The Renaissance Mathematicus

September 10, 2025 · 6:52 am

The history of science in medieval Islamicate culture.

This post is a first attempt to answer a question that came up when I was rude, not for the first time, on social media about Jim Al-Khalili’s book The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave Us the Renaissance (Penguin, 2011). Al-Khalili is an excellent broadcaster, who is very good at presenting modern science, mostly in the form of interviews, to a lay public but he is not a historian of science. Two examples of things that for me disqualify Jim Al-Khalili, as an authority on the history of Arabic science.

Firstly, during a radio panel discussion on the history of Arabic science he said, with reference to the House of Wisdom, “Professor Pormann will tell me that it didn’t exist, but I prefer to believe it did. Professor Peter Pormann is Professor of Classical and Greek-Arab Studies at the University of Manchester and a leading authority on the history of Arabic science. Secondly, Al-Khalili states categorically that because of his experimental programme in optics Ibn al-Haytham (c. 965–c. 1040) is the originator of the modern scientific method. Ibn al-Haytham’s experimental programme was a copy and extension of the very similar programme of Ptolemaeus (fl. 150 CE) so, if Ibn al-Haytham is the originator of the modern scientific method what does that make Ptolemaeus? Secondly, A. Mark Smith, historian of optics and leading authority on Ibn al-Haytham, thinks that most of al-Haytham’s experiments are only thought experiments because using the equipment he describes he would never have achieved the accurate results that he presents.

So having dismissed Al-Khalili’ s book I naturally got asked what I would recommend instead. This is a problem as there is no really good general introduction to the science produced in medieval Islamicate culture, that is in the various areas dominated by Islam from the beginning of the seventh to the end of the sixteenth century. Please nobody recommend Jonathan Lyons’ The House of Wisdom: How the Arabs Transformed Western Civilization (ppb. Bloomsbury, 2010) if anything it’s even worse that Al-Khalili’s book. The blurb from the book contains this little gem: “The Arabs could measure the earth’s circumference (a feat not matched in the West for eight hundred years).” Whoever wrote that has apparently never heard of Eratosthenes (c.276–c. 195 BCE) born in Cyrene, which last time I looked was in the West, about eight hundred years before Muhammad (C. 570–632 CE).

I would recommend the three volume Encyclopedia of the History of Arabic Science, edited by Roshdi Rashed, (Routledge, 1996), which has its own Wikipedia article, but it was almost certainly expensive when it was first published and is now out of print. If, however, you have access to a university library, they might well have a copy or can get one for you via interlibrary loan. Written by an all star cast, it is excellent and reasonably accessible for the layman.

Before I go further, why the circumlocution in my blog post title? There is a major problem about what to call the science under discussion. To simply call it Islamic science is problematic because not all the people who produced it were Muslims. There were also Jews, Christians, Sabians, and possibly Zoroastrians, who contributed to the science flowing out of the areas dominated by the Muslims. The alternative used in the titles of the books I have already mentioned is to call it Arabic science, because most of it was written in Arabic (most but not all), however, many of the authors were not Arabs. From now on I shall refer to it as alternatively as Islamic or Arabic science but with reservations.

One major problem is that the field of Arabic science and technology is very wide as can be seen from the three volumes of the Encyclopedia of the History of Arabic Science and neither AL-Khalili nor Lyons really cover the whole territory. Most of the books that I’m now going to recommend only cover a limited area but as already mentioned apart from the Encyclopedia of the History of Arabic Science there is no book that covers it all.

The first book that I would recommend, but with some reservations, is Stephen P. Blake, Astronomy and Astrology in the Islamic World, (Edinburgh University Press, 2016). This is one of The New Edinburgh Islamic Surveys, series editor Carole Hillenbrand, about which I will say more later.

Blake opens his preface with the following sentence:

To compose a readable, nontechnical account of astronomy and astrology in the Muslim world is challenging. The topic is scientific (dependent on arcane mathematical theories and concepts), the period is long (covering nearly 1,000 years), the geography is extensive (stretching from India in the East to Spain in the West), and the context is crucial. To make sense of the Islamic era (from the middle of the eighth century CE until the middle of the sixteenth century), the narrative must begin three centuries before (with the Egyptians) and continue through the century following ( with Copernicus, Kepler, and Newton).

Having admirably sketched out here the problem that the author faces, one that applies to all the sciences in the Islamic World and not just astronomy and astrology, Blake goes on in his book to give a masterly attempt to meet the challenge and almost succeeds.

The first twenty pages are devoted to a brief, comprehensive but informative survey of the history of astronomy before Islam, covering Ancient Egypt, Mesopotamia, Ancient Greece with a concentration on the works of Ptolemaeus, Ancient India and Persia. Given its brevity a surprisingly good survey. The next sixteen pages covers the birth of Islam and in rapid succession all the usuall figures of early Arabic astronomy and astrology from Abu Ma’shar to al-Biruni giving details of their work and its influences from the sources mentioned in the first capital.

The next ten pages opens with a very brief account of the Abbasids and astronomy in Baghdad, including the House of Wisdom, but emphasising the limited amount of large scale observation and leading up to the first large Islamic observatory in Isfahan founded around 1100 by Umar al-Khayyam (c. 1048–1131). In the early chapters on Islamic astronomy, Blake also goes into detail on the motivations of the Islamic astronomers, time keeping for the daily prayers, the Muslim lunar calendar, determining the direction of Mecca, the qibla, for prayer, and of course astrology.

Blake now goes geographically to the other end of the Islamic domination and devotes a chapter to Astronomy and Astrology in al-Andalus. He examines the similarities but above all the differences in the developments of the disciplines in the western end of the Islamic sphere of influence, to those in Asia. Important because it was al-Andalus that principally introduced Islamic astronomy and astrology into Europe.

We now get chapters on the histories of the large scale observatories from Maragha, over Samarqand, and Istanbul ending in Shajahanabad in India. Each chapter deals with its founding, the astronomers who ran it and the results that they produced.

Despite the brevity of his book, the entire text is only one hundred and fifty pages long, Blakes tour of the history of Islamic astronomy and astrology is very comprehensive. At time I found it perhaps too condensed as a result, letting it read, at times, rather like a telephone book. I also found that Blake lets his own very obvious personal rejection of astrology gets in the way of his historical objectivity. Although he covers the astrology he does so grudgingly, referring to it constantly as a pseudo-science or a superstition.

Despite these minor quibbles I would whole heartedly recommend this book if it wasn’t for the final chapter, Medieval and early-Modern Europe. My friend the HISTSCI_HULK would probably call it a cluster fuck, as it contains errors that a book on this level should not contain.

The book closes with a glossary which gives brief description of the instruments referred to throughout the book. Unfortunately, he repeats the totally erroneous claim that Hipparchus invented the astrolabe.

Each chapter of Blake’s book has end notes that basically just refer the reader to the very extensive bibliography at the end of the book. There is also a good index after the bibliography. There are no illustration in the body of the text but there are eight pages of very nice colour illustration in the middle of the book.

My second book, A Brief Introduction to Astronomy in the Middle East by John M. Steele (SAQI, 2008), I would whole heartedly recommend without reservation.

It is not just about Arabic or Islamic astronomy but delivers exactly what the title says. Steele is leading expert on the history of Mesopotamian science and it is here that his brief account begins, following an introduction that sets out the route that he intends to take.

The opening chapter, The Birth of Astronomy in the Middle East, gives a concise but informative cover of the evolution of astronomical activity in Mesopotamia from the invention of writing in the fourth millennium BCE down to the late Babylonian period in the first millennium BCE. Having followed that evolution Steele now devotes, for this short book, a long chapter to Late Babylonian Astronomy, which is the period in which the mathematical astronomy that is the very recognisable ancestor of our own modern astronomy came into being and evolved. Producing amongst other things the zodiac and mathematic models to predict astronomical phenomena.

The third chapter, Astronomy in the Greek and Roman Middle East, opens with Alexander the Great’s conquest of Babylon in 331BCE, which signalled the start of the transfer of astronomy and its further development to Ancient Greece. A comparatively short capital, it documents the Greek adoption of geometrical models for astronomy to replace the arithmetical-algebraic models of the Babylonians. It also introduces the work of Hipparchus and naturally, above all Ptolemy, closing with his Almagest.

Nearly all of the rest of the book, almost the half, is dedicated to our actual topic Islamic astronomy. The first of three chapters, Astronomy in Medieval Islamic Culture, sketches the religious reasons why astronomy was important to Islamic culture during the medieval period, reasons that led them to adopt and develop the Babylonian and Greek astronomical heritage–the Islamic lunar calendar, the prescribed times of prayer, and the direction of prayer, towards Mecca.

The second of the three chapters, Astronomical Observations and Instruments in the Medieval Islamic World, follows the evolution of the scientific methods that the astronomers undertook to fulfil those religious requirement and to go beyond them in developing a full blown, sophisticated astronomical science.

The third chapter of the three, Medieval Planetary Theory, takes a look at the moves beyond Ptolemy that the Islamic astronomers undertook. New more accurate astronomical tables based on their own more accurate observations made with new improved instruments. Active criticisms of the more unsatisfactory aspects of Ptolemy’s Almagest and the development of new geometrical models with which to track the path of the planets, in particular the models of Nasīr al-Din al-Tūsī and Ibn al-Shatir, both of which, as is well known, later reemerged in the work of Copernicus.

The closing chapter of Steel’s book, Legacies, deals very briefly with exactly this transmission of Islamic astronomical knowledge into Europe and into modern astronomy. The general theme of the transmission of knowledge and especial astronomical knowledge is the golden thread that winds its way through the whole of Steele, all too brief, book.

Although Steele’s book actually has the same number of pages as Blake’s, those pages are noticeably smaller and although he covers in detail a much wider range of material, one doesn’t have Blakes mass of specific detail about individual aspects of Islamic astronomy, making his book an easier and pleasanter read.

Steele’s book closes with a brief explanation of the sexagesimal number system. There are brief biographies for each chapter of the book and a very small number of endnotes giving the sources for the translation used. He also includes a short but usable index. There are black and white illustrations scattered throughout the text.

My third book, Science and Islam: A History by Ehsan Masood (Icon, 2009) is completely different, in that it is a popular book aimed at the general public, which gives an introduction to the whole spectrum of Islamic science.

Whilst by no means perfect, Masood’s book is less flawed than those of Al-Khalili and Lyons. Perhaps its greatest strength lies in that which is implied in the title, this is not a book about Islamic science but about the history of Islamic culture and the science that grew up within it; a subtle but importance difference.

In fact, the first seven chapters are devoted to a sketch of the history of Islam from the very beginning up to the destruction of the Eastern caliphate by the Mongols in the thirteenth and fourteenth centuries. The scientific centres and the leading scientific scholars get mentioned on the way but the emphasis is very much on the religion and the politics and how various aspects of both led to an atmosphere in which science could emerge and grow.

Having first established this framework Masood now turns to a closer look at the science itself. In Part II of the book, he present the Branches of Learning. In The Best Gift From God, the title is part of a quote from Muhammad about good health, Masood presents a short but competent survey of Islamic medicine, covering the principle scholars, their fields and their books. Chapter nine, Astronomy: The Structured Heaven is a reasonably competent brief survey of the Islamic contributions to the study of the heavens and the astronomers, who made those contributions, which covers all the main bases.

We get a similar competent survey of the Islamic world of mathematics in the tenth chapter, Number: The Living Universe of Islam. Starting with the Arabic adoption of the Hindu number system, proceeding from there to algebra with the contributions of al-Khwarizmi and Omar Khayyam. A very brief look at Euclid’s Fifth Postulate and on into the world of geometry. We move on to a handful of pages in chapter eleven, At Home in the Elements in which Masood tries too hard to differentiate between alchemy and chemistry although he admits that at the time the division is not so clear. The final short chapter in this section, Ingenious Devices, is dedicated to technology but only briefly covers the automata and the water clocks.

The third part of Masood’s book, Second Thoughts, starts with a chapter, An Endless Frontier, that looks at the significant Islamic contributions to the development of optics especially in Europe, and then moves on to explain how during the Renaissance, the mood in Europe turned against the Islamic contributions to medicine. It closes with a very brief look at medieval Islamic views on evolution. The penultimate chapter, One Chapter Closes, Another Begins, examines the reasons for the decline of science in Islamicate cultures, finding part of the blame for a continuing decline in Western colonialism. The final chapter, Science and Islam: Lessons From History addresses the themes, Did science need Islam?, Did Islam need science?, and Islam and the new knowledge today.

The book has neither foot nor endnotes and no illustrations. At the end of the book there is a useful timeline and an extensive index. I think that the book, whilst not perfect, is a potentially good introduction for somebody approaching the history of science in the medieval Islamicate cultures for the first time.

When looking at texts on the history of mathematics, I recommended J.L. Berggren’s Episodes in the Mathematics of Medieval Islam, (Springer, 1986, ppb. 2003) as an excellent introduction to the topic, a recommendation that I, naturally, repeat here.

I don’t know of any monographs dedicated to the history of alchemy in the medieval Islamicate cultures but the Encyclopedia of the History of Arabic Science, which I mention above, naturally, has an article on the topic. Lawrence M. Principe in his excellent The Secrets of Alchemy (The University of Chicago Press, 2013), a very readable general history of alchemy that I would recommend to anybody interested in the topic, also, naturally, has a chapter, Development: Arabic al-Kimiyā’.

I stumbled across Stephen Blakes book, one of The New Edinburgh Islamic Surveys, series editor Carole Hillenbrand, fairly recently and discovered that the series contains two further volumes relevant to this review, Donald R. Hill, Islamic Science and Engineering, (Edinburg University Press, 1993) and Peter E. Pormann & Emilie Savage-Smith, Medieval Islamic Medicine (Edinburg University Press, 2007), which I ordered and have arrived in the last few days and which I have started reading. Reviews will follow here in due course. There is also a new monster, 838 pages, Routledge Handbook on the Sciences in Islamicate Societies: Practices from the 2nd/8th to the 13th/19th Centuries eds. Sonja Brentjes, Peter Barker, Rana Brentjes (Routledge , 2025), which I assume is the updated replacement for Encyclopedia of the History of Arabic Science. I have this on order but don’t know when it will turn up here. Will also review, when it does arrive.

4 Comments

Filed under Book Reviews, History of Astrology, History of Astronomy, History of Islamic Science

September 3, 2025 · 9:04 am

A historian of science and New York Times reporter tries his hand at the 17th century Church and heliocentricity controversy and fails dismally.

On 31 August the New York Times published an opinion piece under the following title:

Historians See Autocratic Playbook in Trump’s Attacks on Science

From the headline illustration it was very obvious in which direction at least one part of this essay would go. My friend the HISTSCI_HULK would like to share his, as usual, carefully considered opinion with the author of the piece a William J. Broad, who apparently has a graduate degree in the history of science.

The 1633 trial of Galileo over his backing of the heliocentric theory came to symbolize the church’s hostility to open inquiry. Credit…Universal History Archive/Universal Images Group, via Getty Images

Hulky: The title picture is totally bloody ridiculous; it’s supposed to depict Galileo’s trial but looks more like a third-rate attempt at a Pieter Bruegel depiction of hell!

Just for the record, Galileo’s trial was not a major public spectacle such as depicted here. He was interrogated in a chamber in the Inquisition’s villa with a small number of people present. The whole time, given his age and infirmities, he was treated with care and consideration. His abjuration was a public ceremony but a dignified and solemn one in a church not the circus depicted in the picture.

The article opens with a long screed describing Trump’s attacks on science before our author takes us to his perceived historical parallel starting with:

The Church

A statue of Giordano Bruno in the Campo dei Fiori in Rome, where he was burned at the stake in 1600 for defending Copernicus’s theory of heliocentrism. Credit…Andreas Solaro/Agence France-Presse — Getty Images

Hulky: Another bloody stupid, provocative illustration, a seagull shitting on Bruno’s head! And no, he wasn’t bloody burned at the stake for defending Copernicus’s theory of heliocentrism.

We only have partial records of Bruno’s trial, and it is not known for certain what he was actually found guilty of and sentenced to death for. However, his denial of the Holy Trinity, the divinity of Christ and, the Virgin Birth would certainty have weighed more heavily than anything in his cosmology. Within that cosmology, what the Church found objectionable was not his adoption of Copernicus’ hypothesis but rather his belief that there were other planets out in the infinite cosmos that were inhabited. This caused all sorts of theological conundrums, which I’m not going to go into now.

From the start, modern science faced repression. The backdrop was doctrine: The Roman Catholic Church long held that humans sat at the center of the universe as the stars, planets and sun moved overhead in never-ending tributes.

Hulky: There was absolutely nothing fucking special about the Church’s cosmology. They simply adopted the model that every other normal fucker believed in!

The Church’s view was not that the Earth sat at the center of the universe but rather that the Earth occupied the dregs at the bottom. They certainly did not think that “the stars, planets and sun moved overhead in never-ending tributes.” I think, William J. Broad is making shit up.

Not so, argued Nicolaus Copernicus, a Polish astronomer. In 1543, he laid out evidence showing that the Earth and planets revolve around the sun.

Hulky: How many times do you have to tell people that Copernicus was not bloody Polish or German for that matter.

At the time of his birth Toruń, his birthplace, after being the center of wars between the Teutonic Knights and Poland for many decades, had become an autonomous city under the protection of the Polish crown. Copernicus lived the majority of his life in Ermland, also called Warmia, a self-governing, autonomous prince bishopric also under the protection of the Polish Crown but not part of Poland.

Hulky: Evidence what fucking evidence?

Although Copernicus presented an interesting hypothesis, his major problem, and the reason why he didn’t publish for many years, was that he couldn’t provide any evidence whatsoever that the Earth actually moved. A problem that would dog the heliocentric hypothesis for more than one hundred and seventy years until Bradly produced the first such empirical evidence, stellar aberration, in 1725.

News of his book, 400 pages long and rich in diagrams, moved slowly across Europe. The church in time decided to show its displeasure. In 1600, it had Giordano Bruno, an advocate of Copernicus’s heliocentric theory, burned at the stake.

Hulky: Repeating the bloody Bruno myth doesn’t make it any fucking truer the second time round.

To fight the heresy, the church in 1616 put the Copernican tract on its list of prohibited books. Undeterred, Galileo, an Italian astronomer, in 1632 published his great work, “Dialogue Concerning the Two Chief World Systems.” It backed Copernicus.

Hulky: And took it bloody well off again in 1621

De revolutionibus was only placed on the Index until corrected and, somewhat surprisingly, it was corrected and given free to read again already in 1621. There were in fact only a few minor changes, the removal of the few places where Copernicus said or implied that heliocentrism was a fact rather than a hypothesis. It should also be noted that of the comparatively large number of surviving copies of De revolutionibus very few were actually “corrected” and all of the “corrected” copies, including Galileo’s own copy, were in Italy. Outside of Italy nobody took any notice of the Church’s mild censorship.

Galileo’s trial by the Roman Inquisition in 1633 was a turning point in Western history. The spectacle of the elderly thinker being forced, under threat of torture, to recant came to symbolize the church’s hostility to open inquiry.

Hulky: Galileo pleaded guilty because he was bloody guilty as charged not because somebody threatened him.

As Hulky correctly states, Galileo pleaded guilty, knowing full well that he was guilty as charged, after failing dismally to worm his way out of it with half truths and fudging, not because he was threatened with torture. In fact, he was never really threatened with torture, the threat of torture being a legal formality during a trial for heresy and under the Church’s own rules given his age and infirmities, Galileo would never have been subjected to torture.

Even so, Rome proceeded to adapt churches and cathedrals to serve as solar observatories, which let the church improve the calendar and better fix the date of Easter. The research also gave credence to the Copernican view. Nonetheless, Rome kept its heliocentric ban in place for centuries.

Hulky: Oh, William J. Broad seems to have read his Heilbron, but if he’s going to quote Heilbron he might at least get his fucking facts right!

Hulky is saying is that Broad is obviously referencing John L. Heilbron’s excellent The Sun in the Church: Cathedrals as Solar Observatories (Harvard University Press, 1999). The problems with the calendar and the date of Easter had largely been dealt with in the previous century, although the solar observatories were used to get more precise values for the length of the year and the dates of the solstices and equinoxes. The research gave credence to a heliocentric system, Keplerian rather than Copernican, because the researchers were actively trying to find facts and evidence to prove or disprove either of the prevalent models of the cosmos, Keplerian heliocentricity or Tychonic geo-heliocentricity, usually with diurnal rotation. It should be noted that the research during the seventeenth century was predominantly carried out by Catholic, mostly Jesuit or Jesuit educated, astronomers. A good indication of this is to be found in Giovanni Battista Riccioli’s Almagestum Noum (1651), which gives a detailed survey of that research including his famous list of arguments for and against heliocentricity.

Rome, meaning the Church, never banned heliocentricity. They stated that one could only speak of it as a hypothesis and not as a fact, which was its correct scientific status throughout the seventeenth century. Even that stipulation was quietly forgotten by the middle of the eighteenth century, although it was only formally lifted in the 1830s.

The Catholic Church’s double standard — crushing blue-sky science while enjoying the practical benefits — became a favorite tactic of monarchs, despots and modern autocrats. Today the two categories of exploratory work are known as basic and applied science. The latter can include development, engineering and technology. By nature, basic studies, though risky, tend to yield the most important discoveries.

Hulky: Nobody fucking crushed blue-sky science

The Catholic Church never banned blue-sky research and as stated above Catholic astronomers were very actively involved in pure research throughout the seventeenth century and delivered the results that eventually led to the general acceptance of heliocentricity around 1670, despite the fact that the evidence that the Earth moves first came in 1725.

Monarchs, despots and autocrats had been controlling research into science and technology long before the seventeenth-century dispute over heliocentricity and didn’t need the Church or anybody else, for that matter, to show them the way.

The lopsided approach let rulers curb free thought that threatened their authority while promoting technological spinoffs of applied science that could empower their regimes. For instance, they backed research on celestial navigation, which let fleets of tall ships sail the globe to found colonial empires.

Of course, states and their rulers back research that is of economic advantage to them. It’s a trivial historical truism. They always have and always will so, I’m not quite sure what Broad is trying to say here.

Even enlightened despots such as Catherine the Great in 18th-century Russia, while promoting science and progress, retained absolute power and suppressed ideas they saw as challenging their rule.

Once again, I’ve got no idea what great revelations Broad’s paragraph closing this section is supposed to reveal.

Hulky: Are we all supposed to jump up and fucking down going WOW I never knew that or something?

45 Comments

Filed under History of Alchemy, Myths of Science

August 27, 2025 · 7:34 am

From τὰ φυσικά (ta physika) to physics – LI

In his book, The History of the Barometer (The Johns Hopkins Press, 1964), W. E. Knowles Middleton whilst discussing the contact between Isaac Beeckman (1588–1637) and René Descartes ^[1] (1596–1650) writes:

This brought him into relations, and finally into collision, with René Descartes (1596–1650), a very great philosopher, most of whose ideas about physics have turned out to be wrong.

A harsh judgement but historically correct. A presentist might argue, that being the case we don’t really need to look at Descartes’ ideas about physics, however our presentist would be wrong. Descartes’ mistaken concepts had a massive influence in the second half of the sixteenth century and the first half of the eighteen century. For example, Isaac Newton (1642–1727 os) was initially, as a student, a Cartesian and his intellectual development can be traced by his gradual rejection of Descartes’ ideas. Following the publication of Newton’s Principia in 1687, the main debate in Europe over astronomy and physics up to about 1750 was the embittered struggle between the Cartesians and the Newtonians, with Isaac emerging victorious in the end.

Engraved portrait of Descartes based on painting by Frans Hals the Elder (c. 1582–1666) Source: Wikimedia Commons

Descartes most famous work was his Discours de la méthode (Discourse on the Method) published in 1637, in which he sets out, amongst other things, his rational, deductive methodology of science. To illustrate that methodology with practical example the book has three essays as appendices, La Dioptrique, La Géométrie, and Les Météores.

Descartes presents four precepts to be followed in the acquisition of knowledge and the avoidance of error:

To accept as true only such conclusions as are clearly and distinctly known to be true and to exclude all possibilities of doubt.
To analyse problems under consideration into as many parts as possible.
To reason correctly from the simpler to the more complex elements.
To adopt a comprehensive view which should omit nothing essential to the problem.^[2]

For Descartes the one academic discipline that was beyond doubt was geometry and so La Géométrie is one of his three appendices.

Geometrical optics is, as its name implies, based on geometry, and so La Dioptrique, which contains his theories on optics, is another of the three.

Descartes theories on optics are a mixture of horribly wrong and basically correct. From his time with Beeckman he had adopted his corpuscular, mechanistic theory of the world with one major and, in this context, important difference. Whereas Beeckman was the first natural philosopher in the seventeenth century to accept the existence of the vacuum, Descartes categorically rejected it and would continue to do so all of his life.

For Descartes the world is filled with particles, and his theory of light is derived from this particular theory:

He regarded space as completely filled with perfectly rigid particles of various sizes and shapes. Those of the “third element,” or ordinary matter, are the grossest and have an arbitrary shape. Those of the “second element, “ or “subtle matter,” are round and they fill as much as they can of the space between the former particles. Those of the “first element” are arbitrarily small and they fill the remaining interstices; they are scrapings (raclure) generated during the production of the balls of the second element by mutual attrition of rotating particles; in the process they acquired an intense agitation. The sun and the stars are spherical accumulations of the first element. They are immersed in the subtle matter of the second element. Light is nothing but the pressure (inclination au mouvement or conatus) that the sun and stars exert on the balls of the second element. This pressure is instantaneous and rectilinearly transmitted to the eye, owing to the contiguity of the balls and their perfect rigidity.^[3]

Descartes illustrates the process of seeing by adopting the analogy used by the Stoics in their theory of vision.

It has sometimes doubtless happened to you, while walking in the night without a torch through places that are a little difficult, that it becomes necessary to use a stick in order to guide yourself; and you may have been able to notice that you felt, through the medium of this stick, the diverse objects placed around you, and that you were even able to tell whether they were trees, or stone, or sand, or water, or grass, or mud, or any such thing. True this sort of sensation is rather confused and obscure in those who do not have practice with it; but consider it in those who, being born blind, have made use of it all their lives, and you will find it so perfect and so exact that one might almost say that the see with their hands, or that their stick is the organ of some sixth sense given to them in the place of sight. And in order to draw a comparison from this, I would have you consider light as nothing else, in bodies that we call luminous, than a certain movement or action, very rapid and very lively, which passes towards our eyes through the medium of the air or other transparent bodies, in the same manner that the movement or resistance of the bodies that this blind man encounters in transmitted to his hand through the medium of his stick. (Descartes La Dioptique)

If I’m being honest, I don’t see how either Descartes corpuscular theory of the world, or his theory of light and visual perception in anyway fulfil his four precepts for acquiring knowledge.

Descartes delivers a second analogy for light reaching the eye with wine seeping through a hole in the bottom of a vat full of grapes, whereby the grapes are the second elements and the wine the first elements.

Now consider that, since there is no vacuum in Nature as almost all the Philosophers affirm, and since there are nevertheless many pores in all the bodies that we perceive around us, as experiment can show quite clearly, it is necessary that these pores be filled with some very subtle and very fluid material, extending without interruption from the stars and planets to us. Thus, this subtle material being compared with the wine in that vat, and the less fluid or heavier parts, of the air as well as of other transparent bodies, being compared with the bunches of grapes which are mixed in, you will easily understand the following: Just as the parts of this wine…tend to go down in a straight line through the hole [and other holes in the bottom of the vat]…at the very instant that it is open…without any of those actions being impeded by the others, nor by the resistance of the bunches of grapes in this vat…in the same way, all of the parts of the subtle material, which are touched by the side of the sun that faces us, tend in a straight line towards our eyes at the very instant that we open them, without these parts impeding each other, and even without their being impeded by the heavier particles of transparent bodies which are between the two. (Descartes La Dioptique, Wikipedia)

The multiplicity of Descartes’s analogies suggest his awareness of weaknesses in his deduction of rectilinear propagation. Yet he did not doubt the central tenet of his model of light: light is a pressure instantaneously propagated through contiguous chains of rigid balls. [my emphasis] There can be no delay in the transmission of the pressure because the matter of the balls, being pure extension, is necessarily incompressible. As Descartes wrote to his Dutch mentor Isaac Beeckman: “The instantaneous propagation of light is to me so certain that if its falsity could be shown, I would be ready to admit my complete ignorance of Philosophy.” ^[4]

So, folks you read it here, Descartes admitted in writing that he was completely ignorant of Philosophy, because the propagation of light is not instantaneous. To be fair to Descartes this was at the time a hotly debated issue, is the speed of light finite or infinite? Descartes was with his view on the side of the majority and had been dead for some time when Ole Rømer (1644–1744) made the discovery in the 1670s that led to the determination that the speed of light is finite.

Having set up the basics concerning the nature and propagation of light, Descartes turned his attention to the laws of reflection and refraction, this time with an analogy to the flight of a tennis ball.

Descartes’ mechanistic approach to optics was crucially extended by his derivations of the two central laws of geometrical optics. In his Dioptrics, Descartes proposes to derive the law of reflection by attending to the behaviour of a tennis ball rebounding at an angle off of a hard surface. In reference to Figure 1 below, he postulates that “a ball propelled by a tennis racquet from A to B meets at some point B the surface of the ground CBE, which stops its further passage and causes it to be deflected” (AT VI 93; CSM I, 156).

Figure 1

In order to determine the angle of the ball’s reflection, Descartes suggests that “we can easily imagine that the determination of the ball to move from A towards B is composed of two others, one making it descend from line AF towards line CE and the other making it at the same time go from the left AC towards the right FE” (AT VI 95; CSM I, 157-158). Arguing that the ray’s “encounter with the ground can prevent only one of these two determinations, leaving the other quite unaffected,” Descartes maintains that the horizontal determination of the tennis ball from A to H will remain constant in spite of the ball’s being reflected and thus will be equal to the horizontal determination from H to F (AT VI 95; CSM I, 158). Assuming further that the total speed of the ball is unaffected by reflection, Descartes is able to deduce that the tennis ball must pass through the point F, and that the angle of incidence ABC must be equal to the angle of reflection FBE, in agreement with the accepted law of reflection known since at least the time of the ancient Greeks.^[5]

In the Dioptrics, Descartes next derives the law of refraction in a similar fashion by replacing the hard ground of the previous demonstration with “a linen sheet … which is so thin and finely woven that the ball has enough force to puncture it and pass right through, losing only some of its speed … in doing so” as depicted in Figure 2 (AT VI 97; CSM 1, 158). Descartes further assumes that the total speed of the ball is determined by the resistance of the mediums through which it travels, and that its horizontal speed must remain constant since in passing from the first medium to the second, “it loses none of its former determination to advance to the right” (AT VI 98; CSM I, 158).

Figure 2

Letting HF be twice the length of AH and supposing that the ball travels twice as fast in the incident medium as in the refractive medium, Descartes argues that the ball must reach a point on the circumference of the circle at the same time that it reaches some point on the line FE (since by the first assumption, if the ball travels from A to B in one unit of time, it will travel from B to the circumference in two units of time, and by the second assumption, if the ball travels from A to H in one unit of time, it will travel from H to F in two units of time). Given these assumptions, Descartes concludes that it must be the case that the ball goes “towards I, as this is the only point below the sheet CBE where the circle AFD and the straight line FE intersect,” and therefore that the ratio of the sine of the angle of incidence (BC) to the sine of the angle of refraction (BE) is a constant determined by the ratio of the resistance of the incident medium to the resistance of the refractive medium (AT VI 98; CSM I, 159). To put the same point in more familiar terms, Descartes’ derivation thus purports to establish the law of refraction – published for the very first time in the Dioptrics – that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to a constant determined by the resistances of the mediums involved.^[6]

Descartes presents his readers with a methodological contradiction with his explanations of the laws of reflection and refraction. Having explained that “light is a pressure instantaneously propagated through contiguous chains of rigid balls,” he now uses motion to explain the laws governing light’s behaviour. Descartes answered his critics by stating that he only meant a partial analogy between the two cases: “For it is easy to believe that the action or inclination to move which I have said must be taken for light, must follow in this the same laws as does motion.” So, it seems that belief has replaced: “To accept as true only such conclusions as are clearly and distinctly known to be true.”

Up to the beginning of the seventeenth century, optics meant theory of vision so, having dealt with the basic properties and behaviour of light, Descartes now presented his theory of vision or as Darrigol so aptly puts it, “Descartes went on with a clear and persuasive exposition of Kepler’s theory of vision without caring to name Kepler.” However, he extends the understanding of the brain’s perception of images via the retina and the optic nerve.

L0012003 Descartes: Diagram of ocular refraction. Credit: Wellcome Library, London. Wellcome Images [email protected] http://wellcomeimages.org Diagram of ocular refraction. Woodcut By: Rene DescartesDiscours de la methode… plus la dioptrique… Descartes, Rene Published: 1637 Copyrighted work available under Creative Commons Attribution only licence CC BY 4.0 http://creativecommons.org/licenses/by/4.0/

Descartes initial interest in optics had been triggered by the early telescopes and the problem of spherical aberration. A lens with a spherical surface curvature doesn’t focus the light to a single point but to a messy collection of points spread over a slight distance producing unsharp images. He spent some time in the late 1620s determining that hyperbolic or ellipsoidal lenses would be aplanatic, that is with a single sharp focal point. However, grinding and polishing lenses to these shapes was beyond the technology available at the time. The final sections of La Dioptrique is devoted to this theme with Descartes’ suggestion for a machine to grind hyperbolic lenses.

The third of Descartes appendices Les Météores is Aristotelian in concept in that it deals with the atmosphere and atmospheric phenomena.

It is divided into ten discourses:

De la nature des corps terrestres, Des vapeurs et des exhalaisons, Du sel, Des vents, Des nues, De la neige, de la pluie et de la grêle, Des tempêtes, de la foudre et de tous les autres feux qui s’allument en l’air, De l’arc-en-ciel, De la couleur des nues et des cercles ou couronnes qu’on voit quelquefois autour des astres, De l’apparition de plusieurs soleils.

Descartes started work on his Météores in the late summer of 1629, after reading a description of the parhelia (sundogs or mock suns) observed earlier that year by the Jesuit Scheiner (1573/75–1650). He first gave his own explanation of the same phenomenon, then proceeded to the rainbow and eventually added explanations of wind, rain, and snow, of elementary chemical processes, and of atmospheric phenomena.^[7]

Here it is Descartes analysis of l’arc-en-ciel, the rainbow, carried out with his definition of the laws of reflection and refraction that is of interest.

The history of attempts to understand the rainbow is long and complex^[8]. Aristotle thought it was caused by light reflected from the clouds. Fascinatingly, in the early fourteenth century, both the German Theodoric of Freiberg (c. 1250–c. 1311) in his De iride et radialibus impressionibus (On the Rainbow and the impressions created by irradiance, c. 1304-1311) and the Persian Kamāl ad-Dīn bin ‘Alī bin Ḥasan al-Fārisī (1267–1319) in his Kitab tanqih al-manazir (The Revision of the Optics, 1309)), independently of each other, carried out investigations of the rainbow by observing light through a glass globe filled with water. Both of them working from the Kitab al-Manazir (Book of Optics 1011–1021) of Ḥasan Ibn al-Haytham (c. 965 – c. 1040).

They correctly explained the following:

the colours of the primary and secondary rainbows
the positions of the primary and secondary rainbows
the path of sunlight within a drop: light beams are refracted when entering the atmospheric droplets, then reflected inside the droplets and finally refracted again when leaving them.
the formation of the rainbow: they explained the role of the individual drops in creating the rainbow
the phenomenon of colour reversal in the secondary rainbow

Both works were lost and forgotten. In the late sixteenth century Giambattista della Porta (1535–1615) in his De refractione optices (1589) contended that the rainbow was created by refraction alone. He was not the first to do so but was someone who might well have influenced Descartes. Marco Antonio de Dominis (1560–1624) came close to the correct solution in his Tractatus de radiis visus et lucis in vitris, perspectivis et iride published in 1611. He stated correctly that rainbows are caused by a combination of reflection and refraction but neglects the second refraction and is completely wrong on the formation of the secondary rainbow.

In his Les Météores Descartes became the first person since Theodoric and al-Fārisī to give complete and correct accounts of the formation of both the primary and the secondary rainbow, utilising both the laws of reflection and refraction. Abandoning his fundamental principle that knowledge is won through reasoning he indulged in an empirical experiment, following Theodoric and al-Fārisī in creating an artificial raindrop in the form of a glass sphere filled with rainwater. Descartes announced correctly that the limiting angle of the primary rainbow is 42° and that of the secondary rainbow is 52°. However, he claimed falsely that he was the first to give these correct figures, stating, “This shows what little confidence can be put in the observations which are not accompanied by correct reasoning.”

However, as Boyer write:

Yet the figure of 42° had appeared in a dozen manuscripts and printed works from 1269 to 1611. If Descartes was unaware of any of these anticipations (which appeared in some of the most popular books of the time) one can only conclude that he had a remarkable facility for overlooking in the works of his predecessors anything which might be of value in connection with his own discoveries.

Although Descartes got the theory of the formation of a rainbow correct, his description of the cause of the colours is, to say the least, more that somewhat dubious. Descartes believed that white light was homogeneous, that is monochrome, so, he had to explain the colours of the rainbow or the spectrum in general, as produced by a prism, for example.

Experimenting with a prism Descartes produced the following argument. He stated that the particle of the second element, those that transmitted light, when refracted and rubbing against the particle of the third element, matter, acquired an uneven rotation which manifested itself as colours.

He wrote:

All of this shows that the nature of the colours appearing near consist just in the parts of the subtle matter that transmits the action of light having a much greater tendency to rotate than to travel in a straight line; so that those have a a much stronger tendency to rotate cause the colour red, and those that have only a slightly stronger tendency to rotate cause yellow. The nature of the colours that are seen near H consist just in the fact that these small parts do not rotate as quickly as they would if there were no hindering cause; so that green appears where they rotate just a little more slowly, and blue where they rotate very much slowly.

Note that Descartes rainbow only has four colours.

In La Dioptrique uses the same argument to explain the colour of bodies through the spin communicated to the balls of subtle matter during their impact with surface irregularities.

Of course, Descartes theory of the cause of the colours of the rainbow would be totally torpedoed by the results of Newton’s very extensive prism experiments investigating the spectrum, in which he showed that white light is in fact a heterogeneous mixture of coloured light in which, in fact, every frequency has a different shade of colour. Newton would go on to claim that Descartes contributed nothing to the theory of the rainbow that wasn’t already to be found in the work of De Dominis, indirectly implying that Descartes has plagiarised De Dominis. This was unfair as Descartes’ account was more extensive and substantially corrector than that of De Dominis.

As I hope has become clear Descartes take on optics very much fulfils the Knowles Middleton quote with which I opened this post. He wrote a great deal about the subject, which would remain for some time very influential but a large amount of what he wrote was simply wrong and the theory of vision which he got right was Kepler’s.

^[1] I have dealt with their initial contact and Beeckman’s influence on the young Descartes in 1618 here

^[2] Taken from Boyer see footnote 8

^[3] Olivier Darrigol, A History of Optics: From Greek Antiquity to the Nineteenth Century, OUP, 2012, p. 39

^[4] Darrigol, pp. 40-41.

^[5] Jeffrey K. McDonough, Descartes’ Optics, The Cambridge Descartes Lexicon, ed. Larry Nolan, (CUP, 2015) pp. 2-3

^[6] McDonough pp. 3-4

^[7] Theo Verbeek, Meteors, The Cambridge Descartes Lexicon, ed. Larry Nolan, (CUP, 2015), Summary.

^[8] For an excellent account of the hunt to find the true nature of the rainbow see Carl B. Boyer, The Rainbow: From Myth to Mathematics, Princeton Paperbacks, 1987

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Filed under History of Optics, History of Physics, History of science, Uncategorized

August 20, 2025 · 7:52 am

The Welsh instrument maker who didn’t fit the contemporary mould.

In the last post in this series, we took an in depth look at the career of the English mariner Sir Robert Dudley (1574–1649). This ended with an account of the scientific instruments that he left behind and are now in the Museo Galileo in Florence. These came from several different instrument makers and today I’m going to look at one of the ones I have yet covered in this series, James Kynvyn (c. 1550–1621)^[1].

James Kynvyn just doesn’t really fit the profile of the other English instrument makers in the second half of the sixteenth century. As is often the case, we don’t have any details of Kynvyn’s early life. What we do know is largely patched together from his will in the name of Jacobus Kynvin dated September 1615. Here he is described as citizen and Merchant Taylor of London. This is unusual, Thomas Gemini (c.1510–1562) was an engraver, when he first came to England, and only later turned to instrument making. Humfrey Cole (c. 1530–1591), a goldsmith, was employed at the Royal Mint as an engraver, who became an instrument maker on the side. Both Gemini and Cole also engraved maps. Augustine Ryther (fl. 1576–1593) was more prominent as an engraver than as an instrument maker and is best known for his engraved maps rather than his instruments. We have a total of twelve surviving instruments made by Kynvyn but no maps or other engraving commissions. Also, the relatively poor quality of the calligraphy on his earliest instruments indicates that he was initially not a particularly skilled engraver.

The Court Minutes of the Merchant Taylor Company, dated 26, October 1573 read:

First the Mr [Master] and Wardens aforesaide at the contemplacon of my Lord Mayor his Honour and the request of Sir Thomas Offley Knight have made free James Kynvyn per redempcon gratis.

Offley was a former Lord Mayor meaning Kynvyn was being honoured by the Company on the recommendation of senior City officials. This implies that either he or his family enjoyed high status. This impression is reinforced by the fact the Will of John William Kynvyn, dated 1617, presumably James’ brother, refers to him as ‘gentleman’. James is mentioned in the Will together with another brother, Edward Job. James Kynvyn’s marriage to Elizabeth Coke at the church of St Michael Bassishaw is recorded in the church register for January 1574. His home parish was Llantilio in Monmouthshire and at the time of his death his place of residence was ‘my dwelling house in Fleet Street London. All of this indicates that Kynvyn was different to the other contemporary instrument makers. They were skilled artisans having learnt their trade at an early age. Kynvyn obviously came from a higher strata in the society of the period and there is no indication how or why a tailor became a scientific instrument maker.

There are only twelve surviving instruments made by Kynvyn all made within the ten years following the earliest dated from 1593. This would have made him about forty when he first turned to instrument making and he lived for another twenty years after he stopped. We have no idea why he started when he did or why he then stopped after such a relatively brief period.

The only other source of information on Kynvyn is a note written by the Elizabethan critic and poet, Gabriel Harvey (1545–1631), in his copy of The Mathematical Jewel (1585) by John Blagrave (d. 1611):

His [Blagrave’s] familiar staff, newly published this 1590. The Instrument itself made and sold by M. Kynuin of London neere Powles. A fine workman and mie kind frend: first commended unto me bie M. Digges, & M. Blagraue himself. … Mr Kynvyn selleth ye Instrument in brasse.

In the same note Harvey also refers to ‘old Humfrie Cole.”

‘neere Powels is in the vicinity of St Paul’s Cathedral and M. Digges is, of course, Thomas Digges (c. 1546–1595).

The earliest dated instrument, made by Kynvyn that has survived is a compendium that was made for Robert Devereux, 2^nd Early of Essex (1565–1601), a favourite of Queen Elizabeth and quasi step-brother to Robert Dudley.

*Robert Devereux, 2nd Earl of Essex*, by Marcus Gheeraerts the Younger. Source: Wikimedia Commons

Dudley’s father, Robert Dudley, 1^st Earl of Leicester, who never married his mother, was the second husband of Robert Devereux’s mother Lettice Knollys (1543–1634).

Portrait of Lettice Knollys (1543-1634) attributed to George Gower Source: Wikimedia Commons

The compendium has Devereux’s coat of arms engraved inside the lid.

Quartered arms of Sir Robert Devereux, 2nd Earl of Essex, KG Source: Wikimedia Commons

A compendium is, so to speak, a Swiss Army Penknife amongst the scientific instruments.

‘At the most general level, a compendium is simply an instrument made up of several disparate elements. An astronomical compendium is a mathematical instrument which has a range of astronomical functions incorporated into a single small device.

As a catch-all term there is no standard list of functional elements, but sundials, compasses, calendars, maps, astrolabes, quadrants and various tables are often to be found. Neither is there a standard form for the resulting instrument; astronomical compendia are found in many shapes and formats, whether rectangular, circular, oval or octagonal, and are sometimes constructed to mimic other artefacts, such as books, with covers and leaves. (Stephen Johnston, Epact)

This brass astronomical compendium consists of: a nocturnal; a latitude table for 39 world locations; a magnetic compass; a list of ports and harbours; a perpetual calendar and table of fixed feast days; a high tide computer for several European ports; lunar phase and age indicator volvelles and a planetary aspectarium^[2]. The remains of an equinoctial sundial can also be seen. (British Museum)^[3]

Source: British Museum

Source: British Museum

Source: British Museum

As we saw in the post about Robert Dudley, he owned four of the known Kynvyn instruments, according to the museum in Florence. However, Gerard L’E Turner says eight of them were made by Kynvyn. As well as the four attributed to Kynvyn in my blog post, Turner says that the Astrolabe late 16th cent. Attributed to Charles Whitwell in from Kynvyn as well as the two unattributed nautical compasses and the unattributed Theodolite, late 16th cent., which Turner calls an Altitude semicircle but says might be part of an altazimuth theodolite.

There is an earlier undated compendium in the Historisches Museum in Basel, which Turner attributes to Kynvyn. It has:

Nocturnal
Latitudes of 31 world locations
Equinoctial sundial
Magnetic compass
Establishment of the port: the technical expression for the time that elapses between the moon’s transit across the local meridian at new or full moon at a given place and the time of the next high water at that place.

Source: Gerard L’E Turner, Elizabethan Instrument Makers

The Science Museum Oxford has a sector of the pattern first published by Thomas Hood (c. 1560–1621)

in his 1598 book on The Making and Use of the Geometrical Instrument called a sector.

Finally, there is a surveyor’s Peractor, which Turner attributes tentatively to Kynvyn.

The fact that the majority of Kynvyn’s instruments were seemingly made for Robert Dudley and his earliest dated instrument was made for Robert Devereux, both of whom moved in the highest court circles in the 1590’s would seem to indicate that he was known in the highest circles.

In total, what we know of Kynvyn’s life and instrument production throws up more questions than it answers and he really doesn’t fit in the contemporary mould of the English scientific instrument maker of the second half of the sixteenth century.

^[1] This account of the life and work of James Kynvyn is largely taken from Gerard L’E Turner, Elizabethan Instrument Makers: The Origins of the London Trade in Precision Instrument Making, OUP, 2000, pp. 25-27

^[2] aspectarium: aide-memoire which depicts the astrologically significant angles (or aspects) between the planets.

^[3] For a detailed description of all the instruments go to the British Museum website

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Filed under History of Astronomy, History of Navigation, Renaissance Science, Scientific Instrument Makers

August 13, 2025 · 7:40 am

From τὰ φυσικά (ta physika) to physics – XLIX

We have already looked at the philosophical motivation behind the mathematisation of science in the early modern period as well as the impetus supplied by the mathematical practitioners but there is a third aspect that we also need to address. During the seventeenth century the people developing the natural philosophy were at the same time acquiring and developing a new tool box of mathematical disciplines to replace the almost monopolistic status of Euclidian geometry in the previous centuries.

As noted in several earlier posts on this blog, mathematics did not play a major role in the education available at the medieval universities. Only lip service was paid to the Quadrivium–arithmetic, geometry, music, astronomy–which, if taught at all, was only taught at a very low level. Arithmetic and music, which had very little to do with mathematics, were taught from the very elementary texts of Boethius (c. 480–524), De institutione arithmetica libri duo and De institutione musica libri quinque. Astronomy was taught from De sphaera mundi of Johannes de Sacrobosco (c. 1195–c. 1256), a non-mathematical description of the geocentric astronomy of Ptolemaeus (fl. 150 CE). The only ‘real’ mathematics was The Elements of Euclid (fl. 300 BCE), of which, in theory, only the first six of the thirteen books was taught but in practice, courses often got no further than Book I.

This began to gradually change in the sixteenth century and by the end of the seventeenth the basics of what is still the general school curriculum in mathematics today–algebra, analytical geometry, trigonometry, calculus– was on offer for budding natural philosophers. This didn’t happen overnight but was, as already noted, a gradual evolution in which many played a part.

Algebra, originally thought of as the theory of equations, has roots in antiquity in Mesopotamia, Egypt, India, and China. Although Chinese algebra didn’t play a role in the developments that led to later European algebra. It is often though that ancient Greek didn’t do algebra but in fact they did it geometrically. The meant x is a line segment, x² become a square or quadrate, x³ is a cube, hence quadratic and cubic equations. However, Diophantus of Alexandria (fl. 250 CE) in his Arithmetica produced a quasi-symbolic algebra.

The most advance algebra out of these sources developed in India in the Early Medieval Period, and this was taken over by the early Islamic culture and led to the al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah(The Concise Book of Calculation by Restoration and Balancing) of Muḥammad ibn Musá al-Khwārizmī (c. 780–c. 850), which gave us the name algebra from the Arabic al-Jabr. Al-Khwārizmī’s work was initially translated into Latin by Robert of Chester in 1145.

A page from the *al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah* of Muḥammad ibn Musá al-Khwārizmī Source: Wikimedia Commons

It, however, had more impact through the Liber Abaci (1202) of Leonardo Pisano (c. 1170–after 1240). Following Leonardo’s introduction algebra became what we would call commercial arithmetic in the world of commerce, practiced and taught by reckoning masters rather than a branch of academic mathematics. As was the principle use of algebra in Islamicate culture.

A page of Leonado Pisano’s *Liber Abaci* from the Biblioteca Nazionale di Firenze via Wikimedia Commons

The transition began in the sixteenth century with the Cossist in Germany whose Coss books were algebra rather than commercial arithmetic. Notable here are the Behend und hübsch Rechnung durch die kunstreichen regeln Algebre, so gemeinicklich die Coß genennt werden (Deft and nifty reckoning with the artful rules of Algebra, commonly called the Coss) of Christoff Rudolff published in Straßburg in 1525 and the Arithmetica Integra of Michael Stiffel published in Nürnberg in 1544

Christoff Rudolff *Behend und hübsch Rechnung* Source: Wikimedia Commons

A major development came with the discovery of the general solution of the cubic equation and the dispute that it generated leading to the publication in Nürnberg by Johannes Petreius (c. 1497–1550) of Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra) of Gerolamo Cardano (1501–1576), which contained the general solutions of the cubic and quartic equations. The book has been hyperbolically called ‘the first modern mathematics book’, whether this is true or not is debateable, but it definitely establishes algebra as mathematics and not commercial arithmetic. It was also the book that first introduced imaginary numbers although Cardano didn’t like them.

Cardano’s *Ars Magna* title page Source: Wikimedia Commons

Rafael Bombelli (1526–1572) took up the baton with the publication of his L’Algebra in 1572, a comprehensive algebra textbook, which was the first book in Europe to present the complete rule for operating with negative number and then to do the same with imaginary numbers distinguishing them clearly from real numbers, although he doesn’t use either term, as they first came later. Descarte first used the term ‘imaginary numbers’, as an insult.

Title page of Bombelli’s L’Algebra Source: Wikimedia Commons

The final step was the publication of In artem analyticem isagoge (Introduction to the art of analysis) by Françoise Viète (1540–1603) in 1591. This was the first algebra book that gave the discipline a solid foundation and was largely symbolic rather than rhetorical i.e. all operations expressed in words or syncopated with some abbreviations and symbols. The Jesuit mathematician, Christoph Clavius (1538–1612), who was responsible for introducing mathematics as a primary subject into the Catholic schools and universities wrote a textbook for teaching Viète’s analysis in his pedagogical program.

Thomas Harriot (c. 1560–1621), who famously published almost nothing, wrote an excellent algebra book, Artis Analyticae Praxis, which was published posthumously in 1631. Unfortunately, his editors didn’t really understand the subject and removed all of Harriot’s important innovations. The first Latin translation of Diophantus’ Arithmetica had been published in 1621. Algebra or analysis was now firmly established as an important branch of mathematics.

Title page of the Latin translation of Diophantus’ *Arithmetica* by Bachet (1621). Source: Wikipedia Commons

The next development was the combining of the new analysis, as Viète preferred to call it, with the classical geometry to produce analytical geometry, that is the representation of algebraic equations as geometrical figures on a graph or vice versa geometrical figures as algebraic equations. This development was famously first published by René Descartes (1596–1650) in his La Géométrie as an appendix to his Discours de la méthode (Discourse on the Method) in 1637.

One year earlier, Pierre de Fermat (1601–1665) circulated a manuscript containing the same development, which was never published in his lifetime, but only posthumously in 1679.

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) had done something resembling analytical geometry as had Apollonius of Perga (c. 240–c. 190 BCE) but as neither system was taken up by others so, the analytical geometry of Descartes and Fermat was seen as something new and even revolutionary. One should point out that it actually had its major impact through the expanded Latin translation of La Géométrie published by Frans van Schooten Jr. (1615–1660) in 1649 and further expanded in two volumes in 1659 and 1661. The later two volume edition was the one from which both Leibniz and Newton learnt their analytical geometry. It was also Van Schooten who introduced the signature rectangular or Cartesian coordinate system, which is not present either in Descartes original publication or Fermat’s.

René Descartes, *Geometria à Renato Des Cartes anno 1637 Gallicè edita; postea autem unà cum notis Florimondi de Beaune* (Amsterdam, 1659), Volume I frontispiece

René Descartes, *Geometria à Renato Des Cartes anno 1637 Gallicè edita; postea autem unà cum notis Florimondi de Beaune* (Amsterdam, 1659), Volume I title page. Source

Trigonometry went through a similar historical evolution. It was originally introduced, by Hipparchus (c. 190–c. 120 BCE) in his astronomical work in the form of chords of a circle to define angles. This was developed into spherical trigonometry by Theodosius of Bithynia and Menelaus of Alexandria. Hipparchus’ work was lost but Ptolemaeus (fl. 150 CE) took it up in his Mathēmatikḕ Sýntaxis acknowledging Hipparchus’ priority. Indian astronomers changed the standard to half chords, creating our sine and cosine, this was taken over by the Arabic astronomers. The Arabic mathematicians developed plane trigonometry out of the spherical trigonometry developing the six trigonometrical functions that became standard in European mathematics.

During the High Middle Ages trigonometry gradually began to become established in Europe. This reached a highpoint with the so-called First Viennese School of Mathematics in the work of Georg von Peuerbach (1423–1461) and Johannes Regiomontanus (1436–1476) in the middle of the fifteenth century. Regiomontanus wrote a comprehensive work on trigonometry in 1464, which, however, was first published posthumously by Johannes Schöner (1477–1547) as De Triangulis omnimodis (On Triangles) in Nürnberg in 1533. This was the first account of nearly the whole of trigonometry published in Europe, only the tangent was missing, which Regiomontanus had already presented separately in his Tabulae directionum profectionumque written in 1467 but again first published in print posthumously in 1490. Throughout the sixteenth century, improved trigonometrical tables were calculated and published and by the beginning of the seventeenth century plane trigonometry had become firmly established as a separate discipline.

A new development that was initially combined with trigonometrical functions was the invention of logarithms by John Napier, first published in his Mirifici logarithmorum canonis descriptio… in 1614 are actually logarithms of trigonometric functions. However, Henry Briggs his first work on base ten logarithms Logarithmorum Chilias prima in 1617 and a much more extensive work Arithmetica Logarithmica in 1624.

Cover of *Mirifici logarithmorum canonis descriptio* (1614) Source: Wikimedia Commons

Algebra and trigonometry were old branches of mathematics that were, so to speak, redefined in the Early Modern Period, analytical geometry and logarithms were essentially new developments but the biggest mathematical development for physics in the seventeenth century was calculus. However, it was not invented by Newton and Leibnitz, as is so often claimed, but also had a long history before the seventeenth century and a strong development within that century before Newton and Leibnitz became involved.

The method of exhaustion developed by Eudoxus and expanded upon by Archimedes in antiquity, to determine areas of geometrical figures and centres of gravity, is a form of integration. This was revived in the sixteenth century with the Renaissance in the mathematics of Archimedes. It was used by Kepler in his Nova stereometria doliorum vinarioru (1615) to determine the volume of wine barrels. The method of exhaustion was taken up by Cavalieri and his student Stefano degli Angeli (1623–1697) in his method of indivisibles and further developed and popularised by Evangalista Torricelli (1608–1647). Cavalieri’s method of invisibles was taken up through the influence of Torricelli and degli Angeli by the French mathematicians Jean Beaugrand (1584–1640) and Ismaël Boulliau (1605–1694), the English mathematicians Richard White (1590–1682), John Wallis (1616–1703) and Isaac Barrow (1630–1677), James Gregory (1638–1675) in Scotland, Gottfried Leibniz (1646–1716) in Germany, and Frans van Schooten Jr. in the Netherlands.

Many of these mathematicians also worked on the problem of finding tangents to curves in order to determine rates of change, which became differentiation , most notably Pierre de Fermat, whose work on the topic Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum (1679) was the one in which he introduced his version of analytical geometry. John Wallis combined the indivisibles of Torricelli and the analytical geometry of Frans van Schooten to create his De Sectionibus Conicis published in 1655 and his Arithmetica Infinitorum published in 1656, major steps towards the generalisation of the methodology of calculus.

In his Vera Circuli et Hyperbolae Quadratura published in Padua in 1667, James Gregory finds both areas and tangents and it is obvious that he knows the fundamental theorem of the calculus, i.e. that integration and differentiation are inverse operations.

In his Lectiones Geometricae, published in 1670, Isaac Barrow also uses the fundamental theorem of the calculus.

The work of Wallis and Barrow is known to have influenced both Newton and Leibniz. The collation and formalisation of all these approaches and results into a single disciple by Newton and Leibniz led to an increased application of the methods in various areas of physics

Natural philosophers basically entered the seventeenth century with only Euclidian geometry as a mathematical tool for their work. By the end of the century, they had an extensive mathematical toolbox containing, algebra, plane trigonometry, logarithms, analytical geometry, and the calculus with which to mathematically derive and present their theorem. Ironically, Newton’s Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy) published in 1687, the most important text on physics published in the seventeenth century and some would argue the most important ever, was created entirely using only Euclidian geometry and that despite the fact that Newton had made major contributions to algebra, analytical geometry and above all, the calculus.

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August 9, 2025 · 8:50 am

Conflict what Conflict?

Originally posted as a comment on a Facebook post, a wonderful truth expressed in a few words by the man who was responsible for me starting this blog, historian of biology John Wilkins.

As I have been known to say, there is no conflict between science and religion, largely because there are no such things. There are sciences, and religions, but in the end they have no attitudes, because only individuals within those traditions have attitudes, and these vary like crazy.

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August 6, 2025 · 7:33 am

A book you can count on

Back in 2020, I wrote a very positive review of Benjamin Wardhaugh’s fascination volume, The Book of Wonder: The Many Lives of Euclid’s Elements. This led me to also writing a positive review of Reading Mathematics in Early Modern Europe of which Wardhaugh was both a contributor and an editor. This was followed by a brief blog post on the research project, Reading Euclid, from which both books emerged.

I was recently very pleased to receive an email from Benjamin Wardhaugh asking if I would be interested in receiving a copy of his newest book, Counting: Humans, History and the Infinite Lives of Numbers (William Collins, 2024). Given my more than passing interest in the history of numbers and the excellence of Wardhaugh’s Euclid book, I of course said yes. I have not been disappointed.

I mentioned above my interest in the history of numbers but as Wardhaugh points out that whilst numbers have been and are used in counting they are not necessarily required for the task and there is and has been plenty of counting without numbers. At one point I had to laugh at myself, skimming around the book and getting the general feel of it, I thought this is much more anthropology rather than history. That was when I first bothered to read the quote from The Times on the front cover:

An anthropological sweep through mathematical history from the Stone Age to the cyber age via six continents.

Histories of science or science related topics tend to follow a linear chronological narrative and within limits aim to be comprehensive. Do not expect this with Wardhaugh stimulating, fascinating and at time provocative dive into the history of counting. Why not? Wardhaugh provides the answer in the introduction which itself is not a continuous narrative but a series of fragmentary quotes that illustrate aspects of what counting is or even might be. In answer to the question, “But what is it?” Wardhaugh tells the reader:

‘Counting’ can seem like an unruly grab-bag of almost totally unrelated actions; a label covering a huge set of very different cultural practices. The range of different activities called counting seems too wide for comfort, and at least superficially, it is not clear what they all have in common; or even whether they have anything in common at all.

Almost any definition of counting is problematic, but one of the best is attributed to the seventeenth century German philosopher Gottfried Leibniz. It says counting is repeated attention. Counting is what happens when you think ‘this…this…this…this’, and have some way of keeping track.

Wardhaugh goes on to tell us:

Counting does not have one single history.

[…]

The story of counting is shaped, instead, like a tree. It has several roots, many branches, and innumerable twigs and leaves. Counting has grown and travelled with the human species, ramifying into very nearly every culture past and present. Sometimes it is possible to follow a single branch for some distance: sometimes a branch turns out to cross, to touch (or nearly touch) other branches.

In this book, Wardhaugh takes his readers on a climb first, down into the root system, then up and along some of the branches, across time and across the continents; pausing to examine a twig, pluck a leave, or sample some ripe fruit. The book is not written as a continuous narrative but is presented in eight sections, two of them explorations of the roots, the other six climbing amongst the diverse branches.

Each of the eight sections consists of a group of self-contained essays, which deal with an example or an aspect of the topic announced in the title of the section. In those section which deal with examples of counting, which is most but not all of them, Wardhaugh is very careful to give details of who is doing the counting, what they are counting, how they are counting and not least why they are counting. Through this process he makes it very clear that counting is as he wrote in his introduction, ‘an unruly grab-bag of almost totally unrelated actions’.

The first Roots section is not about counting per say but about Number sense before counting and consists of three essays on how people estimate the size of groups of objects without counting and the question whether this is unique to humans or whether animals possess the same innate ability.

The second Roots section Counting before writing brings three essays on artifacts found in prehistoric Africa that might have been involved in the process of counting. Wardhaugh brings good arguments for such usage but also warns that in the end the claims remain speculative. A fourth essay deals speculatively but well argued with the possible origins of counting words.

Between the Roots sections and the Branches there is an interlude on numbers and their nature.

We now move into the realm of counting in a section titled, Counting with words and symbols in the Fertile Cresent. The first essay here deals with the Sumerians, who developed the oldest number system of which we are aware.

We remain in Mesopotamia for the second essay and an example of a clever device that Wardhaugh uses often in his book. Instead of just dealing with abstract cultures as a whole, he uses the example of an individual within that culture who in some way or another demonstrates counting within that culture. Here we are in Arkadia and the individual is the monarch Tiglath-Pileser I, who is counting his plunder or spoils of war. From Arkadia we move to Egypt and Teianti daughter of Djeho, who is counting coins buying a house. In both these examples Wardhaugh moves on from the specific to the general, looking at number systems in the cultures and what was counted and why.

The second Branch is Counters and opens in Ancient Greece in Athens with Philokleon declaring his judgements in the court, as a member of the jury, by casting counters into an urn. The essay expands into Greek number systems and the extensive use of counters in various contexts. The second essay takes us to Rome and Marcus Aurelius: Counting Years opening up a discussion of the Roman system of indicating numbers with hands and fingers and its survival down the centuries. The final essay in this section introduces the counting board, widespread in both these early cultures but for his example Wardhaugh takes us to the thirteenth century and Blanche of Castile overseeing the accounts. This leads into a general discussion of counting boards and jetons.

On the next Branch we climb into the history of the number symbols from India, an inevitable topic in such a book. For his entry, Wardhaugh choses, from the numerous possibilities, Bhaskara II and the Brahmi numerals, widening out to give a general sketch of the topic. Logic dictates that the next essay covers the use of India numerals in Islamicate culture and Wardhaugh here choses Ibn Mun’im to introduce us to Dust numerals. One of the longer essays, Wardhaugh covers a lot of ground sketching Arabic numerate culture. The third essay covers the early, very gradual transfer of Indian numerals to Europe. The European development continues with an in depth discussion of the seventeenth-century painting The Account Keeperby Nicolaes Maes. This branch closes in the nineteenth century with the weather records of Caroline Molesworth.

Wardhaugh takes a break for a second short but fascination interlude on Number symbols.

We have followed Wardhaugh from the fertile crescent, to the Mediterranean, and from there we followed the Indian numerals from India over the Islamic Empire to Europe. Now on the next Branch, we go to East Asia to look at Machine that count.

The first essay concerns the tax assessment of Hong Gongshou in China and introduced the Chinese use of counting rods, as a calculating aid. The next essay introduces the Chinese suanpan, Japanese soroban, Korean jupan, the wire and bead calculating device known as an abacus in English. This leads to the famous post WWII story of how the Japanese master of the soroban, Kiyoshi Matsuzaki, defeated the GI Thomas Wood using an electric calculating machine in a public calculation contest. This essay ends with the fascinating fact that suanpan and soroban operators can do mental arithmetic without their devices just by going through the moves in their imagination.

The next essay spans Japan and America with an account of the nineteenth-century electric tabulating machines of Kawaguchi Ichitaro and Herman Hollerith used in both lands to tabulate census data. Hollerith would go on to found IBM which became the world’s biggest computer company.

In the following essay we come right up to date, travel to Korea and meet Sia Yoon, who is counting likes. Here Wardhaugh makes his readers aware how much of the Internet world of social media is fixated on counting–like, followers, comments, reposts, …

A new Branch takes as out into the world’s biggest ocean and again back in time with Counting words and more in the Pacific World. In the first essay we count eggs in Ayankidarrba (Groote Eylandt), which lies about 50 kilometres off the Australian coast, leading to a more general discussion of counting in the various indigenous Australian languages. In the following essay we encounter the world of counting of the Oksapmin, one of the isolated peoples of Papua New Guinea. We then count leaves on the island of Tonga before moving into a general discussion of the complexities of the Tongan counting systems.

For the final Branch we leave Oceania to travel to the last continent, to look at Counting in the Americas. The first essay takes us into western Alaska and the tally sticks from the deserted village of Agaligmiut of the ancestors of the Yup’ik people. These found multiple uses. The next essay takes us to the Pacific coast in northern California, and the clam shell beads of the Pomo people. These developed into a trading currency amongst various peoples over a fairly widespread area.

The final essay in this final Branch, takes the reader into the Mayan civilisation of Mesoamerica, where we meet the ruler of the city of Oxwitik, Waxalahun-Ubah-K’awil. Here Wardhaugh introduces his readers to what is probably the most complex system of counting ever developed by humans, the Mayan calendar systems. The Maya counted the days in a complex variety of ways often given dates simultaneously in more than one system. Wardhaugh skilfully guides his readers through the complexities.

There are neither footnotes nor endnote but for each essay there is a brief but highly informative sketch of the sources on which it is based. These sketches refer to the ‘select’ forty-six page bibliography. There is also an extensive and very good index. Many of the essays are illustrated with greyscale images of the artifacts discussed in them.

In his conclusion Wardhaugh summarises his book much better than I could:

The story of counting is a dense tree, with several roots and nearly an infinity of branches. Its end points – as the tree stands today – include elaborate routines with unmarked counters on marked surfaces, highly developed sets of number words, successful – and successfully exported – sets of number symbols, and electric machines whose internal states are, for some purposes, taken to represent numbers. They include smaller sets of words, sometimes accompanied by gestures, tally sticks, and written representations of number words. They include a world in which there is no counting whatever.

There is always a strange alchemy to counting which restlessly transforms one thing into another: days into tally marks; people into counters; books into magnetic tape.

This book has described a few of those processes. There have been thousands more, in all the thousands of languages living and dead, in all the thousands of cultures and hundreds of scripts. No two are alike. The real alchemy, perhaps, is in turning all f these processes into a single thing, and calling it counting.

Wardhaugh is an excellent expedition leader and it is a delight to follow him as he weaves his way through the tree of the history of counting but who should read this book? At first glance it appears to be a very specialised, niche endeavour, something for a select group of historian of mathematics perhaps. But no, stop and think for a moment. Counting is an activity that is firmly embedded is our every day lives. One just needs to stop and think about the long list of everyday expression and clichés that revolve around counting:

One can or one cannot count on, counting down to, counting the days, count the ways, counting calories, count your blessings, count the pros and cons, take a deep beath and count to ten, that doesn’t count, count against, count out, count towards, count up, counting the votes…

Counting is an integral part of our social, cultural, and political existence and its history, presented in Wardhaugh’s excellent book, should appeal to anyone interested in human history.

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Filed under Book Reviews, History of Mathematics

July 30, 2025 · 6:37 am

From τὰ φυσικά (ta physika) to physics – L

Optics is the branch of physics that studies the behaviour, manipulation, and detection of electromagnetic radiation, which for most of its history meant simply light. The study of light begins with three basic phenomena, the propagation of light, reflection, that is the behaviour of light when it meets a non-absorbent surface, and refraction, that is the bending of light when it passes from one medium to another. The scientific and mathematical examination of these three phenomena began in antiquity with the research into geometric optics of the Greeks: Euclid (fl. 300 BCE), Hero of Alexandria (fl. 60 CE), and Ptolemaeus (fl. 150 CE). For the Ancient Greeks, optics consisted of theories of vision and all three of them held an extramission theory of vision, which meant their work was based on visual rays emitted by the eyes and not on light rays. However, Ibn al-Haytham (c. 965–c. 1040) demonstrated that the work they had developed using a theory of visual rays leaving the eyes was equally valid using a theory of light rays entering the eyes.

The earliest work of geometric optics was the Optics of Euclid, which deals with the basic propagation of visual rays. As with his, much more famous, Elements. Euclid first defines a set of postulates, in this case seven:

That rectilinear rays proceeding from the eye diverge indefinitely;
That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;
That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;
That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;
That things seen under more angles are seen more clearly^[1]

Euclid’s geometry describes the visual rays emitted from the eye spreading out in a visual cone. From his seven postulates, Euclid derives fifty-eight propositions. When later applied to light rays by Ibn al-Haytham Euclid’s introduction to the geometry of light rays and their perception remains totally valid.

There is also a Catoptrics, the study of reflection, attributed to Euclid, although some historians dispute the attribution. In his Catoptrics, which begins with six postulates, Euclid analyses the reflection from mirrors whose surfaces take three specific forms: plane, convex spherical, and concave spherical. This time, from the six postulate, Euclid derives the geometric rules for the behaviour of incident and reflected rays, and image formation. Of particular interest is his sixth postulate that deals not with reflection but with refraction and describes the “floating coin” experiment. A coin is placed in the bottom of an opaque vessel and the head is moved back till the coin just disappears from the line of sight. The vessel is then filled with water and the coin again floats mysteriously into view due to refraction.

The editio princeps of Euclid’s Optica and Catoptrica, in Greek with their Latin translations Source

Hero also wrote a Catoptrics, which covers virtually the same material as Euclid. However, whereas Euclid only included a few examples of visual illusions created with mirrors, Hero has a substantially larger number of such cases.

The most important Ancient Greek book on optics was the Optics of Ptolemaeus. It is thought to have been written late in his life after his Mathēmatikē Syntaxis around 160 CE. It only exists in an incomplete Latin translation of a lost Arabic manuscript. However, even in its incomplete form it goes far beyond anything else produced in antiquity and is regarded as one of the most important texts on the topic produced before the seventeenth century. Book I is missing and of the surviving four books, Book II deals with visual perception and, although more extensive and detailed, it is basically that of Euclid. The major difference in that Ptolemaeus doesn’t consider the visual cone to consist of individual visual rays but sees it as continuous. Books III & IV deal extensively with reflection and unlike Euclid and Hero, Ptolemaeus founds his statement on experiments, the details of which he describes.

Finally in Book V, Ptolemaeus presents the earliest known scientific account of refraction, Euclid’s “floating coin” being the only predecessor. There had, however, been discussions of the optical illusions created in astronomy by atmospheric refraction. It is interesting to note that Ptolemaeus did not discuss atmospheric refraction at all in his Mathēmatikē Syntaxis and only introduces it at the end of the surviving Book V, which breaks off before it ends properly.

As with reflection, Ptolemaeus sets up a series of experiments to empirically measure refraction. He makes three series of experiments for refraction from air into water, air into glass, and water into glass. From his acquired data he draws to general conclusions. The first is that “the amount of refraction is the same whichever the direction of passage,” by which he means that when a visual ray passes from a given refractive medium into another, it will be broken by the same amount whichever direction it takes.^[2] The second generalisation, amounts to the claim that if two rays refract into a denser medium, the difference between the angles of incidence will be proportionately greater than the difference between the angles of refraction.^[3]Ptolemaeus had not succeeded in finding a strict mathematical relation in refraction between the angle of incidence and the angle of refraction.

Ptolemaeus’ experiments to measure refraction Source: Museo Galileo Florence

The first person to find the correct mathematical relation between the angle of incidence and the angle of refraction was the Persian mathematician and physicist Ibn Sahl (c. 940–1000) but only sort of. In an optical treatise written around 984, of which only two incomplete and/or damaged manuscripts exist, on burning lenses and mirrors, Ibn Sahl in a section on a plan-convex hyperbolic lens produces a geometrical ray ratio that measures the index of refraction, and by a simple geometrical conversion it yield the sine law of refraction. This was a remarkable achievement but Ibn Sahl work remained unknown both in Islamic culture and in Europe and was only rediscovered in the twentieth century.

Reproduction of Millī MS 867 fol. 7r, showing his discovery of the law of refraction (from Rashed, 1990). The lower part of the figure shows a representation of a plano-convex lens (at the right) and its principal axis (the intersecting horizontal line). The curvature of the convex part of the lens brings all rays parallel to the horizontal axis (and approaching the lens from the right) to a focal point on the axis at the left. Source: Wikimedia Commons.

An obvious candidate to discover the sine law of refraction was the mathematician, astronomer, and physicist Iban al–Haytham, whose Kitab al-Manazir (Book of Optics), written in the 1020s, is the most important text on optics between the Optics of Ptolemaeus in about 160 CE and the Pars Optica of Johannes Kepler (1571–1630) in 1604. As already noted it was al-Haytham, who demonstrated that all the geometrical optics produced by Euclid, Hero, and Ptolemaeus based on visual rays using an extramission theory of vision were equally valid for light rays in an intromission theory of vision. Al-Haytham knew and accepted the work of Ptolemaeus on refraction and although refraction played a central role in his theory of vision and he wrote extensively on atmospheric refraction, he did not discover the sine law of refraction.

Editio princeps *Kitab al-Manazir* (*Book of Optics*) in Latin Friedrich Risner, publ. 1572. *Opticae Thesaurus: Alhazeni Arabis Libri Septem Nunc Primum Editi*, *Eiusdem Liber De Crepusculis Et Nubium Asensionibus* . *Item Vitellonis Thuringopoloni Libri X.* Source: Wikimedia Commons

From the beginning of mathematical optics, with Euclid around 300 BCE down to Kepler in the early sixteenth century, who also didn’t discover it, nobody discovered the sine law of refraction, which is today a standard part of the school physics curriculum. Then in the sixteenth century, it was discovered independently by four different mathematicians, within sixty years.

The first sixteenth-century mathematician to discover the sine law of refraction was Thomas Harriot (c. 1560–1621) in 1602, but as with almost everything he did he never published so, nobody was aware of his discovery.

A portrait believed to be of Thomas Harriot apparently painted during his lifetime. Source: Wikimedia Commons

Truly bizarre was the fact that he began a long, intensive correspondence with Johannes Kepler in 1606 in which one of the main topics they discussed was atmospheric refraction, but Harriot never let on that he had already discovered the sine law. In his Pars Optica in 1604 and then later in 1611 in his Dioptrice, Kepler published the first correct accounts of the behaviour of lenses something totally dependent on the law of refraction. To do so he used the assumption that the angles of refraction are constantly proportional to the angles of incidence. This is, of course wrong but gives a good approximation for small angles.

Chronologically, the next to discover the sine law of refraction was the Netherlander astronomer and mathematician Willebrord Snel van Royen (1580–1626). Snel discovered the law in 1621 but, like Harriot, he didn’t publish the discovery in his lifetime and so his discovery initially remained unknown.

René Descartes (1596–1650) became the first to publish his discovery of the law in his essay Dioptrique, published as one of the appendices to his Discours de la Méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences) in 1637. Descartes proof is based on an analogy to the flight of a tennis ball, which he also uses to derive the laws of reflection.

Page of Descartes’ “La dioptrique” with the tennis ball analogy

Pierre de Fermat (1621–1665) rejected Descartes proof and derived the law instead with the principle of least time. His principle was that light travels between two given points along the path of shortest time showing that the principle predicts the observed law of refraction. Hero had argued similarly in his derivation of the laws of reflection stating that light follows the shortest path in the shortest time.

A portrait of Pierre de Fermat, artist unknown Source: Wikimedia Commons

In 1657, Pierre de Fermat received from Marin Cureau de la Chambre (1594–1669) a copy of newly published treatise, in which La Chambre noted Hero’s principle and complained that it did not work for refraction.Fermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances. His eventual solution, described in a letter to La Chambre dated 1 January 1662, construed “resistance” as inversely proportional to speed, so that light took the path of least time. That premise yielded the ordinary law of refraction, provided that light travelled more slowly in the optically denser medium. (Wikipedia)

Whereas Harriot’s earlier discovery of the sine law of refraction wasn’t rediscovered until the nineteenth century, that of Willebrord Snel resurfaced in the seventeenth century. This led Isaak Vossius (1618–1689), a Netherlander philologist, to accuse Descartes of plagiarism in his De natura lucis et proprietate in1662. The accusation was repeated by both Pierre de Fermat and Christiaan Huygens (1629–1695). However, modern historians do not think that Descartes “stole” the law from Snel.

In his Traité de la Lumière, published in 1678, Huygens showed how the sine law of refraction could be derived from the wave nature of light using the Huygens–Fresnel principle, which states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere.

Wave refraction in the manner of Huygens Source: Wikimedia Commons

Our last independent discoverer of the sine law of refraction was the Scottish mathematician and astronomer James Gregory (1638–1675) who, apparently totally unaware of Descartes’ work in optics, whilst living and working in Aberdeen, independently discovered the law through studying Kepler’s Pars Optica.

Portrait of James Gregory attributed to John Scougal Source: Wikimedia Commons

He wrote up the results of his research in his Optica Promota, famous both for his description of a reflecting telescope and his description of using a transit of Venus to measure the distance between the Earth and the Sun. When he travelled to London to publish the Optica Promota in 1662, he first became aware of Descartes account of the sine law of refraction. He added a preface to the book apologising for claiming discovery of the law of refraction, he had done so “because of the lack of recent mathematical texts in the otherwise excellent library of Aberdeen.”

The sine law of refraction had finally been established and would go on to play a central role in the development of optics, beginning with the work of Isaac Newton (1642–1727 os) on the spectrum produced by white light when refracted, which played a central role in his Opticks, published in London in English in 1704, and in Latin in 1706.

^[1] David C. Lindberg, Theories of Vision from Al-Kindi to Kepler, The University of Chicago Press, 1976, p. 12

^[2] A. Mark Smith, From Sight to Light: The Passage from Ancient to Modern Optics, The University of Chicago Press, 2015, p. 118

^[3] Smith, pp. 118-119

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Filed under History of Optics, History of Physics, History of science, History of Technology, Uncategorized

July 23, 2025 · 7:40 am

The English aristocratic bastard, who advised the Florentines on matters nautical and published the first maritime atlas by an English author.

For historians of Tudor England, the name Robert Dudley immediately evokes the image of the 1^st Earl of Leicester (1532–1588), companion to Edward VI, condemned to death by Mary Tudor for his part in the attempt to declare Lady Jane Grey queen, pardoned by Philip of Spain, and then served as soldier under him, served as courtier under Elizabeth I, whose long year suitor he became, whilst occupying numerous important political posts. Robert Dudley 1^st Earl of Leicester is one of the most colourful creatures in the menagerie of powerful men surrounding England’s Virgin queen.

Portrait of Robert Dudley, 1st Earl of Leicester, attributed to Steven van de Meulen Source: Wikimedia Commons

Although he served as patron to several of the people in this series and was Lieutenant and Captain-General of the Queen’s Armies and Companies during the threat of the Spanish Armada in 1588, he is not the subject of this post but rather his illegitimate son by his lover Douglas, Lady Sheffield (c.1542–1608), daughter of William Howard, 1st Baron Howard of Effingham (c. 1510 – 12 January 1573), who was also named Robert Dudley (1574–1649).^[1]

Artist unknown Source: Wikimedia Commons

unknown artist; Robert Dudley (1574-1649), Son of Robert Dudley, 1st Earl of Leicester, by Lady Douglas Howard, Wife of John Sheffield, 2nd Lord Sheffield; Merthyr Tydfil Leisure Trust; Source: Wikimedia Commons

Following his birth Dudley lived with his mother. At the age of five he was taken up in his father’s household but was allowed to see his mother when he wished. He was given a good education and entered Christ Church College Oxford in 1587, with the status of filius comitis (Earl’s son), he was tutored by Sir Thomas Chaloner (1559–1615). In 1588, just fourteen years old, he joined his father at the Tilbury Camp, where he was commanding the army in preparation for the arrival of the Spanish Armada. The Armada was beaten off without landing but Robert Dudley senior died on 4 September 1588. His son inherited the majority of his estate including Kenilworth castle but the will clearly stated that he was base, that is illegitimate, and so he didn’t inherit his father’s titles.

In the account that he wrote of his voyages for Richard Hakluyt’s Principle Navigations in 1594 Dudley wrote, “Having ever since I could conceive of anything bene delighted with the discovery of Navigation, I fostered in myselfe that disposition till I was of more yeres and better ability to undertake such a matter.” In a later never published Italian sailing manual, Direttorio Marittimo, he described how his passion for seafaring developed. He noted that both his grandfathers the Duke of Northumberland and William Howard, 1st Baron Howard of Effingham and his uncle Lord Charles Howard had all served as Lord Admiral of England. He says that he began studying navigation at the age of seventeen so, whilst at Oxford, most probably stimulated by Thomas Chaloner, who amongst his many talents was knowledgeable about the design and capacity of ships.

At the age of seventeen he also married Margaret the daughter of the mariner Sir Thomas Cavendish (1560–1592), who between 1586 and 1588 had circumnavigated the globe. Cavendish gave his daughter two ships named the Leicester and the Roebuck. She died childless not long after the wedding.

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

In 1594, Dudley, now aged twenty, assembled a fleet of ships including his flagship, the galleon Beare, as well as the Beare’s Whelpe, and the pinnaces Earwig and Frisking. Due to his young age Elizabeth refused to grant him a privateer’s licence but commissioned him instead to sail to Guiana. He sailed on 6 November 1594 having employed the navigator Abraham Kendal (died 1596) as ship’s master, from whom he “learned enough navigation for an Admiral”. After many adventures and the capture of several Spanish ships Dudley returned to England in May 1595. In 1596, he joined Robert Devereux 2^nd Earl of Essex (1565–1601) the son of Lettice Knollys (1543–1634) his father’s second wife, to serve as commander of the Nonpareil in an expedition against Cádiz.

Portrait of Robert Devereux, 2nd Earl of Essex, by Marcus Gheeraerts the Younger Source: Wikimedia Commons

He was knighted for his part in the capture of Cádiz. Shortly after he married Alice Leigh (1579–1669) the daughter of Sir Thomas Leigh, 1^st Baronet of Stoneleigh Abbey. She would bear him seven daughters, five of whom reached adulthood.

In 1601, Dudley supported Robert Devereux’s rebellious march on London, was briefly imprisoned and fell out of favour at court. In 1603, he tried to establish his legitimacy but claiming that his parents had been secretly married. However, he lost the case before the Star Chamber.

In July 1605, apparently dissatisfied with his situation in England, Dudley abandoned his wife and children and left England for Calais. He was accompanied by his cousin Elizabeth Southwell (1584–1631) maid of honour to Queen Elizabeth and, the great granddaughter of Charles Howard, 1^st Earl of Nottingham (1536–1624), Dudley’s uncle. The two converted to Catholicism and were married with a papal dispensation, they were blood relatives, from Pope Paul V in Lyon in 1606. According to English law the marriage was bigamous but the Catholic Church did not recognise Dudley’s earlier Anglican wedding.

Portrait thought to be Elizabeth Southwell, c. 1600 Source: Wikimedia Commons

The couple arrived in Florence in the same year and Robert Dudley quickly became a favourite at the court of Ferdinando I de’ Medici (1549–1609), Grand Duke of Tuscany, advising him on all maritime matters. The English, master-shipwright Matthew Baker (c. 1530–1613) had been approached in 1607 by Lotti the Florentine emissary in London, asking him to come and work in Florence. An offer that Baker declined on grounds of old age.

Sketch by Matthew Baker Source: Wikimedia Commons

Dudley, pupil of Abraham Kendal and experienced mariner, was viewed as a worthy alternative.

By the time Cosimo II (1590–1621) succeeded his father as grand duke in 1609, Dudley was living in Livorno and the first ship of his design had already been launched. Because the port of Pisa was silted up, Cosimo I (1519–1574), had launched a project to turn the fishing village of Livorno into a port under the leadership of the architect Bartolomeo Ammannati (1511-1592) in 1572. The construction of the Porto Mediceo was still an ongoing project when Dudley arrived and he took over leadership. He improved the port’s fortifications, drained the marsh between Livorno and Pisa and took on the task of creating a navy to rid the Mediterranean of Turkish pirates. He also persuaded English merchants to trade with, and even to settle in the new Porto Mediceo and encouraged the Grand Duke to make Livorno open to international business; eventually it became a free port.

A 17th century map of *Porto Mediceo* Source: Wikimedia Commons

Dudley continued to negotiate with both James I and Charles I in England to get his legitimacy recognised and even his marriage to Elizabeth Southwell but I’m not going to detail the years of exchanges here. He built a house in Florence and he and Elizabeth had twelve children, seven sons and five daughters. He was awarded the title of Duke of Northumberland by the Holy Roman Emperor, Ferdinand II (1578–1637), in 1620. He remained over the years a high ranking courtier in the de’ Medici court.

Following the death of Elizabeth in 1631, Dudley withdrew from the court and devoted himself to writing. He had previously produced manuscripts on naval matters including the already mentioned unpublished sailing manual, Direttorio Marittimo but he now devoted himself to his greatest written work the Dell’ Arcano del Mare(Of the Arcanum of the Sea). First published in Florence, in 1646-47, it consists of six books in three volumes dedicated to Grand Duke Ferdinando II de’ Medici.

Book I deals with the various methods of determining longitude ‘invented by the author.’ There is here nothing especially innovative but he detailed and refined the practical methods of calculation with the use of instruments.

Book II contained charts and sailing directions, with sections on tides, currents, and winds.

Book III contains much of the material of the Direttorio Marittimo dealing with naval discipline, and also giving detailed proposals for the creation of a navy with five classes of ship according to tactical function.

Book IV covers shipbuilding and also the building of fortification, containing Dudley’s experiences and practice when working at Livorno.

Book V deals with great circle sailing, building on the work of Pedro Nunes (1502–1578), and again full of practical instructions for the sailor.

BookVI is an atlas of 127 maps, the first atlas to use Mercator’s projection.

Golf of Mexico or Gulf of America? No, Mare de Norte

In his Finding their Way at Sea: the story of portolan charts, the cartographers who drew them and the mariners who sailed by them (Hes & De Graaf Publishers, 2012) Richard Pflederer says this about Dudley’s Dell’ Arcano del Mare:

This book is significant from many points of view. It was the first printed sea atlas to cover the entire world, the first to employ the Mercator projections and the first by an Englishman. Unlike most of the other printed charts, the Arcano made no pretense of duplicating the style of portolan charts. Instead of radiating rhumb lines, the rectangular grid of the Mercator projection formed the backdrop of these charts. These carts are beautifully engraved with flourished of fine calligraphy but displaying only sparse decoration. The elegance of the calligraphy is attributed to the Italian engraver, Antonio Francesco Lucini (1610–after 1660), who stated that he worked on the plates in an obscure Tuscan hilltop village for twelve years and used an amazing 2,200 kg of copper to complete the work. (pp. 142-3)

Dell’ Arcano can be seen as one of the first decisive moves away from the traditional sea chart that had dominated European maritime cartography for almost 400 years, and at last we had a chart which technically and visually presented some commonality with the more modern charts in use today. The use of the Mercator projection was especially significant. When first introduced 75 years earlier, it was touted as being particularly suited for nautical charts since it is the only projection upon which rhumb lines are truly straight lines. In this way Mercator had solved an important technical deficiency of traditional portolan carts, that is, the correct disposition of the rhumb lines. However, for a variety of reasons cartographers avoided the use of this projection on sea charts until Dudley. (p. 143)

Dudley’s Dell’ Arcano del Mare contains nothing new but it brought together in a unified style all the discoveries and developments in maritime knowledge and experience that the preceding century had produced. Paraphrasing slightly, Pflederer here refers to two cartographers:

While the work of Sir Robert Dudley was a technical and commercial success, interestingly it was a dead end from a development point of view. That is to say that the work cannot be seen as contributing directly to the further evolution beyond portolan charts. None of this detracts from his achievements as the producer of a truly groundbreaking work of technical perfection and artistic beauty. (p. 143)

Dudley died in 1649, seventy-fives years old but the Arcano was republished posthumously in 1661. As well as the Arcano, Dudley left a second maritime inheritance in the city of Florence in the form of nineteen scientific instruments, most of which he brought with him from England, and which are now housed in the Museo Galileo:

Thomas Gemini Quadrant Mid 16th-Century

The front of this quadrant carries the windrose, the zodiacal calendar, and the shadow square. On the back are engraved the hour lines, the path of the Sun’s diurnal arc, a zodiacal belt, and the degree scale. Provenance: Robert Dudley bequest to the Medici collections.

Humphrey Cole Surveying Compass 1575

Flat-legged surveying compass signed by Humphrey Cole. There are a degree scale (up to 180°), folding sights, and scales for measuring land plots and timber. The degree scale serves to measure position angles in surveying. The legs also fold out to 180°, enabling the instrument to be used as a ruler or alidade for a plane table. Provenance: Robert Dudley bequest to the Medici collections.

James Kynvyn Folding Rule 1595

Made by James Kynvyn, this rule consists of two legs graduated into 770 parts and a protractor at the swivel joint (180°). The grooved legs contain two flat arms that, when folded out and joined at a right angle, form the shadow square. The folding rule was used to measure angles, heights, and distances. Provenance: Robert Dudley bequest to the Medici collections.

James Kynvyn Nautical Instrument 1597

Instrument made by James Kynvyn, comprising a round plate divided into four quadrants, two with the sine grid, known as the *reduction quadrant*, one with the shadow square, and one with the quadrant for determining the longitude. At the center is a rotating graduated alidade; the circumference is engraved with the degree scale. On the back is a cylindrical section for fitting the instrument on a support. Provenance: Robert Dudley bequest to the Medici collections.

James Kynvyn Plate for nautical use

Used for taking bearings at sea, this instrument, made by James Kynvyn, closely resembles a model illustrated in Robert Dudley’s *Arcano del mare*. Consists of a rectangular brass plate carrying the shadow square and the quadrant for determining the longitude. On the upper right-hand corner was hinged a vane (missing). Provenance: Robert Dudley bequest to the Medici collections.

James Kynvyn [attr.] Quadrant 1595 (?)

The instrument, possibly made by James Kynvyn in 1595, is now incomplete. Of the horizontal circle with the degree scale only one half (from 180° to 360°) remains. However, it does still carry the base for the azimuth compass (missing). A quadrant with a nonius is fitted at right angles to the base. At right angles to this second plate pivoted a diopter (missing) for measuring heights, while the compass served to determine position angles. The instrument was thus used to calculate the coordinates of celestial bodies (altitude above the horizon and position relative to the magnetic meridian). Provenance: Robert Dudley bequest to the Medici collections.

Charles Whitwell [attr.], Astrolabe late 16th cent.

Although unsigned, this astrolabe can be attributed, because of its characteristics, to the maker Charles Whitwell. There are two semicircular sectors that can be attached to the center of the astrolabe, a graduated rule, and a ruler with circle. The instrument was designed for nautical use. Provenance: Robert Dudley bequest to the Medici collections.

Charles Whitwell, Astrolabe, 1595

This universal astrolabe, made by Charles Whitwell, lacks a rule and an alidade. The mater carries a planisphere with an equinoctial stereographic projection. The back of the instrument is just barely roughed out. Provenance: Robert Dudley bequest to the Medici collections. This is one of only two known examples of the astrolabe described by John Blagrave (die 1611) in his publication *The mathematical ievvel* (1585)

Charles Whitwell, Horary Quadrant, 1595

Quadrant made by Charles Whitwell. The shadow square occupies two adjacent sides; the opposite corner carries a pivoting vane with sights. In the center is engraved a Stöffler horary quadrant bounded by the arc of the degree scale. The back carries a zodiacal calendar and the windrose. Provenance: Robert Dudley bequest to the Medici collections.

Charles Whitwell, Nautical Circle late 16th cent.

Designed by Robert Dudley and made by Charles Whitwell, this large disk bears only a superficial resemblance to the astrolabe inv. 1123, 1124, 1127. In fact, it probably belonged to a more complex instrument described in Dudley’s *Arcano del mare*. A ruler complete with circle also forms part of this navigation instrument. Provenance: Robert Dudley bequest to the Medici collections.

Augustine Ryther, Theodolite, 1590

This instrument, made by Augustine Ryther, is one of the oldest models of theodolite extant. It comprises a horizontal circle with a degree scale on the circumference and an inscribed shadow square. At the center pivots a diopter with a magnetic compass. The compass carries a holder for a vertical semicircle with a degree scale, shadow square, and viewer. The plane of the vertical semicircle rotates jointly with the diopter underneath, allowing the simultaneous determination of the azimuth angle and zenith angle of a given point, i.e., its spatial coordinates. The instrument, used for surveying, matches the one introduced in the second half of the sixteenth century by Leonard and Thomas Digges. Highly probable provenance: Robert Dudley bequest to the Medici collections.

Maker unknown, Nautical Compass, late 16th cent.

English nautical compass with an arc graduated from 0° to 90°. The sights are missing. Provenance: Robert Dudley bequest to the Medici collections.

Maker unknown, Nautical Compass, late 16th cent.

Large, incomplete English nautical compass with two sights for each leg and an arc graduated from 0° to 90°. Provenance: Robert Dudley bequest to the Medici collections.

Maker unknown, Nocturnal, 16th cent.

Nocturnal consisting of two concentric circles. The outer circle, divided into twenty-four hours, is fixed and joined to a straight arm serving as a handle. The inner circle rotates freely and carries two indexes pivoting at the center. The index pivot and the band of the rotating circle are connected via a small semi-hollow disk. Highly probable provenance: Robert Dudley bequest to the Medici collections.

Maker unknown, Protractor, late 16th cent.

Protractor consisting of a circle divided into 360°. Inside the upper half of the circle is a graduated semicircle; the lower half contains a rectangular plate with a double graduation into ten equal parts. The instrument was used to plot routes on sea charts and reproduce the bearings taken with the magnetic compass. Highly probable provenance: Robert Dudley bequest to the Medici collections.

Maker unknown, Quadrant, late 16th cent.

Maker unknown, Theodolite, late 16th cent.

Incomplete theodolite. The semicircular sector carries a viewer on the diameter line, the degree scale on the arc of the circumference, and the shadow square at the center. Made in England, it is engraved on one side with the coat of arms of Queen Elizabeth I. Highly probable provenance: Robert Dudley bequest to the Medici collection

Maker unknown, Windrose, late 16th cent.

This windrose probably belonged to a navigational magnetic compass. The back has a wooden base. Provenance: Robert Dudley bequest to the Medici collections.

Francisco de Goes, Portuguese, Mariners Astrolabe 1608

This mariner’s astrolabe, made by Francisco de Goes, is fitted with a suspension ring, a degree scale (only half of which has been engraved), and an alidade. On the alidade are mounted two large squareplates, one of which is perforated. The instrument served to measure the altitude of the Sun, which was indicated by the alidade on the degree scale when a ray passed through the front hole and struck the center of the back plate. Provenance: Robert Dudley bequest to the Medici collections.

^[1] Much of the information in this post is taken from the chapter on Robert Dudley in Gerard L’E Turner, Elizabethan Instrument Makers: The Origins of the London Trade in Precision Instrument Making, OUP, 2000, pp. 77–89

From τὰ φυσικά (ta physika) to physics – XLVIII

In 1618, the same year that Isaac Beeckman (1588–1637) was teaching the principle of inertia to René Descartes (1596–1650), Johannes Kepler (1571–1630) made one of the most significant discoveries in the history of astronomy and physics, his third law of planetary motion. There are epoch making discoveries in the history of science, discoveries that fundamentally change humanities perception of the world they live in, Kepler’s third law is one of them, however, it went almost unnoticed! Why is Kepler’s third law so important and why wasn’t that importance recognised and acknowledged when he first set it loose in the world?

Portrait of Johannes Kepler by August Köhler, c. 1910, after 1620 original Source: Wikimedia Commons

Although force as we know it was never clearly defined before the late seventeenth century, from Aristotle onwards there was an awareness that is one wished to move something one had to apply force. A donkey to pull the cart, an arm to push open a door, a rope and pully system to lift a heavy load of the ground and into a ship perhaps. As already discussed in this series a big problem arose with projectile motion. Why did the thrown ball or the shot arrow continue to fly after it had left the hand or the bow string, when force was no longer being applied. So, in the medieval period the concept of impetus was created. This said that something was put into the projectile that was gradually used up until the projectile fell down. Concepts such as friction and air resistance were unknown.

As we saw in the last episode in this series this problem with projectile motion was the development of the concept of inertia, projectiles don’t need force to keep them moving but rather forces, such as friction or air resistance, were needed to stop them doing so. To come this far, scholars first had to overcome the Aristotelian objection to the existence of a vacuum. The acceptance of the possibility of a vacuum and the concept of inertia go hand in hand.

In the celestial realm things were different. In the Aristotelian philosophy celestial bodies went round in circles because it was their nature to do so. Building up his celestial spheres concept going from the sphere of the stars down to the sphere of the Moon each sphere drove the one inside it, in a sort of friction drive. The motive power came from outside the complete system from the unmoved mover, which Aristotle identified with love. In the Christian medieval period, it was thought that each planet had an angel, who pushed the planet around its orbit. Some medieval thinkers also thought that the planets were propelled by impetus.

By the time Kepler began his journey into the world of astronomy everything had changed. Supralunar comets had metaphorically smashed the Aristotelian spheres and a new concept of celestial motion was needed. As we have seen in his Astronomia Nova, using the copious data collected over the decades by Tycho Brahe (1546–1601), Kepler had deduced his first two laws of planetary motion, elliptical orbits with the Sun at one focus of the ellipse, and a line from the focus of the ellipse to the planet sweeping out equal areas in equal times.

Having got this far, Kepler speculated what could possibly be driving the planets around their orbits. Inspired by the De magnete of William Gilbert (1540?–1603) published in 1600, Kepler hypothesised a force like magnetism radiating from a revolving Sun sweeping the planets around their orbits. This force weakened the further the planet from the Sun thus the longer orbital period. The third law would, of course, deliver the mathematical relation between a planet’s distance from the sun and the period of its orbit.

Jumping ahead chronologically, in his Theoricae mediceorum planetarum ex causis physicis deductae in 1666 Alfonso Borreli (1608–1679) basically explained the shape of the planetary orbits correctly, through a combination of attraction between the planet and the Sun and the inertial tendency of the planet to fly off in a straight line.

Borreli, however, did not have the necessary mathematical apparatus to formally prove this. That would have to wait for Newton’s Principia. One important ingredient, which Borreli ignored, which was central to Newton’s efforts, was Kepler’s third law. Why didn’t people recognise the significance of Kepler’s third law, mainly because he buried it in his bizarre metaphysics.

Kepler’s motivations for his work were anything but scientific in the modern sense. His first publication the Mysterium Cosmographicum (1596) containing his famous model of the cosmos with the planet embedded between the five regular Platonic solids was his answer as to why God had only created six planets. Kepler’s God was mathematics incarnate and highly logical. The model didn’t quite fit and needed fine tuning, which Kepler intended to deliver in his next publication.

On 26 March 1598 in a letter to Herwart von Hohenburg the Bavarian chancellor, Kepler explained that the Mysterium Cosmographicum or Prodromus (forerunner) would serve as the introduction to a series of cosmographical treatises dealing more fully with the subjects of Aristotle’s De caelo and De generatione. […] in his notes for the second edition of the Mysterium Cosmographicum, Kepler remarked that he regarded the Harmonices mundi as “the authentic and appropriate successor” of his Prodromus.

[…]

Then on 14 December 1599 he communicated to Herwart his intention to write a cosmographic dissertation, evidently based on the quadrivium, with the title De harmonice mundi, which would consist of five parts:

A geometrical part on the constructible figures.
An arithmetical part on the solid relations.
A musical part on the origins of harmonies.
An astrological part on the origins of the aspects.
An astronomical part on the origins of the periodic motions of the planets.^[1]

Kepler went to work for Tycho in 1600 to acquire the accurate observational data he required to carry out this work and got sidetracked only managing to finally produce the Harmonices Mundi in 1618, having produced an incredible amount of other high level scientific work in the meantime. The Harmonices mundi when it was finally finished, basically retained this scheme listed above but the original first part was split into two and the original second part was placed at the beginning of the third part.

The book, Kepler’s magnum opus, is his commitment to the Pythagorean theory of cosmic harmony but for Kepler this is geometric and not arithmetical. Two books of geometric theory are followed by a book on music theory the sum of these being applied to astrology in book four. The fifth book brings all of the preceding material together in Kepler’s astronomical/cosmological harmonic theory. Kepler examines all of the mathematical aspects of the planetary orbits looking for ratios that fit with his definitions of the musical intervals. It is here that Kepler hit pay dirt with the discovery of his third law, which he actually terms his harmonic law and with which he is truly delighted:

“After I had discovered true intervals of the orbits by ceaseless labour over a very long time and with the help of Brahe’s observations, finally the true proportion of the orbits showed itself to me. On the 8^th of March of this year 1618, if exact information about the time is desired, it appeared in my head. But I was unlucky when I inserted it into the calculation, and rejected it as false. Finally, on May 15, it came again and with a new onset conquered the darkness of my mind, whereat there followed such an excellent agreement between my seventeen years of work at the Tychonic observations and my present deliberation that I at first believed that I had dreamed and assumed the sought for in the supporting proofs. But it is entirely certain and exact that the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.”

Translated into modern terminology the third law is:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit: i.e. for two planets with P = orbital period and R = semi-major axis P₁²/P₂²=R₁³/R₂³

Although solidly based on geometry, to which Kepler made several important contributions, there appears to be very little that we would consider to be scientific in Kepler’s monumental tour de force. The result was that nobody went digging around in this paean to cosmic harmony for the third law, which for all its significance Kepler didn’t emphasise but simply included along with the rest. Put another way, Harmonices Mundi was a flop and the third law might have gone totally unnoticed if Kepler wasn’t such an incredibly prolific writer.

At the same time as he was writing Harmonices Mundi, when he should have been working on the Tabulae Rudolphinae, Kepler was also writing the first textbook on heliocentric astronomy his Epitome Astronomiae Copernicanae, which despite the title was about the Keplerian elliptical system and not the Copernican system. This book written in the, then, common dialogue style of teacher and student had a total of seven books published in three volumes in 1618, Books I–III, 1620, Books IV–V, and 1621, Books VI–VII. Book IV entitled, Celestial Physics, That Is, Every Size, Motion and Proportion in the Heavens Is Explained by a Cause Either Natural or Archetypal, contains the three laws presented for the first time together.

Like the earlier Astronomia Nova (1609) and the Harmonices Mundi the Epitome initially found few readers. However, this began to change following the publication of the Tabulae Rudolphinae in 1627.

From the time of Ptolemaeus, and even earlier, planetary tables were regarded as the most important aspect of astronomy. In the Early Modern Period the data contained in them was used for astrology, navigation, surveying, and cartography, and the Tabulae Rudolphinae based, as they were on Tycho’s new accurate observational data, were by far and away the most accurate tables that had every been published and were perceived as having been produced using Kepler’s elliptical astronomy. The effect of the combined Epitome– Tabulae Rudolphinae was gradual, but as J. L. Russell wrote:

Between about 1630 and 1650 Kepler’s Epitome Astronomiae Copernicanae (in which all three laws were clearly formulated) was probably the most widely read work on theoretical astronomy in northern and western Europe, while his Rudolphine Tables, which were based on the first two laws, were regarded by the majority of astronomers as the most accurate planetary tables available.^[2]

The reception of the three laws, which together constitute Kepler’s elliptical astronomy, was not a simple all or nothing affair but rather a complex messy either/or.

The laws were taken separately with some simply accepting the first law, elliptical orbits but not the rest. Some accepted first and third laws but not the second, so-called area law, almost on aesthetic grounds; people desired something simpler than what they saw as the clunky, hard to calculate law that Kepler offered, with its highly questionable mathematical derivation. In fact, a whole major debate erupted in Europe over the second law with leading astronomers offering alternatives. Most notable was the debate between the French astronomer Ismaël Boulliau (1605–1694), who accepted and promoted Kepler’s ellipses but rejected the second law substituting his own Conical Hypothesis, and the English astronomer Seth Ward (1617–1689), who rejected Boulliau’s alternative second law whilst presenting his own. The debate was ended when the German mathematician Nikolaus Mercator (c. 1620–1687) delivered a new solid mathematical proof of the second law, (for a fully account of the debate see here).

Both Galileo and Descartes totally rejected Kepler’s system, as did Tycho’s most important disciple, Longomontanus. Philip Landsberg, whose planetary tables were preferred by many because, although not so accurate as the Tabulae Rudolphinae, were simpler to use, also rejected Kepler’s whole system. Bonaventura Cavalieri, whilst not necessarily adopting them, wrote positively about laws one and two. Kepler’s work received a boost when a second edition of the Epitome appeared in Frankfurt in 1635, to quote Russell:

In 1635 Kepler’s theories received a further stimulus with the publication of a second edition of the Epitome at Frankfurt. This work is a substantial volume of nearly a thousand pages; the fact that it was worth republishing some five years after its author’s death is good evidence that there was a lively interest in his ideas at this time.

In England the group centred around Jerremiah Horrocks, who called themselves Nos Keplari, followed the earlier group around Thomas Harriot, in total acceptance of Kepler’s elliptical astronomy but neither group exercised much influence within the European world of astronomy.

In Italy Giovanni Baptista Riccioli gave a full account of Kepler’s astronomy in his Almagestum Nova (1651), although he rejected the basic heliocentric hypothesis, both he and his student, Giovanni Domenico Cassini (1625–1712), later adopted elliptical orbits having proved, using the heliometer in San Petronio, Bologna, that either the Earth’s orbit around the Sun, or the Sun’s orbit around the Earth was an ellipse, but they were unable to prove which was true. As noted above Alfonso Borreli accepted Kepler’s elliptical system in his Theoricae mediceorum planetarum.

In France, as already noted Boulliau was an enthusiastic supporter of Kepler’s work whist rejecting the second law. Both Pierre Gassendi and Marin Mersenne both partially accepted Kepler’s system, most notably elliptical orbits.

After 1650, in a later phase in England there was a general acceptance of elliptical orbits and both Vicent Wing in his Astronomia Britannica (1669) and Thomas Streete in his Astronomia Carolina (1661) accepted Kepler’s full system with first and third laws but with Boulliau’s modified second law. Isaac Newton taught himself astronomy from the works of Wing and Streete and would later take the second from Nicholas Mercator.

During all this time, although quite a lot of astronomers quoted it, nobody before Newton paid very much attention to Kepler’s third law, all of them being much more concerned with the question as to whether the orbits were elliptical and disputing over the unloved second law. It would be left to Newton in his Principia(1687) published almost seventy years after Kepler first discovered it, to demonstrate the fundamental importance of the harmonic law.

^[1] The Harmony of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E. J. Aiton, A. M. Duncan, J. V. Field, Memoirs of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, Volume 209, 1997. Introduction p. XVI

^[2] J. L. Russell, Kepler’s Laws of Planetary Motion: 1609–1666, The British Journal for the History of Science, Vol. 2, No. 1 (Jun., 1964), pp. 1–24, p. 1

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The history of science in medieval Islamicate culture.

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From τὰ φυσικά (ta physika) to physics – XLVIII

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