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

 

Leave a comment

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

A misleading illustration.

 

tumblr_o0k7mkhNSN1uk13a5o1_500

The difference between an easy model and a complicated one.

The gif above, from Malin Christersson’s  Website, has been making the rounds of the Internet to much acclamation but it is in my opinion severely misleading in what it claims to represent. Some people have pointed out that the heliocentric model is false because the orbits should be elliptical. This is my opinion an irrelevance because the eccentricity of the planetary orbits, that is the degree by which the ellipses differ from a circle, is so small that in a diagram of this sizel it wouldn’t be really detectable. In fact illustrations of the heliocentric system tend to exaggerate the eccentricity to make it clear that the orbits are in fact ellipses. My problem is another. The two models are presented side by side as if they were directly comparable but in fact they are two radically different representations.

The heliocentric system is displayed from a bird’s eye, or perhaps a god’s eye, view from a position directly above the sun perpendicular to the plane of the planetary orbits somewhere a couple of billion kilometres out in space. One should point out the sizes of the orbits are not to scale. Opposed to this the presentation of the geocentric system is not something one could actually view in reality. It is a fictitious birds eye view of the system as reconstructed by the astronomers in antiquity based on the activities they saw in the heavens and herein lies the crux of the problem.

Viewed from the earth the moments of the celestial bodies is not the lovely regular circles depicted in the heliocentric model above but a bizarre dance of confusing movements. The sun appears to go around the earth once a year and the moon once every approximately twenty-nine days. The so-called inner planets mercury and venus both also appeared to take a year to orbit the earth never wandering far from the sun, at times to one side and at other times on the other. Often both disappeared for periods of time. This behaviour led some people in antiquity to speculate that they orbit the sun and not the earth, the so-called Egyptian or Heracleidian model. It is however the so-called outer planets mars, jupiter and saturn that display the most puzzling behaviour. They role along in one direction for a lengthy period of time and then appear to stand still for a short period before turning tail and heading back in the opposite direction after a short time remaining stationary again before resuming in the original direction. These apparent loops in the planets progress are known technically as retrograde motion. We now know that this is an illusion created within the heliocentric system as the earth moving faster overtakes one or other of the outer planets. Given the seemingly stationary condition of the earth this was a difficult conception leap for astronomical observers to make. In fact in two thousand or more years of astronomy only two people appear to have made that leap, Aristarchus of Samos in the third century BCE and Copernicus in the fifteenth century CE. Both of these visionaries still had to cope with the very obvious empirical evidence that the earth doesn’t move.

The gif above creates a false impression because it seems to imply that the simplicity of the heliocentric system makes its an obvious choice over the geocentric model but as should be obvious from my description of what you actually see as an observer on the earth, and all observers in the past were on the earth, making that choice is anything but simple or obvious. The creator of the gif includes a short history of the journey from geocentricity to heliocentricity, which unfortunately contains various errors and misconceptions, which I will now highlight.

≈ 350 BC, Aristotle 

Aristotle a pupil of Plato, becomes the tutor of Alexander the Great. Aristotle’s views of the world shape science for centuries. His influence lasts until the enlightenment. In his book On the Heavens (part 14), Aristotle asserts that:

From these considerations then it is clear that the earth does not move and does not lie elsewhere than at the centre.

Aristotle is just one of many scholars from antiquity whose views influenced the future views of the world. He in fact inherited and modified the homocentric geocentric views and models of Eudoxus and Callippus. These models could explain retrograde motion fairly well but not the observable variation in brightness of the planets. This was not the system that medieval Europe inherited from antiquity. See below Ptolemy.

 ≈ 250 BC, Aristarchus

Aristarchus estimates the size of the sun to be much larger than the size of the earth. Based on this observation he then presents the heliocentric model.

The geometrical text, which is attributed to Aristarchus, is for determining both the distance of the sun from the earth and its size relative to the moon. It is a purely geocentric text and has nothing to do with his speculation about a heliocentric cosmos. There are no direct accounts of Aristarchus’ heliocentric model so we don’t actually know what caused him to adopt it.

 ≈ 250 BC, Archimedes

In The Sand-Reckoner, Archimedes estimates the number of sand corns in the universe using the heliocentric model of Aristarchus.

In the Sand Reckoner Archimedes wishes to demonstrate his system for recorded extremely large numbers. He uses Aristarchus’ heliocentric model, which he sketches, because Aristarchus argued that the stars were much further away than hypothesised in the normal geocentric model in order to explain why there was no observable stellar parallax. Archimedes used this model because it would require many more grains of sand to fill thus giving him a much greater number to express with his system. It is only one of two accounts of Aristarchus’ heliocentric system both of which are uninformative.

≈ 150 AD, Ptolemy

In his book Almagest, Ptolemy introduces so called epicycles to explain planetary motions, based on the assumption that the earth is at the centre and does not move. Almagest is considered to be one of the most influential scientific works in history.

The epicycle system of planetary motion, used extensively by Ptolemy in the Almagest in the second century CE, was first introduced by Apollonius of Perga in the third century BCE and used extensively by Hipparchus of Rhodes in the second century BCE.

1543, Nicholaus Copernicus

Just before his death, Copernicus publishes the book De Revolutionibus Orbium Coelestium (On the Revolutions of the Heavenly Spheres) in which he places the sun rather than the earth at the centre of the universe. This book is the beginning of the Copernican Revolution.

In English it’s Nicolaus (no ‘h’) Copernicus and in De revolutionibus the sun is not at the centre of the universe but somewhat off centre. Viewed strictly Copernicus’s system is heliostatic but not heliocentric.

1572, Tycho Brahe

Tyco Brahe observes a star being born and publishes his observation in De nova stella. Brahe’s observation refutes the commonly held view at the time, a view which dates back to Aristotle, that the stars are fix and never changing at the outskirts of the universe. Since Brahe couldn’t observe a stellar parallax, he concluded that the earth did not move. He proposed a model where the planets move around the sun, and the sun moves around the earth. (It was later shown that it wasn’t a star being born Brahe had observed, but the supernova SN 1572, i.e. a star exploding.)

In the first half of this paragraph we have an oft-repeated semi-myth. Although Tycho did indeed observe the nova of 1572 and it did contradict Aristotle’s cosmological theory of an immutable heaven this story is a myth for three different reasons. Firstly Aristotle’s concept of a an immutable heaven had already been seriously challenged in the sixteenth century by several leading astronomers based on their observations of several comets in the 1530s, so the nova of 1572 was not the first problem for Aristotle’s cosmology. Secondly Tycho was by no means the only astronomer to observe and comment on the 1572 nova and Michael Maestlin’s and Christoph Clavius’ acceptance that the nova was supralunar had more impact than Tycho’s. The attribution of this impact to Tycho alone is a version of the lone genius myth and historically false. Thirdly the refutation of Aristotle’s theory of the immutability of heaven actually has no real relevance for the geocentricity/heliocentricity discussion.

1609, Johannes Kepler

Using the observational data collected by Tycho Brahe, Johannes Kepler introduces his first two laws of planetary motion in Astronomia nova. The first law: the planets move in elliptical orbits with the sun at one focus.

Given that it was actually Kepler’s work that led to the acceptance of heliocentricity our author gives him rather short shrift in his chronology. What about the other two laws of planetary motion or the Rudolphine Tables?

 1616, Roman Inquisition

On 24 February 1616 a team of eleven consultants for the Roman Inquisition condemns the Copernican System, stating that the heliocentric system is “foolish and absurd in philosophy and “formally heretical”.

It should be pointed out that the Pope never confirmed the heretical status of heliocentricity thus it never was heretical.

 1633, Galileo Galilei

Galileo Galilei stands trial on suspicion of heresy “ for holding as true the false doctrine taught by some that the sun is the centre of the world”. At the trial he is found guilty and sentenced to formal imprisonment. Galileo spends the rest of his life under house arrest.

 1687, Isaac Newton

Sir Isaac Newton publishes Philosophiæ Naturalis Principia Mathematica (Principia). In Principia, Newton explains Kepler’s laws of planetary motion in terms of universal gravitation. Newton doesn’t consider the sun to be at rest, instead he uses the center of gravity of the solar system.

A small point, but one that irritates me. The man who published the Principia in 1687 was not ‘Sir’ Isaac Newton but just plain Isaac Newton who didn’t get knighted until 1705.

1838, Friedrich Bessel

Friedrich Bessel is the first to accurately measure a stellar parallax. In 1838 he announces that the star 61 Cygni has a parallax of 0.314 arcseconds.

Friedrich Bessel was not the first to accurately measure stellar parallax that honour goes to the Scottish astronomer Thomas Henderson, who measured the parallax of Alpha Centauri. Friedrich Bessel, however, was the first to publish.

1992, Roman Catholic Church

Pope John Paul II closes a 13-year investigation into the church’s condemnation of Galileo in 1633 by declaring that Galileo was right:

 Thanks to his intuition as a brilliant physicist and by relying on different arguments, Galileo, who practically invented the experimental method, understood why only the sun could function as the centre of the world, as it was then known, that is to say, as a planetary system. The error of the theologians of the time, when they maintained the centrality of the earth, was to think that our understanding of the physical world’s structure was, in some way, imposed by the literal sense of Sacred Scripture.

This final paragraph is just a horrible mess. Galileo did not practically invent the experiment method. Also the claim that he “understood why only the sun could function as the centre of the world” is simply bizarre. As I have pointed out in a number of different posts, in Galileo’s time the scientific evidence actually favoured a geocentric system. This also applies to the comment about the theologians, whose belief in a geocentric system was strongly supported by the available scientific evidence and was not just based on Sacred Scripture. It is also interesting to note how a chronology of the geocentric/heliocentric astronomical systems suddenly veers off into an account of Galileo’s troubles with the Catholic Church, which in real terms in the history of astronomy and cosmology is just a small side show.

 

54 Comments

Filed under Uncategorized

Founders of science?

World-renowned wheelchair driver and astrophysicist, Stephen Hawking, recently held the first of this year’s BBC Reith Lectures. This prompted the following tweet from Roger Highfield, science writer and director of external affairs at the Science Museum Group:

If Hawking could time travel, he would like to meet Galileo – ‘founder of modern science’ and ‘a bit of a rebel’ #Reith

Philip Ball, science writer, responded:

Though certainly not, as Hawking claimed, “the first to challenge Aristotle”…

To which I added:

Also not the founder of modern science.

Tom Levenson, another science writer, contributed:

Probably kicked his dog and stiffed his waiter too.

Roger Highfield reacted to this exchange thus:

Ha!

This moderately amusing, or not depending on you point of view, exchange on Twitter prompted Ángel Lamuño, Philosophy & Theology Follower of Bernard J. F. Lonergan SJ (self description), to pose the following question to me:

Who is (are) the founder(s) of modern science?

This whole rather trivial exchange contains several worrying aspects for historians of science, starting with Hawking’s original utterance. This is by no means the first time that Hawking has made such statements in public and in fact I quote one such in my take down of the founder of modern science and similar claims about Galileo – Extracting the stopper – that I wrote more than five years ago and which I’m not going to repeat here. The real problem is here that whatever Hawking’s merits as an astro-physicist he is not a historian of science and this is reflected in the naivety of his history of science comments that are almost invariably false. The problem is that Hawking because of his physical disability has become the most famous scientist in the world instantly recognised and admired whenever he appears in public. Whenever he makes a comment about the history of science then the majority of his audience, who don’t know better, immediately believe him because it’s ‘Stephen Hawking’! People believe Hawking because of who he is and not because his facts are correct, they aren’t. The irony of this situation is that what we have here is knowledge by authority, exactly the non-scientific epistemology that the scholastics supposedly practiced and which Galileo is said to have swept away, making him in Hawking’s words ‘the founder of modern science’.

Equally worrying is Ángel Lamuño’s question, [if Galileo isn’t the founder of modern science] “who is (are) the founder(s) of modern science?” This question is, in my opinion, based on a widespread misconception as to how science has evolved (developed, if you don’t like the word evolved). The misconception is supported in a vast number of texts, many of them written by highly respected historians of science but I think, in the meantime, rejected by a substantial part of the history of science community.

This misconception, or rather set of misconceptions, is that somehow major changes in the history of science are caused by one driving force in mainstream scientific thought and/or brought about by one heroic individual.

The traditional story that I grew up with was that the scientific revolution came about because science became quantified or mathematized when Neo-Platonism replaced Aristotelian scholasticism as the dominant philosophy in Europe. This is, however, not the Neo-Platonism of Plotinus of the third century CE but a Pythagorean Neo-Platonism. This theory was mainly propagated by philosophers. Mathematic historians however challenged this theory accepting that the scientific revolution was a mathematization of science but that this was brought about by an Archimedean renaissance beginning in the fourteenth century. Others have noted that the period also saw both a Euclidean and a Ptolemaic renaissance leading to increases in mathematical activity.

A different popular version of the story is that the scientific revolution was driven by the astronomical revolution brought about singlehandedly by Copernicus publishing his De revolutionibus in 1543. This is somewhat undermined by two facts. Firstly Copernicus’ work is only part of a general reform of astronomy carried out by a fairly large number of astronomers beginning with the first Viennese School of Mathematics in the fifteenth century. Secondly the so-called scientific revolution consists of far more than just astronomy.

There are theories that the astronomical revolution was driven by the renaissance in mathematical cartography sparked by the rediscovery of Ptolemaeus’ Geographia in 1406, alternatively by an attempt to put astrology on a solid empirical footing. At least one Arabic author has argued, with more than a little justification, that the astronomical revolution owes much more to the preceding Islamic astronomy than is usual credited. Another group of historians see the roots of the astronomical revolution in a shift in basic philosophy but not in a Neo-Platonic renaissance but in a Stoic one in the fifteenth and sixteenth centuries.

Another theory sees the scientific revolution being the rise in empirical experimental science, which has its roots in alchemy. An alternative explanation for this rise lies in the development of modern gunpowder based warfare and empirical studies of gunnery. Some see the rise of modern warfare as the driving force behind mathematical cartography, itself the driving force behind astronomical reform.

The above are some, but by no means all, of the theories that have been put forward to explain the emergence of modern science in the early modern period. So which one is the correct one? The answer is, all of them! The emergence of modern science was not caused by one single thing but by a whole range of activities, discoveries, renaissances as well as economic and socio-political developments. As historians we have a strong tendency to oversimplify, to want to find the ‘one’ cause for a given historical development, whereas in fact that development is almost inevitably the result of the interaction of a complex web of causes and it is often very difficult to weight the respective contributions of the individual causes. The mono-causal explanation only occurs if the researcher views the development from one standpoint whilst actively or passively ignoring all other possible standpoints. The same is true of the attribution of the titles ‘father of’ or ‘founder of’ to individuals. If you follow the link above to my earlier post about Galileo you will see how I show that he was only one of several, and sometimes many, making positive contributions to the fields in which he was active in the early seventeenth century and to raise him up on a pedestal is to deny due credit to the others and thus to falsify history. One can do the same with any of the other so-called ‘heroes’ of science, as I did fairly recently with exaggerated claims, contained in a book’s subtitle, for Johannes Kepler.

To repeat my central mantra as a historian of science, the evolution of science is driven by multiple complexly intertwined causes and is realised by the collective efforts of, often large, groups of researches and not by exceptional individuals. One day I hope that people will stop making the sort of statements that Stephen Hawking made, and which sparked off this post, but if I’m honest I’m not holding my breath whilst I wait.

 

30 Comments

Filed under Myths of Science

How papermaking crossed the Alps

Writing has been an important medium for recording and transmitting scientific results throughout history. In order to write you need two basic things, something to write with and something to write on. Carving words or symbols into wood or stone is one possibility that is both hard work and time consuming but does have the advantage that the finished product is highly durable especially if the material used is stone. You can actually also write with some sort of ink on sheets of wood. In fact the word ‘book’ comes from the German Buch, which in turn comes from Buche, which is the German for beech, both the tree and the wood. The ancient Babylonians wrote their Cuneiform script on clay tablets with a blunt reed stylus the result being wedge shaped symbols, the word cuneiform just coming from the Latin for wedge-shaped. The tablets baked when finished also have a relative high level of durability.

Enuma Anu Enlil text This tablet talks about how the planet Venus will appear at certain times in the future.

Enuma Anu Enlil text
This tablet talks about how the planet Venus will appear at certain times in the future.

Sometime in the fourth millennium BCE the ancient Egyptians invented a new material to write on, papyrus. These are mats made of the fibres of the papyrus plant. Unfortunately papyrus lacks flexibility and so can only be rolled and not folded making papyrus scrolls somewhat unwieldy in comparison to the later codex (that’s just Latin for book). Papyrus is also very susceptible to environmental influences, such as damp and excessive dryness, and is therefore not very durable.

Rhind Mathematical Papyrus : detail (recto, left part of the first section) Thebes, End of the Second Intermediate Period (c.1550 BC) Source: Wikimedia Commons

Rhind Mathematical Papyrus : detail (recto, left part of the first section) Thebes, End of the Second Intermediate Period (c.1550 BC)
Source: Wikimedia Commons

The history of parchment and vellum, that is prepared animal skins, which came to replace papyrus in antiquity especially in ancient Greece and Rome is shrouded in legend, myth and confusion but the method probably developed in ancient Egypt in the third millennium BCE. Pergamon in Greece became a major centre for the production of parchment and name of the material derives from the name of the city. Vellum is parchment made from calfskin, which is finer than that of other animals and the name derives from the Latin for calf. Parchment and vellum are difficult and expensive to produce, which amongst other things led to the habit of scraping the writing off of old documents to reuse the material thus producing so-called palimpsests a word derived from the Greek for scraped again, skins being scraped to produce parchment in the first place. If stored properly parchment is considerably less susceptible to environmental influences than papyrus and thus more durable.

A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right

A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right

In order to mass produce writing on a large scale cheaply it is obvious that a new writing material would have to be invented and that material was paper. Paper, which like papyrus is mats of plant fibre, was invented in China, sometime between the third century BCE and the second century CE, came to Europe like many things via the Islamic Empire. Paper is traditionally made of the fibres of flax or hemp, wood pulp paper is largely a product of the industrial nineteenth century, is lighter and much more flexible than papyrus. Originally it was regarded sceptically in Europe because compared to parchment it lacked durability making it, in the opinion of its critics unsuitable for important documents. However, with the invention of moveable type printing and the possibility of producing multiple copies of documents cheaply and quickly paper came into its own. But I run ahead of myself.

According to legend the Arabs discovered the secret of paper making from Chinese prisoners captured at the Battle of Telas in 751 CE. Although probably not true paper mills began to spread fairly rapidly throughout the Islamic Empire in the eighth century CE. The Arabs developed the techniques of mass paper production and introduced the first paper mills into Europe in Spain in the late eleventh century CE. From here papermaking spread to Southern France and Italy by the thirteenth century. Northern Europe relied on expensive imports to supply their paper needs. The first paper mill north of the Alps was set up by Ulman Stormer on the river Pegnitz just outside of Nürnberg in 1390 CE.

Ulman Stromer was born 6 January 1329 the twelfth of eighteen surviving children of the Nürnberger merchant Heinrich Stromer and his second wife Margarete Geusmid. He was apprenticed as a merchant and learnt his trade in Barcelona, Genoa, Madrid and Kraków. In 1370 he took over, together with two of his brothers, the management of the family’s Europe wide trading company. He also served as a local politician in Nürnberg. Stromer a shrewd trader realising the profits being made by the Northern Italian papermakers decide to set up in competition in his own home territory. According to Stromer’s own written account he set up his paper mill on 24 June (St John’s Day) 1390:

“IN the name of Christ, amen. Anno Domini 1390, I, Ulman Stromer, started at making paper on St. John’s day at the Solstice, and began to set up a wheel in the Gleissmühle, and Clos Obsser was the first who came to work.”

There being no papermakers in Germany Stromer brought in the brothers Marco and Francisco di Marchia and their boy Bartolomeo from Italy to set up and run the mill, which became known as the Hadermühle Hadern being Lumpen or in English rags, paper being made not directly from flax plants but from linen rags.

Ulman Stromer's Paper-mill. (From Schedel's Buch der Chroniken of 1493.)

Ulman Stromer’s Paper-mill. (From Schedel’s Buch der Chroniken of 1493.)

Stromer’s paper mill has been recorded for posterity in one of the most famous early books also printed in Nürnberg, the Die Schedelsche Weltchronik better known in English as the Nuremberg Chronicle.

Stromer's paper mill in the Nuremberg Chronicle of 1493. The building complex is at the lower right corner, outside the city perimeter. Source: Wikipedia Commons

Stromer’s paper mill in the Nuremberg Chronicle of 1493. The building complex is at the lower right corner, outside the city perimeter.
Source: Wikipedia Commons

4 Comments

Filed under Early Scientific Publishing

Just another day

A very large number of my Internet acquaintances along with both the English and German language media that I have access to are indulging in their yearly hysteria because today is New Year’s Eve and tomorrow is New Year’s Day, what they all seem to have forgotten is that it’s actually just another day.

A part, but by no means all, of human kind has arbitrarily decided to designate today as the day that they stop counting the days of the sun’s annual journey around the ecliptic and tomorrow start again from one. I say a part but by no means all because tomorrow is only New Year’s Day on the Gregorian calendar but not on many, many other calendars currently in use throughout the world, for example the Jewish, Muslim, Persian, Chinese, Vietnamese and numerous others.

analemma1

The analemma traces the position of the sun in the sky throughout the solar year Source: KBCC Meteorology

The Gregorian New Year’s celebration doesn’t even coincide with a significant day in the annual solar journey, either of the equinoxes when the day and night are equally long or either of the solstices in summer with the longest day or in winter with the shortest day. My favoured candidate for New Year’s would be the winter solstice with, for me in the northern hemisphere, the start of the slow climb to spring and then on to summer, a genuine reason to celebrate and not an arbitrary and artificial one.

January the first wasn’t always the beginning of the calendrical year. Originally the Romans, from whom we inherit our calendar, celebrated the start of the year, as did and do other culture, at the spring equinox around the twenty-fifth of March, also in my opinion a good choice for a calendrical celebration.

When Julius Caesar introduced the solar calendar, that would go on to bear his name, in 46 BCE he moved the start of the year from 25 March to 1 January, the feast of Janus, the Roman god of beginnings and endings. In the Middle Ages some countries, not keen to celebrate a pagan festival, moved New Year’s Day back to 25 March the Christian Feast of the Annunciation, that is the day that Mary supposedly became pregnant with Jesus. This led to two different ways of numbering the days of the year, Circumcision Style starting from 1 January, Circumcision of Our Lord in the Church calendar and Annunciation Style from 25 March. For a time in the Middle Ages the start of the year was counted by some from the 25 December, Nativity Style, or from the Easter Feast, Easter Style. The latter was considered somewhat inconvenient because Easter is a moveable feast.

When Pope Gregory introduced his calendar reform in 1582 his reform committee has settled on 1 January as the unified start of the year. Some countries, most notably Great Britain and its colonies, which initially rejected the Catholic calendar reform retained the Annunciation Style of counting leading to the strange anomaly of Newton’s date of death. On the Gregorian calendar, new style, he not only died eleven days but a whole year later than on the Julian calendar, old style.

So when you set out, to do what ever it is that you plan to do, to celebrate this evening just remember that in reality today is just another day in the sun’s seemingly endless journey along the ecliptic and any other day would do just as well and has done so throughout human history and continues to do so in many other cultures.

However what ever your beliefs and no mater which calendar you follow and on which day you celebrate the start of another round of the loop, I wish you all the best for the next 366 days of your life, as 2016 is a leap year on the Gregorian calendar.

16 Comments

Filed under Uncategorized

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.

 

6 Comments

Filed under History of Computing, History of Mathematics

Christmas Trilogy 2015 Part 2: Understanding the Analytical Engine.

The Acolytes of the Holy Church of Saint Ada still persist in calling her a brilliant mathematician and the ‘first computer programmer’ despite the fact that both are provably wrong. In fact they have now moved into the realm of denialists, similar to evolution or climate denialists, in that they accuse people like myself who point to the historical facts of being male chauvinists who are trying to deny women their rights in the history of science! However the acolytes have gone a step further in the adulation of Lady King in that they now claim that she understood the Analytical Engine better than Babbage! Confronted by this patently ridiculous claim I’m not sure whether to laugh or cry. Babbage conceived, designed and attempted to construct parts of the Analytical Engine whereas Ada Lovelace merely wrote an essay about it based on her exchanges with Babbage on the subject, to suggest that she understood the machine better than its sole creator borders on the insane. I cannot be certain who first set this bizarre claim in the world as nearly all of those who repeat it give neither justification or source for their utterances but the most often quoted in this context is James Essinger and his biography of Ada, which appears to enjoy several different titles[1].

Trial model of a part of the Analytical Engine, built by Babbage, as displayed at the Science Museum (London). Source: Wikimedia Commons

Trial model of a part of the Analytical Engine, built by Babbage, as displayed at the Science Museum (London).
Source: Wikimedia Commons

Before going into detail it should be pointed out the Essinger’s book, which is popular rather then academic and thus lacks sources for many of his claims, suffers from two fundamental flaws. Like much pro Ada writing it doesn’t delve deep enough into the live and work of Charles Babbage. This type of writing tends to treat Babbage as an extra in the film of Ada’s life, whereas in reality in relation to the Analytical Engine it is Ada who is a minor character in Babbage’s life. Also Essinger writes about the translation of the Menabrea essay on the Analytical engine as if the appended notes were exclusively the product of Ada’s brain, whereas it is an established fact from the correspondence that they were very much a co-production between Babbage and Lovelace based on many exchanges both in personal conversations and in that correspondence. This means that in basing any argument on any idea contained in those notes the writer has the job of determining, which of the two would be the more probable source of that idea and not simply blindly attribute it to Ada. As we shall see Essinger’s failure to do this leads to a major flaw in his central argument that Ada understood the Analytical Engine better than Babbage.

Essinger’s approach is two pronged. On the one side he claims that Babbage didn’t understand the future potential of the machine that he, and he alone, conceived and created (on paper at least) and on the other he proposes on the basis of his interpretation of Note A of the essay that Ada, whom he assumes to be the originator of the thoughts this not contains, had a vision of the Analytical engine equivalent to modern computer science. As we shall see Essinger is mistaken on both counts.

Whilst offering absolutely no source for his claim, Essinger states time and again throughout his book that Babbage only every conceived of the Analytical Engine as a device for doing mathematics, a super number cruncher so to speak. If Essinger had taken the trouble to elucidate the origins of Babbage’s inspiration for the Analytical Engine he would know that he is seriously mistaken in his view, although in one sense he was right in thinking that Babbage concentrated on the mathematical aspects of the Engine but for reasons that Essinger doesn’t consider anywhere in his book.

Babbage lived in the middle of the Industrial Revolution and was fascinated by mechanisation and automation throughout his entire life. During the 1820s Babbage travelled throughout the British Isles visiting all sorts of industrial plant to study and analyse their uses of mechanisation and automation. In 1827 his wife, Georgiana, died and Babbage who had married against the opposition of his father out of love was grief stricken. Leaving Britain to escape the scene of his sorrow Babbage, by now having inherited his fathers fortune a rich man, spent many months touring the continent carrying out the same survey of the industrial advances in mechanisation and automation wherever his wanderings took him. It was on this journey that he first learnt of the automated Jacquard loom that would supply him with the idea of programming the Analytical Engine with punch cards. Returning to Britain Babbage now turned all those years of research into a book, On the Economy of Machinery and Manufactures published in 1832, that is a year before he met Ada Lovelace for the first time and ten years before Menabrea essay was written. The book was a massive success going through six editions in quick succession and influencing the work of Karl Marx and John Stuart Mill amongst others. It would be safe to say that in 1832 Babbage knew more about mechanisation and automation that almost anybody else on the entire planet and what it was capable of doing and which activities could be mechanised and/or automated. It was in this situation that Babbage decided to transfer his main interest from the Difference Engine to developing the concept of the Analytical Engine conceived from the very beginning as a general-purpose computer capable of carrying out everything that could be accomplished by such a machine, far more than just a super number cruncher.

analytical_engine

What is true, however, is that Babbage did concentrate in his plans and drafts, and the Analytical Engine never got past the plans and drafts phase, on the mathematical aspects of the machine. This however does not mean that Babbage considered it purely as a mathematical machine. I am writing this post on a modern state of the art computer. I also use the same device to exchange views with my history of sciences peers on Twitter and Facebook, to post my outpourings, such as this one, on my Internet blog. I can telephone, with visual contact if I choose, with people all over the world using Skype. At the touch, or two, of a keyboard key I have access to dictionaries, encyclopaedias and all sorts of other reference tools and through various means I can exchange documents, photographs, sound files and videos with anybody who owns a similar device. I can listen to and watch all sorts of music recordings and videos and with easily accessible software even turn my computer into an unbelievably flexible musical instrument. Finally when I’m done for the day I can settle back and watch television on my large, high-resolution monitor screen. This is only a fraction of the tasks that my computer is capable of carrying out but they all have one thing in common, they can all only be accomplished if they are capable of being coded into an astoundingly banal logical language consisting only of ‘0s’ and ‘1s’. Of course between the activities I carry out on my monitor screen and the electrical circuits that are only capable of reading those ‘0s’ and ‘1s’ there are layer upon layer of so-called sub-routines and sub-sub-routines and sub-sub-sub…, you get the idea, translating an upper layer into a simpler logical form until we get all the way down to those ubiquitous ‘0s’ and ‘1s’. The language in which those ‘0s’ and ‘1s’ exist is a mathematical language, known as Boolean Algebra, and so in the final analysis my super smart ultra modern computer is nothing but a super number cruncher and only two numbers at that.

Babbage, a brilliant mathematician, was well aware that he could only programme his Engine to carry out tasks that could be reduced over a series of steps to a mathematical language and this is the reason he concentrated on the mathematical aspects of his machine but this by no means meant that he only conceived of it only carrying out mathematical tasks, as we will see when addressing Essinger’s second prong.

Essinger quotes the following passage from Note A of the Malebrea translation:

In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated.

It is well to draw attention to this point, not only because its full appreciation is essential to the attainment of any very just and adequate general comprehension of the powers and mode of action of the Analytical Engine, but also because it is one which is perhaps too little kept in view in the study of mathematical science in general. It is, however, impossible to confound it with other considerations, either when we trace the manner in which that engine attains its results, or when we prepare the data for its attainment of those results. It were much to be desired, that when mathematical processes pass through the human brain instead of through the medium of inanimate mechanism, it were equally a necessity of things that the reasonings connected with operations should hold the same just place as a clear and well-defined branch of the subject of analysis, a fundamental but yet independent ingredient in the science, which they must do in studying the engine. The confusion, the difficulties, the contradictions which, in consequence of a want of accurate distinctions in this particular, have up to even a recent period encumbered mathematics in all those branches involving the consideration of negative and impossible quantities, will at once occur to the reader who is at all versed in this science, and would alone suffice to justify dwelling somewhat on the point, in connexion with any subject so peculiarly fitted to give forcible illustration of it as the Analytical Engine.

Attributing its contents to Ada he makes the following comment, “What Ada is emphasising here is the clear distinction between data and data processing: a distinction we tend to take for granted today, but which – like so much of her thinking about computers –was in her own day not only revolutionary but truly visionary”. What is being described here is indeed new in Ada’s day but is a well known development in mathematics know at the time as the Calculus of Operations, a branch of mathematics developed in the first half of the nineteenth century, which differentiates between operators and operations, and in which Babbage worked and to which he made contributions. If the ideas contained in this passage are indeed visionary then the vision is Babbage’s being channelled by Ada and not originating with her. The words might be Ada’s but the thoughts they express are clearly Babbage’s.

Essinger now quotes the next part of the Note:

It may be desirable to explain, that by the word operation, we mean any process which alters the mutual relation of two or more things, be this relation of what kind it may. This is the most general definition, and would include all subjects in the universe. In abstract mathematics, of course operations alter those particular relations which are involved in the considerations of number and space, and the results of operations are those peculiar results which correspond to the nature of the subjects of operation. But the science of operations, as derived from mathematics more especially, is a science of itself, and has its own abstract truth and value; just as logic has its own peculiar truth and value, independently of the subjects to which we may apply its reasonings and processes.

Essinger now reaches maximum bullshit level, “Ada is seeking to do nothing less than invent the science of computing and separate it from the science of mathematics. What she calls ‘the science of operations’ is indeed in effect computing”. As I have already explained what she calls the ‘science of operations’ is in fact the calculus of operation a new but well developed branch of mathematics of which Babbage was fully cognisant. If anybody is inventing the science of computing it is once again Babbage and not Ada.

Essinger now takes up the case further along in Note A:

The distinctive characteristic of the Analytical Engine, […]is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs… […]The bounds of arithmetic [emphasis in original] were however outstepped the moment the idea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere “calculating machines.” It holds a position wholly its own; and the considerations it suggests are most interesting in their nature. In enabling mechanism to combine together general [emphasis in original] symbols in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract [emphasis in original] branch of mathematical science. [Ellipsis in quote by Essinger]

Essinger introduces this quote with the following: “In a terse passage she explains (perhaps better than Babbage ever could, who as designer saw many trees but perhaps no longer the forest itself) the essential relationship between the Analytical Engine and the Jacquard loom and how it is different from the earlier invention”. After the quote he then writes: “In perhaps one of the most visionary sentences written during the nineteenth century [he sure doesn’t hold back on the hyperbole], she lays out what these cards shall be capable of doing by way of programming the machine”

First off, if you put back the bits Essinger removed from this passage it is anything but terse, in fact it’s rather verbose. Is Essinger really trying to tell us that Babbage was not aware of what he was doing when he conceived of programming his Engine with punch cards? Unfortunately for Essinger Babbage himself tells us that this is not the case, writing in his notebook on 10 July 1836, that is 8 years before the original French version of the Malebrea essay was published, he has the following to say:

This day I had for the first time a general but very indistinct conception of the possibility of making the engine work out algebraic developments – I mean without any reference to the value of the letters. My notion is that as the cards (Jacquards) of the calc. engine direct a series of operations and the recommence with the first…[2]

Here we have in Babbage’s own words the germ of the idea contained in the Ada quote, an idea that would naturally mature over the intervening nine years before Ada wrote her piece, so I have problems whatsoever in again attributing the thoughts contained here to Babbage.

I’m not going to go on analysing Essinger’s Ada hagiography for almost all of the things that he attributes to Ada it is not difficult to find its origins in Babbage’s work thus reinforcing the claim in an earlier post that Ada is being used here as Babbage’s mouth piece. Not so much the originator as the parrot. I will however close with one last quote from Note A and Essinger’s comment to demonstrate that his grasp of the history of science in the nineteenth century is apparently almost non-existent. Without really introducing it Essinger quotes the following sentence:

Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connexion together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst: those who thus think on mathematical truth as the instrument through which the weak mind of man can most effectually read his Creator’s works, will regard with especial interest all that can tend to facilitate the translation of its principles into explicit practical forms.

Essinger wonderingly comments on this sentence, “This 158-word sentence is very likely one of the longest sentences in the history of science, but it is also one of the most intriguing. Ada succeeds in this one sentence in linking mathematics, science, religion and philosophy.” Any competent historian of science would immediately recognise this as a rather flowery expression of the basic tenets of natural theology, a philosophy that flourished in the first half of the nineteenth century. This statement could have been made by a very large number of natural philosophers starting with Isaac Newton and going up to and beyond William Whewell and Charles Babbage, for example in the dispute that I outlined on this day last year. What this example clearly illustrates is that Essinger is in no way a real historian who researches and understands his sources but one who thinks he can read the text of Note A and interpret it on the basis of his lack of knowledge rather than on his procession of it.

[1] The copy I read was James Essinger, A Female Genius: how Ada Lovelace, Lord Byron’s daughter started the computer age, London 2015

[2] Babbage notebook quote taken from Dorothy Stein, Ada: A Life and a Legacy, MIT Press, Cambridge Massachusetts &London, 1985 p.102

6 Comments

Filed under History of Computing, Myths of Science