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January 14, 2026 · 7:43 am

From τὰ φυσικά (ta physika) to physics – LVIII

In the previous episode of this series, I took a look at the two English mathematicians, who most influenced the young Isaac Newton (1642–1726 os) in the early stages of his intellectual development, Isaac Barrow (1630–1677) and John Wallis ((1616–1703). Today we take a first of, probably, several looks at Isaac Newton, who played a highly significant role in the evolution of physics, although it still wasn’t called that yet, when he combined terrestrial mechanics with astronomy und the umbrella of universal gravity in his magnum opus, Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy) 1687.  

Source: Wikimedia Commons

The popular hyperbole calls Newton the greatest scientist of all time, which is of course rubbish. Apart from the fact that the use of the term scientist, first coined by William Whewell in 1831, is anachronistic it pays to pause and note that even as late as the end of the seventeenth century there was no such thing as a professional scientist in the modern sense and certainly no preprogrammed career path to become one. If we consider the period from the gradual revival of science in the High Middle Ages to the period of Newton the closest we get to professional scientists are the court astrologers, who were mostly also the astronomers. Even Kepler, who revolutionised astronomy and optics, earned his living mostly as a professional astrologer. 

The medieval university didn’t really take mathematics seriously and there was almost never chairs for mathematics. They were predominantly Aristotelian and what are now the physical sciences were handled philosophically not mathematically. When chairs for mathematics began to be created during the Renaissance in the fifteenth century, first in Krakau and then in the Renaissance universities of northern Italy, there were actually created to teach astrology to medicine students because of the prevailing mainstream astromedicine, or iatromathematics to give it its correct name. To do astrology you need to be able to do astronomy and to do astronomy you need to be able to do mathematics. Even at the beginning of the seventeenth century Galileo, as professor of mathematics in Padua, would have been required to teach astrology to the medical students, although we don’t have a direct record of his having done so. 

Chairs for mathematics and or astronomy gradually spread throughout Europe during the sixteenth century but Britain lagged well behind the continental developments. In England, Henry Savile (1529–1622), who travelled abroad to acquire his own mathematical education, established chairs for geometry and astronomy at Oxford University in 1619. Cambridge had to wait until 1663 before Henry Lucas (c. 1610–1663) bequeathed the funding for a professorship in his will, with Charles II establishing the Lucasian Chair in 1664. Newton was the second Lucasian Professor following in the footsteps of Isaac Barrow. Of course, the Gresham chairs for geometry and astronomy, set up at the beginning of the century, predate both of the university chairs but these were not teaching positions but public lectureships aimed at a general public. Henry Briggs (1561–1630) was both the first Gresham and the first Savilian professor for geometry.

To show that there no such thing as a science career path in the seventeenth century let us briefly recapitulate the life paths of four scholars who have featured in this series. who made serious contributions to the emerging mathematical sciences.

René Descartes (1596–1650) was the son of a minor aristocrat and politician. He was schooled in the Jesuit College of La Flèche meaning he received a first class education including probably the best mathematical education available in Europe at the time. He studied two years at the University of Poitiers graduating with a Baccalaureate and Licence in canon and civil law. However, instead of now becoming a lawyer he set off to become a military engineer but to do that he, a Catholic French aristocrat, went off to Breda in the Netherlands to join the Protestant Dutch States Army. Purely by chance in Breda, he met the Dutch candle maker turned school teacher Isaac Beeckman, who introduced him to both the corpuscular  mechanical theory and mathematical physics. This set him off on a winding path to becoming a mathematician, philosopher and physicist. 

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

Christiaan Huygens (1629–1695) was the son of a powerful aristocratic diplomat who enjoyed an absolutely first class private education before going to Leiden University to study law and mathematics followed by a period at the Orange College in Breda. He had been prepared his whole life to become a diplomat like his father but after one mission he decided the life was not for him he withdrew to the family home and supported by his father became a private scholar studying a wide spectrum of the mathematical sciences. Later he would be become a paid scholar in the new French Académie des sciences. That the Académie employed paid scholars was an advantage over the rival Royal society in London, which only paid Robert Hooke as curator of experiments. 

As we saw John Wallis (1616–1703) had perhaps the weirdest life path for a scientist. The son of a cleric he also became a cleric occupying various church positions. Purely by chance he discovered a talent for cryptography and became the cryptologist of the parliamentary party during the Civil War and Interregnum. In 1649, Cromwell appointed him, a man with no formal education in mathematics, Savilian Professor of Geometry at Oxford, a post he held for fifty years going on to become one of Europe’s leading mathematical authorities having spent his first couple of years in the post teaching himself the full spectrum of mathematics.

Portrait of John Wallis by Godfrey Kneller Source: Wikimedia Commons

Isaac Barrow (1630–1677) the son of a draper born into a family of many prominent scholars and theologians. A graduate and fellow of Trinity College Cambridge he taught himself mathematics and the natural sciences with a small group of like-minded fellows. Leaving England in 1655 because of the rise of puritanism he travelled extensively through Europe and Asia Minor for four year, deepening his impressive linguistic abilities. Returning in 1659 he was appointed both Regius Professor of Greek at Cambridge and three years later Gresham professor of geometry. In 1663, he was appointed the first Lucasian Professor, resigning the Regius and Gresham professorships in 1664. In 1669, he resigned the Lucasian chair in order to devote his time to theology. 

Portrait of a young Isaac Barrow by Mary Beale (1633–1699) Source: Wikimedia Commons

Although their life paths differ substantially, all four of our mathematical scholars have in common that they come from the upper, educated, well off strata of society, two of them were even aristocrats,  and could afford the so-to-speak luxury of pursuing a career in still not really established mathematical disciplines. This, as we will see, was not true for Isaac Newton. 

Born in manor house of the hamlet of Woolsthorpe-by-Colsterworth near Grantham in Lincolnshire on  Christmas Day 1642, on the Julian calendar, Isaac was the son of the yeoman farmer Isaac Newton and his wife Hannah Ayscough. Isaac senior was not only uneducated but could not even sign his own name. He was however not poor and was a successful, prosperous farmer, who unfortunately died three months before his son’s birth.

Woolsthorpe Manor Source: Wikimedia Commons

His mother Hannah, however, came from higher social strata than her husband, from a family that valued education, her brother the Rev William Ayscough MA was a graduate of Trinity College Cambridge.

When Isaac was just three years old, Hannah married the Rev. Barnabus Smith and went to live with him in his parish of North Witham a mile and a half away, leaving Isaac in Woolsthorpe Manor in the care of his maternal grandmother. Eight years later Barnabus died and Hannah returned to Woolsthorpe with Isaac’s three step siblings. Two year later, Isaac, now twelve, was sent off to the grammar school in Grantham, where he lodged with the local apothecary, Mr Clark. Isaac lived an isolated life at school and tended to neglect his studies, which basically consisted just of Latin, but always did just enough to remain school primus.

The grammar school in Grantham, Lincolnshire, attended by Isaac Newton. Engraving, ca. 1820.
Welcome Collection

At the age of sixteen Hannah removed him from the school and by 1659 he was living back in Woolsthorpe, where Hannah tried to make a farmer out of him. This proved to be a dismal failure and the school master Henry Stokes and his uncle William Asycough persuaded Hannah to let him finish his education and go to university. Stokes even remitted his school fees to convince the reluctant widow. 

He graduated school primus and in June 1661 he was admitted to Trinity College Cambridge as a subsizar, this is a student whose fees are partially remitted in return for which he works as a servant for other students. Hannah Ayscough Newton Smith was a very wealthy woman so, why did she force her son to earn his way through college? She also only gave him an allowance of £10 pa. The major theory is that this was her revenge for being pressured into letting him go to university at all but I think there was an element of puritanism, he should not expect to be spoon fed but should learn the value of money. 

575 map showing the King’s Hall (top left) and Michaelhouse (top right) buildings before Thomas Nevile’s reconstruction. Source: Wikimedia Commons

It would seem logical to assume that Isaac went up to Trinity because it had been the college of his maternal uncle, William Ayscough, who had pressured Hannah into sending him to university but there is a second possible source of influence in this issue. There is slight evidence that Isaac served as subsizar to the Trinity fellow, Rev. Humphrey Babington, rector of Boothby Pagnell and brother of Katherine Babington, a friend of Hannah’s and the wife of William Clark the Grantham apothecary where Newton boarded as a schoolboy. Later, Newton stayed with Babington for a time during the summer in 1666-67. It is possible that that the Rev. Babington had recognised Newton’s abilities and taken him under his wing in 1661. 

“Sir Isaac Newton. when Bachelor of Arts in Trinity College, Cambridge. Engraved by B. Reading from a Head painted by Sir Peter Lily in the Possession of the Right Honorable Lord Viscount Cremorne.”  National Portrait Gallery vis Wikimedia Commons

The undergraduate curriculum in Cambridge in the 1660s was little changed from that when the university was founded more than four centuries earlier. This meant Aristotle, Aristotle and more Aristotle, a diet that didn’t appeal to the young Isaac, who remained a mediocre student. Newton was a disciplined note taker all of his life and we know from his own records that he didn’t actually finish any of his set books. By the 1660s standards had fallen so low in Trinity that basically any student who stayed the course for four years could graduate. So, despite his lack of engagement Isaac duly graduated BA in 1664.

The next step was to apply for a scholarship, which would enable him to continue his studies, and this is where his lack of effort almost caused him to stumble. There were a limited number of scholarship and a larger number of excellent potential candidates and it seemed that the lacklustre Isaac was not in the running. However, somebody in the background pulled some strings and he was granted a scholarship on 28 April 1664, enabling him to study for another four years for his MA and making him financially independent for the first time in his life. It is not clear who did the string pulling. It might possibly have been Isaac Barrow who had examined Newton on Euclid for his scholarship and found him wanting or more possibly the Rev Babington, now a highly influential figure in Trinity. In 1667, Babington became one of the eight senior fellow, the group that controlled the college.

What now followed in the years from 1664 up to 1672, when Newton published his first paper, is one of the most impressive period of self-study ever undertaken, including the mythical Annus mirabilis, the year that Newton spent at home in Woolsthorpe Manor having been sent down from Cambridge because of the plague in 1665-66. During this period Newton taught himself the modern mathematics, astronomy, mechanics, and optics utilising the work of the leading scholars in these fields, extending and going beyond them and creating his first contribution to these fields. I’ve written a long blog post outlining all that he did over the second half of the 1660s and am not going to repeat it  here. When he entered the 1670s Isaac stood at the beginning of the process that would see him become the most powerful natural philosopher in Europe. 

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January 9, 2026 · 8:27 am

Much Ado About Nothing

Regular readers will be well aware that a Renaissance Mathematicus book review is usually anything but short. I try as far as possible to give an accurate, informative,  outline sketch of the actual contents of the book under discussion. This leads automatically to a lengthy essay style review, the aim of which is to give potential readers a clear picture of what exactly they can expect if they decide to invest their time and money  in the volume in question. Given this approach to reviewing, how can I produce a Renaissance  Mathematicus style review of a book that is seven hundred and forty pages long and contains thirty nine academic papers covering a very wide array of different aspects of a single topic without it turning into a seemingly never ending essay? The simple answer is, I can’t so, what follows will be far less detailed and informative than is my want. 

So, what is the topic and what is the book that gives this topic so much attention? The topic is one that has fairly often put in an appearance here at the Renaissance Mathematicus, zero and the book is The Origin and Significance of Zero: An interdisciplinary Perspective.[1] 

The book is the result of a cooperation between Closer to Truth, a broadcast and digital media not-for-profit organisation presenting a weekly half-hour television show which airs continuously since 2000 on over 200 PBS and public TV stations, and the Zero Project Foundation, which was set up in the Netherlands in 2015. Closer to Truth is the baby of the book’s one editor Robert Lawrence Kuhn and the Zero Project Foundation was set up by the book’s other editor Peter Gobets, who unfortunately passed away just before the book was published. You can view a Closer to Truth video on the Zero Project here.  and read about Closer to Truth here 

The book opens with a ten page preface in which Kuhn, a philosopher, talks about his life-long obsession with the concept of nothing and discusses a hierarchy of definition of nothing. The twelve page introduction from Gobets explains the motivation behind the Zero Project, its cooperation with Closer to Truth and the structure and intention of the book itself. 

The book is in four parts, whereby Part 0 consists of fifteen papers on Zero in Historical Perspective. Part 1 has sixteen paper on Zero in Religious, Philosophical and Linguistic Perspective, the papers  are as wide ranging as the title suggests. Part 2 Zero in the Arts is very short consisting of a very brief introduction by Gobets to eight art works by the artist British-Indian sculptor Sir Anish Mikhail Kapoor (b. 1924) devoted to Kapoor’s visualisation of the Buddhist concept of the void. Part 3 has seven papers under the title Zero in Science and Mathematics. 

The papers vary considerably, in length, in academic depth, some are fairly general and superficial, some are deeply researched, and writing quality i.e. readability but this is too be expected in a book that tries to pack so many different viewpoints into one volume. At times I got the feeling that some judicious editing would have improved it in general, less would have been more.

As somebody, who is primarily a historian of mathematics it is, of course, Part 0 Zero in Historical Perspectivethat most interested me. The section opens with two papers relating to the multiple appearances of zero as a concept, as a placeholder and as a number in different cultures and the historical problems of trying to establish if, when and how  influences or exchanges took place between those cultures and concepts. Neither paper is particularly helpful and the second Connecting Zeros by Mayank N. Vahia gives prominence to an ahistorical myth. He writes:

Indians were the first to work out the algebra of zero and opened the window to a completely new class of mathematics.

This was not true for the Europeans, to whom life without one was unimaginable. One was the natural smallest number for them. Zero made them uncomfortable. All cultures believed in one form or another, that there exists a Great God. This was the proverbial “One”. This Great Got then created the universe and the many variation in life. The one therefore pervades everything and remains even when all else is gone. 

In early Europe it was forbidden to study zero [my emphasis] as it was considered unnatural and against the working of the Great One who would always be present. 

I could write a whole blog post taking this heap of garbage apart. It comes as no surprise that it was written by a retired engineer who “has become interested in understanding the origin and growth of astronomy and science in India”. He should start by learning something about comparative religion about which he displays an unbelievable ignorance. Perhaps he could explain who the “Great God” is/was in pantheistic Hinduism? Although he doesn’t define what he means by early Europe, one has to assume he means the Middle Ages with its Christian culture, which I’m sorry to tell him, which, despite the widespread myth, never forbade the study of zero. 

Things improve when we get to the histories of zero in the individual cultures. There is an excellent paper, Babylonian Zero on the sexagesimal place-value number system in Mesopotamia and the introduction of a place holder zero and the separate concept of nothing as the result of an arithmetic operation.

There are two good papers on the Egyptian concepts of zero and nothing, Aspects of Zero in Ancient Egypt and The Zero Concept in Ancient Egypt, the latter includes a brief section on the Mayan concept of zero. Followed by an equally good one on zero in ancient Chinese mathematics, On the Placeholder in Numeration and the Numeral Zero in China.

As to be expected India features next with a short paper on the appearance of numerals in Reflection on Early Dated Inscriptions from South India followed by a longer one tracing the path from the religious term Śūnyameaning empty or void to the numeral zero, From Śūnya to Zero – an Enigmatic Journey, which includes section on the Egyptians, the Babylonians, the Incas, the Maya, China, Greece and India with reflection of the reception in Arabic and European culture. The two paragraphs here on the Incas and the Maya are the only mention of the development of zero in Middle America a serious lacuna in the book. This is followed by an essay on The Significance of Zero in Jaina Mathematics an interesting branch of Indian mathematics, somewhat outside the mainstream. 

Now we get the bizarre rantings of Jonathan J. Crabtree, Notes on the origin of the First Definition of Zero Consistent with Basic Physical Laws. Crabtree has been wittering on about his “great discovery” in elementary mathematical pedagogy to my knowledge for at least twenty years and an Internet search shows that it is closer to forty years. Crabtree thinks that English language elementary mathematics teaching is a disaster because it uses an at best ambiguous at worst false definition of multiplication. I write English language because the pesky British spread this abomination through the textbooks it distributed throughout the Empire. Crabtree attributes this pedagogical error to Henry Billingsley’s false translation of Euclid’s definition of multiplication. To this he has added that Europe didn’t understand the true nature of zero because the Arabs mistranslated Brahmagupta.  

Up next we next have a somewhat bizarre four page paper, Putting a Price on Zero about a historian of mathematics asking a class of mathematicians to explain how they would allocate royalties to the various cultures which are claimants for the invention of zero. A waste of printing ink in my opinion. 

Returning to more scholarly realms we now have an interesting article on a famous zero artifact, Revisiting Khmer Stele K-127. This stone stele discovered in1891 on the east bank of the Mekong River in Sambaur contains the date 604 of the śaka era, i.e. 682 CE, and is the oldest known inscription of the numeral zero.

Moving forward in time we get an essay on zero in Arabic arithmetic, The Medieval Arabic Zero. Comprehensive, detailed and highly informative this article meets to highest standards and one wished that it might have been used as a muster for the whole volume. This is followed by an excellent paper on Islamic numerals, Numeration in the Scientific Manuscripts of the Maghreb.

The final paper in Part 0, The Zero Triumphant is about the Tarot. This, however, is not the fortune telling Tarot but the original 15th century Italian card game, which was originally called ‘trionfi’ (i.e., ‘triumphs’ or ‘trumps’). This was played with an amalgamation of two packs of cards, the four-suited deck of playing cards brought into Europe via the Mamluk Empire from the Muslim Near East and a deck of 22 allegorical images originating in medieval Christian iconography. The Islamic deck was numbered with Hindu-Arabic numerals and the European Trumps cards had Roman numerals. The Fool or Crazy One (Il Mato or le Fol) is numbered 0.

A fascinating paper that is however flawed by repeating the myth served up in the second paper Connecting Zeros:

The concept of zero did not exist in the classical mathematics of the Greeks and Romans. And it was an abomination at first to the Christian West. What use did good Christians have for nothingness? God created something not nothing.

As noted above this is ahistorical bullshit. 

Each of the papers as footnotes and its own, oft very extensive, bibliography, and the book has a usable general index. Some but not all of the papers are illustrated. The book closes with an Epilogue by Peter Gobets with more thoughts about the Zero Project and the books role in it. 

Based on what I’ve read, and I admit to not having read the whole volume, I could have titled this review, The Good, The Bad and The Ugly. There are some excellent papers, some that are somewhat iffy and some that probably should not have made it into print. It is actually quite affordable given that it’s a Brill publication the hardback and the PDF both waying in at  €100 plus VAT on the publishers website but I’m not sure I would recommend buying it rather than borrowing it from a library to read the bits that interest the individual reader. I do have one last complaint, the book is so thick, so heavy, and so tightly bound that I literally found it impossible to find a way to read it comfortably. 

[1] The Origin and Significance of Zero: An interdisciplinary Perspective, edited by Peter Gobels and Robert Lawrence Kuhn, Brill, 2024.

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December 26, 2025 · 8:24 am

Christmas Trilogy 2025 Part 2: Pictures of Charles

Yesterday, we took a look at some of the many portraits of Isaac Newton the second Lucasian professor of mathematics at Cambridge, today, we are turning our attention to a nineteenth century occupant of that honourable chair, Charles Babbage (1791–1871). 

Although Babbage came from a very wealthy family with a high social status there are no know childhood portraits. The earliest portraits seem to be from 1833, when he was already forty-two years old and Lucasian Professor. There is a stippled engraving made by the English engraver John Linnell (1792–1863). The son of a carver and guilder he had contact with several painters as a pupil before being admitted to the Royal Academy in 1805. He was only sixteen when left the Academy and went on to long and successful career as painter and engraver.

Self-portrait of John Linnell c. 1860
Linnell’s portrait of Babbage

There is a second stippled engraving of Babbage from 1833 as Lucasian Professor by Richard Roffe (fl. 1805–1827) about who very little is known.

Roffe’s portrait of Babbage

There is an early painted portrait of unknown date and by an unknow artist, now in the National Trust’s collection.

British (English) School; Charles Babbage (1792-1871) ; National Trust

There is a painted portrait in the National Portrait gallery from 1876 by Samuel Lawrence (1812–1884) a British portrait painter, who painted the cream of the mid Victorian society, ncluding the polymath William Whewell, a student friend of Babbage’s.

Samuel Lawrence attributed to Sir Anthony Coningham Sterling, salt print, late 1840s
Samuel Lawrence portrait of Babbage

There is lithographic portrait from 1841 now in the Wellcome Collection by D. Castellini after the pencil drawing Carlo Ernesto Liverati (1805–1844). I can find nothing on either Liverati or Castellini.

L0020480 Charles Babbage
Credit: Wellcome Library, London. Wellcome Images
Portrait bust of Charles Babbage with facsimile
Lithograph By: D. Castellini after: Liverati, C.E.Published: –

Babbage was a man of his times and a major technology fan so we naturally have quite a lot of photographic portraits. There is a daguerreotype from around 1850 made by the French photographer and artist Antoine François Jean Claudet (1797–1867).

Antoine Claudet in 1850
Claudet’s daguerreotype of Babbage

Claudet was active in the Victorian scientific community and was working with Charles Babbage on photographic experiments around the time this compelling portrait of him was made. In it, the pattern of embellished fabric on the side table is picked up in Babbage’s waistcoat. (National Portrait Gallery).

Claudet also took one of the only two surviving photographs of Ada Lovelace in c. 1843 or 1850

Claudet’s daguerreotype of Ada Lovelace

There is a seated photographic portrait of Babbage:

Half-length portrait of Babbage, seated, body turned to the left as viewed, Babbage looking to camera. The image is embossed “J M MACKIE PHOTO”. The reverse has two inscriptions. Top, in ink: “For my dear Aunt Fanny from her affectionate nephew B Herschel Babbage”. [Benjamin Herschel Babbage (1815-1878)]. Below “Copied from a negative taken for the Statistical Society about 1864. Charles Babbage was elected a Fellow of the Royal Society in 1816. (Royal Society)

There is another undated seated photographic portrait of an elder Babbage with the caption,” Charles Babbage (1792-1871). English mathematician and mechanical genius.”

The Illustrated London News published an obituary portrait of Babbage

Obituary portrait of Charles Babbage (1791-1871). The caption is The late Mr Babbage. Illustration for The Illustrated London News, 4 November 1871. This portrait was derived from a photograph of Babbage taken at the Fourth International Statistical Congress which took place in London in July 1860. (Science Museum)

Most of the images shown here were used multiple times in writings about Babbage-.

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October 1, 2025 · 6:11 am

From τὰ φυσικά (ta physika) to physics – LIII

When Isaac Beeckman (1588–1637) was introducing the young René Descartes to mathematical physics in Breda in 1618, he gave him exercises to help him develop his skill in this new discipline. One of those exercises was to derive the hydrostatic paradox, first published by Simon Stevin (1548–1620)  in his De Beghinselen des Waterwichts (Principles on the weight of water) in 1586. Beeckman had been a student of Stevin’s during his time at Leiden University. The discovery of the hydrostatic paradox is often falsely attribute to Blaise Pascal (1623–1662), who did extensive work in the field of hydrostatics and it is to Pascal and this work that we now turn.

Blaise Pascal artist unknown Source: Wikimedia Commons

Pascal led a fascinating life dabbling in many scientific and mathematical fields before turning to religion and philosophy. He was born in Clemont-Ferrand in the Auvergne  Region, the son of the amateur mathematician, jurist, and chief French tax officer, Étienne Pascal (1588–1651) and his wife Antoinette Begon, who died three years after Blaise’s birth in 1626. He had two sister who survived into adulthood, Gilberte (1620–1687) and Jacqueline (1625–1661).

Pascal’s place of birth in Clemont-Ferrand Course Wikimedia Commons

Following the death of his wife, Étienne moved the family to Paris in 1631. Blaise received his education from his father, who initially forbade him from learning mathematics before the age of fifteen. However, Blaise began teaching himself geometry at the age of twelve so, his father gave him a copy of Euclid.

At the age of fourteen, Étienne Pascal began to take his son to the weekly meetings of the Academia Parisiensis in the cell of Marin Mersenne (1588–1648). 

Marin Mersenne Source: Wikimedia Commons

Blaise developed a strong interest in the work of Girard Desargues (1591–1661) on conic sections. At the age of only sixteen, he presented Mersenne with a single sheet of paper  containing several theorem of projective geometry, his Essai pour les conics (Essay on Conics), which included the so-called mystical hexagram. Pascal’s theorem, as it is now called, states that if six arbitrary  points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon, extended if necessary, meet at three points that lie on a straight line, called the Pascal line. 

Pascal line GHK of self-crossing hexagon ABCDEF inscribed in ellipse. Opposite sides of hexagon have the same colour. Source: Wikimedia Commons

Descartes reacted very negatively to the young man’s achievement telling Mersenne that it must be the work of the father. On being assured by Mersenne that it was indeed the work of the son Descartes reaction was dismissive:

“I do not find it strange that he has offered demonstrations about conics more appropriate than those of the ancients,” adding, “but other matters related to this subject can be proposed that would scarcely occur to a 16-year-old child.”

This presaged the future relationship between the two mathematicians that was almost never cordial.

Blaise Pascal would go on to make several important contributions to the development of mathematics. He made contributions to the theory of conics in his The Generation of Conic Sections written over a number of years but never finished. However, Leibniz and Tschirnhaus made notes from it and it is through these notes that a fairly complete picture of the work is now possible. In correspondence with Pierre de Fermat (1601–1665) he laid the foundation of the theory of probability. 

Perhaps most well know from school mathematics lessons is his Traité du triangle arithmétique, written in 1654 but published posthumously in 1665.

Source

One should point out that Pascal’s triangle was already known to Persian, Indian and Chinese mathematicians in the Middle Ages and in Europe to Jordanus de Nemore in the thirteenth century, Levi ben Gershon (1288–1344) in the fourteenth century and to, Peter Apian (1595–1552), who published on the front cover of his Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen, (Ingolstadt, 1527),  

as well as Tartaglia (1500–1577) and Girolamo Cardano (1501–1576) in the sixteenth century. However, Pascal’s publication still had a major impact in the seventeenth century. 

Pascal is also famous for having invented an early mechanical calculator, the Pascaline in 1642, to assist his father with his tax records. Contrary to popular opinion, the Pascaline was neither reliable nor a success and the project was abandoned within a few years.

A Pascaline Source: Wikimedia Commons

Following the involvement of first his father and then his younger sister, Jacqueline, with members of the Jansenist movement, a fringe Catholic movement based on the teachings of the Dutch bishop, Cornelius Jansen, Pascal had a religious revelation on 23 November 1654 and basically gave up physics and mathematics, with one brief exception, and devoted his life to religion and philosophy. He published his Lettres provinciales in 1656 and his Pensées was published posthumously. Both are regarded as masterpieces of French literature.

Portrait of Pascal after his religious retreat from the world artist unknown Source: Wikimedia Commons

During the final religious phase of his life, the one exception to his abandonment of mathematics was his work on the cycloid in 1658. Plagued by toothache he began contemplating several problems concerning the cycloid. His toothache abated and he took this as a signal from God to carry on with this research. Having finished his essay on the topic, he proposed a contest. Pascal posed three questions relating to the centre of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanishdoubloons. The story of the contest is widespread and you can read it on Wikipedia, MacTutor, and probably a dozen different places on the Internet.

Following this all too brief sketch of Pascal’s life and work, we now turn our attention to Pascal’s work on hydrostatic. Pascal’s experiments with the Torricellian tube did not take place in a vacuum, pun intended, but was part of a wider interest in Torricelli’s experiments that took place mostly in France in the 1640s. You can read the sequence of events that led to Torricelli’s experiments here. 

Torricelli never published his experiments but he exchanged a series of letters on the topic with Michelangelo Ricci (1619–1692), his one time colleague. In 1644, the French mathematician Guillaume du Verdus sent a partial copy of Torricelli’s letter to Ricci to Marine Mersenne in Paris. Through this letter Mersenne got some sense of Torricelli’s experiment and that Torricelli thought that the column of liquid stayed up because of the pressure of air. However, he was not clear on Torricelli’s thoughts on the production of the vacuum or on Torricelli’s intention to produce a machine to show the changes in the pressure of air. In October 1644, Mersenne travelled to Italy and in December in Florence, Torricelli demonstrated his experiment to him. From later comments it seems that Mersenne accepted the existence of the vacuum in the tube but not that it was the pressure of air that held the mercury up in the tube.

Mersenne returned to Paris in July 1645, and in the autumn, together with the French Ambassador to Sweden, Pierre Chanut (1601–1662) he tried to recreate Torricelli’s experiment but failed due to the lack of suitable glass tubes. Interestingly, later in the 1640s Chanut would carry out the experiments together with Descartes, when they were living together in Stockholm. 

The first person outside of Italy to succeed in recreating Torricelli’s experiment was the astronomer, physicist, mathematician and instrument maker Pierre Petit (1594–1677), who as a military engineer was Superintendent of Fortifications and who lived in Rouen where there were good glass works. This probably took place in October 1646 because he wrote a long letter to Chanut in Sweden describing his success. This letter was later published. Blaise Pascal was now living in Rouen, having followed his father there, who was appointed king’s commissioner of taxes in Rouen in 1639. Both father and son, Pascal witnessed Petit successful demonstration of the Torricelli tube and Blaise and Petit intensely discussed what they had observed. 

Their discussion centred not on the question of air pressure but on the possible existence of a vacuum at the top of the tube. This would become for a time the central discussion point surrounding the Torricellian tube, because of the strict Aristotelian principle of horror vacui or in English, nature abhors a vacuum, which even the strongly anti-Aristotelian Galileo adhered to. Pascal suggested, hypothetically, that the space was filled with air which had passed through the pores in the glass. Petit countered asking, why did air not continue to enter and the mercury continue to fall. Also, the success of air thermometers, (thermoscopes) shows that air does not permeate glass. If as speculated the air had entered through the mercury, why only a specific amount? This was typical of the exchanges that would take place between supporters and opponents of the presence of a vacuum in the tube.

This demonstration by Petit set Pascal on his way to his experiments with the Torricellian tube in 1646–47. There was a theory that the space at the top of the tube was filled with a ‘spirit’ rising up from the liquid. Pascal set up two tubes in a glass works in Rouen, one filled with wine and one filled with water. He asked those in attendance to predict which fluid would fall further. Because wine  is more ‘spirituous’ they all predicted that it would fall further. Of course, the water fell further because it is heavier. 

The situation was now made more complex by an announcement in July 1647 from Warsaw that the Capuchin missionary Valeriano Magni (1586–1661) had demonstrated the Torricelli experiment at the court of King Władysław IV Vasa (1595–1648).

Valeriano Magni artist unknown Source: Wikimedia Commons

Magni published his results in his Demonstratio occularis loci sine locato: corporis successive moti in vacuo luminis nulli corpori inhaerentis (Petri Elert, 1647). He believed that he was the first to carry out this experiment. In a letter from Gilles de Roberval (1602–1675) Magni was made aware that he was by no means the first, which he then acknowledge in a further publication but stating that he had carried out the experiment following his reading of Galileo’s Discorsi and was not aware of any of his predecessor work. 

Source: Wikimedia Commons

Magni’s publication, which was the first ever on the Torricellian experiment, brought the Jesuits into the discussion. Magni, a confirmed anti-Aristotelian, was already at war with the Jesuits condemning their monopoly on Catholic education, and his assertion that the space at the top of the tube was a vacuum was like a red rag to a bull for the Jesuits, who as strict Thomists rejected the possibility of a vacuum. In 1648 Niccolò Zucchi (1586–1670) published an anonymous letter attacking Magni’s experiment and the earlier one of Gaspero Berti (c. 1600–1644). Meanwhile Pascal had published his own account of his experiments Expérences Nouvelles touchant la vide in October 1647.

Source

This was attacked by Étienne Noël (1581–1659), who had been one of Descartes teachers at the college La Flèche, and was now rector of the Jesuit College de Clermont in Paris both in a letter then in his  Le Plein du Vuide, ou le corps dont le vuide apparent des expériences nouvelles est rempli trouvé par d’autres expériences, confirmé par les mêmes et démontré par raisons physiques, (Paris, 1648). And so, the debate rumbled on.[1]

Source

Pascal had returned to live in Paris in 1647 and Descartes had visited him there twice on 23 and 24 September. Unlike Pascal, Descartes was convinced that the mercury column was held up by air pressure but aggressively rejected the idea of a vacuum. Not unexpectedly, Descartes argued that the space at the top of the tube was filled with “subtle matter.” In a letter to Mersenne, Descartes would later claim that he had suggested the now legendary Puy de Dôme experiment to Pascal. Mersenne had also suggested it in his Reflexiones physico-mathematicae (beginning of October 1647).

On 19 September 1648, Florin Périer (1605–1672), Pascal’s brother-in-law, who lived in Clermont-Ferrand, set out on Pascal’s urging with two Torricellian tubes to the Puy de Dôme a 1,465 metre high lava dome, ten kilometres west of Clermont-Ferrand.

Aerial image of Puy de Dôme (view from the west) Source: Wikimedia Commons

He left one Torricellian tube at the base of the mountain under the supervision of a friend, who had instructions to measure the height of the mercury column at regular timed intervals, noting the results. Périer set off up the mountain with the other Torricellian tube, which he set up to take measurements of the mercury column at several elevations during his ascent, including at the summit. On his way down he repeated the measurements. Back at the base he compared his measurements with those of the control apparatus.

Pascal immediately published a detailed, twenty page account of it, Récit de la grande expérience de l’équilibre des liqueurs projetée par le sieur B. P. pour l’accomplissement du traicté qu’il a promis dans son abbregé touchant le vide et faite par le sieur F. P. en une des plus hautes montagnes d’Auvergne (autumn 1648), consisting principally of Perier’s letter and report.

Source

In a short introduction he presented the experiment as the direct consequence of his Experiences nouvelles, and the text of a letter of 15 November 1647 to Perier, in which he explained the goal of the experiment and the principle on which it was based.[2] Périer’s letter reads:

The weather was chancy last Saturday…[but] around five o’clock that morning…the Puy-de-Dôme was visible…so I decided to give it a try. Several important people of the city of Clermont had asked me to let them know when I would make the ascent…I was delighted to have them with me in this great work…

…at eight o’clock we met in the gardens of the Minim Fathers, which has the lowest elevation in town….First I poured 16 pounds of quicksilver…into a vessel…then took several glass tubes…each four feet long and hermetically sealed at one end and opened at the other…then placed them in the vessel [of quicksilver]…I found the quicksilver stood at 26″ and 3+1⁄2 lines above the quicksilver in the vessel…I repeated the experiment two more times while standing in the same spot…[they] produced the same result each time…

I attached one of the tubes to the vessel and marked the height of the quicksilver and…asked Father Chastin, one of the Minim Brothers…to watch if any changes should occur through the day…Taking the other tube and a portion of the quicksilver…I walked to the top of Puy-de-Dôme, about 500 fathoms higher than the monastery, where upon experiment…found that the quicksilver reached a height of only 23″ and 2 lines…I repeated the experiment five times with care…each at different points on the summit…found the same height of quicksilver…in each case… (Wikipedia)

Pascal repeated the experiment by carrying a Torricellian tube up to the top of the tower of the church at Saint-Jacques.de.la-Boucherie, which was about fifty metre high. The mercury fell about two lines. He determined from both experiments that an ascent of seven fathoms lowers the mercury by half a line.  

Tour St-Jacques the church no longer exists Source: Wikimedia Commons

Robert Boyle (1627–1691) would later become the first person to hail an experiment as an experimentum crucis, with reference to the Puy de Dôme experiment. An experimentum crucis was a concept created by Francis Bacon (1521–1626) in his Novum Organum as instantia crucis, later called experimentum crucis by Robert Hooke (1635–1703)Descartes, who saw his belief that it was air pressure, which held the column of mercury up confirmed, suggested adding a scale to the Torricellian tube, a suggestion that perhaps marks the birth of the barometer. 

At the beginning of 1649 Périer, following Pascal’s instructions, began an uninterrupted series of barometric observations designed to ascertain the possible relationship between the height of a column of mercury at a given location and the state of the atmosphere. The expérience continuelle, which was a forerunner of the use of the barometer as an instrument in weather forecasting, lasted until March 1651 and was supplemented by parallel observations made at Paris and Stockholm.[3] The series in Stockholm had been initiated by Descartes.

Pascal was working on a major treatise summarising all of his work on the Torricellian tube but abandoned this to work a shorter work instead. He seems to have finished working on this version in about 1654, which is when turned to a life of religious contemplation and philosophy. This shorter work final appeared posthumously as Traits de équilibre des liqueurs et de la pesanteur de la masse de l’air… (Paris, 1663). 

This work had far less impact than it would have had if it had been published in 1654, because by 1663 both Boyle and Otto von Guericke (1602–1686) had furthered their own extensive investigations of the vacuum.

One important result in Traits de équilibre des liqueurs is what is known as Pascal’s Law (also Pascal’s Principle or the principle of transmission of fluid pressure) an extension of the hydrostatic paradox:

A change in pressure at any point in an enclosed incompressible fluid at rest is transmitted equally and undiminished to all points in all directions throughout the fluid, and the force due to the pressure acts at right angles to the enclosing walls. (Wikipedia)

This is the principle underlying the hydraulic press, a device that makes the multiplication of forces possible.

Working principle of a hydraulic jack Source: Wikimedia Commons

Pascal sat at the centre of a lively and at time acrimonious debate as to whether a vacuum could exist. He contributed much through his efforts to  show that a vacuum could and, in fact did, exists in the space at the top of a Torricellian tube. He also, through the Puy de Dôme experiment settled the argument as to whether or not it was atmospheric pressure that held the mercury up in the tube. It was, which meant that air was not some nebulous Aristotelian philosophical concept but a real substance with weight. At the same time the experiment showed that that atmospheric pressure decreased with altitude i.e., the higher you go the less air is pressing down on you. The experiments that Pascal instigated led to the creation of the barometer both as an instrument to determine altitude and one to predict weather. Although, it is probably false to attribute the invention of the barometer to any one single individual. Pascal’s achievements were recognised by naming the SI unit of pressure after him. 

[1] For a detailed discussion of the Jesuits involvement in the debates on the Torricellian tube and the possibility of a vacuum see Michael John Gorman, The Scientific Counter Revolution: The Jesuits and the Invention of Modern Science, Chapter 4, The Jesuits and the Vacuum Debate, pp. 125–166

[2] René Taton, Blaise Pascal, DSB

[3] René Taton, Blaise Pascal, DSB

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Filed under History of Mathematics, History of Physics, History of Technology, Uncategorized

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 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.

Source

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.

Source

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.

Source: Wikimedia Commons

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

Source

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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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:

  1. That rectilinear rays proceeding from the eye diverge indefinitely;
  2. 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;
  3. That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;
  4. That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
  5. That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
  6. 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;
  7. 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. 

Source: Wikimedia Commons

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. 

Source: Wikimedia Commons

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.”

Source: Wikimedia Commons

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.

Source: Wikimedia Commons

[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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January 1, 2025 · 9:39 am

On the 1st of Undecember

The Renaissance Mathematicus and the HISTSCI_HULK are taking a break this week

Normal service will be resumed next week.

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November 6, 2024 · 8:59 am

From τὰ φυσικά (ta physika) to physics – XXXIII

Having dealt with the advances in optics, dynamics, statics, and hydrostatics made during the sixteenth century, in this episode we turn to the advances in optics at the beginning of the seventeenth century, which means Johannes Kepler (1571–1630)

August Köhler portrait of Johannes Kepler Source: Wikimedia Commons

and his Ad Vitellionem Paralipomena, Quibus Astronomiae Pars Optica Traditur (Supplement to Witelo, in Which Is Expounded the Optical Part of Astronomy) published in 1604, a book that almost singled handedly redefined the science of optics. 

Kepler’s ‘Astronomiae Pars Optica’ (1604), title page. This work, by German astronomer, mathematician and astrologer Johannes Kepler (1571-1630), is considered the founding text in the science of optics. Kepler wrote about the inverse-square law for light intensity, reflection, pinhole cameras, parallax, the apparent sizes of astronomical bodies, and the human eye. The Latin text includes the title (top), a description of the book, the author’s name, and publication details (bottom) below the printer’s mark. This work was published in Frankfurt by Claudium Marnium and Haeredes Joannis Aubrii, under the patronage of the Holy Roman Emperor Rudolf II.

Although he had almost certainly come across optics as a student of Michael Mästlin (1550–1631) in Tübingen, Kepler’s interest in optics was first truly awakened when he moved to Prague in 1600 to work with Tycho Brahe (1546–1601). In his astronomical work Tycho had discussed two optical themes that would interest Kepler, the first of which was atmospheric refraction. Refraction is the bending of light rays when they pass from one medium to another, a phenomenon that makes lenses function. Because light travelling from the stars, the Moon, or the Sun moves from the vacuum of outer space into the atmosphere when it approaches the Earth it undergoes refraction. The phenomenon was already known in antiquity although the correct explanation wasn’t.

The existence of refraction in water had been known to humanity since the first human had tried to catch fish is a stream, pond, or river. The first written mention of atmospheric refraction was made by Pliny the Elder (23-79 CE) in his vast Naturalis Historia, where quoting Hipparkhos (c. 190–c. 120 BCE) he describes a luna eclipse during which the Sun and the Moon are both seen above the horizon at the same time:

. . . he also discovered for what exact reason, although the shadow causing the eclipse must from sun- rise onward be below the earth, it happened once in the past that the moon was eclipsed in the west while both luminaries were visible above the earth.[1]

Pliny does not give Hipparkhos’ explanation and the majority of his writings are lost. Most of what we know about Hipparkhos is to be found in Ptolemaios’ Mathēmatikē Syntaxis and although he, as we will see, wrote much on atmospheric refraction he doesn’t mention this.

Cleomedes, whose dates are very uncertain but probably the first century CE, wrote about a class of paradoxical eclipses in Chap. 6 Book II of his On the Circular Motions of the Celestial Bodies. During a lunar eclipse the Moon should, of course, be in the Earth’s shadow and not visible in the heavens. Cleomedes, well aware of refraction in water and glass, offers an explanation based on atmospheric refraction. Cleomedes also partially explained the Moon illusion, known to and commented on by Aristotle, that the Moon appears much larger near the horizon, using atmospheric refraction.

Cleomedes’ model for the paradoxical eclipse.

Ptolemaios, naturally had a lot to say about refraction in general and atmospheric refraction in particular. Interestingly there is no discussion of atmospheric refraction in his astronomical tome, Mathēmatikē Syntaxis but in his later Optics he remedies this omission. In Book V of the optics, he writes extensively about refraction in water, glass, and air producing tables of his measurements of the phenomenon. It is obvious that he was trying to find the law of refraction but failed. Immediately following, in paragraphs 23 to 30 of Book V, he then discusses atmospheric refraction. His attention was drawn to atmospheric refraction by actual observations. Firstly:

For stars that rise and set, he observed that they do so farther north than expected: As they approach the horizon, for example, they hesitate to set, and slide a small distance northward along the horizon before disappearing. 

To create a model to explain this Ptolemaios, based on his experience of refraction in a transparent medium, correctly deduced that the refraction was taking place in the air that surrounds the Earth. Unlike his work on refraction in a transparent medium, Ptolemaios gives no measurement for atmospheric refraction, although it should have been possible for him to measure it at the horizon. 

Ptolemy’s model for refraction in air.

Secondly, Ptolemaios attempted to explain the Moon illusion using atmospheric refraction in an explanation that seems to be a copy of Cleomedes’ explanation. In another section of his Optics, he gives a completely different explanation of the Moon illusion based on the psychology of vision that makes no mention of atmospheric refraction. 

In general Ptolemaios’ understanding and explanation of refraction in general and atmospheric refraction in particular was so good that it became the accepted standard model for the next fifteen hundred years. 

Of course, Islamic scholars took up the topic of refraction, when they in turn began to study optics. Famously, Ibn Sahl (c. 940–1000) in his Fī al-‘āla al-muḥriqa (On the burning instruments), the first work on optics to study lenses, was the first to discover the law of refraction, which had eluded Ptolemaios, but it was in a geometrical form and not the sine law as it is usually presented today. However, his work found no resonance and disappeared taking with his ground breaking discovery. 

The most important Islamic scholar of optics, Ibn al-Haytham (c. 965–c. 1040) devoted Chapter VII of his Kitab al-Manazir (Book of Optics), translated into Latin in the twelfth century as De aspectibus, to the study of refraction. On atmospheric refraction Ibn al-Haytham follows Ptolemaios almost exactly. Thanks to the Perspectiva communis of John Peckham (c. 1230–1292) and the De Perspectiva of Witelo (c. 1230–after 1280) both of which are based on De aspectibus, Ibn al-Haytham’s work in optics was the most well known in medieval Europe.

Page from a manuscript of Perspectiva with a miniature of the author Source: Wikimedia Commons
Witelo’s model for the Moon illusion.

Ibn Mu’adh (989–1079) in his Liber De Crepusculis, (On twilight), a work that was for a long time falsely attributed to Ibn al-Haytham and included by Friedrich Risner (c. 1533–1580) in his Opticae thesaurus, attempted to calculate the depth of the Ptolemaic atmosphere:

Mu’adh recognized that the twilight following a sun- set must be caused by illuminated matter high in the sky. This matter must be vapours carried in the highest levels of the atmosphere (the air itself being in- visible because it is entirely transparent). Even if the Sun has long set, the evening skies in the direction of the Sun are lighter than those in opposite direction. The difference ceases to be perceptible only when the Sun has sunk nearly 20 degrees below the horizon. Ibn Mu’adh took the value of 19 degrees below the horizon as the lowest solar elevation for which twilight is still visible, i.e., the lowest elevation for which the Sun’s rays meet the last upper vestiges of vapor-laden air. 

Following his reasoning, let the observer be at A, catching the last glimpse of twilight on the horizon (Fig. 7). Then his line of sight AB must be drawn tangent to the Earth, and at the point B, there should be matter that still is illuminated by the Sun. When the Sun is 19 degrees below the horizon, then the angle AOC must also be 19 degrees, and consequently the angle AOB will be 9.5 degrees. It is then easy to establish that, in modern terms, B should be 88.6 km above the Earth’s surface. Ibn Mu’adh expressed his result in terms of Italian miles, of which 24,000 make up the circumference of the Earth.52 His value of 51.79 Italian miles translates to 86.3 km, which agrees very well with the modern calculation. 

In his De Perspectiva, Witelo presented al-Haytham’s Book VII, but in a more logical order as well as Ptolemaios’ refraction tables for air, water, and glass. Making both widely available in medieval Europe. 

Tycho Brahe (1546–1601) took the decisive step in the history of atmospheric refraction when he became to first investigator to actually measure it:

His model was still the Ptolemaic one, where refraction occurred at the sharp boundary between air and ether. His technique was to use the Sun, whose position was accurately predictable, to determine the refraction. He recorded the differences between his measured and predicted positions for a complete set of solar altitudes to build his tables. He published his findings in 1596 in his Epistolae Astronomicae printed by himself. In his calculations he accepted Ptolemaios’ value for the Sun’s parallax of 3´, which is false. 

He produced separate tables for the refraction of star light because he thought they were much further away than the sun and he needed to take parallax into consideration. 

Tycho’s atmospheric refraction values for the Sun and the fixed stars

When Kepler went to work with Tycho in Prague in 1600 he gained access to Tycho’s studies on atmospheric refraction and this became one on the two reasons why he turned his attention to optics devoting much of his time following the death of Tycho in 1601 to the study of the subject and writing his Astronomiae Pars Optica, hence this part of the title, although the book would eventually contain much more. 

On atmospheric refraction Kepler followed Tycho publishing his refraction tables and adopting his false value for solar parallax. He was the first to note that refraction caused a distortion in the shape of celestial bodies, the setting Sun being elliptical and not round. Kepler used the Ptolemaic model of the atmospheric structure, and assuming, incorrectly, uniform properties of air up to the ether boundary, he reasoned on the basis of refraction that the atmosphere was at most half a German mile thick–approx. 3.7 km. 

Kepler was the first to recognise another atmospheric refraction phenomenon, the mirage and the first to attempt an explanation. In his explanation, too complex to go into here but it is basically a combination of reflection and refraction in thermal layers of the upper atmosphere, he quoted Cleomedes’ work on reflection, showing that he had also read this work. Kepler also showed that the Moon illusion was purely a problem of perception and had nothing to do with optics.

Diagram to the Novaya Zemlya effect, Kepler’s explanation of a mirage

The second trigger for Kepler’s sudden major interest in optics found in Tycho’s work concerns the pinhole camara problem. Observing a partial solar eclipse in 1600 using a pinhole camera, as directed by Tycho, on the market place in Graz, Kepler noted like Tycho that the results showed a false apparent size for the image. This so-called pinhole camara problem had been known since the Middle Ages and many explanations had been attempted by earlier scholars, Roger Bacon (c. 1219–c. 1292) for example. 

Kepler solved the problem, in the process coining the name camera obscura for the pinhole camera, by utilising a trick that he had picked up from Albrecht Dürer’s Underweysung der Messung mit dem Zirckel und Richtscheyt (Instructions for Measuring with Compass and Ruler) (1525). In one of the illustration explaining how to do linear perspective, Dürer shows the artist using a string to simulate the light ray from the object to be drawn onto the drawing.

Source: Wikimedia

Kepler copied Dürer. He drilled a hole in his desk to simulate the camera obscura and suspended a book in the air above the hole. He then stretched strings from the book trough the hole to simulate the light rays and showed that, because of the width of the hole, their wasn’t a single image but a series of overlapping images that distorted the size to the perceived image. 

Astronomical optical themes that Kepler also discussed in his Pars Optica included parallax, his eclipse instruments, and the annual variations in the apparent size of the Sun. Since this variation is dependent on the distance of the Sun from the Earth it would eventually be explained by Kepler’s heliocentric system with its elliptical orbits but in 1604 Kepler hadn’t yet reached that break through theory. However, like Ptolemaios and almost all others before him, Kepler failed to discover the law of refraction.

In proposition 9 of Book 1 in his Pars Optica is one of Kepler’s oft overlooked major contributions to the history of physics the inverse square law of the spreading of light from a point source:

just as [the ratio of] spherical surfaces, for which the source of light is the centre, [is] from the wider to the narrower, so is the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely. For according to 6 & 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there. If, however, there was a difference in the density of the ray or a difference in distribution according to position relative to the centre (which is negated by Prop. 7) it would be a different issue.” This is Proposition 9 of the first book of Kepler’s 1604 Ad Vitellionem para– lipomena, quibus astronomice pars optica traditur. The claim is clear: light expands spherically, with its source as the centre. The surface of a sphere is proportional to the square of its radius, and the amount of light is constant. Thus the *fortitudo seu densitas” of light — its impact on each point of the surface it illuminates — is inversely proportional to the distance of this surface from the source.[2]

S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with increasing distance, where a greater density of flux lines (lines per unit area) means a stronger energy field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area of a sphere increases with the square of the radius. Thus the field intensity is inversely proportional to the square of the distance from the source. Source: Wikimedia Commons

The book contains a major contribution to the evolution of mathematics:

With respect to the beginnings of projective geometry, Kepler introduced the idea of continuous change of a mathematical entity in this work. He argued that if a focus of a conic section were allowed to move along the line joining the foci, the geometric form would morph or degenerate, one into another. In this way, an ellipse becomes a parabola when a focus moves toward infinity, and when two foci of an ellipse merge into one another, a circle is formed. As the foci of a hyperbola merge into one another, the hyperbola becomes a pair of straight lines. He also assumed that if a straight line is extended to infinity it will meet itself at a single point at infinity, thus having the properties of a large circle. (Wikipedia quoting Morris Kline, Mathematical Thought from Ancient to Modern Times, OUP, 1972, p. 299) 

When he started his optical investigations Kepler was a relative novice in the discipline so he acquired and studied  Friedrich Risner’s Opticae thesaurus, which contained his annotated editions of both Ibn al-Haytham’s De aspectibus and Witelo’s De Perspectiva this meant that Kepler was indoctrinated by the two most important perspectivist texts.

Source:

His intensive study of Witelo’s work is the reason for the Ad Vitellionem Paralipomena part of Kepler’s title. The supplement, appendix, additions or however one translates Paralipomena suggests a sort additional text but Kepler’s work is a large format tome running to almost five hundred pages. Having dealt with the astronomical part of optics. Kepler turned to the theory of vision because he considered vision to be an important aspect of the study of astronomy, a rational step, as after all astronomy is fundamentally a visual discipline. Kepler then preceded to dismantle the perspectivist theory of vision. 

Kepler accepted the punctiform theory of light radiation first presented by al-Kindi (c. 801–873) in his De radiis stellarum:

It is manifest that everything in this world, whether it be substance or accident, produces rays in its own manner like a star … Everything that has actual existence in the world of the elements emits rays in every direction, which fill the whole world[3]

However, he rejected the perspectivist solution, first presented by al-Haytham, of how the eye coped with this abundance of light rays during visual perception. Al-Haytham argued that the surface of the cornea is a perfect sphere. He then hypothesised that only those rays that meet the surface of the eye perpendicularly can actually enter the eye. All the other light rays slide or veer off.  Kepler argued that that it would be impossible for the eye to distinguish between a perpendicular ray and one meeting the cornea at 89° and said that all the light rays reaching the eye must take part in visual perception. This meant that Kepler must produce a new theory of vision, which he then preceded to do. 

Kepler starts with the anatomy of the eye. At that point the dominate theory of the anatomy of the eye was that of Andreas Vesalius (1514-1549) from his De Humani Corporis Fabrica Libri Septem (On the Fabric of the Human Body in Seven Books) published in 1543, Kepler turned instead to the Anatomia Pragensis (1601) from his friend Johannes Jessenius (1566–1621)

Source: Wikimedia Commons

and in particular the De corporis humani structura et usu. (1583) of Felix Platter (1536–1614).

Hans Bock portrait of Felix Platter Source: Wikimedia Commons
De corporis humani structura et usu. Source: Wikimedia Commons

Johannes Jessenius had functioned as intermediary between Kepler and Tycho when they had a major falling out when Kepler first arrived in Prague to seek employment and they had remained good friends since. How or why Kepler turned to the work of Platter is not known but it is possible that Jessenius drew his attention to it. 

Vesalius had, like Galen, placed the crystalline humour, or what you and I would call the lens, at the centre of the eye ball, Jessenius and Platter had moved it more to the front. 

Left Vesalius’ image of the eye Right Platter’s image of the eye

However, it is with Platter’s analysis of the retina that the biggest change comes: 

The principal organ of vision, namely the optic nerve, expands through the whole hemisphere of the retina as soon as it enters the eye. This receives and discriminates the form and colour of external objects which together with the light enter the eye through the opening of the pupil and are projected on it by the lens.

All previous theories of vision had placed the act of vision within the crystalline humour, Kepler now moved it to the retina, arguing that the crystalline humour was simply a lens that projected the image created by all of the light rays, not just the perpendicular one, onto the retina in the manner of a camera obscura. Kepler took the analogy of the eye to the camera obscura from the Magia Naturalis (Natural Magic) (first published as four books in 1558 and expanded to twenty book by 1589) of Giambattista della Porta (1535–1615). We now know that Kepler’s theory of vision is basically correct but at the time he was painfully aware that the retinal image was inverted, whereas we perceive the world the right way up. Kepler, correctly surmised that the brain somehow sorts it out. 

Having downgraded the crystalline humour from the seat of visual perception to being merely a lens focussing the image onto the retina, Kepler was now able to publish the first scientific explanation of how eye glasses work since they first began to be used in the late thirteenth century. The lenses in eye glasses simply correct the focal length of crystalline humour when this is in some way deficient. As noted in a previous episode Francesco Maurolico (1494–1575)  had already done this in his Photismi de lumine et umbra and Diaphana. However, Maurolico’s account was first published posthumously in 1611, so Kepler got the credit for this ground breaking discovery. 

Kepler’s Ad Vitellionem Paralipomena, Quibus Astronomiae Pars Optica Traditur’ is renowned for presenting the first correct account of visual perception but it also laid the foundations of moving the discipline of optics away from the study of visual perception towards becoming a  branch of physics concerned with the study of the behaviour and properties of light. This he developed further in his Dioptrice (1611). Although revolutionary, the acceptance of Kepler’s theories was relatively slow but later in the seventeenth century René Descartes (1596–1650) and Robert Hooke (1635–1703), who both made significant contributions to the development of optics, would name Kepler as the leading authority on the topic. 

[1] The sketch of the history of atmospheric refraction presented here is distilled from Waldemar H. Lehn and Siebren van der Werf, Atmospheric refraction: a history, Applied Optics, Vol. 44, No. 27, 2005

[2]  Gal, O. & Chen-Morris, R., The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices, History of Science, vol. 43, p.391-414, p. 397

[3] David C. Lindberg, Theories of Vision: From Al-Kindi to Kepler, University of Chicago Press, 1976, p. 19

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October 30, 2024 · 9:11 am

Picturing the skies

Astronomy is fundamentally at its core a visual scientific discipline. It originated when humans looked up into the night sky and tried to make rhyme or reason out of the twinkling lights visible on a clear night. Lights that when you watched them long enough rotated across the sky. With time those observers noticed that some of those lights wandered across the sky in the opposite direction over periods of many, many days, each at a different speed, stopping occasion to reverse their direction before then continuing on their way in the original direction. The lights that rotated on mass in the one direction became what we now call stars and the errant individuals wandering stars, or planets. 

In an attempt to apply some sort of order to this seemingly uncountable panorama of twinkling lights, almost all ancient cultures began to create pictures by joining up groups or clusters of them with lines, like small children in our age with their join up the dots picture books. They personified these joined up cluster and gave then names filling in the skies around them with pictures that reflected those names. There was a big bear and a little bear, in one culture, an emu in another, and a llama in yet another. There were many, many more dot and line figures mapping the heavenly vaults and giving them structure. 

Some how believing that the nightly spectacle above their heads somehow influenced or even controlled their own lives down below, many cultures turned those errant wanderers into gods. The beginnings of astrology, a significant driving force in the development of those astronomical investigations. As I have explained elsewhere, astrology and astronomy were not separate disciplines but Siamese twins for much of human existence feeding off each other as they developed.

Of course, as well as the twinkling lights and the wanderers the two most obvious things in the sky were the two major luminaries, the Sun providing light during the day and the Moon providing light at night. Given their special status the path of the Moon and the apparent path of the Sun was mapped with a sequence of special joined-upped-clusters, originally seventeen plotting the course of the Moon but later reduced to twelve plotting the path of the Sun, giving the world the zodiac that spread out from its Babylonian-Greek origin to eventually become part of the picture of the sky for cultures from the Atlantic to the Pacific and from the northern polar circle to the equator and beyond heading south. Although their names were retained those twelve dot and line figures of the twinkling lights were replaced with simple thirty degree segment of the ecliptic, the Sun’s apparent path around the Earth. 

The joined up clusters, the wanderers, the Sun and the Moon, the divisions of the zodiac and even the heaven itself were all pictorially manifested in a myriad of different ways. These pictorial manifestations were reproduced in many different ways, in many different contexts, as paintings, as drawings, as three dimensional figures, as reliefs, on single sheets, in books, as title pages or frontispieces, as decoration on buildings, as decorations on a wide range of ritual and domestic objects and in combinations of all the above. The pictorially manifested cosmos­-astronomy-astrology became part of human existence on Earth in almost every conceivable way.

Although this preoccupation with representing the heavens and all that it contains, or was deemed to contain, is dealt with in art history, in cultural history, and political history it has been very seldom handled in the history of astronomy.  I have a modest but fairly extensive collection of books on the history of astronomy, and  have consulted many more via libraries, and whilst they deal extensively with the biographies of the cosmologists, astronomers, and astrologers, who created these disciplines, the attempts of these researchers to produce mathematical models to predict the movement of the celestial bodies, or the instruments they designed and constructed to make their observations,  very little attention is paid to the pictorial manifestations I have outlined above. Occasionally, one or other is reproduced as a nice illustration to break up a monotonous text, often in the form of a book frontispiece from a tome of the astronomer under discussion. The major exception is the literature on star maps, which are by definition themselves pictorial manifestations of the heavens. 

Now a book has been published that does much to redress the balance, Imagining the Heavens Across Eurasia: From Antiquity to Early Modernity,[1] a collection of twenty-three essays that deliver the broad spectrum that the title promises. This book is the product of the research group Visualisation and Material Cultures of the Heavens at the Max Planck Institute for the History of Science in Berlin, in which both Rana Brentjes and Stamatina Mastorakou play a central role. The back cover blurb delivers a succinct and accurate descripts of the contents on offer:

In Imagining the Heavens Across Eurasia, 20 authors [Dieter Blume 3 essays, Antonio Panaino 2 essay: 20 authors = 23 essays] tell in novel ways the histories of astral knowledge through objects and their imagery. These objects include, for instance, caves and buildings, manuscripts and prints, textiles and metal dishes, instruments and paintings, monuments, and sculptures. Each chapter focuses on specific items, analysing their pictorial content and situating them in the contexts of their production and usage. As its main issue, the book addresses the knowledge inscribed in these images and their material carriers. Particular attention is paid to the interconnection between images, materials, themes, and objects across space and time. This approach enables the authors to highlight the numerous cross-cultural relations between the objects, interlinking their chapters with each other. Thus, this book offers a richly illustrated kaleidoscope of astral knowledge in Eurasia to experts and lay people alike.

The introductory material is very detailed, following the contents pages listing the essays and their authors, there is a list of the not particularly extensive figures and drawings. However, this is followed by a thirteen page list, giving full details of the 141 colour plates, the volume’s main material evidence. Whilst the authors all appear to be experts in their respective fields, I did random internet checks on some of those I didn’t know already, there is no short cv list of the authors, which I think is a deficit and really ought to be included in such a volume. The short introduction gives a sweeping account of the contents of the volume.

Beginning in Mesopotamia and Ancient Egypt the volume follows a roughly chronological arc up to the Early Modern Period, veering geographical throughout Eurasia. For the scope that they cover the essays are relative short, varying between ten and twenty pages and are so, in general, densely packed with information.  This means that almost all of the essays have, for their length, very extensive bibliographies for those who wish to follow the topics discussed further. Each essay has footnotes rather than endnotes, which, in my opinion, is a plus point. Given that the book is about pictorially manifestations of celestial objects the extensive collection of colour plates builds an important aspect of the book. Regrettably, these are bound together in a block in the middle of the book and not in the essays to which they respectively refer, I assume to save production costs. Unfortunately, the paperback volume, that I have is so tightly bound that one need both hands to hold the book open so that one can view the plates properly, making jumping back and forth between essay and plates a vexatious process.

The analysis in the individual essays takes place on two levels. Firstly, there is a purely descriptive analysis of the images under discussion, which also describes the context in which it is found. Secondly, there is an iconographical analysis that attempts to interpret the meaning that the image or object was intended to project within the historical, geographical, cultural, and religious contexts in which it was found. Of course, there is no strict division between the two sorts of analysis, which perforce blend into each other. The level of each form of analysis varies from essay to essay, with some heavily concentrating on the descriptive and less on the iconographical and vice versa with others. The level of speculation  also varies from essay to essay, some essays based solidly on factual interpretations, whilst other drift off in to the highly speculative. 

The book closes with an index of names and an extensive subject index, both of which  cover the entire volume enabling a comparison of themes that are touched upon in more than one essay. 

This is a well presented and wide-ranging volume on a subject that has received far too little attention in the study of the astral sciences. As such it is an important addition to the academic literature and should interest a wide range of readers. 

[1] Imagining the Heavens Across Eurasia: From Antiquity to Early Modernity, edited by Rana Brentjes, Sonja Brentjes, and Stamatina Mastorakou, Mimesis International, 2024. 

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