Apian (1495–1552) professor of mathematics at the University of Ingolstadt, cartographer, printer-publisher and astronomer was a third generation representative of the so-called Second Viennese School of Mathematics. A pupil of Georg Tannstetter (1482–1535) a graduate of the University of Ingolstadt who had followed his teachers Johannes Stabius and Andreas Stiborious to teach at Conrad Celtis’ Collegium poetarum et mathematicorum, of which more later. Together Apian and Tannstetter produced the first printed edition of the Optic of Witelo, one of the most important medieval optic texts, which was printed by Petreius in Nürnberg in 1535. The Tannstetter/Apian/Petreius Witelo was one of the books that Rheticus took with him as a present for Copernicus when he visited him in 1539. Already, a brief description of the activities of Schöner and Apian is beginning to illustrate the connection between our three cities.

Apian's Arithmetic Book The Ambassadors Source: Wikimedia Commons

Apian’s Arithmetic Book The Ambassadors
Source: Wikimedia Commons

When Sebastian Münster (1488–1552), the cosmographer, sent out a circular requesting the cartographers of Germany to supply him with data and maps for his Cosmographia, he specifically addressed both Schöner and Apian by name as the leading cartographers of the age. Münster’s Cosmographia, which became the biggest selling book of the sixteenth century, was first published by Heinrich Petri in Basel in 1544. Münster was Petri’s stepfather and Petri was the cousin of Johannes Petreius, who learnt his trade as printer publisher in Heinrich’s printing shop in Basel. The Petri publishing house was also part of a consortium with Johann Amerbach and Johann Froben who had employed Hans Holbein in his time in Basel. Wheels within wheels.

The, mostly astronomical, instruments on the upper shelf are almost certainly the property of the German mathematician Nicolaus Kratzer (1487–1550), who is the subject of the second Holbein portrait who will be looking at.

Nicolas Kratzer by Hans Holbein Source: Wikimedia Commona

Nicolas Kratzer by Hans Holbein
Source: Wikimedia Commona

Born in Munich and educated at the universities of Cologne and Wittenberg Kratzer, originally came to England, like Holbein, to become part of the Thomas More household, where he was employed as a tutor for More’s children. Also like Holbein, Kratzer moved over to Henry VIII’s court as court horologist or clock maker, although the clocks he was responsible for making were more probably sundials than mechanical ones. During his time as a courtier Kratzer also lectured at Oxford and is said to have erected a monumental stone sundial in the grounds of Corpus Christi College. One polyhedral sundial attributed to Kratzer is in the Oxford Museum for the History of Science.

Polyhedral Sundial attributed to Nicolas Kratzer Source: MHS Oxford

Polyhedral Sundial attributed to Nicolas Kratzer
Source: MHS Oxford

In 1520 Kratzer travelled to Antwerp to visit Erasmus and here he met up with Nürnberg’s most famous painter Albrecht Dürer, who regular readers of this blog will know was also the author of a book on mathematics. Dürer’s book contains the first printed instructions, in German, on how to design, construct and install sundials, so the two men will have had a common topic of interest to liven there conversations. Kratzer witnessed Dürer, who was in Antwerp to negotiate with the German Emperor, painting Erasmus’ portrait and Dürer is said to have also drawn a portrait of Kratzer that is now missing. After Kratzer returned to England and Dürer to Nürnberg the two of them exchanged, at least once, letters and it is Kratzer’s letter that reveals some new connections in out nexus.

Albrecht Dürer selfportrait Source: Wikimedia Commons

Albrecht Dürer selfportrait
Source: Wikimedia Commons

In his letter, from 1524, Kratzer makes inquires about Willibald Pirckheimer and also asks if Dürer knows what has happened to the mathematical papers of Johannes Werner and Johannes Stabius who had both died two years earlier.

Willibald Pirckheimer (1470–1530) a close friend and patron of Dürer’s was a rich merchant, a politician, a soldier and a humanist scholar. In the last capacity he was the hub of a group of largely mathematical humanist scholars now known as the Pirckheimer circle. Although not a mathematician himself Pirckheimer was a fervent supporter of the mathematical sciences and produced a Latin translation from the Greek of Ptolemaeus’ Geōgraphikḕ or Geographia, Pirckheimer’s translation provided the basis for Sebastian Münster’s edition, which was regarded as the definitive text in the sixteenth century. Stabius and Werner were both prominent members of the Pirckheimer circle.

Willibald Pirckheimer by Albrecht Dürer Source: Wikimedia Commons

Willibald Pirckheimer by Albrecht Dürer
Source: Wikimedia Commons

The two Johanneses, Stabius (1450–1522) and Werner (1468–1522), had become friends at the University of Ingolstadt where the both studied mathematics. Ingolstadt was the first German university to have a dedicated chair for mathematics. Werner returned to his hometown of Nürnberg where he became a priest but the Austrian Stabius remained in Ingolstadt, where he became professor of mathematics. The two of them continued to correspond and work together and Werner is said to have instigated the highly complex sundial on the wall of the Saint Lorenz Church in Nürnberg, which was designed by Stabius and constructed in 1502.

St Lorenz Church Nürnberg Sundial 1502 Source: Astronomie in Nürnberg

St Lorenz Church Nürnberg Sundial 1502
Source: Astronomie in Nürnberg

It was also Werner who first published Stabius’ heart shaped or cordiform map projection leading to it being labelled the Werner-Stabius Projection. This projection was used for world maps by Peter Apian as well as Oronce Fine, France’s leading mathematicus of the sixteenth century and Gerard Mercator, of whom more, later. The network expands.

Mercator cordiform world map 1538 Source: American Geographical Society Library

Mercator cordiform world map 1538
Source: American Geographical Society Library

In his own right Werner produced a partial Latin translation from the Greek of Ptolemaeus’ Geographia, was the first to write about prosthaphaeresis (a trigonometrical method of simplifying calculation prior to the invention of logarithms), was the first to suggest the lunar distance method of determining longitude and was in all probability Albrecht Dürer’s maths teacher. He also was the subject of an astronomical dispute with Copernicus.

Johannes Werner Source: Wikimedia Commons

Johannes Werner
Source: Wikimedia Commons

Regular readers of this blog will know that Stabius co-operated with Albrecht Dürer on a series of projects, including his famous star maps, which you can read about in an earlier post here.

Johannes Statius Portrait by Albrecht Dürer Source: Wikimedia Commons

Johannes Statius Portrait by Albrecht Dürer
Source: Wikimedia Commons

An important non-Nürnberger member of the Pirckheimer Circle was Conrad Celtis (1459–1508), who is known in Germany as the arch-humanist. Like his friend Pirckheimer, Celtis was not a mathematician but believed in the importance of the mathematical sciences. Although already graduated he spent time in 1489 on the University of Kraków in order to get the education in mathematics and astronomy that he couldn’t get at a German university. Celtis had spent time at the humanist universities of Northern Italy and his mission in life was to demonstrate that Germany was just as civilised and educated as Italy and not a land of barbarians as the Italians claimed. His contributions to the Nuremberg Chronicle can be viewed as part of this demonstration. He believed he could achieve his aim by writing a comprehensive history of Germany including, as was common at the time its geography. In 1491/92 he received a teaching post in Ingolstadt, where he seduced the professors of mathematics Johannes Stabius and Andreas Stiborius (1464–1515) into turning their attention from astrology for medicine student, their official assignment, to mathematical cartography in order to help him with his historical geography.

Conrad Celtis Source: Wikimedia Commons

Conrad Celtis
Source: Wikimedia Commons

Unable to achieve his ends in Ingolstadt Celtis decamped to Vienna, taking Stabius and Stiborius with him, to found his Collegium poetarum et mathematicorum as mentioned above and with it the so-called Second Viennese School of Mathematics; the first had been Peuerbach and Regiomontanus in the middle of the fifteenth century. Regiomontanus spent the last five years of his life living in Nürnberg, where he set up the world’s first scientific publishing house. Stiborius’ pupil Georg Tannstetter proved to be a gifted teacher and Peter Apian was by no means his only famous pupil.

The influence of the Nürnberg–Ingolstadt–Vienna mathematicians reached far beyond their own relatively small Southern German corridor. As already stated Münster in Basel stood in contact with both Apian and Schöner and Stabius’ cordiform projection found favour with cartographers throughout Northern Europe. Both Apian and Schöner exercised a major influence on Gemma Frisius in Louvain and through him on his pupils Gerard Mercator and John Dee. As outlined in my blog post on Frisius, he took over editing the second and all subsequent editions of Apian’s Cosmographia, one of the most important textbooks for all things astronomical, cartographical and to do with surveying in the sixteenth century. Frisius also learnt his globe making, a skill he passed on to Mercator, through the works of Schöner. Dee and Mercator also had connections to Pedro Nunes (1502–1578) the most important mathematicus on the Iberian peninsular. Frisius had several other important pupils who spread the skills in cosmography, and globe and instrument making that he had acquired from Apian and Schöner all over Europe.

Famously Rheticus came to Nürnberg to study astrology at the feet of Johannes Schöner, who maintained close contacts to Philipp Melanchthon Rheticus patron. Schöner was the first professor of mathematics at a school designed by Melanchthon. Melanchthon had learnt his mathematics and astrology at the University of Tübingen from Johannes Stöffler (1452–1531) another mathematical graduate from Ingolstadt.

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica Source: Wikimedia Commons

Kupferstich aus der Werkstatt Theodor de Brys, erschienen 1598 im 2. Bd. der Bibliotheca chalcographica
Source: Wikimedia Commons

Another of Stöffler’s pupils was Sebastian Münster. During his time in Nürnberg Rheticus became acquainted with the other Nürnberger mathematicians and above all with the printer-publisher Johannes Petreius and it was famously Rheticus who brought the manuscript of Copernicus’ De revolutionibus to Nürnberg for Petreius to publish. Rheticus says that he first learnt of Copernicus’s existence during his travels on his sabbatical and historians think that it was probably in Nürnberg that he acquired this knowledge. One of the few pieces of astronomical writing from Copernicus that we have is the so-called Letter to Werner. In this manuscript Copernicus criticises Werner’s theory of trepidation. Trepidation was a mistaken belief based on faulty data that the rate of the precession of the equinoxes is not constant but varies with time. Because of this highly technical dispute amongst astronomers Copernicus would have been known in Nürnberg and thus the assumption that Rheticus first heard of him there. Interestingly Copernicus includes observations of Mercury made by Bernhard Walther (1430–1504), Regiomontanus partner, in Nürnberg; falsely attributing some of them to Schöner, so a connection between Copernicus and Nürnberg seems to have existed.

In this brief outline we have covered a lot of ground but I hope I have made clear just how interconnected the mathematical practitioners of Germany and indeed Europe were in the second half of the fifteenth century and the first half of the sixteenth. Science is very much a collective endeavour and historians of science should not just concentrate on individuals but look at the networks within which those individual operate bringing to light the influences and exchanges that take place within those networks.

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

A double bicentennial – George contra Ada – Reality contra Perception

The end of this year sees a double English bicentennial in the history of computing. On 2 November we celebrate the two hundredth anniversary of the birth of mathematician and logician Georg Boole then on 10 December the two hundredth anniversary of the birth of ‘science writer’ Augusta Ada King, Countess of Lovelace. It is an interesting exercise to take a brief look at how these two bicentennials are being perceived in the public sphere.

As I have pointed out in several earlier posts Ada was a member of the minor aristocracy, who, although she never knew her father, had a wealthy well connected mother. She had access to the highest social and intellectual circles of early Victorian London. Despite being mentored and tutored by the best that London had to offer she failed totally in mastering more than elementary mathematics. So, as I have also pointed out more than once, to call her a mathematician is a very poor quality joke. Her only ‘scientific’ contribution was to translate a memoire on Babbage’s Analytical Engine from French into English to which are appended a series of new notes. There is very substantial internal and external evidence that these notes in fact stem from Babbage and not Ada and that she only gave them linguistic form. What we have here is basically a journalistic interview and not a piece of original work. It is a historical fact that she did not write the first computer programme, as is still repeated ad nauseam every time her name is mentioned.

However the acolytes of the Cult of the Holy Saint Ada are banging the advertising drum for her bicentennial on a level comparable to that accorded to Einstein for the centenary of the General Theory of Relativity. On social media ‘Finding Ada’ are obviously planning massive celebrations, which they have already indicated although the exact nature of them has yet to be revealed. More worrying is the publication of the graphic novel The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer (note who gets first billing!) by animator and cartoonist Sydney Padua. The Analytical Engine as of course not the first computer that honour goes to Babbage’s Difference Engine. More important Padua’s novel is not even remotely ‘mostly’ true but largely fictional. This wouldn’t matter that much if said book had not received major media attention. Attention that compounded the error by conveniently forgetting the mostly. The biggest lie in the work of fiction is the claim that Ada was somehow directly involved in the conception and construction of the Analytical engine. In reality she had absolutely nothing to do with either its conception or its construction.

This deliberate misconception has been compounded by a, in social media widely disseminated, attempt to get support for a Lovelace, Babbage Analytical Engine Lego Set. The promoter of this enterprise has written in his blurb:

Ada Lovelace (1815-1852) is widely credited as the first computer scientist and Charles Babbage (1791-1871) is best remembered for originating the concept of a programmable computer. Together they collaborated on Babbage’s early mechanical general-purpose computer, the Analytical Engine.

Widely credited by whom? If anybody is the first computer scientist in this set up then it’s Babbage. Others such as Leibniz speculated on what we now call computer science long before Ada was born so I think that is another piece of hype that we can commit to the trashcan. Much more important is the fact that they did not collaborate on the Analytical Engine that was solely Babbage’s baby. This factually false hype is compounded in the following tweet from 21 July, which linked to the Lego promotion:

Historical lego [sic] of Ada Lovelace’s conception of the first programmable computer

To give some perspective to the whole issue it is instructive to ask about what in German is called the ‘Wirkungsgeschichte’, best translated as historical impact, of Babbage’s efforts to promote and build his computers, including the, in the mean time, notorious Menabrea memoire, irrespective as to who actually formulated the added notes. The impact of all of Babbage’s computer endeavours on the history of the computer is almost nothing. I say almost because, due to Turing, the notes did play a minor role in the early phases of the post World War II artificial intelligence debate. However one could get the impression from the efforts of the Ada Lovelace fan club, strongly supported by the media that this was a highly significant contribution to the history of computing that deserves to be massively celebrated on the Lovelace bicentennial.

Let us now turn our attention to subject of our other bicentennial celebration, George Boole. Born into a working class family in Lincoln, Boole had little formal education. However his father was a self-educated man with a thirst for knowledge, who instilled the same characteristics in his son. With some assistance he taught himself Latin and Greek and later French, German and Italian in order to be able to read the advanced continental mathematics. His father went bankrupt when he was 16 and he became breadwinner for the family, taking a post as schoolmaster in a small private school. When he was 19 he set up his own small school. Using the library of the local Mechanics Institute he taught himself mathematics. In the 1840s he began to publish original mathematical research in the Cambridge Mathematical Journal with the support of Duncan Gregory, a great great grandson of Newton’s contemporary James Gregory. Boole went on to become one of the leading British mathematicians of the nineteenth century and despite his total lack of formal qualifications he was appointed Professor of Mathematics at the newly founded Queen’s College of Cork in 1849.

Although a fascinating figure in the history of mathematics it is Boole the logician, who interests us here. In 1847 Boole published the first version of his logical algebra in the form of a largish pamphlet, Mathematical Analysis of Logic. This was followed in 1854 by an expanded version of his ideas in his An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probability. These publications contain the core of Boolean algebra, the final Boolean algebra was actually produced by Stanley Jevons, only the second non-standard algebra ever to be developed. The first non-standard algebra was Hamilton’s quaternions. For non-mathematical readers standard algebra is the stuff we all learned (and loved!) at school. Boolean algebra was Boole’s greatest contribution to the histories of mathematics, logic and science.

When it first appeared Boole’s logic was large ignored as an irrelevance but as the nineteenth century progressed it was taken up and developed by others, most notably by the German mathematician Ernst Schröder, and provided the tool for much early work in mathematical logic. Around 1930 it was superseded in this area by the mathematical logic of Whitehead’s and Russell’s Principia Mathematica. Boole’s algebraic logic seemed destined for the novelty scrap heap of history until a brilliant young American mathematician wrote his master’s thesis.

Claude Shannon (1916–2001) was a postgrad student of electrical engineering of Vannevar Bush at MIT working on Bush’s electro-mechanical computer the differential analyzer. Having learnt Boolean algebra as an undergraduate Shannon realised that it could be used for the systematic and logical design of electrical switching circuits. In 1937 he published a paper drawn from his master’s thesis, A Symbolic Analysis of Relay and Switching Circuits. Shannon switching algebra, applied Boolean algebra, would go on to supply the basis of the hardware design of all modern computers. When people began to write programs for the computers designed with Shannon’s switching algebra it was only natural that they would use Boole’s two-valued (1/0, true/false, on/off) algebra to write those programs. Almost all modern computers are both in their hardware and there software applied Boolean algebra. One can argue, as I have actually done somewhat tongue in cheek in a lecture, that George Boole is the ‘father’ of the modern computer. (Somewhat tongue in cheek, as I don’t actually like the term ‘father of’). The modern computer has of course many fathers and mothers.

In George Boole, as opposed to Babbage and Lovelace, we have a man whose work made a massive real contribution to history of the computer and although both the Universities of Cork and Lincoln are planning major celebration for his bicentennial they have been, up till now largely ignored by the media with the exception of the Irish newspapers who are happy to claim Boole, an Englishman, as one of their own.

The press seems to have decided that a ‘disadvantaged’ (she never was, as opposed to Boole) female ‘scientist’, who just happens to be Byron’s daughter is more newsworthy in the history of the computer than a male mathematician, even if she contributed almost nothing and he contributed very much.

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Filed under History of Computing, History of Mathematics, Ladies of Science, Myths of Science

Ohm Sweet Ohm

This is the story of two brothers born into the working class in a small town in Germany in the late eighteenth century. Both of them were recognised as mathematically gifted whilst still teenagers and went on to study mathematics at university. The younger brother was diligent and studious and completed his doctorate in mathematics with a good grade. There followed a series of good teaching jobs before he obtained a lectureship at the then leading university of Berlin, ten years after graduating. In due course, there followed positions as associate and the full professor. As professor he contributed some small but important proofs to the maths cannon, graduated an impressive list of doctoral students and developed an interesting approach to maths textbooks. He became a respected and acknowledged member of the German mathematical community.

The elder brother’s life ran somewhat differently. He started at the local university but unlike his younger brother he was anything but studious preferring a life of dancing, ice -skating and playing billiards to learning mathematics. His father a hard working craftsman was disgusted by this behaviour and forced him to leave the university and take up a teaching post in Switzerland. On the advice of his mathematics professor he taught himself mathematics by reading the greats. He returned to his home university and obtained his doctorate in the same year as his brother. There then followed a series of dead end jobs first as a badly paid university lecturer with little prospect of promotion and then a series of deadbeat jobs as a schoolteacher. In the last of these he had access to a good physics laboratory and began a series of investigations in a relatively new area of physics. At the age of thirty-eight, something of a failure, he published the results of his investigations in a book, which initially failed to make any impact. At the age of forty-four he obtained an appointment as professor at a polytechnic near to his home town and things began to finally improve in his life. At the age of fifty-two his work received acknowledgement at the highest international levels and finally at the age of sixty-three he was appointed professor for physics at a leading university.

The younger brother whose career path had been so smooth, fairly rapidly disappeared from the history of mathematics after his death in 1872, remembered by only a handful of specialists, whereas the much plagued elder brother went on to lend the family name to one of the most frequently used unit of measure in the physical sciences; a name that can be found on multiple appliances in probably every household in the western world.

The two bothers of my story are Georg Simon Ohm (1789–1854), the discover of Ohm’s Law, and his younger brother, the mathematician, Martin Ohm, who was born on 6 May 1792 and the small German town where they were born is Erlangen where I (almost) live.

The Ohm House, Fahrstraße 11, Erlangen Source: Wikimedia Commons

The Ohm House, Fahrstraße 11, Erlangen
Source: Wikimedia Commons

Georg Simon and Martin were the sons of the locksmith Johann Wolfgang Ohm and his wife Maria Elizabeth Beck, who died when Georg Simon was only ten. Not only did the father bring up his three surviving, of seven, children alone after the death of their mother but he also educated his two sons himself. The son of a locksmith he had enjoyed little formal education but had taught himself philosophy and mathematics, which he now imparted to his sons with great success. As Georg Simon was fifteen he and Martin were examined by the local professor of mathematics, Karl Christian von Langsdorf, who, as already described above, found both boys to be highly gifted and spoke of an Erlanger Bernoulli family.

Plaque on the Ohm House

Plaque on the Ohm House

The plaque reads: The locksmith Johann Wolfgang Ohm (1753–1823) brought up and taught in this house as a true master his later famous sons

Georg Simon Ohm (1789–1854) the great physicist and Martin Ohm (1792–1872) the mathematician

I’ve already outlined the lives of the two Ohm brothers so I’m not going to repeat myself but I will fill in some detail.

Martin Ohm Source: Wikimedia Commons

Martin Ohm
Source: Wikimedia Commons

As above I’ll start with Martin, the mathematician. He made no great discoveries as such and in the world of mathematics his main claim to fame is probably his list of doctoral students several of whom became much more famous than their professor. It was as a teacher that Martin Ohm made his mark, writing a nine volume work that attempted a systematic introduction to the whole of elementary mathematics his, Versuch eines vollkommenen, consequenten Systems der Mathematik (1822–1852) (Attempt at a complete consequent system of mathematics); a book that predates the very similar, but far better known, attempt by Bourbaki by one hundred years and which deserves far more attention than it gets. Martin Ohm also wrote several other elementary textbooks for his students. In his time in Berlin Martin Ohm also taught mathematics for many years at both the School for Architecture and the Artillery Academy.

I first stumbled across Martin Ohm whilst researching nineteenth-century algebraic logics. When it was first published George Boole’s Laws of Thought (1864) received very little attention from the mathematical community. With the exception of a small handful of relatively unknown mathematicians who wrote brief papers on it, it went largely unnoticed. One of that handful was Martin Ohm who wrote two papers in German (the first works in German on Boole’s logic). Thus introducing Boole’s ground-breaking work to the German mathematical public. Boole had written and published other mathematical work in German so he was already known in Germany. Later Ernst Schröder would go on to become the biggest proponent of Boolean logic with his three volume Vorlesungen über die Algebra der Logik (1890-1905). It is perhaps worth noting that Boole like the Ohm brothers was the son of a self-educated tradesman who gave his son his first education.

Martin Ohm has one further claim to notoriety; he is thought to have been the first to use the term “golden section” (goldener Schnitt in German) thus opening the door for hundreds of aesthetic loonies who claim to find evidence of this wonderful ration all over the place.  

Georg Simon Ohm

Georg Simon Ohm

We now move on to the man in whose shadow Martin Ohm will always stand, his elder brother Georg Simon.

House in Erlangen just around the corner from their birth house, where both Georg Simon and Martin worked as poorly paid lecturers for physics Photo: Thony Christie

House in Erlangen just around the corner from their birth house, where both Georg Simon and Martin worked as poorly paid lecturers for physics
Photo: Thony Christie

Plaque on house Photo: Thony Christie

Plaque on house
Photo: Thony Christie

The Plaque reads: In this house the physicist Georg Simon Ohm (1789–1854) taught physics in the years 1811 to 1812 and the mathematician Martin Ohm (1792–1872) in the years 1812 to 1817

The school where Georg Simon began the research work into the physics of electricity was the Jesuit Gymnasium in Cologne, which even granted him a sabbatical in 1826 to intensify his researches. He published those researches as Die galvanische Kette, mathematisch bearbeitet (The Galvanic Circuit Investigated Mathematically) in 1827. It was the Royal Society who started his climb out of obscurity awarding him the Copley Medal, its highest award, in 1842 and appointing him a foreign member in the same year. Membership of other international scientific societies, such as Turin followed. Georg Simon’s first professorial post was at the Königlich Polytechnische Schule (Royal Polytechnic) in Nürnberg in 1833. He became the director of the Polytechnic in 1839 and today the school is a technical university, which bears the name Georg Simon Ohm. Georg Simon ended his career as professor of physics at the University of Munich.

The town of Erlangen is proud of Georg Simon and we have an Ohm Place, with an unfortunately rather derelict fountain, the subject of a long political debate concerning the cost of renovation and one of the town’s high schools is named the Ohm Gymnasium. The city of Munich also has a collection of plaques and statues honouring him. Ohm Straße in Berlin, however, is named after his brother Martin.

Statue of Georg Simon Ohm at the Technical University in Munich Source: Wikimedia Commons

Statue of Georg Simon Ohm at the Technical University in Munich
Source: Wikimedia Commons

Any fans of the history of science with a sweet tooth should note that if they come to Erlangen one half of the Ohm House is now a sweet shop specialising in Gummibärs.

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Emmy the student and Emmy the communist!

Emmy Noether’s birthday on 23 March saw her honoured with a Google Doodle, which of course led to various people posting brief biographies of Erlangen’s most famous science personality or drawing attention to existing posts in the Internet.

Emmy Google Doodle

Almost all of these posts contain two significant errors concerning Emmy’s career that I would like to correct here. For those interested I have written earlier posts on Emmy’s family home in Erlangen and the problems she went through trying to get her habilitation, the German qualification required to be able to teach at university.

The first oft repeated error concerns Emmy’s education and I quote a typical example below:

Today she is celebrated for her contributions to abstract algebra and theoretical physics, but in 20th-century Bavaria, Noether had to fight for every bit of education and academic achievement. Women were not allowed to enrol at the University of Erlangen, so Noether had to petition each professor to attend classes.

As a teenager Emmy displayed neither an interest nor a special aptitude for mathematics but rather more for music and dance. She attended the Städtische Höhere Töchterschule (the town secondary girls’ school), now the Marie-Therese-Gymnasium, and in 1900 graduated as a teacher for English and French at the girls’ school in Ansbach. In 1903 she took her Abitur exam externally at the Königlichen Realgymnasium in Nürnberg. The Abitur is the diploma from German secondary school qualifying for university admission or matriculation. Previous to this she had been auditing some mathematics courses in Göttingen as a guest student with the personal permission of the professors whose courses she visited, hence the claim above. However she had become ill and had returned home to Erlangen. In 1903 the laws were changed in Bavaria allowing women to register at university for the first time. Emmy registered as a regular student at the University of Erlangen in 1903 and graduated with a PhD in mathematics in 1907, under the supervision of Paul Gordon, in invariant theory. She was only the second woman in Germany to obtain a PhD in mathematics. In 1908 she became a member of the Circolo Matematico di Palermo and in 1909 a member of the Deutschen Mathematiker-Vereinigung. In 1909 Hilbert and Klein invited her to come to the University of Göttingen, as a post-doc researcher. It was here in 1915 that Hilbert suggested that she should habilitate with the well know consequences.

Emmy remained in Göttingen until the Nazis came to power in 1933. She held guest professorships in Moscow in 1928/29 and in Frankfort am Main in 1930. She was awarded the Ackermann-Teubner Memorial Prize for her complete scientific work in 1932 and held the plenary lecture at the International Mathematical Congress in Zurich also in 1932. In 1933 when the Nazis came to power she was expelled from her teaching position in Göttingen and it is here that the second oft repeated error turns up.

On coming to power the Nazis introduced the so-called Gesetz zur Wiederherstellung des Berufsbeamtentums, (The Law for the Restoration of the Professional Civil Service). This was a law introduced by the Nazis to remove all undesirables from state employment, this of course meant the Jews but also, socialists, communists and anybody else deemed undesirable by the Nazi Party. Like many of her colleges in the mathematics department at Göttingen Emmy was removed from her teaching position under this law. In fact the culling in the mathematics department was so extreme that it led to a famous, possibly apocryphal, exchange between Bernhard Rust (and not Hermann Göring, see comments) and David Hilbert.

Rust: “I hear you have some problems in the mathematics department at Göttingen Herr Professor”.

Hilbert: “No, there are no problems; there is no mathematics department in Göttingen”.

The Wikipedia article on the history of the University Göttingen gives the story as follows (in German)

Ein Jahr später erkundigte sich der Reichserziehungsminister Bernhard Rust anlässlich eines Banketts bei dem neben ihm platzierten Mathematiker David Hilbert ob das mathematische Institut in Göttingen durch die Entfernung der jüdischen, demokratischen und sozialistischen Mathematiker gelitten habe. Hilbert soll in seiner ostpreußischen Mundart (laut Abraham Fraenkel, Lebenskreise, 1967, S. 159) erwidert haben: „Jelitten? Dat hat nich jelitten, Herr Minister. Dat jibt es doch janich mehr.“

The source here is given as Abraham Fraenkel in his autobiography Lebenkreise published in 1967.

This translates as follows:

One year later [that is after the expulsions in 1933] the Imperial Education Minister Bernhard Rust, who was seated next to the mathematician David Hilbert at a banquet, inquired, whether the Mathematics Institute at Göttingen had suffered through the removal of the Jewish, democratic and socialist mathematicians. Hilbert is said to have replied in his East Prussian dialect” Suffered? It hasn’t suffered, Herr Minister. It doesn’t exist anymore”

It is usually claimed that Emmy lost her position because she was Jewish, a reasonable assumption but not true. Emmy lost her position, like many other in Göttingen, because the Nazis thought she was a communist. Like many European universities in the 1920s and 30s Göttingen was a hot bed of radical intellectual socialism. Emmy had been a member of a radical socialist party in the early twenties but changed later to the more moderate SPD, who were also banned by the Nazis. However it was her guest professorship in Moscow that proved her undoing. Because she reported positively on her year in Russia the Nazis considered her to be a communist and this was the reason for her expulsion from the university in 1933.

Initially Emmy, after her expulsion, actually applied for a position at the University of Moscow but the attempts by the Russian topologist Pavel Alexandrov to get her a position got bogged down in the Russian bureaucracy and so when, through the good offices of Hermann Weyl, she received the offer of a guest professorship in America at Bryn Mawr College she accepted. In America she taught at Bryn Mawr and the Institute of Advanced Studies in Princeton but tragically died of cancer of the uterus in 1935.

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A Swiss Clockmaker

We all have clichéd images in our heads when we hear the names of countries other than our own. For many people the name Switzerland evokes a muddled collection of snow-covered mountains, delicious superior chocolates and high precision clocks and watches. Jost Bürgi who was born in the small town of Lichtensteig, in the  Toggenburg region of the canton of St. Gallen on 28 February 1552 fills this cliché as the most expert clockmaker in the sixteenth century. However Bürgi was much more that just a Swiss clockmaker, he was also an instrument maker, an astronomer, a mathematician and in his private life a successful property owner and private banker, the last of course serving yet another Swiss cliché.

As we all too many figures, who made significant contributions to science and technology in the Renaissance we know next to nothing about Bürgi’s origins or background. There is no known registration of his birth or his baptism; his date of birth is known from the engraving shown below from 1592, in which the portrait was added in 1619 but which was first published in 1648. That the included date is his birthday was confirmed by Bürgi’s brother in law.

Bramer1648

His father was probably the locksmith Lienz Bürgi but that is not known for certain. About his education or lack of it nothing is known at all and just as little is known about where he learnt his trade as clockmaker. Various speculations have been made by historians over the years but they remain just speculations. The earliest documentary proof that we have of Bürgi’s existence is his employment contract when he entered the service of the Landgrave Wilhelm IV of Hessen-Kassel as court clockmaker, already twenty-seven years old, on 25 July 1579. Wilhelm was unique amongst the German rulers of the Renaissance in that he was not only a fan or supporter of astronomy but was himself an active practicing astronomer. In his castle in Kassel he constructed, what is recognised as, the first observatory in Early Modern Europe.

Wilhelm IV. von Hessen-Kassel Source: Wikimedia Commons

Wilhelm IV. von Hessen-Kassel
Source: Wikimedia Commons

He also played a major role in persuading the Danish King Frederick II, a cousin, to supply Tycho Brahe with the necessary land and money to establish an observatory in Denmark. In the 1560s Wilhelm was supported in his astronomical activities by Andreas Schöner, the son of the famous Nürnberger cartographer, globe and instrument maker, astronomer, astrologer and mathematician Johannes Schöner. He also commissioned the clockmaker Eberhard Baldewein (1525-1593) to construct two planet clocks and a mechanical globe.

 

Eberhart Baldewein Planet clock 1661 Source: Wikimedia Commons

Eberhart Baldewein Planet clock 1661
Source: Wikimedia Commons

The planet clock shows the positions of the sun, moon and the planets, based on Peter Apian’s Astronomicom Caessareum, on its various dials.

 

Eberhard Baldewein Mechanical Celestial Globe circa 1573

Eberhard Baldewein Mechanical Celestial Globe circa 1573 The globe, finished by Heinrich Lennep in 1693, was used to record the position of the stars mapped by Wilhelm and his team in their observations.

These mechanical objects were serviced and maintained by Baldewein’s ex-apprentice, Hans Bucher, who had helped to build them and who had been employed by Wilhelm, for this purpose, since 1560. When Bucher died in 1578-1579 Bürgi was employed to replace him, charged with the maintenance of the existing objects on a fixed, but very generous salary, and commissioned to produce new mechanical instruments for which he would be paid extra. Over the next fifty years Bürgi produced many beautiful and highly efficient clocks and mechanical globes both for Wilhelm and for others.

Bürgi Quartz Clock 1622-27 Source: Swiss Physical Society

Bürgi Quartz Clock 1622-27
Source: Swiss Physical Society

 

 

 

 

 

Bürgi Mechanical Celestial Globe 1594 Source: Wikimedia Commons

Bürgi Mechanical Celestial Globe 1594
Source: Wikimedia Commons

 

 

Jost Bürgi and Antonius Eisenhoit: Armillary sphere with astronomical clock made 1585 in Kassel, now at Nordiska Museet in Stockholm. Source Wikimedia Commons

Jost Bürgi and Antonius Eisenhoit: Armillary sphere with astronomical clock made 1585 in Kassel, now at Nordiska Museet in Stockholm.
Source Wikimedia Commons

Bürgi was also a highly inventive clockmaker, who is credited with the invention of both the cross-beat escapement and the remontoire, two highly important improvements in clock mechanics. In the late sixteenth century the average clocks were accurate to about thirty minutes a day, Bürgi’s clock were said to be accurate to less than one minute a day. This amazing increase in accuracy allowed mechanical clocks to be used, for the first time ever, for timing astronomical observations. Bürgi also supplied clocks for this purpose for Tycho’s observatory on Hven. In 1592 Wilhelm presented his nephew Rudolph II, the German Emperor, with one of Bürgi’s mechanical globes and Bürgi was sent to Prague with the globe to demonstrate it to Rudolph. This was his first contact with what would later become his workplace. Whilst away from Kassel Bürgi’s employer, Wilhelm died. Before continuing the story we need to go back and look at some of Bürgi’s other activities.

As stated at the beginning Bürgi was not just a clockmaker. In 1584 Wilhelm appointed the Wittenberg University graduate Christoph Rothmann as court astronomer. From this point on the three, Wilhelm, Rothmann and Bürgi, were engaged in a major programme to map the heavens, similar to and just as accurate, as that of Tycho on Hven. The two observatories exchanged much information on instruments, observations and astronomical and cosmological theories. However all was not harmonious in this three-man team. Although Wilhelm treated Bürgi, whom he held in high regard, with great respect Rothmann, who appears to have been a bit of a snob, treated Bürgi with contempt because he was uneducated and couldn’t read or write Latin, that Bürgi was the better mathematician of the two might have been one reason for Rothmann’s attitude.

In the 1580s the itinerant mathematician and astronomer Paul Wittich came to Kassel from Hven and taught Bürgi prosthaphaeresis, a method using trigonometric formulas, of turning multiplication into addition, thus simplifying complex astronomical calculations. The method was first discovered by Johannes Werner in Nürnberg at the beginning of the sixteenth century but he never published it and so his discovery remained unknown. It is not known whether Wittich rediscovered the method or learnt of it from Werner’s manuscripts whilst visiting Nürnberg. The method was first published by Nicolaus Reimers Baer, who was then accused by Tycho of having plagiarised the method, Tycho claiming falsely that he had discovered it. In fact Tycho had also learnt it from Wittich. Bürgi had expanded and improved the method and when Baer also came to Kassel in 1588, Bürgi taught him the method and how to use it, in exchange for which Baer translated Copernicus’ De revolutionibus into German for Bürgi. This was the first such translation and a copy of Baer’s manuscript is still in existence in Graz. Whilst Baer was in Kassel Bürgi created a brass model of the Tychonic geocentric-heliocentric model of the cosmos, which Baer claimed to have discovered himself. When Tycho got wind of this he was apoplectic with rage.

In 1590 Rothmann disappeared off the face of the earth following a visit to Hven and for the last two years of Wilhelm’s life Bürgi took over as chief astronomical observer in Kassel, proving to be just as good in this work as in his clock making.

Following Wilhelm’s death his son Maurice who inherited the title renewed Bürgi’s contract with the court.

 

Kupferstich mit dem Porträt Moritz von Hessen-Kassel aus dem Werk Theatrum Europaeum von 1662 Source: Wikimedia Commons

Kupferstich mit dem Porträt Moritz von Hessen-Kassel aus dem Werk Theatrum Europaeum von 1662
Source: Wikimedia Commons

However Maurice did not share his father’s love of astronomy investing his spare time instead in the study of alchemy. Bürgi however continued to serve the court as clock and instrument maker. Over the next eight years Bürgi made several visits to the Emperor’s court in Prague and in 1604 Rudolph requested Maurice to allow him to retain Bürgi’s services on a permanent basis. Maurice acquiesced and Bürgi moved permanently to Prague although still remaining formally in service to Maurice in Kassel. Rudolph gave Bürgi a very generous contract paying him 60 gulden a month as well as full board and lodging. As in Kassel all clocks and globes were paid extra. To put that into perspective 60 gulden was a yearly wage for a young academic starting out on his career!

In Prague Bürgi worked closely with the Imperial Mathematicus, Johannes Kepler. Kepler, unlike Rothmann, respected Bürgi immensely and encouraged him to publish his mathematical works. Bürgi was the author of an original Cos, an algebra textbook, from which Kepler says he learnt much and which only saw the light of day through Kepler’s efforts. Kepler was also responsible for the publication of Bürgi’s logarithmic tables in 1620.

 

Bürgi's Logarithmic Tables Source: University of Graz

Bürgi’s Logarithmic Tables
Source: University of Graz

This is probably Bürgi’s greatest mathematical achievement and he is considered along side of John Napier as the inventor of logarithms. In many earlier historical works Bürgi is credited with having invented logarithms before Napier. Napier published his tables in 1614 six years before Bürgi and is known to have been working on them for twenty years, that is since 1594. Bürgi’s fan club claim that he had invented his logarithms in 1588 that is six years earlier than Napier. However modern experts on the history of logarithms think that references to 1588 are to Bürgi’s use of prosthaphaeresis and that he didn’t start work on his logarithms before 1604. However it is clear that the two men developed the concept independently of each other and both deserve the laurels for their invention. It should however be pointed out that the concept on which logarithms are based was known to Archimedes and had already been investigated by Michael Stifel earlier in the sixteenth century in a work that was probably known to Bürgi.

Through his work as clock maker Bürgi became a very wealthy man and invested his wealth with profit in property deals and as a private banker lending quite substantial sums to his customers. In 1631 Bürgi, now 80 years old, retired and returned ‘home’ to Kassel where he died in January of the following year shortly before his 81st birthday. His death was registered in the Church of St Martin’s on the 31 January 1632. Although now only known to historians of science and horology, in his own time Bürgi was a well-known and highly respected, astronomer, mathematician and clock maker who made significant and important contributions to all three disciplines.

 

 

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