From τὰ φυσικά (ta physika) to physics – XXIX

In the most recent episodes, we have been looking at developments in mechanics during the sixteenth century. Today we are going to turn to a different branch of what would become physics and cast a light on the development of optics in the sixteenth century.

We left optics in the high middle ages with the introduction of the geometric optics conform, intromission theory of Ibn al-Haytham (c. 965–c. 1040) by Roger Bacon (c. 1219–c. 1292) and the text books based of this approach to optics produced by John Peckham (c. 1230–1292) and Witelo (c.1230– between 1275 and 1314). This direction in optics came to be known as the science of perspectiva.

At the end I briefly introduced the discovery of linear perspective at the beginning of the fifteenth century by Lorenzo Ghiberti (c. 1378–1455), Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–1470). I pointed out that the claims that this was dependent of knowledge of al-Haytham’s optical theories doesn’t hold water because linear perspective is constructed entirely with Euclidean optics/geometry and as Alberti pointed out in linear perspective it is irrelevant with you hold an extramission or intromission theory of vision.

Throughout the fifteenth century artists continued to develop the use of linear perspective in their art culminating in the work of Leonardo da Vinci (1452–1519) and Albrecht Dürer (1471–1528) going into the sixteenth century. The latter introducing the techniques of linear perspective into Northern Europe. 

Linear perspective is of course applied optics and there were in the period between the thirteenth century, with Bacon, Peckham, and Witelo, and the sixteenth century no real advances in the scientific theories of optics. At the beginning of the seventeenth century Johannes Kepler (1571–1630) would revolutionise the theory of optics but before he did so four others made significant contributions to the discipline during the sixteenth century and it is to these four that we now turn.

The first of our four is the Sicilian mathematician and astronomer Francesco Maurolico (1494–1574).

Engraved portrait of F. Maurolico by Bovis after Caravaggio
Credit: Wellcome Library, London. Wellcome Images via Wikimedia Commons

He was born in Messina, one of the seven sons of the Antonio Marulì, a Greek physician from Constantinople, who had fled from the Ottoman invaders. His father was a pupil of Constantine Lascaris (1434–1501) the Greek scholar and grammarian, a promoter of the revival of Greek learning in Italy during the Renaissance. Francesco also received a Lascarian education through his father and Francesco Faraone and Giacomo Genovese two other pupils of Lascaris. In 1534, Francesco Marulì changed his surname to Mauro Lyco (with the meaning of occult wolf). 

Ordained a priest in 1521 and the recipient of various ecclesiastical benefits, he was appointed abbot of the monastery of Santa Maria del Parto (in Castelbuono) in 1550 by Simone Ventimiglia (1485–1544) marquis of Geraci, pupil and patron of Maurolico. 

Given his Lascarian education, it is no surprise that in the mathematical sciences for his improved manuscripts of the mathematical texts of Theodosius of Bithynia, Menelaus of Alexandria, Autolycus of Pitane, Euclid, Apollonius of Perga and Archimedes, which played a significant role in the Renaissance of Ancient Greek mathematics.

His own mathematical output included in his Compaginationes solidorum regularium (1537), an unpublished statement of Euler’s formula V – E + F = 2 for the Platonic solids long before Euler’s more general formulation for convex polyhedra in 1752. His Arithmeticorum libri duo (1575) includes the first known proof by mathematical induction. His De momentis aequalibus (completed in 1548, but first published only in 1685) attempted to calculate the barycentre of various bodies (pyramid, paraboloid, etc.).

His De Sphaera Liber Unus (1575) contains a fierce attack against Copernicus’ heliocentrism, in which Maurolico writes that Copernicus “deserved a whip or a scourge rather than a refutation”. In his Cosmographiahe described a methodology for measuring the earth, which was later employed by Jean Picard in measuring the length of a degree of longitude in 1670.

However, Maurolico was not just a mathematician but a true polymath. He was commission by the Senate of Messina in 1553 and paid 100 gold pieces a year for two years to write a history of Sicily, his Sicanicarum rerum compendium, which also contains some autobiographical details. In other activities for the city of Messina, Maurolico also served as head of the mint and was responsible (with the architect Antonio Ferramolino da Bergamo) for maintaining the fortifications of the city.

Added to all this Maurolico published an edition of Aristotle’s Mechanics, and a work on music. He summarized Ortelius’s Theatrum orbis terrarum and also wrote Grammatica rudimenta (1528) and De lineis horariis. He made a map of Sicily, which was published in 1575.

Leaving this impressive display of intellectual activity, what interests us here is his work on optics. He wrote  Photismi de lumine et umbra and Diaphana. Photismi de lumine et umbra, on light and shadow as the title says, was written in 1521, the first part of  Diaphana in 1523 and the second and third parts in 1552. All four parts were first published together, posthumously, in 1611. On the whole his work is a fairly unspectacular presentation of the perspectivist tradition in optics in which he quotes both Bacon and Peckham amongst his sources, although he does introduce two important first into the perspectivist literature but for which he gets no credit because of the late publication of his work, the credit, as we shall see in a later episode, going instead to Kepler. 

Maurolico’s first innovation is the first geometrical theory of the camera obscura in which he solves the so-called pinhole camera problem which had puzzled perspectivists since Roger Bacon. Put simply the image of a luminous object, the Moon for example, viewed through a camera obscura is bigger than it should be according to the mathematics. Maurolico was able to demonstrate that in fact one doesn’t have a single image but a series of overlapping images due to the width of the pinhole. 

Maurolico’s solution of the pinhole camera problem Lindberg (see footnote 1) p. 179

Maurolico’s second innovation is that he was the first perspectivist to attempt to explain scientifically how spectacle lenses correct error in vision. This is particularly interesting given the time it took for somebody to do so. Eye glasses, convex lenses to correct presbyopia, farsightedness caused by aging, first appeared in Europe in the late thirteenth century and were a purely technical invention. This could also be used for hyperopia. The accepted theory of their invention, from Rolf Willach, is that monks grinding and polishing precious stone to decorate reliquaries, the containers for holy relics realised that they could be used to improve their eyesight. Concave lenses to cure myopia (near-sightedness) began to appear in the fifteenth century. Making eyeglasses became a major industry throughout Europe, however before Maurolico nobody had offered a scientific explanation of how and why they functioned. Different lenses for different degrees of visual handicap were produced by trial and error. Lenses for presbyopia were produced in graded strengths and sold by age category, over forty, over fifty, over sixty, etc. A system still found in the cheap reading glasses sold in chain stores and supermarkets today. 

First Maurolico deduced that a double convex lens produces convergence of the rays, whereas a double concave lens causes the rays to diverge.

Lindberg (see footnote 1) p. 180

He knew that the lens of eye is a double convex lens, which must refract and transmit radiation according to the “law and covenant” of refraction. As a consequence of this he rejected the perspectivist theory that only the rays that meet the lens of the eye perpendicularly enter the eye. For Maurolico it was clear that all rays enter the eye. The lens being double convex the rays are thus convergent when exiting the lens inside the eye.

From this the causes of myopia and hypermetropia are clear as well as the function of eye glasses in correcting them. Myopia occurs when the lens is excessively curved, so that the rays passing through it converge too much or too soon. In hypermetropia the lens is insufficiently curved and convergence is delayed. The spectacle lenses correct the defects of nature.

Although Maurolico had given the first geometrical explanation of the camera obscura he did not take the next step and view the eye as a camera obscura with the lens focusing the image on the retina, as Kepler was later to do. He still, like al-Haytham and all the other perspectivist, regarded the eye lens as being the seat of vision.^[1]

Although Maurolico covered much the same territory as Kepler would cover after him, including his analysis of the camera obscura and how eye glasses function, because his work in optics was first published posthumously in 1611 it had no influence on Kepler.

Our second sixteenth-century, Italian perspectivist is the Neapolitan, aristocrat Giovanni Battista Della Porta (1535(?)–16159.

Giambattista della Porta Source: Wikimedia Commons

I have already in the past written an extensive post about all of della Porta’s wide ranging activities, so I will confine myself here to his contributions to optics. His major work on optics was De refraction optics parte libri novem published in 1593.

Image from De refraction optics parte libri novem

As the title suggests he examines refraction in all of its ramifications. His theories are all mainstream perspectivist and would have been accepted without comment by his medieval predecessors. On the anatomy of the eye, he basically follows Vesalius (1514–1564). His major innovation concerns the camera obscura. He appears to have been the first to describe the insertion of a lens into the hole in the camera obscura in his Magia naturalis and from there he, unlike Maurolico, concludes that the eye is in fact itself a camera obscura.

Kepler owned a copy of Magia naturalis and even credited della Porta with the invention of the telescope based on the description of an optical device he found in there. However, he never read De refraction and even complained that he was unable to find a copy.

Source: Wikimedia Commons

Our third sixteenth century perspectivist was the German mathematician Friedrich Risner (c. 1533–1580). Almost nothing is known about Risner other than that he was born in Hersfeld in Hesse. Seemingly he moved to Paris in 1565 where he became assistant to and protégé of Petrus Ramus (Pierre de la Ramée 1515–1572), the infamous, Huguenot, anti-Aristotelian philosopher. Under Ramus’ direction, Risner edited the editio princeps of Ibn  al-Haytham’s Optics (De aspectibus) together with an edition of Witelo’s Perspectiva, “Opticae thesaurus: Alhazeni Arabis libri septem, nuncprimum editi; Eiusdem liber De Crepusculis et nubium ascensionibus, Item Vitellonis Thuringopoloni libri X” (Optical Treasure: Seven books of Alhazen the Arab, published for the first time; His book On Twilight and the Rising of Clouds, Also of Vitello Thuringopoloni book X), which was published in 1572. The Witelo had been published twice before in Nürnberg in 1535 and 1551 by Johannes Petreius but Risner’s was an improved edition. This publication made the work, in particular of al–Haytham, available to the seventeenth century optical physicists such as Kepler, Snel, Descartes, and Huygens.

Source

Famously, Ramus was murdered during the St Bartholomew’s Day Massacre, 23–24 August 1572. In his will Ramus had established a chair for mathematics at the Collège Royal de France and specified that Risner should be its first occupant. Following the settlement of legal disputes in 1576, Risner accepted the salary but never actually lectured, resigning a few months later and returning to Hersfeld where he died after long illness in 1580. 

Gravestone from Friedrich Risner in Hersfeld Abbey Source: Wikimedia Commons

In cooperation with Ramus, as with the Opticae thesaurus, Risner wrote the Opticae libri quatuor, an optics text book largely based on Witelo. Risner’s only notable contribution to optics was the invention of the portable camera obscura.

Before leaving the sixteenth century we need to briefly look at the anatomical study of the eye made by the Swiss physician Félix Platter (1536–1614).

Portrait of Félix Platter by Hans Bock Source: Wikimedia Commons

Platter was born in Basel, the son of the Lutheran, humanist, schoolmaster, and printer Thomas Platter (1499–1582).

Engraving of Thomas Platter Source: Wikimedia Commons

In 1552, he went to the University of Montpellier, then the leading university for medicine in Europe, to study under Guillaume Rondelet (1507–1566) the teacher of a long list of famous physicians and biologists.

Portrait of Guillaume Rondelet artist unknown Source: Wikimedia Commons

He graduated Doctor of Medicine in 1557 and returned to Basel where he quickly established himself as a physician and was appointed professor for practical medicine at the University of Basel. In 1571 he was appointed chief physician of the city. During his career he published several important works on human pathology and gained a reputation as an important anatomist.

However, what interests us here is a comparatively short work intended to popularise the work of Vesalius, Colombo, and Falloppio,  his De corporis human structura et usu … libri III (1663), which consist of fifty plates and a text, entirely tabula in form.

Describing the anatomy of the eye he wrote of the optic nerve:

The primary organ of vision, namely the optic nerve, expands when it enters the eye into a hollow retiform hemisphere. It receives and judges the species and colours of external objects, which, along with brightness, fall into the eye through the pupil and are manifest to it through its looking glass as will be described. 

Thus Platter made the optic nerve, together with its expansion in the eye (the retina), the principle organ of vision. As for the crystalline humour (the lens), it

is the looking glass of the optic nerve; and, placed before the nerve and the pupil, it collects the species passing into the eye as rays and, spreading them over the whole of the retiform nerve, presents them enlarged in the manner of an interior looking glass, so that the nerve can more easily perceive them.^[2]

Plate showing the structure of the eye from De corporis human structura

So, for Platter the crystalline humour is nothing but an optical lens and the retina is the seat of vision. Platter offers no proof of this but it is at the core of the new theory of vision that Kepler would later deliver.

After a couple of centuries of slumber following the medieval reception of Ibn al-Haytham optics was now slowly developing new momentum. 

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^[1] The account of Maurolico’s optics is largely taken from David C. Lindberg, Theories of Vision: From Al-Kindi to Kepler, The University of Chicago Press, 1976, ppb. pp. 178-182

^[2] Lindberg p. 176

Describing the world with mathematics.

The word map in modern English is taken from the medieval mappa mundi, whereby the term mappa is the Latin for napkin, cloth, table cloth on which these philosophical representations of the Medieval world were first drawn or painted. In general, maps were representative pictures that expressed in some way the possessive concept of ours, we belong here, or mine, I rule over this. Whilst often reasonably recognisable as representations of the areas they depicted they weren’t particularly accurate. Although early cultures often had methods of surveying, these were mostly only used for depictions of comparatively small areas. The Ancient Greeks, most notably Ptolemaeus, introduced the concept of applying measurement and coordinates to make maps more accurate. However, it was first in the Early Modern Period that the strict application of mathematics to surveying and mapmaking became the norm. So it is that we now think of maps as a mathematically correct depiction of the world in which we live. 

In the first place, we think of maps as something geographical but the term mapping has been extended over the centuries to cover the mathematical depictions of various aspects of the world in which we live. In her book, Mapmatics: How We Navigate the World Through Numbers^[1], Paulina Rowińska, self-confessed lover of maps and a mathematician, who has a PhD in Mathematics of Planet Earth, takes he readers on a step by step historical journey through how different branches of mathematics came to describe different aspects of the world and explains in layman’s terms the methodology of how they do it. 

In the introduction Rowińska explains how she fell in love with cartography the first time she saw a globe and then how she later became aware that it is mathematically impossible  to make a flat map of the Earth, a sphere. The first chapter of her book, Curved: How to Describe the Earth, takes up this theme and the mathematical problems inherent in projection, the process of turning a curved surface into a flat one. We get to meet to Gemma Frisius (1508–1555), who introduced triangulation, which enabled large scale accurate surveys, into the world of cartography and Carl Friedrich Gauss (1777–1855), who used triangulation to survey the Kingdom of Hanover, and, more significantly, proved that you can’t project a curved surface onto a flat sheet of paper without distortion.

Unfortunately, in this opening chapter, Rowińska drops the only major historical clangour that I detected in her book. She tackles the debate that started in the late seventeenth century as to whether the earth is an oblate or a prolate  spheroid–grapefruit or egg shaped is how Rowińska describes it. She explains that it became a political dispute with the French supporting  the Cartesian egg shape and the English the Newtonian grapefruit. This is already historically false, as the grapefruit shape was also propagated by Christiaan Huygens (1629–1695) the leading Cartesian philosopher of the period. She then correctly introduces the decision by the French Academy of Science to settle the argument empirically by measurement. Her source for this section is Larrie D Ferreiro author of Measure of the Earth: The Enlightenment Expedition That Reshaped Our World (Basic Books, 2011). She produces a relatively brief version of Ferreiro’s account of the expedition to the equator in Peru to measure one degree of longitude with all of its trials and tribulations. Then says that the result compared with the determination one degree of longitude of Jean Picard (1620­–1682) for Paris, he doesn’t get a name check at all, showed that Newton was right. However, Rowińska doesn’t appear to be aware that there were two expeditions, one to Peru and one to Lapland, and it was the comparison of these two results that confirmed that Newton and Huygens were correct.

The succeeding chapter, Flat: How to Make a Map, takes us from the curved Earth to the flat sheet of paper and the world of map projection. This chapter is centred around the work of Gerard Mercator and his famous, or as Rowińska explains, in our times, infamous, projection. I must admit I was somewhat surprised, or maybe disappointed, that Ptolemaeus (fl 150 CE), who basically introduced map projection into the world and whose rediscovery in the early fifteenth century led in an almost straight line to Mercator and his work, doesn’t even get a mention. Mercator actually spent a lot of time and effort production  a new edition of Ptolemaeus’ Geographia, as the first volume of his atlas. 

The 1569 Mercator map of the world Source: Wikimedia Commons Not in Rowińska’s book

She first explains the geometry of the various basic types of projection, before going on to the genesis of Mercator’s, Made for Navigation, map projection that through no fault of its own became the standard, ubiquitous world map that hung on so many classroom walls. She then takes the reader through the, not so easy, maths of the Mercator projection and the history of how other people had to derive it, because Mercator never explained how he did it. We then move on to a discussion of the area distortion inherent in the Mercator projection and the discussion that flared up in the second half of the twentieth century about how it discriminated against the southern continents and the third world. This leads to a discussion of the famed, proposed alternative Peters projection and the various other attempts to save the situation. The detailed presentation of this discussion is excellent and I loved cartographer Arthur H. Robinson’s comparison of the landmasses on the Peters projection to “ragged, long, winter underwear hung out to dry on the Artic circle.”^[2]

Peters projection Source: Wikimedia Commons Not in Rowińska’s book

Leaving the controversial world of map projection in her third capital, Scaled: , Rowińska introduces her readers to another problem inherent in all mapmaking, How to Measure a Line. After a discussion of the concept of scales in mapmaking, Rowińska introducers her readers to the problematic fact that the length of lines on a map, borders or coastlines for example, vary massively depending on the scale that is used to represent them, this is the measuring stick used to measure them. She tells us:

We call this phenomenon of measurements’ dependence on the measuring stick’s length the coastline paradox.^[3]

She takes her readers through the historical discovery of this paradox and then on to Benoit B. Mandelbrot (1924–2010) and his discovery/invention^[4] of fractals the mathematics behind the coastline paradox. Having explained the mathematics of fractals, Rowińska takes a look at some political disputes that arose because of the coastline paradox.

Chapter 4, Distanced: How to Navigate, opens with a discussion of London’s iconic Tube Map.

First Tube Map 1908 not topological Source: Wikimedia Commons In Rowińska’s book but in black and white

Iconic topological tube map Source BBC Not in Rowińska’s book

Rowińska introduces the discipline of topology and explains that the Tube Map is a topological map, that is it keeps the properties crucial for commuters, the connections and the order but ignores factors such as distance or the real geographical positions of the stations. She explains the history of how this iconic map came to be created and outlines the advantages and disadvantages of such a topological map. Rowińska also introduces a fascinating topological map produced by the Native American, Catawba peoples in c. 1724.

She points out that the even older Peutinger road map of the Roman Empire is another example of a topological map. Moving on, Rowińska takes a look at urban street maps and the mathematics of distance between points and the different ways of measuring them, knowns as metrics in mathematics.

Chapter 5, Connected: How to Simplify a Map, stays in the discipline of topology and opens with Leonhard Euler (1707–1783) and the famous Seven Bridges of Königsberg problem. Unfortunately, she starts with a somewhat inaccurate statement:

AN ALMOST COMPLETELY BLIND [emphasis in original] father of thirteen, Swiss-born and Russia Empire-based Leonard Euler wasn’t the most likely candidate to become one of the most prolific mathematicians in history.^[5]

That opening four word blast is a misleading oversimplification of one of the most fascinating stories in the history of mathematics. In 1738 following illness, Euler almost lost the sight in his right eye but retained that in his left eye. First in 1766, he began to lose the sight in his left eye due to a cataract. Through couching of the cataract, he regained the sight in his left eye but over time it deteriorated and he final became almost totally blind. He continued to work in his head dictating his work to scribes. 

Rowińska tells the story of the Seven Bridges of Königsberg problem and how Euler approached the problem by stripping the map of Königsberg of all of its details and reducing it to just the river and the bridges.

He had created what is now known as a graph.

Rowińska now takes her readers into the world of graph theory and the travelling salesperson problem, how to work out the most efficient delivery route for a given number of destinations. She goes into great detail on the various methods to find approximate solution to the problem since the middle of the twentieth century. Moving on we have the equally famous four colour map problem that was also solved through graph theory as Rowińska explains. She ends with a discussion of what constitutes a proof provoked by the computer based solution to the four colour map problem.

Chapter 6, Divided: How to Shape Society opens with a discussion of the hot political topic, gerrymandering; how politicians divide up electoral districts to ensure that their side wins an election. Rowińska deals with the history of gerrymandering in American politics and then takes her readers on a tour of the mathematics of dividing up geographical areas into electoral districts to achieve either a biased or a fair vote.

It’s a more than somewhat complex topic and she goes into it in depth. Following American electoral districts, we have American school districts another thorny area in everyday American social politics. 

Chapter 7, Found: How to Save a Life, once again takes on a legendary story of mapping, how the doctor John Snow stopped a cholera epidemic in nineteenth century London. Rowińska takes us through the history of the nineteenth century cholera epidemics in the UK and the well known story of John Snow and the Broad Street pump. How he located the source of the outbreak by mapping the occurrence of the cases.

She then takes the trouble to separate the myths from what really happened to show that Snow was not quite the super-efficient hero that he is usually presented as. Having dealt with the beginnings of epidemic mapping, Rowińska now takes her readers on a tour of the subject of geographic mapping of disease and its importance in the modern world. She then explains how the same methodology can and is used to fight crime or to find a missing airplane. 

The final chapter, Deep: How to Map the Invisible is the one I probably enjoyed most as it delt with a topic of which I was previously totally ignorant, mapping the oceans using sonar and the interior of the earth using seismographs. It is truly fascinating to learn about the problems that have to be surmounted and the methodology that has been developed to map those parts of the earth that simply can’t be seen.

How seismographic waves are used to map the earth’s core

As far as notes are concerned the book has the worst of all option, what I term hanging endnotes, that is endnotes that aren’t referenced in the text. You read something interesting and then go a look to see if maybe there’s a note about it. Please publisher, get read of this abomination. The notes themselves are mainly to sources. There are short biographies for each chapter at the end of the book and an adequate index! The book is illustrated with a limited number of black and white diagrams and occasional maps. 

As is often the case, my blow by blow account of the individual chapters doesn’t even come close to describing the large quantity of material packed into this volume on the different ways of mapping the world and human existence.  Rowińska writes in a pleasant almost chatty style, sprinkled with personal anecdotes. Reading her book, I got the feeling of sitting in a comfortable coffee shop chatting to someone whose passion for maps and mathematics spilled out in a torrent of fascinating twist and turns. Definitely a book for a wide general audience of readers if they are prepared to put in the work to understand the mathematics of mapping in its numerous variations.  

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^[1] Paulina Rowińska, How We Navigate the World Through Numbers, Picador, 2024

^[2] Rowińska p. 64

^[3] Rowińska p. 93

^[4] Choose your preferred term according to your philosophy of mathematics

^[5] Rowińska p. 153

From τὰ φυσικά (ta physika) to physics – XXVIII

It is one of the ironies of the history of science that Galileo’s most important and most  solid piece of science writing^[1],  his Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) was published almost at the end of his life, when the research that he carried out, on which it was based, took place in his younger years, in the late 1590s and early 1600s.  

His initial  interest in dynamics began in Pisa between 1589 and 1592  during his first appointment as a very insignificant professor of mathematics hired to teach Euclid as a sort of proof theory to philosophy students and astrology to medical students.

Galileo Galilei – Porträt von Domenico Tintoretto, ca. 1602–1607 Source: Wikimedia Commons

During this early phase his work consisted of philosophical speculation and contained none of the mathematical analysis or experiments for which he became famous. His thoughts on mechanics from this period, which were in his lifetime never published^[2] and don’t appear to have been in anyway circulated as manuscripts, have survived as a single collective manuscript, De motu antiquiora. It is not known who bundled the contents of De motu together, or who gave it this name. 

De motu actually consists of four separate pieces of work, a dialogue on motion (D), a ten chapter essay (E10), a twenty-three chapter essay (E23), and a rewrite of the first two chapters of E23 (E2).^[3] Internal evidence strongly suggest that the texts were written in the order D, E23, E2, E10. Most sources suggest that none of the text is complete but Meli thinks E23 is.

The dialogue is between Alessandro (Galileo), and Domenico (thought to be the Platonic philosopher  Jacopo Mazzoni (1548–1598)), as in all Galileo’s dialogues the stooge, who gets convinced that Galileo is right. Domenico starts from De motu gravium et levium (1575) of Girolamo Borro (1512–1592), who had probably been one of Galileo’s teachers in his time as a student in Pisa, and wishes to discuss six questions:

1. Whether there is rest at the turning point of motion.
2. Why, when two bodies of the same size but different materials, such as wood and iron, are dropped from a height, the light moves more swiftly than the heavier, if this truly does happen.
3. Why natural motion continuously  becomes swifter while forced motion continuously slows down.
4. Why bodies fall more swiftly in air than in water.
5. Why cannonballs travel farther when the cannon is fired at right angles to the horizon (vertically) than when fired parallel to it (horizontally).
6. Why heavier balls shot from a cannon travel more swiftly that lighter ones, even though the latter are easier to move.

During the dialogue, which is almost certainly not complete, Alessandro (Galileo) devote considerable energy to answering questions 4, 1, and 3 in that order. 

Obviously unhappy with the way the dialogue was developing, Galileo abandoned it in an obviously unfinished state and wrote instead the twenty-three chapter essay. This addresses the same question but extends the scope of the discussion into the area of statics and mechanics in the sense of the five simple machines.

In all of this work we have to consider Galileo’s situation. He is still a relatively young man, in his late twenties, lacking a university degree himself but employed as the lowliest and worst paid professor^[4] at a thoroughly Aristotelian university. As noted in the last post in this series the two dominant professors of philosophy in Pisa were both strident Aristotelians, Girolamo Borro (1512–1592) an adherent Averroes interpretation of Aristotle and his aggressive opponent Francesco Buonamici (1533–1603), who insisted on a pure Greek interpretation of Aristotle. Galileo received lecture notes from the Jesuits of the Collegio Romano, who were, of course, strict Thomists, to assist him in his first teaching endeavours. 

This is not the infamous, strident anti-Aristotelian warrior of his later years but an unknown scholar still half-trapped in an Aristotelian corset, who is setting out to criticise and replace Aristotle’s theories of motion, fall and projectile motion. Something, that others had been striving to do since, at the latest, John Philoponus in the sixth century CE.

I’m not going to give a blow by blow detailed account of Galileo’s thoughts and ideas as expressed in the dialogue and essay contained in De motu but merely pick out a few salient points.  

Firstly and fundamentally, Galileo rejects Aristotle’s dichotomy between gravitas and levitas. A theory that attributed the property to either fall, gravitas, or rise, levitas, to all material bodies. For Galileo there is only gravitas, the tendency to fall that varies from body to body, sometimes more sometimes less. He also, like many others, rejects Aristotle’s theory of fall , which says that the rate of fall of an object in a medium in is directly proportional to their weight and inversely proportional to the density of the medium. For Galileo the rate of fall of a body through a fluid medium is directly proportional to the arithmetical difference between the specific weight (gravitas in specie) of the body and the specific weight of the fluid. This is an application of Archimedes’ hydrostatics to the theory of fall. A natural consequence for Galileo is that bodies have a rate of fall in a vacuum, i.e. when the arithmetical difference is zero, contradicting Aristotle’s argument that the rate of fall in a  vacuum would be infinite therefore a vacuum cannot exist.

In the eighth section of the essay, Galileo discusses the rates of fall of different materials in different media and what happens to those rates when you combine the materials. During this discussion brought for the first time the argument that would end up in Discorsi:

Galileo then returns to Aristotle’s claim that heavier bodies of the same material fall faster: if two bodies of the same material but different sizes (and likewise weights) fall with different speeds, then when connected together, the assumption leads us to believe that the combined bodies will have an intermediate speed; however, the combination of the two bodies will have a total weight that is greater than any of the standalone bodies. Therefore, according to Aristotle, the combined weight should fall even faster than either of the standalone bodies, which leads to self-contradiction. The only way to correct the contradiction is to reject Aristotle’s claim and assume that the two bodies of same material but different size (and weight) fall at the same speeds. (Wikipedia)

One aspect of Galileo’s theory of fall at this time is that he regarded uniform speed and not uniform acceleration to be the norm for a falling body. The observed acceleration was perturbation caused by impressed force, (virtus impressa) that was generated by the force holding the body in equilibrium before the fall–for example the hand holding it up. This appears to be the same as or at least similar to the impetus theory, which raises the question of Galileo’s source for such a theory. Most likely he had it from the work of Borro, with which he was well acquainted, because Borro discusses the theories of Ibn Bājja, known to Borro and Galileo as Avempace, against which Averroes had argued. 

In the essay Galileo turns to discussion of motion in terms of statics and the five simple machine, a hot topic in the sixteenth century especially for Galileo’s earliest patron Guidobaldo del Monte (1545–1607), a topic I will deal with in a later episode. Galileo first turns his attention to motion respective the balance, which I’m not going to discuss here. Later he considers motion on an inclined plane, which is interesting given his later series of experiments to prove the laws of fall using one.

Galileo initially addressed the question, why a heavy body falls more swiftly along a steeper inclined plane than on a less steep one. He moves on to consider the case of the force required to move a body on a horizonal plane. He argued that if the plane and a ball were very smooth and perfectly hard then it would require very little force to set the ball in motion and, “indeed by a force less than any given force.” Galileo conceptualised this motion as parallel to the horizon, so in theory the ball would circle the earth and not sheer off at a tangent. One can sense Galileo’s theory of circular inertia emerging in these thoughts. He also goes on to consider the case of constructing a plane with an inclination such that two bodies of equal size and different materials would fall along  it and down the vertical in equal times. Here he is already formulating the famous inclined plane experiments that would later form the core of the Discorsi.

The essay closes with a discussion of projectile motions, where Galileo’s theories are very close to those of Tartaglia, which should not come as a surprise as Galileo’s the man who taught Galileo mathematics in Florence, Ostilio Rici (1540–1603), had been a student of Tartaglia.

Page from De Motu displaying Tartaglia’s theory of projectile motion

The strong similarity between the theories that Galileo propagates in De motu and those propagated by Girolamo Benedetti in his various publications immediately poses the question, was Galileo parroting Benedetti or even plagiarising him? There appears to be little hard evidence and historians have divided opinion of the relationship between Galileo’s work and that of Benedetti. However, the majority opinion is that Galileo had no knowledge of Benedetti’s work at the time he wrote the various manuscripts contained in his De motu. 

The main argument is that although Galileo mentions both Borro and Buonamici in his writings at this time, he makes absolutely no mention of Benedetti. Some argue, unconvincingly, that Benedetti’s work was unknown in general, so it’s not surprising that Galileo had never read it. If Benedetti’s work was unknown how come the Wallonian musician, mathematician and astrologer, Jean Taisnier (1508–1562),  plagiarised the 1^stedition of Benedetti’s Demonstratio in 1562. A plagiarism that Simon Stevin quoted in 1586 and Richard Eden (c. 1520–1577) translated into English in 1579. 

Portrait of Taisnier by N. de Larmessin, 1682 based on the 1562 woodcut Source: Wikimedia Commons

The strangest argument is that because Benedetti’s work was published in Venice it wouldn’t have been known in Pisa, where Galileo was working at the time. This argument is particularly bizarre because Galileo’s friend and colleague in Pisa, Jacopo Mazzoni, wrote a philosophical treatise , Praeludia, comparing the philosophies of Aristotle and Plato, of which Galileo owned a copy, which on the subject of motion often quotes Benedetti’s Diversarum speculationum mathematicarum, et physicarum, liber. This cannot be viewed as the potential source for Galileo borrowing from Benedetti, as it was first published in 1597, long after Galileo had abandoned De motu and moved to Padua. 

Jacopo Mazzoni Source: Wikimedia Commons

One final, interesting thesis is that Galileo wrote De motu unaware of Benedetti’s work and then somebody on reading his manuscripts pointed out to him that Benedetti had already formulated and published a very similar theory of motion forty years earlier. This revelation leading to Galileo abandoning his work on De motu.

Whatever the case maybe respective Galileo’s knowledge, or lack of it, of Benedetti’s work on the theory of motion, what we see, in the various manuscripts of De motu, is a young Galileo groping his way towards his own ant-Aristotelian theories of motion, an aim that he would first realise in the famous inclined plane experiments he conducted after he had moved to Padua and which he first wrote up and published in his Discorsi almost at the end of his life, 

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^[1] Lagrange agrees with me on this ‘Galileo made first this important step forward [the law of free fall and the parabolic trajectory of projectiles], disclosing therefore a new and vast pathway for the advancement of mechanics. These discoveries are exposed and developed in the work entitled: Dialoghi delle Scienze nuove, etc., appeared for the first time in Leiden in 1637; they did not give Galileo, while still alive, the same celebrity that his discoveries on the system of the world gave to him, but nowadays they represent the most solid and real part of the glory of this great man.’’ Joseph-Louis de Lagrange, Me ́canique Analytique, 2 vols. (1788; Paris: Courcier, 1811), 158–59 

As did both Newton & Leibnitz 

^[2] De motu first appeared in print in 1890

^[3] The letter-number labels for the four texts are taken from Domenic Bertoloni Meli. Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century, Johns Hopkins University Press, 2006, pp. 50-65, the main, but not the only, source for this post .

^[4] As I like to quip, the cleaning staff had a higher status and were better paid than a professor for mathematics at a late medieval university. 

The Man from Wales who founded the English School of Mathematics

The English School of Mathematics is not some sort of institution that exists or existed in the real world but an artificial construct used by historians of mathematics to collectively describe the mathematical practitioners, who established the study of the mathematical disciplines in England in the period from the middle of the sixteenth century till the end of the seventeenth century and did so largely in the vernacular, English hear meaning the English language. The mathematical practitioners usually assigned to this group were in fact often well connected with each other through networks that extended over generations, so collecting them under one identity is not so odd.

Regular readers will know that I generally have a very strong aversion to anyone being described as ‘the father of’ or ‘the founder of’ anything in the history of science. However, I might almost make an exception in the case of Robert Recorde (c. 1512–1558), who is often presented as the founder of the English School of Mathematics, as he was in fact the first named author of a mathematics book printed and published in English, a book, as we will see, that remained a key vernacular textbook for more that one hundred and fifty years. 

Robert Recorde Memorial in St Mary’s Church in Tenby There is no known portrait of Robert Recorde

Following the appearance of the Liber Abbaci by Leonardo Pisano (c. 1170–c. 1240–50) in 1202, and particularly after the introduction of double-entry bookkeeping into Europe at the end of the thirteenth century, vernacular libri d’abbaco used by the teachers in scuola d’abbaco to teach arithmetic, using the Hindu-Arabic place value decimal system, elementary  algebra, and elementary geometry to apprentice merchant traders, began to appear in Italy and spread  fairly rapidly throughout Europe. This development did not, however, reach the British Isles. 

Leonardo Pisano Liber Abbaci Source

With the invention of printing using moveable type in the fifteenth century, these abbacus book began to appear in print, the earliest known one being the Treviso Arithmetic, or Arte dell’Abbaco written in vernacular Venetian and published in 1478. It was soon followed by many others. Most famous was the Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions and proportionality) by Luca Pacioli (c. 1447–1517) published in 1494. The first one in German was published in 1482, in French and Spanish in 1512, and in Portuguese in 1519. 

Treviso Arithmetic, or Arte dell’Abbaco Source

England continued to lag behind the continent, as it did in almost all things mathematical–cartography, navigation, instrument making–and the first arithmetic textbook printed in England was De Arte Supputandi By Cuthbert Tunstall (1474–1559), based on Pacioli’s Summa, published in 1522, but as the title clearly indicated written in Latin. The first vernacular printed arithmetic book was the anonymous An Introduction for to learne to recken with the pen or with the counters published by a John Herford in St. Albans in 1537 and reprinted in London in 1539, 1546, 1552, 1574, 1581, and 1629. It appears that this was a composite text put together from translations from French and Dutch sources. 

The first truly English, printed arithmetic book was Robert Recorde’s The Ground of Artes, published in 1543 with an expanded edition appearing in 1552. In total it went through at least forty-five editions under various editors, or whom the first was John Dee (1527–c. 1609), up to 1699. Dee also contributed a poem on arithmetic which is printed on the reverse of the title page.

Title page

On this reverse of the title page, there is a poem by John Dee (1527-1609) dealing with arithmetic.

On this page there is an outline of the first dialogue in this book. (Recorde wrote his book in the form of a dialogue between student and master.) This dialogue deals with some of the elements of arithmetic, including the basic operations and the use of the rule of three (or the Golden rule)

A page from Robert Recorde’s The Grounde of Artes (1543) dealing with various “rules of practice” in arithmetic. Source

Robert Recorde was born in the small harbour town of Tenby within Carmarthen Bay on the coast of Pembrokeshire in West Wales, the second son of Thomas Recorde and Rose, daughter of Thomas Johns of Machynlleth in Montgomeryshire. His paternal grandfather had moved to Tenby from Kent.

Bradforth, Eric; Tenby in 1586; Tenby Museum & Art Gallery; http://www.artuk.org/artworks/tenby-in-1586-181875

Nothing is known of his upbringing but he graduated BA at Oxford University in 1531 and became a fellow of All Souls College in the same year. It is claimed that he taught mathematics at Oxford and began his medical studies there but he vacated his scholarship in 1535. At an unknown date he transferred to Cambridge University and graduated MD there in 1545. His only known medical work, The Urinal of Physick was published in 1547. The work in dedicated to the warden and Company of Surgeons in London and indicates that he was working in London at this time. The work was apparently very popular and went through many editions the last one appearing under title  The Judgement of Urines in 1679. 

Source: Internet Archive

Many popular sources state that he was a court physician but there is no evidence to support this claim. However, he did hold several government appointments over the years that would eventually prove his downfall. 

Following the accession of Edward VI in 1547, Recorde was appointed a commissioner to investigate the running of the Bristol Mint and in January 1549 he was appointed comptroller of it. He was also appointed comptroller of the new mint at Durham House in the Strand. He took over the Bristol Mint entirely in June but it was closed down in October. He was confined to court for sixty days for supporting Protector Somerset against the King and refusing to divert money to Lord John Russel (1485–1555) and Sir William Herbert (c. 1501–1570).

Despite this set back, Recorde was appointed surveyor of the mines and moneys in Ireland and given control of the mint in Dublin and the projected silver mines in Wexford in 1551. The silver mines failed and Recorde was recalled in 1553 having clashed with William Herbert, now Earl of Pembroke, over the management of the mint. In 1556, Recorde brough a suit of malfeasance (misconduct in public office) against Pembroke, who in turn sued for libel in October of that year. Recorde had chosen the wrong adversary. In January 1557, Pembroke won his case and Recorde was ordered to pay £1000 in damages. He was subsequently committed to the king’s bench prison, presumably for debt, where he died. 

William Herbert, 1st Earl of Pembroke in 1567 Source: Wikimedia Commons

Recorde was a man of great learning, a student of Greek and Old English and a notable historian but it is his mathematical endeavours that interest us here. As already noted he published The Ground of Artes in 1543, which was intended to be the first in a series of elementary maths textbook written for the simple man in particular the autodidact. In his dedication to the first edition Recorde wrote:

Suche as shall lacke enstructers, for whose sak I haue playnely set for the the examples, as no boke (that I haue sene) hath done hitherto, which thing shall be great ease to y^e rude reader. Therefore good M. Whalley, though this booke can be vnto youtr selfe but small ayde, yet shall it be some help unto young children, whose futheraunce you desire no less than your owne. 

Unlike the continental libri d’abbaco, which only taught arithmetic using the Hindu-Arabic place value number system, Recorde taught both this and the use of the counting board to do calculations. Recorde’s first edition from 1543 only included arithmetic using whole numbers, but the expanded edition of 1552 included the use of fractions. The Ground of Artes, and all of his subsequent textbooks, was written in the form of a dialogue between a Master and a Scholar. The instruction was a lively exchange between the two participants. The Scholar was not a passive absorber of instruction but made suggestion for solutions that were wrong and were then corrected by the Master.

Recorde’s second book The Pathway to Knowledge, published in 1551, took up the topic of geometry and was a translation and rearrangement of the first four books of Euclid’s Elements. Following Proclus, Recorde separated the constructions from the theorems. No proofs were given but explanations and examples were provided. New editions were published in 1574 and 1602). This elementary text was intended to be followed by two further volumes on practical applications covering surveying and map-making but these were never written.

This is the first page of Pathway to Knowledge by Robert Recorde Source

Recorde appears to have followed the order of the quadrivium with his first books. Having covered arithmetic and geometry he now turned to astronomy with his third and fourth books. The Gate of Knowledge was apparently a treatise on measurement and the use of the quadrant. There is no extant copy and it was probably never published although it is referred to as complete in The Castle of Knowledge, containing the Explication of the Sphere both Celestiall and Materiall, etc. published in 1556 and reissued in 1596. 

Title Page Source

An illustration of the Ptolemaic system is given on page 9.

The Castle is based primarily on Ptolemaeus, Proclus, Sacrobosco and Oronce Fine but Recorde provides a critical analysis of his sources, pointing out errors in the Greek authors and suggesting that the mistakes of Sacrobosco and others were due to their lack of Greek. It should be remembered that Recorde was an expert on ancient Greek.

Although fundamentally a presentation of geocentric cosmology, Recorde makes favourable references to Copernicus in his text. However, he warns the student that this is advanced material and he must first master the geocentric system before turning his attention to Copernicus. It is possible that he intended to deal with heliocentric astronomy in the planned, but never realised, The Treasure of Knowledge.

It is interesting that just thirteen years after the publication of De revolutionibus, we find positive references to Copernicus in an introductory, vernacular textbook. This is by no means the only, early, positive reference to Copernicus by the English School of Mathematics. It should be remembered that John Dee travelled to Louvain in 1547 to study under Gemma Frisius (1508–1555), just two years after  Frisius issued the first ever response to Copernicus in print in his De radio astronomico et geometrico a booklet of a multipurpose astronomical and geometrical instrument published in 1545. Dee’s pupil John Feild (c. 1525–1587) published an ephemerides, in English, based on De revolutionibus in 1557, to which Dee wrote a positive introduction. In 1576, Thomas Digges (c. 1546–1595), who will get his own post in this series, published an edition of his father’s A Prognostication Everlasting that contained an appendix, a Perfit description of the caelestiall orbes, that is a relatively free  English translation of the cosmological part of Book I of De revolutionibus. 

For his last published book, Recorde abandoned the sequential titles of his series with a book on advanced arithmetic, which introduced algebra into England for the first time, his The Whetstone of Witte, published in 1557. This is the only one of Recorde’s book that didn’t have at least two editions. Possibly because it was an advanced text and thus less useful than his elementary text for craftsmen, navigators and other practical users of mathematics. 

Title Page Source

On this page (Sig. S, f. i v & f. 2 r), Recorde explains the notation for a unknown and its various powers.  Note that the owner of this particular copy wrote notes to help him understand the various names and abbreviations for the powers.

Recorde explains subtraction of polynomials by use of a poem (Sig. X, f. ii r).

On these pages (Sig. Ii, f. iv r & v and Sig. Kk, f. i r) is Recorde’s attempt to design a real problem whose solution requires a quadratic equation.  This problem is entitled a “question of jorneying” and requires knowledge of the formula for the sum of an arithmetic progression.

The algebra in Recorde’s Whetstone is largely taken from German authors in the Coss tradition, mainly Johann Scheubel (1494–1570) and Michael Stifel (1487–1567). Coss was the fifteenth century German name for algebra. In the Italian libri d’abbaco, which presented a rhetorical algebra the unknown was represented by the word ‘cosa’ (thing), often abbreviated to co. Algebra was known as l’arte della cosa. In German this became first Cosse and then Coss in the work of Christoph Rudolff (c. 1500–before 1543) Behend und hübsch Rechnung durch die kunstreichen regeln Algebre so gemeinicklich die Coss genent werden (Nimble and beautiful calculation via the artful rules of algebra [which] are so commonly called “coss”) published in 1525. In English this became the cossike arte, or the rule of coss.

Michael Stifel reissued Rudolff’s book as Die Coss in 1553

Micael Stifel Source: Wikimedia Commons

Micael Stifel’s own Arithmetica Integra from 1544

Title page of Scheubel’s Algebrae Compendiosa from the collection of Dr. Sid Kolpas. The full translation of the title from the Latin is Concise Algebra easily described, which brings forth the great wonders of arithmetic. Source

Recorde did not use negative numbers but did allow negative terms in quadratic equations. He took over the cossic symbols including the + and – first introduced in print by Johannes Widman (c. 1460–1498) in his in his book Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft published in Leipzig in 1489 in reference to surpluses and deficits in business problems.

The introduction of + and – in Widman’s Mercantile Arithmetic

If people know anything about Robert Recorde it is usually that he introduced the equals symbol (=) in The Whetstone of Witte with the famous quote: 

And to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle.

And to avoid the tedious repetition of these words: “is equal to” I will set as I do often in work use, a pair of parallels, or duplicate lines of one [the same] length, thus: =, because no 2 things can be more equal.

On this page (Sig. Ff, f. i r), we see Recorde introducing, for the first time, the “equal” sign.  He explains that he picked two parallel lines to represent this concept “because no two things can be more equal.” He then gives various examples of the use of the equal sign in algebraic equations.

Unfortunately, as is often the case in the history of science, the claim that Recorde invented the equals sign is not strictly true. On page thirty-seven of his paper Hesitating Progress the slow development towards algebraic symbolization in abbacus-and related manuscripts, c. 1300 to c. 1550, the Danish historian of mathematics, Jens Høyrup, draws the reader’s attention to one aspect of a marginal note in the manuscript Vatican, Ottobon. lat.3307 that he is discussing:^[1]

We observe that equality is indicated by a double line.

To this simple comment he adds a footnote:

The double line is also used for equality in a Bologna manuscript from the mid-sixteenth century reproduced in (Cajori, 1928, I, p. 129); whether Recorde’s introduction of the same symbol in 1557 was independent of this little known Italian tradition is difficult to decide. In any case the combination with the geometric symbols indicates that the present example (and thus the Italian tradition) predates Recorde by at least half a century or so.

Turning to Florian Cajori’s A History of Mathematical Notation (Two Volumes Bound As One), Dover, New York, 1993, original The Open Court Publishing Company, La Salle, Illinois, 1928-1929,^[2] we find on page 126:

A manuscript, kept in the Library of the University of Bologna, contains data regsardingregarding the sign of equality (=). These data have been communicated to me by Professor E. Bortolotti and tend to show that (=) as a sign of equality was developed at Bologna independently of Robert Recorde and perhaps earlier. 

This is particularly interesting as Cajori is one of the main sources quoted for Recorde being the inventory of the equals sign.^[3]

As we have already seen. Recorde’s career as a maths textbook author came to a sticky end just one year after the publication of The Whetstone of Witte. However, as already noted his books did not die with him, The Ground of Artes going through numerous editions up to the end of the seventeenth century. We know from marginalia that copies of the book continued to be handed down through the generations well into the eighteenth century. 

The first posthumous edition of The Ground of Artes was edited by John Dee, who as we saw contributed a poem to the first edition. The two men were obviously connected. Dee, as we have seen from an earlier post was the mathematical advisor to the Muscovy company. Recorde’s The Castle of Knowledge was written and specifically printed in 1556 for the use of the Muscovy Company’s navigators and The Whetstone of Witte was dedicated to the Governors of the Muscovy Company. Dee was also the technical advisor to Frobisher’s Cathey Company expedition to find the Northwest Passage, as well as being a shareholder in the Cathey Company. Recorde’s Castle of Knowledge was one of the books in Frobisher’s on bord library.

Robert Recorde the physician from Tenby set the study of mathematics in the vernacular in England in motion in the middle of the sixteenth century and as we will see he found many others prepared to follow his example.

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^[1] Høyrup, J. (2010). Hesitating progress: the slow development toward algebraic symbolization in abbacus-and related manuscripts, c. 1300 to c. 1550. In A. Heeffer, & M. Van Dyck (Eds.), Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics (pp. 3-56). College Publications. 

^[2] Everyone has a copy on their bookshelf, don’t they?

^[3] I own the information that Recorde was not the first to use two parallel lines as an equal sign to Michael J Barany currently Seniour Lecturer in the History of Science at the University of Edinburgh.

From τὰ φυσικά (ta physika) to physics – XXVII

Regular readers of this blog will know that I challenge the big names, big events version of the history of science going into battle for the less well-known figures, who often made highly significant contributions to the progress of science. I often argue that many of these figures should be pulled out from under the shadows of the giants of #histsci and be much better known than they are. If I had to choose a figure in the history of physics in the Early Modern Period, who I thought most deserved to be much better known than he is, then that choice might well fall on the Italian mathematician, Giambattista Benedetti (1530–1590), who formulated much of the theory of falling bodies attributed to Galileo fifty years before Galileo did so. 

Very little is known about Benedetti’s origins. He appears to have been born into a wealthy family in Venice. The Italian astrologer and astronomer, Luca Gaurico (1475–1558), famous for his negative horoscope of Martin Luther, said that Benedetti’s father was a Spaniard, philosopher, and physicus, probably meaning student of nature but also possibly meaning physician. According to Gaurico he was mainly educated by his father, who made him a philosopher, musician and mathematician by the time he was eighteen.

Luca Gaurico Source: Wikimedia Commons

In one of the few autobiographical records that we have, he stated that he had received no formal education beyond the age of seven, except that he studied the first four books of Euclid under Niccolò Tartaglia (c. 1499–1557), probably about 1546–1548. The don’t appear to have been close as Tartaglia makes no mention of him as a pupil; Benedetti only mentioning him in 1553 “to give him his due.” In later years Benedetti would severely criticise Tartaglia’s writings. 

Niccolò Tartaglia Source: Wikimedia Commons

In 1558, Benedetti became court mathematicus to Ottavio Farnese (1524–1586), Duke of Parma, where he remained for about eight years, serving as court astrologer and engineer. He also made astronomical observations and constructed sundials. In the winter of 1559/60, he lectured in Rome on the science of Aristotle creating a good impression. 

Tommaso Manzuoli, called Maso da San Friano (1531–1571) portrait of Ottavio Farnese (1525-86), 2nd Duke of Parma and Piacenza – CC446 – Cobbe Collection Source: Wikimedia Commons

In 1567, he became court mathematicus to Emanuele Filiberto (1528–1580) Duke of Savoy, in Turin. Here he taught mathematics and science, and advised the Duke on appointments to the university, although he was never appointed to a professorship himself.

Portrait of Emmanuel Philibert, attributed to Giorgio Soleri Source: Wikimedia Commons

He remained in Turin until his death, as in Parma undertaking engineering projects and constructing sundials. In 1574, he published a treatise on the construction of sundials his De gnomonum umbrarumque solarium usu Liber, the most comprehensive volume on the topic at that time. He also published his De temporum emendatione on the correction of calendars in 1578 but it is his work on the laws of fall that interests us here. 

De gnomonum umbrarumque solarium usu Liber Source

Source

Benedetti’s first excursion into the world of anti-Aristotelian mechanics was surprisingly in a book on Euclidian geometry, his Resolutio omnium Euclidis problematum published in 1553, when he was just twenty-two years old.

Source: Wikimedia Commons

This was the book that contained his grudging acknowledgement of Tartaglia. The Resolutio concerns the general solution of all problems in Euclid’s Elements using only a compass of fixed opening. His book displayed his mathematical talent and was superior to similar volumes by Tartaglia and Ludovico Ferrari (1522–1565), a pupil of Cardano. Included in this book was a letter of dedication addressed to the Spanish, Dominican priest, Gabriel de Guzman with whom he had conversed in Venice in 1552. Guzman was interested in Benedetti’s theory of free fall and asked him to publish it. Apparently to avoid his idea being stolen Benedetti now outlined it in the dedicatory letter, although it had nothing to do with Euclidian geometry.

Seemingly based on Archimedes’ work on hydrostatics, On Floating Bodies, which almost certainly came to his attention through Tartaglia’s translation published in Venice in 1551, Benedetti held that bodies of the same material, regardless of weight, would fall through a given medium at the same speed. This contradicted Aristotle’s theory that they would fall at speeds proportional to their weights. 

Benedetti asks his reader to imagine two spheres of the same material, one of which has four times the volume of the other. He then says they should reconstitute the material of larger sphere as four smaller spheres each one equal in volume to the small  sphere  but joined together by a fine wire. The four joined together spheres would fall at the same rate as the single small sphere, end of argument. Astute readers will recognise this is the same argument as the famous thought experiment that Galileo published in his Discorsi in 1638, as a part of the dialog on the First Day:

Salviati. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?

Simplicio. You are unquestionably right.

Salviati. But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly. 

This is universally hailed as a sign of Galileo’s scientific genius and very often presented as the prime example of a thought experiment. Wikipedia also describes it as “a significant step forward in the history of modern science.” Galileo first published it, as already noted, in 1638 that’s eighty-five years after Benedetti had published it. To be fair Galileo probably first formulated it around 1590 when he was composing his unpublished manuscript De motu, to which we will return later, but that was still almost forty years later than Benedetti. However, you almost never hear of Benedetti’s formulation let alone hear him being praised for having taken “a significant step forward in the history of modern science.” 

In his dedicatory letter to Guzman, Benedetti wrote that he was in the process of writing a book on the topic that he intends to publish soon. In fact, said book, his Demonstratio proportionum motuum localium contra Aristotilem et omes philosophos was published to the beginning of 1554. Benedetti mentions in his dedicatory letter to this book that there was talk of Rome of his work and that as Aristotle could not err his theory must be false. These discussions might explain why another statement on freefall , Giovanni Battista Bellaso of Brescia, question why a ball of iron and one of wood will fall to the ground at the same time, in his II vero modo di scrivere in cifra, mentioned in the previous post in this series, was also published in 1553.  

The Demonstratio repeats the arguments made in the dedicatory letter to his Resolutio going into more detail, above all he elucidated the passages in the works of Aristotle that he was contradicting. In the Demonstratio, he also stated that objects in a vacuum would all fall at the same speed, thus contradicting Aristotle’s claim that they would accelerate to an infinite speed. Once again Benedetti is anticipating Galileo by decades with a statement for which Galileo gets praised to the heavens. Spectacular modern demonstrations of this fact in vacuum chambers or on the Moon are always accompanied by the commentary, “look, just like Galileo predicted!” 

In 1562, the Wallonian musician, mathematician and astrologer Jean Taisnier (1508–1562) plagiarised the 1^stedition of Benedetti’s Demonstratio together with the Epistola de magnete Petrus Peregrinus de Maricourt in his Opusculum perpetua memoria dignissimum, De Natura Magnetis et ejus effectibus, Item De Motu Continuo (“A little work worthy of preservation, On the Nature of the Magnet and its Effects, and another On Perpetual Motion.” He makes no mention of either author and in the dedication talks of “hoc meum parvulum opusculum” – this my little work. 

Portrait of Taisnier by N. de Larmessin, 1682 based on the 1562 woodcut Source: Wikimedia Commons

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Benedetti drew attention to the plagiarism in the Ad lectorem to his 1574 De gnomonum:

“…ut fecit impurissimus omnium Iohannes Taisnerus Hannonius. Qui opusculum nostrum… ita integrum sibi desumpsit, ut nihil praeter authoris nomen immutaverit; quid enim mutavisset, qui nec percipere poterat quae in ea disputatione continerentur? Homo vanus ab omni mathematica facultate alienus, qui merito propter crassissimam ignorantiam verebatur, ne vel aliqua Syllaba sublata aut addita totius tractationis inficeretur substantia. Credidit (ut opinor) me iam vita functum qui furti nunquam argui posse confidit…” (“as John Taisnier Hannonius did, the most unwholesome of all of them. Who so completely took for himself our little work, that he altered nothing except the name of the author – for what could he have changed, this vain man devoid of all mathematical capability, who was not able to grasp the things contained in that discourse? who justly feared, on account of his very gross ignorance, that by the addition or removal of a single syllable he might undo the meaning of the entire argument. I think he believed that I was already dead, and trusted that I would never be able to denounce his theft…”) Wikipedia

Ironically the Taisnier plagiarism became better known that the Benedetti original. It was even translated into English by the alchemist and cosmographer, Ricard Eden  (c. 1520–1577), which was published posthumously in 1579. Simon Stevin quotes and criticises Benedetti on the laws of fall from the Taisnier plagiarism, in the appendix to his book on statics, De Beghinselen der Weeghconst ( “The Principles of the Art of Weighing”) in 1586, where is describes his own experimental confirmation of the laws of fall. 

Frontpage of De Beghinselen der Weeghconst by Simon Stevin, 1586 Source: Wikimedia Commons

Interestingly, Benedetti published a second modified edition of the Demonstratio on Valentine’s Day, on 14 February 1554. The biggest change in this second edition is that whereas Benedetti had stated in the first edition that different sized bodies fall at the same rate both in a medium and in a vacuum, he now states that this is only valid for a vacuum. This is exactly the point that Stevin, only having read the first edition in the plagiarism by Taisnier, would criticise in 1586. Benedetti who replaced the Archimedean buoyancy with the term resistance also discussed how shape and surface area affected the rate of fall.

Benedetti’s final contribution on the topic took place in his collected work Diversarum speculationum mathematicarum, et physicarum, liber published in Turin in 1585 then reissued as Speculationum mathematiucarum, et physicarum, fertilissimus, pariterque utilissimus tractatus... in Venice a year later.  It was published a third time, posthumously as Speculationum liber: in quo mira subtilitate haec tractata continentur... in Venice in  1599. Here he discusses the acceleration of falling bodies in terms of increments of impetus. 

Source:

For all his inventiveness in contradicting Aristotle and anticipating Galileo, it should be noted that Benedetti’s work on the laws of fall was purely philosophical. He is not known to have carried out any experiments and he makes no mathematical analysis of the topic. However, he very much deserves to be better known and to be given more credit than has been the case in the majority of writings on the topic. 

Johannes Kepler’s preoccupation with the concept of harmony.

Johannes Kepler was an incredibly prolific writer, as well as more than sixty books and pamphlets, he corresponded with a wide range of people writing a huge number of oft, very long letters, on an extensive assortment of serious topics. In the field of astronomy, the discipline for which he is best known, his magnum opus was the voluminous Harmonices mundi libri V (The Harmony of the World), published in 1619. 

The jewel at the centre of this massive tome is his Third Law of Planetary Motion, known as his Harmony Law. That the term harmony is used both in the title of the book and as the name for his law is deliberate, because the concept of harmony was central to every aspect of Kepler’s extensive world of thought, be it in science, theology, politics, or even the whole of human existence. Why this was so and what he meant or better understood when using the term harmony is the topic of Aviva Rothman’s excellent book, The Pursuit of Harmony: Kepler on Cosmos, Confession, and Community.^[1]

Having read Rothman’s book, the first thought that occurred to me when thinking about writing a review was that this is not a book for the faint hearted. This sounds negative but it is not intended to be so and to explain what I mean; I find myself needing to wax lyrical or maybe not so lyrical about historiography. 

I am on record as having stated that I don’t like historiography, because historiography becomes dogma and dogma makes people blind. Historiography is a word with multiple definitions but a simple working definition might be that history is the study of the past and historiography is the study of how we write about the past. There are prescriptive historiographies that seek to impose the way that we approach studying and writing about the past. 

Most notoriously in the history of science there is the divide between internal and external histories, whereby internal histories concern themselves mostly or even exclusively with the scientific results that a scholar in the past was involved in producing. In opposition to this, external histories look at the context or contexts within which said scholar produced those results, which for the internalist are largely irrelevant. I think we should do both, whereby I lay emphasis on the contextual history of a scientific discipline. Rothman’s book is, of course, external history and that at its finest. 

Another prescriptive historiography is Marxist historiography that demands that the historian examines all aspects of the past from an economic standpoint. What were the economic forces that shaped the situation that the historian has taken under their investigatory magnifying glass. For me a not insignificant external factor but only one of several and often not the most important. 

Different prescriptive historiographies emphasise different aspects of the evolution of a scientific discipline with their adherents often claiming that their approach or methodology is the all important one. This is what I mean when I say that historiography become dogma.

I want now to look at a different aspect of history and historiography. There is a famous quote from L.P. Hartley’s novel The Go-Between, “The past is foreign country; they do things differently there,” and the similar common saying “the past is another country, we can’t go there anymore.” Writing history is not in any way an exact discipline, what the two quotes state or even emphasise is that we have no direct empirical access to the past. It’s over, it’s gone, it’s the past and it no longer exists. Historians can’t enter their TARDIS^[2] and go back to a specific era, occurrence, incidence in the past that they wish to investigate and make video and sound recordings, write protocols, take stills or in any way whatsoever capture in original that which they wish to investigate. 

It’s as if the past is a giant jigsaw puzzle that somehow has lost a very large number of its pieces and a lot of those that still exist are in addition damaged in some way or another. Those surviving puzzle pieces, both the undamaged and the damaged ones are the evidence, consisting of written and material sources, that have somehow survived the ravages of time that provide the material with which the historian has to work. That work consists of constructing, or better said reconstruction, a plausible picture or narrative of what took place in that aspect of the past that they are researching.  As new pieces of evidence appear over time, that reconstruction gets modified, in extreme case even overturned. This is the much denigrated “rewriting of history” that conservative commentators and right wing politicians constantly and vehemently attack but is actually what historians are supposed to do. 

There are in historiography different levels of reconstruction of any given piece of history depending on the context in which the reconstruction is being presented. The presentation of Newton’s Principia, for example, is very different in a school textbook or in a popular magazine article to the presentation in an academic article. Even within academic presentations  the level of reconstruction can vary substantially depending on the point that the author is trying to get across to their potential readers. These reconstructions vary according to the width and depth that the historian investigates the available evidence, the complexity of the reconstruction increasing with the greater amount of evidence that is utilised and also with the level of scrutiny that the evidence is subjected to. Each level of reconstruction has its validity within the context that it is presented in but every one of them must be true to the evidence used, it may not contradict the given facts. This last statement raises the thorny questions, what counts as evidence and what exactly are facts? Whole books have been written on either or both of these questions and I’m not going down that road here but they are questions that every serious historian should give some thought to.

When I wrote above that Rothman’s book was not for the faint hearted what I meant was that she has dived into the available historical material to a very great depth and subjected the vast quantity of evidence that she has accumulated to an incredibly intense scrutiny. To utilise a cliché, she has almost literally left no stone unturned or potential piece of evidence unexamined in her attempt to elucidate what Johannes Kepler meant with his continuous use of the term harmony across all aspects of his life and work. The result of this is a historiographical tour de force, a master piece of historical reconstruction but one that makes its readers work hard in following the twists and turns of Herr Kepler’s world of thought. 

The introduction, delivers what its title, Kepler and the Harmonic Idea, promises, summed up in the simple sentence:

Harmony was the cause to which Kepler devoted his life; it was both the intellectual bedrock and the crucial goal for his seemingly disparate endeavors.^[3]

Having very briefly sketched the areas where Kepler sought and/or applied his concept of harmony, Rothman takes us on a brief guided tour of the Pythagorean mathematical-musical concept of harmony and the celestial or cosmic version of it presented and publicised by Plato in his writing, that lies at the heart of Kepler’s own concept of harmony. She shows how it was taken up and developed by a diverse range of thinkers, particularly during the Renaissance in philosophy, science, politics, and religion, and how it changed and evolved in the works of those thinkers. 

As a small footnote, but not a criticism, she missed one of my favourite applications of the Pythagorean concept and one that has a strong connection to Kepler through his relationship with Tycho Brahe. Tycho designed Uraniborg, his palace and observatory on the island of Hven, himself. All the dimensions of this extraordinary structure are laid out in harmonic Pythagorean ratios, from the ground plan, over the floor plans of the rooms, down to the dimensions of the windows and the doorways. 

The book is divided into six capitals, each of which deals with a different aspect of Kepler’s world of thought and how his central harmonic idea played out within that within that given area. 

Chapter one, “The Study of Divine Things”: Kepler as Astronomer-Priest, takes us deep into the world of Kepler’s mathematical astronomy and his adherence to the Lutheran faith, which were, as Rothman shows, intimately linked. Unfortunately, in the middle of the very first page of this chapter is one of the very few historical errors that I detected reading Rothman’s illuminating book. She writes of the young Johannes:

If he could not speak to God, then he would speak for him; he would become a Lutheran priest. Kepler pursued this dream for the next thirteen years, until, while he was completing his theology degree [my emphasis] at the University of Tübingen, a letter arrived that was to change the course of his life.^[4]

I wrote a whole blog post explaining that the widespread claim that Kepler was studying for a theology degree is simply false, he wasn’t and never did. Rothman’s error is somewhat puzzling as in her bibliography she actually lists Charlotte Methuen’s Kepler’s Tübingen: Stimulus to a Theological Mathematics (1998), which is the academic article that debunks the myth.

However, as Rothman correctly notes Kepler, who was in a programme to produce schoolteachers and village pastor (small quibble, Lutherans are not priests), wanted to become a pastor and not a schoolteacher but was reluctantly sent off to Graz as a maths teacher. 

This very minor opening error aside, Rothman then delivers up an excellent in depth analysis of Kepler’s discovery that he could worship his God by revealing the geometrical structure of God’s creation. His discovery was the solution to the problem why, in the Copernican world system, there are only six planets? Kepler’s God was a rational God so there must be a reason for this. Kepler’s answer was that there are only five regular Platonic solids to fill out the gaps. He famously present this discovery in his first book Mysterium Cosmographicum (1596). 

In this tome he included his belief that God was not just a geometer but the very embodiment of geometry, and his fervent belief that mathematics and the Copernican world system was the key to reuniting the divided and quarrelling Christian community. He was still a servant of the Duchy of  Württemberg and the University of Tübingen and required their permission to publish his book. His teacher Michael Mästlin was delighted by Kepler’s contribution and passed it on with recommendation to the theologian Matthias Hafenreffer, who would decide whether it could be published or not. Rothman takes her readers through the thickets of Lutheran, Calvinist, and Catholic doctrine with particular reference to the Eucharist, a major point of dissent between them. She skilfully outlines Kepler’s own heterodox view on the topic and the three way debate by correspondence between Kepler, Mästlin and Hafenreffer concerning Kepler’s theological-astronomical views as expressed in the book. 

This chapter which follows the twist and turns of oft complex and confusing doctrinal debates is a tour de force. It cleverly positions Kepler, with his rather unique viewpoint,  within the theological debates of the intensive phase of the Reformation and Counter Reformation and is alone worth the price of the book. 

In the two sentence popular version of Kepler’s excommunication, it is stated that that his views in general were too ecumenical and too accommodating to the Calvinists in particular. Both statements are to some extent true but as Rothman explains in great detail in her second chapter far too simplistic. Having established to his own satisfaction that God was geometry and that through cosmology and astronomy the three main Christian denominations could find common ground and reunite in harmony. Kepler had entered into a deep theological discussion with his teachers from Tübingen. They told him very clearly not to talk about things that didn’t concern him and that he was not qualified to judge and instead to stick to his mathematics. In her second chapter, “Maters of Conscience”: Kepler and the Lutheran Church, Rothman take another deep dive into the theological divide between Kepler and his church and why that church was in no way prepared to extend the hand of forgiveness and welcome their lost sheep back into the fold. Rothman delivers up a master class in early seventeenth century Lutheran theological doctrine and what it was that Kepler couldn’t accept in it and why the Lutheran theologian couldn’t accept his personal criticisms and wishes.

Contrary to popular belief during the hot phase of the reformation and counterreformation scientists on both sides of the divide did not stop communicating and sharing with each other and Kepler, who as a Lutheran spent his formative years as an astronomer living and working in Catholic Prague, had strong personal contacts to Catholic, and indeed Jesuit, mathematicians and astronomers. In her chapter three, Of God and His Community”: Kepler and the Catholic Church, Rothman takes a close look at those contacts and the wish from the side of the Jesuits that Kepler would convert and Kepler’s reactions to those wishes and his general attitude to the Catholic Church. As might be expected Kepler’s attitude towards Catholicism was neither acceptance nor total rejection but like his position within his own religious community highly complex. Within both religious communities Kepler wanted to define his own theological standpoint accepting some items of dogma and rejecting other. Basically, he keeps saying we all believe in God, so why can’t I just do it my way but saying it with great sophistication. As in the previous chapters Rothman guides he readers through the complex theological thickets with verve.

Throughout all of the three opening chapters we experience Kepler’s life long struggle to establish harmony amongst the three major Christian confessions. 

In chapter four, “An Ally in the Search for Truth”: Kepler and Galileo, we leave the thickets of early seventeenth century theological disputes and turn to Kepler’s scientific endeavours and his forlorn attempts to win Galileo as a companion in his battle for the Copernican world system. 

The Humanist Renaissance was kicked off by the rediscovery of the great orators of classical antiquity in particular Marcus Tullius Cicero (106–43 BCE) and Marcus Fabius Quintilianus (c. 35–c. 100 CE), usually simple known as Cicero and Quintilian. Both were masters of rhetoric and Kepler, very much a Renaissance scholar turned to rhetoric in his attempt to entice Galileo into joining him in his crusade to convince the world to accept Copernican heliocentricity in the interest of cosmic harmony, setting up a Copernican scientific community. Rothman opens this capital with a brief but comprehensive master class in the history and methodologies of rhetoric before moving on to the rhetorical strategies, including subtle lying, that Kepler used in his campaign to gain Galileo’s support. A campaign, which as is well known failed. In this context, Rothman also delivers an excellent analysis of Kepler’s dispute with Martin Horky the young Bohemian scholar who ridiculed Galileo’s failed attempt to demonstrate his telescope to Giovanni Antonio Magini (1555–1617) in Bologna.

In chapter five, “Political Digression(s)”: Kepler and the Harmony of State, Rothman turns to an area that one normally does not associate with Kepler, politics. Rothman, of course, points out that as imperial mathematicus, read astrologer, to the Holy Roman Emperor, Kepler functioned as a political advisor and that politics is an area where the term harmony, or at least the desire for it,  was historically very much at home. There follows an excurse on the role of Tacitus’ Histories as a major source of advice on political behaviour during the Renaissance. Having introduced Tacitus, we now get The Astrologer as Politician, and Kepler as Tacitist. Astrologers have always functioned as political advisors and Kepler is infamous for his admission that in this role he relied more on applied psychology that on astrological forecast in giving advice to Rudolf II, a passionate advocate of astrology. A succinct analysis is followed by an equally succinct one of Kepler’s Tacitist attitude. 

Up next an analysis of the politics of patronage of astronomy/astrology that contains the second minor historical error that I detected, Rothman writes:

These are some of the reasons why Rudolf II agreed to sponsor the new set of planetary tables–begun by Tycho Brahe and finished by Kepler–that would ultimately bear his name.^[5]

The Tabulae Rudolphinae, although based on the data collected by Tycho, were not started by him but were alone the result Herculean efforts of Kepler.

The quote in the title of this capital “Political Digression(s)” refers to a section of Kepler’s Harmonice mundi in which Kepler grapples with the mathematical political laws of the French philosopher Jean Bodin (c. 1530–1596) based on the mathematical concepts of Petrus Ramus (1515–1572). Yet another masterful short analysis by Rothman.

The final chapter “The Christian Resolution of the Calendar”: Kepler as Impartial Mathematician returns us to the world of science but a world embroiled in religious and political dispute over the Gregorian calendar reform. During his debates over the decades with Matthias Hafenreffer, the latter kept telling Kepler that he was not a theologian but a mathematician and he should stick to that which he knows and not meddle in affairs that don’t concern him. Kepler countered again and again with his God as geometer and mathematics as theology. Kepler claimed that mathematical studies are impartial and this chapter looks at the claim and how Kepler interpreted it with respect to the dispute over the calendar reform. 

The term impartial gives Rothman another chance to display her talent for deep historical research. We get informed that the Latin term partialis had no direct opposite and the concept impartial, for Kepler in German unpartheylich, roughly not taking side, was a very recent coinage. Rothman now delivers a whole battery of 17^th century dictionary definitions of the term unpartyisch. Sometimes I get the impression that she left no intellectual alleyways unilluminated whilst researching this unbelievably rich piece of historical writing. 

We now turn to the calendar reform and Kepler’s attitude towards it. In her brief description of the reform, Rothman in which she brings her third minor historical error, she writes:

And the reform process was finally completed in 1582, when Pope Gregory XIII created a commission, headed by the Jesuit mathematician Christoph Clavius, to reform the calendar on the basis of the recommendations of astronomer Aloysius Lilius.^[6]

Gregory XIII did not set up the commission in 1582, but that is when they finished their deliberations. We don’t know exactly how long they deliberated but it was at least ten years.  I wrote a whole blog post explaining that of the nine members of the commission who signed their deliberation in 1582, Clavius was far from being the head but was in fact the least significant member. It was only after the introduction of the new calendar that he was appointed to explain and defend it against its critics and thus became Mister Calendar, so to speak.

Rothman gives a brief history of the initial reception, acceptance and rejection, of the new calendar before introducing Kepler’s stance. As a young astronomer in 1597 he took a cautious approach outline where he agreed and disagreed with his teacher Michael Mästlin, the official, Lutheran, technical spokesman on the topic. However, in 1604, he wrote a long, unpublished dialogue on calendar reform between a political and a theological representative for each side in the debate and himself as Mathematicus in the middle as impartial referee. Rothman’s presentation and analysis of this complex and fascinating text is quite simply brilliant.

The book closes with a twenty-five page conclusion, Perspective, Perception, and Pluralism, of which the opening paragraph summarises Rothman’s entire endeavour far better than my entire, feeble attempt at a review:

Kepler’s vision of a better world, I’ve argued, rested on harmony. He believed that heavenly harmony should serve as a blueprint for earthly harmony, particularly given the increasing confessional and political dissonance around him. He understood harmony to be active and changeable, as were the planets in their cosmic symphony; like their harmonies, earthly harmony too should be multi-voiced and required ´difference–and even discord–to give it life. Many of the themes I’ve explored in the previous chapters–polyphony, tolerance, accommodation, diversity, and dialogue among them–were all, in some sense coextensive with harmony as Kepler understood it. A harmonious community, like a harmonious cosmos, was one that embraced many perspectives rather than just one.^[7]

As the title of the conclusion states, here Rothman takes a brief look at Kepler’s views on perspective and perception through his work on optics in particular his use, both real and metaphorical, of the camera obscura. Not important, but Rothman neglects to mention that Kepler coined the term camera obscura. She does however point out that Kepler likens God to a human architect and whilst, as she tells us, Alberti expounds on architectural harmony, Kepler’ most famous frontispiece,  from the  Rudolphine Tables, is architectural disharmony written large.

The book has extensive, mostly bibliographical endnotes and the extensive bibliography to go with them. It also has a very good index. There are only a handful of greyscale images and unfortunately the portrait of Kepler at the front of the book  is the one now no longer considered to be Kepler.

Although Rothman writes smoothly and elegantly and has an excellent literary style, this is very much an academic and not a popular book. However, having said that, I think it is a must read for all, who have more than a superficial interest in Johannes Kepler. Over the years I have read an incredible amount of biographical literature about Johannes Kepler since I first discovered him in Arthur Koestler’s The Sleepwalkers more than fifty years ago and this counts as one of the very best that I have read. Unfortunately, there is no good modern, general biography of Johannes Kepler, maybe Aviva Rothman could be persuaded to write one. On the evidence of this work, it would almost certainly be excellent. 

---

^[1] Aviva Rothman, The Pursuit of Harmony: Kepler on Cosmos, Confession, and Community, The University of Chicago Press, Chicago, 2017.

^[2] “Time And Relative Dimension In Space” see Doctor Who

^[3] Rothman p. 5

^[4] Rothman p. 33

^[5] Rothman p.197

^[6] Rothman p. 230

^[7] Rothman p. 257

From τὰ φυσικά (ta physika) to physics – XXVI

In this series we have seen that various scholars over the centuries have questioned, challenged and even rejected Aristotle’s theories of free fall and projectile motion. Just as Tartaglia extended the challenges to  his theory of projectile motion with his work on ballistics other scholars took up the challenge to his law of fall in the sixteenth century, decades before Galileo did so.  

The standard, unfortunately widespread, myth is that Galileo was the first to challenge the standard Aristotelian theory on fall. As we have already seen John Philoponus (c. 490–c. 570) had challenged both Aristotle’s theories of fall and projectile motion. As did several Islamic scholars. In the sixteenth century we find several scholars continuing those challenges and openly contradicting the central claims of Aristotelian physics.

This anti-Aristotelian tendence is clearly stated in the Questioni sull’Alchimia of the Italian humanist, historian and poet Benedetto Varchi (c. 1502–1564) written before 1544.

Benedetto Varchi, by Titian Source: Wikimedia Commons

Here Varchi discussing the experimental evidence relating to the motion of heavy bodies, mentions the findings of Francesco Beato, a Dominican philosopher at Pisa, and Luca Ghini (1490–1556) an Italian physician and botanist, who created one of the first university botanical gardens and the first ever herbarium at Pisa.

Luca Vachi Source: Wikimedia Commons

Also, Giovan Battista Bellaso (1505–?) of Brescia, best known as a cryptographer, in his work Il vero modo di scrivere in cifra, published in 1553, asked why it is that a ball of iron and one of wood fall to the ground at the same time.

Source: Wikimedia Commons

More interesting is the case of the Spanish theologian Domingo de Soto (1494–1560) like Beato, a Dominican. De Soto was a hardcore Thomist who wrote extensively on international law. Theologically he argued for a strong adherence to Aristotelian philosophy opposing Renaissance Humanism. His Aristotelian philosophy is, however, Aristotle filtered through Albertus Magnus and Thomas Aquinas. 

Escultura de Domingo de Soto en Segovia (España) Source: Wikimedia Commons

Given his attitude to theology and philosophy, it comes as more than somewhat of a surprise that in his Physicorum Aristotelis quaestiones, published in 1551, he simply contradicts Aristotle’s theory of fall stating that the motion of bodies in free fall is uniformly accelerated, in other words, that their motion is uniformiter difformis with respect to time. This come eighty years before Galileo made the same statement. Even more surprisingly de Soto presents it in a matter-of-fact manner as if it were common knowledge. Having defined motion that is uniformly difform  with respect to time, he notes that this motion is:

properly found in object that move naturally and in projectiles. … For when a heavy object falls through a homogeneous medium from a height, it moves with greater velocity at the end than at the beginning. The velocity of projectiles, on the other hand, is less at the end than at the beginning. And what is more, the first increases uniformly difformly, whereas the second decreases uniformly difformly.

Soto goes on to explain that the falling body will cover the same distance during its fall as another body moving at half the velocity with uniform speed, which he calculates out to yield the correct distance of fall.^[1]

De Soto’s statements lead almost automatically to two questions. Firstly, were his comment purely theoretical or were they based in some way on experimentation, and if then who’s, his own or somebody else’s. Secondly, did his text have any influence on Galileo. 

We simply don’t know the answer to the first question. It is, however, known that apart from Francesco Beato, another Spanish Dominican, Petrus Arches had told Giambattista Benedetti (1530–1590), whom I shall deal with in the next episode, that criticisms of Aristotle’s dynamics were being discussed in Rome in the summer of 1554, so it is possible that de Soto might have acquired knowledge of experimental investigations through a Dominican network.

On the second question, de Soto’s Physicorum Aristotelis quaestiones was quite popular and went through nine editions in the second half of the sixteenth century. The eighth edition was published in Venice in 1582 just as Galileo was beginning his own studies of Aristotle’s physics, at the University of Pisa. Galileo also mentions de Soto’s Physicorum Aristotelis quaestiones in an early notebook. However, there is no real evidence that he took the concept of uniformly accelerated fall directly from de Soto. 

However, several Jesuit scholars had studied de Soto’s work and integrated his ideas into their own work. When he began teaching in Pisa, Galileo borrowed Jesuit teaching notes to help develop his own courses and there could well have been an indirect influence there.

A certain source on the subject for Galileo was the very lively dispute between Girolamo Borro (1512–1592) and Francesco Buonamici (1533–1603), who both taught philosophy at Pisa during Galileo’s time there, as a student and then later as a professor.

Girolamo Borro was born in Arezzo and probably studied at the University of Padua.

Source: Wikimedia Commons

Around 1537 he entered the service of the diplomate Cardinal Giovanni Salviati (1490–1553) as a theologian, whom he served for sixteen years. Travelling and working all over Europe.

Portrait, oil on canvas, of Giovanni Salviati (1490–1553) by Pier Francesco Foschi (1502–1567) Source: Wikimedia Commons

Following Salviati’s death, he began lecturing at the University of Pisa but left again in 1559. Over the next decades he was involved in various accusations of heresy. In 1561, he published a book explaining the motion of the tides based on Aristotelian principles Del flusso e reflusso del mare.

Source: Wikimedia Commons

In 1575, he began teaching philosophy at Pisa again, where he remained until 1586, when he was dismissed following pressure on the university authorities by his opponents, including Buonamici. During this period, he published his De motu gravium et levium (1575) and his De peripatetiva docendi atque addtscendi methodo (1584), an exposition of scientific method according to Aristotelian principles.

Source

Borro was a hardcore Aristotelian, who rejected the new Renaissance Humanist developments. He strongly rejected the use of mathematics in philosophy and all forms of Platonism. His basic approach was the Aristotelian interpretations of Ibn Rushd (1126–1198), known in Renaissance Europe as Averroës. Interesting in our context is his emphasis on an experiential approach to the study of natural philosophy in opposition to a mathematical one. 

In De motu gravium et levium(pp. 214–217) he described a protoexperiment performed at his home with the aid of students. In order to resolve a dispute over whether a body of heavy material will fall faster than one of light material, he “took refuge in experience [experientia] the teacher of all things.” He described how lead and wood balls were dropped from a “high window” to resolve this question: “The lead descended more slowly, namely [it descended) above the wood, which had fallen first to the ground: however many times we were all there waiting for the result of this occurrence, we saw the latter [the wood) fall downward [before the lead]. Not only once but many times we tried it with the same results.” The conclusiveness and somewhat puzzling nature of his results parallel those described by Galileo in De motu (I, 333–337)^[2]. 

Galileo also owned a copy of Borro’s work on tides which he quotes in his Dialogo. 

Francesco Buonamici was probably born in Florence and studied philosophy and medicine at the University of Pisa. In 1565 he was appointed assistant professor (extraordinarius) for philosophy at Pisa and in 1571 he was promoted to full professor (ordinarius) where he remained all of his life. As ordinarius he taught Aristotle’s De caelo, De anima, and Physica in three-year cycles. Buonamici’s most important publication was his more than one thousand pages long De motu libri X, which covers all aspects of Aristotle’s theories of motion but also all aspects of his theories of change. For Aristotle motion, change of place was just one form of change. Buonamici delivers a careful and exhaustive study of the topic with extended discussions of Aristotle’s views as well as that of a long list of commentators both ancient and modern. These include, Lucretius, Proclus, Plutarch, Pseudo-Aristotle, Pappus, John Philoponus, Theon, Archimedes, Nicolaus Copernicus, Pereira, Ludovico Boccadiferro, Christoph Clavius, Zabarella, and Toletus, as well as many others.^[3]

Source: Wikimedia Commons

Buonamici had an excellent knowledge of Greek and only accepted Aristotelian texts and commentaries translated from the original Greek. Like many Renaissance scholars he rejected texts that had been translated into Latin from Arabic and he also virulently rejected Arabic commentators  such Averroës (Ibn Rushd). He attacked the Averroës views on the motion of elements, which was the source of his bitter dispute with Borro.

Galileo had studied under Buonamici and his debate with Borro is traceable in his earliest work on motion, De motu antiquiora.

In particular, there are implicit references to Buonamici’s work in the discussion of the questions of falling bodies and of Archimedean extrusion.

As for the first subject, Galileo, in chapter 22 of his treatise De motu antiquiora, discusses the problem of the fall of bodies of different matters (wood, lead, and iron), in direct reference to the accounts already provided by Buonamici and by his rival, Borro. All of them (Borro, Buonamici, and Galileo) resorted to experimental evidence as a way to corroborate their own theories. From this point of view the famous leaning tower experiment, allegedly performed by Galileo at the time in which he composed his early writings on dynamics, seems to be rooted in a tradition of experimental research shared also by Borro and Buonamici, as well as by other Pisan professors, such as Jacopo Mazzoni and Giorgio Coresio.^[4]

As can be clearly seen, there are serious debates on Aristotle’s theories of motion, in particular his theories on falling bodies, that predate Galileo’s own work and also influenced it significantly. Developments that in the popular accounts are attributed to Galileo often preceded him by significant periods of time. 

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^[1] Quoted from William A Wallace, Domingo de Soto and the Iberian Roots of Galileo’s Science, in Domingo de Soto and the Early Galileo: Essays on Intellectual History, Routledge, 2004

^[2] Charles B. Schmitt DSB

^[3] Michele Camerota DSB

^[4] Michele Camerota DSB

John Dee navigational advisor

After our longer discourse on the history of magnetic variation, we return today to the history of navigation in England during the second half of the sixteenth century. John Dee (1527–c. 1608) was a central figure in the English mathematical world during this period. His name has already turned up in several of the earlier posts in this series. I have also in the past written posts about other aspects of Dee’s mathematical activity. In this post I want to gather together his contribution to the developments in navigation and cartography during this period.

According to Charlotte Fell Smith, this portrait was painted when Dee was 67. It belonged to his grandson Rowland Dee and later to Elias Ashmole, who left it to Oxford University. Source: Wikimedia Commons

It is worth looking at the beginnings of Dee’s path through life to see how he became England’s leading authority on all things mathematical, as at that time the land was actually largely a mathematical desert. Dee’s family came from Wales, where they lived on the Radnorshire border with England and the family name was Ddu, the Welsh for black. John’s father, Roland, like other members of the family, emigrated to London where the family name was pronounced Dye and eventually mutated to Dee. Roland was a mercer, that is a cloth trader at a time when England’s wealth was based on the cloth trade. 

In 1524 John married the fifteen-year-old Jane Wilde the heiress of William Wilde of Milton-next-Gravesend in Kent. A daughter was born in 1525 and John in 1527, followed by three other sons. The Wilde’s were connected to the court of Henry VIII and Roland would become a “gentleman sewer” to the king and in the 1540s a Packer to the Strangers, assessing customs on exports by foreigners, and charging fees for packing them. He also became a member of the Worshipful Company of Mercers on 19 February 1543. His social rise would end disastrously in 1547, but we are running ahead of our narrative, which is actually about John and not his father.

As noted above John was born into a prospering and aspiring family. John was enrolled in Chelmsford Chantry School in 1535, which was rebranded the King Edward VI Grammar School in 1551 and still going strong today. Here he was taught Latin, a prerequisite for attending university. In November 1542, aged fifteen, he entered St. John’s College Cambridge, which had only been founded thirty years before, and graduated BA in 1546. . St John’s had a strong Protestant fraction, and Dee was Catholic, so he was denied a fellowship. However, Henry VIII made Dee a founding junior Fellow of Trinity College in December 1546. He taught logic and sophistry for two years in the University Schools before graduating MA in 1548, now twenty-one-years-old. 

Dee had already developed a favour for mathematics and  although nominally taught, mathematics was largely ignored at Cambridge. Roger Ascham (c. 1515–1568), a Fellow of St. John’s who lectured on mathematics on mathematics in 1539-40, dismissed excessive devotion to such manual studies, which rendered gentlemen ‘unapt to serve in the world.’ At Trinity Dee found a patron in the Greek scholar John Christopherson (died 1558), who arranged for him to study at the University of Louvain in the summer of 1547 with a commendatory letter from Christopherson and money from Trinity. Louvain became Dee’s mathematical home. He returned there when he graduated MA in 1548, once again financed by Trinity College.^[1]

Dee was by no means the only scholar from Britain who sought a mathematical education on the continent. The Scotsmen, John Craig (died 1620) and Duncan Liddel (1561–1613) both sought their mathematical education in the universities of Northern Germany.

Source: Wikimedia Commons

Henry Saville (1549–1622), who endowed the first chairs for astronomy and geometry at an English university, at Oxford in 1619, deepened his mathematical knowledge with an extensive tour of Europe.

Henry Savile in 1621. School of Marcus Gheeraerts the Younger Source: Wikimedia Commons

In Louvain Gemma Frisius (1508–1555) was actually professor for medicine but he was also the leading cosmographer in Europe. He both worked in and taught mathematics, astronomy, astrology, geography, cartography, surveying, and globe and instrument making. He edited and published numerous new updated and expanded editions of the Cosmographia of Peter Apian (1495–1552).

Source: Wikimedia Commons

He was a notable teacher with Johannes Stadius (c. 1527–1579), Andreas Vesalius (1514–1564), and Rembert Dodoens (1517–1585) amongst his student. He became Dee’s teacher in cosmography ably aided by his own most famous pupil Gerard Mercator (1512–1594), who remained Dee’s close friend and correspondent.

The Frans Hogenberg portrait of 1574, showing Mercator pointing at the North magnetic pole Source: Wikimedia Commons

Through Frisius and Mercator, Dee also became friends with Abraham Ortelius (1527–1598), Mercator’s cartographical friend and rival. Through his time in Louvain Dee was already established in the inner circle of the European cosmographical elite. 

Abraham Ortelius by Peter Paul Rubens Source: Wikimedia Commons

When John Dee had returned from Louvain in 1547, he had brought with him besides ‘sea-compasses of divers sortes,’ what were comparative novelties in England, ‘rare and exquisitely made instruments Mathematical,’ ‘two great globes of Gerardus Mercator’s making,’ and astronomical instruments. These he had given to Trinity College, Cambridge, for ‘the use of the Fellows and Scholars.’^[2]

In 1550, Dee returned to the continent, this time to Paris, carrying with him a letter of introduction from Mercator. He held lectures at the university on Euclid and became friends with the Huguenot, anti-Aristotelian  mathematician Pierre de la Ramée (1515–1572).

Source: Wikimedia Commons

He claimed to have been offered a chair for mathematics in Paris, but there is some doubt about this claim. In 1563, Dee visited Italy where he met with Federico Commandino (1509–1575) noted from his numerous translations of mathematical treatises from Greek into Latin. Dee presented him with a manuscript that was published in Pesaro as De superficierum divisionibus liber Machometo Bagdedino ascriptus nunc primum Joannis Dee Londinensis et Federici Commandini Urbinatis opera in lucem editus. Federici Commandini de eadem re libellus in 1570. 

Source: Wikimedia Commons

The most puzzling of John Dee’s continental connections concerns the Portuguese cosmographer and mathematician Pedro Nunes (1502–1578), famous for being the mathematician, who determined that a course of constant bearing would be a loxodrome or rhumb line, the basis on which the Mercator projection is based. In 1558, Dee published his first book, Propaedeumata aphoristica, an astrological/astronomical work. This contains a dedicatory letter addressed to Mercator in which Dee writes:

You should know that, besides the extremely dangerous illness from which I have suffered during the whole year just past, I have also borne many other inconveniences (from those who, etc.) which have very much hindered my studies, and that my strength has not yet been able to sustain the weight of such exertion and labor as the almost Herculean task will require for its completion. And if my work cannot be finished or published while I remain alive, I have bequeathed it to that most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us, D. D. Pedro Nuñes, of Salácia, and not long since prayed him strenuously that, if this work of mine should be brought to him after my death, he would kindly and humanely take it under his protection and use it in every way as if it were his own: that he would deign to complete it, finally, correct it, and polish it for the public use of philosophers as if it were entirely his. And I do not doubt that he will himself be a party to my wish if his life and health remain unimpaired, since he loves me faithfully and it is inborn in him by nature, and reinforced by will, industry, and habit, to cultivate diligently the arts most necessary to a Christian state.^[3]

It would seem obvious from this text that Dee was closely acquainted with Nunes but there exists no known evidence, apart from this letter to Mercator, that the two even met or even corresponded. However, for the history of cosmography the friendship would obviously  be highly significant. 

Image of Portuguese mathematician Pedro Nunes in Panorama magazine (1843); Lisbon, Portugal. Source: Wikimedia Commons

In 1552, Dee met up with Gerolamo Cardano (1501–1576) when the latter was in London. Nothing is recorded of their exchange but I strongly suspect it had more to do with astrology than with mathematics.

Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews Source: Wikimedia Commons

Unlike Craig and Liddel, who both did, Dee never visited Tycho Brahe (1546–1601) on the island of Hven and Dee was in Prague at the court of Rudolf II eighteen years before Tycho arrived there. However, the two did correspond on the subject of how to correctly measure the parallax of comets in order to determine if there were sublunar, as Aristotle, believed or supralunar as they actually are. This was a very hot topic in both astronomy and astrology in the sixteenth century. 

Source: Wikimedia Commons

In popular presentations, John Dee is almost exclusively presented as the arch-occultist, the man who consulted angels, wrote complex occult tomes and is discussed as the role model for Marlow’s Doctor Faustus  and/or Shakespeare’s Prospero. So, I have presented Dee’s mathematical pedigree in a fair amount of detail in order to show that he was a fully integrated member of the sixteenth-century European mathematical elite and not just away with the fairies.

Dee’s most significant mathematical work was the ‘Mathematicall Praeface’ to Sir Henry Billingsley’s English translation of the Element of Euclid, published in 1570. Both the translation and Dee’s preface had a major impact on the development of mathematics in sixteenth century England. In his preface Dee gives the first definition by an English scholar of navigation at the same time demonstrating that he is a cosmographer in the mould of Frisius and Nunes by coupling it with definitions of hydrography, astronomy, astrology and horometry.

THE ARTE OF NAVIGATION, demonstrateth how, by the shortest good way, by the aptest Directiō, & in the shortest time, a sufficient Ship, betwene any two places (in passage Nauvigable,) assigned : may be cōnducted : and in all stormes, & naturall disturbances chauncyng, how, to vse the best possible meanes, whereby to recouer the place first assigned. What need the Master Pilote, hath of other Artes, here before recited, it is easie to know : as of Hydrographie, Astronomie, Astrologie, and Horometrie. Pre-supposing continually, the common Base, the foudacion of all : namely. Arithmeticke and Geometrie. So that, he be hable to vnderstand, and Iudge his own necessary Instrumentes, and furniture Necessary.

The definition carries on for several hundred more words, detailing what is necessary to determine latitude and longitude and various other things, which we don’t need to go into here. I found his emphasis on the return journey, ‘to recouer the place first assigned’ interesting. Let us look at some of his other definitions. Not mentioned above but highly relevant is his definition of geography:

…GEOGRAPHIE teacheth wayes, by which, in sūndry forms, (as Sphaerike, Plaine or other), the Situation of Cities, Townes, Villages, Fortes, Castells, Mountaines, Woods, Hauens, Riuers, Crekes, & such other things, vpō the outface of the earthly Globe (either in the whole, or in some principall meber and portion thereof cōntayned) may be described and designed, in cōmensurations Analogicall to Nature and veritie : and most aptly to our view. May be represented. Of this Arte how great pleasure, and how manifolde commodities do come vnto vs, daily and hourly : of most men, is perceaued. 

[…]

To conclude, some, for one purpose : and some, for an other, liketh , loueth, getteth, and vseth, Mappes, Chartes and Geographicall Globes. Of whose vse, to speake sufficiently, would require a booke peculiar.

He describes hydrography in analogy to geography:

HYDROGRAPHIE, deliuereth to our knowledge, on Globe or in Plain, the perfect Analogicall description of the Ocean Sea coastes, through the whole world : por in the chief and principle partes thereof : with the Iles and chiefe particular places of daungers, conteyned within the boundes, and Sea coastes described : as, of Quicksandes, Bankes, Pittes, Rockes, Races, Countertides, Whorlepooles, etc. This dealeth with the Element of water chiefly : as Geographie did take principally the elements of the Earthes description (with his appurtenances) to taske.

As with the definitions of navigation and geography this goes on in more detail but we will now turn to Dee’s definitions of astronomy, astrology and horometry.

…ASTRONOMIE; is an Arte Mathematicall which demonstrateth the distance, magnitudes, and all naturall motions apparences, and passions propre to the Planets andfixed Sterres : for any time past, present and to come : in respect to a certaine Horizon, or without respect to any Horizon. By this Arte we are certified of the distance of the Starry Skye, and of eche Planete from the centre of the Earth : and of the greatnes of any Fixed starre sene, or Planete, in respect to the Earthes greatnes … 

Of ASTROLOGIE, here I make an Arte, seuerall from Astronomie : not by new diuise, but by good reason and authoritie : for, Astrologie, is an Arte Mathematicall, which reasonably demonstrateth the operations and effectes, of the naturall beames, of light, and secrete influence : of the Sterres and Planets : in euery elementall body, at all times, in any Horizon assigned…

What today seems strange, the inclusion of astrology in Dee’s navigation’s definition, would have seemed perfectly normal in the sixteenth century. What he doesn’t mention here in this definition is that astrological almanacs were used by navigators because of the actual astronomical data, such as luna phases,  that they contained to which in this period tide tables had already begun to be added. 

HOROMETRIE; is an Arte Mathematicall, which demōnstrateth, how at all times appointed, the precise vsvall demoninatiō of time, may be knowen, for any place assigned…^[4]

Dee worked as an advisor, teacher, and supplier of charts and instruments for ships masters and pilots. He mostly did this work for the Muscovy Trading Company, so we need to know something about this organisation.

Seal of the Muscovy Company Source: Wikimedia Commons

The Company of Merchant Adventurers to New Lands was an early joint stock association, set up by Richard Chancellor (c. 1521–1556), Sebastian Cabot (c. 1474–1557), and Sir Hugh Willoughby (fl. 1544–died 1554), in order to search for a Northeast Passage to China. Willoughby and Chancellor set of on an expedition to search for the Northeast Passage in 1553, in which Stephen Borough (1525–1585) took part. 

Vladimir Kosov. 1553 expedition of Richard Chancellor Source: Wikimedia Commons

Chancellor’s reception in Moscow, as depicted in the Illustrated Chronicle of Ivan the Terrible Source: Wikimedia Commons

The Company of Merchant Adventurers to New Lands was rechartered as the Muscovy Company by Mary I of England in 1555. A second expedition to Russia led by Chancellor took place in the same year. They continued to trade with Russia over the next years. In 1556, Stephen Borough led a new expedition to attempt to find the Northeast Passage.

In 1576, the Muscovy Company licenced the expedition led by Martin Frobisher (c. 1535–1594) to find the Northwest Passage. In 1577, Frobisher launched a second attempt to find the Northwest Passage, this time under the auspices of the Cathay Company, newly founded by Frobisher and Michael Lok (c. 1532–c.1621). In 1578, Frobisher undertook his greatest expedition to the northwestern Artic waters this time planning on establishing a colony. Like the previous two expeditions this one also failed. 

Full-length life-size oil painting portrait of English explorer Martin Frobisher commissioned by the Company of Cathay to commemorate his 1576 Northwest Passage voyage and promote the planned follow-up expedition of 1577. It is the only surviving painting of a series of fifteen that Netherlandish artist Cornelis Ketel made in England between 1576 and 1578 for the Company. The fourteen lost paintings depicted English and Inuit people involved with Frobisher’s three Northwest Passage voyages, as well as the ship Gabriel. Source: Wikimedia Commons

The Muscovy Company continued to explore the Artic waters, trade with Russia, and hunt whales into the seventeenth century. From the very first expedition of Willoughby and Chancellor, for which he wrote his TheAstronomicall and Logisticall Rules and Canons to calculate the Ephemerides to be used on the voyage, John Dee acted as cosmographical advisor to the pilots and masters of the Muscovy Company up till 1583 when he left England for the continent. He also instructed Frobisher and Christopher Hall, his master, in the use of navigational instruments and the mathematics of navigation, as well as advising them which books, charts, and instruments the expedition should purchase. Dee was also a shareholder in the Cathey Company.

Dee advised, taught and supplied, books, charts and instruments over thirty years to Chancellor, Stephen Borough, with whom he developed his  Paradoxal compass, a circumpolar chart for sailing in Artic waters where the distortions of a Mercator projection are too great, William Borough (1536–1598), Humphrey Gilbert (c. 1539–1583), and John Davis (c. 1550–1605). These were the leading English master pilots in the first decades during which England tried to make ground good in marine exploration following the Dutch, Spanish and Portuguese, who all had a substantial lead in this new endeavour. 

During this early period, the third quarter of the sixteenth century, the Muscovy Company was the only formal, chartered English trading in existence. The Levant Company was first chartered by Elizabeth I in 1592 following the merger of the Venice Company, chartered 1583, and the Turkey Company, chartered 1581, The most famous company, The East India Company was first chartered in 1600 and the Hudson’s Bay Company wasn’t chartered until 1670. In those first three decades, John Dee haven’t acquired his education largely on the continent was the only cosmographical advisor working in England and his services were eagerly sought after. When he departed for the continent to follow his, much better known, occult ambitions. He was superceded by Thomas Digges (1546–1595), his adoptive son, Thomas Harriot (1560–1621), who became Walter Raleigh’s chief cosmographical advisor and the ‘first scientist in America,’ Edward Wright (1561–1615), and others. 

In the 1570s, Dee began preparing a major text on his involvement  in marine expeditions, which he planned to publish in four volumes. In 1576, Dee published the first of the four volumes, his General and Rare Memorials pertayning to the Perfect Arte of Navigation, which despite its title has nothing to do with navigation as discussed above.

This book was an appeal to Queen Elizabeth to support the founding of British colonies and is the first book to coin the phrase British Empire. Dee produced some complex, spurious, historical arguments to justify England’s supposed claim to North America. The second volume of this work was intended to collect and publish all of his manuscripts on navigation but he was unable to find a sponsor for what would have been an expensive project because of all the tables and diagrams that the book would require and the manuscript has been lost.

Dee’s important contributions to the early decades of England’s deep sea explorations, which were very important, tend to get overlooked in the shrill comments about his later occult activities. 

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^[1] The details of Dee’s family background and his education are largely taken from Glyn Parry, The Arch-Conjuror of England: John Dee, Yale University Press, 2011

^[2] David Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, 1958 quoting E. G. R. Taylor, Tudor Geography, ( 1485–1583), Routledge, 1930.

^[3] Bruno Almeida, On the origins of Dee’s mathematical programme: The John Dee–Pedro Nunes connection, Studies in History and Philosophy of Science Part A, Volume 43, Issue 3, September 2012, pp. 460-469 

^[4] All the definitions are taken from Waters pp 521–524

The most stupid, effin take on Newton, the apple, and the theory of gravity ever

My mate the HISTSCI_HULK, known to his friends as Hulky, was perusing our email website this morning, which as well as being the repository for our electronic mail is a sort of online newspaper, unfortunately rather close to the gutter press. He had just begun reading an editorial on the heatwaves, caused by the climate crisis, that the world is currently suffering and the resulting high number of deaths caused by heat exhaustion, when he began to splutter, his agitation quickly turning to an outbreak of HISTSCI rage!

“BLEEDIN’ BRAIN DEAD IMBICILE!” he screamed, threatening to throw the computer out of the window.

“DON’T THE IDIOTS LEARN ANTHING ABOUT SCIENCE AND ITS HISTORY AT SCHOOL?” he shouted as I gently prised him away from the computer screen.

“IT’S 2024 CE FOR FUCK’S SAKE! NOT BLEEDIN’ 2024 BCE!” he howled, before storming off to pulverise some rocks.

Cautiously, I looked at the computer monitor to see what had provoked the good beast into such an explosive fury and read the following mindbogglingly ludicrous statement.

„Ein fallender Apfel soll es gewesen sein, der einst den Naturwissenschaftler Isaac Newton auf die zündende Idee brachte, dass es eine Schwerkraft gibt.“

For those of you who don’t read German in English it’s:

“It was supposedly a falling apple that once brought Isaac Newton to the explosive idea that gravity exists.”

This was written by a leading conservative political commentator and not some BRAIN DEAD IMBICILE, to use Hulky’s choice phrase, and has got to be the most stupid take on the Newton’s apple and the theory of gravity story that has ever crossed my path. 

Does the author of this totally absurd statement, which is potentially going to be read by literally tens of millions of people, who use this email service, really believe that nobody realised that gravity existed before the young Isaac, sitting in his mother’s garden in Woolsthorpe during an outbreak of the plague in 1665, observed an apple falling from a tree? 

Alone the etymology of the word gravity, Latin gravitatum from gravis meaning heavy, used in Latin translations to express Aristotle’s concept that naturally heavy things, i.e. those with gravity, are attracted to the Earth and fall downwards when dropped, should have given our author a clue that people have always known that gravity exists and that Newton did not to realise that gravity exists in 1665 but twenty years later he provided a scientific explanation for why it exists.

Sometimes I despair.

HISTSCI_HULK ANGRY, HISTSCI_HULK SMASH!!!!!!!!!!!!

From τὰ φυσικά (ta physika) to physics – XXV

At the beginning of the first episode of this series I wrote the following:

In popular histories of science in Europe the history of physics is all too often presented roughly as follows, in antiquity there was Aristotle, whose writings also dominated the Middle Ages, until Galileo came along and dethroned him, following which Newton created modern physics.

We have already seen that already beginning in the sixth century CE with John Philoponus, the impetus theory that would be taken up and developed by both Islamic and medieval European scholars offered a strong alternative to Aristotle’s theory of projectile motion. Philoponus, and many others, also questioned his theories of fall and as we have seen in the fourteenth century, the Oxford Calculatores and the Paris Physicists did quite a lot on the laws of fall that is usually credited to Galileo.

What is very often ignored in that Galileo was very much aware of substantial work done on both projectile motion and fall in the sixteenth century on which he built his own theories. 

The first scholar to make an important contribution to the physics of motion during the sixteenth century was Niccolò known as Tartaglia (1499–1557). Although, often referred to as Niccolò Fontana, his actual surname is not known for certain. He came from simple circumstances and suffered much tragedy in his childhood. He was born in Brescia, in the Lombardy, the son of Michele a dispatch rider, who was murdered when he was just six years old. In 1512, French troops invaded Brescia and although his family sought refuge in the cathedral, the French troops entered the building and the young Niccolò was slashed across the face with a sabre slicing open his jaw and palate and leaving him for dead. His mother nursed him back to life but he was left with a speech impediment , which earned him the nickname Tartaglia, the stammerer. He grew a beard to cover his scars. 

Source: Wikimedia Commons

Largely self-taught, he moved to Verona around 1517 and then to Venice in 1534. He earned his living teaching practical mathematics in abbacus schools. A Maestro d’abaco or reckoning master Tartaglia was one of the first to transcend the world of practical mathematics that was common for the period and in which mathematicians were viewed not as scholars but as craftsmen, and in many senses became a mathematician in the modern meaning of the term. This transition of mathematicians from craftsmen to scholars was only truly completed a century later thanks largely to the contributions of Kepler and Galileo. 

Tartaglia is, of course, best known as the second mathematician after Scipione del Ferro to discovery a general solution for some forms of the cubic equations, a solution wider ranging than that of Scipione. I wrote about this briefly in the last episode and more fully in two earlier posts, here and here, so I won’t repeat it here. It’s also not directly relevant to the topic of this episode. 

As well as his work on the cubic equation, Tartaglia also wrote a typical reckoning mater guide to elementary mathematics his General trattato di numeri et misure, 6 pts. (Venice, 1556–1560).

General trattato de’ numeri et misure, 1556 Source: Wikimedia Commons

In the second part of which he includes the triangle of binomial coefficients, known generally as Pascal’s Triangle, who first published it a hundred years later.

Tartaglia’s triangle from General Trattato di Numeri et Misure, Part II, Book 2, p. 69 Source: Wikimedia Commons

One should point out that Peter Apian (1495–1552) had already published it in his Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen, in 1527. It was earlier published by al-Karaji (953–1029), by Omar Khayyám (1048–1131). In Arabic it’s known as Kayyám’s  Triangle. In China it was first published by Jia Xian (1010–1070) in the early eleventh century, and by Yang Hui (1238–1298) in the thirteenth century were it is known as Yang Hui’s Triangle. 

In 1543, Tartaglia produced the first translation of the Elements of Euclid into Italian, which was also the first translation into the vernacular. There was a second edition in 1565 and a third in 1585. It appears that he translated from Latin not Greek and in his second edition he mentions the first translation by Campano, that is the Latin edition of Campanus of Novara ( c. 1220–1296), which was based on the translation of Robert of Chester (12^th century) and which became the first printed edition, published by Erhard Ratdolt (1442–1528) in Venice in 1482. However, there is reason to believe that Tartaglia’s translation is actually based on the 1505 Latin translation direct from the Greek of Bartolomeo Zamberti (c.1473–after 1543) published in Venice. 

In 1543,Tartaglia also produced a seventy-one page edition of the Latin translation of the works of Archimedes by William of Moerbeke ( between 1215 & 1235–c. 1286), Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi containing Archimedes’ works on the parabola, the circle, centres of gravity, and floating bodies. In 1551, he published an Italian translation of parts of the Archimedes text, part of Book I of De insidentibus aquae. His active interest in Archimedes would have an influence on his one time pupil Giambattista Benedetti (1530­–1590), as we will see in a later episode. 

Tartaglia’ contribution to the laws of motion were not made in what we would recognise as a work of physics but in what was the first ever mathematical treatise on gunnery or ballistics, which of course is the study of projectile motion. The earliest known mention of gunpowder in European literature is by Roger Bacon in his Opus Major in 1267. Depiction of guns begin to appear in the early fourteenth century. The use of cannons in warfare, particularly during sieges, developed over the fourteenth and fifteenth century. The invention of the gun carriage at the end of the fifteenth century saw the introduction of field artillery and the need for a science of gunnery or ballistics. A need that Tartaglia became the first to fulfil. 

Tartaglia’s first published book Nova Scientia (Venice,1537), a tome on ballistics, was in the words of historian Matteo Valleriani:

In 1537, a mathematician from Brescia, Nicolò Tartaglia (1500–1557) published a work entitled Nova scientia. It is this work that established the modern science of ballistics, as characterized by the search for a mathematical understanding of the trajectory of projectiles. Tartaglia’s intentions were to create a science based on axioms and more geometrico, fundamental to the entire subject of mechanics, starting from a limited number of principles and arriving at a series of propositions through a process of rigid deduction. The methodological model Tartaglia intended to follow was the one he was able to extrapolate from works like Euclid’s Elements.

[…]

However, from a wider perspective, more specifically from the perspective of the entire history of the development of mechanics during the Renaissance, Tartaglia’s most important achievement is having demonstrated in 1537 that an exact science of ballistics was possible, based on the application of mathematical and geometrical methods. Challenged by the knowledge and experience of the bombardier, Tartaglia made an enormous contribution to the field of mathematical physics.^[1]

Nova Scientia frontispice

The book was very successful and very widely read. There was a second edition published in 1550 with reprints in 1551 and 1558. Further reprints were made in 1562, 1583, and 1606. There were translations into French and English. 

Tartaglia wrote a second more widely ranging book including the topic of artillery his Quesiti et Inventioni Diversi published in 1554. 

In this work Tartaglia dealt with algebraic and geometric material (including the solution of the cubic equation), and such varied subjects as the firing of artillery, cannonballs, gunpowder, the disposition of infantry, topographical surveying, equilibrium in balances, and statics.^[2]

Galileo was influenced by the works of Tartaglia and owned a richly annotated copies of his works on ballistics. These contained the first statement of the theorem that

…the maximum range, for any given value of the initial speed of the projectile, is obtained with a firing elevation of 45°. The latter result was obtained through an erroneous argument, but the proposition is correct (in a vacuum) and might well be called Tartaglia’s theorem. In ballistics Tartaglia also proposed new ideas, methods, and instruments, important among which are “firing tables.”^[3]

Fig. 2.1: Representation of a cannon positioned at a 45-degree angle of elevation as verified by means of the bombardier’s quadrant. From Tartaglia 1558.

Although he rejected the theories of Aristotle, Tartaglia’s work was informed by the impetus theory and his projectiles did not fly along parabolic trajectories. Tartaglia’s trajectories were in three segments,  first a straight line upwards, then a curve, and finally a fall straight down to earth when the impetus was exhausted and gravity took over. 

Nova Scientia 1606 Ballistic curve

Through his work on ballistics Tartaglia had a major impact on the physics of projectile motion in the first half of the sixteenth century. It continued to be a major influence in the practical field of gunnery well into the eighteenth century.

Addendum:

Jacopo Bertolotti, Associated Professor of Physics at the University of Exeter posted the following on social media inspired by this post:

If you ever studied any Physics in school you probably know that the trajectory of an object in a uniform gravitational field will be a parabola. But if the drag is not negligible, the trajectory will be much more skewed, and it will fall almost vertically.

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^[1] Matteo Valleriani, Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia, Berlin, 2013.

^[2] Arnaldo Masotti, DSB

^[3] Arnaldo Masotti, DSB