Category Archives: Mediaeval Science

Our medieval technological inheritance.

“Positively medieval” has become a universal put down for everything considered backward, ignorant, dirty, primitive, bigoted, intolerant or just simply stupid in our times. This is based on a false historical perspective that paints the Middle Ages as all of these things and worse. This image of the Middle Ages has its roots in the Renaissance, when Renaissance scholars saw themselves as the heirs of all that was good, noble and splendid in antiquity and the period between the fall of the Roman Empire and their own times as a sort of unspeakable black pit of ignorance and iniquity. Unfortunately, this completely false picture of the Middle Ages has been extensively propagated in popular literature, film and television.

Particularly in the film and television branch, a film or series set in the Middle Ages immediately calls for unwashed peasants herding their even filthier swine through the mire in a village consisting of thatch roofed wooden hovels, in order to create the ‘correct medieval atmosphere’. Add a couple of overweight, ignorant, debauching clerics and a pox marked whore and you have your genuine medieval ambient. You can’t expect to see anything vaguely related to science or technology in such presentations.

Academic medieval historians and historians of science and technology have been fighting an uphill battle against these popular images for many decades now but their efforts rarely reach the general lay public against the flow of the latest bestselling medieval bodice rippers or TV medieval murder mystery. What is needed, is as many semi-popular books on the various aspects of medieval history as possible. Whereby with semi-popular I mean, written for the general lay reader but with its historical facts correct. One such new volume is John Farrell’s The Clock and the Camshaft: And Other Medieval Inventions We Still Can’t Live Without.[1]


Farrell’s book is a stimulating excursion through the history of technological developments and innovation in the High Middle Ages that played a significant role in shaping the modern world.  Some of those technologies are genuine medieval discoveries and developments, whilst others are ones that either survived or where reintroduced from antiquity. Some even coming from outside of Europe. In each case Farrell describes in careful detail the origins of the technology in question and if known the process of transition into European medieval culture.

The book opens with agricultural innovations, the deep plough, the horse collar and horse shoes, which made it possible to use horses as draught animals instead of or along side oxen, and new crop rotation systems. Farrell explains why they became necessary and how they increased food production leading indirectly to population growth.

Next up we have that most important of commodities power and the transition from the hand milling of grain to the introduction of first watermills and then windmills into medieval culture. Here Farrell points out that our current knowledge would suggest that the more complex vertical water mill preceded the simpler horizontal water mill putting a lie to the common precept that simple technology always precedes more complex technology. At various points Farrell also addresses the question as to whether technological change drives social and culture change or the latter the former.


Having introduced the power generators, we now have the technological innovations necessary to adapt the raw power to various industrial tasks, the crank and the camshaft. This is fascinating history and the range of uses to which mills were then adapted using these two ingenious but comparatively simple power take offs was very extensive and enriching for medieval society. One of those, in this case an innovation from outside of Europe, was the paper mill for the production of that no longer to imagine our society without, paper. This would of course in turn lead to that truly society-changing technology, the printed book at the end of the Middle Ages.


Along side paper perhaps the greatest medieval innovation was the mechanical clock. At first just a thing of wonder in the towers of some of Europe’s most striking clerical buildings the mechanical clock with its ability to regulate the hours of the day in a way that no other time keeper had up till then gradually came to change the basic rhythms of human society.

Talking of spectacular clerical buildings the Middle Ages are of course the age of the great European cathedrals. Roman architecture was block buildings with thick, massive stonewalls, very few windows and domed roofs. The art of building in stone was one of the things that virtually disappeared in the Early Middle Ages in Europe. It came back initially in an extended phase of castle building. Inspired by the return of the stonemason, medieval, European, Christian society began the era of building their massive monuments to their God, the medieval cathedrals. Introducing architectural innovation like the pointed arch, the flying buttress and the rib vaulted roof they build large, open buildings flooded with light that soared up to the heavens in honour of their God. Buildings that are still a source of wonder today.


In this context it is important to note that Farrell clearly explicates the role played by the Catholic Church in the medieval technological innovations, both the good and the bad. Viewed with hindsight the cathedrals can be definitely booked for the good but the bad? During the period when the watermills were introduced into Europe and they replaced the small hand mills that the people had previously used to produce their flour, local Church authorities gained control of the mills, a community could only afford one mill, and forced the people to bring their grain to the Church’s mill at a price of course. Then even went to the extent of banning the use of hand mills.

People often talk of the Renaissance and mean a period of time from the middle of the fifteenth century to about the beginning of the seventeenth century. However, for historians of science there was a much earlier Renaissance when scholars travelled to the boundaries between Christian Europe and the Islamic Empire in the twelfth and thirteenth centuries in order to reclaim the knowledge that the Muslims had translated, embellished and extended in the eight and ninth centuries from Greek sources. This knowledge enriched medieval science and technology in many areas, a fact that justifies its acquisition here in a book on technology.

Another great medieval invention that still plays a major role in our society, alongside the introduction of paper and the mechanical clock are spectacles and any account of medieval technological invention must include their emergence in the late thirteenth century. Spectacles are something that initially emerged from Christian culture, from the scriptoria of the monasteries but spread fairly rapidly throughout medieval society. The invention of eyeglasses would eventually lead to the invention of the telescope and microscope in the early seventeenth century.

Another abstract change, like the translation movement during that first scientific Renaissance, was the creation of the legal concept of the corporation. This innovation led to the emergence of the medieval universities, corporations of students and/or their teachers. There is a direct line connecting the universities that the Church set up in some of the European town in the High Middle Ages to the modern universities throughout the world. This was a medieval innovation that truly helped to shape our modern world.

Farrell’s final chapter in titled The Inventions of Discovery and deals both with the medieval innovations in shipbuilding and the technology of the scientific instruments, such as astrolabe and magnetic compass that made it possible for Europeans to venture out onto the world’s oceans as the Middle Ages came to a close. For many people Columbus’ voyage to the Americas in 1492 represents the beginning of the modern era but as Farrell reminds us all of the technology that made his voyage possible was medieval.

All of the above is a mere sketch of the topics covered by Farrell in his excellent book, which manages to pack an incredible amount of fascinating information into what is a fairly slim volume. Farrell has a light touch and leads his reader on a voyage of discovery through the captivating world of medieval technology. The book is beautifully illustrated by especially commissioned black and white line drawing by Ryan Birmingham. There are endnotes simply listing the sources of the material in main text and an extensive bibliography of those sources. The book also has, what I hope, is a comprehensive index.[2]

Farrell’s book is a good, readable guide to the world of medieval technology aimed at the lay reader but could also be read with profit by scholars of the histories of science and technology and as an ebook or a paperback is easily affordable for those with a small book buying budget.

So remember, next time you settle down with the latest medieval pot boiler with its cast of filthy peasants, debauched clerics and pox marked whores that the paper that it’s printed on and the reading glasses you are wearing both emerged in Europe in the Middle Ages.

[1] John W. Farrell, The Clock and the Camshaft: And Other Medieval Inventions We Still Can’t Live Without, Prometheus Books, 2020.

[2] Disclosure: I was heavily involved in the production of this book, as a research assistant, although I had nothing to do with either the conception or the actual writing of the book that is all entirely John Farrell’s own work. However, I did compile the index and I truly hope it will prove useful to the readers.


Filed under Book Reviews, History of science, History of Technology, Mediaeval Science

Mathematics and natural philosophy: Robert G socks it to GG

In my recent demolition of Mario Livio’s very pretentious Galileo and the Science Deniers I very strongly criticised Livio’s repeated claims, based on Galileo’s notorious Il Saggiatore quote on the two books, that Galileo was somehow revolutionary in introducing mathematics into the study of science. I pointed out that by the time Galileo wrote his book this had actually been normal practice for a long time and far from being revolutionary the quote was actually a common place.

Last night whilst reading my current bedtime volume, A Mark Smith’s excellent From Sight to Light: The Passage from Ancient to Modern Optics,(University of Chicago Press, 2015) I came across a wonderfully appropriate quote on the topic from Robert Grosseteste (c.1175–1253). For those that don’t know Grosseteste was an English cleric who taught at Oxford University and who became Bishop of Lincoln. He played an important and highly influential role in medieval science, particularly in helping to establish optics as a central subject in the medieval university curriculum.


An early 14th-century portrait of Grosseteste Source: Wikimedia Commons

Of course, this is problematic for Livio, who firmly labelled the Catholic Church as anti-science and who doesn’t think there was any medieval science, remember that wonderfully wrong quote:

Galileo introduced the revolutionary departure from the medieval, ludicrous notion that everything worth knowing was already known.

If this were true then medieval science would be an oxymoron but unfortunately for Livio’s historical phantasy there was medieval science and Grosseteste was one of its major figure. If you want to know more about Grosseteste then I recommend the Ordered Universe website set up by the team from Durham University led by Giles Gasper, Hannah Smithson and  Tom McLeish

I already knew of Grosseteste’s attitude towards natural philosophy and mathematics but didn’t have a suitable quote to hand, so didn’t mention it in my review. Now I do have one. Let us first remind ourselves what Galileo actually said in Il Saggiatore:

Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

And now what Grosseteste wrote four hundred years earlier in his De lineis, angulis et figuris (On lines, angles and figures) between 1220 and 1235:

“…a consideration of lines, angles and fugures is of the greatest utility because it is impossible to gain a knowledge of natural philosophy without them…for all causes of natural effects must be expressed by means of lines, angles and figures”

Remarkably similar is it not!






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

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

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

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


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

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


Robert Bellarmine artist unknown Source: Wikimedia Commons

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


Astronomia Nova title page Source: Wikimedia Commons

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

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


William Gilbert Source: Wikimedia Commons

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


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

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


Ismaël Boulliau portrait by Pieter van Schuppen Source: Wikimedia Commons

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


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


Giovanni Alfonso Borelli Source: Wikimedia Commons

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

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







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

It’s all a question of angles.

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


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

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


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

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


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

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


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

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


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

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


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

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

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


Abū al-Wafā Source: Wikimedia Commons

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


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

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

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

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

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


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

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


Source: Wikimedia Commons

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


Source: Wikimedia Commons

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

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


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


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

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

One of the most well known popular stories told about Galileo is how he dropped balls from the Leaning Tower of Pisa to disprove the Aristotelian hypothesis that balls of different weights would fall at different speeds; the heavier ball falling faster. This event probably never happened but it is related as a prelude to his brilliant experiments with balls and inclined planes, which he carried out to determine empirically the correct laws of fall and which really did take place and for which he is justifiably renowned as an experimentalist. What is very rarely admitted is that the investigation of the laws of fall had had a several-hundred-year history before Galileo even considered the problem, a history of which Galileo was well aware.

We saw in the last episode that John Philoponus had actually criticised Aristotle’s concept of fall in the sixth century and had even carried out the ball drop experiment. However, unlike his impulse concept for projectile motion, which was taken up by Islamic scholars and passed on by them into the European Middle Ages, his correct criticism of Aristotle’s fall theory appears not to have been taken up by later thinkers.

As far as we know the first people, after Philoponus, to challenge Aristotle’s concept was the so-called Oxford Calculatores.


Merton College in 1865 Source: Wikimedia Commons

This was a group of fourteenth-century, Aristotelian scholars at Merton College Oxford, who set about quantifying various theory of nature. These men–Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–c. 1372), Richard Swineshead (fl. c. 1340–1354) and John Dumbleton (c. 1310–c. 1349)–studied mechanics distinguishing between kinematics and dynamics, emphasising the former and investigating instantaneous velocity. They were the first to formulate the mean speed theorem, an achievement usually accredited to Galileo. The mean speed theorem states that a uniformly accelerated body, starting from rest, travels the same distance as a body with uniform speed, whose speed in half the final velocity of the accelerated body. The theory lies at the heart of the laws of fall.

The work of the Oxford Calculatores was quickly diffused throughout Europe and Nicole Oresme (c. 1320–1382), one of the so-called Parisian physicists,


Portrait of Nicole Oresme: Miniature from Oresme’s Traité de l’espère, Bibliothèque Nationale, Paris, France, fonds français 565, fol. 1r. Source: Wikimedia Commons

and Giovanni di Casali (c. 1320–after 1374) both produced graphical representation of the theory.


Oresme’s geometric verification of the Oxford Calculators’ Merton Rule of uniform acceleration, or mean speed theorem. Source: Wikimedia Commons

We saw in the last episode how Tartaglia applied mathematics to the problem of projectile motion and now we turn to a man, who for a time was a student of Tartaglia, Giambattista Benedetti (1530–1590). Like others before him Bendetti turned his attention to Aristotle’s concept of fall and wrote and published in total three works on the subject that went a long way towards the theory that Galileo would eventually publish. In his Resolutio omnium Euclidis problematum (1553) and his Demonstratio proportionum motuum localium (1554) he argued that speed is dependent not on weight but specific gravity and that two objects of the same material but different weights would fall at the same speed.


Source: Wikimedia Commons

However, in a vacuum, objects of different material would fall at different speed. Benedetti brought an early version of the thought experiment, usually attributed to Galileo, of viewing two bodies falling separately or conjoined, in his case by a cord.  Galileo considered a roof tile falling complete and then broken into two.

In a second edition of the Demonstratio (1554) he addressed surface area and resistance of the medium through which the objects are falling. He repeated his theories in his Demonstratio proportionum motuum localium (1554), where he explains his theories with respect to the theory of impetus. We know that Galileo had read his Benedetti and his own initial theories on the topic, in his unpublished De Motu, were very similar.

In the newly established United Provinces (The Netherlands) Simon Stevin (1548–1620) carried out a lot of work applying mathematics to various areas of physics. However in our contexts more interesting were his experiments in 1586, where he actually dropped lead balls of different weights from the thirty-foot-high church tower in Delft and determined empirically that they fell at the same speed, arriving at the ground at the same time.


Source: Wikimedia Commons

Some people think that because Stevin only wrote and published in Dutch that his mathematical physics remained largely unknown. However, his complete works published initially in Dutch were translated into both French and Latin, the latter translation being carried out by Willebrord Snell. As a result his work was well known in France, the major centre for mathematical physics in the seventeenth century.

In Italy the Dominican priest Domingo de Soto (1494–1560) correctly stated that a body falls with a constant, uniform acceleration. In his Opus novum, De Proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum. Item de aliza regula (1570) Gerolamo Cardano (1501–1576) demonstrates that two balls of different sizes will fall from a great height in the same time. The humanist poet and historian, Benedetto Varchi (c. 1502–1565) in 1544 and Giuseppe Moletti (1531–1588), Galileo’s predecessor as professor of mathematics in Padua, in 1576 both reported that bodies of different weights fall at the same speed in contradiction to Aristotle, as did Jacopo Mazzoni (1548–1598), a philosopher at Padua and friend of Galileo, in 1597. However none of them explained how they arrived at their conclusions.

Of particular relevance to Galileo is the De motu gravium et levium of Girolamo Borro (1512–1592) one of Galileo’s teachers in Pisa. In a dispute with his colleague Francesco Buonamici (1533–1603), another Pisan professor, Borro carried out experiments in which he threw objects of different material and the same weights out of a high window to test Aristotle’s theory, which he describes in his book. Borro’s work is known to have had a strong influence on Galileo’s early work in this area.

When Galileo started his own extensive investigations into the problem of fall in the late sixteenth century he was tapping into an extensive stream of previous work on the subject of which he was well aware and which to some extent had already done much of the heavy lifting. This raises the question as to what extent Galileo deserves his reputation as the man, who solved the problem of fall.


Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

We saw in the last episode that his much praised Dialogo, his magnum opus on the heliocentricity contra geocentricity debate, not only contributed nothing new of substance to that debate but because of his insistence on retaining the Platonic axioms, his total rejection of the work of both Tycho Brahe and Kepler and his rejection of the strong empirical evidence for the supralunar nature of comets he actually lagged far behind the current developments in that debate. The result was that the Dialogo could be regarded as superfluous to the astronomical system debate. Can the same be said of the contribution of the Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) (1638) to the debate on motion? The categorical answer is no; the Discorsi is a very important contribution to that debate and Galileo deserves his reputation as a mathematical physicist that this book gave him.


Source: Wikimedia Commons

What did Galileo contribute to the debate that was new? It not so much that he contributed much new to the debate but that he gave the debate the solid empirical and mathematical foundation, which it had lacked up till this point. Dropping weights from heights to examine the laws of fall suffers from various problems. It is extremely difficult to ensure that the object are both released at the same time, it is equally difficult to determine if they actually hit the ground at the same time and the whole process is so fast, that given the possibilities available at the time, it was impossible to measure the time taken for the fall. All of the previous experiments of Stevin et al were at best approximations and not really empirical proofs in a strict scientific sense. Galileo supplied the necessary empirical certainty.

Galileo didn’t drop balls he rolled them down a smooth, wooden channel in an inclined plane that had been oiled to remove friction. He argued by analogy the results that he achieved by slowing down the acceleration by using an inclined plane were equivalent to those that would be obtained by dropping the balls vertically. Argument by analogy is of course not strict scientific proof but is an often used part of the scientific method that has often, as in this case, led to important new discoveries and results.  He released one ball at a time and timed them separately thus eliminating the synchronicity problem. Also, he was able with a water clock to time the balls with sufficient accuracy to make the necessary mathematical calculations. He put the laws of falls on a sound empirical and mathematical footing. One should also not neglect the fact that Galileo’s undoubtable talent as a polemicist made the content of the Discorsi available in a way that was far more accessible than anything that had preceded it.


Galileo’s demonstration of the law of the space traversed in case of uniformly varied motion. It is the same demonstration that Oresme had made centuries earlier. Source: Wikimedia Commons

For those, who like to believe that Catholics and especially the Jesuits were anti-science in the seventeenth century, and unfortunately they still exist, the experimental confirmation of Galileo’s law of fall, using direct drop rather than an inclined plane, was the Jesuit, Giovanni Battista Riccioli(1598–1671).


Giovanni Battista Riccioli Source: Wikimedia Commons

The Discorsi also contains Galileo’s work on projectile motion, which again was important and influential. The major thing is the parabola law that states that anything projected upwards and away follows a parabolic path. Galileo was not the only natural philosopher, who determined this. The Englishman Thomas Harriot (c. 1560–1621) also discovered the parabola law and in fact his work on projectile motion went well beyond that of Galileo. Unfortunately, he never published anything so his work remained unknown.  One of Galileo’s acolytes, Bonaventura Cavalieri (1598–1647),


Bonaventura Cavalieri Source: Wikimedia Commons

was actually the first to publish the parabola law in his Lo Specchio Ustorio, overo, Trattato delle settioni coniche (The Burning Mirror, or a Treatise on Conic Sections) 1632.

This brought an accusation of intellectual theft from Galileo and it is impossible to tell from the ensuing correspondence, whether Cavalieri discovered the law independently or borrowed it without acknowledgement from Galileo.

The only problem that remained was what exactly was impetus. What was imparted to bodies to keep them moving? The answer was nothing. The solution was to invert the question and to consider what makes moving bodies cease to move? The answer is if nothing does, they don’t. This is known as the principle of inertia, which states that a body remains at rest or continues to move in a straight line unless acted upon by a force. Of course, in the early seventeenth century nobody really knew what force was but they still managed to discover the basic principle of inertia. Galileo sort of got halfway there. Still under the influence of the Platonic axioms, with their uniform circular motion, he argued that a homogenous sphere turning around its centre of gravity at the earth’s surface forever were there no friction at its bearings or against the air. Because of this Galileo is often credited with provided the theory of inertia as later expounded by Newton but this is false.

The Dutch scholar Isaac Beeckman (1588–1637) developed the concept of rectilinear inertia, as later used by Newton but also believed, like Galileo, in the conservation of constant circular velocity. Beeckman is interesting because he never published anything and his writing only became known at the beginning of the twentieth century. However, Beeckman was in contact, both personally and by correspondence, with the leading French mathematicians of the period, Descartes, Gassendi and Mersenne. For a time he was Descartes teacher and much of Descartes physics goes back to Beeckman. Descartes learnt the principle of inertia from Beeckman and it was he who published and it was his writings that were Newton’s source. The transmission of Beeckman’s work is an excellent illustration that scientific information does not just flow over published works but also through personal, private channels, when scientists communicate with each other.

With the laws of fall, the parabola law and the principle of inertia the investigators in the early seventeenth century had a new foundation for terrestrial mechanics to replace those of Aristotle.









Filed under History of Physics, History of science, Mediaeval Science, Renaissance Science

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

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

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

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

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

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

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

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

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


Avicenne Portrait (1271) Source: Wikimedia Commons

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


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

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


Jean Buridan Source

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

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


Niccolò Fontana Tartaglia Source: Wikimedia Commons

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


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


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


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

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







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Filed under History of Islamic Science, History of Mathematics, History of Physics, Mediaeval Science, Renaissance Science

Renaissance Heavy Metal

One of the most fascinating and spectacularly illustrated Renaissance books on science and technology is De re metallica by Georgius Agricola (1494–1555). Translated into English the author’s name sounds like a figure from a game of happy families, George the farmer. In fact, this is his name in German, Georg Pawer, in modern German Bauer, which means farmer or peasant or the pawn in chess. Agricola was, however, anything but a peasant; he was an extraordinary Renaissance polymath, who is regarded as one of the founders of modern mineralogy and geology.


Georg Bauer was born in Glauchau on 24 March 1494, the second of seven children, to Gregor Bauer (born between 1518 and 1532) a wealthy cloth merchant and dyer. He was initially educated at the Latin school in Zwickau and attended the University of Leipzig, where he studied theology, philosophy and philology from 1514 to 1517. From 1518 to 1522 he worked as deputy director and then as director of schools in Zwickau. In 1520 he published his first book, a Latin grammar. The academic year 1522-23 he worked as a lecturer at the University of Leipzig. From 1523 to 1526 he studied medicine, philosophy and the sciences at various Northern Italian university graduating with a doctorate in medicine. In Venice he worked for a time for the Manutius publishing house on their edition of the works of Galen.

From 1527 to 1533 Agricola worked as town physician in St. Joachimsthal*, today Jáchymov in the Czech Republic. In those days Joachimsthal was a major silver mining area and it is here that Agricola’s interest in mining was ignited.


Silver mining in Joachimsthal (1548) Source: Wikimedia Commons

In 1530 he issued his first book on mining, Bermannus sive de re metallica, published by the Froben publishing house in Basel. It covered the search for metal ores, the mining methods, the legal framework for mining claims, the transport and processing of the ores. Bermannus refers to Lorenz Bermann, an educated miner, who was the principle source of his information. The book contains an introductory letter from Erasmus, who worked as a copyeditor for Froben during his years in Basel.

In 1533 he published a book on Greek and Roman weights and measures, De mensuris et ponderibus libri V, also published Froben in Basel.


From 1533 to his death in 1555 he was town physician in Chemnitz. He was also district historian for the Saxon aristocratic dynasty. From 1546 onwards he was a member of the town council and served as mayor in 1546, 1547, 1551 and 1553. In Chemnitz he also wrote a book on the plague, De peste libri tres, his only medical book,  as ever published by Froben in 1554.


Source: Internet Archive

Having established himself as an expert on mining with the Bermannus, Agricola devoted more than twenty years to studying and writing about all aspects of mining and the production of metals. He wrote and published a series of six books on the subject between 1546 and 1550, all of them published by Froben.

De ortu et causis subterraneorum libri V, Basel 1546

The origin of material within the earth

De natura eorum, quae effluunt ex terra, Basel 1546

The nature of the material extruded out of the earth

De veteribus et novis metallis libri II, Basel 1546

Ore mining in antiquity and in modern times

De natura fossilium libri X, Basel 1546

The nature of fossils whereby fossils means anything found in the earth and is as much a textbook of mineralogy

De animantibus subterraneis liber, Basel 1549

The living underground

De precio metallorum et monetis liber III, 1550

On precious metals and coins

At the same time he devoted twenty years to composing and writing his magnum opus De re metallica, which was published posthumously in 1556 by Froben in Basel, who took six years to print the book due to the large number of very detailed woodcut prints with which the book is illustrated. These illustrations form an incredible visual record of Renaissance industrial activity. They are also an impressive record of late medieval technology. Agricola’s pictures say much more than a thousand words.

De re metallicahas twelve books or as we would say chapters. What distinguishes Agricola’s work from all previous writings on mineralogy and geology is the extent to which they are based on empirical observation rather than philosophical speculation. Naturally this cannot go very far as it would be several hundred years before the chemistry was developed necessary to really analyse mineralogical and geological specimens but Agricola’s work was a major leap forward towards a modern scientific analysis of metal production.


Book I: Discusses the industry of mining and ore smelting

Book II: Discusses ancient mines, finding minerals and metals and the divining rod

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Book III: Discusses mineral veins and seams and plotting with the compass

Book IV: Discusses the determination of mine boundaries and mine organisation

Book V: Discusses the principles of mining, the metals, ancient mining and mine surveying

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Book VI: Discusses mining tools and equipment, hoists and pumps, ventilation and miners’ diseases

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Book VII: Discusses assaying ores and metals and the touchstone

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Book VIII: Discusses preparing ores for roasting, crushing and washing and recovering gold by mercury

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Book IX: Discusses ores and furnaces for smelting copper, iron and mercury and the use of bellows

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Book X: Discusses the recovery of precious metals from base metals as well as separating gold and silver by acid

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Book XI: Discusses the recovery of silver from copper by liquidation as well as refining copper

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Book XII: Discusses salts, solvents, precipitates, bitumen and glass

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Agricola’s wonderfully illustrated volume became the standard reference work on metal mining and production for about the next two hundred years. The original Latin edition appeared in Basel in 1556 and was followed by a German translation in 1557, which was in many aspects defective but remained unchanged in two further editions. There were further Latin editions published in 1561, 1621, and 1657 and German ones in 1580, and 1621, with an improved German translation in 1928 and 1953. There was an Italian translation published in 1563.

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Of peculiar interest is the English translation. This was first published in 1912 in London, the work of American mining engineer Herbert Hoover (1874–1964) and his wife the geologist Lou Henry (1874–1944). A second edition was published in 1950. Hoover is, of course, better know as the 31stPresident of the USA, who was elected in 1928 and served from 1929–1933.


Herbert Hoover in his 30s while a mining engineer Source: Wikimedia Commons


Lou Henry, circa 1930 Source: Wikimedia Commons

Agricola’s tome also represents an important development in the history of trades and professions. Before De re metallicaknowledge of trades and crafts was past from master to apprentice verbally and kept secret from those outside of guild, often on pain of punishment. Agricola’s book is one of the first to present the methods and secrets of a profession in codified written form for everyone to read, a major change in the tradition of knowledge transfer.

*A trivial but interesting link exists between St. Joachimsthal and the green back. A silver coin was produced in St. Joachimsthal, which was known as the Joachimsthaler. This got shortened in German to thaler, which mutated in Dutch to daalder or daler and from there in English to dollar.

All illustrations from De re metallica are taken from Bern Dibner, Agricola on Metals, Burndy Library, 1958






Filed under Early Scientific Publishing, History of Technology, Mediaeval Science, Renaissance Science

The Seven Learned Sisters

I have suffered from a (un)healthy[1]portion of imposter syndrome all of my life. This is the personal feeling in an academic context that one is just bluffing and doesn’t actually know anything and then any minute now somebody is going to unmask me and denounce me as an ignorant fraud. I always thought that this was a personal thing, part of my general collection of mental and emotional insecurities but in more recent years I have learned that many academics, including successful and renowned ones, suffer from this particular form of insecurity. On related problem that I have is the belief that anything I do actually know is trivial, generally known to everyone and therefore not worth mentioning[2]. I experienced an example of this recently on Twitter when I came across the following medieval illustration and its accompanying tweet.


Geometria Source: Wikimedia Commons

Woman teaching geometry to monks. In the Middle Ages, it is unusual to see women represented as teachers, in particular when the students appear to be monks. Euclid’s Elementa, in the translation attributed to Adelard of Bath, 1312.

I would simply have assumed that everybody knew what this picture represents and not commented. It is not a “women teaching geometry to monks” as the tweeter thinks but a typical medieval personification of Geometria, one of the so-called Seven Learned Sisters. The Seven Learned Sisters are the personifications of the seven liberal arts, the trivium (grammar, rhetoric and dialectic)

and the quadrivium (arithmetic, geometry, music and astronomy),

which formed the curriculum in the lower or liberal arts faculty at the medieval university. The seven liberal arts, however, have a history that well predates the founding of the first universities. In what follows I shall only be dealing with the history of the quadrivium.

As a concept this four-fold division of the mathematical sciences can be traced back to the Pythagoreans. The mathematical commentator Proclus (412–485 CE) tells us, in the introduction to his commentary on the first book of Euclid’s Elements:

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving.

The earliest know written account of this division can be found at the beginning of the late Pythagorean Archytas’ book on harmonics, where he identifies a set of four sciences: astronomy, geometry, logistic (arithmetic) and music. Archytas’ dates of birth and death are not known but he was, roughly speaking, a contemporary of Plato. He was the teacher of Eudoxus (c.390–c.337 BCE) Harmonics, by the way, is the discipline that later became known as music in the quadrivium.

Without mentioning Archytas, Plato (428/427 or 424/423 – 348/347 BCE), who was highly influenced by the Pythagoreans,takes up the theme in his Republic (c.380 BCE). In a dialogue with Glaucon, Plato explains the merits of learning the “five” mathematical sciences; he divides geometry into plane geometry (two dimensional) and solid geometry (three dimensional). He also refers to harmonics and not music.

In the CE period the first important figure is the Neo-Pythagorean, Nicomachus of Gerasa (c.60–c.120 CE), who wrote an Introduction to Arithmeticand a Manual of Harmonics, which are still extant and a lost Introduction to Geometry. The four-fold division of the mathematical sciences only acquired the name quadrivium in the works of Boethius (c.477–524 CE), from whose work the concept of the seven liberal arts was extracted as the basic curriculum for the medieval university. Boethius, who saw it as his duty to rescue the learning of the Greeks, heavily based his mathematical texts on the work of Nicomachus.

Probably the most influential work on the seven liberal arts is the strange De nuptiis Philologiae et Mercurii (“On the Marriage of Philology and Mercury“) of Martianus Capella (fl.c. 410-420). The American historian H. O. Taylor (1856–1941) claimed that On the Marriage of Philology and Mercurywas “perhaps the most widely used schoolbook in the Middle Ages,” quoted from Martianus Capella and the Seven Liberal Artsby William Harris Stahl.[3]Stahl goes on to say, “It would be hard to name a more popular textbook for Latin reads of later ages.”

Martianus introduces each of the members of the trivium and quadrivium as bridesmaids of the bride Philology.

“Geometry enters carrying a radius in her right hand and a globe in her left. The globe is a replica of the universe, wrought by Archimedes’ hand. The peplos she wears is emblazoned with figures depicting celestial orbits and spheres; the earth’s shadow reaches into the sky, giving a purplish hue to the golden globes of the sun and moon; there are gnomons of sundials and figures showing intervals weights, and measures. Her hair is beautifully groomed, but her feet are covered with grime and her shoes are worn to shreds with treading across the entire surface of the earth.”[4]


A 16th century Geometria in a printed copy of the Margarita Philosophica

“As she enters the celestial hall, Arithmetic is even more striking in appearance than was Geometry with her dazzling peplos and celestial globe. Arithmetic too wears a robe, hers concealing an “intricate undergarment that holds clues to the operations of universal nature.” Arithmetic’s stately bearing reflects the pristine origin, antedating the birth of the Thunder God himself. Her head is an awesome sight. A scarcely perceptible whitish ray emanates from her brow; then another ray, the projection of a line, as it were, coming from the first. A third ray and a fourth spring out, and so on, up to a ninth and a tenth ray–all radiating from her brow in double and triple combinations. These proliferate in countless numbers and in a moment are miraculously retracted into the one.”[5]An allusion to the Pythagorean decade.


Gregor Reisch Margarita Philosophica: Arthimetica presiding over a computing competition between Hindu-Arabic numerals and a reckoning board

“Astronomy like her sister Geometry, is a peregrinator of the universe. She has traversed all the heavens and can reveal the constellations lying beneath the celestial arctic circle. […] Astronomy tells us that she is also familiar with the occult lore of Egyptian priest, knowledge hoarded in their sanctums; she kept herself in seclusion in Egypt for nearly forty thousand years, not wishing to divulge those secrets. She is also familiar with antediluvian Athens.”[6]


“Harmony herself is ineffably dazzling and Martianus is stricken in his efforts to describe her. A lofty figure, her head aglitter with gold ornaments, she walks along between Apollo and Athena. Her garment is tiff with incised and laminated gold; it tinkles softly and soothingly with every measured step She carries in her right hand what appears to be a shield, circular in form. It contains many concentric circles, and the whole is embroidered with striking figure. The circular chords encompass one another and from them pours forth a concord of all tones: Small models of theatrical instruments, wrought of gold, hang suspended from Harmony’s left hand. No know instrument produces sounds to compare with those coming from the strange rounded form.”[7]



I have included Stahl’s passages of Martianus’ descriptions of the quadrivium to make clear then when I talk of the disciplines being personified as women I don’t just mean that they get a female name but are fully formed female characters. This of course raises the question, at least for me, why the mathematical disciplines that were taught almost exclusively to men in ancient Greece, the Romano-Hellenistic culture and in the Middle Ages should be represented by women. Quite honestly I don’t know the answer to my own question. I assume that it relates to the nine ancient Greek Muses, who were also women and supposedly the daughters of Zeus and Mnemosyne (memory personified). This however just pushes the same question back another level. Why are the Muses female?


Having come this far it should be noted that although the quadrivium was officially part of the curriculum on medieval universities it was on the whole rather neglected. When taught the subjects were only taught at a very elementary level, arithmetic based on the primer of Boethius, itself an adaption of Nicomachus, geometry from Euclid but often only Book One and even that only partially, music again based on Boethius and astronomy on the very elementary Sphere of Sacrobosco. Often the mathematics courses were not taught during the normal classes but only on holidays, when there were no normal lectures. At most universities the quadrivium disciplines were not part of the final exams and often a student who had missed a course could get the qualification simple by paying the course fees. Mathematics only became a real part of the of the university curriculum in the sixteenth century through the efforts of Philip Melanchthon for the protestant universities and somewhat later Christoph Clavius for the Catholic ones. England had to wait until the seventeenth century before there were chairs for mathematics at Oxford and Cambridge.

[1]On the one hand imposter syndrome can act as a spur to learn more and increase ones knowledge of a given subject. On the other it can lead one to think that one needs to know much more before one closes a given research/learn/study project and thus never finish it.

[2]To paraphrase some old Greek geezer, the older I get and the more I learn, the more I become aware that what I know is merely a miniscule fraction of that which I could/should know and in reality I actually know fuck all.

[3]William Harris Stahl, Martianus Capella and the Seven Liberal Arts: Volume I The Quadrivium of Martianus Capella. Latin Traditions in the Mathematical Science, With a Study of the Allegory and the Verbal Disciplines by Richard Johnson with E. L. Burge, Columbia University Press, New York & London, 1971, p. 22

[4]Stahl pp. 125–126

[5]Stahl pp. 149–150

[6]Stahl p. 172

[7]Stahl p. 203


Filed under History of Astrology, History of Astronomy, History of Mathematics, Mediaeval Science, Renaissance Science, Uncategorized

Open shelved serendipity

One of my favourite radio science programmes is BBC Radio 4’s Science Stories presented by Philip Ball and Naomi Alderman. Yesterday was the first episode of the fifth series of this excellent piece of popular history of science broadcasting. Last week whilst advertising the new series on Twitter Philip Ball let drop the fact that next weeks episode would be about the medieval theologian and scholar Robert Grosseteste, featuring the physicist of the fascinating interdisciplinary University of Durham research project Ordered Universe, Thom McLeish. This brief Internet exchange awoke in me memories of my own first encounter with the medieval Bishop of Lincoln.

14th-Century Portrait of Robert Grosseteste, Bishop of Lincoln by unknown scribe
Source: Wikimedia Commons

I studied mathematics, philosophy, English philology and history with a strong emphasis on the history and philosophy of science, as a mature student, at the University of Erlangen between 1981 and 1991. It was this period of my life that converted me from an enthusiastic amateur into a university educated and trained researcher into the history of science (For more on this see my post next Monday). When I started this decade of formal studies I held a fairly standard, conservative view of the Scientific Revolution; this started with the publication of Copernicus’ De revolutionibus in 1543 and was completed with the publication of Newton’s Principia Mathematica in 1687. What disrupted, one could even say exploded, this idealised picture was my first encounter with Grosseteste.

Erlangen University is a comparatively large university and its main library is, like that of almost all such institutions, closed shelf. However the department libraries are almost all open shelf and as a student I developed the habit of browsing library bookshelves with no particular aim in view. The Bavarian State university library system has for book purchases an emphasis policy. Each Bavarian university library has a collecting emphasis so that specialist books in a particular discipline are only bought/collected by one university but are available to all the others through the interlibrary loan system. This is a method of making the available funds go further. Erlangen’s collection emphasis is philosophy, including the history and philosophy of science, so the philosophy department library is particularly well stocked in this direction.

One day fairly early in my time as a student in Erlangen I was cruising the history and philosophy of science bookshelves in the philosophy department library when my eyes chanced upon a rather unimposing, fairly weighty book by some guy called Alistair Crombie (I had know idea who he was then) with the title Robert Grosseteste and the origins of experimental science: 1100 – 1700. I have no idea what motivated me to take that volume home with me but I did and once I started reading didn’t stop until I had reached the end. This was a whole new world to me, the world of medieval science, of whose existence I had been blissfully unaware up until that point in time. Reading Crombie’s book radically changed my whole understanding of the history of science.

Here was this twelfth/thirteenth century cleric, lecturer at Oxford University (and possibly for a time chancellor of that august institution), who went on to become Bishop of Lincoln, teaching what amounted to empirical mathematical science.

Grosseteste’s Tomb and Chapel in Lincoln Cathedral
Source: Wikimedia Commons

It should be pointed out that whilst Grosseteste was strong on mathematical empirical science in theory, his work was somewhat lacking in the practice of that which he preached. Crombie has Grosseteste standing at the head of a chain of scholars that include Roger Bacon in the thirteenth century, the Oxford Calculators (about whom there is a good podcast from History of Philosophy without any gaps) and the Paris Physicists in the fourteenth century and so on down to Isaac Newton at the end of the seventeenth century. Unknown to me at the time Crombie was presenting a modernised version of the Duhem Thesis that the scientific revolution took place in the thirteenth and fourteenth centuries and not as the standard model has it, and as I had believed up till I read Crombie’s book, in the sixteenth and seventeenth centuries.

This was the start of a long intellectual journey for me during which I read the works of not only Crombie but of Edward Grant, Marshal Clagett, John Murdoch, David Lindberg, A. Mark Smith, Toby Huff and many other historians of medieval science. This journey also took me into the fascinating world of Islamic science, which in turn led me to the histories of both Indian and Chinese science although I still have the impression that in all these areas medieval European science, Islamic science, and Indian and Chinese science I have till now barely scratched the surface.

As I said above this journey started with Crombie’s book and Robert Grosseteste discovered whilst aimlessly browsing the shelves in the department library. This is by no means the only important and influential book that I have discovered for myself by this practice of browsing in open shelf department libraries. On one occasion I went looking for one specific book on map projection in the geography department library and, after a happy hour or two of browsing, left with an armful of books on the history of cartography. On another occasion I discovered, purely by accident, The Life and Letters of Sir Henry Wotton edited by Logan Pearsall Smith in the English Department Library. Wotton a sixteenth/seventeenth century English diplomat was a passionate fan of natural philosophy, who sent the first copies of Galileo’s Sidereus Nuncius, fresh off the printing press to London on its day of publication in 1610.

There are many other examples of the scholarly serendipity that my habit of browsing open shelf library shelves has brought me over the years but I think I have already made the point that I wanted to when I set out to write this post. Libraries are full of wonderful, vista opening books, so don’t wait for somebody to recommend them to you but find an open shelf library and go and see what chance throws you way, it might just change your life.







Filed under Autobiographical, History of science, Mediaeval Science

The Astrolabe – an object of desire

Without doubt the astrolabes is one of the most fascinating of all historical astronomical instruments.

Astrolabe Renners Arsenius 1569 Source: Wikimedia Commons

Astrolabe Renners Arsenius 1569
Source: Wikimedia Commons

To begin with it is not simply one object, it is many objects in one:


  • An astronomical measuring device
  • A timepiece
  • An analogue computer
  • A two dimensional representation of the three dimensional celestial sphere
  • A work of art and a status symbol


This Medieval-Renaissance Swiss Army penknife of an astronomical instrument had according to one medieval Islamic commentator, al-Sufi writing in the tenth century, more than one thousand different functions. Even Chaucer in what is considered to be the first English language description of the astrolabe and its function, a pamphlet written for a child, describes at least forty different functions.

The astrolabe was according to legend invented by Hipparchus of Nicaea, the second century BCE Greek astronomer but there is no direct evidence that he did so. The oldest surviving description of the planisphere, that two-dimensional representation of the three-dimensional celestial sphere, comes from Ptolemaeus in the second century CE.

Modern Planisphere Star Chart c. 1900 Source: Wikimedia Commons

Modern Planisphere Star Chart c. 1900
Source: Wikimedia Commons

Theon of Alexandria wrote a thesis on the astrolabe, in the fourth century CE, which did not survive and there are dubious second-hand reports that Hypatia, his daughter invented the instrument. The oldest surviving account of the astrolabe was written in the sixth century CE by John Philoponus. However it was first the Islamic astronomers who created the instrument, as it is known today, it is said for religious purposes, to determine the direction of Mecca and the time of prayer. The earliest surviving dated instrument is dated 315 AH, which is 927/28 CE.

The Earliest  Dated Astrolabe Source: See Link

The Earliest Dated Astrolabe
Source: See Link

It is from the Islamic Empire that knowledge of the instrument found its way into medieval Europe. Chaucer’s account of it is based on that of the eight-century CE Persian Jewish astrologer, Masha’allah ibn Atharī, one of whom claim to fame is writing the horoscope to determine the most auspicious date to found the city of Baghdad.

So-called Chaucer Astrolabe dated 1326, similar to the one Chaucer describes, British Museum Source: Wikimedia Commons

So-called Chaucer Astrolabe dated 1326, similar to the one Chaucer describes, British Museum
Source: Wikimedia Commons

However this brief post is not about the astrolabe as a scientific instrument in itself but rather the last point in my brief list above the astrolabe as a work of art and a status symbol. One of the reasons for people’s interest in astrolabes is the fact that they are simply beautiful to look at. This is not a cold, functional scientific instrument but an object to admire, to cherish and desire. A not uncommon reaction of people being introduced to astrolabes for the first time is, oh that is beautiful; I would love to own one of those. And so you can there are people who make replica astrolabes but buying one will set you back a very pretty penny.

That astrolabes are expensive is not, however, a modern phenomenon. Hand crafted brass, aesthetically beautiful, precision instruments, they were always very expensive and the principal market would always have been the rich, often the patrons of the instrument makers. The costs of astrolabes were probably even beyond the means of most of the astronomers who would have used them professionally and it is significant that most of the well know astrolabe makers were themselves significant practicing astronomers; according to the principle, if you need it and can’t afford it then make it yourself. Other astronomers would probably have relied on their employers/patrons to supply the readies. With these thoughts in mind it is worth considering the claim made by David King, one of the world’s greatest experts on the astrolabe, that the vast majority of the surviving astrolabes, made between the tenth nineteenth centuries – about nine hundred – were almost certainly never actually used as scientific instruments but were merely owned as status symbols. This claim is based on, amongst other things, the fact that they display none of the signs of the wear and tear, which one would expect from regular usage.

Does this mean that the procession of astrolabes was restricted to a rich elite and their employees? Yes and no. When European sailors began to slowly extend their journeys away from coastal waters into the deep sea, in the High Middle Ages they also began to determine latitude as an element of their navigation. For this purpose they needed an instrument like the astrolabe to measure the elevation of the sun or of chosen stars. The astrolabe was too complex and too expensive for this task and so the so-called mariners astrolabe was developed, a stripped down, simplified, cheaper and more robust version of the astrolabe. When and where the first mariner’s astrolabe was used in not known but probably not earlier than the thirteenth century CE. Although certainly not cheap, the mariner’s astrolabe was without doubt to be had for considerably less money than its nobler cousin.


Mariner’s Astrolabe Francisco de Goes 1608 Source: Istituto e Museo di Storia della Scienza, Firenze

Another development came with the advent of printing in the fifteenth century, the paper astrolabe. At first glance this statement might seem absurd, how could one possibly make a high precision scientific measuring instrument out of something, as flexible, unstable and weak as paper? The various parts of the astrolabe, the planisphere, the scales, the rete star-map, etc. are printed onto sheets of paper. These are then sold to the customer who cuts them out and pastes them onto wooden forms out of which he then constructs his astrolabe, a cheap but serviceable instrument. One well-known instrument maker who made and sold printed-paper astrolabes and other paper instruments was the Nürnberger mathematician and astronomer Georg Hartmann. The survival rate of such cheap instruments is naturally very low but we do actually have one of Hartmann’s wood and paper astrolabes.

Hartmann Paper Astrolabe Source: Oxford Museum of History of Science

Hartmann Paper Astrolabe
Source: Oxford Museum of History of Science

In this context it is interesting to note that, as far as can be determined, Hartmann was the first instrument maker to develop the serial production of astrolabes. Before Hartmann each astrolabe was an unicum, i.e. a one off instrument. Hartmann standardised the parts of his brass astrolabes and produced them, or had them produced, in batches, assembling the finished product out of these standardised parts. To what extent this might have reduced the cost of the finished article is not known but Hartmann was obviously a very successful astrolabe maker as nine of those nine hundred surviving astrolabes are from his workshop, probably more than from any other single manufacturer.

Hartmann Serial Production Astrolabe Source: Museum Boerhaave

Hartmann Serial Production Astrolabe
Source: Museum Boerhaave


If this post has awoken your own desire to admire the beauty of the astrolabe then the biggest online collection of Medieval and Renaissance scientific instruments in general and astrolabes in particular is the Epact website, a collaboration between the Museum of the History of Science in Oxford, the British Museum, the Museum of the History of Science in Florence and the Museum Boerhaave in Leiden.

This blog post was partially inspired by science writer Philip Ball with whom I had a brief exchange on Twitter a few days ago, which he initiated, on our mutual desire to possess a brass astrolabe.






Filed under History of Astrology, History of Astronomy, History of science, History of Technology, Mediaeval Science, Renaissance Science